Geometric Aspects of Dwork Theory

Geometric Aspects of Dwork Theory Volume I

Bernie in Erice, October 1994

Bernard M. Dwork 19231998

Geometric Aspects of Dwork Theory Editors Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, and Franc¸ois Loeser Volume I

≥ Walter de Gruyter · Berlin · New York

Editors Alan Adolphson Department of Mathematics Oklahoma State University Stillwater, OK 74078 USA e-mail: [email protected]

Francesco Baldassarri Dipartimento di Matematica Universita` di Padova Via Belzoni 7 35131 Padova Italy e-mail: [email protected]

Nicholas Katz Department of Mathematics Princeton University Princeton, NJ 08544-1000 USA e-mail: [email protected]

Pierre Berthelot IRMAR Universite´ de Rennes 1 Campus de Beaulieu 35042 Rennes cedex France e-mail: [email protected]

Franc¸ois Loeser E´cole Normale Supe´rieure De´partement de mathe´matiques et applications UMR 8553 du CNRS 45 rue d’Ulm 75230 Paris Cedex 05 France e-mail: [email protected]

Mathematics Subject Classification 2000: 14-06; 14Fxx, 14Gxx, 11Gxx, 11Lxx Keywords: p-adic cohomologies, zeta functions, p-adic modular forms, D-modules

P Printed on acid-free paper which falls within the guidelines of the E ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Geometric aspects of Dwork theory / edited by Alan Adolphson … [et al.]. p. cm. Includes bibliographical references. ISBN 3-11-017478-2 (cloth : alk. paper) 1. Geometry, Algebraic. 2. Number theory. 3. p-adic analysis. I. Adolphson, Alan, 1951 QA564.G47 2004 516.315dc22 2004011345

ISBN 3-11-017478-2 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de. ” Copyright 2004 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Cover design: Thomas Bonnie, Hamburg. Typeset using the authors’ TEX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen.

Preface

Bernard Dwork (New York 5/27/1923 – Princeton 5/9/1998) stunned the algebrogeometric world in 1958 with his proof of rationality of the zeta function of an algebraic variety over a finite field and with the p-adic methods he introduced in his proof. He then went on to create a completely new p-adic cohomology theory for hypersurfaces of characteristic p > 0, and to study the p-adic variation, in families, of the zeta function, creating out of whole cloth a p-adic theory of Picard–Fuchs equations and their relation to zeta functions. He established a general theory of p-adic differential equations and deep results on the relation between the Hodge and slope filtrations of Picard–Fuchs equations. In the course of this work, he introduced ideas that have been fundamental in the development of p-adic arithmetic geometry as it presently stands. Among these are the notions of Frobenius structure for p-adic differential equations, and of the slope and growth filtrations on their solution spaces. Equally fundamental has been his insistence on the role of monodromy of Picard–Fuchs equations around supersingular points, and on the still mysterious notion of excellent lifting of Frobenius. In order to explore and clarify the intrinsic geometric content of Dwork’s arithmetic results and his p-adic analytic methods a group of Dwork’s mathematical heirs organized an extended cycle of conferences which was entitled “The Dwork Trimester in Italy” by reason of its duration and venue. Moreover, during this intensive trimester new developments were presented via a series of mini-courses dedicated to some of the most salient aspects of Dwork’s theory. The Dwork Trimester took place from May to July of 2001. The principal site was at the University of Padova. There were also two one-week special conferences. The first, on p-adic modular forms, p-adic L-functions and p-adic integration, was organized by M. Bertolini, and held at the Villa Monastero in Varenna on Lake Como from June 3 to June 9, 2001. The second, which comprised the final week of the Dwork Trimester, was held at the mountain resort Bressanone from July 1 to July 7, 2001. The Dwork Trimester brought together over a hundred mathematicians from many countries to pay a fitting scientific and personal homage to the remarkable mathematician and man who was Bernard M. Dwork. The organizers of the Dwork Trimester wish to extend their heartfelt thanks to the Istituto Nazionale di Alta Matematica (INdAM), Rome, and to the European Network “Arithmetic Algebraic Geometry” which co-sponsored the events, as well as to the Dipartimento di Matematica Pura ed Applicata of the Università di Padova, to the staffs of Villa Monastero in Varenna and the Scuola Estiva of the Università di Padova at Bressanone whose unstinting cooperation made possible the Dwork Trimester. As one might infer from the fact that the Dwork Trimester took place in northern Italy, Bernie had strong and enduring ties with Italy. Already in the fall of 1966 he

vi

Preface

had accepted an invitation from Francesco Gherardelli to be a visiting professor at the University of Florence. It was an eventful year: early November rains brought the Arno to historically unprecedented levels while Bernie, Shirley and their three children were visiting Rome for a very wet All Saints holiday. The heavy rains and flooding had blocked the roads back to Florence, thus forcing the family to make an overnight stop in Cortona. From there Bernie sent his brother Leo a memorable two-word telegram (“Glug glug”), clearly showing both that the family was safe, and that his sense of humor was intact. (His brother sent a two word reply: “Use Listerine”.) The havoc produced by the flood made a return to Florence by car unthinkable. Following the family’s return to Rome the threat of typhoid fever in Florence ruled out even the train trip to Florence that Bernie and his son Andrew had planned, hoping to retrieve warm clothing for the family. The planned sabbatical in Italy seemed ruined. The Dworks were on the verge of returning to the United States, but the timely efforts of Aldo Andreotti enabled Bernie and his family to move to the University of Pisa for the rest of the academic year. In the fall of 1975 Francesco Baldassarri, recently “laureato” at the University of Padova under the direction of I. Barsotti, went to Princeton University as a postdoctoral fellow. Dwork soon became his mentor, and a lasting collaboration and friendship ensued. A few years later another young mathematician from Padova, Bruno Chiarellotto, also went to Princeton to study with Dwork. As a result, both Bernie and Shirley Dwork became frequent visitors to Padova in the years that followed. During their many visits throughout the 1980s they established firm friendships both within the mathematical community at Padova and also beyond it. Thus, it was natural (although by no means bureaucratically trivial) that when Bernie retired from Princeton University he was “called” to the University of Padova where he served as “Professore per Chiara Fama” from 1992 until his death in 1998. During this period his influence on mathematics in Padova was vibrant and deep. Among his other students and collaborators from this period are Maurizio Cailotto, Lucia DiVizio, Giovanni Gerotto, Frank Sullivan, and Francesca Tovena. But Bernie’s interest in Italy was not limited to its mathematicians. Indeed, Bernie had a keen interest in Italian life, culture, and politics. His openness and joie de vivre won him, throughout his life, close friendships in many places, and Italy was no exception. His friends in Italy were honored to organize a mathematical trimester dedicated to the developments which came out of his work and out of the tools he invented. The result of that trimester is the present publication. April 2004

A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz, and F. Loeser

Table of Contents of Volume I

Preface

v

Table of Contents of Volume II

ix

The Mathematical Publications of Bernard Dwork

xi

Alan Adolphson Exponential sums and generalized hypergeometric functions. I: Cohomology spaces and Frobenius action

1

Alan Adolphson and Steven Sperber Exponential sums and free hyperplane arrangements

43

Yves André Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

55

Fabrizio Andreatta and Eyal Z. Goren Hilbert modular varieties of low dimension

113

Francesco Baldassarri and Pierre Berthelot On Dwork cohomology for singular hypersurfaces

177

Francesco Baldassarri and Andrea D’Agnolo On Dwork cohomology and algebraic D-modules

245

Laurent Berger An introduction to the theory of p-adic representations

255

Vladimir G. Berkovich Smooth p-adic analytic spaces are locally contractible. II

293

Jean-Benoît Bost Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems

371

Gilles Christol Thirty years later

419

Robert F. Coleman and William A. Stein Approximation of eigenforms of infinite slope by eigenforms of finite slope

437

viii

Table of Contents of Volume I

Richard Crew Crystalline cohomology of singular varieties

451

Andrea D’Agnolo and Pietro Polesello Stacks of twisted modules and integral transforms

463

Jan Denef and François Loeser On some rational generating series occuring in arithmetic geometry

509

Mladen Dimitrov Compactifications arithmétiques des variétés de Hilbert et formes modulaires de Hilbert pour
1 (c, n)

527

Table of Contents of Volume II

Mladen Dimitrov and Jacques Tilouine Variétés et formes modulaires de Hilbert arithmétiques pour
1 (c, n)

555

Lucia Di Vizio Introduction to p-adic q-difference equations

615

Matthew Emerton and Mark Kisin An introduction to the Riemann–Hilbert correspondence for unit F -crystals

677

Jean-Yves Etesse Introduction to L-functions of F -isocrystals

701

Ofer Gabber Notes on some t-structures

711

Haruzo Hida Non-vanishing modulo p of Hecke L-values

735

Luc Illusie On semistable reduction and the calculation of nearby cycles

785

Nicholas M. Katz and Rahul Pandharipande Inequalities related to Lefschetz pencils and integrals of Chern classes

805

Kiran S. Kedlaya Full faithfulness for overconvergent F -isocrystals

819

Bernard Le Stum Frobenius action, F -isocrystals and slope filtration

837

Shigeki Matsuda Conjecture on Abbes–Saito filtration and Christol–Mebkhout filtration

845

Christine Noot-Huyghe Transformation de Fourier des D-modules arithmétiques I

857

Tomohide Terasoma Boyarsky principle for D-modules and Loeser’s conjecture

909

Nobuo Tsuzuki Cohomological descent in rigid cohomology

931

x

Table of Contents of Volume II

Isabelle Vidal Monodromie locale et fonctions Zêta des log schémas

983

Anne Virrion Trace et dualité relative pour les D-modules arithmétiques

1039

Daqing Wan Geometric moment zeta functions

1113

The Mathematical Publications of Bernard Dwork

Books [1] Generalized hypergeometric functions. The Clarendon Press, Oxford University Press, New York 1990. [2] Lectures on p-adic differential equations. Grundlehren Math. Wiss. 253, Springer-Verlag, New York, Berlin 1982. [3] (with G. Gerotto, and F. J. Sullivan), An Introduction to G-functions. Ann. of Math. Stud. 133, Princeton University Press, Princeton, NJ, 1994.

Articles [1]

Detection of a pulse superimposed on fluctuation noise. Proc. I. R. E. 38 (1950), 771–774.

[2]

On the root number in the functional equation of the Artin-Weil L-series. PhD thesis, Columbia University, 1954.

[3]

The local structure of the Artin root number. Proceedings of the National Academy of Science, 41 (1955), 754–756.

[4]

On the Artin root number. Amer. J. Math. 78 (1956), 444–472.

[5]

Norm residue symbol in local number fields. Abh. Math. Sem. Univ. Hamburg 22 (1958), 180–190.

[6]

On the congruence properties of the zeta function of algebraic varieties. J. Reine Angew. Math. 203 (1960), 130–142.

[7]

On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82 (1960), 631–648.

[8]

A deformation theory for the zeta function of a hypersurface defined over a finite field. In Proceedings of the International Congress of Mathematics, Stockholm, 1962.

[9]

On the zeta function of a hypersurface. Inst. Hautes Études Sci. Publ. Math. 12 (1962), 5–68.

[10] On the zeta function of a hypersurface, part II. Ann. of Math. 80 (1964), 227–299. [11] Analytic theory of zeta functions of algebraic varieties. In Arithmetical Algebraic Geometry (O. F. Schilling, editor), Harper and Row, New York 1965, 18–32. [12] On p-adic analysis. In Proceedings Annual Science Conference Belfer Graduate School, Yeshiva University, 1965–1966, volume 2, Belfer Graduate School of Science, Yeshiva University, New York, 129–154.

xii

The Mathematical Publications of Bernard Dwork

[13] On the rationality of zeta functions and L-series. In Proceedings of the Conference on Local Fields, Driebergen, Springer-Verlag, Berlin 1966, 40–55. [14] On the zeta function of a hypersurface, part III. Ann. of Math. 83 (1966), 457–519. [15] On zeta functions of hypersurfaces. In Les tendances géométriques en algèbre et théorie des nombres, Editions du Centre National de la Recherche Scientifique, Paris 1966, 77–82. [16] A deformation theory for singular hypersurfaces. In Algebraic Geometry, Bombay Colloquium 1968, Oxford University Press, 1969, 85–92. [17] On the zeta function of a hypersurface, part IV: A deformation theory for singular hypersurfaces. Ann. of Math. 90 (1969), 335–352. [18] P -adic cycles. Inst. Hautes Études Sci. Publ. Math. 39 (1969), 327–415. [19] Normalized period matrices I. Ann. of Math. 94 (1971), 337–388. [20] On Hecke polynomials. Invent. Math 12 (1971), 249–256. [21] Normalized period matrices II. Ann. of Math. 98 (1973), 1–57. [22] On p-adic differential equations II. Ann. of Math. 98 (1973), 366–376. [23] On p-adic differential equations III. Invent. Math 29 (1973), 35–45. [24] On p-adic differential equations IV. Ann. Sci. École Norm. Sup. 6 (1973), 295–316. [25] On the Up operator of Atkin modular functions of level 2 with growth conditions. In Modular functions of one variable (W. Kuyk and J. P. Serre, editors), Lecture Notes in Math. 350, Springer-Verlag, Berlin 1973, 57–67. [26] Bessel functions as p-adic functions of the argument. Duke Math. J. 41 (1974), 711–738. [27] On p-adic differential equations I. Bull. Soc. Math. France 39–40 (1974), 711–738. [28] On ordinary linear p-adic differential equations with algebraic function coefficients. Groupe d’étude d’Analyse Ultrametrique, 10 pages, exp. 18, 1975/76. [29] (with P. Robba), On linear p-adic differential equations. Trans. Amer. Math. Soc. 231 (1977), 1–46. [30] (with P. Robba), Majorations effective. Groupe d’étude d’Analyse Ultrametrique, 1978/79. exp. 18, 2pp. [31] (with F. Baldassarri), On second order linear differential equations with algebraic solutions. Amer. J. Math. 101 (1979), 42–76. [32] (with P. Robba), On natural radii of p-adic convergence. Trans. Amer. Math. Soc. 256 (1979), 199–213. [33] (with S. Bosch and P. Robba), Un théorème de prolongement pour les fonctions analytiques. Math. Ann. 252 (1980), 165–173. [34] (with P. Robba), Effective p-adic bounds for solutions of homogeneous linear differential equations. Trans. Amer. Math. Soc. 259 (1980), 559–577. [35] Maurizio Boyarsky (pseudonym for B. Dwork). p-adic gamma functions and Dwork cohomology. Trans. Amer. Math. Soc. 257 (1980), 359–369.

xiii

The Mathematical Publications of Bernard Dwork

[36] Nilpotent second order linear differential equations with Fuchsian singularities. Groupe d’étude d’Analyse Ultrametrique, 7 pages, exp. 19, 1980/81. [37] On Apery’s differential operator. Groupe d’étude d’Analyse Ultrametrique, 10 pages, exp. 25, 1980/81. [38] Arithmetic theory of differential equations. In Symposia Mathematica XXIV (Sympos., INDAM, Rome, 1979), Academic Press, London, New York 1981. [39] Majoration effective et application. Groupe d’étude d’Analyse Ultrametrique, 5 pages, exp. 1, 1981/82. [40] A note on the p-adic gamma function. Groupe d’étude d’Analyse Ultrametrique, 10 pages, exp. 15, 1981/82. [41] (with A. Adolphson and S. Sperber), Growth of solutions of linear differential equations at a logarithmic singularity. Trans. Amer. Math. Soc. 271 (1982), 245–252. [42] Differential equations which come from geometry. Groupe d’étude d’Analyse Ultrametrique, 6 pages, exp. 9, 1982/83. [43] Puiseux expansions. Groupe d’étude d’Analyse Ultrametrique, 6 pages, exp. 14, 1982/83. [44] Singular residue classes which are ordinary for F (a, b, c, d). Groupe d’étude d’Analyse Ultrametrique, 11 pages, exp. 23, 1982/83. [45] On the Boyarsky principle. Amer. J. Math. 105 (1983), 115–156, [46] On the Tate constant. Groupe d’étude d’Analyse Ultrametrique (1983/84), 14 pages, exp. 11. [47] On Kummer’s twenty four solutions of the hypergeometric equation. Trans. Amer. Math. Soc. 285 (1984), 497–521. [48] Maurizio Boyarsky (pseudonym for B. Dwork). The Reich trace formula. p-adic cohomology. Astérisque 119–120 (1984), 129–150. [49] (with S. Chowla and R. Evans), On the mod p2 determination of




(p − 1)/2 (p − 1)/4



. J. Number

Theory 24 (1986), 188–196. [50] (with A. Ogus), Canonical liftings of Jacobians. Compositio Math. 58 (1986), 111–131. [51] On the Tate constant. Compositio Math. 61 (1987), 43–59. [52] (with F. Baldassarri and F. Tovena), On singular projective structures on Riemann surfaces. J. Differential Equations 80 (1989), 364–376. [53] On the uniqueness of the Frobenius operator on differential equations. Adv. Stud. Pure Math. 17 (1989), 89–96. [54] Differential operators with nilpotent p-curvature. Amer. J. Math. 112 (1990), 749–786. [55] Work of Philippe Robba. In p-adic Analysis Proceedings, Trento 1989, Lecture Notes in Math. 1454, Springer-Verlag, Berlin 1990, 1–10. [56] (with G. Christol), Effective p-adic bounds at regular singular points. Duke Math. J. 62 (1991), 689–720. [57] (with S. Sperber), Logarithmic decay and overconvergence of the unit root and associated zeta functions. Ann. Sci. École Norm. Sup. 24 (1991), 575–604.

xiv

The Mathematical Publications of Bernard Dwork

[58] (with G. Christol), Differential modules of bounded spectral norm. Contemp. Math. 133 (1992), 39–58. [59] (with A. van der Poorten), The Eisenstein constant. Duke Math. J. 65 (1992), 23–43. [60] Cohomological interpretation of hypergeometric series. Rend. Sem. Mat. Univ. Padova 90 (1993), 239–263, [61] (with F. Loeser), Hypergeometric series. Japanese Journal of Mathematics 19 (1993), 81–129. [62] (with Gilles Christol), Modules différentiels sur des couronnes. Ann. Inst. Fourier (Grenoble) 44 (1994), 669–672. [63] (with F. Loeser), Hypergeometric series and functions as periods of exponential modules. In Barsotti Symposium in Algebraic Geometry (Abano Terme 1991), Perspect. Math. 15, Academic Press, San Diego, CA, 1994, 153–174 [64] (with Alfred J. van der Poorten), Corrections to “the Eisenstein constant”. Duke Math. J. 65 (1994), 669–672. [65] (with Alan Adolphson), Contiguity relations for generalized hypergeometric functions. Trans. Amer. Math. Soc. 347 (2) (1995), 615–625. [66] On exponents of p-adic differential modules. J. Reine Angew. Math. 484 (1997), 85–126. [67] On systems of ordinary differential equations with transcendental parameters. J. Differential Equations 156 (1) (1999), 18–25. [68] On the size of differential modules. Duke J. Math. 96 (2) (1999), 225–239. [69] Cohomology of singular hypersurfaces. Pacific J. Math. 195 (1) (2000), 81–89.

Exponential sums and generalized hypergeometric functions. I: Cohomology spaces and Frobenius action Alan Adolphson∗

Abstract. We discuss the variation of p-adic cohomology of families of exponential sums on the n-torus Tn . This variation is described by p-adic analogues of classical generalized hypergeometric functions. 2000 Mathematics Subject Classification: 11

1 Introduction In [4], Dwork studied generalized Jacobi sums, i.e., sums of the form n1 

 n x∈(F× q) 1

i=1

n2  

χi (xi )

 χn1 +j (f (j ) (x)) ,

(1.1)

j =1

where Fq is a finite field of characteristic p, the χi are multiplicative characters on F× q, (j ) and the f are forms in n1 variables with coefficients in Fq . More precisely, he studied how such sums vary p-adically as functions of the χi and of the coefficients of the f (j ) and explained how this variation is described by p-adic analogues of classical hypergeometric functions. As explained in the introduction to [4], up to simple factors the sum (1.1) equals  n +n x∈(F× q) 1 2

+n2  n1 i=1

n2    χi (xi )  xn1 +j f (j ) (x1 , . . . , xn1 ) ,

(1.2)

j =1

where xn1 +1 , . . . , xn1 +n2 are additional variables and  is a nontrivial additive character of Fq . From the viewpoint of Dwork theory this latter sum is somewhat more natural than (1.1), hence sums of the form (1.2) are the basic objects of study in [4]. ∗ Partially supported by NSF Grant no. DMS-0070510

2

Alan Adolphson

The work of Sperber and myself ([2, 3]) indicates that one should be able to prove analogues of the results of [4] without restricting the polynomial that appears in the additive character  to have the special form of (1.2). That is, one could study exponential sums of the form n   n x∈(F× q)

 χi (xi ) (f (x)),

(1.3)

i=1

where f is a Laurent polynomial in n variables over the field Fq . The p-adic variation of such sums as functions of the χi and of the coefficients of f would then be described in terms of p-adic analogues of classical hypergeometric functions, now possibly confluent. (Sums of the form (1.2) are related to nonconfluent hypergeometric functions.) This article carries out part of such a program. We extend chapters 1–4 and chapter 6 of [4] to exponential sums of the form (1.3). This involves defining and studying the cohomology space Wa,λ associated to (1.3), its dual space Ka,λ , and the Frobenius map ∗ αa,a ,λ : Ka ,λp → Ka,λ .

(The p-adic n-vector a parametrizes the multiplicative characters {χi }ni=1 , the p-adic vector λ parametrizes the coefficients of f , and a is a preimage of a modulo Zn under multiplication by p.) An important role is played by the “contiguity map” x u : Wa,λ → Wa−u,λ , which is an algebraic reflection of the classical contiguity relationship between two hypergeometric functions of the variables λ, one with parameters a, the other with parameters a − u (where u ∈ Zn ). In the second paper of this series, we plan to extend chapter 5 of [4] (deformation theory and action of Frobenius on solutions of p-adic differential equations) to this setting. The deformation equations that arise from the variation of λ in Ka,λ are p-adic versions of generalized hypergeometric equations, and one obtains an action of Frobenius on p-adic analytic solutions of these equations. We follow rather closely the methods of [4]. Our main results are the analyticity of the Frobenius matrix (Theorem 9.12) and the description of the determinant of the contiguity matrix (Theorem 8.7). By “analyticity of the Frobenius matrix” we mean ∗ that the matrix representing αa,a ,λ (relative to fixed bases and a fixed choice of a ) is an analytic function of a and λ in a certain region (described in Theorem 9.12). Theorem 8.7 implies immediately Theorem 8.1, which, together with Theorem 5.2, gives a necessary and sufficient condition for the contiguity map to be an isomorphism. This settles in the affirmative [4, Conjecture 6.3.1]. In order to make this paper more accessible, we have included some results of an expository nature. In sections 2 and 3 we review the p-adic cohomology of exponential sums of the form (1.3). Proposition 4.6 then explains the reason for studying the spaces denoted Wa and Wa,λ later in the paper. In Remark 9.9 we explain the precise

Exponential sums and generalized hypergeometric functions

3

relationship between exponential sums, the “dual spaces” Ka,λ , and the “Frobenius ∗ map” αa,a ,λ . This article is based on lectures I gave at Princeton University in Spring, 1991, and I am grateful for their hospitality. It was originally intended to be part of a larger joint project with Professor Dwork, which was to extend more of [4] to sums of the form (1.3) and also improve some of the results of that book. Unfortunately, circumstances prevented its completion.

2 Preliminaries We begin by fixing notation and defining the exponential sums that will be studied. Let p be a prime and q a power of p. Recall that the series (where π = (−p)1/(p−1) ) p

θ(t) = exp π(t − t ) =

∞ 

cj t j

j =0

satisfies the estimate



 p−1 ord cj ≥ j . p2

(2.1)

Furthermore, θ (1) is a primitive p-th root of unity and if z lies in an extension of Qp r and satisfies zp = z, then θ(1)z+z

p +···+zp r−1

= θ(z)θ (zp ) · · · θ (zp

r−1

).

Let f ∈ Fq [x1 , x1−1 , . . . , xn , xn−1 ] and write f in the form  f = fν x ν ,

(2.2)

(2.3)

ν∈

ˆ where  ⊆ Zn is finite and fν ∈ F× q . Let fν ∈ Qp (ζq−1 ) be the Teichmüller lifting of fν and define  F0 (x) = θ(fˆν x ν )θ(fˆνp x pν ) · · · θ (fˆνq/p x qν/p ) ν∈

∈ Qp (π, ζq−1 )[[x1 , x1−1 , . . . , xn , xn−1 ]]. By (2.1), the series F0 (x) converges on a polyannulus containing the polycircle |x| = 1 m (i.e., |xi | = 1 for i = 1, . . . , n), and if x ∈ Qp (ζq m −1 )n satisfies x q = x (i.e., qm xi = xi for i = 1, . . . , n), then by (2.2) m−1  i=0

i

TraceFq m /Fp (f (x)) ¯

F0 (x q ) = θ(1)

,

(2.4)

4

Alan Adolphson

where x¯ ∈ (Fq m )n is the reduction of x modulo p. Thus the function F0 gives a p-adic analytic representation of a nontrivial additive character on Fq . × Let ω : F× q → Qp (ζq−1 ) be the Teichmüller character, i.e., ω(x) is the Teichmüller lifting of x. Since ω generates the character group of F× q , every multiplicative −a(q−1) , where character of F× can be written in the form ω q a∈

1 Z, q −1

0 ≤ a < 1.

(2.5)

Let a = (a1 , . . . , an ) ∈ Qn , where each ai satisfies (2.5). Exponential sums of the form (1.3) can then be represented p-adically as Sm (f, a) =



x −a(q

m −1)

m−1 

i

F0 (x q ).

(2.6)

i=0

x∈Qp (ζq m −1 ) m x q −1 =1

Its associated L-function is L(f, a; t) = exp

∞  m=0

Sm (f, a)

tm  . m

(2.7)

We describe the cohomology of such exponential sums. Let f be as in (2.3). The Newton polyhedron of f , denoted (f ), is defined to be the convex hull in Rn of  ∪{(0, . . . , 0)}. Let C(f ) be the cone over (f ), i.e., the union of all rays emanating from the origin and passing through (f ). For u ∈ C(f ), we define the weight of u, denoted w(u), to be the least nonnegative real number such that u ∈ w(u)(f ), where w(u)(f ) denotes the dilatation of (f ) by the factor w(u). The weight function is easily seen to have the following properties. Lemma 2.1. (a) If c ∈ R, c ≥ 0, and u ∈ C(f ), then w(cu) = cw(u). (b) If u, u ∈ C(f ), then w(u + u ) ≤ w(u) + w(u ) with equality holding if and only if u and u lie over a common face of (f ). (c) There exists a positive integer M such that w(u) ∈ M −1 Z for all u ∈ Zn ∩C(f ). Let K be an extension of Qp . Until section 4, we assume that the residue field of K is Fq . Let “ord” denote the p-ordinal, normalized by ord p = 1. For b, c ∈ R, b > 0, define     L(b, c) = Au x u Au ∈ K, ord Au ≥ bw(u) + c u∈Zn ∩C(f )

L(b) =



c∈R

L=



L(b, c) 

u∈Zn ∩C(f )

  Au π w(u) x u Au ∈ K, Au → 0 as w(u) → ∞ .

5

Exponential sums and generalized hypergeometric functions

Note that these sets depend on K but we omit K from the notation. Note also that when writing π w(u) in the definition of L we are implicitly assuming that a fixed M-th root of π has been chosen (see Lemma 2.1(c)) and that this M-th root lies in K. One sees that L(b) ⊆ L for b > 1/(p − 1) and L ⊆ L(1/(p − 1)). The spaces L(b) are Banach spaces under the norm ξ  =

sup

u∈Zn ∩C(f )

|Au /p bw(u) |,

u where ξ = u∈Zn ∩C(f ) Au x ∈ L(b). We give L the norm inherited from the inclusion L ⊆ L(1/(p − 1)): ξ  =

sup

u∈Zn ∩C(f )

|Au |

for ξ = u∈Zn ∩C(f ) Au π w(u) x u ∈ L. It is straightforward to check from (2.1) and 0). Lemma 2.1(b) that F0 ∈ L((p

− 1)/pq, u For any formal series u∈Zn Au x , define    ψ Au x u = Apu x u . (2.8) u∈Zn

u∈Zn

Note that ψ(L(b)) ⊆ L(pb). Let a ∈ Qn be such that (q − 1)a ∈ Zn and −a ∈ C(f ). Write q = ps and define α(q−1)a = ψ s  x (1−q)a F0 (x) : L(b) → L(b) for 0 < b < (p − 1)/p, i.e., α(q−1)a is the composition ψs

x (1−q)a F0 (x)

L(b) −−−−−−−→ L(min{b, (p − 1)/pq}) −→ L(min{qb, (p − 1)/p}) → L(b). The following result is often referred to as the Dwork trace formula (see, for example, [1]).

Theorem 2.2. 

(q m − 1)n Trace((α(q−1)a )m |L(b)) = xq

m −1

x −a(q

m −1)

=1

m−1 

i

F0 (x q ).

i=0

Remark 2.3. This theorem remains true when L is substituted for L(b) provided that 1/(p − 1) < (p − 1)/p (i.e., p = 2) because then for  > 0, L(1/(p − 1) + ) ⊆ L ⊆ L(1/(p − 1)) and α(q−1)a is stable on L.

6

Alan Adolphson

By (2.6), (2.7), and Theorem 2.2 we have  ∞ m  (−1)n−1 n−1 m n mt L(f, a; t) = exp (−1) (q − 1) Trace (α(q−1)a ) m m=1       ∞ n m  n j j m Trace (α(q−1)a ) = exp − . (−1) (q t) m j m=1

j =0

But

 det(I − tα(q−1)a ) = exp

−

∞ 

Trace (α(q−1)a )m

m=1

 tm , m

so if we define an operator φ on formal power series with constant term 1 by g(t)φ = g(qt), then we have L(f, a; t)(−1)

n−1

n

= det(I − tα(q−1)a )(1−φ) .

(2.9)

To interpret (2.9) cohomologically, we introduce certain differential operators on our p-adic Banach spaces. For f as in (2.3), let fˆ be its Teichmüller lifting:  fˆν x ν ∈ Qp (ζq−1 )[x1 , x1−1 , . . . , xn , xn−1 ] fˆ = ν∈

and set fˆi = xi ∂ fˆ/∂xi . For a ∈ K n , define differential operators Di,a by Di,a = xi

∂ + ai + π fˆi . ∂xi

The Di,a operate on L and on the L(b) and commute with one another. Choose µ ∈ Zn such that −µ ∈ C(f ) and set a = (a + µ)/q. Lemma 2.4. As operators on L or L(b), αµ  Di,a = qDi,a  αµ . Proof. By continuity and linearity, it suffices to check the action of each side on a   single monomial x u with u ∈ Zn ∩ C(f ). This is a straightforward calculation. This lemma admits the following interpretation. Let K• (L, {Di,a }ni=1 ) be the Koszul complex on L formed by the Di,a . Define a map α˜ µ : K• (L, {Di,a }ni=1 ) → n K (L, {D }n ) as being q j α acting on the j -th term K (L, {D }n ) = L(j ) •

i,a i=1

of the Koszul complex.

µ

j

i,a i=1

Corollary 2.5. The map α˜ µ is a chain map from the complex K• (L, {Di,a }ni=1 ) to the complex K• (L, {Di,a }ni=1 ).

7

Exponential sums and generalized hypergeometric functions

Now let a ∈ Qn be such that (q − 1)a ∈ Zn and −a ∈ C(f ) and consider the L-function L(f, a; t) defined in (2.7). Take µ = (q − 1)a, so that a = a. Then by Corollary 2.5, α˜ µ is an endomorphism of the complex K• (L, {Di,a }ni=1 ), and from (2.9) we have L(f, a; t)(−1)

n−1

=

n 

(−1)j (nj)

det(I − tq j α(q−1)a |L)

(2.10)

j =0

= det(I

− t α˜ (q−1)a |K• (L, {Di,a }ni=1 )).

If we let Hj (K• (L, {Di,a }ni=1 )) denote the homology of this Koszul complex, then we have by [7] L(f, a; t)(−1)

n−1

=

n  j =0

j

det(I − tq j α(q−1)a |Hj (K• (L, {Di,a }ni=1 )))(−1) .

(2.11)

3 Calculation of Hj (K• (L, {Di,a }ni=1 )) We first use a theorem of Kouchnirenko to calculate the homology of a related Koszul complex in characteristic p and then use a theorem of Monsky to lift the result to characteristic 0. Let R = Fq [x u | u ∈ Z ∩ C(f )]. The weight function w : Zn ∩ C(f ) → M −1 Z (see Lemma 2.1(c)) defines a filtration on R by setting for each nonnegative integer i    bu x u ∈ R  w(u) ≤ i/M if bu = 0 . Ri/M = (i/M) be the associated graded ring, where Let gr(R) = ∞ i=0 gr(R) gr(R)(i/M) = Ri/M /R(i−1)/M . Then R and gr(R) are identical as Fq -vector spaces, but, by Lemma 2.1(b), in gr(R) x u+u if u and u lie over a common face of (f ), u u x x = 0 otherwise.

For σ a face of (f ), set fσ = ν∈σ ∩ fν x ν . We say that f is nondegenerate (relative to (f )) if for every face σ of (f ) that does not contain the origin, the n ¯ polynomials {∂fσ /∂xi }ni=1 have no common zero in (F¯ × q ) , where Fq denotes the algebraic closure of Fq . Let σ be a polytope in Rn , containing the origin, with vertices in Zn . Let Lσ be the smallest linear subspace of Rn containing σ . By the dimension dσ of σ we mean the dimension of Lσ . Define V (σ ) to be the dσ -dimensional volume of σ relative to Zn , i.e., V (σ ) is the volume of σ with respect to Haar measure on Lσ normalized so that a fundamental domain for the lattice Zn ∩ Lσ has volume 1. (So if dσ = n, V (σ )

8

Alan Adolphson

is ordinary volume with respect to Lebesgue measure on Rn .) Set Nσ = dσ !V (σ ), a nonnegative integer. In the special case where σ = (f ), we write df , Lf , V (f ), Nf in place of d(f ) , L(f ) , V ((f )), N(f ) , respectively. For i = 1, . . . , n, set fi = xi ∂f/∂xi ∈ R1 and let gr(fi ) ∈ gr(R)(1) be its image in the associated graded. If the exponents ν ∈  appearing in f all lie in a n hyperplane i=1 bi νi = 0 (where ν = (ν1 , . . . , νn )), then ni=1 bi f¯i = 0 also. Since dim (f ) = df , it follows that there exists a subset S ⊆ {1, . . . , n} with |S| = df such that each gr(fj ) is an Fq -linear combination of {gr(fi )}i∈S . The following result is a variation of [5, Théorème 2.8]. For a proof, see [2, Theorem 2.14]. Theorem 3.1. If f is nondegenerate, the Koszul complex K• (gr(R), {gr(fi )}i∈S ) is acyclic in positive dimension and  gr(fi )gr(R) = Nf . dimFq gr(R) i∈S

Remark 3.2. This result remains true for a polynomial f with coefficients in any field, i.e., the finiteness of Fq is not used in the proof. For c ∈ Fnq , define differential operators Di,c acting on gr(R) by Di,c = xi

∂ + ci + gr(fi ). ∂xi

Corollary 3.3. If f is nondegenerate, the Koszul complex K• (gr(R), {Di,c }i∈S ) is acyclic in positive dimension and  Di,c gr(R) = Nf . dimFq gr(R) i∈S

Proof. The grading on gr(R) makes K• (gr(R), {Di,c }i∈S ) into a filtered complex whose associated graded complex is K• (gr(R), {gr(fi )}i∈S ). By standard facts in commutative algebra, the assertion of the lemma then follows immediately from Theorem 3.1.   Remark 3.4. It also follows from standard facts

in commutative algebra that any elements of gr(R) that form a basis for gr(R)/ i∈S gr(fi )gr(R) also form a basis for gr(R)/ i∈S Di,c gr(R). Remark 3.5. Note that if c lies in the subspace of Fnq generated by the reductions mod p of the elements of , then as operators {Di,c }ni=1 are linear

n on gr(R) the combinations of the {Di,c }i∈S . In this case, i=1 Di,c gr(R) = i∈S Di,c gr(R). Let L(0) be the unit ball of L, i.e.,     Au π w(u) x u  |Au | ≤ 1 for all u and Au → 0 as u → ∞ . L(0) = u∈Zn ∩C(f )

Exponential sums and generalized hypergeometric functions

9

In the obvious way, L(0) is a module over OK , the ring of integers of K. We now assume that π 1/M is a uniformizer for OK (for example, take K = Qp (ζq−1 , π 1/M )). The map   A¯ u x u , (3.1) Au π w(u) x u → where A¯ u denotes the reduction of Au modulo π 1/M OK , induces an identification L(0) /π 1/M L(0) = gr(R) of Fq -vector spaces. In fact, more is true. The space L(0) is a ring and, using Lemma 2.1(b), one checks that (3.1) is a homomorphism of rings. Thus L(0) /π 1/M L(0) and gr(R) are identified as Fq -algebras. n , we may form the Koszul complex K (L(0) , {D } If a = (a1 , . . . , an ) ∈ OK • i,a i∈S ). 1/M ˆ OK of this complex Since π fi → gr(fi ) under the map (3.1), the reduction mod π is the Koszul complex K• (gr(R), {Di,a¯ }i∈S ), where a¯ = (a¯ 1 , . . . , a¯ n ) ∈ Fnq and a¯ i denotes the reduction of ai modulo the maximal ideal of OK . To “lift” the result of Corollary 3.3 to characteristic 0 and compute the cohomology of the complex K• (L, {Di,a }ni=1 ), which describes the L-function L(f, a; t), we need the following result of Monsky[6, Theorem 8.5]. For a proof, one can also see the appendix to [2]. Theorem 3.6. Let O be a complete discrete valuation ring with uniformizer πO and let K• = {· · · → K1 → K0 → 0} be a complex of O-modules satisfying (i) multiplication by πO is injective on each Ki ,  l (ii) ∞ l=0 πO Ki = 0 for all i, (iii) each Ki is complete in the metric |x| = |πO |l(x) , where l(x) = max{l | x ∈ πOl Ki }. Let K¯ • be the complex obtained by reducing K• modulo πO . Then (a) for any l, Hl (K¯ • ) = 0 implies Hl (K• ) = 0, (b) if dimO/(πO ) H0 (K¯ • ) = r < ∞ and H1 (K¯ • ) = 0, then H0 (K• ) is a free O-module of rank r and any lifting of any basis for H0 (K¯ • ) is a basis for H0 (K• ). Applying this theorem to the Koszul complex K• (L(0) , {Di,a }i∈S ) (and Corollary n, 3.3 to its reduction modulo π 1/M ), we conclude that if f is nondegenerate and a ∈ OK then Hj (K• (L(0) , {Di,a }i∈S )) = 0 for j > 0 while H0 (K• (L(0) , {Di,a }i∈S )) is a free OK -module of rank Nf . Since L = L(0) ⊗OK K and K is a flat OK -module, the following result is an immediate consequence.

10

Alan Adolphson

n , then Theorem 3.7. If f is nondegenerate and a ∈ OK

Hj (K• (L, {Di,a }i∈S )) = 0 for j > 0 and



dimK L

Di,a L = Nf .

i∈S

From (2.11) we get immediately the following. Corollary 3.8. Let a ∈ Qn be such that (q − 1)a ∈ Zn and −a ∈ C(f ). Suppose that f is nondegenerate and that df = n (hence S = {1, . . . , n}). Then L(f, a; t)(−1)

n−1

n    = det I − tα(q−1)a | L Di,a L . i=1

(−1)n−1

In particular, L(f, a; t)

is a polynomial of degree ≤ Nf .

Remark 3.9. It follows from Theorem 9.8 and equation 9.5 that the degree of n−1 L(f, a; t)(−1) equals Nf . For completeness, we describe the L-function when df < n. We assume that n ∩ (L ⊗ K). This implies that as operators on L the {D }n a ∈ OK f i,a i=1 are linear Q combinations of the {Di,a }i∈S . Since |S| = df , a standard result in commutative algebra relating the homology of K• (L, {Di,a }ni=1 ) to the homology of K• (L, {Di,a }i∈S ) gives the following. n ∩ (L ⊗ K), then Theorem 3.10. If f is nondegenerate and a ∈ OK f Q

Hj (K• (L, {Di,a }ni=1 )) = 0 for j > n − df , Hj (K• (L, {Di,a }ni=1 )) is isomorphic to the direct sum of

copies of L/ ni=1 Di,a L for 0 ≤ j ≤ n − df , and n 

dimK L


n−df  j

Di,a L = Nf .

i=1

When a satisfies the hypothesis of Corollary 3.8, these Koszul complexes have an action of Frobenius α˜ (q−1)a and the isomorphisms of Theorem 3.10 respect that action. Applying (2.11) we get the following result. Corollary 3.11. If f is nondegenerate, (q − 1)a ∈ Zn , and −a ∈ C(f ), then L(f, a; t)(−1)

n−1

n  (1−φ)n−df  = det I − tα(q−1)a | L Di,a L . i=1

Exponential sums and generalized hypergeometric functions

11

4 Generalization of the cohomology spaces

Under the hypothesis of Corollary 3.11, the quotient space L/ ni=1 Di,a L carries the essential information about L(f, a; t). This quotient space depends on the rational variable a and the Teichmüller lifting fˆ of f that were used to define the Di,a . To discuss p-adic variation of cohomology, we need to replace a and fˆ by p-adic variables. The purpose of this section is to construct such a generalization of the Di,a and show that the corresponding quotient spaces behave nicely. Let K be an arbitrary extension of Qp containing π and let  gν x ν ∈ K[x1 , x1−1 , . . . , xn , xn−1 ]. g= ν∈

If g has coefficients in OK , we let g¯ be its reduction modulo the maximal ideal. Lemma 4.1. If g and g¯ have the same Newton polyhedron, then g¯ nondegenerate implies g nondegenerate. Proof. Suppose there is a face σ of (f ) not containing the origin such that ∂gσ /∂x1 , . . . , ∂gσ /∂xn have a common zero (α1 , . . . , αn ) ∈ (K¯ × )n , where K¯ denotes the algebraic closure of K. Since gσ = ν∈σ ∩ gν x ν , we have  ∂gσ  = gν νi α ν = 0 xi  ∂xi (α1 ,...,αn ) ν∈σ ∩

for i = 1, . . . , n. Let ord αi = ri and set βi = p−ri αi , so that ord βi = 0 for all i. Then  gν νi β ν p r1 ν1 +···+rn νn = 0 (4.1) ν∈σ ∩

for i = 1, . . . , n. Let τ be the face of σ on which the linear form r1 x1 + · · · + rn xn attains its minimum as function on σ and let c be that minimum. Dividing (4.1) by p c gives   gν νi β ν = − gν νi β ν p r1 ν1 +···+rn νn −c ν∈τ ∩

ν∈(σ \τ )∩

for all i. Reducing modulo the maximal ideal, we have  g¯ ν νi β¯ ν = 0 ν∈τ ∩

for all i, where g¯ ν , β¯ denote reductions modulo the maximal ideal. But this says that n   β¯ ∈ (F¯ × q ) is a common zero of ∂ g¯ τ /∂x1 , . . . , ∂ g¯ τ /∂xn . We use the notation of Section 3 and apply some of those results to the present situation. Let RK = K[x u | u ∈ Z ∩ C(g)]. As in section 3, the weight function

12

Alan Adolphson

w defines a filtration on RK , and we let gr(RK ) be its associated graded. Let gi = xi ∂g/∂xi ∈ (RK )1 and let gr(gi ) ∈ gr(RK )(1) denote the image of gi in the associated graded. By Theorem 3.1 and Remark 3.2 we have the following. Proposition 4.2. If g is nondegenerate, then the Koszul complex K• (gr(RK ), {gr(gi )}i∈S ) is acyclic in positive dimension and



dimK gr(RK )

gr(gi )gr(RK ) = Ng .

i∈S

Let a ∈ K n and define Di,a = xi

∂ + ai + πgi ∂xi

for i = 1, . . . , n. Although we use the same notation as in sections 2 and 3, these differential operators generalize those of the earlier sections in that the Teichmüller lifting fˆ of f is replaced by an arbitrary Laurent polynomial g. Theorem 4.3. If g is nondegenerate, the Koszul complex K• (RK , {Di,a }i∈S ) is acyclic in positive dimension and  Di,a RK = Ng . dimK RK i∈S

Proof. Relative to the filtration w on RK , the associated graded of the complex K• (RK , {Di,a }i∈S ) is the complex K• (gr(RK ), {π gr(gi )}i∈S ), so the assertions of the theorem follow from Proposition 4.2.   Remark 4.4. By standard facts in commutative algebra,

it follows from the proof that that form a basis for gr(R )/ any elements of R K K i∈S gr(gi )gr(RK ) also form a

basis for RK / i∈S Di,a RK . Remark 4.5. If a ∈ Lg ⊗Q K, then as operators on RK the {Di,a }ni=1 are linear combinations of the {Di,a }i∈S . Hence when g is nondegenerate dimK RK

n 

Di,a RK = Ng .

i=1

Finally, we want to show that the spaces RK / ni=1 Di,a RK really generalize the spaces L/ ni=1 Di,a L of Theorem 3.10. Suppose the hypothesis of Theorem 3.10 holds and take g = fˆ. The inclusion RK → L induces a map RK

n  i=1

n 

Di,a RK → L

i=1

Di,a L.

13

Exponential sums and generalized hypergeometric functions

Proposition 4.6. Under the hypothesis of Theorem 3.10, this map is an isomorphism.

Proof. The proof of Theorem 3.7 shows that L/ ni=1 Di,a L has a basis consisting of monomials (see Theorem 3.6(b)), hence the map is surjective. But by Theorem 3.10, Lemma 4.1, and Remark 4.5, both quotients have the same dimension.  

5 Contiguity mapping For the remainder of this article, we assume a ∈ Lg ⊗Q K (see Remark 4.5). Set Wa = RK

n 

Di,a RK .

i=1

Let u ∈ Zn ∩ C(g). Since multiplication by x u maps RK into itself and satisfies Di,a−u  x u = x u  Di,a , it induces a map x u : Wa → Wa−u . We shall determine when this map is an isomorphism. For this, it is necessary to study the faces of (g) and C(g). Let σ1 , . . . , σs be the codimension-one faces of (g) that contain the origin. Then C(σ1 ), . . . , C(σs ) are the codimension-one faces of C(g). Define linear forms l1 , . . . , ls on Lg by the following conditions: lj = 0

on Lσj ,

(5.1)

lj (Z ∩ Lg ) = Z, lj ≥ 0 on C(g).

(5.2) (5.3)

n

Condition (5.1) determines lj up to a scalar multiple, while conditions (5.1) and (5.2) together determine lj up to sign. Equivalently, one may choose a basis v1 , . . . , vdg for Zn ∩ Lg such that v1 , . . . , vdg −1 is a basis for Zn ∩ Lσj and such that vdg lies on the same side of Lσj as C(g) and then define lj by the conditions lj (v1 ) = · · · = lj (vdg −1 ) = 0

(5.4)

lj (vdg ) = 1.

(5.5)

The following lemma is straightforward. Lemma 5.1. For u ∈ Lg , one has u ∈ C(g) if and only if lj (u) ≥ 0 for j = 1, . . . , s. Furthermore, no proper subset of {l1 , . . . , ls } has this property. By abuse of notation, we shall also write lj for any extension of lj to a linear form on Rn . Thus n  lj = αij xi (5.6) i=1

14

Alan Adolphson

with αij ∈ Q. Define differential operators lj (Da ) by lj (Da ) =

n 

αij Di,a .

(5.7)

i=1

As an operator on RK , lj (Da ) is independent of the choice of extension of lj . To see this, let u ∈ Zn ∩ Lg . Then n 

αij Di,a (x u ) =

i=1

n 

αij (ui + ai + πgi )x u

i=1 u

= lj (a + u)x +



πgν lj (ν)x

u+ν

(5.8) .

ν∈

This is independent of the extension of lj to Rn because a, u, ν ∈ Lg ⊗Q K. Recall the Pochhammer notation (for j a nonnegative integer): 1 if j = 0, (a)j = a(a + 1) . . . (a + j − 1) if j > 0. Theorem 5.2. Suppose g is nondegenerate and let u ∈ Zn ∩ C(g). If s 

(lj (a − u))lj (u) = 0,

j =1

then

xu

: Wa → Wa−u is an isomorphism.

Proof. By Remark 4.5, Wa and Wa−u have the same dimension Ng , so it suffices to show the map is surjective. Since Wa−u has a basis of monomials, it suffices to show that, given v ∈ Zn ∩ C(g), there exists ξ ∈ RK such that v

u

x ≡x ξ

(mod

n 

Di,a−u RK ).

i=1

If lj (v) ≥ lj (u) for all j , then v − u ∈ Zn ∩ C(g) by Lemma 5.1 and we may take ξ = x v−u . Otherwise, we may suppose, for example, that l1 (u) > l1 (v). Then by (5.8)  πgν l1 (ν)x v+ν . (5.9) l1 (Da−u )(x v ) = l1 (a − u + v)x v + ν∈

Since x ν is a monomial appearing in g, we have l1 (ν) ≥ 0. Thus all monomials x v+ν appearing in the sum on the right-hand side of (5.9) with nonzero coefficient satisfy l1 (v + ν) > l1 (v). Since l1 (a − u) ≤ l1 (a − u + v) < l1 (a),

Exponential sums and generalized hypergeometric functions

15

our hypothesis implies that l1 (a − u + v) = 0. Hence by (5.9), 

xv ≡

bw x w

(mod

n 

Di,a−u RK )

(5.10)

i=1

l1 (w)>l1 (v)

where bw ∈ K and the sum is over a finite set of w. This argument may be reapplied to each x w on the right-hand side of (5.10) for which l1 (w) < l1 (u). After finitely many iterations, we will arrive at a relation v

x ≡

 w

c x w

w

(mod

n 

Di,a−u RK )

i=1

where l1 (w ) ≥ l1 (u) for all w . Furthermore, it is straightforward to check that the w produced by this procedure satisfy lj (w ) ≥ lj (v) for j = 2, . . . , s. Applying the   argument successively to l2 , . . . , ls establishes the theorem. By Theorem 5.2, x u : Wa → Wa−u is an isomorphism for all u ∈ Zn ∩ C(g) if lj (a) is not a positive integer for any j = 1, . . . , s. We give a geometric interpretation of this condition. Proposition 5.3. The value lj (a) is not a positive integer if and only if a ∈ (Zn ∩ (C(g) \ C(σj ))) + Lσj .

(5.11)

Proof. Since lj assumes positive integral values at all points lying in the right-hand side of (5.11), the “only if” part of the proposition is clear. So suppose lj (a) is a positive integer. By (5.2) there exists u ∈ Zn ∩ Lg such that lj (u) = lj (a), i.e., a − u ∈ Lσj . Note that for j = j , there exist elements v of Zn ∩ C(σj ) for which lj (v) is arbitrarily large. (Otherwise lj would vanish on C(σj ), contradicting the construction of l1 , . . . , ls .) By adding such elements of Zn ∩ C(σj ) to u, we may assume that u ∈ Zn ∩C(g). It follow that a ∈ u+Lσj , where u ∈ Zn ∩(C(g)\C(σj )).   We shall call a semi-nonresonant if lj (a) is not a positive integer for j = 1, . . . , s. Corollary 5.4. If g is nondegenerate and a is semi-nonresonant, then the map x u : Wa → Wa−u is an isomorphism for all u ∈ Zn ∩ C(g).

n = K[x u | u ∈ Zn ∩ L ]. We want to describe W = R / Let RK g a i=1 Di,a RK . K Note that the approach of Theorem 4.3, namely, approximating Di,a by gr(gi ) will not /(g R + g R ) = 0, while we shall work. For example, if g = x1 + x2 , then RK 1 K 2 K see that Wa = 0. Lemma 5.5. If g is nondegenerate and a is semi-nonresonant, then the natural map is an isomorphism. Wa → Wa induced by the inclusion RK → RK

16

Alan Adolphson

. Then x u ξ ∈ R for some u ∈ Zn ∩ C(g). By Corollary 5.4, Proof. Let ξ ∈ RK K there exists η ∈ RK such that

xuη ≡ xuξ

(mod

n 

Di,a−u RK ).

i=1

Dividing this equation by x u gives η≡ξ

(mod

n 

Di,a (x −u RK )),

i=1

which proves the surjectivity of Wa → Wa . . There exists Now let ξ ∈ RK and suppose ξ = ni=1 Di,a (ηi ) with all ηi ∈ RK n −u u ∈ Z ∩ C(g) such that ηi = x ηi with ηi ∈ RK for all i. Then ξ=

n

n  i=1

Di,a (x −u ηi )

=

n  i=1

x −u Di,a−u (ηi ),

i=1 Di,a−u (ηi ).

u = This says that ξ lies in the kernel of the map

n x : Wa → Wa−u , hence by Corollary 5.4 there exist ηi ∈ RK such that ξ = i=1 Di,a (ηi ),   proving the injectivity of the map Wa → Wa .

i.e., x u ξ

Corollary 5.6. If g is nondegenerate, then dimK Wa = Ng . , hence Proof. For all u ∈ Zn ∩ Lg , multiplication by x u is an automorphism of RK u there is an induced isomorphism x : Wa → Wa−u . In particular, dimK Wa = dimK Wa−u .

(5.12)

Given a ∈ Lg , we can always choose u ∈ Zn ∩ C(g) such that lj (a − u) is not a positive integer for any j , i.e., such that a − u is semi-nonresonant. (For example, pick u so that a − u is in the relative interior of −C(g).) Then by Lemma 5.5, . dimK Wa−u = dimK Wa−u

(5.13)

The assertion of the corollary now follows from (5.12), (5.13), and Remark 4.5.

 

For future reference we record some related results. If we shift the indexing so u u that Wa is the target of multiplication sby x rather than the source,u then the map x : Wa+u → Wa is an isomorphism if j =1 (lj (a))lj (u) = 0. Thus x : Wa+u → Wa is an isomorphism for all u ∈ Zn ∩ C(g) if lj (a) is not a nonpositive integer for any j . We call a nonresonant if lj (a) ∈ Z for any j . Proposition 5.7. The value lj (a) is not an integer if and only if a ∈ (Zn ∩ Lg ) + Lσj .

Exponential sums and generalized hypergeometric functions

17

Proof. Since lj assumes integral values on (Zn ∩Lg )+Lσj , the “only if” part is obvious. Suppose lj (a) ∈ Z. By (5.2) there exists u ∈ Zn ∩ Lg such that lj (u) = lj (a), i.e., lj (a − u) = 0. Thus a − u ∈ Lσj and a = u + (a − u) ∈ (Zn ∩ Lg ) + Lσj .

 

Remark 5.8. Suppose that g is nondegenerate and a is nonresonant. Then by are isomorphisms for all u ∈ Lemma 5.5, both Wa → Wa and Wa−u → Wa−u n Z ∩ Lg . Thus when a is nonresonant we can define for all u ∈ Zn ∩ Lg an isomorphism x u : Wa → Wa−u as the composition xu

→ Wa−u , Wa → Wa −→ Wa−u

(5.14)

where the first arrow is the isomorphism of Lemma 5.5 and the third arrow is the inverse of the isomorphism of Lemma 5.5. When u ∈ Zn ∩ C(g), this map coincides with the map induced by multiplication by x u .

6 Cohomology with parameters In this section, we consider what happens when the coefficients of our polynomial g vary with parameters. Let λ1 , . . . , λN be indeterminates over K and let  gν (λ)x ν ∈ K[λ][x1 , x1−1 , . . . , xn , xn−1 ], (6.1) g(λ, x) = ν∈

where we write λ for λ1 , . . . , λN . We also put gi (λ, x) = xi ∂g/∂xi and set Di,a,λ = xi

∂ + ai + πgi (λ, x). ∂xi

(6.2)

We shall study how Wa,λ = RK(λ)

n 

Di,a,λ RK(λ)

i=1

varies with λ. We introduce some notation. Let B be a subring of K[λ] containing Z and the gν (λ). Set RB = B[x u | u ∈ Zn ∩ C(g)]. Let RB,m/M be the B-submodule of RB generated by all x u with w(u) ≤ m/M and (m/M) be the B-submodule generated by all x u with w(u) = m/M. let RB N

g . To insure the theory is not vacuous, Choose a collection of monomials {x uj }j =1 (0) we assume that for some specialization λ ∈ K¯ N , g(λ(0) , x) is nondegenerate and

18

Alan Adolphson

n

(0) i=1 gr(gi (λ , x))gr(RK(λ(0) ) ). Let N g , let WB,m/M be the BWB be the B-submodule of RB generated by the {x uj }j =1 (m/M) u be the submodule generated by those x j with w(uj ) ≤ m/M, and let WB u B-submodule generated by those x j with w(uj ) = m/M.

these monomials form a basis for gr(RK(λ(0) ) )/

N

g Remark 6.1. We observe that all the monomials {x uj }j =1 satisfy w(uj ) ≤ dg . From Proposition 4.2 (acyclicity of the Koszul complex on gr(RK(λ(0) ) ) defined by {gr(gi (λ(0) , x))}i∈S , |S| = dg ), it follows

that the dimension of the subspace of elements of degree m/M in gr(RK(λ(0) ) )/ ni=1 gr(gi (λ(0) , x))gr(RK(λ(0) ) ) depends only on (g). By induction on the dimension and simplicial subdivision, we can reduce to the case where (g) is a simplex. Let v1 , . . . , vdg be its vertices (in addition to the origin). Then x v1 , . . . , x vdg can be written as linear combinations of {gr(gi (λ(0) , x))}i∈S .

dg ci vi with ci ∈ Q, ci ≥ 0. If now v ∈ Zn ∩ C(g) with w(v) ≥ dg , write v = i=1

dg Since i=1 ci = w(v) ≥ dg , we have ci ≥ 1 for some i, say, i = 1. Put

v = (c1 − 1)v1 +

dg 

ci vi ∈ Zn ∩ C(g).

i=2

Then v = v1

+ v ,

hence

x v = x v1 x v ∈

 i∈S

gr(gi (λ(0) , x))gr(RK(λ(0) ) ).

Consider the map (m/M−1) n

φm : (RB

(m/M)

) → RB

defined by φm (η1 , . . . , ηn ) = terms of weight m/M in

n 

gi (λ, x)ηi .

i=1

Relative to monomial bases, this map is represented by a matrix Am with entries in B. By Remark 6.1, we know that for some specialization λ(0) this map is surjective when m/M > dg . It follows that in this case, Am has a square submatrix A m (whose size is the number of monomials of weight m/M in RB ) such that bm/M := det A m is not (m/M) there exist ζ ∈ RB,(m−1)/M , zero. By Cramer’s Rule, we conclude that for ξ ∈ RB (m/M−1) ηi ∈ RB , such that bm/M ξ = ζ +

n  i=1

gi (λ, x)ηi .

(6.3)

19

Exponential sums and generalized hypergeometric functions

We show that (6.3) can be achieved using only finitely many of the bm/M . For m/M > dg , we can write uniquely m s = dg + t + , M M where s, t are integers, 0 < s ≤ M, t ≥ 0.

(6.4)

(d +t+s/M)

Proposition 6.2. Let m, s, t be as in (6.4) and suppose ξ ∈ RB g exist ζ ∈ RB,dg +t+(s−1)/M , ηi ∈

(d +t−1+s/M) RB g

bdg +s/M ξ = ζ +

n 

. Then there

such that

gi (λ, x)ηi .

i=1

Proof. The proof is by induction on t. For t = 0 the proposition holds by (6.3), so we assume that t ≥ 1 and that the proposition is valid for t − 1. We may assume ξ is a monomial, say, ξ = x v with w(v) = dg + t + s/M. The point v lies over some (dg − 1)-simplex on some (dg − 1)-dimensional face of (g). Let v1 , . . . , vdg be the vertices of this simplex. The argument of Remark 6.1 show that x v = x v1 x v , where w(v1 ) = 1, w(v ) = w(v) − 1, and v1 , v ∈ Z ∩ C(g). By the induction hypothesis,

bdg +s/M x v = ζ +

n 

gi (λ, x)ηi ,

(6.5)

i=1 (d +t−2+s/M)

where ζ ∈ RB,dg +t−1+(s−1)/M , ηi ∈ RB g

bdg +s/M x v = x v1 ζ +

n 

. Multiplying by x v1 gives

gi (λ, x)x v1 ηi .

(6.6)

i=1 (d +t−1+s/M)

Write x v1 ηi = ηi + ζi , where ηi ∈ RB g

, ζi ∈ RB,dg +t−1+(s−1)/M . Then

n n     bdg +s/M x v = x v1 ζ + gi (λ, x)ζi + gi (λ, x)ηi ,

with x v1 ζ

+

n

i=1

i=1 gi (λ, x)ζi

(6.7)

i=1

∈ RB,dg +t+(s−1)/M . This establishes the proposition.  

We now treat the case m/M ≤ dg . Consider the map (m/M)

φm : WB

(m/M−1) n

⊕ (RB

(m/M)

) → RB

defined by φm (ζ, η1 , . . . , ηn ) = terms of weight m/M in ζ +

n  i=1

gi (λ, x)ηi .

20

Alan Adolphson

Relative to monomial bases, it is represented by a matrix with entries in B. If we take bm/M to be the determinant of any square submatrix of maximal size, then we have (m/M) , by Cramer’s Rule that for ξ ∈ RB bm/M ξ =



Aj x u j + ζ +

n 

gi (λ, x)ηi ,

i=1

w(uj )=m/M (m/M−1)

. Since we are assuming there exists where Aj ∈ B, ζ ∈ RB,(m−1)/M , ηi ∈ RB a specialization λ(0) which makes φm surjective, there exists a choice of submatrix such that bm/M = 0. We assume that this choice has been made. We have proved the following result. Proposition 6.3. For 0 ≤ m ≤ dg M, there exists a nonzero element bm/M ∈ B such (m/M) , that for ξ ∈ RB bm/M ξ =



Aj x u j + ζ +

n 

gi (λ, x)ηi ,

i=1

w(uj )=m/M (m/M−1)

with Aj ∈ B, ζ ∈ RB,(m−1)/M , ηi ∈ RB

.

Remark 6.4. Since B ⊆ K[λ], we may regard the bm/M as polynomials in λ. ¯ N is such that g(λ(0) , x) has the same Newton The proof shows that if λ(0) ∈ (K) M(dg +1) Ng polyhedron as g(λ, x) and m=0 bm/M (λ(0) ) = 0, then {x uj }j =1 is a basis for

n (0) RK(λ(0) ) / i=1 gi (λ , x)RK(λ(0) ) and hence also a basis for Wa,λ(0) := RK(λ(0) )

n  i=1

Di,a,λ(0) RK(λ(0) ) .

We may also regard the Aj , ζ , and ηi in Propositions 6.2 and 6.3 as polynomials in λ. We shall need bounds for their degrees. Set (m/M) M(dg +1) }m=0 .

κ = degλ g(λ, x) · max{dimK RK

Proposition 6.5. For Aj , ζ , ηi as in Proposition 6.2 or 6.3, degλ Aj , degλ ζ, degλ ηi ≤ κ + degλ ξ. Proof. The polynomials Aj , ζ , ηi were obtained by using Cramer’s Rule to solve a certain system of equations. The coefficients of this system of equations are polynomials in λ of degree ≤ degλ g(λ, x), so the desired bound follows immediately from Cramer’s Rule. Since bm/M is the determinant of this coefficient matrix, we also have degλ bm/M ≤ κ

for m = 0, 1, . . . , M(dg + 1).

(6.8)  

21

Exponential sums and generalized hypergeometric functions

We conclude this section with a result on the b-invariant that will be used in section 8. M(d +1) bm/M (λ(0) ) = 0. Then Proposition 6.6. Suppose λ(0) ∈ K¯ N satisfies m=0g (0) g(λ , x) is nondegenerate relative to (g) and dimK(λ(0) ) Wa,λ(0) = Ng . In particN

g is a basis for Wa,λ(0) . ular, {x uj }j =1

Proof. If g(λ(0) , x) is nondegenerate relative to (g), then Theorem 4.3 implies that dimK(λ(0) ) Wa,λ(0) = Ng . Furthermore, it follows immediately from Propositions 6.2 M(d +1) Ng bm/M (λ(0) ) = 0, then {x uj }j =1 spans Wa,λ(0) , hence it is a baand 6.3 that if m=0g M(d +1) sis for Wa,λ(0) . Thus it suffices to prove that the nonvanishing of m=0g bm/M (λ(0) ) implies the nondegeneracy of g(λ(0) , x). Let τ be a face of (g) that does not contain the origin and let gr(RK(λ(0) ) )τ be the subring of gr(RK(λ(0) ) ) generated by those monomials whose exponents lie in C(τ ).

Let gτ = ν∈τ ∩ gν (λ(0) )x ν , a homogeneous element of degree 1 in the graded ring gr(RK(λ(0) ) )τ . If for every such τ we have dimK(λ(0) ) gr(RK(λ(0) ) )τ

n  i=1

gτ,i gr(RK(λ(0) ) )τ < ∞

(6.9)

(where gτ,i = xi ∂gτ /∂xi ), then [5, Théorème 6.2] implies that g(λ(0) , x) is nondegenerate. M(d +1) By Propositions 6.2 and 6.3, the nonvanishing of m=0g bm/M (λ(0) ) implies that

N g {x uj }j =1 spans gr(RK(λ(0) ) )/ ni=1 gr(gi (λ(0) , x))gr(RK(λ(0) ) ). We establish (6.9) by showing that the subset {x uj }uj ∈C(τ ) spans the quotient gr(RK(λ(0) ) )τ

n  i=1

gτ,i gr(RK(λ(0) ) )τ .

Let ξ be a homogeneous element of degree m/M in gr(RK(λ(0) ) )τ . We have ξ=

 uj ∈C(τ ) w(uj )=m/M

cj x u j +

n  (gr(gi (λ(0) , x))ηi )τ ,

(6.10)

i=1

where ci ∈ K(λ(0) ), ηi ∈ gr(RK(λ(0) ) )(m/M−1) , and the subscript τ appearing on the right-hand side means that we select only those terms from the product gr(gi (λ(0) , x))ηi that lie in gr(RK(λ(0) ) )τ . Consider a monomial x u appearing in gr(gi (λ(0) , x)) and a monomial x v appearing in ηi . Their product in gr(RK(λ(0) ) ) equals either 0 or x u+v . If it equals x u+v , then u and v lie over a common face τ1 of (g). If also u + v ∈ C(τ ),

22

Alan Adolphson

then u + v ∈ C(τ ) ∩ C(τ1 ) = C(τ ∩ τ1 ). But since τ ∩ τ1 is a face of τ1 , the conditions u, v ∈ C(τ1 ) and u + v ∈ C(τ ∩ τ1 ) imply u, v ∈ C(τ ∩ τ1 ) (⊆ C(τ )). It follows that (gr(gi (λ(0) , x))ηi )τ = gτ,i (ηi )τ .  

Substituting this into (6.10) gives (6.9).

7 Dual theory We define spaces of formal series   ∗ RK(λ) = ∗ = RK(λ)





Au x −u | Au ∈ K(λ)

u∈Zn ∩C(g)

 Au x −u | Au ∈ K(λ) .



u∈Zn ∩Lg

∗ × RK(λ) → K(λ) by Define a pairing RK(λ)  

  Au x −u , Bv x v = A u Bu . u

v

(7.1)

u

(This makes sense because the second sum on the left-hand side is finite.) This pairing induces identifications ∗ RK(λ) = HomK(λ) (RK(λ) , K(λ)),

∗ RK(λ) = HomK(λ) (RK(λ) , K(λ)).

For i = 1, . . . , n, let ∗ Di,a,λ = −xi

∂ + ai + πgi (λ, x). ∂xi

∗ , ξ ∈ R For ξ ∗ ∈ RK(λ) K(λ) , one checks that ∗ (ξ ∗ ), ξ . ξ ∗ , Di,a,λ (ξ ) = Di,a,λ ∗ ∗ Let γ− : RK(λ) → RK(λ) be the natural projection:     Au x −u = γ− u∈Zn ∩L

g

Au x −u .

(7.2)

(7.3)

u∈Zn ∩C(g)

∗ , ξ ∈ RK(λ) , we have For ξ ∗ ∈ RK(λ) ∗ (ξ ∗ ), ξ . ξ ∗ , Di,a,λ (ξ ) = γ− Di,a,λ

(7.4)

Exponential sums and generalized hypergeometric functions

23

∗ , ξ ∈ R We observe for future reference that if ξ ∗ ∈ RK(λ) K(λ) , then

(I − γ− )(ξ ∗ ), ξ  = 0,

(7.5)

where I is the identity map.

n

nIt follows from (7.2) and (7.4) that the annihilators of i=1 Di,a,λ RK(λ) and i=1 Di,a,λ RK(λ) are, respectively, ∗ ∗ | γ− Di,a,λ (ξ ∗ ) = 0 for i = 1, . . . , n} Ka,λ = {ξ ∗ ∈ RK(λ) ∗ ∗ = {ξ ∗ ∈ RK(λ) | Di,a,λ (ξ ∗ ) = 0 for i = 1, . . . , n.} Ka,λ

Thus there are identifications Ka,λ = HomK(λ) (Wa,λ , K(λ)) = HomK(λ) (Wa,λ , K(λ)). Ka,λ

(7.6) (7.7)

Proposition 7.1. If g(λ, x) is nondegenerate (as polynomial in x with coefficients in K(λ)), then dimK(λ) Ka,λ = dimK(λ) Ka,λ = Ng .

Proof. The proposition follows immediately from Remark 4.5, Corollary 5.6, (7.6), and (7.7).   into Ka,λ . In fact, the One can check from the definitions that γ− maps Ka,λ ∗ ∗ projection γ− : RK(λ) → RK(λ) is adjoint under the pairing (7.1) to the inclusion , hence γ− : Ka,λ → Ka,λ is adjoint to the map Wa,λ → Wa,λ RK(λ) → RK(λ) induced by the inclusion. Lemma 5.5 then implies the following result.

Lemma 7.2. If g(λ, x) is nondegenerate and a is seminonresonant, then the induced →K map γ− : Ka,λ a,λ is an isomorphism. ∗ ∗ Since x u  Di,a−u,λ = Di,a,λ  x u , multiplication by x u induces an isomorphism for all u ∈ Zn ∩ L . If u ∈ Zn ∩ C(g), multiplication by x u → Ka,λ x u : Ka−u,λ g composed with γ− induces a map γ−  x u : Ka−u,λ → Ka,λ .

Lemma 7.3. If g(λ, x) is nondegenerate and a is seminonresonant, then γ−  x u : Ka−u,λ → Ka,λ is an isomorphism for all u ∈ Zn ∩ C(g). Proof. The lemma follows immediately from Corollary 5.4 and the fact that γ−  x u ∗ acting on RK(λ) is adjoint to multiplication by x u acting on RK(λ) .   For u ∈ Zn ∩ C(g), we define bw(u) as follows. For 0 ≤ w(u) ≤ dg , put b

w(u)

=

Mw(u)  i=0

bi/M ,

24

Alan Adolphson

where the bi/M are as in Proposition 6.3. For w(u) > dg , write w(u) = dg + t + s/M, with t ≥ 0 and 0 < s ≤ M. Then put bw(u) =

Mdg



bi/M ·

i=0

M 

bdg +i/M

s t  · bdg +i/M ,

i=1

i=1

where the bi/M are as in Propositions 6.2 and 6.3. N

∗ } g of K Theorem 7.4. Suppose g(λ, x) is nondegenerate. The basis {ξi,a,λ a,λ dual i=1 N

g to the basis {ξi = π w(ui ) x ui }i=1 of Wa,λ satisfies

∗ ξi,a,λ =



π −w(u) x −u

u∈Zn ∩C(g)

Gi,u (a, λ) , bw(u)

where Gi,u ∈ Z[a, λ] and dega Gi,u ≤ w(u) degλ Gi,u ≤ κMw(u).

(7.8) (7.9)

Proof. We show that the coefficient of x −u has the asserted form by induction on w(u). If w(u) = 0, then u = (0, . . . , 0) and x u = 1 is one of the basis elements ξi , so the coefficient of x −u is either 0 or 1. Thus the assertion is valid in this case. If 0 < w(u) ≤ dg , put h = w(u). Otherwise, write w(u) = dg + t + s/M with t ≥ 0, 0 < s ≤ M, and put h = dg + s/M. By Propositions 6.2 and 6.3 we can write bh x u =



Aj,u π −w(u) ξj + ζ +

n 

gi (λ, x)ηi ,

(7.10)

i=1

w(uj )=w(u)

where Aj,u ∈ B,

ζ ∈ RB,w(u)−1/M ,

(w(u)−1)

ηi ∈ RB

,

(7.11)

and degλ Aj,u , degλ ζ, degλ ηi ≤ κ.

(7.12)

It follows that ∗ ∗ , x u  = π −w(u) Ai,u + ξi,a,λ ,ζ + bh ξi,a,λ

n  ∗ ξi,a,λ , gk (λ, x)ηk  k=1

∗ , ζ  − π −1 = π −w(u) Ai,u + ξi,a,λ

n  ∗ ξi,a,λ , (xk ∂/∂xk + ak )(ηk ), k=1

25

Exponential sums and generalized hypergeometric functions

∗ annihilates the image of Dk,a,λ = xk ∂/∂xk + ak + πgk (λ, x) under the since ξi,a,λ ∗ pairing. Thus the coefficient of x −u in ξi,a,λ is −1 ∗ −1 −1 π −w(u) b−1 h Ai,u + bh ξi,a,λ , ζ  − π bh

n  ∗ ξi,a,λ , (xk ∂/∂xk + ak )(ηk ). k=1

The theorem follows from (7.11) and (7.12) by applying the induction hypothesis to each term in this expression.   N

g is a basis for both Wa,λ and If g(λ, x) is nondegenerate, then {ξi = π w(ui ) x ui }i=1 n Wa−u,λ . For u ∈ Z ∩ Lg and a nonresonant, let M(a, a − u, λ) be the matrix of → W x u : Wa,λ → Wa−u,λ (or x u : Wa,λ a−u,λ ) relative to this basis (see Remark 5.8). n When u ∈ Z ∩ C(g), the contiguity mapping is defined for all a, so in this case we can define M(a, a − u, λ) for all a as the matrix of x u : Wa,λ → Wa−u,λ relative to this basis. By Corollary 5.4 we have the following result.

Proposition 7.5. If g(λ, x) is nondegenerate and a is seminonresonant, then det M(a, a − u, λ) = 0 for all u ∈ Zn ∩ C(g). Using Theorem 7.4, we can give more precise information about this matrix. Proposition 7.6. For u ∈ Zn ∩ C(g), the entries of M(a, a − u, λ) are polynomials in a and rational functions in λ. The irreducible factors of the denominators of these rational functions are factors of bdg +1 . Proof. By definition, u

x ξj ≡

Ng 

M(a, a − u, λ)ij ξi

(mod

i=1

n 

Dk,a−u,λ RK(λ) ),

k=1

hence ∗ , x u ξj . M(a, a − u, λ)ij = ξi,a−u,λ

(7.13)

Thus if ξj = π w(uj ) x uj , then ∗ , π w(uj ) x u+uj  M(a, a − u, λ)ij = ξi,a−u,λ

= π w(uj )−w(u+uj ) Gi,u+uj (a − u, λ)/bw(u+uj )

(7.14)  

by Theorem 7.4.

Proposition 7.7. Suppose that g(λ, x) is nondegenerate and a is nonresonant. The Ng dual to the basis {ξ = π w(ui ) x ui }Ng of W is given by }i=1 of Ka,λ basis {ξi,a,λ i a,λ i=1  = x −u Hi,u (a, λ), ξi,a,λ u∈Zn ∩Lg

26

Alan Adolphson

where for u ∈ Zn ∩ C(g) Hi,u (a, λ) = π −w(u) Gi,u (a, λ)/bw(u) and for u, v ∈ Zn ∩ Lg Hi,u (a, λ) =

Ng 

M(a − v, a, λ)ij Hj,u+v (a − v, λ).

j =1

Proof. When u ∈ Zn ∩ C(g), the formula for Hi,u follows from Theorem 7.4 and the →K fact that γ− : Ka,λ a,λ is adjoint to Wa,λ → Wa,λ . n −v Suppose u, v ∈ Z ∩ Lg . The map x : Ka,λ → Ka−v,λ is dual to the map −v −v x : Wa−v,λ → Wa,λ , so the matrix of x : Ka,λ → Ka−v,λ is the transpose of M(a − v, a, λ). Thus x −v ξi,a,λ

=

Ng  j =1

M(a − v, a, λ)ij ξj,a−v,λ .

We have , xu Hi,u (a, λ) = ξi,a,λ

=

Ng  j =1

=

Ng 

M(a − v, a, λ)ij ξj,a−v,λ , x u+v 

M(a − v, a, λ)ij Hj,u+v (a − v, λ).

j =1

 

Remark 7.8. For v ∈ Zn ∩ C(g), this gives a description of Hi,u for all u ∈ −v + (Zn ∩ C(g)) in terms of the Gi,u+v , bw(u) , and M(a − v, a, λ). Remark 7.9. The proof shows that if we only assume a to be seminonresonant, then the conclusion of Proposition 7.7 is still true if we make the additional hypothesis that v ∈ Zn ∩ C(g).

8 Dual of the contiguity mapping Throughout this section we assume u ∈ Zn ∩ C(g). The purpose of this section is to apply the dual theory to prove the converse of Theorem 5.2. More precisely, the dual of Theorem 5.2 is the assertion that, for g(λ, x) nondegenerate, if si=1 (li (a))li (u) = 0,

27

Exponential sums and generalized hypergeometric functions

then γ−  x u : Ka,λ → Ka+u,λ is an isomorphism. We shall prove the following result. Theorem 8.1. If

s

i=1 (li (a))li (u)

= 0, then ker(γ−  x u | Ka,λ ) = (0).

Recall that M(a + u, a, λ) is the matrix of x u : Wa+u,λ → Wa,λ relative to the Ng basis {ξi = π w(ui ) x ui }i=1 . It follows that the transpose M(a + u, a, λ)t is the matrix N

N

g ∗ } g , {ξ ∗ of γ−  x u : Ka,λ → Ka+u,λ relative to the bases {ξi,a,λ i=1 i,a+u,λ }i=1 . We shall sprove Theorem 8.1 by establishing a series of lemmas that show the vanishing of i=1 (li (a))li (u) implies the vanishing of det M(a + u, a, λ). Let σi be the face of (g) lying in the hyperplane li = 0 and let (as in section 3) dσi = dim σi , Nσi = (dσi )!V (σi ), a positive integer.

Remark 8.2. The following observation will be useful in the proof of the next lemma. Ng Recall that in section 6 we chose a set of monomials {x uj }j =1 which is a basis for n 

gr(RK(λ) )

gr(gi (λ, x))gr(RK(λ) )

(8.1)

i=1

and hence also a basis for Wa,λ = RK(λ)

n 

Di,a,λ RK(λ) .

(8.2)

i=1

The matrix M(a + u, a, λ) is defined relative to (scalar multiples of) this basis. ConNg . We get two change-of-basis sider what happens if we choose a different basis {x vj }j =1 matrices, one for each of the spaces (8.1) and (8.2): x

uj

≡

Ng 

αij x

vi

(mod

x

≡

Ng  i=1

gr(gi (λ, x))gr(RK(λ) ))

(8.3)

Di,a,λ RK(λ) ),

(8.4)

i=1

i=1 uj

n 

βij x

vi

(mod

n  i=1

where the αij depend only on λ but the βij depend on λ and a. Since the vector space (8.1) is graded, it follows that αij = 0 if w(vi ) = w(uj ). Furthermore, since (8.1) is the “associated graded” to (8.2), βij = αij for w(vi ) ≥ w(uj ). Thus the matrix (αij ) has “block diagonal” form (where each block corresponds to the basis elements of a given weight) and the matrix (βij ) has “block triangular” form with the same diagonal blocks as (αij ). We conclude that det(βij ) = det(αij ), and, in particular, det(βij ) depends only on λ, not on a. Lemma 8.3. Suppose that li (u) > 0 for some i. det M(a + u, a, λ) is divisible by li (a)Nσi .

Then (as polynomials in a)

28

Alan Adolphson

Proof. We prove a stronger statement, namely, that there are at least Nσi rows of M(a + u, a, λ) with the property that all of their entries are divisible by li (a). To establish this, we claim that it suffices to show that there are at least Nσi rows of M(a + u, a, λ) with the property that all of their entries (as functions of a) vanish whenever li (a) = 0. For by Proposition 7.6, the entries of M(a + u, a, λ) all have the form  b−t (dg +1) Pi1 ···iN (a)λi11 · · · λiNN , i1 ,...,iN ≥0

where t is a positive integer, the sum is finite, and each Pi1 ···iN (a) is a polynomial. If this expression vanishes whenever li (a) = 0, then each Pi1 ···iN (a) vanishes whenever li (a) = 0. But li (a) is irreducible as polynomial in a, so the Nullstellensatz implies that each Pi1 ···iN (a) is divisible by li (a). So fix a with li (a) = 0. We denote by Kσi ,a,λ (resp. Wσi ,a,λ ) the K-space (resp. W -space) defined using gσi in place of g. (It is here that we use the assumption li (a) = 0: the definitions of Kσi ,a,λ and Wσi ,a,λ require that a ∈ Lσi ⊗Q K.) We first show that Kσi ,a,λ ⊆ Ka,λ . Let ξ ∈ Kσi ,a,λ , i.e., γ− (xj ∂/∂xj − aj − πgσi ,j (λ, x))(ξ ) = 0

(8.5)

for j = 1, . . . , n. We need to show γ− (xj ∂/∂xj − aj − πgj (λ, x))(ξ ) = 0

(8.6)

for j = 1, . . . , n. The difference between the left-hand sides of (8.5) and (8.6) is γ− (πgj − πgσi ,j )ξ . But the exponents of all monomials appearing in gj − gσi ,j lie in the region li > 0 and the exponents of all monomials appearing in ξ lie in the hyperplane li = 0. Thus the exponents of all monomials appearing in (πgj −πgσi ,j )ξ lie in the region li > 0. This implies γ− ((πgj − πgσi ,j )ξ ) = 0, hence (8.5) implies (8.6). We may assume that our monomial basis for Wa,λ contains a basis for Wσi ,a,λ . For if not, we can replace it by a new basis with this property and, by Remark 8.2, det M(a + u, a, λ) will be multiplied by a rational function in λ. This does not affect Nσ the conclusion of the lemma. Let {ξσ∗i ,j,a,λ }j =1i be the dual basis for Kσi ,a,λ . Then Nσ

N

g ∗ {ξσ∗i ,j,a,λ }j =1i ⊆ {ξj,a,λ }j =1 , the dual basis for Ka,λ . If li (u) > 0, then the exponents ∗ u of all monomials in x ξσi ,j,a,λ lie in the region li > 0, hence

γ− (x u ξσ∗i ,j,a,λ ) = 0 for j = 1, . . . , Nσi . This says that the Nσi columns of the matrix M(a + u, a, λ)t that Nσ

correspond to {ξσ∗i ,j,a,λ }j =1i all vanish.

Let B be the set of irreducible factors of bdg +1 .

 

Exponential sums and generalized hypergeometric functions

29

Lemma 8.4. Let u ∈ Zn ∩ C(g). Then det M(a + u, a, λ) =





b β(b ,u)

b ∈B

s li (u)−1   (li (a) + j )αi (u,j ) , i=1 j =0

where the αi (u, j ) are nonnegative integers and the β(b , u) are integers. Proof. By Proposition 7.6 we have Pu (a, λ) b(dg +1)t for some positive integer t, where Pu (a, λ) is a polynomial  in a and λ. Suppose (λ(0) , a (0) ) is a specialization with bdg +1 (λ(0) ) = 0 and si=1 (li (a (0) ))li (u) = 0. By Proposition 6.6, g(λ(0) , x) is nondegenerate, so we may apply Theorem 5.2 to conclude (0) , λ(0) ) = 0. Equivalently, the vanishing of P (a, λ) implies the vanishing that Pu (a u s (l (a)) . Thus, by the Nullstellensatz, Pu (a, λ) divides a power of of bdg +1 i l (u) i i=1  bdg +1 si=1 (li (a))li (u) . It follows that the irreducible factors of Pu (a, λ) lie among   the elements of B and the li (a) + j , i = 1, . . . , s, j = 0, . . . , li (u) − 1. det M(a + u, a, λ) =

Set Bu (λ) =





b β(b ,u) ,

b ∈B s li (u)−1   (li (a) + j )αi (u,j ) , ρ(a + u, a) = i=1 j =0

so Lemma 8.4 becomes det M(a + u, a, λ) = Bu (λ)ρ(a + u, a).

(8.7)

Since x u+v : Wa+u+v,λ → Wa,λ is the composition of x v : Wa+u+v,λ → Wa+u,λ and x u : Wa+u,λ → Wa,λ , we have det M(a + u + v, a, λ) = det M(a + u + v, a + u, λ) det M(a + u, a, λ), which implies Bu+v (λ)ρ(a + u + v, a) = Bv (λ)ρ(a + u + v, a + u)Bu (λ)ρ(a + u, a).

(8.8)

By unique factorization of polynomials, we have (up to a constant factor) s li (u+v)−1   (li (a) + j )αi (u+v,j ) = i=1

j =0 s li (v)−1 s li (u)−1     (li (a + u) + j )αi (v,j ) · (li (a) + j )αi (u,j ) . i=1 j =0

i=1 j =0

(8.9)

30

Alan Adolphson

Comparing exponents of common factors on the left-hand and right-hand sides of this equation, we see that we must have for 0 ≤ j ≤ li (u) − 1, αi (u, j ) (8.10) αi (u + v, j ) = αi (v, j − li (u)) for li (u) ≤ j ≤ li (u + v) − 1. We extend the definition of αi (u, j ) to all j ∈ Z by defining αi (u, j ) = 0 if j < 0 or j ≥ li (u). Let χli (u) be the characteristic function of the interval [0, li (u) − 1]. Lemma 8.5. There exists a function βi : Z → Z such that αi (u, j ) = βi (j )χli (u) (j ). Proof. Fix j ∈ Z. We must show that for all u1 , u2 ∈ Zn ∩ C(g) with 0 ≤ j ≤ min(li (u1 ) − 1, li (u2 ) − 1) we have αi (u1 , j ) = αi (u2 , j ). One then defines βi (j ) to be this common value. Choose v1 , v2 ∈ Zn ∩ C(g) such that u1 + v1 = u2 + v2 . Then by (8.10), αi (u1 , j ) = αi (u1 + v1 , j ) = αi (u2 + v2 , j ) = αi (u2 , j ), which is the desired assertion.

 

We immediately sharpen Lemma 8.5 by showing that each βi may be taken to be a constant function. Lemma 8.6. There exists δi ∈ Z such that αi (u, j ) = δi χli (u) (j ). Proof. We show that for all j ≥ 0, one has βi (j ) = βi (j + 1), which implies the lemma. For u, v ∈ Zn ∩ C(g), we have by (8.10) αi (u + v, j + li (u)) = αi (v, j ) for 0 ≤ j ≤ li (v) − 1. By Lemma 8.5, this means that βi (j + li (u))χli (u+v) (j + li (u)) = βi (j )χli (v) (j ).

(8.11)

It is not hard to check from the definitions that there exists w ∈ Zn ∩ C(g) with li (w) = 1. Let t be a positive integer. Replacing u by w and v by tw in (8.11) gives βi (j + 1)χt+1 (j + 1) = βi (j )χt (j ). For t sufficiently large we have χt+1 (j + 1) = χt (j ) = 1, which implies βi (j + 1) = βi (j ).   We summarize our analysis of det M(a + u, a, λ) in the following result.

31

Exponential sums and generalized hypergeometric functions

Theorem 8.7. For u ∈ Zn ∩ C(g), det M(a + u, a, λ) = Bu (λ)

s  ((li (a))li (u) )δi . i=1

Furthermore, the rational functions Bu (λ) satisfy Bu+v (λ) = Bu (λ)Bv (λ) for u, v ∈ Zn ∩ C(g). Proof. The first equation follows from equation (8.7) and Lemma 8.6. Lemma 8.6 also implies that the two sides of (8.9) are equal (not just equal up to a constant factor). The second equation then follows from equation (8.8).   Remark 8.8. Lemma 8.3 implies that δi ≥ Nσi (> 0). Thus Theorem 8.7 implies Theorem 8.1. It would be interesting to know if δi = Nσi for all i.

9 Action of Frobenius Instead of regarding λ as an indeterminate, in this section we regard it as a (variable) element of K N . For b ∈ R, b > 0, define    Au x −u | Au ∈ K, ord Au ≥ −bw(u) + c L∗ (b, c) = u∈Zn ∩C(g)

L∗ (b) =



L∗ (b, c).

c∈R

For a ∈

K n,

let ord a = inf i {ord ai }, ord λ = inf i {ord λi }, and set M(d +1)

cλ = −κM inf(0, ord λ) + M sup(0, {ord bm/M (λ)}m=0g 1 − inf(0, ord a). ca = p−1

)

Previously we defined B to be a subring of K containing Z and the gν (λ). Here, we assume in addition that B ⊆ OK . Lemma 9.1. If bdg +1 (λ) = 0, then Ka,λ ⊆ L∗ (ca + cλ ). Proof. By Theorem 7.4, it suffices to check that  ∗ = π −w(u) x −u Gi,u (a, λ)/bw(u) (λ) ξi,a,λ u∈Zn ∩C(g)

32

Alan Adolphson

lies in L∗ (ca + cλ ) for all i. But this is immediate from the definitions since (6.8), (7.8), and (7.9) imply ord

π −w(u) Gi,u (a, λ) ≥ −(ca + cλ )w(u). bw(u) (λ)

(9.1)  

Define

 L ∗ (b) = ξ ∗ =



 Au x −u | for all v ∈ Zn ∩ Lg , γ− (x −v ξ ∗ ) ∈ L∗ (b) .

u∈Zn ∩Lg

We derive some simpler conditions for checking whether a series lies in L ∗ (b). Lemma 9.2. Suppose that for every v ∈ Zn ∩ Lg there exists v ∈ Zn ∩ C(g) such that γ− (x −v−v ξ ∗ ) ∈ L∗ (b). Then ξ ∗ ∈ L ∗ (b).

n and v ∈ Zn ∩C(g) are such that γ− (x −v−v ξ ∗ ) ∈ L∗ (b). Proof. Suppose

v ∈ Z ∩Lg−u ∗ Write ξ = u∈Zn ∩Lg Au x . Then  γ− (x −v−v ξ ∗ ) = Au x −u−v−v ∈ L∗ (b), u∈Zn ∩Lg u+v+v ∈C(g)

so ord Au ≥ −bw(u + v + v ) + c. One also has γ− (x −v ξ ∗ ) =

 u∈Zn ∩L

(9.2)

Au x −u−v . g

u+v∈C(g)

By equation (9.2) and Lemma 2.1(b), if u + v, v ∈ C(g), then ord Au ≥ −bw(u + v) − bw(v ) + c, hence γ− (x −v ξ ∗ ) ∈ L∗ (b). Thus ξ ∗ ∈ L ∗ (b).

 

Corollary 9.3. If γ− (x −v ξ ∗ ) ∈ L∗ (b) for all v ∈ Zn ∩ C(g), then ξ ∗ ∈ L ∗ (b). Proof. Given v ∈ Zn ∩Lg , one can always find v ∈ Zn ∩C(g) such that v+v ∈ C(g).   By similar reasoning, one can prove the following result. Corollary 9.4. If γ− (x −v ξ ∗ ) ∈ L∗ (b) for all v ∈ pZn ∩ C(g), then ξ ∗ ∈ L ∗ (b). ⊆ L ∗ (c + c ). Corollary 9.5. If bdg +1 (λ) = 0, then Ka,λ a λ

Exponential sums and generalized hypergeometric functions

33

and let u ∈ Zn ∩C(g). Then x −u ξ ∗ ∈ K −u ∗ Proof. Let ξ ∗ ∈ Ka,λ a−u,λ , so γ− (x ξ ) ∈ ∗ Ka−u,λ . By Lemma 9.1, Ka−u,λ ⊆ L (ca−u + cλ ). Since ca−u = ca , this shows that   γ− (x −u ξ ∗ ) ∈ L∗ (ca + cλ ). Hence by Corollary 9.3, ξ ∗ ∈ L ∗ (ca + cλ ). ∗ → R ∗ by Define  : RK K

(ξ ∗ (x)) = ξ ∗ (x p ). Lemma 9.6. (a) (L ∗ (b)) ⊆ L ∗ (b/p). (b) For η ∈ L(b ), b > b > 0, multiplication by η defines an endomorphism of L ∗ (b).

Proof. Let ξ ∗ = u∈Zn ∩Lg Au x −u ∈ L ∗ (b). We show that (ξ ∗ ) ∈ L ∗ (b/p). By Corollary 9.4 it suffices to show that γ− (x −pv (ξ ∗ )) ∈ L∗ (b/p) for all v ∈ Zn ∩C(g), i.e., (γ− (x −v ξ ∗ )) ∈ L∗ (b/p). But ξ ∗ ∈ L ∗ (b) implies γ− (x −v ξ ∗ ) ∈ L∗ (b), so we ∗ follows easily from the definitions. are reduced to proving (L∗ (b))

⊆ L (b/p). This To prove part (b), let η = v∈Zn ∩C(g) Bv x v ∈ L(b ) and choose c ∈ R such that

ord Bv ≥ b w(v) + c for all v ∈ Zn ∩ C(g). Let ξ ∗ = u∈Zn ∩Lg Au x −u ∈ L ∗ (b) and let t ∈ Zn ∩ Lg . The coefficient of x −t in ηξ ∗ is  Ct = Bv Av+t . (9.3) v∈Zn ∩C(g)

Since γ− (x t ξ ∗ ) ∈ L∗ (b), there exists ct ∈ R such that ord Av+t ≥ −bw(v) + ct for all v ∈ Zn ∩ C(g), hence ord Bv Av+t ≥ b w(v) + c − bw(v) + ct = (b − b)w(v) + c + ct . Since b − b > 0, this proves the series (9.3) hence ηξ ∗ is a well-defined el converges,−t ∗ ∗ ement of RK . We now show that ηξ = t∈Zn ∩Lg Ct x ∈ L ∗ (b). By Corollary 9.3, it suffices to prove that for all s ∈ Zn ∩ C(g) we have     γ− Ct x −s−t = Ct x −s−t ∈ L∗ (b). t∈Zn ∩Lg

s+t∈Zn ∩C(g)

Since γ− (x −s ξ ∗ ) =



Au x −u−s ∈ L∗ (b),

u∈Zn ∩Lg u+s∈C(g)

there exists cs ∈ R such that ord Au ≥ −bw(u + s) + cs

(9.4)

34

Alan Adolphson

for u + s ∈ C(g). Suppose s + t ∈ C(g). Then v + t + s ∈ C(g) for v ∈ C(g), so we may put u = v + t in (9.3). It follows from (9.3) that ord Ct ≥ b w(v) + c − bw(v + t + s) + cs ≥ b w(v) + c − bw(v) − bw(s + t) + cs ≥ −bw(s + t) + c + cs .

But this says that s+t∈Zn ∩C(g) Ct x −s−t ∈ L∗ (b), which was the desired result. For g(λ, x) =

ν∈

 

gν (λ)x ν ∈ K[x1 , x1−1 , . . . , xn , xn−1 ], write  gσ ν Mσ ν (λ), gν (λ) = σ ∈Sν

where Mσ ν (λ) is a monomial in λ1 , . . . , λN and define   F (λ, x) = θ(gσ ν Mσ ν (λ)x ν ) ∈ K[[x1 , x1−1 , . . . , xn , xn−1 ]]. ν∈ σ ∈Sν

It follows from (2.1) that this definition makes sense provided, for example, that ord gσ ν ≥ 0 for all σ, ν and that (degλ g)(ord λ) > −(p − 1)/p 2 , in which case
p − 1

F (λ, x) ∈ L

p2

 + (degλ g) inf(0, ord λ) .

(9.5)

Lemma 9.7. Let µ ∈ Zn ∩ Lg and λ ∈ K N with 0
p−1 + p(degλ g) inf(0, ord λ). p

The map δλ,µ defined by δλ,µ (ξ ∗ (x)) = x −µ F (λ, x)ξ ∗ (x p ) is an injection of L ∗ (b) into L ∗ (b/p). Proof. It follows from (9.5) and our hypothesis that F (λ, x) ∈ L(b/p), hence Lemma 9.6 implies that δλ,µ maps L ∗ (b) into L ∗ (b/p). It is injective because it has a left inverse γλ,µ : L ∗ (b/p) → L ∗ (b) defined by γλ,µ (ξ ∗ (x)) = ψ(F (λ, x)−1 x µ ξ ∗ (x)), where ψ : L ∗ (b/p) → L ∗ (b) is defined by (2.8).

 

Define eλ = cλp − p(degλ g) inf(0, ord λ), a nonnegative real number. Note that together the conditions ord λ ≥ 0 and ord bdg +1 (λ) = 0 imply eλ = 0.

Exponential sums and generalized hypergeometric functions

35

Theorem 9.8. Let a ∈ K n and µ ∈ Zn ∩ Lg be such that inf(1, ord (a + µ)) > eλ +

1 1 + . p p−1

Put a = (a + µ)/p. Then the restriction of δλ,µ to Ka ,λp defines an isomorphism ∗ αa,a ,λ : Ka ,λp → Ka,λ .

Proof. By Corollary 9.5, Ka ,λp ⊆ L ∗ (ca + cλp ). Thus by Lemma 9.7, δλ,µ is injective on Ka ,λp provided c a + cλ p <

p−1 + p(degλ g) inf(0, ord λ), p

i.e., provided (p − 1)/p − ca > eλ . But p−1 1 1 − ca + + = 1 + inf(0, ord a ) p p p−1 = inf(1, ord (a + µ)), hence our hypothesis implies δλ,µ is injective. have the same dimension, so to show that By Proposition 7.1, Ka ,λp and Ka,λ ∗ . This follows αa,a ,λ is an isomorphism it suffices to show that δλ,µ (Ka ,λp ) ⊆ Ka,λ immediately from the commutativity of the diagram L ∗ (b)

δλ,µ

∗ Di,a ,λp

 L ∗ (b)

/ L ∗ (b/p) ∗ Di,a,λ

pδλ,µ

 / L ∗ (b/p)

under the hypothesis that b<

p−1 + p(degλ g) inf(0, ord λ). p

The proof of this commutativity is completely analogous to the proof of Lemma 2.4: writing out both compositions, one reduces to showing that xi ∂F /∂xi = πgi (λ, x) − pπgi (λp , x p ), F (λ, x) which follows from the definition of F (λ, x).

 

∗ Remark 9.9. We explain the connection between the maps αa,a ,λ and exponential sums. Let λ1 , . . . , λN ∈ k and let  f (λ, x) = fν (λ)x ν ∈ Fq [x1 , x1−1 , . . . , xn , xn−1 ], ν∈

36

Alan Adolphson

where fν (λ) = σ ∈Sν fσ ν Mσ ν (λ), fσ ν ∈ Fq , and Mσ ν (λ) is a monomial in the λi . We assume that f (λ, x) is nondegenerate as polynomial in x. Take the polynomial g to be the Teichmüller lifting of f (λ, x):  ˆ ν, ˆ x) = gν (λ)x g(λ, ν∈ s

ˆ = σ ∈S fˆσ ν Mσ ν (λ). ˆ If q = ps , then λˆ i satisfies λˆ p = λˆ i for i = where gν (λ) i ν s ˆ Let a ∈ Q with (q − 1)a ∈ Z. 1, . . . , N. We abbreviate this by writing λˆ p = λ. Then

a = (a0 + a1 p + · · · + as−1 p s−1 )(1 + p s + p 2s + · · · ), where 0 ≤ ai ≤ p − 1 for i = 0, 1, . . . , s − 1. If we choose µ0 = −a0 , µ1 = −a1 , …, µs−1 = −as−1 , then inductively a (i) =

a (i−1) + µi−1 = (ai + ai+1 p + · · · + ai−1 ps−1 )(1 + p s + p 2s + · · · ). p

In particular, a (s) = a. In this situation, the composition Ka (s) ,λˆ ps

α∗

a (s−1) ,a (s) ,λˆ p

s−1

−−−−−−−−−−→

K (s−1) ˆ ps−1 ,λ a

→ ··· →

Ka ,λˆ p

α∗

a,a ,λˆ

−−−→ Ka, λˆ

is an endomorphism of K ˆ . Denote it by β ∗ ˆ . a,λ a,λ From the definitions, we see that ∗ = δλˆ ,µ0  δλˆ p ,µ1  · · ·  δλˆ ps−1 ,µ βa, λˆ

= (x

−µ0

F (λˆ , x)  )  (x

= x −µ0 −µ1 p−···−µs−1

p s−1

−µ1

s−1

F (λˆ p , x)  )  · · ·  (x −µs−1 F (λˆ p p s−1

F (λˆ , x)F (λˆ p , x p ) · · · F (λˆ

,x

p s−1

s−1

, x)  )

)  s

ˆ x)  s , = x (1−q)a F0 (λ,  ˆ pi pi where F0 (λˆ , x) = s−1 i=0 F (λ , x ) as in section 3. Under the pairing (7.1),  is dual ˆ x) is self-dual. Equation (7.7) then implies to ψ and multiplication by x (1−q)a F0 (λ, that the dual of β ∗ ˆ acting on K ˆ is the operator ψ s  x (1−q)a F0 (λˆ , x) acting on a,λ

W ˆ . In particular,

a,λ

a,λ

∗ ˆ x) | W ). | Ka, ) = det(I − tψ s  x (1−q)a F0 (λ, det(I − tβa, λˆ λˆ a,λˆ

If we assume also that a is seminonresonant, Lemma 5.5 implies that we can replace W ˆ by Wa,λˆ : a,λ

∗ ˆ x) | W ˆ ). | Ka, ) = det(I − tψ s  x (1−q)a F0 (λ, det(I − tβa, a,λ λˆ λˆ

37

Exponential sums and generalized hypergeometric functions

By Proposition 4.6 we get det(I

∗ − tβa, λˆ

|

Ka, ) λˆ



s

= det I − tψ  x

(1−q)a

n 

ˆ x) | L F0 (λ,

i=1

 Di,a,λˆ L .

ˆ x) on L was denoted α(q−1)a . CorolIn section 2, the operator ψ s  x (1−q)a F0 (λ, lary 3.11 thus implies L(f, a; t)(−1)

n−1

∗ = det(I − tβa, | Ka, )(1−φ) λˆ λˆ

n−df

.

(9.6)

This equation expresses the relation between the L-function of the exponential sum ∗ and the “Frobenius map” αa,a ,λ . From Corollary 9.5 we had Ka,λ ⊆ L ∗ (ca + cλ ). But under the hypothesis of Theorem 9.8 we have
 Ka,λ ⊆ δλ,µ (L ∗ (ca + cλp )) ⊆ L ∗ (ca + cλp )/p , which may be a stronger assertion. We make this precise. Corollary 9.10. Suppose that B ⊆ Z[λ], cλ = 0, ord a ≥ 0, and ν is a nonnegative integer. If there exists µ ∈ Zn ∩ Lg such that ord (a + µ)/pν ≥ 0, then ⊆ L ∗ (1/(p − 1)p ν ). Ka,λ

Proof. The proof is by induction on ν. Our hypothesis implies cλ = 0 and ca = 1/(p − 1), so the case ν = 0 follows from Corollary 9.5. Assume the result true for ν − 1 and put a = (a + µ)/p. Then ord

a + (0, . . . , 0) ≥ 0. p ν−1

The hypothesis that B ⊆ Z[λ] implies that cλp = 0 also. The induction hypothesis then implies that Ka ,λp ⊆ L ∗ (1/(p − 1)p ν−1 ). Applying Theorem 9.8, we get = δλ,µ (Ka ,λp ) ⊆ δλ,µ (L ∗ (1/(p − 1)p ν−1 )) ⊆ L ∗ (1/(p − 1)p ν ). Ka,λ

Proposition 9.11. Suppose that u, v, µ ∈ Zn ∩ Lg and inf(1, ord (a + µ)) > eλ +

1 1 + . p p−1

Put a = (a + µ)/p. Then there is a commutative diagram Ka ,λp

xv

∗ αa,a ,λ

 Ka,λ

/ K a +v,λp ∗ αa+u,a +v,λ

xu

 / Ka+u,λ

 

38

Alan Adolphson

Proof. Just compute both compositions:

∗ u −µ x u  αa,a F (λ, x)   = x u+a−pa F (λ, x)  , ,λ = x  x



∗ v a+u−p(a +v) αa+u,a F (λ, x)    x v = x u+a−pa F (λ, x)  . +v,λ  x = x

 

N

g }i=1 of When bdg +1 (λ) = 0 and a is nonresonant, we constructed a basis {ξi,a,λ K a,λ (Proposition 7.7). Assume the hypothesis of Theorem 9.8 is satisfied. Then a Ng is also nonresonant so we get a basis {ξi,a ,λp }i=1 of Ka ,λp . We define γ (a, a , λ) to ∗ be the transpose of the matrix of the isomorphism αa,a ,λ relative to these bases. Since

N

g }i=1 a − u is also nonresonant for all u ∈ Zn ∩ L(g), we also have bases {ξi,a−u,λ for the Ka−u,λ . We recall that M(a, a − u, λ) is the transpose of the matrix of relative to these bases. u x : Ka−u,λ → Ka,λ

Theorem 9.12. Assume B ⊆ Z[λ]. For fixed µ ∈ Zn , γ (a, (a + µ)/p, λ) extends to a function of (a, λ) meromorphic in the region inf(1, ord (a + µ)) > eλ +

1 1 + p p−1

with polar factor det M(a , a − u0 , λp ), where a = (a + µ)/p and u0 is any element of Zn ∩ C(g) such that   µ Zn ∩ u0 − + C(g) ⊆ C(g). p In particular, the polar factor is trivial if  µ  Zn ∩ − + C(g) ⊆ C(g). p Proof. Write F (λ, x) =



Au (λ)x u .

u∈Zn ∩C(g)

We know that ord Au (λ) ≥

p − 1 p2

 + (degλ g) inf(0, ord λ) w(u).

From the definitions we have ∗ w(uj ) uj x , γ (a, a , λ)ij = αa,a ,λ (ξi,a ,λp ), π

Exponential sums and generalized hypergeometric functions

39

hence by Proposition 7.7 π −w(uj ) γ (a, a , λ)ij = x −µ F (λ, x) =



 v∈Zn ∩L

x −pv Hi,v (a , λp ), x uj  g

Au (λ)Hi,v (a , λp ),

where denotes a summation over all u ∈ Zn ∩ C(g) and v ∈ Zn ∩ Lg such that −µ + u − pv + uj = 0. This condition says that v=−

 µ  µ u + uj + ∈ Zn ∩ − + C(g) . p p p

Choose u0 ∈ Zn ∩ C(g) such that   µ Zn ∩ u0 − + C(g) ⊆ C(g). p Then for v occurring in the above summation, we have u0 + v ∈ Zn ∩ C(g). By Proposition 7.7,

Hi,v (a , λp ) =

Ng 

M(a − u0 , a , λp )ik Hk,u0 +v (a − u0 , λp ).

k=1

Substituting in the above we get

π −w(uj ) γ (a, a , λ)ij =

Ng 

M(a − u0 , a , λp )ik



Au (λ)Hk,u0 +v (a − u0 , λp ).

k=1

Since u0 + v ∈ Zn ∩ C(g), we have by Proposition 7.7 and equation (9.1) ord Au (λ)Hk,u0 +v (a − u0 , λp ) ≥ p − 1  + (deg g) inf(0, ord λ) w(u) − (cλp + ca )w(u0 + v). λ p2 Write v = (uj − µ)/p + u/p. Note that the definition of u0 does not imply that u0 + (uj − µ)/p ∈ C(g). However, we can choose u1 ∈ C(g) such that u0 + u1 + (uj − µ)/p ∈ C(g). Then w(u0 + v) ≤ w(u0 + u1 + v), so we may

40

Alan Adolphson

conclude ord Au (λ)Hk,u0 +v (a − u0 , λp ) p − 1  ≥ + (deg g) inf(0, ord λ) w(u) − (cλp + ca )w(u0 + u1 + v) λ p2  p − 1 + (deg g) inf(0, ord λ) w(u) ≥ λ p2  uj − µ u  − (cλp + ca ) w(u0 + u1 + ) + w( ) p p  w(u) p − 1 + p(degλ g) inf(0, ord λ) − cλp − ca = p p  uj − µ  − (cλp + ca )w u0 + u1 + p   w(u) p − 1 uj − µ  − c a − eλ − (cλp + ca )w u0 + u1 + = p p p  w(u)  1 1 + inf(0, ord a ) − eλ = 1− − p p−1 p  uj − µ  − (cλp + ca )w u0 + u1 + p  1 1  w(u) = inf(1, ord (a + µ)) − eλ − − p p−1 p  uj − µ  − (cλp + ca )w u0 + u1 + . p

It follows that the series Au (λ)Hk,u0 +v (a − u0 , λp ) converges when inf(1, ord (a + µ)) > eλ +

1 1 + . p p−1

Since M(a −u0 , a , λp ) is the inverse of M(a , a −u0 , λp ), its polar locus is contained in the zero set of det M(a , a −u0 , λp ). (Note that the polar locus of M(a , a −u0 , λp ) is excluded from the region of convergence by the hypothesis of the theorem.)   The matrix version of the commutative diagram of Proposition 9.11 is the following. Proposition 9.13. Assume B ⊆ Z[λ] and let µ, u, v ∈ Zn , a ∈ K n . Put a = (a + µ)/p. As meromorphic functions on the region inf(1, ord (a + µ)) > eλ +

1 1 + p p−1

we have γ (a + u, a + v, λ) = M(a + v, a , λp )−1 γ (a, a , λ)M(a + u, a, λ).

Exponential sums and generalized hypergeometric functions

41

We now analyze the condition given in Theorem 9.12 for the polar factor to be trivial. Proposition 9.14. The condition 
µ Zn ∩ − + C(g) ⊆ C(g) p is equivalent to the condition li (µ) ≤ p − 1 for i = 1, . . . , s. Proof. Note that the first condition is equivalent to

µ  li Zn ∩ − + C(g) ≥ 0 for i = 1, . . . , s. p Suppose that li (µ) ≤ p − 1. Then li (−µ/p) ≥ −(p − 1)/p, so for all v ∈ C(g) 
µ
µ p−1 + li (v) ≥ − . li − + v = li − p p p But if (−µ/p) + v ∈ Zn , then li ((−µ/p) + v) ∈ Z, hence li ((−µ/p) + v) ≥ 0. Suppose conversely that li (µ) ≥ p for some i. Choose v ∈ Zn ∩ C(g) such that li (v) = 0, lj (v) > 0 for all j = i. Choose w ∈ Zn ∩ Lg with li (w) = −1 and consider z = µ + pw + ptv, where t is a positive integer chosen large enough so that lj (z) = lj (µ + pw) + ptlj (v) > 0 for all j = i. Then z ∈ C(g). But 
µ z µ − + = w + tv ∈ Zn ∩ − + C(g) p p p and li (−µ/p + z/p) = li (w) + tli (v) = −1, which shows that   µ Zn ∩ − + C(g) ⊆ C(g). p

 

References [1]

A. Adolphson, On the Dwork trace formula. Pacific J. Math. 113 (1984), 257–268.

[2]

A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra: Cohomology and estimates. Ann. Math. 139 (1989), 367–406.

[3]

A. Adolphson and S. Sperber, Twisted exponential sums and Newton polyhedra. J. Reine Angew. Math. 443 (1993), 151–177.

42

Alan Adolphson

[4]

B. Dwork, Generalized Hypergeometric Functions. Oxford Mathematical Monographs, Oxford University Press, 1990.

[5]

A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor. Invent. Math. 32 (1976), 1–31.

[6]

P. Monsky, p-Adic Analysis and Zeta Functions. Lectures in Mathematics, Kyoto University, Kinokuniya Bookstore, Tokyo 1970.

[7]

J.-P. Serre, Endomorphismes complètement continus des espaces de Banach p-adiques. Inst. Hautes Études Sci. Publ. Math. 12 (1962), 69–85.

AlanAdolphson, Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, U.S.A. E-mail: [email protected]

Exponential sums and free hyperplane arrangements Alan Adolphson∗ and Steven Sperber

Abstract. We consider exponential sums on An /Fq defined by a polynomial f (x) ∈ Fq [x1 , . . . , xn ]. Here we assume the leading form of f defines a hypersurface Z in An such that Zred is a free hyperplane arrangement (or a 3-arrangement) and we assume also that the next-to-leading form of f is “generic with respect to Z”. With a (mild) restriction on the characteristic of Fq , we prove the vanishing of all but middle-dimensional p-adic cohomology for these exponential sums. 2000 Mathematics Subject Classification: Primary 11L07, 11T23, 14F20, 14F30

1 Introduction We consider here certain exponential sums on An /Fq where Fq is a finite field of characteristic p having q = pa elements. Let f (x) ∈ Fq [x1 , . . . , xn ] and let  : Fq → C∗ be a nontrivial additive character. The exponential sums of concern are defined by   Tr Fq m /Fq (f (x1 , . . . , xn )) Sm (An , f ) = (x1 ,...,xn )∈(Fq m )n

where Tr denotes the trace map. The associated L-function is  ∞  Tm n n L(A , f, T ) = exp . Sm (A , f ) m m=1

We write 

f = f (δ) + f (δ ) + · · · + f (0) ∗ Partially supported by NSF Grant #DMS-0070510

Key words and phrases: exponential sums, hyperplane arrangements, p-adic cohomology, -adic cohomology

44

Alan Adolphson and Steven Sperber

for the decomposition of f by degree into homogeneous forms. (Note that in our no tation we are assuming f (δ−1) = · · · = f (δ +1) = 0.) The case in which the vanishing of the leading form, f (δ) (x) = 0, defines a nonsingular projective hypersurface Zδ in Pn−1 has been studied using both -adic and p-adic methods ([4], [9], [1]) and is fairly well understood. In [5], García López calculated -adic cohomology (under some mild restriction on the characteristic p) for the L-function above in the case in which δ  = δ − 1 and Zδ has a finite number of quasi-homogeneous isolated singularities none of which lie on the projective hypersurface Zδ−1 (which is defined by the vanishing of the “next-to-leading” form f (δ−1) .) In a series of papers [1], [2], [3], the authors have taken this point of view and studied (via p-adic cohomology) exponential sums on An in which Zδ is a singular hypersurface with a very prescribed singular locus which behaves “well” in some sense when Zδ and the next-to-leading hypersurface Zδ  intersect. The present work is yet another case of this type. We recall some of the definitions and main results of [2]. It will be convenient to work at first over an arbitrary field K. Let kK[x] denote the module of differential k-forms of K[x1 , . . . , xn ] over K. Every ω ∈ kK[x] can be written uniquely in the form  ω(i1 , . . . , ik )dxi1 · · · dxik ω= 1≤i1 <···
with ω(i1 , . . . , ik ) ∈ K[x1 , . . . , xn ]. If each ω(i1 , . . . , ik ) in the form ω above is homogeneous of degree we say that ω is homogeneous and write degcoeff (ω) = , deg(ω) = + (n − k)(δ − 1). We consider the complex •K[x] with coboundary φf where φf (ω) = df ∧ ω. This complex has an increasing filtration by subcomplexes. Set F kK[x] = K-span of homogeneous k-forms ω with deg(ω) ≤ . • . The associated Then F •K[x] is a subcomplex of (•K[x] , φf ) and F • ⊆ F +1 spectral sequence has E1 -term

E1r,s = H r+s (Fr /Fr−1 , φf ) ⇒ H r+s (•K[x] , φf ). Clearly, the complex ((Fr /Fr−1 )• , φf ) is isomorphic to the r-th graded piece of the graded complex (•K[x] , φf (δ) ). In [2], we considered exponential sums with f (x) ∈ Fq [x] in which this spectral sequence degenerates (i.e., there exists a positive integer e such that Eer,s = 0 for r + s  = n). We recall some of these results. Let K = Fq and f ∈ Fq [x1 , . . . , xn ]. Following Dwork’s approach, we associate to f a complex (•C(b) , D) of length n depending on a rational parameter b satisfying 0 < b < p/(p − 1). Each iC(b) , i = 0, . . . , n, is a

45

Exponential sums and free hyperplane arrangements

˜ 0 (which itself is a finite extension of Qp , the field p-adic Banach space over a field  i of p-adic numbers). Each C(b) is equipped with a Frobenius operator αi commuting with the coboundary D of the complex. Dwork’s trace formula then states L(An , f, T ) =

n i=0

det(I − T αi | H i (•C(b) , D))(−1)

n+1

.

Among the results of [2] is Theorem 1.1. Suppose that there exist integers e and m such that Eer,s = 0 for all r, s such that r + s = m. Then for b in the range pδ δ
that L(An , f, T )(−1)

n+1

is a polynomial of degree Mf .

We note that the range for b is non-empty if and only if   p δ > 1+ (e − 1). (p − 1)2 In [2], we applied this result in the following case. Theorem 1.2. Suppose that f (δ) = f1a1 . . . frar where for each subset {i1 , . . . , ik } ⊆ {1, 2, . . . , r}, the system of equations fi1 = fi2 = · · · = fik = 0 defines a smooth complete intersection of codimension k in Pn−1 . Assume that if either k ≥ 2 or k = 1 and ai1 > 1 then the system of equations 

f (δ ) = fi1 = · · · = fik = 0 defines a smooth complete intersection of codimension k + 1 in Pn−1 . Assume also that (p, δδ  a1 a2 · · · ar ) = 1. Then for e = δ − δ  + 1, Eer,s = 0 for r + s  = n; so the conclusions of Theorem 1.1 hold in this case.

46

Alan Adolphson and Steven Sperber

2 Main results 

We write f = f (δ) + f (δ ) + · · · + f (0) with δ > δ  > · · · > 0 and each f (i) homogeneous of degree i. We assume f (δ) = f1a1 · · · frar

 with each fi a linear form in K[x1 , . . . , xn ]. Let Q = ri=1 fi and let A be the arrangement of hyperplanes defined by the vanishing of Q. Let DerK K[x1 , . . . , xn ] be the K[x]-module of derivations of K[x] over K. Define the submodule of DerK K[x] Der(A) = {θ ∈ DerK K[x] | θ(Q) ∈ QK[x1 , . . . , xn ]} Recall that A is said to be a free arrangement if Der(A) is a free K[x]-module. Let (H1 ) be the following hypothesis: (H1 )

The hyperplane arrangement A defined by Q = 0 is a free arrangement.

It is known, for example, that “generic” hyperplane arrangements and arrangements in 1 and 2 variables (so called 1- and 2-arrangements) are free. Let K¯ be an algebraic closure of K. Let C be the singular set of the affine variety Zδ defined by f (δ) = 0: C = {c ∈ K¯ n | fi (c) = 0 for some i with ai > 1 or fi (c) = fj (c) for some i  = j }. Of course, the {fi }ri=1 being linear forms, the singular locus C of Zδ is the union of a finite number of linear subspaces of An . For c ∈ C, denote by Wc the minimal linear subspace of Zδ containing c. Definition 2.1 (cf. [8, Definition 2.1]). A form g(x) ∈ K[x1 , . . . , xn ] is generic with respect to Zδ if for every c ∈ C = Sing(Zδ ), c  = (0, . . . , 0), d(g|Wc )(c) = 0. Let (H2 ) be the following hypothesis: (H2 )



f (δ ) is generic with respect to Zδ .

Theorem 2.2. Suppose f satisfies (H1 ) and (H2 ) above. Assume in addition that (H3 )

char K > δ.

(Note that this implies in particular that (char K, a1 a2 . . . ar δδ  ) = 1). Then Eer,s = 0 for e = δ − δ  + 1 and r + s  = n. Corollary 2.3. If f ∈ Fq [x1 , . . . , xn ] and p := char Fq satisfy hypotheses (H1 ), (H2 ), and (H3 ) above, then the hypotheses of Theorem 1.1 are satisfied. Hence H i (•C(b) , D) = 0 for i  = n

Exponential sums and free hyperplane arrangements

and L(f, An , T )(−1)

n+1

47

is a polynomial of degree Mf .

We recall  [7, Definition 5.22] that an arrangement A defined by the vanishing of Q = ri=1 fi ∈ K[x1 , . . . , xn ] is said to be generic if the hyperplanes of every subarrangement B ⊆ A with |B| ≤ n are linearly independent. It is worth noting that if the hypothesis (H1 ) in Theorem 2.2 and Corollary 2.3 is replaced by the stronger hypothesis that the arrangement is generic and the hypothesis (H3 ) is replaced by the weaker hypothesis (char K, δδ  a1 a2 . . . ar ) = 1, then the resulting modified versions of Theorem 2.2 and Corollary 2.3 are consequences of Theorem 1.2. (Note that in the case of generic hyperplane arrangements,  the hypothesis (H2 ) is equivalent to the hypothesis in Theorem 1.2 concerning f (δ ) ). It remains to prove Theorem 2.2 (hence Corollary 2.3) above. Consider the complex (•K[x] , df (δ) ∧). It is useful to denote the k-cocycles and k-coboundaries of this complex respectively by Z k and B k . Also, let H k = H k (•K[x] , df (δ) ∧). We make  H • itself into a complex using df (δ ) ∧ as the coboundary. Let Z˜ k and B˜ k respec tively denote the k-cocycles and k-coboundaries of the complex (H • , df (δ ) ∧). Let  H˜ k = H k (H • , df (δ ) ∧). To establish Theorem 2.2 it suffices (by the definition of a spectral sequence) to prove H˜ k = 0 for k < n. We recall the following result ([8, Theorem 5.8]). Theorem 2.4. Let d0

d1

C • : 0 → C 0 −→ C 1 − → ··· be a cochain complex of finite K[x1 , . . . , xn ]-modules with each coboundary map d i a K[x]-linear map. Assume also that each H i (C • ) is finite-dimensional over K. If i is a non-negative integer satisfying pd(C k ) < n + k − i for all k, then H i (C • ) = 0. (Here “pd” denotes projective dimension). 

We will apply this result to the complex (H • , df (δ ) ∧). The coboundary map is clearly K[x]-linear. First we note the following corollary. Corollary 2.5. Assume pd(H k ) ≤ k and dimK H˜ k < ∞ for all k. Then H˜ k = 0 for k < n. To establish Theorem 2.2, it thus suffices to show that the hypotheses of Corollary 2.5 are a consequence of the hypotheses of Theorem 2.2. The remainder of this article is devoted to this demonstration. It is useful to define the space of k-forms with logarithmic poles along A. Let kK(x) denote rational k-forms with coefficients in K(x1 , . . . , xn ). Then klog (A) = {ω ∈ kK(x) | Qω ∈ kK[x] and Qdω ∈ k+1 K[x] }.

48

Alan Adolphson and Steven Sperber

We recall that the K[x]-modules 1log (A) and Der(A) are dual to each other ([7, Theorem 4.75]), so that A is a free arrangement if and only if 1log (A) is a free K[x1 , . . . , xn ]-module. Therefore, assuming (H1 ), klog (A) is a free K[x1 , . . . , xn ]module ([7, Proposition 4.81]). In particular, pd(klog (A)) = 0 for all k if A is a free arrangement. In the following lemma we do not need to assume the fi are linear nor do we assume the arrangement is free. Lemma 2.6. Let f (δ) = f1a1 . . . frar have degree δ. Assume the fi are homogeneous, irreducible, and pairwise distinct. Assume (char K, a1 a2 · · · ar δ) = 1. As above,  Q = ri=1 fi and klog (Q) = {ω ∈ kK(x) | Qω ∈ kK[x] and Qdω ∈ k+1 K[x] }. If pd(klog (Q)) ≤ k for all k, then pd(H k ) ≤ k for all k. It is useful to recall the terminology of Terao–Yuzvinsky[10]. A hyperplane arrangement A is tame if pd(klog (A)) ≤ k for all k. It is natural  to say that an arrangement of hypersurfaces defined by the vanishing of Q = fi (where the fi are no longer assumed to be linear) is tame if pd(klog (Q)) ≤ k for all k. Proof. Set r

 dfi df (δ) η = (δ) = ai . fi f i=1

We consider the complex (•log (Q), η∧) and denote its k-coboundaries and k-cocycles by Bˆ k and Zˆ k , respectively. Let Hˆ k = H k (•log (Q), η∧). By the argument of [7, Proposition 4.86], if (char K, δ) = 1, then this complex is acyclic. We prove first (by induction on k) that pd(Zˆ k ) ≤ k − 1. Clearly Zˆ 0 = 0. Suppose k ≥ 1. In the short exact sequence η∧ 0 → Zˆ k → klog (Q) −→ Bˆ k+1 → 0

we know pd(Zˆ k ) ≤ k−1 and by hypothesis pd(klog (Q)) ≤ k. But then pd(Bˆ k+1 ) ≤ k by standard properties of the projective dimension [6, Chapter VII, Proposition 1.8]. But Hˆ k+1 = 0 then implies pd(Zˆ k+1 ) ≤ k for k ≥ 0 as desired.

Exponential sums and free hyperplane arrangements

49

Consider the injective map (multiplication by Q) Q → Zk . Zˆ k −

It is also surjective: If ω ∈ Z k then Q−1 ω ∈ Zˆ k , which follows at once provided (δ) ∧ ω = 0 implies Qd(Q−1 ω) ∈ k+1 K[x] . But df r  i=1

Then, for each i,

 j  =i

ai

dfi ∧ ω = 0. fi

 fj dfi ∧ ω ∈ fi k+1 K[x]

since ai = 0 in K. But K[x] is a UFD and the fj ’s are irreducible and pairwise distinct so dfi ∧ ω ∈ fi k+1 K[x] for each i. In particular Qd(Q−1 ω) = −

r  dfi i=1

fi

∧ ω + dω,

so it belongs to k+1 K[x] . The isomorphism Zˆ k ∼ = Z k implies pd(Z k ) ≤ k − 1. In the short exact sequence 0 → Z k → kK[x] → B k+1 → 0, we also know kK[x] is a free K[x1 , . . . , xn ]-module so pd(kK[x] ) = 0. It follows that pd(B k ) ≤ k − 1 for all k. Finally, in the short exact sequence 0 → B k → Z k → H k → 0, pd(B k ) ≤ k − 1 and pd(Z k ) ≤ k − 1, so that pd(H k ) ≤ k for all k as desired.

 

Lemma 2.6 shows that the first hypothesis of Corollary 2.5 is satisfied when (H1 ) and (H3 ) hold. It remains to show that the second hypothesis of Corollary 2.5, namely, the finite-dimensionality of H˜ k for all k, is a consequence of (H1 ), (H2 ), and (H3 ). 

Lemma 2.7. Let θ ∈ Der K (K[x1 , . . . , xn ]) be such that θ (f (δ) ) = 0. Then θ (f (δ ) ) annihilates H˜ k for all k.

50

Alan Adolphson and Steven Sperber

Proof. A cohomology class in H˜ k is represented by a differential form ω ∈ kK[x] satisfying df (δ) ∧ ω = 0 df

(δ  )

(2.1)

∧ ω = df (δ) ∧ ξ

(2.2)

for some homogeneous form ξ ∈ kK[x] . We will show 



θ(f (δ ) )ω = df (δ ) ∧ ζ + df (δ) ∧ γ for some homogeneous (k − 1)-forms ζ and γ with df (δ) ∧ ζ = 0. Recall for every k there is a bilinear map of K[x]-modules ,

DerK K[x] × kK[x] −→ k−1 K[x] which we denote θ, ω. This “inner product” satisfies θ, ω1 ∧ ω2  = θ, ω1  ∧ ω2 + (−1)k ω1 ∧ θ, ω2  for all ω1 ∈ kK[x] , ω2 ∈  K[x] , and θ, dh = θ(h)

for all h ∈ K[x1 , . . . , xn ].

Now df (δ) ∧ ω = 0, so 0 = θ, df (δ) ∧ ω = θ, df (δ)  ∧ ω − df (δ) ∧ θ, ω

(2.3)

= θ(f (δ) )ω − df (δ) ∧ θ, ω = −df (δ) ∧ θ, ω. By a similar calculation, 





θ, df (δ ) ∧ ω = θ(f (δ ) )ω − df (δ ) ∧ θ, ω, so that 





θ(f (δ ) )ω = df (δ ) ∧ θ, ω + θ, df (δ ) ∧ ω.

(2.4)

By (2.2) above, 

θ, df (δ ) ∧ ω = θ, df (δ) ∧ ξ  = θ, df (δ)  ∧ ξ − df (δ) ∧ θ, ξ  = θ(f (δ) )ξ − df (δ) ∧ θ, ξ  = −df (δ) ∧ θ, ξ .

(2.5)

Exponential sums and free hyperplane arrangements

51

Hence (2.4) and (2.5) yield 



θ(f (δ ) )ω = df (δ ) ∧ θ, ω − df (δ) ∧ θ, ξ . So taking ζ = θ, ω and γ = −θ, ξ  (and observing that, by (2.3), df (δ) ∧ ζ = 0) we obtain the desired result.   We now return to the case of interest in which the fi are linear forms. We will need the following lemma. Lemma 2.8. Assume char K > δ. For each c ∈ C, c  = (0, . . . , 0), there exists  θc ∈ Der K K[x] such that θc (f (δ) ) = 0 and θc (f (δ ) )(c)  = 0. Proof. Here we assume the fi are linear forms. Let c = (c1 , . . . , cn ) ∈ C with some cj = 0. We assume that for some s, 1 ≤ s ≤ r, s of the forms, say, f1 , . . . , fs , vanish at c and fs+1 (c), . . . , fr (c) are all non-zero. The space Wc of Definition 2.1 is then the solution space of the linear equations f1 (x) = f2 (x) = · · · = fs (x) = 0. It is convenient to make a coordinate change so that Wc is defined by the vanishing of coordinates, say, xk+1 = · · · = xn = 0 for some k. Then ck+1 = · · · = cn = 0 and f1 , . . . , fs are forms in xk+1 , . . . , xn alone. Since cj = 0 for some j we have k ≥ 1; since s ≥ 1 and the fi are non-trivial,  we have k ≤ n − 1. The differential of the restriction of f (δ ) to Wc is 



d(f (δ ) |Wc ) = d(f (δ ) (x1 , . . . , xk , 0, . . . , 0))   k   ∂f (δ )

= dxi .
∂xi xk+1 =···=xn =0 i=1

a

s+1 Let H = fs+1 · · · frar . By hypothesis (H2 ) we have 

∂f (δ ) (c)  = 0 ∂xi

(2.6)

∂f (δ) ∂ ∂f (δ) ∂ − ∈ DerK K[x], ∂xi ∂xj ∂xj ∂xi

(2.7)

H (c) for some i, 1 ≤ i ≤ k. Consider the derivations ϕij =

52

Alan Adolphson and Steven Sperber

where 1 ≤ i ≤ k and k + 1 ≤ j ≤ n. Write f (δ) = f1a1 · · · fsas H and substitute into (2.7). Then     s  ∂fm /∂xj ∂ ∂H a1 as ∂H ∂ ϕij = f1 · · · fs − +H am , (2.8) ∂xi ∂xj ∂xj fm ∂xi m=1

where we have used the observation above that the f , 1 ≤ ≤ s, do not depend on xi for 1 ≤ i ≤ k. Multiplying (2.8) by xj and summing over j ∈ {k + 1, . . . , n}, we obtain ϕi :=

n 

xj ϕij

j =k+1

=

f1a1

· · · fsas



    n s n  ∂ ∂H ∂H  ∂ xj − xj +H am , ∂xi ∂xj ∂xj ∂xi j =k+1

(2.9)

m=1

j =k+1

again using that the fi for 1 ≤ i ≤ s are linear forms in xk+1 , . . . , xn alone. Finally, dividing ϕi by f1a1 . . . fsas we obtain derivations θi = f1−a1 · · · fs−as ϕi

   s n n  ∂H  ∂ ∂H ∂ = xj − H am + xj . ∂xi ∂xj ∂xj ∂xi j =k+1

m=1

(2.10)

j =k+1

Since ϕij (f (δ) ) = 0 for all i, j , it follows that θi (f (δ) ) = 0 for all i, 1 ≤ i ≤ k. Recall cj = 0 for k + 1 ≤ j ≤ n so that  s   ∂(f (δ ) ) θi (f )(c) = − am H (c) (c) ∂xi m=1  for each i, 1 ≤ i ≤ k. Since char K > δ, sm=1 am  = 0 in K. It now follows from  (2.6) that for at least one i, 1 ≤ i ≤ k, θi (f (δ ) )(c)  = 0. This completes the proof of the lemma.  

(δ  )

It remains to prove the finite dimensionality of H˜ k . If c  ∈ C then the hypersurface Zδ is nonsingular at c. In fact by the Euler relation and our hypothesis on char K, the set C is precisely the set of points where df (δ) vanishes. In particular, at any point z at which df (δ) does not vanish, at least one ∂f (δ) /∂xi is a unit in the local ring K[x]z at z; so the Koszul complex on K[x]z defined by {∂f (δ) /∂xi }i=1,...n is acyclic. But then H k and a fortiori H˜ k have their support in the set of points in K¯ n where df (δ) vanishes. If c ∈ C, c = (0, . . . , 0), then by Lemmas 2.7 and 2.8  there exists a derivation θc ∈ DerK K[x] such that θc (f (δ ) ) annihilates H˜ k for all k  but θc (f (δ ) )(c)  = 0. This implies that c does not lie in the support of H˜ k so that H˜ k is supported only at (0, . . . , 0). The H˜ k are finitely generated K[x]-modules with support only at the origin so the only prime associated ideal is (x1 , . . . , xn ). So

Exponential sums and free hyperplane arrangements

53

(x1 , . . . , xn ) is the radical of the module by primary decomposition. But then clearly the module H˜ k is a finite-dimensional K-space. It is worth noting that in Lemma 2.6 above  we did not assume deg fi = 1 or that the hypersurface arrangement A defined by i=1 fi = 0 is free. We assumed only that A is a tame arrangement. Furthermore, in Lemma 2.7 no hypothesis at all is made concerning the arrangement A. In Lemma 2.8 again no hypothesis at all is made concerning the arrangement, but we do use in an essential way the hypothesis that the fi are linear forms for i = 1, 2, . . . , r. By [8, Lemma 5.14], 3-arrangements are tame. Recall that 3-arrangements need not be free (see [7, Example 4.34]). Theorem 2.9. Assume that f (x) ∈ K[x1 , x2 , x3 ] and that hypotheses (H2 ) and (H3 ) hold. Then Eer,s = 0 for e = δ − δ  + 1, r + s = n. Furthermore, if, in addition, K = Fq and p = char Fq satisfies (H3 ), then the conclusions of Theorem 1.1 apply here as well. In particular, H i (•C(b) , D) = 0 for i  = 3 and L(f, A3 , T ) is a polynomial of degree Mf . Remark 2.10. It is not unreasonable to expect that in the cases treated in Corollary 2.3 and Theorem 2.9 above, the only non-vanishing -adic cohomology is middledimensional and this cohomology is pure.

References [1]

A. Adolphson and S. Sperber, Exponential sums on An . Israel J. Math. 120 (2000), Part A, 3–21.

[2]

A. Adolphson and S. Sperber, Exponential sums on An , II. Trans. Amer. Math. Soc. 356 (2004), 345–369.

[3]

A. Adolphson and S. Sperber, Exponential sums on An , III. Manuscripta Math. 102 (2000), 429–446.

[4]

P. Deligne, Applications de la formule des traces aux sommes trigonometriques. In SGA 4 1/2, Cohomologie Etale, Lecture Notes in Math. 569, Springer-Verlag, Berlin, 1977.

[5]

R. García-López, Exponential sums and singular hypersurfaces. Manuscripta Math. 97 (1998), 45–58.

[6]

E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry. Birkhäuser, Boston, 1985.

[7]

P. Orlik and H. Terao, Arrangements of Hyperplanes. Springer-Verlag, Berlin, Heidelberg, 1992.

[8]

P. Orlik and H. Terao, Arrangements and Milnor fibers. Math. Ann. 301 (1995), 211–235.

54

Alan Adolphson and Steven Sperber

[9]

S. Sperber, On the p-adic theory of exponential sums. Amer. J. Math. 108 (1986), 255–296.

[10]

H. Terao and S. Yuzvinsky, Logarithmic forms on affine arrangements, Nagoya Math. J. 139 (1995), 129–149.

Alan Adolphson, Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078, U.S.A. E-mail: [email protected] Steven Sperber, School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A. E-mail: [email protected]

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork Yves André

Table des matières Introduction

56

I

59

II

La propriété de Grothendieck–Katz. Application d’un critère d’algébricité 1

L’algèbre de Lie de Galois différentielle . . . . . . . . . . . . . . . . . . . . . . 59

2

p-courbures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3

L’algèbre de Lie des p-courbures . . . . . . . . . . . . . . . . . . . . . . . . .

4

La propriété de Grothendieck–Katz . . . . . . . . . . . . . . . . . . . . . . . . 67

5

Un critère d’algébricité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

6

Preuve de 4.3.4 et 4.3.6 (cas d’un corps de nombres) . . . . . . . . . . . . . . .

79

Analogue de la conjecture de Grothendieck en équicaractéristique nulle

64

82

7

Énoncé des résultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8

Réduction au cas projectif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

9

Espaces de modules de connexions (rappels) . . . . . . . . . . . . . . . . . . .

85

10 Une application du théorème de Jordan . . . . . . . . . . . . . . . . . . . . . .

86

III Connexions d’origine géométrique

88

11 Isotrivialité, et horizontalité de la filtration de Hodge (rappels) . . . . . . . . . .

88

12 Cycles motivés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

13 Anneaux semi-simples motiviques . . . . . . . . . . . . . . . . . . . . . . . . . 94 14 Motifs et algèbre de Lie de Galois différentielle . . . . . . . . . . . . . . . . . . 97 15 Une application du théorème de Mazur–Ogus . . . . . . . . . . . . . . . . . . . 102 16 Conjecture de Grothendieck–Katz et problème de Dwork pour les connexions d’origine géométrique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 A Conjecture de Grothendieck et théorie des champs conformes

109

Références

110

56

Yves André

Résumé. This is a study of the interplay between properties of an integrable algebraic connection in characteristic zero, and properties of its reductions modulo p for large primes p (with special emphasis on the case of connections of geometric origin). 1991 Mathematics Subject Classification: 12H25, 14C99, 14G99, 34M15

Introduction 0.1. Cet article traite des liens entre les propriétés d’une équation différentielle linéaire à coefficients dans Q(x) et celles de ses réductions modulo p. L’origine de cette problématique est une question classique de Fuchs et Schwarz, reprise par Klein dans son livre sur l’icosaèdre : comment reconnaître si toutes les solutions d’une équation différentielle linéaire donnée à coefficients dans Q(x) sont des fonctions algébriques sur Q(x) ? Question à laquelle Schwarz a apporté une réponse complète dans le cas particulier des équations hypergéométriques de Gauss, via la classification des triangulations régulières de la sphère. Une approche arithmétique de cette classification a ensuite été inaugurée par Landau [L04]1 , qui a mis l’accent sur les dénominateurs des coefficients des solutions formelles. Dans le cas général, Grothendieck a proposé une réponse “arithmétique” conjecturale à cette question : une équation différentielle linéaire à coefficients dans Q(x) admet une base de solutions algébriques si et seulement si il en est de même par réduction modulo p pour presque tout p. Deux aspects remarquables de cette conjecture méritent d’être soulignés d’emblée : 1) une équation différentielle linéaire à coefficients dans Fp (x) admet une base de solutions algébriques si et seulement si elle admet une base de solutions rationnelles, et cela se vérifie par un algorithme très simple : nullité de la p-courbure, 2) la conjecture de Grothendieck peut être vue comme une généralisation “différentielle” d’un cas particulier, dû à Kronecker, du théorème de Chebotarev : les racines d’un polynôme à coefficients dans Q sont dans Q si et seulement si les racines de ses réductions modulo p sont dans Fp pour presque tout p. 0.2. Cette conjecture a été étudiée en profondeur et popularisée par N. Katz [38]. Plus récemment [39], il a proposé une conjecture plus générale prédisant que la composante neutre du groupe de Galois différentiel est déterminée par les p-courbures : plus précisément, l’algèbre de Lie de sa forme générique, qui est une algèbre de Lie algébrique sur Q(x) et contient modulo presque tout p les p-courbures, est minimale pour cette propriété. On retrouve la conjecture de Grothendieck comme cas particulier où presque toutes les p-courbures sont nulles. 0.3. Toujours dans le même esprit, B. Dwork [29] a décrit très précisément le lien entre p-courbures et dénominateurs des solutions formelles, dans le cas des équations 1 référence communiquée par D. Bertrand

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

57

“d’origine géométrique” (ou plus généralement, dans le cadre de la théorie des Gfonctions). Dans ce cadre, il a posé le problème de décrire la forme de l’ensemble des nombres premiers p pour lesquels la p-courbure s’annule. Illustrons cela comme dans [29] par deux exemples : 1) L’équation différentielle hypergéométrique de Gauss à paramètres a, b, c ∈ Q (comme chez Schwarz) : les p-courbures sont alors nilpotentes pour tout p ne divisant pas le dénominateur commun N de a, b, c. Pour analyser leur nullité éventuelle, on ne perd pas de généralité (grâce aux relations de contiguïté) à supposer a, b, c dans [0, 1[, c = a, c = b. Alors la p-courbure est nulle si et seulement si p ≡ u−1 mod. N pour tout unité u de Z/NZ telle que les représentants ua, ub, uc de ua, ub, uc dans [0, 1[ vérifient ua ≥ uc > ub ou bien ub ≥ uc > ua2 . Peut-on s’attendre à ce que, pour des équations différentielles “d’origine géométrique” plus générales, l’ensemble des p pour lesquels la p-courbure s’annule ait une densité rationnelle, ou même soit un ensemble “de congruence généralisée” (à un nombre fini d’exceptions près) ? Comme le remarque Dwork, l’exemple suivant montre qu’il convient de prendre des précautions (imposer par exemple une hypothèse de semisimplicité). 2) L’équation différentielle du logarithme d’une courbe elliptique X définie sur Q : c’est une équation d’ordre deux provenant d’une équation inhomogène d’ordre 1. La p-courbures est nulle si et seulement si X est supersingulière modulo p. D’après [31], si X est sans multiplication complexe, cela arrive pour un ensemble infini de p de densité 0 (qui ne peut être un ensemble de congruence généralisé d’après Chebotarev). 0.4. Le présent article est consacré à l’étude de la conjecture de Grothendieck–Katz et au problème de Dwork, dans le cadre plus général des connexions intégrables sur une variété lisse géométriquement connexe S sur un corps k de caractéristique nulle. Il comprend trois chapitres. Les résultats principaux se trouvent aux paragraphes 4, 7, et 16.2. 0.5. Le leitmotiv du premier chapitre (et en partie aussi du troisième) est que dans l’étude de ces questions, il y a intérêt à remplacer le module à connexion intégrable donné M par un autre, LG(M), qui possède en outre un crochet de Lie horizontal, et dont les fibres sont les algèbres de Lie des groupes de Galois différentiels attachés aux points-base correspondants. 0.5.1 Théorème (cf. 4.3.1). M vérifie la conjecture de Katz si et seulement si tout quotient de Lie simple non-abélien de LG(M) (dans la catégorie des modules à connexion) a une infinité de p-courbures non nulles. Nous nous ramènerons au cas particulier suivant : si l’algèbre de Lie de Galois différentielle est résoluble (de sorte qu’il n’y a aucun quotient de Lie simple nonabélien), M vérifie la conjecture de Katz. Ce cas particulier figurait déjà dans [2] et [3] 2 Katz [38] a vérifié que cette condition est satisfaite pour presque tout p si et seulement si (a, b, c) est

dans la liste de Schwarz, ce qui prouve la conjecture de Grothendieck dans ce cas

58

Yves André

(dans le cas où k est un corps de nombres), en généralisant un résultat de Chudnovsky [18]. Nous présenterons une variante raffinée de l’argument de [3] tenant compte de résultats plus récents en théorie des G-fonctions, et basé sur un avatar du critère de rationalité de Borel–Dwork. 0.6. Dans ces questions, la réduction d’un corps de base k quelconque de caractéristique nulle à un corps de nombres n’est pas formelle. Ce qui est en jeu, c’est un analogue de la conjecture de Grothendieck en équicaractéristique nulle, traité au second chapitre : 0.6.1 Théorème (cf. 7.2.2). Dans une famille de connexions intégrables (M(t) )t∈T paramétrée par une k-variété T , si M(t) est isotriviale pour tout point fermé t de T , alors il en est de même de la fibre générique M(η) . Rappelons qu’“isotrivial” veut dire trivialisé par un revêtement fini étale de la base. La difficulté est bien entendu de “borner” les groupes de Galois différentiels finis qui interviennent. Nous prouvons cet énoncé en utilisant les espaces de modules de connexions construits par C. Simpson, par un argument dont la clé est le théorème classique de Jordan sur les sous-groupes finis de GLn . Dans un manuscrit récent [34], E. Hrushovsky prouve un énoncé de ce type au moyen de la théorie des modèles. 0.7. Le troisième chapitre traite de la conjecture de Grothendieck–Katz et du problème de Dwork dans le cas d’une connexion d’origine géométrique M, c’est à dire telle qu’il existe S → S étale dominant tel que MS soit extension successive de sous-quotients de connexions de Gauss–Manin attachées à des morphismes lisses f de but S . Rappelons que dans le cas des connexions de Gauss–Manin (et de certains facteurs directs très particuliers), la conjecture de Grothendieck a été prouvée par Katz [38]. Sa méthode repose sur une formule remarquable reliant la p-courbure à l’application de Kodaira–Spencer. Pour aborder la conjecture de Katz, nous mettons à profit le théorème 0.5.1 qui nous ramène à prouver la conjecture de Grothendieck pour les quotients simples de l’algèbre de Lie de Galois différentielle. Ceux-ci sont de nature “motivique”, ce qui permet d’utiliser la formule de Katz. Toutefois, en raison d’une lacune actuelle de la théorie motivique, nous n’obtenons de résultat définitif que sous une hypothèse technique de connexité (conjecturalement toujours satisfaite) : 0.7.1 Théorème (cf. 16.2.1 et corollaires). Soit un morphisme projectif lisse f : X → S, de base S une k-variété lisse géométriquement connexe. Supposons que le groupe de Galois motivique3 d’au moins une fibre géométrique de f soit connexe. Soit Hf la connexion de Gauss–Manin attachée à f , et soit M un module à connexion sousquotient d’une construction tensorielle sur Hf . Alors : 1) M vérifie la conjecture de Katz, 2) Il existe un corps de nombres totalement réel E galoisien sur Q (ne dépendant que de f ), et un ensemble C(M) de classes de conjugaison de Gal(E/Q), tels que 3 voir 12.2 pour la définition

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

59

l’ensemble des p tels que les p-courbures de M s’annulent soit égal, à un nombre fini d’exceptions près, à l’ensemble des p tels que la classe de conjugaison de Frobenius en p soit dans C(M). En particulier, cet ensemble a une densité rationnelle. Le point 2) répond affirmativement, dans ce cas particulier, à la question de Dwork. Le mystérieux corps E apparaît comme clôture galoisienne d’un corps d’endomorphismes de motifs. Dans l’exemple hypergéométrique 0.3 1), c’est Q(cos 2π N ). Nous discutons brièvement en appendice d’une application intéressante de la conjecture de Grothendieck à la classification des théories des champs conformes : le problème de l’“algébricité” des blocs conformes dans les théories rationnelles. 0.8. Cet article est une version élaguée, corrigée, et entièrement réécrite du manuscrit [7]. Nous remercions D. Bertrand, J.-B. Bost, A. Chambert-Loir et L. Di Vizio de nous avoir encouragé, à maintes reprises, à entreprendre cette révision. Nous remercions aussi le referee de sa lecture vigilante. Si l’article est directement inspiré par les travaux de N. Katz [38] [39] et (dans une moindre mesure) de D. et G. Chudnovsky [18], les travaux de B. Dwork touchent aussi par bien des points aux thèmes abordés ici : critère de Borel–Dwork, p-courbures et taille des G-connexions [29], lieu de nilpotence des p-courbures [28], structures de Frobenius, et bien sûr le problème auquel le titre fait allusion. Quant à l’appendice, il est né d’une conversation avec C. Itzykson, qui nous a fait découvrir la classificationA.D.E.

I La propriété de Grothendieck–Katz. Application d’un critère d’algébricité 1 L’algèbre de Lie de Galois différentielle 1.1. Soit S une variété algébrique lisse géométriquement connexe sur un corps k. Soit M = (M, ∇) la donnée d’un OS -module M localement libre de rang r < ∞ et d’une connexion intégrable ∇ : M → 1S ⊗OS M. Sur un ouvert U ⊂ S où M est libre, fixons des coordonnées locales x1 , . . . , xd , et une base e1 , . . . , er de M(U ). Notons A(h) la “matrice” de ∇ ∂ dans cette base : ∇ de sorte que ∇(ei ) =



h,i

∂ ∂xh

(ej ) =

∂xh

A(h)ij ei ,

i

dxh ⊗ A(h)ij ei .

1.2. On peut aussi exprimer la situation en termes de solutions de M à valeurs dans une extension différentielle R de O(U ) ad libitum (par exemple un anneau de série

60

Yves André

formelles, complété de l’anneau local en un point de U ) ; par “solution”, nous entendons un homomorphisme O(U )-linéaire commutant à la connexion. Voici le dictionnaire. On sait que dans la base duale e1∨ , . . . , er∨ , la connexion duale ∇ ∨ contractée par ∂x∂ h est représentée par −t A(h) . Écrivons alors notre solution y sous  la forme yj ej∨ où yj ∈ R. La condition d’horizontalité ∇ ∨ (y) = 0 se traduit alors par le système linéaire intégrable aux dérivées partielles

∂yj = yi A(h)ij , ∂xh i

c’est-à-dire ∂ y = y A(h) , ∂xh en notant y le vecteur-ligne de composantes yi . 1.2.1 Remarque. Cette façon, chère à B. Dwork, d’écrire les systèmes différentiels matriciels “à droite” semble plus naturelle que l’écriture “à gauche” (elle fait l’économie d’une transposition - ou d’un signe, si l’on compare à la connexion duale). Elle présente d’autre part l’avantage, dans la situation complexe, d’être compatible avec l’écriture à gauche usuelle de la monodromie. 1.3. Supposons car k = 0. Les modules cohérents à connexion intégrable sur S forment une catégorie tannakienne MICS sur k. À partir d’un module à connexion intégrable M, on peut former ses puissances tensorielles mixtes Tnm M = M⊗m ⊗(M ∨ )⊗n . Les sous-quotients des sommes finies de Tnm M (pour divers m, n) forment une souscatégorie tannakienne de MICS , notée M⊗ . Tout point s de S définit un foncteur fibre ωM,s : M⊗ → Vecκ(s) à valeurs dans les vectoriels sur le corps de définition de s. On pose Gal(M, s) := Aut⊗ ωM,s . C’est un κ(s)-schéma en groupe affine, appelé groupe de Galois différentiel de M, basé en s ([40], [8]). Si s est un point k-rationnel de S, i.e. si κ(s) = k, ce groupe peut se décrire à la Picard–Vessiot–Kolchin, et ωM,s s’enrichit en une équivalence de catégories M⊗ ∼ = Repk Gal(M, s). Si s = η est le point générique, Gal(M, s) coïncide avec le groupe de Galois “générique” ou “intrinsèque” considéré dans [39] (voir aussi [14]) : il se décrit comme stabilisateur dans GL(Mη ) des fibres génériques des sous-objets des sommes finies de Tnm M (cf. [8, 3.2.2]).

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

61

1.3.1 Remarque. D’après [40, 1.3.2], la formation de Gal(M, s) commute à l’extension des scalaires k /k. 1.4. Par ailleurs, on peut considérer le foncteur oubli de la connexion : ωM : M⊗ → FibVecS à valeurs dans les fibrés vectoriels sur S. On pose G(M) := Aut⊗ ωM . C’est un S-schéma en groupe affine et plat, loc. cit.. Sa fibre en s n’est autre que Gal(M, s). Compte tenu du fait que pour tout objet N = (N, ∇) ∈ M⊗ et tout sous-objet N de N , le fibré vectoriel sous-jacent N est localement facteur direct de N , on montre aisément que G(M) est le sous-schéma en groupe fermé de GL(M) qui stabilise les (fibrés vectoriels sous-jacents aux) sous-objets des sommes finies de Tnm M (voir [22, II.1.3.6] pour la définition d’un tel stabilisateur ; voir aussi [8, 2.1]). En prenant des puissances extérieures à la Chevalley, on montre en fait que G(M) est le stabilisateur dans GL(M) d’un sous-objet D de rang un dans une somme finie convenable ⊕ Tnm M. Nous n’aurons à considérer que son algèbre de Lie LG(M) := Lie G(M), qui est un objet de M⊗ , que nous appellerons algèbre de Lie4 de Galois différentielle de M. Pour un point quelconque s ∈ S, LG(M) est l’unique sous-objet de T11 M = EndM (“End interne”) de fibre en s égale à LieGal(M, s) ⊂ EndMs . Elle peut aussi se décrire comme la sous-algèbre de Lie maximale de gl(M) qui stabilise les (fibrés vectoriels sous-jacents aux) sous-objets des sommes finies de Tnm M (resp. qui stabilise D). Bien entendu, il n’est pas vrai, réciproquement, que tout sous-fibré d’un ⊕ Tnm M stabilisé par LG(M) soit horizontal, i.e. sous-jacent à un sous-objet de ⊕ Tnm M (heuristiquement : l’horizontalité est une propriété k-linéaire, et non OS -linéaire). 1.4.1 Lemme. 1) Pour tout objet M de M⊗ , on a un épimorphisme canonique LG(M) → → LG(M ). 2) Si LieGal(M, η) est semi-simple, l’épimorphisme LG(M) → → LG(LG(M)) est un isomorphisme. Démonstration. Puisqu’on a affaire à des objets de MICS , il suffit de le vérifier sur une fibre. Par la théorie tannakienne, on a un épimorphisme canonique Gal(M,s)Gal(M ,s), d’où 1). Pour 2), on remarque que Gal(M, s) → Gal(LG(M), s) ⊂ GL(LieGal(M, s)) n’est autre que la représentation adjointe, qui induit un isomorphisme LieGal(M, s) → LieGal(LG(M), s) puisque LieGal(M, s) est semi-simple.
4 on trouvera plus bas (14.3) une discussion des algèbres de Lie dans une catégorie tannakienne

62

Yves André

1.4.2 Lemme. La formation de LG(M) est compatible aux changements de base dominants. Démonstration. Soient X une autre k-variété lisse géométriquement connexe, et f : X → S un morphisme dominant. Il suffit de prouver la propriété en la fibre générique, donc de montrer que f ∗ (LieGal(M, ηS )) = LieGal(f ∗ M, ηX ). On a l’inclusion Gal(f ∗ M, ηX ) ⊂ f ∗ Gal(M, ηS ) (ce dernier étant le stabilisateur des images inverses sur X des sous-objets des ⊕ Tnm M), et il suffit de montrer qu’on a l’égalité des composantes neutres. Comme l’énoncé ne dépend de f que via l’extension de corps k(X)/k(S), on se ramène à traiter d’une part le cas d’un morphisme étale (qui est clair), et d’autre part le cas d’une projection X = S × S → S. Par la remarque 1.3.1, on peut d’ailleurs supposer que S possède un point k-rationnel, d’où une rétraction i de f . Comme les suites exactes courtes dans la catégorie tannakienne f ∗ M⊗ sont localement scindées eu égard aux fibrés sous-jacents, l’endofoncteur idempotent (i  f )∗ de f ∗ M⊗ est exact, donc finalement égal à l’identité puisque (i  f )∗ M = M. Il s’ensuit que f ∗ : M⊗ → f ∗ M⊗ est une équivalence de catégories, ce qui implique que
Gal(f ∗ M, ηX ) = f ∗ Gal(M, ηS ) dans ce cas. 1.4.3 Lemme. Les conditions suivantes sont équivalentes : a) LG(M) = 0, b) pour tout s ∈ S, Gal(M, s) est fini, c) localement pour la topologie étale, M est engendré par ses sections horizontales, d) M est isotrivial, i.e. il existe un revêtement étale fini S → S au-dessus duquel M devient trivial (en tant que module à connexion). Démonstration. Comme la fibre de LG(M) en s s’identifie à LieGal(M, s), l’équivalence de a) et b) est claire. Celle de c) et d) est laissée au lecteur, de même que l’implication c) ⇒ b) (qui est d’ailleurs un cas particulier du lemme précédent). L’implication b) ⇒ c) découle de la théorie de Galois différentielle5 : en effet, on peut remplacer k par une extension finie et S par un voisinage affine Spec A d’un point k-rationnel s de S ; alors le torseur Isom⊗ (ωM , ωM,s ⊗k S) sous Gal(M, s) ⊗k S est un S-schéma fini étale (tout comme Gal(M, s) ⊗k S), et l’image inverse de M sur ce torseur est triviale, cf. [8, 3.4.2]. 1.5. Nous aurons besoin de la notion de “radical de LG(M)”. C’est l’unique sousobjet RadLG(M) de LG(M) dont la fibre en un point s (ou en tout point s) est le radical de LieGal(M, s) (noter que ce dernier est un idéal Gal(M, s)-invariant). C’est donc automatiquement un idéal de Lie de LG(M).

5 dans [39] on trouve un argument beaucoup plus indirect utilisant à la fois le passage à la caractéristique

p et l’équivalence de Riemann–Hilbert

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

63

2 p-courbures 2.1. Supposons maintenant que car k = p > 0. Alors les constantes différentielles ne se réduisent plus au corps de base k, mais forment le sous-faisceau k.(OS )p de OS . Un autre fait nouveau est que Der OS = (1S )∨ est muni d’une p-structure, ce qui permet de définir l’opérateur de p-courbure RM,p : pour toute section ∂ de (1S )∨ au-dessus d’un ouvert U , RM,p (∂) = (∇∂ )p − ∇∂ p . RM,p (∂) est un endomorphisme O(U )-linéaire de M(U ), additif et p-linéaire en la dérivation ∂ [37] ; autrement dit, Rp = RM,p définit une section globale de FS∗ 1S ⊗ EndM, où FS désigne l’endomorphisme de Frobenius de S. 2.2. Écrivons la connexion sous forme matricielle comme en 1.2, en remplaçant l’indice (h) par le d-uplet dont toutes les composantes sont nulles sauf la h-ième qui vaut 1. Par dérivations “formelles” successives du système ∂ y = y A(0,...,0,1,0,...,0) , ∂xh on obtient des équations ∂m y = y A[m] ∂x m pour tout multi-indice m à d composantes (avec les conventions usuelles sur les multiindices). La p-courbure RM,p ( ∂x∂ h ) est alors représentée par la matrice A[(0,...,0,p,0,...,0)] (p en h-ème position). 2.3. Une autre particularité de la caractéristique p est qu’on dispose d’un moyen algorithmique simple de détecter si une connexion intégrable est Zariski-localement triviale : un théorème de P. Cartier affirme en effet que M est engendré sur OS par M ∇ := Ker ∇ si et seulement si l’opérateur de p-courbure RM,p est nul (cf. [37]). En particulier, cette propriété est locale pour la topologie étale sur S, ce qui contraste fortement avec le cas de caractéristique nulle. 2.4. La théorie de Galois différentielle en caractéristique p est décrite, sous deux points de vue différents (générique et local, respectivement), dans [50] et dans [8, 3.2.2.5]. Dans [50], on montre que la catégorie Mη ⊗ est tannakienne neutre sur k.(k(S))p ; le groupe tannakien est infinitésimal abélien de hauteur 1, et de p-algèbre de Lie engendrée par les p-courbures RM,p (∂). Dans [8], on choisit un point s ∈ S(k) (supposé exister), et on remplace S par le schéma local non réduit Ss,p = Spec OS,s /(mS,s )p , où mS,s est l’idéal maximal de OS,s , de sorte que l’anneau des constantes différentielles redevienne k. On a un

64

Yves André

foncteur fibre ωM,s,p : MSs,p ⊗ → Veck pd

pd

donné par N → (N ⊗ OSs,p )∇ où OSs,p est le complété à puissances divisées de OSs,p . Le k-groupe affine Aut⊗ ωM,s,p est infinitésimal abélien de hauteur 1, et de p-algèbre de Lie engendrée sur k par les p-courbures RM,p (∂). Par ailleurs, on peut considérer comme plus haut le foncteur oubli de la connexion : ωMSs,p : MSs,p ⊗ → FibVecSs,p . Le Ss,p -schéma en groupe Aut⊗ ωM est en fait constant : Aut⊗ ωM ∼ = (Aut⊗ ωM,s,p ) ⊗k Ss,p , cf. [8, 3.2.2.6]. Sa p-algèbre de Lie, qui est un objet de MSs,p ⊗ , est donc engendrée (en tant que p-algèbre de Lie) par les RM,p (∂).

3 L’algèbre de Lie des p-courbures 3.1. Revenons au cas d’un corps de base k de caractéristique nulle. Comme dans [37] et [39], “épaississons” la situation en choisissant un sous-anneau o de k de type fini sur Z, un o-schéma connexe S à fibres géométriquement connexes, et un OS -module localement libre à connexion intégrable M, tels que S et M se déduisent de S et M par extension des scalaires o → k. On suppose aussi que l’objet D de M⊗ considéré en 1.3.1 provient d’un sousobjet D de ⊕ Tnm M, et que le fibré sous-jacent à D est localement facteur direct. Dans ces conditions, LG(M) provient d’une sous-algèbre de Lie LG(M) de gl(M) (le stabilisateur de D), dont la fibre en tout point fermé v de Spec o contient les pcourbures RM⊗κ(v),p (∂) de la fibre de M en v (ici p désigne la caractéristique résiduelle de v), cf. [39, 9.3]. On notera que tous les nombres premiers sauf un nombre fini interviennent ici. Comme dans loc. cit., nous dirons abusivement que “LieGal(M, η) contient les p-courbures modulo presque tout p”. 3.1.1 Remarque. On appellera parfois abusivement v-courbures de M les RM⊗κ(v),p (∂) (surtout lorsque k est un corps de nombres). Elles dépendent bien sûr de o et des modèles, mais cette dépendance est “innocente” : étant donné deux ensembles de données (o1 , S1 , M1 , D1 ) et (o2 , S2 , M2 , D2 ), il en existe un troisième (o3 , S3 , M3 , D3 ) tel que o3 contienne o1 et o2 , et que (S3 , M3 , D3 ) s’obtienne à partir de (Si , Mi , Di ) par extension des scalaires oi → o3 , i = 1, 2. On a par ailleurs oi ⊗ Fp → o3 ⊗ Fp , i = 1, 2, pour presque tout p, cf. [39, 6.1].

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

65

3.2. Soit P un ensemble de nombres premiers, et soit L ⊂ gl(Mη ) une sous-algèbre de Lie. Nous dirons que L contient les p-courbures modulo presque tout p ∈ P si pour un choix (ou pour tout choix - cela revient au même par la remarque ci-dessus) de données (o, S, M, D) comme ci-dessus et d’une OS -sous-algèbre de Lie L ⊂ gl(M) étendant L, et pour tout p ∈ P sauf un nombre fini, la fibre de L en tout point fermé v de Spec o de caractéristique résiduelle p contient les p-courbures RM⊗κ(v),p (∂) de la fibre de M en v. De même, si F ⊂ N = ⊕Tnm M est un sous-fibré localement facteur direct d’une somme finie de puissances tensorielles mixtes sur M, nous dirons que F est stable sous les p-courbures modulo presque tout p ∈ P si pour un choix (ou pour tout choix) de données (o, S, M) comme ci-dessus et d’un sous-fibré F ⊂ N = ⊕Tnm M localement facteur direct étendant F , et pour tout p ∈ P sauf un nombre fini, la fibre de F en tout point fermé v de Spec o de caractéristique résiduelle p est stable sous les p-courbures RN⊗κ(v),p (∂) de la fibre de N en v. Nous appellerons algèbre de Lie des P -courbures de M et noterons P -C(M, η) la plus petite sous-algèbre de Lie algébrique de gl(Mη ) “contenant les p-courbures modulo presque tout p ∈ P ”. 3.2.1 Proposition. P -C(M, η) est fibre générique d’un unique sous-module horizontal de LG(M) (noté P -C(M)). 3.2.2 Corollaire. P -C(M) est un idéal de Lie de LG(M). Démonstration. Il suffit de faire voir que P -C(M, η) est un sous-k(S)-espace horizontal de T11 Mη . Par le lemme de Chevalley, P -C(M, η) peut se décrire comme la sousalgèbre de Lie de EndMη qui stabilise une certaine droite Dη dans un espace de tenseurs mixtes sur Mη . Soit Nη le plus petit sous-k(S)-espace horizontal de Mη contenant Dη . Ainsi Nη est la fibre générique d’un objet N de M⊗ , et quitte à remplacer S par un ouvert dense, Dη est la fibre générique d’un fibré en droite D localement facteur direct de N. Alors D est stable sous les p-courbures modulo presque tout p ∈ P . Bien plus, l’argument de [K82]10.2, basé sur l’horizontalité des p-courbures, montre que les p-courbures agissent par multiplication par une fonction à dérivée nulle sur N modulo presque tout p ∈ P . Ceci entraîne que le sous-espace P -C(M, η) de EndMη est formé des éléments qui agissent par homothéties sur Nη (en d’autres termes, on a P -C(M, η) = Ker(LieGal(M, η) → End(End(Nη ))) ). Il est alors clair que ce sous-espace est stable sous les ∇(∂).
3.2.3 Remarque. L’idéal P -C(M) croît avec P , et si P et P sont deux ensembles de nombres premiers, on a (P ∪ P )-C(M) = P -C(M) + P -C(M). 3.2.4 Lemme. 1) Pour tout objet M de M⊗ , l’épimorphisme canonique → P -C(M ). LG(M) → → LG(M ) induit un épimorphisme P -C(M) →

66

Yves André

2) Si LieGal(M, η) est semi-simple, alors l’homomorphisme P -C(M) → → P -C(LG(M)) est un isomorphisme. Démonstration. Par 3.2.1, il suffit de le vérifier sur les fibres génériques. 1) Il est immédiat que P -C(M , η) est contenue dans l’image de P -C(M, η). Réciproquement, il est clair que l’image inverse de P -C(M , η) dans LG(M, η) contient les p-courbures modulo presque tout p ∈ P. 2) découle de 1) (surjectivité) et de 1.4.1 (injectivité).   3.2.5 Remarque. En général o ⊗ Fp n’est pas connexe, et on peut vouloir préciser la composante connexe des points v dont les v-courbures nous intéressent. Par exemple, si la fermeture algébrique de Q dans oQ contient un corps de nombres E, on peut introduire des variantes de P -C(M) en considérant les v-courbures pour tout v audessus d’un ensemble fixé P E de places finies de E (à un ensemble fini près). Ces variantes P E -C(M) vérifient les mêmes propriétés 3.2.1 à 3.2.4. Elles nous seront utiles en 16.2.1. 3.2.6 Lemme. La formation de P -C(M) est compatible aux changements de base dominants. Démonstration. On raisonne comme dans la preuve du lemme 1.4.2, dont on reprend les notations. Il suffit de montrer que f ∗ (P -C(M, ηS )) = P -C(f ∗ M, ηX ). On a l’inclusion ⊃, en raison de la formule de changement de base pour les p-courbures (O. Gabber, cf. [39, app.]). L’inclusion opposée se démontre exactement comme dans loc. cit..   3.2.7 Proposition ([37]). Supposons que P -C(M) = 0. 1) Si P est infini, alors M est régulier (i.e. à singularités régulières “à l’infini” ). 2) Si en outre P est de densité 1, alors il existe un revêtement fini étale S → S tel MS s’étende en un module à connexion sur toute compactification lisse S¯ telle que S¯ \ S¯ soit un diviseur à croisements normaux. 3.3. Si M est isotrivial (i.e. s’il existe un revêtement étale fini S → S au-dessus duquel M devient trivial), on a LG(M) = 0, d’où P -C(M) = 0 pour tout P . La conjecture de Grothendieck prédit la réciproque : 3.3.1 Conjecture. Soit P l’ensemble des nombres premiers. Si P -C(M) = 0, alors M est isotrivial. 3.3.2 Remarque. Par 3.2.7.2, on peut supposer dans cette conjecture que S est projectif (en passant à une compactification d’un revêtement fini étale convenable).

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

67

Compte tenu du théorème de Cartier, cf. 2.3, il est facile de voir que cette conjecture s’exprime aussi comme suit, en prenant un modèle (o, S, M) de (k, S, M) comme en 3.1 (l’énoncé ne dépend pas du choix du modèle) : 3.3.3 Conjecture (équivalente). Si pour tout point fermé v d’un ouvert dense de Spec o, la fibre de M en v est (iso)triviale6 , alors M est isotrivial. Cette conjecture est encore largement ouverte (en revanche, un analogue pour les équations aux q-différences est démontré dans [25]).

4 La propriété de Grothendieck–Katz Dans tout ce paragraphe, P désigne l’ensemble des nombres premiers. 4.1. Dans [39], Katz conjecture que pour tout module à connexion intégrable sur S, la plus petite sous-algèbre de Lie algébrique de LieGal(M, η) qui contient les p-courbures pour presque tout p est LieGal(M, η) elle-même. Dans le langage précédent, cela se traduit alors par : 4.1.1 Conjecture. Pour tout M ∈ MICS , P -C(M) = LG(M). Cette conjecture entraîne, via 1.4.3, la précédente (cas où P -C(M) = 0). Réciproquement : 4.1.2 Proposition ([39]). P -C(M) = LG(M) 3.3.1.

⇐⇒ tout objet de M⊗ vérifie

4.2. Dans la suite, nous dirons que M a la propriété de Grothendieck–Katz si P -C(M) = LG(M). 4.2.1 Lemme. La propriété de Grothendieck–Katz est stable par changement de base dominant. Cela découle de 1.4.2 et 3.2.6. Nous ignorons en revanche si la propriété de Grothendieck–Katz est stable par restriction à un fermé (lisse géométriquement connexe). L’étude de la stabilité de la propriété de Grothendieck–Katz par images directes supérieures (cohomologie de de Rham à coefficients) est délicate, et ne sera abordée au chapitre III que dans le cas de la connexion triviale (cohomologie de de Rham à coefficients constants). 6 rappelons que “trivial” équivaut à “isotrivial” en caractéristique p > 0

68

Yves André

4.3. L’énoncé suivante précise 4.1.2 : 4.3.1 Théorème. Les conditions suivantes sont équivalentes : 1) M a la propriété de Grothendieck–Katz, 2) Pour toute algèbre de Lie simple non-abélienne L dans M⊗ qui est quotient de LG(M) par un idéal de Lie, on a P -C(L)  = 0. 4.3.2 Corollaire. M a la propriété de Grothendieck–Katz si et seulement s’il en est de même de son semi-simplifié Mss . En effet, la théorie de Galois différentielle montre que LG(M) est extension de LG(M ss ) par un idéal nilpotent, donc LG(M) et LG(Mss ) ont mêmes quotients simples (non abéliens). 4.3.3 Corollaire. La sous-catégorie pleine de MICS formée des objets ayant la propriété de Grothendieck–Katz est tannakienne et stable par extension. Le fait qu’elle soit tannakienne est clair, compte tenu de 3.2.4. Qu’elle soit stable par extension découle du corollaire précédent. Un cas très particulier, qui résout une question posée dans [45, 2.5], est le suivant : 4.3.4 Corollaire. Supposons que N ∈ MICS s’inscrive dans une suite exacte 0 → N → N → N

→ 0 où N et N

sont isotriviaux, et que P -C(N ) = 0. Alors N est isotrivial. Voici des exemples où la condition 2) de 4.3.1 est facile à vérifier. 4.3.5 Corollaire. Tout M ∈ MICS tel que LieGal(M, η) soit résoluble a la propriété de Grothendieck–Katz. En effet, LG(M) n’a alors aucun quotient de Lie simple non-abélien. Un cas très particulier (essentiellement dû à Chudnovsky–Chudnovsky [18]) est le suivant : 4.3.6 Corollaire. Tout objet N ∈ MICS de rang 1 vérifie la conjecture de Grothendieck 3.3.1. 4.3.7 Corollaire. Supposons que tout sous-objet simple de EndM soit ou bien isotrivial, ou bien irrégulier (i.e. à singularités irrégulières “à l’infini” ). Alors M a la propriété de Grothendieck–Katz. (Utiliser 3.2.7.1).

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

69

4.3.8 Corollaire. Supposons que S soit le complémentaire d’une réunion de diviseurs Di dans l’espace projectif, que M soit régulier, et que les différences des exposants7 de M autour de chaque Di soit irrationnelles ou entières. Alors M a la propriété de Grothendieck–Katz. (Utiliser 3.2.7.2 ; la monodromie de L autour de chaque Di est triviale, donc L est trivial sous nos hypothèses). C’est par exemple le cas, pour un paramètre t suffisamment général, de la connexion de Knizhnik–Zamolodchikov donnée par la 1-forme

d(xi − xj ) t A(i,j ) xi − x j i<j

sur le fibré trivial sur (A1 )d \ diagonales, où les matrices constantes A(i,j ) vérifient les conditions d’intégrabilité usuelles (cf. [43]). 4.4 Réduction de 4.3.1 à 4.3.4 + 4.3.6. Remarquons que 1) ⇒ 2) dans 4.3.1 découle immédiatement de l’implication directe ⇒ dans 4.1.2, compte tenu de ce que dans le diagramme commutatif d’épimorphismes canoniques LG(M)  LG(LG(M))

/L ι

 / LG(L)

ι est un isomorphisme puisque L est simple. Nous allons déduire l’implication réciproque 2) ⇒ 1) de 4.3.1 des corollaires 4.3.4 et 4.3.6, qui seront prouvés ultérieurement (et simultanément, cf. §6 complété par 7.1.3). Considérons l’algèbre de Lie J = LG(M)/P -C(M) dans la catégorie tannakienne M⊗ (qu’on peut supposer neutre quitte à remplacer k par une extension finie). Il est clair que les p-courbures de J s’annulent pour presque tout p, i.e. que P -C(J) = 0. Par ailleurs, considérons le quotient Q = LG(M)/(P -C(M) + RadLG(M)) de J par l’idéal RadLG(M)/(P -C(M) ∩ RadLG(M)). Cette algèbre de Lie de M⊗ est produit de ses idéaux simples non-abéliens dans M⊗ (voir plus bas, 14.3, pour les détails sur la notion d’“algèbre de Lie semi-simple” dans une catégorie tannakienne). Or par hypothèse, toute algèbre de Lie simple non-abélienne L dans M⊗ qui est quotient de LG(M) par un idéal de Lie (en particulier, tout facteur simple de Q), on a P -C(L) = 0. On a donc Q = 0. A fortiori, les fibres de J sont des algèbres de Lie résolubles. Considérons à nouveau l’objet N de M⊗ introduit dans la preuve de 3.2.1 : J n’est autre que l’image de LG(M) dans N := End(EndN ), qui s’identifie aussi à LG(N ). Donc la composante neutre du groupe de Galois différentiel de N (en n’importe quel 7 on trouvera une discussion détaillée des exposants dans [9, I.6.5]

70

Yves André

point) est résoluble. La théorie de Galois différentielle montre que, quitte à remplacer S par un revêtement étale fini (ce qui est loisible), N est extension itérée d’objets de rang 1 dans MICS . Puisque P -C(LG(N )) = 0, on conclut de 4.3.4 et 4.3.6 que LG(N ) = J = 0, d’où LG(M) = P -C(M).


5 Un critère d’algébricité Nous démontrons, dans le cadre de la théorie des G-fonctions (cf. [3]), un critère d’algébricité dont la formulation est proche du critère de rationalité bien connu de Borel–Dwork [26], et qui s’avère particulièrement adapté à l’étude de la conjecture de Grothendieck. 5.1 Densités. Soit k un corps de nombres. Pour toute place finie v de k, on note pv [kv :Qpv ] sa caractéristique résiduelle, kv le complété de k en v. On pose εv = [k:Q] et on

−εv normalise  la valeur absolue v-adique par |pv | = pv ; on note N(v) le degré résiduel. On a v|p εv = 1. Soit V un ensemble de places finies de k. On introduit les “densités” inférieure et supérieure   pv ≤n,v∈V εv p ≤n,v∈V εv , δ(V ) = lim sup v . δ(V ) = lim inf  n n pv ≤n εv pv ≤n εv

En notant V c le complémentaire de V dans l’ensemble des places finies de k, on a δ(V ) = 1 − δ(V c ). Pour k = Q (et seulement dans ce cas), on retrouve les notions usuelles de densités naturelles inférieure et supérieure ; plus généralement, si V est l’ensemble des places du corps de nombres k au-dessus d’un ensemble P de nombres premiers rationnels, on retrouve les densités naturelles inférieures et supérieures de P . Les densités δ et δ sont adaptées à la théorie des G-fonctions, notamment sous l’expression “logarithmique” que leur donne le lemme suivant : 5.1.1 Lemme. δ(V ) = lim inf n

1 n

log |pv |−1 , δ(V ) = lim sup n

pv ≤n,v∈V

1 n

pv ≤n,v∈V

Démonstration. On a 1 n

log |pv |−1 ≤

pv ≤n,v∈V

et log n n

pv ≤n,v∈V

log n n

pv ≤n,v∈V

 εv ∼

p ≤n,v∈V

v

pv ≤n εv

εv

εv ,

log |pv |−1 .

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

71

d’après le théorème des nombres premiers. Dans l’autre sens, on a, pour tout η ∈]0, 1[ et tout n > 1/η, 

log ηn 1 ηn≤pv ≤n,v∈V εv −1  log |pv | ≥ εv ∼ n n pv ≤n εv ηn≤pv ≤n,v∈V

ηn≤pv ≤n,v∈V

toujours d’après le théorème des nombres premiers. Le lemme s’ensuit en faisant η → 0.    log |pv |−1 5.1.2 Lemme. Si v∈V pv −1 < ∞ , alors δ(V ) = 0 (ce qui équivaut à dire que l’ensemble des premiers résiduels de v est de densité nulle). −1  v| < η, d’où En effet, pour tout η > 0, il existe n tel que n ≤pv ,v∈V logp|p v −1  log |pv |−1 1 −1 ≤ n ≤pv ≤n,v∈V pv −1 < η. Grâce au lemme précén ≤pv ≤n,v∈V log |pv | n  dent, on obtient alors δ(V ) = lim supn n1 pv ≤n,v∈V log |pv |−1 < η.


Un lien entre ces densités et les densités naturelles usuelles définies par   N(v)≤n,v∈V 1 N(v)≤n,v∈V 1 , d(V ) = lim sup  d(V ) = lim inf  n n N(v)≤n 1 N(v)≤n 1 est donné par : 5.1.3 Lemme. Soit V déc le sous-ensemble de V formé des places décomposées sur Q, 1 . Alors i.e. telles que εv = [k:Q] δ(V déc ) =

d(V ) d(V ) , δ(V déc ) = . [k : Q] [k : Q]

Démonstration. Quand n → ∞, on a

1∼ v|N(v)≤n

v décomposée|N(v)≤n

1 = [k : Q]

εv .

v décomposée|pv ≤n

 Par la relation v|p εv = 1, et compte tenu de ce que la densité des premiers p décomposés dans k est 1/[k : Q] (cas particulier du théorème  de Chebotarev qui se ramène immédiatement au cas galoisien), on voit que [k : Q] v décomposée|pv ≤n εv ∼  déc ) = d(V ), d(V déc ) = v|pv ≤n εv . Le lemme s’ensuit, compte tenu de ce que d(V d(V ).
On obtient alors une variante du théorème de Chebotarev (cas où le corps de base est Q) qui distingue entre les éléments d’une même classe de conjugaison : 5.1.4 Proposition. Supposons l’extension k/Q galoisienne, et soit σ ∈ Gal(k/Q). Soit Vσ l’ensemble des places finies de k non ramifiées sur Q telles que l’automorphisme

72

Yves André

de Frobenius (v, k/Q) en v soit σ . Alors δ(Vσ ) = δ(Vσ ) =

1 . [k : Q]

Démonstration. On reprend l’argument de Deuring usuel pour se ramener au cas cyclique. Soit Wσ l’ensemble des places finies w de k σ non ramifiées sur Q telles que l’élément de Frobenius en w dans k/k σ soit σ . Alors Vσ est l’ensemble des places d(Wσ ) de k au-dessus de Wσdéc . On en déduit que δ(Vσ ) = δ(Wσdéc ) qui est aussi [k σ :Q] par le lemme précédent. De même avec les densités supérieures. Or par le théorème de
Chebotarev (dans le cas cyclique), d(Wσ ) = d(Wσ ) = [k:k1 σ ] . 5.2 Les invariants ρ, σ, τ . Soit S une variété lisse géométriquement connexe sur un corps de nombres k. Considérons un module à connexion intégrable M = (M, ∇) sur ouvert U non vide de S. Dans un repère, et en termes de coordonnées locales xh , la connexion correspond par le dictionnaire 1.2 à un système linéaire intégrable aux dérivées partielles8 . Par dérivations successives du système ∂ y = y A(0,...,0,1,0,...,0) , ∂xh on obtient des équations 1 ∂m y = y Am m! ∂x m pour tout multi-indice m à d composantes, Am ∈ GLr (k(S)). Choisissons un modèle plat S de S sur Ok . Pour toute place finie v de k de bonne réduction (i.e. telle que S ⊗ Okv soit lisse au-dessus de Okv ), on définit la norme de Gauss v-adique sur k(S) comme étant la valeur absolue étendant | |v sur k, associée à l’anneau local du point générique de la fibre spéciale de S ⊗ Okv , qui est un anneau de valuation discrète. Elle dépend du choix du modèle, mais étant donné deux modèles, il n’y a qu’un nombre fini de places v pour lequelles les normes de Gauss v-adiques associées à chaque modèle diffèrent. On pose h(n, v) =

sup log+ ||Am ||v ,

m,|m|≤n

où || ||v désigne le maximum des normes de Gauss des coefficients, et log+ = admet une limite quand n tend vers l’infini, qui max(0, log). On montre que h(n,v) n n’est autre que le log+ de l’inverse du rayon de solubilité v-adique générique de la 8 la situation que nous considérons est toutefois un peu plus générale que celle de 1.2, en ce que M n’est pas supposé défini au point de coordonnées 0 (qui n’est pas supposé appartenir à U ), ce qui est la cause des pôles apparaissant dans les matrices

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

73

connexion ([3, p. 230], [24]). On définit alors les invariants

h(n, v)

h(n, v) lim sup , σ (M) = lim sup , ρ(M) = n n n n v finie

v finie

τ (M) = inf lim sup p

n

v, pv ≥p

h(n, v) , n

qui ne dépendent pas du choix du repère auxiliaire ni du modèle de S choisi, et sont invariants par extension finie du corps de base k et par remplacement de S par un ouvert dense. Si l’un de ces trois invariants est fini, les deux autres le sont aussi, et on a σ (M) = ρ(M) + τ (M) ; on dit alors que l’on a affaire à une G-connexion ([9], [24]). 5.2.1 Remarque. Dans ce cas, les p-courbures sont toujours nilpotentes au moins −1 v| ≥ logp|p pour un ensemble de p de densité égale à 1 ; en effet, on a limn h(n,v) n −1 v si la v-courbure n’est pas nilpotente, et on conclut grâce au lemme 5.1.2. On montre que toute connexion d’origine géométrique (cf. ci-dessous, 16.1) est une G-connexion (cf. [3, IV],[9]). On conjecture en fait qu’être une G-connexion, avoir ses p-courbures nilpotentes pour un ensemble de p de densité 1, et être d’origine géométrique, sont des notions équivalentes9 . Rappelons aussi qu’une connexion dont presque toutes les p-courbures sont nulles est une G-connexion ; c’est une question ouverte de savoir s’il en est de même en supposant seulement les p-courbures nulles pour un ensemble de premiers de densité 1. Comme l’a démontré B. Dwork, la quantité τ (M) est intimement liée à la distribution de l’ordre de nilpotence des p-courbures de M, lorsque p parcourt les nombres premiers [29] : 5.2.2 Proposition. On se place dans le cas d’une G-connexion. Soit V (M) l’ensemble des places finies v telles que la réduction de M modulo v soit non triviale (ou, ce qui revient au même, telles que les v-courbures soient non nulles). Alors  1 1  δ(V (M)) ≤ τ (M) ≤ 1 + + · · · + δ(V (M)). 2 r −1 A fortiori, les p-courbures de M s’annulent pour un ensemble de premiers de densité 1 si et seulement si τ (M) = 0. Via 2.1.1, c’est démontré dans [29]. Dwork se limite au cas d = 1 et k = Q, mais on trouvera la généralisation au cas de plusieurs variables et d’un corps de nombres arbitraire dans [24]).
9 cette conjecture remplace depuis les années 1980 la croyance opposée, le point d’oscillation se trouvant

dans l’article de Dwork [27], au cours duquel l’opinion de l’auteur semble évoluer. Rappelons toutefois que Dwork a toujours préféré, semble-t-il, s’abstenir de proposer une définition formelle des systèmes différentiels d’origine géométrique

74

Yves André

5.2.3 Remarque. Dwork établit aussi (loc. cit., cor. 6) la minoration suivante. Soit Vi (M) l’ensemble des places finies v de k où les v-courbures sont nilpotentes d’échelon exact i. Alors

 1  1 δ(Vi (M)). 1 + + ··· + τ (M) ≥ 2 i−1 i=2,...,r

Est-ce une égalité ? Il serait intéressant d’autre part de savoir si δ(Vi (M)) = δ(Vi (M)), et si ces densités sont rationnelles. En ce sens, la conjecture suivante précise des suggestions de [3] et [29] : 5.2.4 Conjecture. Soit M une G-connexion. Alors i) l’ensemble des places finies v de k telles que les v-courbures soient nilpotentes d’échelon fixé a une “densité” (au sens de 5.1) qui est un nombre rationnel, ii) τ (M) ∈ Q. iii) (conjecture de Grothendieck légèrement renforcée)10 . τ (M) = 0 si et seulement si M est isotriviale. Voir §16 pour un résultat partiel. 5.3 Invariants ρ, σ, τ de séries. Normalisons les valeurs absolues archimédiennes w :R] et où | |∞ est la valeur absolue w de k comme suit : |ξ |w = |ξ |ε∞w , où εw = [k[k:Q] euclidienne. La “formule du produit” s’écrit alors v placede k |ξ |v = 1 (pour x ∈ k ∗ ).  Soit y = am x m ∈ k r [[x1 , . . . , xd ]] un vecteur-ligne de séries formelles. On définit

1 lim sup sup log+ || am ||v , ρ( y) = n |m|≤n n v placede k

σ ( y ) = lim sup n

v placede k

τ ( y ) = inf lim sup p

n

1 sup log+ || am ||v , n |m|≤n

1 am ||v , sup log+ || n v, p ≥p |m|≤n v

quantités invariantes par extension finie du corps de base, et liées par l’inégalité σ ( y ) ≤ ρ( y ) + τ ( y) . 5.3.1 Corollaire. Supposons que y ∈ k r [[x]] soit solution d’une G-connexion, et supposons
que le point 0 ne soit  pas sur le lieu singulier (i.e. soit dans U ). Alors on 1 δ(V (M)). En particulier, si les p-courbures s’annulent a τ ( y ) ≤ 1 + 21 + · · · + r−1 pour un ensemble de premiers p de densité 1, alors τ ( y ) = 0. 10 “pour presque tout p” étant remplacé par “pour une infinité de p de densité 1”

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

75

Démonstration. Pour toute place finie v, on a l’inégalité am ||v ≤ h(n, v) + cv , sup log+ ||

|m|≤n

y ) ≤ τ (M), et on applique où cv est une constante nulle pour presque tout v, d’où τ ( la proposition précédente.
5.3.2 Remarque. Il est immédiat que τ ( y ) = 0 si les coefficients des composantes de y sont entiers, ou plus généralement “S-entiers” (pour un ensemble fini convenable S de places de k). Dans le cas où y est solution d’un système différentiel comme cidessus, sans singularité en 0, une (variante d’une) conjecture de G. Christol prédit que si τ ( y ) = 0, alors y est algébrique sur k(x). Le corollaire précédent montre que cette conjecture entraîne la conjecture de Grothendieck (sur le corps de base k). Voici une réciproque partielle à 5.3.1 (qui sera utilisée dans l’appendice) : 5.3.3 Proposition. Plaçons-nous dans le cas d’une seule variable. Supposons que M admette une k-base de solutions yi dont les coefficients de chaque composante sont des S-entiers. Alors les p-courbures de M s’annulent pour presque tout p. Démonstration. Choisissons un vecteur cyclique pour M sur k(x), ce qui nous permet de traiter d’un opérateur   différentiel L, et de remplacer les vecteurs yi par des solutions yi de L dans o N1 [[x] (pour N convenable), linéairement indépendantes sur le corps de constantes k de k((x)). Leur wronskien W (y1 , . . . , yr ) est non nul, et le reste par réduction modulo v pour presque toute place finie v. Le lemme du wronskien – valable en toute caractéristique11 – permet de conclure que pour presque tout v, les réductions modulo v des yi sont linéairement indépendantes sur le corps des constantes κ(v)((x p )) de κ(v)((x)), ce qui entraîne la nullité de la p-courbure d’après [38, 6.0.7].
Considérons à présent le vecteur ysym n dont les composantes sont les monômes de degré n en les composantes de y (écrits dans l’ordre lexicographique). 5.3.4 Lemme.

 1 1 y ), τ ( y sym n ) ≤ 1 + + · · · + τ ( y ), ρ( y sym n ) ≤ ρ( 2 n  1 1 τ ( y ). y) + 1 + + · · · + σ ( y sym n ) ≤ ρ( 2 n

Démonstration. lim supn n1 sup|m|≤n log+ || am ||v n’est autre que le log+ de l’inverse du rayon de convergence de y, ce qui rend immédiate la première inégalité. Pour  la seconde, écrivons une composante arbitraire bm x m ∈ k r [[x1 , . . . , xd ]] de 11 voir par exemple [23, App. B] qui traite même le cas de plusieurs variables

76

Yves André

j =n  ysym n comme monôme yi1 yi2 . . . yin . On a donc bm = mj =m j =1 aij ,mj , et j =n + + log |bm |v ≤ max m =m j =1 log |aij ,mj |v (plus un terme O(log |m|) si v est archij médienne). Notons (. . . , m j , . . . ) le n-uplet qui réalise le maximum. Quitte à permuter les facteurs yij , on peut supposer que |m j | décroît avec j , de sorte que |m j | ≤ |m|/j . j =n On obtient log+ |bm |v ≤ j =1 max|m j |≤|m|/j log+ |aij ,mj |v (plus un terme O(log |m|) si v est archimédienne), et il n’est alors pas difficile d’en déduire la deuxième inégalité. La troisième découle des deux précédentes. optimale. Par exemple, on peut 5.3.5 Remarque. La borne pour τ ( y sym n ) est souvent 1 montrer que c’est le cas pour y = en suivant [3, p.150]. log(1 + x) 5.4 Le critère. Il s’agit d’un critère d’algébricité inspiré à la fois de celui de Chudnovsky [18] (lui-même inspiré du critère de transcendance bien connu de Schneider– Lang), et du critère de rationalité de Borel–Dwork. Ce critère figurait sous une forme plus générale dans [3, VIII 1.2] ; nous en reproduisons ci-dessous le cas particulier (légèrement raffiné) qui nous intéresse, pour la commodité du lecteur. Pour une réécriture de ce critère dans le style arakelovien de J.-B. Bost [15][16], voir [17]. Rappelons le critère de rationalité Borel–Dwork, sous forme légèrement générali sée : une série y ∈ k[[x]] est rationnelle si (et seulement si) τ (y) = 0 et v finie Mv (y) > 1 (où Mv désigne le rayon de méromorphie de y vue comme fonction v-adique). Notre critère d’algébricité est d’aspect semblable, mais remplace le rayon de méromorphie de y(x) par le rayon de méromorphie d’une uniformisation simultanée de y et x par une nouvelle variable z. 5.4.1 Définition. Soit y ∈ k[[x1 , . . . , xd ]] et soit v une place de k. Une uniformisation v-adique simultanée de y et x dans le polydisque non-circonférencié D(0, Rv )d est la donnée de d + 1 fonctions méromorphes v-adiques hv (zv ), hv,i (zv ) (i = 1, . . . , d) de d variables z1,v , . . . , zd,v dans le polydisque non circonférencié D(0, Rv )d , vérifiant : 1) hv,i (0) = 0 pour i = 1, . . . , d,
∂hv,i  2) la matrice jacobienne à l’origine ∂zv,j (0) est la matrice identité, 3) y(hv,1 (zv ), . . . , hv,d (zv )) est le germe en l’origine de la fonction méromorphe hv (zv ). 5.4.2 Exemple. zi,v = xi , hv = y est une uniformisation v-adique simultanée (dite triviale) de y et x dans le polydisque D(0, Rv (y))d . 5.4.3 Théorème. Soit y ∈ k[[x]] telle que τ (y) = 0 et ρ(y) < ∞. Supposons que pour toute place v de k, il existe une uniformisation v-adique simultanée de y et x dans un polydisque D(0, Rv )d , et que Rv > 1. Alors y est algébrique sur k(x).

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

77

5.4.4 Remarques. 1) Le cas où l’uniformisation v-adique est l’uniformisation triviale pour tout v correspond au critère de Borel–Dwork ; on obtient dans ce cas la conclusion plus forte que y ∈ k(x). L’hypothèse est à rapprocher de la notion d’uniformisation adélique simultanée > 1 était affaiblie en introduite en transcendance dans [5], où la condition R v Rv ≥ 1. Nous y reviendrons dans l’appendice. 2) Le théorème admet une réciproque partielle (qui justifie notamment la généralisation à plusieurs variables) : si y ∈ k[[x]] est algébrique sur k(x) (une seule variable), alors pour d convenable, et pour toute place complexe de k, il existe une uniformisation simultanée de x1 , . . . , xd et de toute fonction symétrique de y(x1 ), . . . , y(xd ) sur Cd (uniformisation d’Abel–Jacobi). L’idée (dûe à Chudnovsky [18]) d’utiliser l’uniformisation d’Abel–Jacobi dans ce contexte apparaîtra en 6.4. Démonstration. Nous procéderons en plusieurs étapes (en suivant [3, VIII]). 1er pas. Fixons un entier r > 0 (destiné à tendre ultérieurement vers ∞) et un paramètre η ∈]0, 1[ (destiné à tendre ultérieurement vers 0). Notons y = am x m le vecteur de composantes (1, y, . . . , y r−1 ). La finitude de τ (y) et ρ(y) entraîne celle de σ ( y ). On construit une suite de vecteurs-polynômes pN ∈ k r [x], vérifiant : a) M := ord0 pN . y ≥ N, b) deg pN ≤ ( 1r (1 + η1 ))1/d N + o(N ), y )N +o(N ) (où h(pN ) désigne la hauteur logarithmique invariante c) h(pN ) ≤ ησ ( des coefficients de pN ). est standard : la condition a) définit un système linéaire  homogène à  La construction N −1+d deg pN + d inconnues (les coefficients des composantes de pN ) et r d d équations, et on applique le lemme de Siegel). 2ème pas. On suppose pN . y non identiquement nul (i.e. M fini, autrement le problème est résolu). On choisit alors un multi-indice m, avec |m| = M, tel que α :=

1 dm (pN . y )  = 0. m! dx m

Pour toute place finie v de k, on a l’estimation ()

am ||v . log |α|v ≤ log+ |pN |v + sup log+ || |m|≤M

3ème pas. On considère un ensemble fini V (destiné à augmenter) de places de k contenant les places archimédiennes. Pour chaque v dans V , on considère les uniformisations v-adiques simultanées de y et x dans le polydisque D(0, Rv )d ; on écrit hv (zv ) = fv (zv )/gv (zv ), hv,i (zv ) = fv,i (zv )/ev (zv ), quotients de fonctions analytiques dans D(0, Rv )d avec gv (zv ), ev (zv ) sans zé v (zv ) le vecteur ros dans D(0, Rv )d et valant 1 en l’origine. En notant Z de composantes (fv (zv )r−1 , fv (zv )r−2 , . . . , 1), et ψ(zv ) la fonction analytique  v (zv ), on voit que v-adique ev (zv )deg pN .pN (fv,1 (zv )/ev (zv ), . . . , fv,s (zv )/ev (zv )).Z

78

Yves André

pN . y (hv,1 (zv ), . . . , hv,s (zv )) est le germe en l’origine de la fonction méromorphe
∂hv,i  gv (zv )1−r .ev (zv )− deg pN ψ(zv ) dans D(0, Rv )d . Comme ∂zv,j (0) = Id, et que gv (0) = ev (0) = 1, on a α =

1 dm m! dx m

ψ(0).

4ème pas. Pour chaque v dans V , choisissons Rv assez proche de Rv par défaut. Les estimations de Cauchy donnent    1 dm 

−M   sup |ψ(z)|v ,  m! dx m ψ(0) ≤ Rv |z|v =Rv v d’où log |α|v ≤ −M log Rv + log+ |pN |v + deg pN .χv + o(N),

() où l’on a posé

χv = max(log sup |fi,v (zv )|v , . . . , log sup |ev (zv )|v ). |z|v =Rv

|z|v =Rv

5ème pas. On additionne les inégalités () pour v ∈ / V et () pour v ∈ V , et on applique la formule du produit, ce qui donne

M log Rv ≤ h(pN ) + sup log+ || am ||v + deg pn χv + o(N). |m|≤m

v∈V

v∈V

On divise cette inégalité par M(≥ N ) et on fait tendre N vers l’infini, ce qui donne, compte tenu des estimations pour la hauteur et le degré de pN  

1 1 1 1/d

+ log Rv ≤ ησ ( y )+lim sup am ||v + χv . sup log || 1+ M |m|≤M r η M v∈V

v∈V

v∈V

6ème pas. Comme τ (y) = 0, on peut remplacer σ ( y ) par ρ(y), et

1 sup log+ || am ||v lim sup M |m|≤M M v∈V  + −1 (par l’argument de 5.3.4 ; cette dernière série converge puisque par v ∈V / log Rv (y) ρ(y) < ∞). Ceci permet de faire tendre r vers l’infini, puis η vers 0, et enfin Rv vers Rv , ce qui donne

log Rv ≤ log+ Rv (y)−1 . v∈V

v ∈V /

En prenant V assez grand, ceci contredit l’hypothèse Rv > 1. On conclut de là que y = 0, donc y est algébrique.
pour N assez grand, pN . 5.4.5 Corollaire. Supposons que y ∈ k r [[x]] soit solution d’une G-connexion de rang r dont les p-courbures s’annulent pour un ensemble de premiers p de densité 1. Supposons en outre que pour toute place v de k, il existe une uniformisation v-adique

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

79

simultanée d’une certaine composante y de y et x dans un polydisque D(0, Rv )d , et que Rv > 1. Alors y est algébrique sur k(x). Démonstration. Puisque les p-courbures de M s’annulent pour un ensemble de premiers p de densité 1, on peut, d’après 3.2.7, après passage de U à un revêtement fini étale U , supposer que MU s’étend en un module à connexion sur une compactification lisse S¯ , donc n’ait pas de singularité “à l’infini”. On peut alors appliquer 5.3.1 et 5.4.3 pour conclure.
5.4.6 Remarque. Dans la situation où Rv > 0 et où Rv est infini pour l’une au moins des places archimédiennes de k, ce critère devient un cas particulier du critère de J.-B. Bost formulé dans le cadre beaucoup plus général des feuilletages algébriques.

6 Preuve de 4.3.4 et 4.3.6 (cas d’un corps de nombres) 6.1. Il s’agit de prouver les énoncés suivants (le premier étant légèrement plus fort que 4.3.4), dans le cas où k est un corps de nombres12 (voir aussi [3, VIII], et [18] pour le second énoncé). 6.1.1 Proposition. Supposons que N ∈ MICS s’inscrive dans une suite exacte 0 → N → N → N

→ 0 où N et N

sont isotriviaux, et supposons que les p-courbures de N s’annulent pour une infinité de p de densité 1. Alors N est isotrivial. 6.1.2 Proposition. Tout objet N ∈ MICS de rang 1 vérifie la conjecture de Grothendieck 3.3.1. Il est commode de se ramener au cas où S est une courbe. Comme dans les deux cas, il suit de l’hypothèse sur les p-courbures que N est régulier (3.2.7), tout est affaire de montrer la finitude de la monodromie (k étant plongé dans C). Or on sait d’après Lefschetz que pour une courbe affine lisse “assez générale” C tracée sur S (quitte à remplacer k par une extension finie), π1 (C(C)) → π1 (S(C)) est surjectif. 6.1.3 Remarque. Par ailleurs, l’hypothèse sur les p-courbures implique que N s’étend à la complétion projective lisse de C (3.2.7), ce qui permet de supposer que S est une courbe projective lisse. Nous n’aurons pas à nous en servir. 12 le cas d’un corps de base de caractéristique nulle quelconque sera ramené au cas d’un corps de nombres

au chapitre suivant ; du reste le fragment 10.2 suffirait à effectuer cette réduction

80

Yves André

6.2. Il est suggestif et utile de traduire 6.1.1 et 6.1.2 en termes de différentielles (en nous plaçant de nouveau dans la situation d’une variété lisse géométriquement connexe quelconque S définie sur un corps de nombres13 k). Comme l’isotrivialité est une propriété locale pour la topologie étale sur S, de même que l’hypothèse sur les pcourbures, on se ramène à supposer dans 6.1.1 que M et M

sont des connexions triviales, puis que M = M

= (OS , d). L’extension N correspond alors à un élément 1 (S) dont la nullité équivaut à la trivialité (et à l’isotrivialité) de M (cf. [45, [ω] de HDR 2.5]). Quitte à localiser S, on peut supposer que [ω] est la classe d’un élément ω de H 0 (S, 1S ). Ceci ramène 6.1.1 à ceci : 6.2.1 Proposition. Soit ω une 1-forme fermée sur S telle que ω modulo v soit localement exacte (ou, ce qui revient au même, annulée par l’opérateur de Cartier [38, §7]) pour tout place finie v de k au-dessus d’un ensemble de premiers rationnels p de densité 1. Alors ω est exacte. Quant à 6.1.2, l’énoncé se ramène au suivant, dans le cas où le fibré de rang 1 sous-jacent M est isomorphe à OS (ce qu’on peut supposer en localisant), cf. [38, introduction, et 7.4] : 6.2.2 Proposition. Soit ω une 1-forme fermée sur S telle que ω modulo v soit localement logarithmiquement exacte (ou, ce qui revient au même, fixée par l’opérateur de Cartier) pour toute place finie v de k au-dessus d’un ensemble de premiers rationnels p de densité 1. Alors il existe n entier non nul tel que nω soit logarithmiquement exacte (i.e. de la forme df/f ). Cette proposition donne une réponse positive au problème étudié dans [38, §7] sous divers avatars. 6.3 Réduction au cas où S est un schéma en groupe commutatif. Comme on l’a vu plus haut, on peut supposer que S est une courbe lisse, et il s’agit de prouver 6.1.1 et 6.1.2. On peut aussi supposer que S(k)  = ∅ et on fixe un point base s0 ∈ S(k). Soit S¯ la complétion projective lisse de S, et étendons ω en une section (encore notée ω) de 1S¯ (−D) pour un diviseur effectif convenable de S (par exemple, le “diviseur des ¯ D) la jacobienne généralisée JD de S¯ de module D, qui pôles” de ω). On attache à (S, paramètre les fibrés inversibles sur S¯ “rigidifiés” au-dessus de D. On a un morphisme ϕs0 : S → JD ,

s → [OS¯ (s) ⊗ OS¯ (s0 )∨ ],

et il existe une unique forme différentielle invariante ωJD telle que ω = ϕs∗0 ωJD , cf. [46, p. 97]. Pour tout entier n > 0, ϕs0 induit un morphisme ϕs(n) : S (n) → JD , 0

s1 + · · · + sn  → [⊗(OS¯ (si ) ⊗ OS¯ (s0 )∨ )]

13 on verra au chapitre suivant comment généraliser au cas d’un corps de base de caractéristique nulle

quelconque

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

81

(n)

(n) ∗ où  S ∗ désigne la puissance symétrique n-ième de S, et (ϕs0 ) (ωJD ) n’est autre que pr i ω. (n) . Il suit que la condition que Pour n ≥ dim JD , on sait que ϕs0 est dominante14 l’opérateur de Cartier annule (resp. fixe) ω - donc aussi pri∗ ω - modulo v, implique (n) la condition analogue pour (ϕs0 )∗ (ωJD ). Par ailleurs, si ωJD est exacte (resp. logarithmiquement exacte) il en est de même de ω = ϕs∗0 ωJD . On peut donc remplacer, dans 6.1.1 et 6.1.2, S par le schéma en groupe commutatif JD .

6.4 Cas où S est un schéma en groupe commutatif. Supposons donc que S soit un schéma en groupe commutatif. 6.1.1 (resp. 6.1.2) équivaut à l’existence, sous l’hypothèse d’annulation des p1/n courbures, d’une solution y1 ∈ H 0 (S, OS ) (resp. de la forme y2 avec y2 ∈ H 0 (S, OS )) du système différentiel (∗)1 dy = ω ⊗ 1,

resp.

(∗)2 dy = ω ⊗ y.

En fait, il est équivalent de trouver une solution algébrique sur k(S) de (∗)1 (resp. (∗)2 ) : cela vient de ce que le groupe de Galois différentiel attaché à (∗)1 est a priori un sous-groupe de Ga (resp. Gm ), donc il est équivalent de dire que ce groupe est fini ou qu’il est trivial (resp. cyclique). Choisissant des coordonnées locales x1 , . . . , xd au voisinage de l’origine de S, il s’agit donc de montrer que toute solution formelle y ∈ k[[x1 , . . . , xd ]] de (∗)1 (resp. (∗)2 ) est algébrique, sous l’hypothèse d’annulation des p-courbures. Pour appliquer le critère 5.4.5, il nous faut savoir que i) la connexion sous-jacente est une G-connexion, ii) pour toute place v de k, il existe une uniformisation v-adique simultanée de y et x dans un polydisque D(0, Rv )d , et que Rv > 1. Pour i), c’est clair dans le cas de (∗)1 puisque la connexion sous-jacente est extension de connexions triviales. Dans le cas de (∗)2 , cela découle de l’hypothèse d’annulation des p-courbures pour presque tout p (et non seulement pour une infinité de densité 1, cf. 5.2.3). A fortiori, l’invariant ρ de la connexion est fini. Pour ii), il suffit de prendre l’uniformisation triviale pour toute place finie, et pour tout plongement complexe k → C, l’uniformisation donnée par l’exponentielle du groupe de Lie complexe S(C) → S(C) ∼ expS(C) : Lie S(C) ∼ = Cd → = Lie S(C)/π1 (S(C), 0) qui donne un rayon d’uniformisation simultanée infini. La finitude de l’invariant ρ de la R connexion implique, par spécialisation, la finitude de ρ(y), donc que v finie v > 0.
On a finalement v place de k Rv = ∞.

14 et d’ailleurs birationnelle pour n = dim J D

82

Yves André

II Analogue de la conjecture de Grothendieck en équicaractéristique nulle 7 Énoncé des résultats 7.1. Soit M ∈ MICS . En 3.1, nous avons “épaissi” la situation en choisissant un sousanneau o de k de type fini sur Z, un o-schéma connexe S à fibres géométriquement connexes, et un OS -module localement libre à connexion intégrable M, tels que S et M se déduisent de S et M par extension des scalaires o → k. Il est facile de voir que si M est isotrivial, alors pour tout point fermé t d’un ouvert dense de Spec oQ , la fibre de MQ en t est isotriviale. Nous nous proposons de démontrer la réciproque15 , analogue en équicaractéristique nulle de la conjecture de Grothendieck 3.3.3 : 7.1.1 Proposition. Si pour tout point fermé t d’un ouvert dense de Spec oQ , la fibre de MQ en t est isotriviale, alors M est isotrivial. 7.1.2 Remarque. Dans un manuscrit récent [34], E. Hrushovsky propose une autre approche de ce résultat par la théorie des modèles. 7.1.3 Corollaire. M vérifie la conjecture de Grothendieck si et seulement si (MQ )t vérifie la conjecture de Grothendieck pour tout point fermé t d’un ouvert dense de Spec oQ . Comme κ(t) est un corps de nombres, cela ramène l’étude de la conjecture de Grothendieck au cas où le corps de base k est un corps de nombres, démontrant ainsi une assertion non justifiée de [38, Intro.], et complétant la preuve de 4.3.4 et 4.3.6 dans le cas général. 7.1.4 Remarque. On peut se limiter, dans l’étude de la conjecture de Grothendieck, au cas où S est une courbe affine sur un corps de nombres k. D’après le théorème de Belyi [12], il existe un revêtement étale π : S → P 1 − {0, 1, ∞}. Au lieu de M, il suffit de traiter π∗ (M), ou même d’après ce qui précède, les quotients de Lie simples horizontaux de LG(π∗ (M)). On est donc ramené au cas d’un fibré à connexion irréductible N sur P 1 − {0, 1, ∞} (muni d’un crochet de Lie horizontal si l’on veut). Les hypothèses sur les p-courbures impliquent la régularité de N en 0, 1, ∞. Or il est connu que tout fibré à connexion régulière irréductible sur P 1 − {0, 1, ∞} admet 
A Bij  une base de sections ej telle que ∇ ∂ (ej ) = i xij + x−1 ei , où A et B sont des ∂x matrices constantes (cf. [11, §5]). Dans le cas “universel” où A et B sont vues comme des indéterminées non-commutatives, la condition d’annulation des p-courbures se traduit par des identités polynômiales universelles en A et B modulo p. 15 qui était implicite dans [7, §5]

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

83

7.2. Soient k un corps de caractéristique nulle, T un k-schéma séparé de type fini, f : S → T un T -schéma lisse à fibres géométriquement connexes. Soit M = (M, ∇) la donnée d’un OS -module M localement libre de rang r < ∞ et d’une connexion relative intégrable ∇ : M → 1S/T ⊗OS M, qu’on voit comme une famille de connexions intégrables paramétrées par T . Dans ce cadre relatif, diverses difficultés se présentent : – la catégorie des fibrés à connexion relative intégrable n’est pas abélienne si dim T > 0, – la notion de connexion triviale soulève divers problèmes, cf. [8, 3.1.1]. Suivant loc. cit., on dit que M est trivial (du point de vue de la connexion, pas du fibré sous-jacent) s’il est de la forme (OS , dS/T )⊗f −1 OT f −1 N , où N est un OT -module localement libre. Du fait que les anneaux locaux de S soient “différentiellement simples par couches” au sens de loc. cit., on a (cf. [8, 3.1.2.3, 3.1.3.2]) : 7.2.1 Lemme (non utilisé par la suite). Les conditions suivantes sont équivalentes : i) M est trivial, ii) M est Zariski-localement engendré par ses sections horizontales, iii) l’homomorphisme canonique M ∇ ⊗f −1 OT OS → M est un isomorphisme, En fait, nous éviterons ces difficultés en supposant T intègre et en travaillant au point générique de T . 7.2.2 Théorème. MηT est isotrivial si et seulement si pour tout point fermé t d’un ouvert dense de T , M(t) est isotrivial. La proposition 7.1.1 s’en déduit immédiatement, en localisant o et en prenant T = Spec oQ , S = SQ , M = MQ . 7.2.3 Remarque. Il ne suffirait pas, dans 7.2.2, de demander l’existence d’une ensemble Zariski-dense de points fermés t tels que M(t) est isotrivial, comme le montre 1 l’exemple de la connexion ∇(1) = t dx x sur OA1 ×Gm relative à T = A (coordonnée t 1 sur A ), dont la fibre est isotriviale pour tout t ∈ Q. 7.3. L’implication directe de 7.2.2 est facile. Occupons-nous de la réciproque. Nous commencerons par nous ramener au cas où S est projectif sur T .

8 Réduction au cas projectif 8.1. Comme MηT est isotrivial si et seulement si Mη¯ T l’est, il est loisible de remplacer T par T génériquement fini et dominant sur T . En particulier, on se ramène à supposer k algébriquement clos, et T géométriquement connexe. Il est aussi loisible de remplacer S par S étale dominant sur S, ce qui permet de supposer Sη quasiprojectif.

84

Yves André

8.1.1 Lemme. Si M(t) est régulier pour un ensemble dense de points t ∈ T , il en est de même pour tout point d’un ouvert dense de T . Pour le voir, on peut par exemple utiliser le critère de régularité par restriction à des courbes (relatives dans S/T ) [10, I.5], puis, en appliquant le lemme du vecteur cyclique, d’utiliser le critère de régularité en termes de valuation des coefficients de l’opérateur différentiel associé, voir par exemple [10, II. 4.2] pour des détails. 8.1.2 Remarque. On ne peut remplacer “pour tout point d’un ouvert dense” par “pour tout point” à cause du phénomène de confluence. 8.2. Comme toute connexion isotriviale est régulière, on déduit de l’hypothèse de 7.2.2 et du lemme que MηT est régulière. Quitte à remplacer T par un ouvert dense, on peut trouver une compactification projective lisse relative S¯ → T de S → T , où ∂S/T := S¯ \ S est un diviseur à croisements normaux strict relatif à T (Hironaka). La méthode algébrique de [10, I.4]16 permet d’étendre M, quitte à remplacer T par un ouvert étale, en un module à connexion M¯ à pôles logarithmiques le long de ∂S/T . Pour chaque composante Di de ∂S/T , on a alors la notion de résidu de ∇ le long de Di : c’est un endomorphisme de M¯ Di , dont les valeurs propres appartiennent à l’image d’une section fixée : τ : k(S) → k(S)/Z. Si M(t) est isotrivial pour tout point fermé t ∈ T , la fibre en t de ResDi (∇) est semi-simple, et de valeurs propres ∈ τ (Q/Z). Il s’ensuit que ResDi (∇) est lui-même semi-simple, de valeurs propres “constantes” ∈ τ (Q/Z). En particulier, pour tout point t ∈ T et tout plongement κ(t) → C (s’il en est), la monodromie locale de M(t) ⊗ C autour de (Di )t ⊗ C est d’ordre fini indépendant de t. 8.2.1 Lemme. Il existe un diagramme commutatif ? _ S S¯ Go G GG GG GG GG # T

/S  /T

où T est étale dominant sur T , S est étale dominant sur S, S → T est lisse à fibres géométriquement connexes, S¯ → T est projectif lisse, S¯ \ S est un diviseur de S¯ à croisements normaux relatifs sur T , et MS s’étend en un fibré à connexion intégrable sur S¯ relativement à T . Démonstration. On peut supposer k de cardinal inférieur ou égal à celui du continu, ¯ ) dans C. Par un descente standard, il suffit de montrer ce qui permet de plonger k(T l’existence d’un revêtement étale fini connexe SC de SC = S ⊗T Spec C tel que MS C s’étende en un fibré à connexion intégrable sur une compactification projective lisse S¯C de SC telle que S¯C \ SC soit à croisements normaux. L’existence d’extensions à 16 appliquée en prenant pour corps de base K une clôture algébrique k(S) de k(S)

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

85

pôles logarithmiques ramène la question à trouver un revêtement étale fini SC de SC tel que les monodromies locales autour des images inverses des Di,C dans SC , qui sont d’ordre fini, soient en fait triviales. Fixant un point base fermé s ∈ SC , il suffit de définir SC par un sous-groupe d’indice fini de π1 (SC , s) dont l’image par la représentation de monodromie est un sous-groupe sans torsion de GL((MSC )s ). L’existence d’un tel sous-groupe est garantie par le lemme classique de Selberg.
8.3. Ce lemme ramène la preuve de 7.2.2 au cas où S est projectif lisse sur T . Quitte à remplacer T par T étale dominant sur T , on peut supposer par ailleurs qu’il existe une section T → S.

9 Espaces de modules de connexions (rappels) 9.1. Soient k un corps algébriquement clos de caractéristique nulle, T un k-schéma séparé de type fini, f : S → T un T -schéma projectif lisse à fibres géométriquement  connexes. Soit MDR (S/T , r) le foncteur contravariant qui associe à tout T -schéma séparé de type fini T l’ensemble des classes d’isomorphismes de OS×T T -modules localement libres de rang r munis d’une connexion intégrable relative à T . Dans [49], C. Simpson construit un T -schéma quasi-projectif MDR (S/T , r)  qui coreprésente MDR (S/T , r) : il existe un morphisme canonique de foncteurs  ϕ : MDR (S/T , r) → MDR (S/T , r) tel que si Y est un T -schéma séparé de type  fini, tout morphisme de foncteurs MDR (S/T , r) → Y se factorise de manière unique à travers ϕ. Cette propriété détermine MDR (S/T , r) à isomorphisme unique près. Simpson décrit beaucoup de propriétés intéressantes de MDR (S/T , r) (voir loc. cit., II, 6.1317 ). Notamment, c’est un schéma de modules grossier : les points géométriques de MDR (S/T , r) représentent les classes d’équivalences de modules à connexion intégrable de rang r sur les fibres géométriques de f - deux modules à connexion étant décrétés équivalents si leur semi-simplifiés sont isomorphes. La fibre de MDR (S/T , r) en tout point t ∈ T s’identifie canoniquement à MDR (St , r). 9.2. Le fibré à connexion (OS , dS/T )r correspond à une section, dite neutre, du morphisme structural MDR (S/T , r) → T . D’ailleurs, toute connexion relative triviale18 de rang r donne lieu à la même section, comme on le voir sur les points géométriques.   Le morphisme MDR (S/T , 1)×T r → MDR (S/T , r) “somme directe de r connexions de rang 1” donne lieu à un morphisme de schémas MDR (S/T , 1)×T r → MDR (S/T , r) qui nous sera utile (MDR (S/T , 1)×T r désigne la puissance r-ième fibrée sur T ). 9.3. Pour r = 1, ces espaces de modules sont classiques : le produit tensoriel des connexions munit MDR (S/T , 1) d’une structure de schéma en groupe commutatif 17 Simpson énonce ses résultats avec k = C, mais la construction est purement algébrique et vaut sur un corps algébriquement clos de caractéristique nulle quelconque 18 voire “nilpotente”

86

Yves André

sur T (la section neutre s’identifiant dans ce cas à la section nulle), et l’oubli de la connexion induit un homomorphisme de MDR (S/T , 1) vers le schéma de Picard relatif Pic0 (S/T ). Cet homomorphisme est fidèlement plat, de noyau le schéma en groupe vectoriel attaché à 1S/T , de sorte que MDR (S/T , 1) s’identifie à l’extension vectorielle universelle de Pic0 (S/T ). Les connexions isotriviales de rang 1 sur les fibres géométriques St correspondent aux points de torsion de MDR (S/T , 1)t = MDR (St , 1).

10 Une application du théorème de Jordan 10.1. On se place de nouveau dans le cadre de 7.2.2, en supposant désormais que f est projectif lisse et admet une section σ . 10.1.1 Lemme. Il existe un revêtement étale fini S /S tel que les fibres de S → T soient géométriquement connexes, et tel que pour tout t ∈ T (k) et tout objet M(t) ∈ MICSt isotrivial, le point de MDR (S /T , r) au-dessus de t représentant M(t)St est l’image d’un point de torsion du k-groupe MDR (St , 1)r . Démonstration. Plongeons k dans C, et fixons t ∈ T (k). Pour tout objet M(t) ∈ MICSt isotrivial, l’image de π1 (St (C), σ (t)) par représentation de monodromie attachée à M(t) est un sous-groupe fini de GLr (C), bien défini à conjugaison près. D’après un résultat classique de Jordan (cf. e.g. [19] pour un exposé “moderne”), un tel sous-groupe admet un sous-groupe normal abélien d’indice fini borné par une fonction j (r) de r seul. Comme π1 (St (C), σ (t)) est finiment engendré, l’intersection  de ses sous-groupes normaux d’indice ≤ j (r) est un sous-groupe caractéristique d’indice fini. Considérons la suite exacte de groupes fondamentaux {1} → π1 (St (C), σ (t)) → π1 (S(C), σ (t)) → π1 (T (C), t) → {1} qui est d’ailleurs scindée par σ∗ . Le sous-ensemble .σ∗ (π1 (T (C), t)) de π1 (S(C), σ (t)) est en fait un sous-groupe, qui définit un revêtement étale fini SC /SC tel que les fibres de SC → TC soient géométriquement connexes, revêtement qui descend automatiquement de C à k. On a alors pour tout t ∈ T (k) la suite exacte {1} → π1 (St

(C), σ (t )) → π1 (S (C), σ (t )) → π1 (T (C), t ) → {1} et l’intersection de tout sous-groupe fini G de π1 (S(C), σ (t )) avec π1 (St

(C), σ (t )) est abélien. Ceci montre que pour tout objet M(t ) ∈ MICSt isotrivial, le groupe de Galois différentiel de M(t )S

est fini abélien, donc M(t )S

est somme directe d’objets de t t rang 1, nécessairement isotriviaux. D’où le lemme.
10.2. Terminons la preuve de 7.2.2. On peut remplacer S par S comme dans 10.1.1, et M par son image inverse sur S . Cette connexion définit une section τ de

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

87

MDR (S/T , r) → T . D’après l’hypothèse de 7.2.2 et 10.1.1, pour tout point t ∈ S(k), τ (t) est dans l’image de MDR (S /T , 1). Considérons le produit fibré MDR (S /T , 1)×T r ×MDR (S /T ,r) T (construit via τ ). Il existe un morphisme h : T → T étale dominant tel que la seconde projection MDR (S /T , 1)×T r ×MDR (S /T ,r) T admette une section τ . La composée de τ et de la projection sur MDR (S /T , r) n’est autre que τ  h. Quitte à remplacer T par T , MDR (S /T , r) par MDR (S ×T T /T , r) etc. . . , on peut donc supposer que τ se relève en une section τ de MDR (S /T , 1)×T r → T . Il découle du lemme précédent qu’en fait τ (t) ∈ ((MDR (S /T , 1)×T r )tors )t pour tout t ∈ T (k). On en déduit que τ est une section de torsion de MDR (S /T , 1)×T r . Le semisimplifié de MηT est donc une connexion isotriviale. Pour terminer, on remarque que MηT est semi-simple en vertu du lemme suivant : 10.2.1 Lemme. Si M(t) est semi-simple pour un ensemble dense de points t ∈ T , il en est de même pour tout point d’un ouvert dense de T . Démonstration. Il suffit de prouver que si M s’inscrit dans une suite exacte (∗) 0 → M → M → M

→ 0, dont la fibre en un ensemble dense de points est scindée, alors cette suite est scindée sur un ouvert dense de T . L’extension (∗) définit une section λ de 1 (S/T , Hom(M

, M )). Comme S est projective lisse sur T , on sait qu’en remHDR 1 (S/T , Hom(M

, M )) = plaçant T par un ouvert dense si nécessaire, HDR
derechef  ∗ 1

R f∗ S/T (Hom(M , M )) est localement libre de type fini et commute à tout changement de base (cf. [37, 8.0]). De plus, la fibre de λ en tout point t est la classe de l’extension (∗)(t) . L’hypothèse se traduit par le fait que la section λ s’annule sur un sous-ensemble dense de T , donc λ = 0.
10.2.2 Remarques. 1) On pourrait raffiner 7.2.2, en affaiblissant l’hypothèse sur l’ensemble  considéré de points fermés t tels que M(t) soit supposé isotrivial. Par exemple si dim T = 1, dim S = 2, l’argument des exposants de 8.2 vaut dès que  n’est pas image de Q par une fonction algébrique (qu’on peut préciser). L’argument de la section de torsion de 10.2 vaut sous des hypothèses de hauteurs bien étudiées en géométrie diophantienne. La méthode alternative de Hrushovsky le conduit à proposer d’autres types de conditions, dans l’esprit de l’élimination des quantificateurs. 2) Même pour r = 1, cas où l’on dispose de MDR (S/T , r) en inégales caractéristiques (l’extension vectorielle universelle de la composante neutre du schéma de Picard), une approche dans la veine ci-dessus ne fournirait pas une preuve alternative de la conjecture de Grothendieck en rang 1. En effet, il n’est plus vrai, en caractéristique p > 0, que les connexions isotriviales de rang 1 correspondent aux points de torsion de MDR (S/T , 1). 3) Comme nous l’a signalé J.-B. Bost, c’est précisément pour établir un résultat sur les équations différentielles à solutions algébriques dans la veine du lemme 10.1.1

88

Yves André

ci-dessus19 que Jordan a démontré son théorème sur les sous-groupes finis du groupe linéaire [36].

III Connexions d’origine géométrique N. Katz a démontré la conjecture de Grothendieck pour la connexion de Gauss–Manin attachée à un morphisme lisse quelconque f : X → S, ainsi que pour certains de ses facteurs directs découpés par un groupe fini de S-automorphismes de X. Sa méthode repose sur une formule remarquable reliant la p-courbure à l’application de Kodaira– Spencer [38]. Nous allons généraliser ce résultat à tout sous-quotient d’une telle connexion de Gauss–Manin, du moins sous une hypothèse (conjecturalement toujours vérifiée) de connexité des groupes de Galois motiviques. Par les résultats du chapitre I, cela suffira à établir la conjecture de Grothendieck–Katz pour toute connexion d’origine géométrique (sous une hypothèse analogue). Outre la formule de Katz, les ingrédients essentiels de la démonstration sont le théorème de Mazur–Ogus [13], et la théorie des motifs purs, sous la forme inconditionnelle présentée dans [6]. La stratégie est de remplacer, dans l’esprit du chapitre I, les facteurs arbitraires de la connexion de Gauss–Manin attachée à un morphisme projectif lisse, qui ont “peu de structure” en général, par les facteurs de l’algèbre de Lie de Galois différentielle, qui ont une certaine interprétation motivique. Si la démonstration s’avère un peu technique, cela tient en grande partie, comme nous le verrons, à une lacune dans la théorie des motifs : on ne sait pas prouver que les groupes de Galois motiviques sur un corps algébriquement clos de caractéristique nulle sont connexes.

11 Isotrivialité, et horizontalité de la filtration de Hodge (rappels) 11.1 Rappels sur la filtration de Hodge et l’application de Kodaira–Spencer. Soient k un corps, S une k-variété lisse géométriquement connexe, et f : X → S un k-morphisme projectif et lisse. Pour alléger quelques notations, et sans perte de généralité, nous supposerons S affine. Le OS -module localement libre gradué Hf = ⊕q Rq f∗ ∗X/S est muni de sa connexion de Gauss–Manin ∇. La filtration de Hodge de Hf est la filtration décroissante donnée par Fili Hf = Im(Rf∗ ≥i X/S → Hf ). Elle n’est pas horizontale, mais on i a la “transversalité de Griffiths” : ∇Fil Hf ⊂ 1S ⊗Fili−1 Hf . Pour tout i et tout champ de vecteurs D sur S, on a donc une application OS -linéaire Gr i ∇(D) : Gr i Hf → Gr i−1 Hf , dite de Kodaira–Spencer. 19 mais formulé de manière assez imprécise, cf. [32, p. 142]

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

89

Considérons l’hypothèse suivante : ij i+j f ∗ (∗)f la suite spectrale de Hodge–De Rham E1 = R j f∗ ≥i ∗ X/S X/S ⇒ R ij

dégénère en E1 (i.e. ⊕j E1 = Gr i Hf pour tout i ), et les Gr i H sont localement libres sur S. Elle entraîne que i) pour tout sous-module à connexion M de Hf , la filtration de M induite par la filtration de Hodge est horizontale (i.e. les Fili M sont des sous-modules à connexion) si et seulement si pour tout i et tout champ de vecteurs D, les applications de Kodaira– Spencer Gr i ∇(D) sont nulles [38, 1.4.1.9] ; ii) (∗)f m est vérifiée, en notant f m la puissance fibrée m-ième de f au-dessus de S, et l’isomorphisme de Künneth Hf⊗m ∼ = Hf m est compatible aux connexions et filtrations de Hodge (en munissant le membre de gauche de la connexion, resp. filtration, puissance tensorielle). Par ailleurs, la dualité de Poincaré identifie le dual de Hf à Hf comme module à connexion, et cette identification est compatible aux filtrations à un décalage près (par la dimension relative de f ). 11.2 Isotrivialité et filtration de Hodge : le cas de caractéristique nulle. L’hypothèse (∗)f est satisfaite lorsque car k = 0 en vertu de la théorie de Hodge. 11.2.1 Proposition. i) Soit M un sous-module à connexion isotrivial20 de Hf . Alors la filtration de Hodge induite sur M est horizontale. De plus, si k est un sous-corps algébriquement fermé de C, il existe un sous-module à connexion isotrivial N de Hf contenant M et provenant d’une sous-variation de structures de Hodge rationnelles

de q R q f∗an Q. ii) Réciproquement, supposons (avec k ⊂ C) que M soit un sous-module à connexion de structures de Hodge rationnelles de

q ande H provenant d’une sous-variation ∗ q R f∗ Q, et que la filtration Fil M soit horizontale. Alors M est isotrivial. Démonstration. i) Comme la condition de nullité de l’application de Kodaira–Spencer est invariante par revêtement étale fini de S, on peut supposer M trivial. Alors d’après le théorème de la partie fixe [20, 4.1], le plus grand sous-module à connexion trivial de Hf provient d’une sous-variation de structures de Hodge rationnelles de R q f∗an Q. ii) Voir [38, 4.2.1.3]. Le point est que la monodromie en s ∈ S(C) de Ms ∩ H ∗ (Xs (C), R) est compacte par un argument de polarisation, mais aussi contenue dans le groupe discret Aut(Ms ∩ H ∗ (Xs (C), Z)/torsion), donc finie. Dans ii), l’hypothèse de rationalité est essentielle (cf. [4] app.). 11.3 Rappels sur la filtration conjuguée et l’isomorphisme de Cartier. On suppose maintenant car k = p > 0. La filtration conjuguée de Hf est la filtration croissante donnée par Fili Hf = Im(Rf∗ (τ≤i ∗X/S ) → Hf ). C’est la filtration sur l’aboutissement de la suite spectrale conjuguée ij c E2

= R i f∗ H j (∗X/S ) ⇒ Ri+j f∗ ∗X/S .

20 i.e. qui devient constant sur un revêtement étale fini de S

90

Yves André ij

La connexion de Gauss–Manin agit sur cette suite spectrale, et le terme c E2 est de pcourbure nulle. En particulier, les p-courbures ψp (D) envoient Fili Hf dans Fili−1 Hf . On note Gri ψp (D) : Gri Hf → Gri−1 Hf l’application OS -linéaire induite sur les gradués. Soit FX/S : X → X(p) le morphisme de Frobenius relatif. On dispose de l’isomorphisme de Cartier inverse C −1 : iX(p) /S → H i (FX/S∗ ∗X/S ), caractérisé par

sa multiplicativité et la formule locale suivante : si x (p) est la coordonnée locale sur X (p) correspondant à x sur X, C −1 (dx (p) ) = x p−1 dx. Il induit un isomorphisme ji ij C −1 : FS∗ E1 ∼ = c E2 (cf. ([38, 2.3.1.2]). Sous l’hypothèse (∗)f , la suite spectrale conjuguée dégénère en E2 ([38, 2.3.2]), d’où un isomorphisme C −1 : FS∗ Gr i Hf ∼ = Gri Hf . 11.4 (Iso)trivialité et filtration de Hodge : le cas de caractéristique > 0. Rappelons qu’en caractéristique p > 0, l’isotrivialité d’un module à connexion équivaut à sa trivialité, ou encore à la nullité des p-courbures. Comme l’a montré N. Katz, la trivialité d’un sous-module à connexion M de Hf est reliée à l’horizontalité de la filtration de Hodge non de M lui-même, mais d’un certain “tordu par Cartier”. Son résultat technique principal [38, 3] peut se formuler de la manière suivante (cf. aussi [44]). 11.4.1 Théorème. Outre (∗)f , supposons qu’il existe un sous-module à connexion M de Hf tel que pour tout i, C −1 envoie FS∗ Gr i M isomorphiquement sur Gri M. Alors le diagramme FS∗ Gr i M

FS∗ Gr i ∇(D)

/ F ∗ Gr i−1 M S

C −1

 Gri M

C −1

Gri ψp (D)

 / Gri−1 M

est anticommutatif : C −1  FS∗ Gr i ∇(D) = −Gri Rp (D)  C −1 . En particulier, si M est un module à connexion trivial, la filtration de Hodge de M est horizontale. Réciproquement, si la filtration de Hodge de M est horizontale, on voit que les Rp (D) envoient Fili M dans Fili−2 M, mais il n’est pas clair que les Rp (D) s’annulent. Nous verrons en 15.5.2 un résultat partiel dans cette direction. Katz remarque que l’existence de M est vérifiée pour certains facteurs M découpés sur Hf par l’action d’un groupe fini d’automorphismes de f . En général, c’est une hypothèse difficile à satisfaire. Notre fil conducteur sera de remplacer (en caractéristique nulle) Hf par la connexion LG(Hf ) dont les fibres sont les algèbres de Lie des groupes de Galois différentiels de Hf , et de montrer l’existence de M lorsque M est la réduction modulo p d’un quelconque facteur direct de LG(Hf ).

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

91

11.4.2 Remarque (Question de signe). La preuve “cristalline” de 11.4.1 que donne A. Ogus (via le théorème de Mazur) est plus dans le fil de la suite du présent chapitre. Elle donne bien l’anticommutativité, comme énoncé ci-dessus ([44, 2.9]). Chez Katz, on trouve l’affirmation que le diagramme ci-dessus commute au signe (−)i−1 près, au lieu du signe − ([38, 3.4.1.6], [44, rem. 2.12]) ; il semble que l’erreur de [38] soit l’interprétation 3.4.3 du calcul de cocycles (3.4.3.0) : le calcul (3.4.3.0) démontre en fait l’anticommutativité du diagramme ci-dessus.

12 Cycles motivés 12.1 Introduction. Dans tout ce paragraphe, on suppose de nouveau que k est de caractéristique nulle, et même plongeable dans C pour simplifier. Les motifs interviennent dans notre contexte par le biais suivant : il s’avère que l’algèbre de Lie du groupe de Galois différentiel générique (LieGalHf )|k(S) est la réalisation de De Rham d’un motif sur k(S) au sens de [6]. Parmi les diverses définitions des motifs purs dont on dispose et qui mènent à une théorie non conjecturale, celle de [6], très proche de la définition originale de Grothendieck, est la plus algébrique et se prête à la réduction modulo p. Nous commencerons par quelques rappels sur cette théorie. Nous supposons le lecteur familier avec la construction et les propriétés élémentaires de la catégorie monoïdale Q-linéaire des motifs de Grothendieck, définis en termes de correspondances algébriques modulo l’équivalence homologique. L’équivalence homologique des cycles algébriques se définit par la nullité des classes de cohomologie associées dans une cohomogie de Weil classique (Betti, De Rham, étale), et ne dépend pas du choix de cette dernière en vertu des isomorphismes de comparaison. Chaque cohomologie classique définit un foncteur fibre, appelé réalisation, sur cette catégorie des motifs homologiques. Le problème est que faute de savoir démontrer les conjectures standard, on ignore si c’est une catégorie abélienne. Le remède proposé dans [6] consiste à inverser formellement certains morphismes de la catégorie dont les réalisations sont des isomorphismes. Les morphismes de cette catégorie sont appelés correspondances motivées dans loc. cit. 12.2 Cycles et correspondances motivées. On fixe une classe V de k-schémas projectifs lisses, stable par produit, somme disjointe. Soit X ∈ V. On introduit dans loc. cit. le Q-espace gradué A∗mot (X) des cycles motivés sur X (modelés sur V), et pour toute cohomologie classique H , un homomorphisme gradué A∗mot (X) → H 2∗ (X)(∗), dont l’image s’explicite ainsi : c’est l’ensemble des sommes finies de classes de co−1 homologie de la forme pr XY X∗ (α ∩ Lef XY (β)), où Y ∈ V (arbitraire), α et β sont des −1 cycles algébriques sur X × Y , et Lef XY est l’inverse de l’isomorphisme de Lefschetz attaché à des polarisations de X et Y : Lef XY ∈ ⊕r Hom(H 2r (X × Y )(r), H 2(d+dim Y −r) (X × Y )(d + dim Y − r)).

92

Yves André

Étant donné un plongement ι : k → C, les images des cycles motivés dans les différentes cohomologies classiques se correspondent par les isomorphismes de comparaison canoniques compB,DR : H (XC , C) = HB (X) ⊗Q C → HDR (X) ⊗k C , compB,ét.p,¯ι : HB (X) ⊗Q Qp → Hét (Xk¯ , Qp ) =: Hét.p (X) (qui dépend du choix d’un plongement ι¯ : k¯ → C d’une clôture algébrique fixée de k, compatible à ι). Les images des cycles motivés en cohomologie de Betti sont des cycles de Hodge, ¯ les images en cohomologie étale sont invariants par Gal(k/k) (cela découle de la propriété analogue pour les cycles algébriques, cf. loc. cit. 2.5). On définit à partir des cycles motivés la notion de correspondance motivée (exemple : les projecteurs de Künneth), et leur composition. Cela permet de bâtir la catégorie monoïdale Q-linéaire des motifs MV modelés sur V en suivant la procédure usuelle, mais en remplaçant correspondances algébriques par correspondances motivées. On note h(X) le motif attaché à X. 12.2.1 Proposition ([6]). Cette catégorie MV est tannakienne neutre, semi-simple, graduée, polarisée. Toute cohomologie classique (Betti, De Rham, étale) donne lieu à un foncteur fibre (réalisation). Pour tout plongement ι : k ⊂ C, la sous-catégorie tannakienne MV (M) de MV engendrée par un objet M arbitraire est donc équivalente, via la réalisation de Betti, à la catégorie tannakienne des représentations d’un sous-Q-groupe réductif Gmot (M) de GL(HB (MC )), le groupe de Galois motivique de M (relatif à ι). La catégorie des motifs d’Artin est la sous-catégorie abélienne (qui est tannakienne) engendrée par les h(Spec k ), où est une extension finie de k. Tout comme pour la catégorie des motifs homologiques de Grothendieck, les réalisations de MV s’enrichissent naturellement en des foncteurs fibres à valeurs dans des catégories d’espaces vectoriels munis de structures supplémentaires : – la réalisation de Betti s’enrichit en un foncteur à valeurs dans les structures de Hodge rationnelles polarisables, – la réalisation de De Rham s’enrichit en un foncteur à valeurs dans les k-vectoriels filtrés (filtration de Hodge), – la réalisation étale p-adique s’enrichit en un foncteur à valeurs dans les ¯ Qp -vectoriels munis d’une action continue de Gal(k/k). 12.3 Frobenius cristallin. Supposons à présent k de type fini sur Q. Pour X ∈ V fixé, soit o une Z-algèbre lisse intègre (donc de type fini), de corps des fractions k, telle que X provienne d’un o-schéma projectif lisse X (il en existe). Pour tout m ≥ 0, Xm provient alors du o-schéma projectif lisse m-ième puissance fibrée de X sur Spec o. Si M est un motif découpé sur une puissance de X, on dira alors abusivement que M “a bonne réduction” sur o.

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

93

Soit v un idéal maximal de o, de caractéristique résiduelle p = pv . Soient Wv l’anneau de Witt du corps résiduel κv , Kv son corps de fractions, σv son automorphisme de Frobenius, K¯ v une clôture algébrique fixée de Kv . Comme o est lisse sur Z, l’homomorphisme o → κv se relève en un homomorphisme injectif ιv : o → Wv (non unique). La cohomologie cristalline de Hcris (X ⊗ κv , Wv ) est munie d’une action σv -linéaire de Frobenius (Frobenius cristallin). Le Kv -espace qui s’en déduit en inversant p, muni de Frobenius, dépend de v et de X mais est indépendant du choix du modèle X (et de ι) ; on le note (Hcris,v (X), ϕX,v ) (ϕX,v est d’ailleurs bijectif). Pour tout choix de ιv , l’isomorphisme de Berthelot HDR (X) ⊗o Wv → Hcris (X ⊗ κv , Wv ), munit HDR (X) ⊗o Wv d’une action σv -linéaire par transport de structure. On note compDR,cris,ιv : HDR (X) ⊗k Kv ∼ = Hcris,v (X) l’isomorphisme qui s’en déduit, et ϕX,ιv l’endomorphisme σv -linéaire de HDR (X) ⊗k Kv déduit de ϕX,v (il dépend de ιv et de X, mais pas du modèle X). 12.3.1 Proposition ([6, 2.5.2]). Pour tout cycle motivé ξ ∈ Armot (X) et pour tout ιv 2r (X)(r) ⊗ K est invariante sous ϕ comme ci-dessus, l’image de ξ dans HDR k v X,ιv (on rappelle que le “twist” (r) multiplie ϕX,ιv par un facteur p −r ). La difficulté réside dans l’éventualité21 où les variétés auxiliaires Y ∈ V intervenant dans la définition de ξ n’ont pas bonne réduction en o. La preuve de loc. cit. utilise la Kv -algèbre de Fontaine Bcris,Kv (muni de son action de Gal(K¯ v /Kv ), de son Frobenius et de sa filtration canonique), et l’isomorphisme de comparaison compatible à ces structures compét,DR,ι¯v : Hét (Xk¯ , Qp ) ⊗Qp Bcris,Kv → HDR (X) ⊗k Bcris,Kv qui dépend du choix d’un plongement ι¯v : k¯ → K¯ v au-dessus de ιv . Au paragraphe suivant, on utilisera le fait que compét,DR,ι¯v envoie le Qp -espace ¯

(Hét2r (Xk¯ , Qp )(r))Gal(Kv /Kv ) bijectivement sur 2r (X)(r)) ⊗k Kv )ϕX,ιv . (Fil0 (HDR

12.3.2 Corollaire. Sur la sous-catégorie de MV formée des motifs à bonne réduction sur o, on peut définir pour tout idéal maximal v de o un foncteur fibre22 Hcris,v à valeurs dans les Kv -vectoriels munis d’un endomorphisme σv -linéaire, vérifiant Hcris,v (h(X)) = Hcris,v (X) (pour tout X à bonne réduction sur o).
21 pour nos applications, on peut l’éviter car ces variétés auxiliaires interviendront en nombre fini, et on peut toujours rétrécir o en conséquence 22 bien défini à isomorphisme canonique près

94

Yves André

13 Anneaux semi-simples motiviques 13.1 Digression sur les anneaux semi-simples dans une catégorie tannakienne neutre. Soit T une catégorie tannakienne neutre sur un corps K de caractéristique nulle. Soit A un anneau de T . Il est automatiquement artinien. De même, l’anneau ordinaire F (A) est artinien, pour tout foncteur fibre F (à valeurs dans les vectoriels sur une extension de K). Un idéal simple de A est un idéal (bilatère) I qui, en tant qu’anneau de T , n’admet pas d’idéal non nul distinct de I . 13.1.1 Lemme. Les conditions suivantes sont équivalentes : a) A n’a pas d’idéal nilpotent non nul, b) A est produit de ses idéaux simples, c) F (A) est semi-simple pour un (resp. pour tout ) foncteur fibre F . Si ces conditions sont vérifiées, on dit alors que A est semi-simple. Démonstration. Notons a)F et b)F les propriétés analogues à a) et b) respectivement, avec F (A) au lieu de A. Il est bien connu que a)F ⇐⇒ b)F ⇐⇒ F (A) est semi-simple. Cette dernière condition est indépendante de F : deux foncteurs fibres deviennent isomorphes après extension des corps des coefficients, et la semi-simplicité des algèbres est insensible à l’extension des scalaires puisque car K = 0. Fixons donc un F neutralisant T , de sorte que les idéaux de A correspondent aux idéaux de F (A) stables sous le groupe tannakien Aut⊗ F . Il est clair que a) ⇐⇒ a)F , du fait que le radical nilpotent de F (A) est stable sous Aut⊗ F . Par ailleurs, b)F ⇒ b) : en effet Aut⊗ F permute les idéaux simples de A. Donc tout idéal J de F (A) stable sous Aut⊗ F et minimal pour cette propriété est produit d’idéaux simples, et F (A) est produit de ces idéaux J . Enfin, b) ⇒ a) est clair.
On prendra garde toutefois à ce qu’un idéal simple de F (A) n’est pas nécessairement égal à l’image par F d’un idéal simple de A (même si F neutralise T ). 13.1.2 Remarques. 1) Si A est commutative, le groupe tannakien Aut⊗ F agit nécessairement à travers un quotient fini G (un groupe fini étale non nécessairement constant) sur la K-algèbre F (A), qui est un produit fini d’extensions finies de K : pour toute extension K /K, on a G(K ) ⊂ AutK -alg (F (A) ⊗K K ). En particulier, A est un objet semi-simple de T , et cet objet est isomorphe à 1n si Aut⊗ F est connexe (1 désignant l’unité de T ). 2) Sans supposer A commutative, il n’est plus vrai qu’un anneau semi-simple dans T soit un objet semi-simple de T (considérer par exemple le End interne de la représentation standard de dimension 2 du groupe additif Ga ). 3) Un cas particulier intéressant d’anneau commutatif semi-simple se présente lorsque Aut⊗ F est fini : à savoir A = O(π(T )), l’anneau correspondant au groupe fondamental “interne” π(T ) [21]. On a F (A) = O(Aut⊗ F ), sur lequel Aut⊗ F agit par conjugaison (loc. cit.). En particulier, A ∼ = 1n (en tant qu’objet de T ) si et seulement si Aut⊗ F est abélien.

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

95

13.2 Anneaux commutatifs semi-simples dans MV . Soit A un anneau commutatif semi-simple dans la catégorie tannakienne neutre MV . D’après ce qui précède, la réalisation de Betti AB relative à un plongement k → C (resp. de De Rham ADR , resp. étale p-adique Aét.p , resp. cristalline Acris,v ) est une algèbre commutative semi-simple de dimension finie sur Q (resp. k, resp. Qp , resp. Kv ). On choisit o intègre de type fini sur Z, de corps des fractions k, tel que A ait bonne réduction sur o (cf. 12.3). plus précisément, représentons A comme découpé par une correspondance motivée e sur une variété Y ∈ V admettant un modèle projectif lisse Y sur o. Comme on l’a vu, AB est un produit de corps de nombres. On note E le compositum des clôtures galoisiennes dans C des facteurs de AB (relatives à Q). 13.2.1 Proposition. On suppose que k contient E. Alors : 1) ADR = Fil0 ADR . 2) Tout idempotent de la k-algèbre commutative semi-simple ADR est, pour tout ιv k → Kv comme ci-dessus, combinaison linéaire à coefficients 23 dans E d’invariants sous le Frobenius cristallin ϕA,ιv dans ADR ⊗k Kv . 3) Si en outre ADR est scindé, i.e. ADR ∼ = k n , alors le complété Ev de E (pour la place induite par le plongement dans Kv ) scinde la Qp -algèbre (ADR ⊗k Kv )ϕA,ιv , et ϕA,ιv permute les idempotents minimaux de ADR . 4) Si les idempotents minimaux de ADR s’étendent en des endomorphismes de HDR (Y) (ce qui se produit toujours après localisation convenable de o), et si ce dernier est sans torsion, alors dans 3), la permutation induite par ϕA,ιv ne dépend que de v (et de A, mais pas de ιv ). ¯ Démonstration. Prouvons d’abord 2). L’action de Gal(k/k) sur Aét.p respecte la structure de Qp -algèbre, donc se factorise à travers un sous-groupe du groupe symétrique Sn alg (avec n = [AB : Q]). Par Hermite–Minkowski, les homomorphismes π1 (Spec o) → Sn sont en nombre fini. Il est loisible de remplacer k = F rac(o) par l’extension finie correspondant à l’intersection des noyaux de ces homomorphismes. Cela nous ramène ¯ au cas où Gal(k/k) agit trivialement sur chaque Aét.p . En particulier, pour tout plongement ιv : k → Kv comme ci-dessus, et pour tout plongement ι¯v : k¯ → K¯ v au-dessus de ιv , le groupe de Galois local Gal(K¯ v /Kv ) agit trivialement sur Aét.p . Considérons les isomorphismes de comparaison compB,ét.p,¯ι ⊗1

compét,DR,ι¯v

AB ⊗Q Bcris,Kv −−−−−−−→ Aét.p ⊗Qp Bcris,Kv −−−−−−−→ ADR ⊗k Bcris,Kv . Comme ceux-ci respectent la structure d’algèbre, l’inverse du composé envoie tout idempotent e de ADR sur un idempotent de AB ⊗Q Bcris,Kv qui se trouve nécessairement déjà dans AB,ι ⊗Q E (E étant plongé dans Kv via ιv ). On conclut que e est dans le E-sous-espace de ADR ⊗k Bcris,Kv engendrée par l’image de AB . Comme 23 dépendant, a priori, de ι

v

96

Yves André ¯

compB,et.p,¯ι (AB ) ⊂ (Aét.p )Gal(Kv /Kv ) , cette image est contenue dans Fil0 (ADR ) ⊗ Kv et est fixe sous ϕA,ιv . Prouvons 1). On peut supposer, quitte à remplacer k par une extension finie, que ADR ∼ = k [AB :Q] . Les idempotents de ADR engendrent alors ADR sur k, et sont dans Fil0 d’après la première assertion24 . Prouvons 3). Si k scinde ADR , les idempotents de ADR engendrent ADR comme k-vectoriel, donc ADR ⊗Qp Ev comme Ev -vectoriel. Or par le point 2), ces idempotents sont dans (ADR ⊗k Kv )ϕA,ιv ⊗Qp Ev . On a un isomorphisme canonique de Ev -algèbres HomQp (ADR ⊗k Kv )ϕA,ιv ,Ev )

(ADR ⊗k Kv )ϕA,ιv ⊗Qp Ev = Ev et l’assertion en découle.

Prouvons enfin 4). Il suffit de montrer que la bijection Homk-alg (ADR , k) → HomKv -alg (Acris,v , Kv ) induite par compDR,cris,ιv ne dépend que de v, et non de ιv . Pour cela, on remarque que cette dernière se factorise comme suit : Homk (ADR , k) → Homo (eHDR (Y), o) → Homκv (eHDR (Y ⊗ κv ), κv ) → Homκv (eHcris,v (Y ⊗ κv ), κv ) → HomWv (eHcris,v (Y), Wv ) → HomKv (Acris,v , Kv ), et que chacune de ces bijections ne dépend que de v.




Ces résultats sont malheureusement insuffisants pour nos applications. Pour aller plus loin dans l’étude des Frobenius cristallins d’un anneau commutatif semi-simple A dans MV , nous sommes amenés à supposer que A est un motif d’Artin. C’est conjecturalement25 toujours le cas (quitte à agrandir V), puisque le groupe de Galois motivique Gmot (A) est fini. Supposons que ADR soit scindé, d’où un isomorphisme canonique d’algèbres ADR = k Hom(ADR ,k) . Via compB,DR , Hom(AB , E) s’identifie alors à Homk (ADR , k), donc ADR = k Hom(AB ,E) . On indexera en conséquence26 les idempotents minimaux eχ de ADR par χ ∈ Hom(AB , E) : eχ (χ ) = δχ,χ . On a une action canonique à droite de Gal(E/Q) sur ces idempotents, donnée par (eχ )σ := eσ −1 χ . 13.2.2 Proposition. Supposons que A soit un motif d’Artin. Alors ni AB ni E ne dépendent du plongement ι : k → C choisi. Pour tout idéal maximal v de o, notons pv le premier de E correspondant. 24 on peut aussi déduire la seconde assertion de la connexité des groupes de Mumford-Tate 25 cela découlerait tant de la conjecture de Hodge que de la conjecture de Tate 26 la motivation pour cette indexation qui privilégie la réalisation de Betti est l’application ultérieure de

11.2.1 ii), qui fait appel à la rationalité en réalisation de Betti

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

97

Alors pour tout ιv : o → Wv comme ci-dessus, la permutation des eχ induite par le Frobenius cristallin ϕA,ιv est donnée par ϕA,ιv (eχ ) = e(pv ,E/Q)χ , où (pv , E/Q) ∈ Gal(E/Q) est le symbole d’Artin. En particulier, ϕA,ιv respecte les Gal(E/Q)-orbites parmi les eχ . Démonstration. Quitte à remplacer k par une extension finie, on peut supposer que A - et par suite tout objet de MV (A) - est isomorphe à une somme de copies de 1 en tant qu’objet de MV , donc que Gmot (A) = {1}. Pour tout autre plongement ι : k → C, les foncteurs fibres HB,ι et HB,ι sur MV (A) sont alors isomorphes (via un unique isomorphisme), donc AB = HB (ι∗ A) ne dépend pas de ι (ni a fortiori E). Sur MV (A), on a par ailleurs une suite d’isomorphismes de foncteurs fibres comp−1 B,ét.p,¯ι

compDR,ét.p,¯ιv

compB,DR

HB ⊗Q Kv −−−−−−−→ HDR ⊗k Kv −−−−−−−→ Hét,p ⊗Qp Kv −−−−−−−→ HB ⊗Q Kv dont la composée est Gmot (A)(Kv ), donc égale à l’identité. La composée des deux derniers isomorphismes, appliquée à A, identifie (ADR ⊗k Kv )ϕA,ιv à AB ⊗Q Qp . Or par le point 3) de la proposition précédente, on a un isomorphisme canonique de Ev -algèbres HomQp (ADR ⊗k Kv )ϕA,ιv ,Ev )

(ADR ⊗k Kv )ϕA,ιv ⊗Qp Ev = Ev qu’on peut aussi écrire (via compDR,cris,ιv ) ϕ

ϕ

A,v HomQp (Acris,v ,Ev )

A,v ⊗Qp Ev = Ev Acris,v

,

ϕ

A,v ⊗Qp Ev , l’action de ϕA,v est celle de (pv , E/Q) sur les coefficients (le et sur Acris,v facteur ⊗Ev ). D’où l’assertion.


14 Motifs et algèbre de Lie de Galois différentielle 14.1 Groupe de monodromie et groupe de Galois motivique. On reprend la situation et les notations f : X → S, Hf de 11.1, en supposant k plongeable dans C. Soit s ∈ S(C). On note Gmono (Hf , s) l’adhérence de Zariski de la représentation de monodromie (rationnelle) en s : π1 (S(C), s) → GL(HB (Xs )) = GL(H (Xs , Q)). Sa composante neutre G0mono (Hf , s) est un groupe semi-simple d’après [20, 4.4] (voir aussi [4]). 14.1.1 Théorème ([6, 5.2]). Supposons k = C. Quitte à agrandir la classe V, il existe un système local (s )s∈S(C) de sous-groupes algébriques réductifs de GL(HB (Xs )), tel que

98

Yves André

a) pour tout s, G0mono (Hf , s) est un sous-groupe normal de s , b) pour tout s, s contient Gmot (Xs ), c) pour tout s hors d’une partie maigre de S(C), s = Gmot (Xs ). C’est l’élaboration du fait que si un cycle motivé est invariant par monodromie, son transport parallèle en tout autre point est encore un cycle motivé (loc. cit.). (Le point c) ne sera utilisé qu’en 14.4.2 3)). 14.1.2 Corollaire. Pour tout s ∈ S(C), LieGmono (Hf , s) est normalisée par Gmot (Xs ), donc est la réalisation de Betti d’un facteur direct LGs du motif Endh(Xs ). Ce motif LGs est une algèbre de Lie dans la catégorie tannakienne MV . L’existence de ce motif LGs jouera un rôle fondamental dans la suite. Il provient en fait d’un motif défini sur une extension finie du corps de rationalité de s (cf. [6, 2.5, scolie]). 14.2 Réalisation de De Rham de LGs . La réalisation de De Rham du motif h(Xs ) n’est autre que la fibre en s de Hf . Du fait de la régularité de la connexion de Gauss– Manin, l’isomorphisme de comparaison compB,DR induit un isomorphisme Gmono (Hf , s) ⊗Q C ∼ = Gal(Hf , s) entre le groupe de monodromie complexe en s et le groupe de Galois différentiel pointé en s de Hf (c’est un avatar de l’équivalence de Riemann–Hilbert de Deligne, cf. [39]). Comme Gmono (Hf , s) est réductif, il suit que Hf est un module à connexion (intégrable) semi-simple. De même, compB,DR induit un isomorphisme entre LieGmono (Hf , s)C et la fibre en s de l’algèbre de Lie de Galois différentielle LG(Hf ) bâtie sur Hf . Avec la notation du théorème précédent, on a en fait LG(Hf )s = HDR (LGs ). (Ceci vaut encore sur le sous-corps de C sur lequel le motif LGs est défini). 14.3 Digression : algèbres de Lie semi-simples dans une catégorie tannakienne neutre. Soit T une catégorie tannakienne neutre sur un corps K de caractéristique nulle. On note 1 l’objet unité. Soit L une algèbre de Lie dans T . Le crochet de Lie L ⊗ L → L correspond à un morphisme L → L∨ ⊗L = End(L) (représentation adjointe).  La composition des endomorphismes “internes” End(L) ⊗ End(L) → End(L) induit alors un morphisme L ⊗ L → End(L) ∼ = L ⊗ L∨ , d’où finalement, en composant avec ∨ l’évaluation L ⊗ L → 1, une forme bilinéaire β : L ⊗ L → 1. Pour tout foncteur fibre F (à valeurs dans les vectoriels sur une extension de K), F (β) n’est autre que la forme de Killing de F (L). Un idéal simple de L est un idéal (de Lie) I qui, en tant qu’algèbre de Lie de T , est non-commutative et n’admet pas de d’idéal non nul distinct de I . L’orthogonal de I eu égard à β est alors un idéal de L dont l’intersection avec I est réduite à 0 (cela se vérifie au moyen d’une foncteur fibre F ).

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

99

14.3.1 Lemme. Les conditions suivantes sont équivalentes : a) β est non-dégénérée (i.e. identifie L à L∨ ), b) L est le produit de ses idéaux simples, c) F (L) est une algèbre de Lie semi-simple pour un (resp. pour tout) foncteur fibre F . Si ces conditions sont vérifiées, on dit alors que L est semi-simple. Démonstration. Même principe de démonstration que pour le lemme 13.1.1. On vient de voir que b) ⇒ a). La condition c) est indépendante de F , ce qui permet de choisir F neutralisant T . On a a) ⇐⇒ a)F ⇐⇒ b)F ⇐⇒ F (L) est semi-simple. Enfin
b)f ⇒ b) se voit comme en 13.1.1. On prendra garde toutefois à ce qu’un idéal simple de F (L) n’est pas nécessairement l’image par F d’un idéal simple de L (même si F neutralise T ). 14.3.2 Remarques. 1) On pourrait aussi définir la notion de radical d’une algèbre de Lie L de T (plus grand idéal R tel que le dérivé n-ième D n R = 0 pour n >> 0) et montrer que L est semi-simple si et seulement si R = 0. 2) En général, une algèbre de Lie semi-simple L n’est pas un objet semi-simple de T . Par exemple, l’algèbre de Lie semi-simple sl2 , vue comme représentation de Ga ⊂ SL2 par l’action adjointe, est un objet indécomposable, mais non semi-simple, dans la catégorie des représentations de Ga . 3) Voici un cas particulier intéressant : si T est algébrique, on peut considérer l’algèbre de Lie du groupe fondamental “interne” π(T ) (cf. [21]). Si c’est une algèbre de Lie semi-simple de T , T est semi-simple, et tout idéal simple de Lieπ(T ) est un objet irréductible de T . Réciproquement, si T est semi-simple, alors Lieπ(T ) est une algèbre de Lie semisimple si et seulement si pour un (ou pour tout) foncteur fibre F , F (Lieπ(T )) est une algèbre de Lie semi-simple au sens usuel. En revanche, si Lieπ(T ) est simple, F (Lieπ(T )) n’est pas nécessairement une algèbre de Lie simple au sens usuel. Soit L une algèbre de Lie semi-simple. On note EndLie L le sous-anneau de L∨ ⊗ L = End(L) respectant le crochet de Lie. L opère sur EndLie L (action adjointe). 14.3.3 Définition. On définit l’anneau A(L) de T comme le commutant de L dans EndLie (L). 14.3.4 Lemme. Soit L une algèbre de Lie semi-simple (resp. simple) dans T . Alors A(L) est un anneau commutatif semi-simple (resp. simple) dans T . Pour tout foncteur fibre F , F (A(L)) = A(F (L)). Démonstration. La seconde assertion est immédiate. Via 13.1.1, elle ramène la première assertion (cas semi-simple) à l’assertion correspondante pour l’algèbre de Lie ordinaire F (L). Quitte à étendre le corps K des coefficients de F , on peut supposer que dans la

100

Yves André j =n

décomposition en idéaux simples de la K -algèbre de Lie ordinaire F (L) = ⊕j =1 L(j ) , n les L(j ) sont absolument simples.Alors F (A(L)) = A(F (L)) = ⊕A(F (L(j ) )) ∼ = K . Enfin, comme L est produit de ses idéaux de Lie simples et A est produit de ses idéaux simples, il est clair que “L simple” équivaut à “A simple”.
14.4 L’anneau semi-simple motivique A(LGs ). Appliquons ce qui précède à la catégorie tannakienne MV sur Q. Il suit de 14.2 et du lemme précédent que LGs est une algèbre de Lie semi-simple de MV , et que A(LGs ) est un anneau commutatif semi-simple de MV . 14.4.1 Remarque. Il n’est pas difficile de montrer, à l’aide du théorème 14.1.1, que l’anneau motivique A(LGs ) est indépendant de s à isomorphisme près. Nous n’en aurons pas besoin. Avant d’appliquer les résultats de 13.2, il nous faut redescendre du corps de base C à un sous-corps k de type fini sur Q. Quitte à passer à une extension finie d’un corps de définition pour f et à remplacer S par un revêtement étale fini, on peut supposer, et nous supposerons que (∗)f,1 il existe un point s ∈ S(k) tel que Gal(Hf , s) soit connexe (ou, ce qui revient au même, que Gmono (Hf , s) soit connexe, condition qui ne dépend pas de s). On fixe alors s ∈ S(k). Quitte à remplacer derechef k par une extension finie, et à agrandir V, on peut supposer, et nous supposerons que les conditions suivantes sont satisfaites : (∗)f,2 LGs est un motif défini sur k, i.e. un objet de MV ) ; donc A(LGs ) aussi, (∗)f,3 les idéaux simples de la k-algèbre de Lie LG(Hf )s = HDR (LGs ) sont absolument simples (ou, ce qui revient au même, HDR (A(LGs )) ∼ = k n ), (∗)f,4 k contient le compositum E des clôtures galoisiennes27 des facteurs de la Q-algèbre semi-simple A(LGs )B := HB (A(LGs )). (∗)f,5 le groupe de Galois motivique Gmot (Xs,C ) est connexe.
 Sous ces hypothèses, A(LGs )DR = End∇ LG(Hf ) s ∼ = k n , et s’identifie canoniquement à End∇ LG(Hf ). Les idempotents minimaux eχ de A(LGs )DR découpent les idéaux de Lie simples de LG(Hf ) (en tant qu’algèbre de Lie semi-simple dans la catégorie tannakienne - neutralisée par le foncteur fibre en s - des modules à connexion intégrables sur S). En outre, comme Gmot (Xs,C ) agit sur A(LGs,C ) à travers un quotient fini, il agit trivialement, i.e. Gmot (A(LGs,C )) = {1} ; il en est alors de même en remplaçant C par une extension finie de k convenable, et il s’ensuit que A(LGs ) est un motif d’Artin. A fortiori, lorsque  parcourt les Gal(E/Q)-orbites d’idempotents eχ , les idempotents 27 dans C, relativement à Q

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

101

 e := σ ∈ eσ χ sont des correspondances motivées, qui découpent les idéaux simples de LGs (en tant qu’algèbre de Lie semi-simple dans la catégorie tannakienne neutre de motifs MV ). La proposition suivante résume les propriétés qui nous servirons par la suite. 14.4.2 Proposition. 1) Les endomorphismes horizontaux de LG(Hf ) préservent la filtration de Hodge. 2) Soit o une Z-algèbre lisse intègre de corps de fractions k telle que Xs provienne d’un o-schéma projectif lisse Xs . Alors pour tout idéal maximal v de o et tout , tout plongement ιv : o → Wv = W (κv ) relevant l’application canonique o → κv , et tout χ ∈ , le Frobenius cristallin de e (LGs ) vérifie la formule e(pv ,E/Q)χ  ϕe (LGs ),ιv = ϕe (LGs ),ιv  eχ , où pv est le premier de E induit par v. 3) Le compositum E est un corps de nombres totalement réel. Démonstration. 1) découle de 13.2.1.1), ou plus simplement, du fait que A(LGs )DR = Fil0 A(LGs )DR puisque A(LGs ) est un motif d’Artin.
 2) La formule s’écrit aussi e(pv ,E/Q)χ = ϕe A(LGs ),ιv eχ , ce qui résulte de 13.2.2.3) (appliqué à l’anneau simple A = e A(LGs ), et Y = Xs ×k Xs ). 3) On a HB (LGs ) = LieGmono (Hf , s), et A(LGs )B = EndGmono (Hf ,s) HB (LGs ). Lorsque s varie, il forment des systèmes locaux de Q-algèbres de Lie et de Q-algèbres respectivement. Ils ne changent donc pas, à isomorphisme près, lorsqu’on change s. Quitte à changer s, on peut alors supposer par le point c) de 14.1.1 que LieGmono (Hf , s) est un idéal de l’algèbre de Lie dérivée (semi-simple) LieGmot (Xs,C )der . Comme MV (Xs ) est polarisée, tout idéal de Lie simple de LieGmot (Xs,C )der C est défini sur 28 R . Il en est donc de même des idéaux simples de HB (LGs )C , ce qui entraîne que tout homomorphisme (d’anneaux unitaires) A(LGs )B → C se factorise à travers R, donc que E est totalement réel.
14.4.3 Remarque. Il n’est pas difficile de montrer, à l’aide du théorème 14.1.1, que si la condition (∗)f,5 est satisfaite pour s, elle l’est aussi pour tout point de S(C) hors d’une partie maigre. Nous n’en aurons pas besoin. Le point est le suivant : soit π0 (s ) le schéma en groupe fini des composantes connexes de s , pour s ∈ S(C). Alors l’action de Gmot (Xs ) ⊂ s sur O(π0 (s )) est l’action par conjugaison, qui se factorise par π0 Gmot (Xs ). Sous (∗)f,1 (qu’il est loisible de supposer), il découle de 14.1.1 que l’anneau semi-simple de MV dont O(π0 (s )) est la réalisation de Betti est indépendant de s ∈ S(C), à isomorphisme près. 28 cela découle par exemple du fait que la conjugaison complexe agit sur le graphe de Dynkin de LieGmot (Xs,C )der par l’involution d’opposition, cf. [47, 8.5]

102

Yves André

15 Une application du théorème de Mazur–Ogus L’objectif de ce paragraphe est d’appliquer le théorème de Katz 11.4.1 aux réductions modulo p des connexions eχ (LG). Le théorème de Mazur–Ogus nous permettra de comprendre la relation entre eχ et l’isomorphisme de Cartier inverse C −1 . 15.1 Préparation. Plaçons-nous dans la situation 11.1, avec k de type fini sur Q. On peut supposer f à fibres connexes de dimension d. Sans perte de généralité, on suppose les conditions (∗)f,1 à (∗)f,4 satisfaites. Par la dualité de Poincaré et Künneth, on identifie EndHf à Hf 2 (d) de manière compatible aux connexion de Gauss–Manin et aux filtrations de Hodge29 , ce qui permet de voir LG comme un sous-objet de Hf 2 (d). On identifie de même le motif LGs à un sous-motif de h(Xs2 )(d). Les sorites usuels sur les limites d’objets de présentation finie permettent de construire une Z-algèbre lisse intègre o de corps de fractions k, avec les propriétés suivantes : (∗∗)f,1 le morphisme f : X → S provient d’un morphisme projectif lisse f : X → S de o-schémas lisses, avec S affine, et s s’étend en un o-point s de S. (∗∗)f,2 Les OS -modules H j (X, iX/S ) sont localement libres. Cette condition entraîne que la suite spectrale de Hodge–De Rham dégénère en E1 , et que Hf := ⊕q Rq f∗ ∗X/S est localement libre ; il est muni de sa connexion de Gauss–Manin, et de sa filtration de Hodge (dont les gradués sont localement libres par hypothèse). Il en découle aussi que l’isomorphisme choisi EndHf ∼ = Hf 2 (d) s’étend en End Hf ∼ = Hf2 (d). (∗∗)f,3 Tout eχ s’étend en un endomorphisme de EndHf , encore noté eχ . Il est nécessairement idempotent, horizontal, et préservant la filtration de Hodge puisque c’est vrai sur k (cf. théorème
14.4.2, point 2). Il découle de (∗∗)f,3 que LG  provient du facteur direct LG := ⊕eχ EndHf de EndHf . 15.2 Frobenius relatif. Fixons un plongement ιv : o → Wv = W (κv ). Notons pv la ˆ le complété pv -adique de S⊗o Wv . On a un isomorphisme caractéristique résiduelle, S ˆ [13]. canonique entre Hf ⊗OS OSˆ et la cohomologie cristalline de X ⊗ κv relative à S ˆ ˆ Pour tout relèvement σv -linéaire ϕ : S → S du Frobenius absolu FS⊗κv , l’action du Frobenius relatif cristallin induit alors un endomorphisme horizontal injectif  : Hf ⊗OS ϕ ∗ OSˆ → Hf ⊗OS OSˆ . Par ailleurs, quitte à rétrécir le schéma affine S autour de s, on sait qu’il existe ϕ tel que s soit un point de Teichmüller pour ϕ : σv  s∗ = s∗  ϕ ∗ [44, 3.11]. C’est celui qu’on choisira. La fibre en s de  coïncide alors, après inversion de pv , avec le Frobenius cristallin sur HDR (Xs ) ⊗k Kv . 29 le twist est là pour éviter le décalage de la filtration de Hodge

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

103

Appliquons  au End interne de Hf (en notant que eχ agit sur Hf ⊗OS ϕ ∗ OSˆ par transport de structure). Avec les notations de 14.4.2 3) : 15.2.1 Proposition.  induit des endomorphismes horizontaux (LG) ⊗OS ϕ ∗ OSˆ → (LG) ⊗OS OSˆ ,

e (LG) ⊗OS ϕ ∗ OSˆ → e (LG) ⊗OS OSˆ ,

eχ (LG) ⊗OS ϕ ∗ OSˆ → e(pv ,E/Q)χ (LG) ⊗OS OSˆ .

Démonstration. Comme  est horizontal, il suffit de vérifier ces assertions au point fixe s de ϕ, et après inversion de pv . Ces deux premières assertions sont alors conséquence, par 12.3.1, de ce que LGs et e (LGs ) sont des motifs. La dernière résulte du point 2)
de 14.4.2 (qui s’applique grâce à l’hypothèse (∗∗)f,3 ). 15.3 Le théorème de Mazur–Ogus. Il dit ceci (avec les notations de 11.3) : 15.3.1 Théorème ([13, 8.28.3]). Pour tout i, le diagramme
 i−1 H / F∗ −1 pvi Hf ⊗OS OSˆ f⊗κv S⊗κv Gr pv−i 



C −1


 pv−i Im  ∩ pi Hf ⊗OS OSˆ

 / Gri−1 Hf⊗κv

est commutatif (les applications horizontales étant induites par la réduction modulo pv et le passage aux gradués). 15.4 Application. On l’applique en substituant f 2 à f (ou, si l’on préfère, en prenant le End interne). Par la proposition précédente, on en déduit que l’isomorphisme de Cartier inverse C −1 envoie le facteur
 ∗ Gr i eχ (LG) ⊗ κv FS⊗κ v sur le facteur


 Gri e(pv ,E/Q)χ (LG) ⊗ κv .

15.4.1 Remarque. Cela donne une nouvelle preuve que pv ne dépend pas de ιv . Par ailleurs, cela est valable pour le schéma affine S d’origine, même si on l’a établi en localisant provisoirement pour disposer de ϕ. On peut alors appliquer le théorème de Katz 11.4.1 (en remplaçant k par κv , f par le carré de f ⊗ κv et en notant que l’hypothèse (∗) modulo v figurant dans 11.4.1 découle de l’hypothèse (∗∗)f,2 ), en prenant M = e(pv ,E/Q)χ (LG) ⊗ κv , M = eχ (LG) ⊗ κv , ce qui donne :

104

Yves André

15.4.2 Théorème. Sous les hypothèses (∗)f,i et (∗∗)f,i , les conditions suivantes sont équivalentes, pour tout idéal maximal v de o : a) les Gri Rpv sur e(pv ,E/Q)χ (LG) ⊗ κv s’annulent, b) les Gr i ∇ sur eχ (LG) ⊗ κv s’annulent,
c) la filtration de Hodge de eχ (LG) ⊗ κv est horizontale. (Rappelons que E ⊂ k est le compositum des clôtures galoisiennes des facteurs de A(LGs )B , et que pv est le premier de E induit par v). 15.5 Quelques sous-ensembles de Gal(E/Q). Il est commode de reformuler ce résultat, dans le cadre global de 15.1, en introduisant les sous-ensembles suivants de Gal(E/Q), indexés par un ensemble V arbitraire non vide d’idéaux maximaux de o : V = {σ | les Gri Rpv s’annulent sur eσ χ (LG) ⊗ κv pour tout v ∈ v} TV = {τ | les Gr i ∇ s’annulent sur eτ χ (LG) ⊗ κv pour tout v ∈ v} Supposons que tous les (pv , E/Q) soient égaux (v ∈ V ), et notons θV cet élément. Alors 15.4.2 se reformule comme une double inclusion : θV−1  V ⊂ TV , θV  TV ⊂ V . Par ailleurs, posons T = {τ | les Gr i ∇ s’annulent sur eτ χ (LG)} = {τ | la filtration de Hodge de eτ χ (LG) est horizontale }, et remarquons que TV = T dès que V est Zariski-dense dans Spec o. On en déduit 15.5.1 Corollaire. θV  T ⊂ V . Si V est Zariski-dense dans Spec o, θV−1  V ⊂ T. (Si k est un corps de nombres, “Zariski-dense” équivaut à “infini”). On peut raffiner la première inclusion en introduisant le sous-ensemble V = {σ | Rpv s’annule sur eσ χ (LG) ⊗ κv pour tout v ∈ V} ⊂ V . 15.5.2 Variante. θV  T ⊂ V . Si V est Zariski-dense dans Spec o, V = V . (Ceci répond partiellement à la question évoquée après 11.4.1). Démonstration. On a vu (14.4) que les idempotents eχ découpent les idéaux de Lie simples de LG(Hf ) (en tant qu’algèbre de Lie semi-simple dans la catégorie tannakienne semi-simple de modules à connexion S engendrée par elle-même). Comme LG(Hf ) n’est autre que l’algèbre de Lie du groupe tannakien, on déduit de la remarque 14.3.2 3) que les eχ LG(Hf ) sont simples en tant que modules à connexion30 . 30 en termes plus concrets, on exprime ici le fait que les idéaux de Lie simples de LieG

des représentations irréductibles de Gmono (Hf , s)

mono (Hf , s) sont

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

105

Il s’ensuit que si τ ∈ T, la filtration de Hodge de eτ χ (LG(Hf )) n’a qu’un cran : Fil0 = eτ χ LG(Hf ), Fil1 = 0. Alors par 15.5.1,il en est de même de filtration conjuguée de eθV τ χ LG(Hf ) ⊗ κv pour tout v ∈ V , et on conclut que θV  τ ∈ V . Si V est Zariski-dense dans Spec o, on a alors V = θV  (θV−1  V ) ⊂ V , d’où finalement V = V .


16 Conjecture de Grothendieck–Katz et problème de Dwork pour les connexions d’origine géométrique 16.1 Connexions d’origine géométrique. Rappelons la définition que nous adoptons. Soit S une variété algébrique lisse géométriquement connexe sur un corps k de caractéristique nulle. Soit M un module à connexion (intégrable) sur S. 16.1.1 Définition. M est dit d’origine géométrique s’il existe S → S étale dominant tel que MS soit extension successive de sous-quotients de connexions de Gauss– Manin attachées à des morphismes lisses f de but S . On note MCgéom (S) la catégorie k-linéaire formée de ces connexions. 16.1.2 Remarques. 1) La définition ne dépend que de la fibre générique géométrique de M. 2) Dans cette définition, on peut supposer que les morphismes f sont projectifs. Commençons par nous ramener au cas propre. Quitte à modifier S , on sait d’après Hironaka, que f : X → S s’étend en un morphisme propre lisse f¯ : X¯ → S et ∂X = X¯ \ X est un diviseur à croisements normaux stricts relativement à S . On raisonne alors par récurrence sur la dimension relative de f et sur le nombre de composantes irréductibles de ∂X , en utilisant la suite exacte du résidu (Gysin), cf. [3, ch. 2]31 . Pour passer du cas propre au cas projectif, on peut, quitte à remplacer S par un ouvert dense, trouver par le lemme de Chow et la résolution des singularités ε un S-schéma projectif X¯

→ X¯ → S qui domine X¯ , et ε∗ : Hf¯ → Hf¯ est alors injective. 3) Via Künneth et dualité de Poincaré (disponible compte tenu de la remarque précédente), il n’est pas difficile de voir que MCgéom (S) est une sous-catégorie tannakienne de la catégorie tannakienne des connexions intégrables sur S. 4) (non utilisé dans ce texte) Tout morphisme (resp. morphisme lisse) de type fini g : S1 → S2 induit des foncteurs g ∗ : MCgéom (S2 ) → MCgéom (S1 ) (resp. i g : MC RDR ∗ géom (S1 ) → MCgéom (S2 )), loc. cit. 5) Si k est un corps de nombres, toute connexion d’origine géométrique est une G-connexion (cf. [3, IV], [9]). La réciproque est une (variante d’une) conjecture de Bombieri–Dwork. 31 la définition ci-dessus est légèrement plus générale que celle adoptée dans [3], mais les mêmes arguments

s’appliquent

106

Yves André

16.2 Conclusion. Supposons k de type fini sur Q. Soit f : X → S un morphisme projectif lisse tel que (•) pour une fibre complexe Xs au moins, le groupe de Galois motivique Gmot (Xs ) soit connexe. Rappelons qu’on a attaché à f (après une éventuelle extension finie de k) le corps de nombres totalement réel E ⊂ k, galoisien sur Q. Pour tout point fermé v ∈ Spec o, on continue à noter pv (resp. pv ) le premier de E induit par v (resp. la caractéristique résiduelle). 16.2.1 Théorème. Soit M un objet dans la catégorie tannakienne des modules à connexion sur S engendrée par Hf . Quitte à localiser o, on peut supposer que M se prolonge en un module localement libre à connexion M sur un modèle de S sur o. 1) Soit θ un élément Gal(E/Q). Supposons qu’il existe un ensemble Zariski-dense de points fermés v ∈ Spec o tels que (pv , E/Q) = θ et Rpv (M ⊗ κv ) = 0. Alors (quitte à localiser davantage o) Rpv (M⊗κv ) = 0 pour tout v tel que (pv , E/Q) = θ . 2) Supposons que pour tout θ ∈ Gal(E/Q), il existe un ensemble Zariski-dense de points fermés v ∈ Spec o tels que (pv , E/Q) = θ et Rpv (M ⊗ κv ) = 0. Alors M est isotrivial. Démonstration. Comme la catégorie tannakienne des modules à connexion sur S engendrée par Hf est semi-simple, par Künneth dualité de Poincaré, au cas d’un facteur direct M = eHf de Hf . Soit P E,θ l’ensemble des premiers p non ramifiés de E tels que (p, E/Q) = θ et considérons l’algèbre de Lie P E,θ -C(M) dans MICS , définie comme dans la remarque 3.2.5. Comme LieGal(M, η) est semi-simple, on a P E,θ C(M) ∼ = P E,θ -C(LG(M)) en vertu de la proposition 3.2.4 et de 3.2.5. Il revient donc au même de démontrer 1) (resp. 2)) ou de démontrer l’assertion correspondante pour chaque facteur simple de LG(M) (ces derniers sont des facteurs simples eχ LG(Hf ) de LG(Hf )). On peut se placer dans la situation où toutes les hypothèses (∗)f,i et (∗∗)f,i sont satisfaites. Fixons donc χ, dont on note  l’orbite sous Gal(E/Q), et examinons le facteur simple eχ LG(Hf ). Preuve de 1). Soit V θ l’ensemble des points fermés v ∈ Spec o tels que (pv , E/Q) = θ et 1 ∈ V θ . Par hypothèse, V θ est Zariski-dense. Par 15.5.1, on a donc θ −1 ∈ T, et par 15.5.2, 1 ∈ V θ . Ceci établit l’assertion. Preuve de 2). D’après ce qu’on vient de voir, l’hypothèse de 2) entraîne que T = Gal(E/Q). On conclut en appliquant 11.2.1 que eχ LG(Hf ), donc M, est isotrivial.
16.2.2 Corollaire. Il existe un sous-ensemble  ⊂ Gal(E/Q) ayant les propriétés suivantes : i) Soit P l’ensemble des nombres premiers non ramifiés dans E tels que (p, E/Q) soit contenu dans la classe de conjugaison de . Alors P -C(M) = 0, et tout ensemble P de nombres premiers tel que P -C(M) = 0 est contenu, à un ensemble fini près, dans P . En particulier, P a une densité qui est un nombre rationnel.

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

107

ii) Si k est un corps de nombres, l’ensemble des places finies v telles que les vcourbures de M s’annulent est, à un ensemble fini près, l’ensemble des places finies telles que (pv , E/Q) ∈  . En particulier, l’ensemble de ces places a une “densité” au sens de 5.1 qui est un nombre rationnel (5.1.4). Cela répond positivement à la question de Dwork, dans le cas des connexions semi-simples d’origine géométrique (du moins sous l’hypothèse technique (•)).

(S) la plus petite sous-catégorie tannakienne de 16.2.3 Notation. Notons MCgéom MCgéom (S) stable par extension et contenant les Hf avec f vérifiant (•).

16.2.4 Corollaire. Tout objet de MCgéom (S) a la propriété de Grothendieck–Katz.

Démonstration. Par la remarque 16.1.2.2 et le corollaire 4.3.3, la conjecture de Gro (S) se ramène à prouver la propriété de Grothendieck– thendieck–Katz pour MCgéom Katz pour les Hf avec f : X → S projectif lisse vérifiant (•), et ensuite (par le théorème 4.3.1) à prouver que les facteurs simples eχ (LG(Hf )) de LG(Hf ) ont une infinité de p-courbures non nulles. Il suffit donc de montrer que pour tout M comme dans 16.2.1, s’il existe un ensemble P de nombres premiers de densité 1 tel que Rpv (M ⊗ κv ) = 0 pour tout v tel que pv ∈ P , l’hypothèse de 16.2.1 est alors satisfaite ; cela suit immédiatement du théorème de Chebotarev (pour M comme en 1 suffirait. L’exemple suivant montre que cette borne 16.2.1, P de densité > 1 − [E:Q] est optimale).
16.3 Exemple. La condition du point 2) du théorème est optimale. En effet, soit F une extension quadratique totalement imaginaire d’un corps E totalement réel galoisien sur Q (et distinct de Q). D’après G. Shimura [48], il existe des schémas abéliens polarisés f : X → S non isotriviaux de dimension relative [F : Q], de fibre générique absolument simple, tels que EndS X = F , et tels que pour tout automorphisme τ de F distinct de l’identité et de la conjugaison complexe, R 1 f∗an Q ⊗F,τ C soit purement de type de Hodge (1, 0) ou (0, 1). Il n’est pas difficile de voir que Gmono (Hf ), supposé connexe, est alors de la forme ResE/Q SU2 (restriction des scalaires à la Weil), de sorte Gal(E/Q) que E coïncide avec le corps noté E plus haut (et LG(Hf )s ∼ ). = sl2 La condition (•) est réalisée, du moins si V est assez grand, car alors les groupes de Galois motiviques des variétés abéliennes (les fibres complexes de f ) coïncident avec leurs groupes de Mumford-Tate, qui sont connexes, cf. [6]. Comme dans la preuve du théorème, on montre que pour p assez grand, la p1 (X/S) où F agit à travers l’identité est nulle si et courbure du facteur de Hf1 = HDR seulement si pour tout p au-dessus de p, l’automorphisme de Frobenius (p, E/Q) est 1 d’après Chebotarev. distinct de l’identité. La densité de tels p est 1 − [E:Q] Dans le cas où F est un corps cyclotomique, on pourrait du reste construire de telles familles à partir de la jacobienne d’un pinceau de courbes “de type hypergéométrique” convenable sur Q(x) (d’équation affine v N = ua (u − 1)b (u − x)c ), ce qui nous ramènerait à l’exemple 0.3.1) ).

108

Yves André

16.3.1 Remarques. 1) Une variante de la démonstration du point 1) de 16.2.1 donne l’énoncé suivant, pour tout entier naturel n : supposons qu’il existe un ensemble Zariskidense de points fermés v ∈ Spec o tels que (pv , E/Q) = θ et les composés Gri+n Rpv (M ⊗ κv )  · · ·  Gri Rpv (M ⊗ κv ) s’annulent. Alors ces composés s’annulent pour tout v tel que (pv , E/Q) = θ . En particulier, si k est un corps de nombres, l’ensemble de ces places v a une “densité” au sens de 5.1 qui est un nombre rationnel. En revanche, nous ne savons pas ce qu’il en est de la nilpotence d’échelon n des pcourbures elles-mêmes (sans passer aux gradués). Cela irait dans le sens de la conjecture 5.2.4. 2) Un peu d’effectivité : on peut grouper les applications (potentielles) de la conjecture de Grothendieck en deux champs : i) les applications (exceptionnelles) où l’on connaît a priori l’existence d’une base de solutions formelles à coefficients entiers, dont on cherche à montrer l’algébricité (voir l’appendice) ; ii) les applications où une équation différentielle est donnée (dépendant éventuellement de paramètres), dont on cherche à tester l’isotrivialité en calculant un certain nombre de p-courbures (voir [50]). Les variantes de la conjecture de Grothendieck adaptées à ce second champ sont des variantes effectives : il s’agit de déterminer un entier q = q(M) tel que si pour tout premier p < q tel que la réduction de M modulo p soit bien définie, les p-courbures de M s’annulent, alors M est isotrivial. Pour une connexion semi-simple M provenant de la géométrie sur un corps de nombres k, la méthode ci-dessus permet en principe de déterminer un tel entier q. Supposons d’abord M de la forme eχ (LG(Hf )) comme plus haut. Soit q tel que toute place finie de k de caractéristique résiduelle > q soit dans Spec o (les conditions (∗)f,i et (∗∗)f,i étant supposées remplies). Soit d’autre part q

> q un majorant de la hauteur (non logarithmique) de la matrice de l’application de Kodaira–Spencer de EndHf dans des bases convenables. Alors la nullité de cette application de Kodaira–Spencer équivaut à sa nullité modulo une quelconque place v de caractéristique résiduelle > q

. Enfin les formes effectives du théorème de Chebotarev [41] donnent un entier q > q

tel que l’intervalle ]q

, q[ contient au moins un premier dans chaque classe de conjugaison de Gal(E/Q). On peut alors prendre pour q(M) cet entier q. On obtiendra une variante effective de la propriété de Grothendieck–Katz pour tout facteur de Hf , du moins sous l’hypothèse (•), en se ramenant à calculer les p-courbures de M pour tout facteur de EndHf et pour tout p < q(M). 3) Via le corollaire 16.2.4, la conjecture de Bombieri–Dwork et la conjecture ¯ entraînent la conjecture de de connexité des groupes de Galois motiviques (sur k) Grothendieck–Katz pour toute connexion intégrable. En effet, cette dernière se ramène à la conjecture de Grothendieck, dont l’hypothèse implique que la connexion en question est une G-connexion.

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

109

A Conjecture de Grothendieck et théorie des champs conformes A.1. L’objet de ces remarques, qui précisent la conclusion de [1], est de mettre en lumière l’intérêt de la conjecture de Grothendieck dans le problème de classification des théories conformes du champ dites rationnelles. Là se présente la situation rare de systèmes différentiels linéaires dont on sait a priori qu’ils admettent une base de solutions formelles à coefficients entiers, donc sont à p-courbures nulles pour tout p. A.2. Dans le programme de classification des théories conformes en dimension 2, le domaine des théories rationnelles (RCFT) est le plus accessible. Dans ces théories apparaissent des représentations projectives de dimension finie de divers groupes de tresses ou de “mapping class groups”, et l’un des problèmes qui se posent alors est celui de la finitude de l’image de ces représentations. En d’autres termes, il s’agit de déterminer quand les “blocs conformes” de la théorie sont algébriques sur les espaces de modules de courbes de genre g à N points marquées, pour divers (g, N ) ; il resterait ensuite à dresser les “listes de Schwarz” correspondantes. Ce programme a été mené à bien dans le cas c < 1 (c est la charge centrale qui apparaît dans le multiplicateur e2π ic/24 de l’extension centrale du mapping class group, ou de l’algèbre de Virasoro intervenant), du moins pour (g, N ) = (1, 1), et du modèle WZNW pour SU2 . La “classification A.D.E.” donne les valeurs c = 1 − 6/m(m + 1) et les multiplicités de la fonction de partition (voir la discussion de [35]). A.3. Examinons le cas (g, N) = (1, 1). L’espace de Hilbert H des états d’une théorie conforme de charge centrale c est une représentation d’un produit de deux algèbres de Virasoro, de la forme H = ⊕i,j V (hi , c) ⊗ V¯ (h¯ j , c), où V (hi , c) est la représentation irréductible de plus haut poids hi ≥ 0 et de niveau c, et où les barres marquent les conjugués complexes. Les fonctions de corrélation de la théorie, et la fonction de  ¯ partition en particulier Z(t) = T rq L0 −c/24 q¯ L0 −c/24 = i,j Nij χ (hi , c)χ¯ (h¯ j , c) sont des invariants modulaires sur le demi-plan de Poincaré (q = e2πiτ ). Ce sont donc 1+q n 8 des fonctions du paramètre λ = 16q 1/2 ∞ 1 ( 1+q n−1/2 ) de Legendre. Les théories rationnelles sont caractérisées par la finitude du rang de la matrice à coefficients entiers naturels (Nij ). G. Anderson et G. Moore ont montré que c et les plus hauts poids hi de l’algèbre de Virasoro sont alors rationnels [1]. Ils s’appuient sur la remarque que Z(t) s’écrit comme une somme finie fk , où les fk = Nkj χ (hk , c) (resp. les gk ) engendrent des représentations de dimension finie du groupe modulaire ; ici g¯ k =  ¯ h¯ j , c), où les coefficients Dj k satisfont à Nij = k=r χ¯ (h¯ k , c) + Dj k χ( k=1 Dj k Nik pour tout i et tout j > r. Il en découle que, vu comme fonction holomorphe multiforme en la variable λ, le vecteur f (resp. g ) de composantes les fk (resp.gk ) vérifie un système différentiel linéaire sur C \ {0, 1}. A.4. En choisissant r assez grand, on peut supposer que les Dj k sont entiers. Il en découle que les développements de Puiseux en la variable q des fonctions fk et gk sont à coefficients entiers. Comme q 1/2 et λ/16 s’expriment mutuellement comme séries

110

Yves André

à coefficients entiers l’un de l’autre, on en déduit, via la proposition 5.3.3, que les systèmes différentiels satisfaits par f et g sont à p-courbures nulles pour presque tout p. Que les fk et gk soient des fonctions algébriques de la variable λ découlerait donc de la conjecture de Grothendieck. A.5. En fait, cette situation représente un cas très spécial d’application de la conjecture de Grothendieck : c’est le cas limite du critère 5.4.5 dans lequel il existe une uniformisation v-adique simultanée de f, g, et x = λ dans un disque D(0, Rv ), avec Rv = 1 (au lieu de Rv > 1). Notons que ce critère ne s’étend pas sans restriction à ce cas limite, comme le montre l’exemple de la fonction hypergéométrique F ( 21 , 21 , 1; x) (uniformisation par les fonctions thêta, cf. [5]). Il faut donc tenir compte en outre de ce que les monodromies locales sont semi-simples dans la situation (du fait que les p-courbures sont presque toutes nulles). Cela suggère de rechercher des uniformisations de norme > 1 de (produits de) surfaces de Riemann compactes par des (poly)disques unité.

Références [1] Anderson, G., Moore, G., Rationality in conformal field theory. Commun. Math. Phys. 117 (1988), 441–450. [2] André, Y., Quatre descriptions des groupes de Galois différentiels. In Séminaire d’algèbre (M.-P. Malliavin, ed.), Lecture Notes in Math. 1296, Springer-Verlag, Berlin 1986, 28–41. [3] André, Y., G-functions and Geometry. Aspects of Math. E13, Vieweg, Braunschweig 1989. [4] André,Y., Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part. Compositio Math. 82 (1992), 1–24. [5] André, Y., G-fonctions et transcendance. J. Reine Angew. Math. 476 (1996), 95–125. [6] André, Y., Pour une théorie inconditionnelle des motifs. Inst. Hautes Études Sci. Publ. Math. 83 (1996), 5–49. [7] André, Y., Sur la conjecture de Grothendieck-Katz. manuscrit (1997). [8] André, Y., Différentielles non commutatives et théorie de Galois différentielle ou aux différences. Ann. Sci. École Norm. Sup. 34 (2001), 685–739. [9] André,Y., Baldassarri, F., Geometric theory of G-functions. In Proceedings of the Conference in Arithmetic Geometry, Cortona, 17–21 October 1994, Cambridge University Press, Cambridge 1997. [10] André, Y., Baldassarri, F., De Rham cohomology of differential modules on algebraic varieties. Progr. Math. 189, Birkhäuser, Basel 2000. [11] Beauville, A., Monodromie des systèmes différentiels linéaires à pôles simples sur la sphère de Riemann (d’après A. Bolibruch). Exp. 765, Séminaire Bourbaki, Mars 1993. [12] Belyi, G., On Galois extensions of the maximal cyclotomic field. Izv. Akad. Nauk SSSR Ser. Mat. 43 2 (1979) 269–276.

Sur la conjecture des p-courbures de Grothendieck–Katz et un problème de Dwork

111

[13] Berthelot, P., Ogus, A., Notes on crystalline cohomology. Math. Notes 21, Princeton University Press, Princeton, N.J, 1978. [14] Bertrand, D., Groupes algébriques et équations différentielles linéaires. Exp. 750, Sém. Bourbaki (Fév. 1992). [15] Bost, J.-B., Algebraic leaves of algebraic foliations over number fields. Inst. Hautes Études Sci. Publ. Math. 93 (2001), 161–221. [16] Bost, J.-B., Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems. In Geometric Aspects of Dwork Theory (A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz and F. Loeser, eds.), Volume I, Walter de Gruyter, Berlin 2004, 371–418. [17] Chambert-Loir, A., Théorèmes d’algébricité en géométrie diophantienne. Exp. 886, Sém. Bourbaki (2001). [18] Chudnovsky, D., Chudnovsky, G., Applications of Padé approximations to the Grothendieck conjecture on linear differential equations. In Number theory, Lecture Notes in Math. 1135, Springer-Verlag, Berlin 1985, 52–100. [19] Curtis, C., Reiner, I., Representation theory of finite groups and associative algebras. Interscience, 1962. [20] Deligne, P., Théorie de Hodge II. Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57. [21] Deligne, P., Catégories tannakiennes. In The Grothendieck Festschrift, vol. 2, Progr. Math. 87, Birkhäuser, Boston, MA, 1990, 111–198. [22] Demazure, M., Gabriel, P., Groupes algébriques 1. North Holland, Amsterdam 1970. [23] Di Vizio, L., Sur la théorie géométrique des G-fonctions (le théorème de Chudnovsky à plusieurs variables). Math. Ann. 319 (2001), 181–213. [24] Di Vizio, L., On the arithmetic size of linear differential equations. J. Algebra 242 (2001), 31–59. [25] Di Vizio, L., Arithmetic theory of q-difference equations. The q-analogue of Grothendieck-Katz’s conjecture on p-curvatures. Invent. Math. 150 (2002), 517–578. [26] Dwork, B., On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82 (1960), 631–648. [27] Dwork, B., Arithmetic theory of differential equations. In Symposia Mathematica XXIV, Academic Press, London, New York 1981, 225–243. [28] Dwork, B., Differential operators with nilpotent p-curvature. Amer. J. Math. 112 (1990), 749–786. [29] Dwork, B., On the size of differential modules. Duke Math. J. 96 (1999), no.2, 225–239. [30] Dwork, B., Gerotto, G., Sullivan, F., An introduction to G-functions. Ann. of Math. Stud. 133, Princeton, 1994. [31] Elkies, N., The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89 (1987). 561–567. [32] Gray, J., Linear differential equations and group theory from Riemann to Poincaré. Birkhäuser, Boston 1986. [33] Honda, T., Algebraic differential equations. In Symposia Mathematica XXIV, Academic Press, London, New York 1981, 169–204. [34] Hrushovsky, E., Computing the Galois group of a linear differential equation. Manuscrit.

112

Yves André

[35] Itzykson, C., From the harmonic oscillator to the A-D-E classification of conformal models. In Integrable systems in quantum field theory and statistical mechanics, Adv. Stud. Pure Math. 19, Academic Press, London, New York 1989, 287–346. [36] Jordan, C., Mémoire sur les équations différentielles linéaires à intégrale algébrique (1878). Oeuvres II, 13–140. [37] Katz, N., Nilpotent connections and the monodromy theorem. Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232. [38] Katz, N., Algebraic solutions of differential equations (p-curvature and the Hodge filtration). Invent. Math. 18 (1972), 1–118. [39] Katz, N., A conjecture in the arithmetic theory of differential equations. Bull. Soc. Math. France 110 (1982), 203–239 ; Corrig. : Bull. Soc. Math. France 111, 347–348. [40] Katz, N., On the calculation of some differential Galois groups. Invent. Math. 87 (1987), 13–61. [41] Lagarias, J. C., Odlyzko, A., Effective versions of the Chebotarev density theorem. In Algebraic number fields, Proc. Durham Symp. (1977), 409–464. [42] Landau, E., Eine Anwendung des Eisensteinsche Satz auf die Theorie der Gaussche Differentialgleichung. J. Reine Angew. Math. 127 (1904), 92–102. [43] Mathieu, O., Équations de Knizhnik-Zamolodchikov et théorie des représentations. Sém. Bourbaki. Exp. 777, nov. 1993. [44] Ogus, A., F-crystals and Griffiths transversality. In International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya, Tokyo 1978, 15–44. [45] Ogus, A., Hodge cycles and crystalline cohomology. In Hodge cycles, Motives and Shimura varieties (P. Deligne et al, eds.). Lecture Notes in Math. 900, Springer-Verlag, Berlin 1982, 357–414. [46] Serre, J.-P., Groupes algébriques et corps de classes, Hermann, Paris 1959. [47] Serre, J.-P., Propriétés conjecturales des groupes de Galois motiviques et des représentations -adiques. In Motives, Proc. Symp. Pure Math. 55, part 1, Amer. Math. Soc., Providence, RI, 1994, 377–400. [48] Shimura, G., Moduli and fibre systems of abelian varieties. In Algebraic Groups and Discontinuous Subgroups Proc. Symp. Pure Math. 9, Amer. Math. Soc., Providence, RI, 1966, 312–332. [49] Simpson, C., Moduli of representations of the fundamental group of a smooth projective variety I. Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47–129 ; II, Inst. Hautes Études Sci. Publ. Math. 80 (1995), 5–75. [50] Van der Put, M., Reduction modulo p of differential equations. Indag. Math. 7 (3) (1996), 367–387. Yves André, Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France E-mail: [email protected]

Hilbert modular varieties of low dimension Fabrizio Andreatta and Eyal Z. Goren

Abstract. We study in detail properties of Hilbert modular varieties of low dimension in positive characteristic p; in particular, the local and global properties of certain stratifications. To carry out this investigation we develop some new tools in the theory of displays, intersection theory on a singular surface and Hecke correspondences at p. 2000 Mathematics Subject Classification: 11F41, 11G18, 11G35

Contents 1

Introduction

114

2

Definitions and notations

115

3

Stratification of Hilbert modular varieties 116 3.1 p unramified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.2 p maximally ramified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4

Background on the singularities of Hilbert modular varieties 4.1 Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Local models . . . . . . . . . . . . . . . . . . . . . . 4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Singular points . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

121 121 122 124 127

5 The display of an abelian variety with RM 5.1 Recall . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Factorizing according to primes . . . . . . . . . . . . . 5.3 The setting in which the theorems are proved . . . . . . 5.4 Further decomposition of the local model . . . . . . . . 5.5 The display over the special fiber and its trivial extension 5.6 The main results on displays . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

130 132 134 135 135 136 137

6

Some general results concerning strata in the maximally ramified case 139 6.1 Foliations of Newton polygon strata . . . . . . . . . . . . . . . . . . . . . . . 139 6.2 Connectedness of T1 and T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.3 Irreducibility results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

114

Fabrizio Andreatta and Eyal Z. Goren

7

Intersection theory on a singular surface 153 7.1 Definition of the intersection number . . . . . . . . . . . . . . . . . . . . . . . 154 7.2 Pull-back and intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.3 Adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8

Hilbert modular surfaces 158 8.1 The inert case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8.2 The split case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8.3 The ramified case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

9

Hilbert modular threefolds 166 9.1 Points of type (1, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.2 Points of type (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 9.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

1 Introduction This paper studies Hilbert modular varieties of low dimension. Besides the interesting geometric problems it raises, we also feel that such a detailed study is bound to play a valuable role in future applications to number theory. For example, the Hilbert modular varieties of dimension one are the modular curves that have been studied extensively and their geometric properties are intimately connected with the theory of modular forms. We consider here mainly the case of dimension 2 and 3. To carry out this study we had to further develop existing tools and these results are of independent interest. One is intersection theory on a surface with isolated normal singularities, developed in § 7; the other is methods to calculate the universal display of a PEL problem. Regarding the latter, some of the details will appear, under a much more general setting, in a future work [AG4]. Let L be a totally real field of degree g over Q, let OL be its ring of integers, let p be a rational prime and let M be the moduli space parameterizing abelian varieties of dimension g, in characteristic p, endowed with an action of OL . Some further conditions are imposed – see § 2. The properties of M that we study are mostly defined using the Frobenius morphism on various objects that are OL ⊗ Fp -modules. For example, the Hodge bundle E and the cohomology group H 1 (A, OA ) of an abelian variety A. Hence, the analysis is divided according to the prime decomposition of p in OL . In § 3 we recall the stratifications defined in [AG1, GO] and their main properties. In § 4 we discuss the singularities of Hilbert modular varieties. We recall the theory of local models, introduced by Deligne–Pappas [DP], de Jong [deJ] and Rapoport– Zink [RZ], and illustrate the results for the Hilbert and Siegel moduli varieties. The singularities in the Hilbert case are local complete intersections. Given a closed M,x parafactorial. A question of interest point x ∈ Msing we determine when is O

Hilbert modular varieties of low dimension

115

here is when the pair (M, Msing ) is parafactorial. This is motivated by the question of whether certain automorphic line bundles, giving rise to Hilbert modular forms, initially defined on the non-singular locus in M, actually extend to M. We show that (M, Msing ) is not parafactorial in the presence of ramification. See Theorem 4.4.3, and its corollaries, for applications as indicated. Section 5 discusses the display of an abelian variety with real multiplication. After some preparatory work, we provide two main theorems. The first, Theorem 5.6.1, gives the universal display with real multiplication. It uses Theorem 5.6.2 that provides a criterion for a display to be universal. Both theorems can be generalized considerably, i.e., to the setting of PEL problems, (hopefully) even with level involving p. Details will appear in [AG4]. The results are applied in the sequel to study the local properties of the strata. See, for instance, § 8.3.1 and § 9. In § 6 we provide some general results concerning our stratification in the maximally ramified case. This continues our investigation in [AG1]. Some of our results are the following. In § 6.1 we show that each stratum W(j,n) of M is quasi-affine and we describe the foliation structure, as defined by Oort [Oo4], on the Newton polygon stratification of M. In § 6.2 we show that certain of the strata Ta , i.e., where the anumber is greater or equal to a, are connected. In § 6.3 we show (a striking result) that the non-ordinary locus is irreducible for g ≥ 3. Section 7 develops intersection theory on a complete surface with isolated normal singularities, building on [RT1, RT2]. Our approach is very concrete and suitable for the calculations we need to perform. This approach can be developed further [Arc]. One of the applications we give is determining in Theorem 8.1.1, for p inert, which automorphic line bundles (yielding Hilbert modular forms of, usually, non-parallel weight) are ample. Finally, in § 9 we study in some detail Hilbert modular threefolds in the maximally ramified case.

2 Definitions and notations Let L be a totally real field of degree g over Q with ring of integers OL . Let DL be its different ideal and dL its discriminant. Let p be a rational prime and p a prime of OL dividing p. We let Fp denote the residue field OL /p. Let a1 , . . . , ah+ be ideals of L forming a complete set of representatives for the strict class group cl+ (L) of L. By an abelian variety with RM we shall mean a triple (A → S, ι, λ) consisting of an abelian scheme A of relative dimension g over a scheme S; an embedding of rings ι : OL → EndS (A); an isomorphism of OL -modules with a notion of positivity λ : ai → MA := HomOL (A, At )symm , where At is the dual abelian variety (for some, necessarily unique, i). One imposes the condition A ⊗OL ai ∼ = At . By a µN -level structure we mean an embedding of OL -S-group schemes µN ⊗Z OL → A. Let F be the composite of the fields Fp for every p dividing p. The moduli problem

116

Fabrizio Andreatta and Eyal Z. Goren

of abelian varieties with RM over F-schemes and µN -level structure is a rigid moduli problem for N ≥ 4. We let M be the moduli space of abelian varieties with RM defined over F-schemes and level N ≥ 4 (N prime to p); we let N be the moduli space obtained by taking an additional level structure consisting of a connected OL -group scheme of order p. We refer to [AG1, AG2, DP] for details. Note that our N is slightly different from the one appearing in these references (in that we assume the subgroup to be connected). See § 5.1.1 for the definition of a p-divisible group with RM. The following notation is used: Fq denotes a field with q elements; Z, Q, Zp , Qp denote the integers, rationals, p-adic integers and p-adic numbers; W(k) denotes the ring of infinite Witt vectors, with respect to a prime p, over a ring k, and Wt (k) the truncated Witt vectors (a0 , . . . , at−1 ). If C ⊂ k is any subset, we let W(C) (resp. Wt (C)) denote the vectors in W(k) (resp. Wt (k)) all whose coordinates belong to C. We denote by F w, V w the Frobenius and Verschiebung maps on W(k), cf. [Zin, pp. 127-8]. For a Dedekind ring R and a prime ideal p, we let fp = dimFp (R/p). In the case of OL , we also let ep be the absolute ramification index of p and we define gp = ep fp . For a prime p|p of OL , we let OL,p be the localization of OL at the multiplicative L,p be the completion, Lp its field of fractions, and O ur be the set OL \ p, we let O L,p ring of integers of the maximal unramified sub-extension of Lp over Qp . Let k be a perfect field of characteristic p. A p-divisible group over k is called ordinary if all its slopes are zero and one. An abelian variety over k is called ordinary if its p-divisible group A(p) is; it is called supersingular if the slopes of its Newton polygon are all equal 1/2, equivalently, if it is isogenous to a product of supersingular elliptic curves [Oo1, Thm. 4.2]; it is called superspecial if it is isomorphic over k¯ to a product of supersingular elliptic curves, equivalently, if F : H 1 (A, OA ) → H 1 (A, OA ) is zero [Oo2, Thm. 2]. We denote by Ck the category of local artinian kalgebras (R, m) equipped with an identification R/m = k. We denote the closure of a set Z in a topological space by Z c . Let Ag be the Siegel moduli space of principally polarized abelian varieties of dimension g in characteristic p, § 4.2.1 – often with a rigid prime-to-p level n structure that is not explicitly specified; Xuni → Ag (or Xuni → M) will denote the universal object with section e and E = e∗ 1Xuni /A (or E = e∗ 1Xuni /M ) denotes the Hodge g bundle. It is a locally free sheaf of rank g. We let ω = det E.

3 Stratification of Hilbert modular varieties We shall be concerned primarily with the geometry of the moduli space M. The moduli space N will provide us with a ‘Hecke correspondence’ at p that we shall utilize to study certain strata in M. Two particular cases will be considered in detail: when p is unramified and when p is maximally ramified, i.e., decomposes as (p) = pg in OL .

Hilbert modular varieties of low dimension

117

3.1 p unramified ∼ ⊕p|p Fp is a sum of fields. Let A/k be a RM abelian variety In this case OL ⊗ Fp = over a perfect field k ⊇ Fp . It is known that H 1 (A, OA ) is a free OL ⊗Z k module of rank 1. The kernel of Frobenius F : H 1 (A, OA ) → H 1 (A, OA ) is a k-subspace of dimension a = a(A). Let us assume that for every p|p an embedding Fp → k is given, thus a decomposition Fp ⊗ k = ⊕p|p k. The action of Fp on every OL -eigenspace i of H 1 (A, OA ) is either zero or is given by x acting as multiplication by x p for some 1 ≤ i ≤ fp . The structure of the OL ⊗ k-module Ker(F : H 1 (A, OA ) → H 1 (A, OA )) is therefore uniquely determined by a vector (τp )p|p = (τp )p|p (A) of sets, with τp ⊂ {1, . . . , fp }. There is a natural partial order, induced from inclusion of sets in each component, on the set of possible vectors (τp )p|p . Given any vector (τp )p|p , where each τp ⊂ {1, . . . , fp }, we can define a closed subset D(τp )p|p of M by the property that for each geometric point x ∈ D (τp )p|p we have (τp )p|p (Ax ) ≥ (τp )p|p . This is a regular subvariety of codimension p|p |τp |. For further properties see [Go1, GO]. Consider vectors S of the form (τp )p|p with all τp = ∅, except for a single p for which τp is a singleton. For each such S one can define a Hilbert modular form hS whose divisor is DS . Each stratum D(τp )p|p is the transversal intersection of the divisors DS for S as above satisfying S ≤ (τp )p|p . Furthermore, with respect to a suitable cusp, the kernel of the q-expansion map is given by the ideal (hS −1 :S a set as above). See [Go2, Thm. 2]. Example 3.1.1. For g = 1 (so L = Q) the vector (τp )p|p (A) has a single component and there are only two possibilities. Either (τp )p|p (A) = (∅), which corresponds to A being an ordinary elliptic curve, or (τp )p|p (A) = ({1}), which corresponds to A being supersingular. The locus D(∅) is the whole moduli space (of codimension 0), and the locus D({1}) is the supersingular locus (of codimension 1). Example 3.1.2. For g = 2 (L is a real quadratic field) we have two cases: p is inert in L. In this case the possibilities for (τp )p|p (A) are the vectors of sets (∅), ({1}), ({2}), ({1, 2}). The case (∅) corresponds to ordinary abelian surfaces, the cases ({1}), ({2}) to supersingular, but not superspecial abelian surfaces, and the case ({1, 2}) to superspecial abelian surfaces. The variety D(∅) is the whole moduli space, the varieties D1 = D({1}) , D2 = D({2}) are (usually reducible) divisors, and D({1,2}) = D1
D2 is the finite set of superspecial points. We also know that each Di is a disjoint union of non-singular rational curves and that D1 and D2 intersect transversely. See Figure 3.1. See [BG] for details. p is split in L. In this case the possibilities for (τp )p|p (A) are (∅, ∅), (∅, {1}), ({1}, ∅), and ({1}, {1}). The case (∅, ∅) corresponds to ordinary abelian surfaces, the cases (∅, {1}) and ({1}, ∅) to non-ordinary (but not supersingular) abelian surfaces (they are in fact simple abelian surfaces), and the case ({1}, {1}) to superspecial abelian surfaces.

118

Fabrizio Andreatta and Eyal Z. Goren

superspecial

P1 P1

supersingular

D1

D2

Figure 3.1. Hilbert modular surface – inert case.

In this case, the divisors D1 = D(∅,{1}) , D2 = D({1},∅) are also each a disjoint union of non-singular curves but, in contrast with the situation of inert prime, we have no real information on these curves: they are not the reduction of Shimura curves, we do not know their genera. We do know, however, that D1 and D2 intersect transversely and that D1
D2 = D({1},{1}) is precisely the set of superspecial points, and in § 8.2 we provide an argument that suggests that the components of the Di have usually genus 2.

3.2 p maximally ramified Let k ⊇ Fp be a field. In this case OL ⊗ k ∼ = k[T ]/(T g ), where T may be chosen to be an Eisenstein element of the discrete valuation ring OL ⊗ Zp . It is known 1 (A/k) is a free k[T ]/(T g )-module of rank 2 [Rap, Lem. 1.3]. We have a that HdR sequence of k[T ]/(T g ) modules 1 (A/k) −→ H 1 (A, OA ) −→ 0. 0 −→ H 0 (A, 1A/k ) −→ HdR

We let i = i(A), j = j (A) be the elementary divisors of H 0 (A, 1A/k ), normalized 1 (A/k) so that j ≤ i. Note that j = g −i. Thus, there is a k[T ]/(T g )-basis α, β to HdR such that H 0 (A, 1A/k ) ∼ = (T i )α ⊕ (T j )β.

119

Hilbert modular varieties of low dimension

An easy calculation shows that a(A) ≥ 2j and we let n := n(A) = a(A) − j (A). Then j ≤ n ≤ g − j . We let J = {(j, n) ∈ Z2 : 0 ≤ j ≤ n ≤ g − j }. For every (j, n) ∈ J one proves [AG1, §5] that there is a locally closed subvariety W(j,n) of M, whose geometric points parameterize the abelian varieties A with RM such that (j (A), n(A)) = (j, n). We know [AG1, Thm. 10.1] that W(j,n) is a pure dimensional, non-singular variety of dimension g−(j +n), that the Newton polygon is constant on W(j,n) , consisting of two slopes (n/g, (g −n)/g) with equal multiplicities (unless n ≥ g/2 and then the Newton polygon has one slope equal to 1/2), and that the collection {W(j,n) : (j, n) ∈ J } is a stratification of M. The description of the order defined by “being in the closure” is complicated to write, but is easy to describe pictorially. We provide the graphs for g = 1, 2, 3, 4 and 8 in Diagram A. The convention is that if a point a is above a point b in the graph, and a is connected to b by a strictly decreasing path, then the strata corresponding to a is in the closure of the strata corresponding to b. Diagram A: g=1

g=2

(0, 1)

(1, 1)

(0, 0)

(0, 2) QQQ QQQ Q (0, 1)

(0, 0)

g=3 (0, 3) QQQ QQQ Q (1, 1) Q QQQ (0, 2) QQQ

(1, 2)

(0, 1)

(0, 0)

g=4 (2, 2)

3) (0, 4) OXOXOXXXX(1, OOO XXXXXX XXX (1, 2) Q QQQ (0, 3) QQQ (1, 1) Q QQQ (0, 2) QQQ (0, 1)

(0, 0)

g=8 (4, 4)

5) WW (0, 8) W (2, 6) XXX (1, 7) OWOWOWWWW(3, OOO WWWWWWWWWWWWWWWXXXXXXXXX XXX WW WW (3, 4) WW 5) XXX (1, 6) (0, 7) OOOWWWW(2, X X X X W XXXXX OOO WWWWW XXX WW (3, 3) 5) (0, 6) OOO (2, 4) XOOXOXXXX(1, OOO OOO XXXXXX XXX (2, 3) OOO (1, 4) QQQQ (0, 5) OOO QQQ (2, 2) (1, 3) Q OOO QQQ (0, 4) OOO QQQ (1, 2) Q QQQ (0, 3) QQQ (1, 1) Q QQQ (0, 2) QQQ (0, 1)

(0, 0)

120

Fabrizio Andreatta and Eyal Z. Goren

c We know that W(1,1) = ∪(j,n),j ≥1 W(j,n) is the singular locus of M, and, in a sense, j c is a measure for severity of the singularities. More precisely, put Sj := W(j,j ) = ∪(s,t),s≥j W(s,t) , then, by [DP, §4] sing

Sj +1 = Sj

.

(3.1)

We provide a diagram for the case g = 2; See Figure 3.2. The lower part of the diagram

N

P1 M *

(1,1)

(0,1) (0,2)

Figure 3.2. Hilbert modular surface – ramified case.

depicts the modular surface M with a description of the local structure around a point of type (1, 1). The completion of the local ring is a cone, and the supersingular locus, c , has p + 1 branches at such a point. equal to W(0,1) One of the main tools used in [AG1] is the correspondence defined by the moduli c the morphisms πi space N and its two projections π1 , π2 to M. In fact, over W(1,1) 1 are P -bundles. In Figure 3.2 we provide a picture for g = 2; in this case the morphisms π1 , π2 : N → M are blow-ups at the points of type (1, 1) and the p + 1-branches of c get separated; cf. Proposition 8.3.1. We can trace the invariants of the locus W(0,1)

the image π2 π1−1 (x) of a point x of type (j, n) under this correspondence. Again, the formal description is cumbersome and we content ourselves with providing Diagram

121

Hilbert modular varieties of low dimension

B, referring the reader to [AG1] for more details. The convention is that the invariants along π2 π1−1 (x) of a point x of type (j, n) are the pairs (j  , n ), connected to, and in distance one from the pair (j, n) (whether above or below; a loop is considered distance 1). Diagram B: g=1 (0, 1) (0, 0)

g=2 (1, 1)

g=3

MMM (0, 2)

g=4

(1, 2)

MMM (0, 3) (1, 1) MMM (0, 2)

(0, 1)

(2, 2)

(0, 1)

(0, 0)

III (1, 3) MMM (0, 4) (1, 2) MMM (0, 3) (1, 1)

MMM (0, 2) (0, 1)

(0, 0)

(0, 0)

g=8 (4, 4)

III (3, 5) III (2, 6) III (1, 7) MMM (0, 8) (3, 4) III (2, 5) III (1, 6) MMM (0, 7) (3, 3)

III (2, 4) III (1, 5) MMM (0, 6) (2, 3) III (1, 4) MMM (0, 5) (2, 2)

III (1, 3) MMM (0, 4) (1, 2) MMM (0, 3) (1, 1)

MMM (0, 2) (0, 1) (0, 0)

4 Background on the singularities of Hilbert modular varieties 4.1 Cusps Let Xuni → M be the universal abelian scheme with RM and let e : M → Xuni be the identity section. The Hodge bundle E is the locally free sheaf of rank g over M defined by e∗ 1Xuni /M . Let ω = det E; it is an ample invertible sheaf on M. This follows from the ampleness of ω on Ag , cf. [FC, V.2 Thm. 2.3] and from the finiteness of the morphism M → Ag . The Satake compactification MS of M is

122

Fabrizio Andreatta and Eyal Z. Goren

n S defined as Proj(⊕∞ n=0 (M, ω )); it is a projective normal variety and M \ M is a finite set of points, called cusps. Though ω extends to the Satake compactification, we do not know if the Hodge bundle itself extends. The set MR = M \ Msing is the largest open set S over which the Hodge bundle is a locally free OL ⊗ OS -module. One has MR = M if and only if p is unramified [DP, Thm. 2.2]. Let k be a big enough finite field so that OL ⊗ k is a direct sum of local artinian rings with residue field k. Let I OL ⊗ k be an ideal and let I E be the sub-sheaf of E corresponding to I , defined over MR ⊗ k. In general I E does not extend as a locally free sheaf to the cusps. We illustrate the obstruction below for g = 2 and p split.

Example 4.1.1. 1. If p = p1 . . . pg is a product of split primes, then the Hodge bundle is a direct sum E = Ep1 ⊕ · · · ⊕ Epg of line bundles over M. Since we shall refer to that case later, we introduce the simpler notation E = L1 ⊕ · · · ⊕ Lg . Assume, to fix ideas, that g = 2. If L1 , say, extended to the cusp as an invertible p−1 sheaf, then so would L1 ⊗ω(p−1)n for every n. Recall that we have two Hilbert modular form h1 , h2 in this situation (the divisor of h1 being D({1},∅) , of h2 being p−1 D(∅,{1}) ). The Hilbert modular form h1 (h1 h2 )n is a section of L1 ⊗ ω(p−1)n and is not a cusp form. Since the compactification of M is normal and the cusps are of codimension 2, h1 (h1 h2 )n will extend to a section of the extension p−1 of L1 ⊗ ω(p−1)n to the compactification. Usual base-change arguments, p−1 using the vanishing of H 1 (MS , L1 ⊗ ω(p−1)n ) for large enough n, show that the mod p Hilbert modular form h1 (h1 h2 )n will lift to a Hilbert modular form in characteristic 0, which is not a cusp form and has non-parallel weight ((p − 1)(n + 1), (p − 1)n). This is a contradiction, see [Fre, I, Rmk. 4.8]. g

2. If p is an inert prime in OL then OL ⊗ Fpg = ⊕i=1 Fpg , and the Hodge bundle is again a direct sum of line bundles E = L1 ⊕ · · · ⊕ Lg over M. 3. If p = pg is maximally ramified, we get a quotient line bundle L of E defined over MR . We remark that in this case the complement of MR is of codimension 2 in M [DP] and it is not a priori clear whether L can be extended to a line bundle on M. We shall discuss this problem in § 4.4.

4.2 Local models Many of the results we stated above require a detailed understanding of the local (infinitesimal) structure of the moduli space M. Such information may be obtained by the technique of local models. The theory of local models constructs for a moduli space B of abelian varieties another scheme Bloc , typically a flag variety, such that

Hilbert modular varieties of low dimension

123

for every geometric point x ∈ B, there exists a geometric point y ∈ Bloc and an isomorphism of completed local rings Bloc ,y . B,x ∼ O =O We shall use the following notation for Grassmann varieties. Let k be an algebraically closed field, B be a k-algebra, and let a < b be positive integers. Assume that a ring homomorphism B → Mb (k) is given. Assume also that a bilinear alternating pairing  ,  on k b is given. We shall use Grass(a, b) (resp. Grass( , , a, b); resp. Grass(B,  , , a, b)) to denote the Grassmannian of a-dimensional subspaces of k b (resp. isotropic; resp. isotropic and B-invariant). Often implicit in the notation Grass(B,  , , a, b) is a connection between the pairing and the action of B, e.g., the elements of B are self-adjoint with respect to the pairing. 4.2.1 The idea of local models. Let Ag be the moduli space of principally polarized abelian varieties of dimension g in characteristic p. We shall assume that on Ag , or M, there is a given prime-to-p level structure, which we omit from the notation. 1 in a Given a point x ∈ M, or x ∈ Ag , one can trivialize the locally free sheaf HdR Zariski open neighborhood U of x. Then, the locally free, locally direct summand of rank g given by the Hodge bundle E, provides a morphism U → Grass( , , g, 2g) (resp. Grass(OL ,  , , g, 2g)), where the Grassmannian is of isotropic g-dimensional (and OL -invariant) subspaces of a 2g-dimensional space with a perfect alternating pairing. The idea of local models is to show that this is an isomorphism on the level of completed local rings. There is a shortcoming to this result in that the morphism is not canonical and therefore it is not a priori clear how to define the strata coming from the moduli space on the local model (even in an infinitesimal neighborhood of a point). The crystalline theory makes this morphism somewhat more canonical. But, in fact, the proof that this is an isomorphism on the completed local rings often requires an auxiliary scheme and a dimension count. Let f : A → S be an abelian scheme and let D∗ (A) be the associated Grothendieck– Messing crystal, defined on the nilpotent crystalline site of S [Gro, §V.4]. This crystal is defined by D∗ (A) = R 1 fcrys,∗ (OAcrys ). The value of this crystal on S is the de Rham sheaf D∗ (A)S = R 1 f∗ (•A/S ), hence it provides us with a locally free direct summand of rank g, EA ⊂ D∗ (A)S , which is f∗ A/S . The crucial theorem here is due to Grothendieck [Gro, p. 116]. Theorem 4.2.1. Let S → S  be a nilpotent thickening with a divided powers structure. The filtered Dieudonné functor gives an equivalence of categories between 1. the category of abelian schemes over S  , and 2. the category of pairs (A, E), where A is an abelian scheme over S and E ⊂ D∗ (A)S  is a locally free direct summand which lifts EA ⊂ D∗ (A)S .

124

Fabrizio Andreatta and Eyal Z. Goren

Morphisms are homomorphisms f : A1 → A2 such that the induced morphism f ∗ : D∗ (A2 )S  → D∗ (A1 )S  satisfies f ∗ (E2 ) ⊂ E1 . Let S be the spectrum of an algebraically closed field k. Let S ⊂ S  be a PD thickening such that S  is a local artinian k-algebra. Let A → S  be the trivial 1 (A/S) ⊗ O  . Then, given deformation of A over S  for which D∗ (A)S  = HdR k S 1 (A /S  ) ∼   any other deformation A of A to S , the canonical isomorphism HdR = 1 1 HdR (A/k) ⊗k OS  provides us with a submodule EA ⊂ HdR (A) ⊗k OS  lifting 1 (A). Thus we get a morphism from the functor of deformations over the EA ⊂ HdR 1 (A/k)). nilpotent crystalline site of S to the functor Grass(g, HdR M,x and let f : Xuni → T be the universal object. Let T to be the spectrum of O 2g • 1 ∼  Trivialize R f∗ (Xuni /T ) = O M,x with respect to a basis horizontal for the Gauss–

Manin connection. Considering the submodule EXuni /T ⊂ R 1 f∗ (•Xuni /T ), we obtain a morphism T → Grass(OL ⊗k,  , , g, 2g). Similar constructions can be made with endomorphism and polarization structures. Using this map and the crystalline theory, one obtains [DP, Thm. 3.3], [deJ] the following theorem (recall the tacit assumption of rigid level structure):

Theorem 4.2.2. 1. In the Siegel case, there is an isomorphism A ,x ∼ G,y , O =O g where G is the Grassmannian variety Grass( , , g, 2g) that parameterizes g1 (A/k) and y is the point corresponding dimensional isotropic subspaces of HdR 1 (A/k). to the Hodge filtration H 0 (A, 1A/k ) ⊂ HdR 2. In the Hilbert case, there is an isomorphism M,x ∼ G,y , O =O where G is the Grassmannian variety Grass(OL ⊗ k, , , , g, 2g) that param1 (A/k) and y eterizes g-dimensional isotropic OL -invariant subspaces of HdR 1 (A/k). 0 is the point corresponding to the Hodge filtration H (A, A/k ) ⊂ HdR Remark 4.2.3. The theorem holds, for a suitably formulated Grassmannian problem, without the restriction to characteristic p. See [DP, deJ]

4.3 Examples We only consider deformations in characteristic p. 4.3.1 The Siegel case. Let V be a 2g-dimensional vector space, let  ⊂ V be a g-dimensional subspace of V and choose a complementary subspace W ⊂ V such

125

Hilbert modular varieties of low dimension

that V =  ⊕W . Then an affine chart of Grass(g, V ) about  is given by Hom(, W ). Given t ∈ Hom(, W ) we associate to it its graph. Suppose that V has a symplectic pairing and  is isotropic. Choose a basis a1 , . . . , ag to  and complete it to a standard symplectic basis by b1 , . . . , bg . Take W to be the span of b1 , . . . , bg . We may identify t with a g × g matrix (ti,j ) such that aj → aj + i ti,j bi . The graph of t is isotropic if and only if for each j, k we have       ti,j bi ∧ ak + ti,k bi = 0. (4.1) aj + 

i



i

Since (aj + i ti,j bi ) ∧ (ak + i ti,k bi ) = tj,k − tk,j , Equation (4.1) is equivalent to (ti,j ) being a symmetric matrix. This is of course in accord with Ag (with a rigid level structure prime to p) being a non-singular variety of dimension g(g + 1)/2. 4.3.2 The Hilbert case. We again consider two cases. g

• The inert case. In this case we have a decomposition OL ⊗ k = ⊕i=1 k. We denote the projection of OL on the i-th component by σi . One may assume that Frob  σi = σi+1 . We then have g

1 (A/k) = ⊕i=1 D(i), HdR

where each D(i) is a two dimensional k-vector space with a perfect alternating pairing, on which OL acts via σi . There is a compatible decomposition g

H 0 (A, A/k ) = ⊕i=1 H (i), where each H (i) is a one dimensional k-vector space on which OL acts via σi . The Grassmannian is therefore isomorphic to Grass(1, 2)g ∼ = (P1k )g . Note that the completed local ring of every point x on M is isomorphic to the completed power series ring k[[t1 , . . . , tg ]], where ti is canonical up to an element of k[[ti ]]× . • The maximally ramified case. In this case 1 (A/k) ∼ HdR = k[T ]/(T g ) ⊕ k[T ]/(T g ).

The Grassmannian Grass(OL ⊗k,  , , g, 2g) is that of parameterizing isotropic g-dimensional subspaces that are OL -invariant. One can show [DP] that one can replace the k-valued pairing, for which the action of OL is self-adjoint, by a k[T ]/(T g )-valued pairing, which is k[T ]/(T g )-linear. 1 (A/k) such that Given A/k we can find a basis α, β of HdR

H 0 (A, A/k ) = (T i )α ⊕ (T j )β,

α ∧ β = 1,

126

Fabrizio Andreatta and Eyal Z. Goren

where j = j (A), i = g − j , i ≥ j . This determines i, j uniquely. We choose the complementary subspace to be i−1 

T s kα ⊕

s=0

The deformations f of termined as follows. Under f , i

T α → T α+

i−1 

s

as T α+

s=0

j −1 

T s kβ.

s=0

H 0 (A, A/k )

i

j −1 

s

1 (A/k) that are O -linear are dein HdR L

j

j

bs T β, T β → T β+

s=0

i−1 

s

cs T α+

s=0

j −1 

ds T s β.

s=0

We write that in shorthand notation as T i α → T i α + aα + bβ,

T j β → T j β + cα + dβ,

with a=

i−1 

as T s ,

b=

s=0

c=

i−1 

j −1 

bs T s ,

s=0 s

cs T ,

d=

j −1 

s=0

ds T s .

s=0

To have an isotropic subspace we must require (T i α + aα + bβ) ∧ (T j β + cα + dβ) = 0. This is equivalent to ad − bc + aT j + dT i = 0. It then follows that the OL ⊗ k-span of T i α + aα + bβ, T j β + cα + dβ is a g-dimensional isotropic OL -invariant subspace. Example 4.3.1 (j = 0 (non-singular points)). In this case i = g. We get immedig−1 ately b = d = 0 and hence also a = 0. It follows that c = s=0 cs T s is unobstructed and we conclude that the completed local ring is isomorphic to k[[c0 , . . . , cg−1 ]]. Example 4.3.2 (g = 2, i = j = 1). In this case we find the equation a0 d0 − b0 c0 + a0 T + d0 T = 0. We get the relations a0 = −d0 and a0 d0 − b0 c0 = 0. This gives that the completed local ring is isomorphic to k[[a0 , b0 , c0 ]]/(a02 + b0 c0 ).

Hilbert modular varieties of low dimension

127

Example 4.3.3 (g = 3, j = 1, i = 2). We have a = a0 + a1 T ,

b = b0

c = c0 + c1 T ,

d = d0 .

with the equation (a0 d0 − b0 c0 ) + (a0 + a1 d0 − b0 c1 )T + (a1 + d0 )T 2 = 0. This yields d0 = −a1 , a0 = a12 +b0 c1 and that the completed local ring R is isomorphic to k[[a1 , b0 , c0 , c1 ]]/(a13 + a1 b0 c1 + b0 c0 ), which is 3-dimensional with a tangent cone at the origin defined by b0 c0 = 0. The singular locus of Spec(R) is given by b0 = c0 = 0 (which implies a1 = 0) and is hence one dimensional, isomorphic to Spec(k[[c0 ]]).

4.4 Singular points Using the local models one can show [DP, Thm. 2.2] that M is singular if and only if p is ramified in OL and that the singular locus is of codimension 2. However, the singularities are local complete intersections, hence Cohen–Macaulay and so normal, by Serre’s criterion. We remark that, in particular, the completed local rings are domains, i.e., the moduli space is locally (formally) irreducible. In local commutative algebra a property which is subtle and of interest is the property of parafactoriality. The definition is motivated by its relation to factoriality and representability of the local Picard functor of invertible sheaves. For this we refer the interested reader to the references below and to [Lip1]. A noetherian local ring (R, m) is called parafactorial if it is of depth at least 2 and if Pic(R − {m}) = 0. A global definition follows: Definition 4.4.1. Let (X, Z) be a pair consisting of a ringed space X and a closed subset Z. Let U = X \ Z. One says that (X, Z) is parafactorial if, for every open set V of X, the restriction functor M  → M|U ∩V , from the category of invertible OV modules to the category of invertible OU ∩V -modules, is an equivalence of categories. We refer the reader to [EGA IV, §21.13], [SGA 2, Exp. XI] for details. In particular, [EGA IV, §21.13.8] gives the equivalence of the definitions for local rings. Lemma 4.4.2. Let k be a field. Let R be the ring k[[a0 , . . . , ag−2 , b0 , c0 , . . . , cg−2 , d0 , x1 , . . . , xN ]]/ (a0 d0 − b0 c0 , ag−2 + d0 , {ai d0 + ai−1 − b0 ci : 1 ≤ i ≤ g − 2}).

(4.2)

128

Fabrizio Andreatta and Eyal Z. Goren

The closed set Spec(R)sing is defined by the ideal (a0 , b0 , c0 , d0 ) of R. The pair (Spec(R), Spec(R)sing ) is not parafactorial. Proof. First, one proves that R is isomorphic to the ring k[[b0 , c0 , . . . , cg−2 , d0 , x1 , . . . , xN ]]/




g−2

b0 c0 − d0 b0 c1 + d02 b0 c2 − d03 b0 c3 + · · · + (−1)g−2 d0

g b0 cg−2 + (−1)g−2 d0 , (4.3)

cf. proof of Lemma 6.3.4. Then, a direct application of the Jacobi criterion gives that Spec(R)sing is defined by the ideal (a0 , b0 , c0 , d0 ). Let Uab (resp. Ucd ) denote the open set where either a0 or b0 (resp. c0 or d0 ) are not zero. Note that U := Uab ∪ Ucd = Spec(R) − Spec(R)sing . We consider the closed subscheme given on Uab ∪ Ucd by (b0 , d0 ). Note that by Equation (4.2), this closed subscheme is an irreducible reduced Weil divisor D0 on Uab ∪ Ucd , automatically locally principal. Now, consider the closed subscheme D of Spec(R) defined by the same ideal (b0 , d0 ). There is unique extension of D0 as a Weil divisor to Spec(R) which is just D (because Spec(R)sing has codimension 2). If the pair (Spec(R), Spec(R)sing ) is parafactorial then the invertible sheaf OU (D0 ) extends to an invertible sheaf F over Spec(R). By [Har, Prop. II.6.15] F ∼ = O(D  ), where D  is locally principal and,  without loss of generality, D |U = D0 and so D  = D. Thus, it remains to prove that D is not locally principal. We follow the argument of [Har, II 6.5.2]. Assume that D is locally principal. Let mR be the maximal ideal of R. Then D is given by a unique equation in mR /m2R . But mR /m2R is just the k-vector space with basis b0 , c0 , . . . , cg−2 , d0 , x1 , . . . , xN . On the other hand, clearly D is given in mR /m2R by b0 , d0 . Contradiction.  Theorem 4.4.3. Assume that p ramifies in OL . Then the pair (M, Msing ) is not parafactorial. In fact, there is an invertible subsheaf L of the Hodge bundle that does not extend to any open set strictly containing MR = M − Msing . If p is maximally ramified, then L = pg−1 E, and L extends to an invertible sheaf over N. Proof. Assume that (p) = pe11 . . . per r in OL and that e1 > 1. Let k be an algebraically f (p /p) closed field of characteristic p. Write OL ⊗ k = ⊕r=1 ⊕m=1 km [T ]/(T e ), with km = k for all m. Consider a k-rational point x on M with the property that H 0 (Ax , Ax /k ) = f (p /p) r ⊕=1 ⊕m=1 U,m with U,m = km [T ]/(T e ), except for U1,1 , which is taken to be the km [T ]/(T e1 ) module given by (T ) ⊕ (T e1 −1 ). The closure of the collection of such points is a closed subscheme Z of M. Cf. §§ 3.2, 4.3.2. The completed local ring S of x is by the theory of local models isomorphic to r  f (p /p)  ⊗=1 ⊗m=1 R,m , with R,m a power series ring over k, except for R1,1 , which is isomorphic to the ring R in Equation (4.2) with g = e1 and N = 0. That is, the ring S is itself of the form given in Equation (4.2) with Z = Spec(S)sing .

Hilbert modular varieties of low dimension

129

Suppose that (M, Msing ) is parafactorial. Recall that over MR = M − Msing the relative cotangent space of the universal abelian scheme is a locally free  r  f (p /p) ⊕=1 ⊕m=1 km [T ]/(T e ) ⊗k OMR module. Consider the invertible sheaf L f (p /p) defined on MR by the ideal ⊕r=1 ⊕m=1 I,m with I,m equal to 0, except for I1,1 , which is equal to T e1 −1 k1 [T ]/(T e1 ). Since the pair (M, Msing ) is parafactorial, it follows that the invertible sheaf L can be extended from U := Spec(S) − Z to Spec(S). The de Rham sheaf corresponds under the theory of local models to a free OL ⊗k S module with generators α, β, and L is then the submodule generated over U by T e1 −1 a0 α + T e1 −1 b0 β and T e1 −1 c0 α + T e1 −1 d0 β. Let Uab (resp. Ucd ) denote the open set where either a0 or b0 (resp. c0 or d0 ) are not zero. We have a trivialization of L over Uab (T e1 −1 a0 α + T e1 −1 b0 β is a basis) and over Ucd (T e1 −1 c0 α+T e1 −1 d0 β is a basis). Note that on S we have the relation a0 d0 = b0 c0 . The transition function between the trivializations is d0 /b0 = c0 /a0 . Let D be the divisor on Spec(S) defined by the ideal (b0 , d0 ). The divisor D is defined on Uab by b0 and on Ucd by d0 and so has the same transition function as L. Parafactoriality implies that D must be locally principal, cf. the proof of Lemma 4.4.2. But we have shown in that proof that this is not the case. To show that L cannot be extended outside MR we argue as follows: Let K be a closed set that contains every point x as constructed above (where we allow a different choice of , m as long as e > 1). Such points are dense in Msing as follows from [DP, §4]. Therefore, K ⊃ Msing . Hence, if U is an open set strictly containing MR then U contains such a point x. But we have shown that L cannot be extended as an invertible sheaf over the completed local ring of x. Assume now that p is maximally ramified. The first claim was already proven. To prove the second claim, consider the Lie algebra of the subgroup defining the moduli problem N. It provides us with a locally free quotient sheaf H of the Hodge bundle E over N. We claim that when we restrict H to NR = MR then H is isomorphic to L. This follows from the fact that over MR the Hodge bundle E has a canonical filtration E ⊃ pE ⊃ · · · ⊃ pg−1 E ⊃ 0, with successive graded pieces being isomorphic under multiplication by T .  Corollary 4.4.4. Assume that p is maximally ramified in OL . The section of the morphism N → M, A  → (A, T g−1 Ker(FA )), defined on MR , does not extend to any open set strictly containing MR . Proposition 4.4.5. Let x be a (scheme theoretic) point of M of codimension at least 4. Then the local ring of x is parafactorial. If p is unramified in OL , the local ring of x is parafactorial for any x. Proof. Let x be a (scheme theoretic) point of M. By [SGA 2, Exp. XI, Cor. 3.7], M,x is to show that the local ring OM,x is parafactorial it is enough to show that O  parafactorial. If x is of codimension at least 4, the ring OM,x is of dimension ≥ 4

130

Fabrizio Andreatta and Eyal Z. Goren

and is a complete intersection by the theory of local models (see [DP, Prop. 4.4]). It follows from [SGA 2, Exp. XI, Thm. 3.13] that it is parafactorial. It is known that a regular noetherian local ring of dimension at least 2 is parafactorial – a result due to Auslander–Buchsbaum – cf. [SGA 2, Thm. 3.13], [EGA IV, §21.13.9 (ii)].  The parafactoriality of the completed local rings of closed points on M is completely covered by the results above except for the situation g = 3 and (p) = p2 q. In this case, the completed local ring of any non-singular point is of the form k[[t]]. Such a ring is parafactorial [Bou, III, Prop. 1.2]. k[[x, y, z]]/(z2 + xy)⊗

5 The display of an abelian variety with RM We wish to study the local deformation theory of abelian varieties with RM in characteristic p > 0. In this paper we only study equi-characteristic deformations. Our main tools are the theory of local models and the theory of displays, both available in the arithmetic setting as well. One thus hopes that the methods below will extend to the arithmetic setting. Let x ∈ M be a k-valued point, where k is an algebraically closed field of charM,x , and acteristic p. The theory of local models allows us to determine the ring O even the behavior of the strata Sj , but falls short of describing the behavior of the strata W(j,n) . As we shall explain, the local deformation theory factors according to the prime ideals dividing p in OL and that allows us, essentially, to assume that the p-divisible group Ax (p) is either ordinary, or local-local. The first case is studied very effectively using Serre–Tate coordinates but is of no interest to us in this paper. In order to study the second case, we make use of the theory of displays as reformulated and developed by Zink [Zin]. Our main idea, which is similar to [Zin, §2.2], is the following. Suppose, for simplicity, that the abelian variety Ax has a local-local p-divisible group. Then, the M,x ), whose fiber over the display associated to the abelian scheme A → Spec(O closed point is Ax , is universal with respect to the problem of deformations over local artinian k-algebras (R, m) with R/m = k of the polarized OL -display associated to Ax . Indeed, the universality is one of Zink’s main results. We denote it by P uni . On the other hand, the theory of local models provides us with a concrete model R M,x , which is the completion of the local ring of a point on a suitable Grassmann for O variety. We view the universal display P uni as lying over R. We explicitly construct a display P over R that we want to show is universal. By the universal property, P is obtained from P uni by base change coming from a unique map ϕ : R → R. At least over R/m2R , the Hodge filtrations defined by P uni and P produce two maps (that are unique) ψ1 , ψ2 : R → R, coming from the interpretation of R as a completed local

Hilbert modular varieties of low dimension

131

ring of a point on a Grassmannian, and the crystalline nature of displays. One gets a commutative diagram ϕ  ψ1 = ψ2 . We then argue that, in fact, ψ1 and ψ2 are isomorphisms, hence so is ϕ. The universality of P ensues. We next discuss the connection to a well known result that gives the universal display for the Siegel case [Oo3, pp. 412-414],[Zin, Eqn. (86)]. Let (X, λ)/k be a principally polarized abelian variety over an algebraically closed field k of charac
teristic  p > 0 and choose a symplectic basis for the display of X to yield a matrix A B , as explained in [Zin, pp. 128-9]. Let R := k[[t : i, j = 1, . . . , g]]/(t − t ) ij ij ji C D be the completed local ring provided by the theory of local models (cf. § 4.3.1). It is identified, non-canonically, with the completed local ring of the k-rational point of Ag corresponding to (X, λ) (the usual choice of auxiliary rigid level structure prime to p is required for that). Let Tij be the Teichmüller lift of tij and let T be the square g × g matrix (Tij ). The universal display for the universal infinitesimal equi-characteristic deformation of (X, λ) is then given by

A+TC B +TD . (5.1) C D Note, for example, that A+T C is the “universal Hasse–Witt matrix” and thus the nonordinary locus is infinitesimally defined by the equation det(A + T C) = 0 (mod p). This determinant can be interpreted as the Hasse invariant – a Siegel modular form of weight p − 1 that vanishes exactly along the non-ordinary locus. Equation (5.1) is a red herring of a sort. In that expression the Hodge filtration “seems constant”; namely, in the specified basis e1 , . . . , e2g , with respect to which the display is given, the kernel of Frobenius modulo p is the span of eg+1 , . . . , e2g . As such, its behavior is exactly the opposite of the behavior expected from the crystalline theory and the theory of local models. However, consider the automorphism of the module of the display   underlying
I T   A pB 
I T  A+T C pB+T pD = 0I provided by 0 I and write C pD C pD . One checks that with respect to a suitable basis (see below) the Frobenius operator is given by



A AT σ + pB A pB I Tσ . (5.2) = C CT σ + pD 0 I C pD The kernel
−T of  the Frobenius operator modulo p is now spanned by the columns of the matrix I , which indeed has the “desired maximal variation” dictated by the local model. The point is, the basis in which Equation (5.1) is given is not horizontal with respect to the Gauss-Manin connection, whereas the basis in which Equation (5.2) is written is, at least over R/m2R . As will become apparent from the discussion below (§ 5.6), this is enough to conclude that this display is a universal display. We make all this more precise. Consider the composition φ  τ of two operators, φ being a σ -linear map and τ being a linear automorphism. Here the operators are operating on the underlying module of the display of the special fibre, extended trivially to a display over R. We take
φ to be the Frobenius operator and τ the automorphism expressed in a basis B by I0 TI . Let [φ  τ ]B be the expression of φ  τ as a matrix

132

Fabrizio Andreatta and Eyal Z. Goren

  A AT σ +pB with respect to the basis B. Then [φ τ ]B = C . Let C be the basis τ −1 (B). σ  CT +pD C pB+T pD Then [φ  τ ]C = [τ ]B [φ]B = A+T . C pD

Furthermore, let IR be the augmentation ideal of W(R), IR =V W(R). Let K be the kernel of φ (mod IR ) so τ −1 (K) is the kernel of φ  τ (mod IR ). Let [K]B be the set of coordinate vectors expressing K in the
basis B (mod IR ). Then we have [τ −1 (K)]C = [K]B , but of course [τ −1 K]B = I0 −T [K]B . I According to Zink’s  (see [Zin, Thm. 44] and §5.1.3 below) the display P0
Atheory B gives a crystal D over k determined by C P0 over the nilpotent crystalline site of D Spec(k). To conclude our discussion, it remains to show that there is a display P over R, whose Frobenius operator is given by (5.2), such that the isomorphism from 0 ⊗ W(R/m2 ), dictated by the crystalline theory, is simply the  (mod m2 ) to P P R R
σ identity. In essence, that follows from the fact that the operator I0 TI is the identity when reduced modulo IR and then modulo m2R .

5.1 Recall In this section we review the theory of displays, developed in [Zin], discussing a variant where a real multiplication is considered. Having in mind applications to local models, we recall the connection between displays and crystals as developed in [Zin].

5.1.1. The deformation theory of abelian varieties is equivalent, by Serre–Tate, to the deformation theory of their p-divisible groups. One wishes to isolate the type of p-divisible groups on which OL acts as endomorphisms that arise in this fashion from RM abelian varieties. To illustrate the problem, note that if p splits in OL then OL acts as endomorphisms of any one dimensional p-divisible group, but does not act on any elliptic curve. To rule out such possibilities we make the following definition:

Definition 5.1.1. Let B be a finitely generated Zp -algebra. Let k be a field of characteristic p. Let G be a p-divisible over k on which B acts as endomorphisms. We say that G has RM by B if the Dieudonné module of G ⊗k k alg is a free B ⊗Zp W(k alg )module of rank 2. We say that G has RM by OL if it has RM by OL ⊗Z Zp in the sense just defined. Let R be a local Noetherian ring with residue field k as above. A p-divisible group G over R is said to have RM by B if B acts as endomorphisms of G and G ⊗ k has RM by B in the sense defined above. Ibid. for RM by OL . A polarized p-divisible group with RM over a ring R as above, is a pair (G, λ) where G is a p-divisible group over R with RM by B and λ : G → Gt is a B-linear symmetric isomorphism.

Hilbert modular varieties of low dimension

133

5.1.2. Let R be an Fp -algebra. Let W(R) be the Witt vectors over R and let IR be the kernel of the ring homomorphism W(R) → R given by projection on the first coordinate. A polarized display
P over R with real multiplication by OL , a RM display for short, is a quintuple P , Q, F, V −1 , _ , _  consisting of: 1. a projective OL ⊗Z W(R)-module P of rank 2; 2. a finitely generated OL ⊗Z W(R)-submodule Q of P such that IR P ⊂ Q ⊂ P and P /Q is a direct summand of the R-module P /IR P ; 3. additive maps F : P → P and V −1 : Q → P , which are  with respect
linear to OL and σ -linear with respect to W(R), and satisfy V −1 V wy = wF (y) for any w ∈ W(R) and any y ∈ P . One imposes a further nilpotence condition [Zin, Def. 2]; 4. an OL ⊗ W(R)-bilinear map _ , _  : P × P → DL−1 ⊗ W(R) satisfying the identity V V −1 (x), V −1 (y) = x, y for every x and y in Q. Define DP := P /IR P ,

HP := Q/IR P .

The filtration HP ⊂ DP is called the Hodge filtration of P . Replacing OL with its completion OLp and DL with its completion at p, one gets the notion of a polarized display with OLp -action. The main example of a display is the Dieudonné module. Let k be a perfect field of characteristic p and let G be a connected polarized p-divisible group with RM by OL over k. Then the Dieudonné module of G, say P , equipped with its Frobenius and Verschiebung morphisms and OL -bilinear pairing, gives the RM display (P , V P , F, V −1 , _ , _ ). A variant of [Zin, Thm. 9] is the following: Theorem 5.1.2. Let R be an excellent local ring or a ring such that R is an algebra of finite type over a field k. Assume that p = 0 in R. Then there is a natural equivalence of categories between the category of polarized connected p-divisible groups over R with RM by OL (resp. OLp ) and the category of displays over R with RM by OL (resp. OLp ). 5.1.3. The following is a consequence of [Zin, Thm. 44]. Let S → R be a surjective homomorphism of rings such
that p is nilpotent inS and its kernel a is equipped with divided powers. Let P := P , Q, F, V −1 , _ , _  be an RM display (or a polarized 
display with OLp -action) over R. Let Pi = Pi , Qi , Fi , Vi−1 , _ , _ 1 , i = 1, 2, be i be RM displays (or polarized displays with OLp -action) over S reducing to P . Let Q

134

Fabrizio Andreatta and Eyal Z. Goren

i . The theorem the inverse image of Q via Pi → P . Then, Vi−1 extends uniquely to Q states that there is a unique isomorphism,  

1 , F1 , V −1 −→ P 2 , F2 , V −1 , 2 := P2 , Q 1 := P1 , Q α: P 1 2 reducing to the identity on P and commuting with the OL -action (or OLp -action).
 Therefore, the sheaf P Spec(R) ⊂ Spec(S) := P1 on the crystalline site (of pd-thickenings with kernel a nilpotent ideal) of Spec(R) defines a crystal. AnaloTherefore, the sheaf gously, DP1 and DP2 are canonically isomorphic.
D Spec(R) ⊂ Spec(S) := DP1 on the crystalline site of Spec(R) defines a crystal called the covariant Dieudonné crystal. Let A be an abelian variety over R with RM, let G be its p-divisible group and let P be the associated display. The crystal DP is canonically isomorphic to the crystal D∗ (At ). See [Zin, Thm. 6] and [MM, II (1.5)].

5.2 Factorizing according to primes 5.2.1 The local deformation theory and displays. Lemma 5.2.1. Let k be an algebraically closed field of positive characteristic p. Let x ∈ M be a k-valued point. Then, 1. the RM p-divisible group Ax (p) factors canonically as the product of the OLp -polarized p-divisible groups, denoted Ax (p); 2. for each p, the OLp -polarized p-divisible group Ax (p) is either ordinary or local-local. Its Dieudonné module is a free OLp ⊗Z W(k)-module of rank 2; 3. the functor of deformations of Ax (p) on Ck as an OL -polarized p-divisible group is naturally equivalent to the direct product, over p dividing p, of the functors of deformations of Ax (p) on Ck as an OLp -polarized p-divisible group. One considers RM displays as in §5.1 and polarized displays with OLp action. It is easy to see that the first category is naturally isomorphic to the direct product of the categories of polarized displays with OLp action, where p runs over primes factor of pOL . Under the equivalence of categories stated in Theorem 5.1.2 between deformations of connected p-divisible groups and displays, the decomposition according to primes is respected. 5.2.2 The associated local model. Let D0 be the OL -module OL ⊗ k ⊕ OL ⊗ k, let  ,  : D0 × D0 → ∧2OL D0 = OL ⊗ k be the wedge product, and let H0 ⊂ D0 be an isotropic OL ⊗ k-submodule of D0 having dimension g over k. Let R be the complete local ring pro-representing the moduli problem of associating to a local

Hilbert modular varieties of low dimension

135

artinian k-algebra (S, mS ) an OL ⊗ S-submodule H of D := D0 ⊗k S, such that H is free as a S-module, is a direct summand of D, is totally isotropic with respect to the pairing  , , and reduces to H0 modulo mS . The ring R is isomorphic to the completion of the local ring of the point corresponding to (H0 , D0 ) in the appropriate Grassmann variety Grass(OL ⊗ k,  , , g, 2g). The Grassmann variety Grass(OL ⊗ k,  , , g, 2g) is canonically isomorphic to the product, over p|p, of the Grassmann varieties Grass(OL,p ⊗ k,  , p , gp , 2gp ). In particular, writing D0 = ⊕p D0 (p), H0 = ⊕p H0 (p), using the decomposition OL ⊗ k = ⊕p|p OLp ⊗ k, and noting that the pairing decomposes accordingly, we find p|p R(p), where R(p) is the completed local ring of the point (H0 (p), D0 (p)) that R = ⊗ on the Grassmann variety Grass(OL,p ⊗ k,  , p , gp , 2gp ) and the completed tensor product is taken over k.

5.3 The setting in which the theorems are proved Using the decomposition above, one sees that the construction of the universal RM display (for deformations of a given RM display over k) may be considered “one prime at a time”, and therefore, for notational convenience, one may assume that pOL = pe . The results in this section will be formulated under this assumption, from which the more general assertions follow immediately. We set the following notation: pOL = pe , f = [OL /p : Fp ]. Let σ1 , . . . , σf ur → W(k), ordered such that F (·)  σi = σi+1 . Note denote the embeddings of O L,p f

that OL ⊗W(k) = ⊕i=1 B(i), where the decomposition is induced by the isomorphism f W(Fpf )⊗Zp W(k) ∼ = ⊕i=1 W(k), a⊗λ → (. . . , σi (a)λ, . . . ). We also have OL ⊗k = f e ∼ i=1 B(i) with the obvious notation. Note that B(i) = k[T ]/(T ), where T is the ur L,p /O  . reduction of an Eisenstein element for the extension O L,p F V For any k-algebra S denote by · and · the maps on OL ⊗ W(S) given by F ( ⊗ w)  →  ⊗ F w and V ( ⊗ w)  →  ⊗ V w for all  ∈ O and w ∈ W(S). L

5.4 Further decomposition of the local model For r = 1, . . . , f let D0 (r) := B(r) ⊕ B(r) and denote by  ,  : D0 (r) × D0 (r) → B(r) the wedge product. Let H0 (r) ⊂ D0 (r) be an isotropic B(r)-submodule of D0 (r) having dimension e over k. There exist a basis {α(r), β(r)} of D0 (r), as a B(r)-module, such that α(r), β(r) = 1 and H0 (r) = (T i(r) )α(r) ⊕ (T j (r) )β(r)

136

Fabrizio Andreatta and Eyal Z. Goren

for uniquely determined integers e ≥ i(r) ≥ j (r) ≥ 0 satisfying i(r) + j (r) = e. Let R(r) be the complete local ring pro-representing the moduli problem of associating to an object (S, mS ) of Ck a B(r) ⊗ S-submodule H (r) of D(r) := D0 (r) ⊗k S such that H (r) is free as a S-module, is a direct summand of D(r), is totally isotropic with respect to the pairing  , , and reduces to H0 (r) modulo mS . Then, R(r) ∼ = k[[a(r)0 , . . . , a(r)i(r)−1 , b(r)0 , . . . , b(r)j (r)−1 , c(r)0 , . . . , c(r)i(r)−1 , d(r)0 , . . . , d(r)j (r)−1 ]]/
 a(r)d(r) − b(r)c(r) + a(r)T j (r) + d(r)T i(r) , where a(r) := a(r)0 +· · ·+a(r)i(r)−1 T i(r)−1 , b(r) := b(r)0 +· · ·+b(r)j (r)−1 T j (r)−1 , c(r) := c(r)0 + · · · + c(r)i(r)−1 T i(r)−1 and d(r) := d(r)0 + · · · + d(r)j (r)−1 T j (r)−1 . The universal flag H (r) ⊂ D(r) over R(r) is defined by the B(r)-span of T i(r) α(r) + a(r)α(r) + b(r)β(r) and T j (r) β(r) + c(r)α(r) + d(r)β(r). Note that the Grassmann variety Grass(OL ⊗ k,  , , g, 2g) decomposes as the product of the Grassmann varieties Grass(B(r),  , r , e, 2e). Hence, fr=1 R(r). R∼ =⊗

5.5 The display over the special fiber and its trivial extension
 Let P0 := P0 , Q0 , F0 , V0−1 ,  , 0 be a RM display over k with an OL ⊗ k-linear isomorphism of the Hodge filtration HP0 ⊂ DP0 with H0 ⊂
D0 , compatible with the pairings on P0 and D0 . Choose a decomposition P0 = ⊕r B(r)α(r) ⊕ B(r)β(r) as OL ⊗ W(k)-module so that P0 /pP0 = D0 , Q0 /pP0 = H0 and α(r), β(r)0 = 1. Note that F0 = ⊕F0 (r), a direct sum of F -linear maps, and F0 (r) [B(r)α(r) ⊕ B(r)β(r)] ⊂ [B(r + 1)α(r + 1) ⊕ B(r + 1)β(r + 1)] . The matrix of F0 (r) with respect to the bases {α(r), β(r)} and {α(r + 1), β(r + 1)} is of the form  j (r) T g1,1 (r) T i(r) g1,2 (r) F0 (r) := . (5.3) T j (r) g2,1 (r) T i(r) g2,2 (r) To state in the next section the main theorem we need some
more  notation. Let ˆ t , c(r) ˆ t be the Teichmüller lifts in W R(r) of a(r)s , b(r)t , ˆ s and d(r) a(r) ˆ s , b(r) c(r)s and d(r)t for 1 ≤ r ≤ f , 0 ≤ s ≤ i(r) − 1 and 0 ≤ t ≤ j (r) − 1. Define i(r)−1 j (r)−1 i(r)−1 ˆ ˆ s T s , c(r) a(r) ˆ := s=0 a(r) ˆ s T s , b(r) := s=0 b(r) ˆ := s=0 c(r) ˆ s T s and
 j (r)−1 s ˆ s T ; these are elements of B(r) ⊗W(k) W R(r) . Let ˆ d(r) := s=0 d(r) j (r) i(r) ˆ ˆ − b(r) ˆ c(r) + d(r)T . n(r) := a(r) ˆ d(r) ˆ + a(r)T ˆ

137

Hilbert modular varieties of low dimension

F Lemma 5.5.1. Let M(r) be
 the maximal ideal of R(r). Then, the element n(r) lies e in T B(r) ⊗W(k) W M(r) . Let

ur := 1 + T −e F n(r).
 Then ur is a unit in B(r) ⊗W(k) W R(r) .

 Proof. Note that T e B(r) ⊗W(k) W M(r) is equal to pB(r) ⊗W(k) W M(r) . Since multiplication by p coincides
 with the composition of Verschiebung and Frobenius, we conclude that pW M(r) consists of the Witt vectors (a0 , a1 , . . . ) with a0 = 0 and ai ∈ F M(r).
The assertion concerning F n(r) follows. Note that u(r) lies   in 1 + B(r) ⊗W(k) W M(r) . It is a unit by Lemma 5.5.2. Lemma 5.5.2. Let S be a k-algebra. Let v ∈ B(r) ⊗W(k) W(S). Assume that the image v of v via the composition B(r) ⊗W(k) W(S) → B(r) ⊗W(k) S → S is a unit. Then, v is a unit. Proof. Let Norm on B(r) ⊗W(k) W(S) (resp. B(r) ⊗W(k) S) be the norm as a W(S)module (resp. a S-module). Then, v (resp. v) is a unit if and only if Norm(v) is a unit. Hence, we may assume OL = Z. Let u be an element (resp.
Norm(v))  of W S such that uv = 1 − i with i ≡ 0 in W1 (S) = S. Notethat i n ≡ 0 in Wn (S). n Since W(S) = lim Wn (S), we get that the element z = n i exists in W(S). ←− Hence, v(uz) = 1.  Let
 F (r) : B(r) ⊗W(k) W R ⊕ B(r) ⊗W(k) W(R(r)) −→ B(r + 1) ⊗W(k) W(R) ⊕ B(r + 1) ⊗W(k) W(R) be the F -linear operator whose matrix with respect to the bases {α(r), β(r)} and {α(r + 1), β(r + 1)} is: F (r) := u(r)−1 × j (r) F

F ˆ ˆ g1,1 (r) + (d(r))g 1,1 − (b(r))g 1,2 j (r) F F ˆ ˆ T g2,1 (r) + (d(r))g 2,1 − (b(r))g 2,2

T

F ˆ T i(r) g1,2 (r) − F (c(r))g ˆ 1,1 + (a(r))g 1,2

T i(r) g

2,2

(r) − F (c(r))g ˆ

2,1

+ F (a(r))g ˆ

2,2

.

(5.4)

5.6 The main results on displays Theorem 5.6.1. Let P := P0 ⊗W(k) W(R) and let Q be the inverse image of H via the projection P → D. Let F : P → P be the F -linear map whose matrix form with respect to the decomposition P = ⊕r B(r) ⊗W(k) W(R)α(r) ⊕ B(r) ⊗W(k) W(R)β(r)

138 is

Fabrizio Andreatta and Eyal Z. Goren



0 F (1)   ... 0

0 0 ... 0

... ... ... ...

0 0 ... F (g − 1)

 F (g) 0  , ...  0

with F (r) given in Equation (5.4). Then, there exist a unique F -linear homomorphism V −1 : Q → P and an OL ⊗ W(R)-bilinear map  ,  : P × P → DL−1 ⊗ W(R) so that P := (P , Q, F, V −1 ,  , ) is a RM display. Moreover, 1. its base change via R/m = k coincides with P0 as RM display; 2. (R, P ) is the universal pro-representing object and the universal RM display for the moduli problem of deforming P0 to objects of Ck as a RM display; 3. the projection P → D identifies HP ⊂ DP with H ⊂ D compatibly with the pairings on P and D.   Proof. Let ψ be the map P ⊗Z Q → P ⊗Z Q defined as diag ψ(1), . . . , ψ(f ) , where the map ψ(r) is defined with respect to the basis {α(r), β(r)} by the matrix  −i(r) −j (r) c(r)T ˆ 1 + a(r)T ˆ . −i(r) −j (r) ˆ ˆ b(r)T 1 + d(r)T Note that F is the composition F0  ψ −1 of the F -linear base change of F0 to P = P0 ⊗W(k) W(R) with the inverse of ψ. One proves that, indeed, F is well defined. One defines V −1 := Fp on P ⊗ Q and one proves that V −1 restricted to Q is well defined, it is compatible with  ,  and V −1 (Q) spans P . By definition V −1 is compatible with F . See [AG4] for details. Claims (1) and (3) follow immediately from the construction. Claim (2) follows from the following theorem.  
Theorem 5.6.2. Let P := P , Q, F, V −1 ,  ,  be a RM display over R and let τ : DP → D be an isomorphism as OL ⊗ R-modules, compatible with pairings, such that τ (HP ) = H and τ is a horizontal map mod m2 . Here, we consider the connection on DP ⊗R R/m2 induced by the fact that DP is a crystal and we consider on D ⊗R R/m2 the connection having D0 ⊂ D as horizontal sections. Then, (R, P ) is the universal pro-representing object and the universal RM display for the moduli problem of deforming the special fiber P0 of P to local artinian k-algebras as RM display. 
Proof. Let P uni := P uni , Quni , F uni , (V uni )−1 ,  , uni be the universal RM display deforming the special fiber P0 . By the theory of local models [DP, Thm. 3.3] and the equivalence of categories between deformations of displays and of formal p-divisible groups [Zin, Thm. 9] it exists over R.

Hilbert modular varieties of low dimension

139


Let φ : Spec(R) → Spec(R) be the unique homomorphism such that P = φ ∗ P uni . Since R pro-represents a Grassmannian moduli problem, we get unique


  maps ψi : Spec R/m2 → Spec R/m2 , such that ψ1∗ (H
⊂ D) ∼ = HP
uni ⊂ DP uni  and ψ1∗ (D) ∼ = DP uni is horizontal, and ψ2 such that ψ2∗ H ⊂ D) = HP ⊂ DP ∗ ∼ and ψ2 (D) = DP is horizontal. Moreover, ψ1  φ = ψ2 - all the maps appearing being canonical. By [DP, Lem. 3.5] the map ψ1 is an isomorphism. Hence, φ is an isomorphism on tangent spaces. Let Gr(R) be the graded ring ⊕n mn /mn+1 associated to R. The induced map Gr(φ  ) : Gr(R) → Gr(R) is then surjective on each graded piece and, hence, by dimension considerations it is injective. Since Gr(φ  ) is an isomorphism, we conclude that φ  is an isomorphism as well [AtM, Lem. 10.23]. Hence, φ is an isomorphism as claimed.  Corollary 5.6.3. Let p be maximally ramified. Let x ∈ M be a geometric point of type (j, n). 1. The deformations to Sj  , where j  ≤ j , are parameterized by the closed subscheme defined by the ideal ai , bi , ci , di : 0 ≤ i ≤ j  − 1. 2. The deformations to W(j  ,n ) , where j  ≤ j , are parameterized by the closed subscheme of deformations to Sj  intersected with the closed subscheme (with   the reduced structure) given by the relations T j +n |F 2 .

6 Some general results concerning strata in the maximally ramified case 6.1 Foliations of Newton polygon strata In this section we complete the analysis, started in [AG1], of the strata {W(j,n) }. For their definition see §3.2. We prove that each stratum W(j,n) is quasi-affine. We proceed as follows. First, by an explicit normalization of the display over the completed local ring of a point of type (j, j ), we prove that for every m the p m -torsion of the universal RM abelian scheme over W(j,j ) can be trivialized over a finite cover of W(j,j ) (depending on m). Using the “Raynaud trick”, we conclude that W(j,j ) is quasi-affine. We deduce the quasi-affineness of W(j,n) by showing that it is the image of W(n,n) if n > g2 (resp. W(g−n,n) if n ≤ g2 ) via iterated Hecke correspondences at p. We also describe the analogue of the foliations of the Newton polygon strata introduced by [Oo4] in the Siegel case. Recall that the stratification {W(j,n) } refines the Newton polygon stratification; [AG1, Thm. 10.1]. Since the universal RM p-divisible group over W(j,j ) is geometrically constant, W(j,j ) is the central leaf at any of its points; cf. Definition 6.1.1. The foliation on the loci W(j,n) is then described using the Hecke correspondence linking W(j,n) and W(n,n) if n > g2 (resp. W(g−n,n) if n ≤ g2 ).

140

Fabrizio Andreatta and Eyal Z. Goren

Definition 6.1.1 ([Oo4, §2]). Let P be a RM display over a perfect field k of positive characteristic p. Let T be a noetherian scheme over k. Let A → T be a RM abelian scheme. Define
 CP A → T as the subset of T consisting of the geometric points t ∈ T for which there exists an isomorphism of RM displays between P ⊗W (k) W (k(t)) and the display associated to At . If P is the RM display associated to a geometric point x ∈ T , we write CAx instead of CP and we call it the central leaf at x. Note that the Newton polygon of the geometric points of CP are those of P and hence constant. By [Oo4, Thm. 2.2] the set CP is a closed subset of the locally closed subscheme of T consisting of the points having the same Newton polygon as P . Definition 6.1.2. Let k be a perfect field of characteristic p and let S be a k-algebra. Let s = t (s)g + r(s) with t (s) ∈ N and 0 ≤ r(s) ≤ g − 1. Consider the exact sequence of W(S)-modules ϕ

0 −→ S −→ Wt (s)+1 (S) −→ Wt (s) (S) −→ 0. ϕ

The map ϕ is the t (s)-th power of Verschiebung r  → (0, . . . , 0, r). It identifies S with the W(S)-module whose additive structure is that of S and multiplication of r ∈ S p t (s)

by a = (a0 , a1 . . . ) ∈ W(S) is given by a · r := a0

r. The sequence

1⊗ϕ

0 −→ OL ⊗Z S −→ OL ⊗Z Wt (s)+1 (S) −→ OL ⊗Z Wt (s) (S) −→ 0 is an exact sequence of OL ⊗ W(S)-modules. Since S is of characteristic p, we have that OL ⊗Z S ∼ = Fp [T ]/(T g ) ⊗Fp S. Consider the OL ⊗Z W(S)-submodule of OL ⊗Z S defined by In := T n S ⊕ · · · ⊕ g−1 T S. Let

 Zs (S) := OL ⊗Z Wt (s)+1 (S) / 1 ⊗ ϕ(Ir(s) ) . By construction we have an exact sequence of OL ⊗Z W(S)-modules 0 −→ S −→ Zs+1 (S) −→ Zs (S) −→ 0, where S is a W(S)-module as above and OL acts on S via the quotient OL /(T ). We note that OL ⊗ W(S) = lim Zs (S). ←− s

Note that T g is equal to p up to a unit in OL ⊗Z Zp . Thus, multiplication by T g on OL ⊗Z W(S) is, up to a unit, multiplication by p. The latter coincides with the composite of Verschiebung and Frobenius on W(S). In particular, if S is reduced for

Hilbert modular varieties of low dimension

141

every positive integer i the kernel of multiplication by T i on Zs+i (S) coincides with the kernel of Zs+i (S) → Zs (S). Remark 6.1.3. Let A be an abelian variety with RM by OL over a perfect field of characteristic p, p totally ramified in L. It is proven in [AG1, Prop. 4.10] that one can choose an OL ⊗ W(k) basis α, β for the Dieudonné module (or “display the display”) of A such that, if A is not superspecial, Frobenius is given with respect to this basis by a matrix n

T c3 T i , (6.1) Tj 0 with c3 ∈ (OL ⊗W(k))× . Furthermore, it follows from [AG1, Prop. 7.2] that W(j,j ) is regular of dimension g −2j and that for any geometric point x of W(j,j ) , the completed W ,x is isomorphic to k[[fj , . . . , fi−1 ]]. Moreover, the isomorphism can local ring O (j,j ) be chosen so that Frobenius on the universal display over this ring is of the form

(1 + w(fj ) + w(fj +1 )T + · · · + w(fi−1 )T i−j −1 )T j c3 T i , (6.2) Tj 0 where w(fh ) denotes the Teichmüller lift of fh . Remark 6.1.4. Let j ≥ g/2 be an integer. If A is superspecial then, in fact, one can choose the basis for the Dieudonné module of A so that the matrix of Frobenius is

0 Ti . (6.3) Tj 0 The locus W(j,g−j ) is zero dimensional. Since Frobenius of the Dieudonné module of each of its points has the canonical form described by the matrix in Equation (6.3), it follows that for every m ∈ N the OL -group scheme A[pm ] ×M W(j,g−j ) is constant. 
Proposition 6.1.5. Let R be an Fp -algebra. Let P , Q, F, V −1 be an OL -display over R such that F (α) = d T j α + T j β,

F (β) = c3 T i α,

(6.4)

where d and c3 are invertible elements of OL ⊗ W(R) and we require i > j > 0. Then, there exist ring extensions R = R0 ⊂ · · · ⊂ Rs ⊂ Rs+1 ⊂ · · · , and elements As and Bs in Zs (Rs ), such that defining the elements of P ⊗OL ⊗W(R) Zs (Rs ) 

αs := As α + Bs β, βs := dAσs − As + Bsσ c3 T i−j α + Aσs − Bs β, (6.5) the following properties hold: 1. we have F (αs ) = T j αs + T j βs ,

F (βs ) = T i αs ;

(6.6)

142

Fabrizio Andreatta and Eyal Z. Goren

2. the elements αs and βs generate P ⊗OL ⊗W(R) Zs (Rs ) as a Zs (Rs )-module; 3. As+1 and Bs+1 map to As and Bs respectively, viewing As and Bs as lying in Zs (Rs+1 ) via the inclusion Zs (Rs ) ⊂ Zs (Rs+1 ); 4. Rs+1 is a finite free Rs -module; 5. for every s ∈ N the extension R ⊂ Rs satisfies the following universal property. s+i be OL ⊗ W(S)-generators of Let S be a reduced R-algebra. Let  αs+i and β P ⊗OL ⊗W(R) Zs+i (S) satisfying (6.6). Then, there exists a unique R-algebra s+i αs+i and fs (βs ) = β homomorphism fs : Rs → S such that fs (αs ) =  in P ⊗OL ⊗W(R) Zs (S). Proof. First of all, we reformulate property (5) in a way which is more convenient for the proof. As remarked above, since S is reduced, the kernel of multiplication by T i in Zs+i (S) coincides with the kernel of the reduction map Zs+i (S) → Zs (S). In particular, it factors via Zs (S) and Zs (S) embeds in Zs+i (S) via multiplication by T i . Thus, property (5) is equivalent to the existence of a unique R-algebra homomorphism s+i in P ⊗O ⊗W(R) αs+i and T i fs (βs ) = T i β fs : Rs → S such that T i fs (αs ) = T i  L Zs+i (S). This is the actual identity we verify below. Put formally αs := As α + Bs β,

βs := Gs α + Hs β.

(6.7)

Then F (αs ) = Aσs F (α) + Bsσ F (β) = dAσs T j α + Aσs T j β + Bsσ c3 T i α. Since T j αs + T j βs = As T j α + Bs T j β + T j βs , the first equality of (6.6) gives that j As T j α + Bs T j β +
T j βs = dAσs T j α + Aσs T j β +
Bsσ c3 T i α.  j Hence, T βs = j j σ σ i−j j σ Gs T α + Hs T β = dAs − As + Bs c3 T T α + As − Bs T β, and therefore,

 T j Gs = T j dAσs − As + Bsσ c3 T i−j , T j Hs = T j Aσs − Bs . (6.8) The second equality of (6.6) now gives As T i α + Bs T i β = T i αs = F (βs ) = Gσs F (α) + Hsσ F (β)
 = dGσs + c3 Hsσ T i−j T j α + Gσs T j β   2 2 = (d d σ Aσs − Aσs + Bsσ c3σ T i−j   2 + Aσs c3 T i−j − Bsσ c3 T i−j )T j α 
2 2 + d σ Aσs − Aσs + Bsσ c3σ T i−j T j β This is equivalent to the following two equations:
 2 2 T j · d σ Aσs − Aσs + Bsσ c3σ T i−j − Bs T i−j = 0,

(by (6.7)) (by (6.6)) (by (6.7)) (by (6.4))

(by (6.8))

(6.9)

Hilbert modular varieties of low dimension

143


2 T i · Aσs c3 − Bsσ c3 − As + d Bs = 0.

(6.10)

For As , Bs , Gs and Hs , Equation (6.6) holds if and only if Equations (6.8), (6.9) and (6.10) hold. Put also d σ Aσs − Aσs + Bsσ c3σ T i−j − Bs T i−j = 0,

(6.11)

Aσs c3 − Bsσ c3 − As + d Bs = 0.

(6.12)

2

2

2

We construct rings Rs and elements As and Bs of Zs (Rs ), such that if we choose Gs = dAσs − As + Bsσ c3 T i−j ,

Hs = Aσs − Bs ,

then properties (1)–(5) of the proposition hold and, moreover, also Equations (6.11) and (6.12) hold. We proceed by induction on s. Start with s = 1. Let c3 , d, be the reduction of c3 and d in R = Z1 (R). Let  
 p−1 R1 := R A1 , u / (dA1 − 1)p , up−1 − dc−1 3 . Putting α1 and β1 as in the proposition, with A1 the given element of R1 and B1 := d −1 A1 + u, one checks that Equations (6.11) and (6.12) hold in Z1 (R1 ) = R1 . Furthermore, property (2) is equivalent to requiring that the element

p
p 
p  A1 dA1 − A1 p = A1 A1 − B1 − B1 dA1 − A1 = A1 (A1 − dB1 ) det p B1 A1 − B1 is invertible. This holds since A1 and dB1 − A1 = u are invertible. Let S be an R-algebra as in property (5) with s = 1. In particular, Equations (6.9) 1+i in Z1+i (S) = OL /(T i+1 ) ⊗Z S. Note 1+i and B and (6.10) have solutions A
1+i )p + 1+i − d −1 A that, using Equation (6.9), Equation (6.10) becomes T i −c3 (B  1+i ) = 0. Since  1+i − d −1 A 1+i generate P ⊗O ⊗W(R) Z1+i (S), α1+i and β d(B L 1+i ) is T i p (A 1+i − d B a similar argument using Equation (6.8) gives that T i · A 1+i times a unit. Thus, we can define f1 : R1 → S as the R-algebra homomorphism 1+i = T i f1 (A1 ) and T i B 1+i = T i d −1 A 1+i +T i f1 (u). This concludes satisfying T i A the base step of the induction. Assume that the induction hypothesis holds for a given s ∈ N. Let As and Bs be elements in Zs+1 (Rs) reducing to As and Bs respectively in Zs (Rs ). Let Rs be  the polynomial ring Rs λ, µ . Let λs := (0, . . . , 0, λ) and µs := (0, . . . , 0, µ)
in Ker Zs+1 (Rs ) → Zs (Rs ) . Let As+1 := As + λs and Bs+1 := Bs + µs . Then, Equation (6.11) becomes d σ λp − λp + Ps = 0, 2

(6.13)


 

σ i−j + where Ps is the element of Rs defined by Ps = d σ As − Aσ s + Bs c3 − Bs T  i−j

σ2 σ . Since T i−j kills Ker Zs+1 (Rs ) → Zs (Rs ) , we have Ps = µ s c 3 − µs T σ 2

σ 2

144

Fabrizio Andreatta and Eyal Z. Goren

 i−j
σ 2 σ 2 σ  d σ Aσ . For the same reason Ps is independent of the s − As + Bs c3 − Bs T choice of Bs . Finally, Equation (6.12) becomes µp c3 − d µ + Qs = 0,

(6.14) σ2

where Qs is the element of Rs defined by Qs = (Bs )p c3 − dBs − As+1 c3 + As+1 . Equations (6.13), (6.14) define an ideal Js in Rs . Let Rs+1 := Rs /Js . The ring Rs+1 is an extension of Rs , finite and free as Rs -module. Define αs+1 and βs+1 as in the statement of the proposition. By construction Equation (6.6) holds and As+1 and Bs+1 reduce to As and Bs in Zs (Rs+1 ). Property (2) is equivalent to the invertibility of

σ c T i−j As+1 d Aσs+1 − As+1 + Bs+1 3 det . Bs+1 Aσs+1 − Bs+1 Since such element is invertible in Z1 (Rs+1 ), we deduce from 5.5.2 that it is indeed invertible. Let S be an R-algebra as in property (5) with s + 1. Using the induction hypothesis on Rs , we know that there exist a unique map of R-algebras fs : Rs → S such that s+1+i in P ⊗O ⊗W(R) Zs (S). Let  αs+1+i and fs (βs ) = β αs+1+i = fs (αs ) =  L     As+1+i α + Bs+1+i β. Equations (6.9) and (6.10) hold for As+1+i and Bs+1+i . Thus, to Rs is fs there exists a unique fs+1 : Rs+1 →
map of R-algebras
S whose restriction   s+1+i and T i fs+1 (Bs+1 ) = T i B s+1+i . By and such that T i fs+1 (As+1 ) = T i A the reformulation of property (5) given at the beginning of the proof one concludes  that Rs+1 , αs+1 and βs+1 satisfy property (5). Until the end of this subsection we assume that the base field k over which the moduli space M lives is algebraically closed. Corollary 6.1.6. Let A → W(j,j ) be the universal RM abelian scheme. Let y ∈ W(j,j ) [m] be k-valued point. For every m ∈ N there exists a scheme W(j,j ) finite and dominant [m] ∼ [m] m over W(j,j ) such that A[pm ] ×W(j,j ) W(j,j ) = Ay [p ] ×k W(j,j ) .

Proof. The case j = 0, corresponding to the ordinary case, is easy and is left for the reader. The case i = j = g/2, occurring only for g even, is covered by Remark 6.1.4, [m] where we define W(j,j ) := W(j,j ) . We now assume i > j > 0. For every n ∈ N, the T over W(j,j ) the group 
functor, associating to a scheme of isomorphisms Isom A[p n ] ×W(j,j ) T , Ay [pn ] ×k T as group schemes over T endowed with an OL -action, is represented by a scheme Isom(pn ), affine and of finite [m] type over W(j,j ) (see [Oo4, Lem. 2.4]). Let W(j,j ) be the scheme theoretic image m+2 m ) → Isom(p ). It follows from Proposition 6.1.5 that for every geof Isom(p ometric point x of W(j,j ) one can trivialize Frobenius on the Dieudonné module

Hilbert modular varieties of low dimension

145

of Ax [pm+2 ]. Hence, one can trivialize the Dieudonné module of Ax [pm ]. We con[m] clude that the reduced fiber of W(j,j ) over x is non-empty. Using Dieudonné theory and properties (4) and (5) of Proposition 6.1.5 for R = S = k and s = g(m + 2), [m] we deduce that the reduced fibers of W(j,j ) → W(j,j ) are of finite cardinality. Thus,

[m] W(j,j ) is quasi-finite over W(j,j ) . We now apply the valuative criterion of properness to prove that the morphism [m] W(j,j ) → W(j,j ) is proper. Let Py be the RM display associated to Ay . Let R be a complete dvr which is also a k-algebra. Let K be its fraction field. Suppose we [m] are given a map φ : Spec(R) → W(j,j ) and a K-valued point of W(j,j ) over it. It follows from Remark 6.1.3 that the Frobenius of the RM display P associated to the formal p-divisible group G over Spec(R) defined by φ, admits an OL ⊗ W(R)basis α and β such that Frobenius is of the form given in Equation (6.4). Using Dieudonné theory and our assumption, the base change of the display P to an algebraic such that Frobenius satisfies α and β closure K alg of K admits an OL ⊗W(K alg )-basis   and F (β)  ≡ T i F ( α) ≡ T j  α + T jβ α modulo pm+1 . We deduce from properties (4) and (5) of Proposition 6.1.5, applied to K alg and s = g(m + 2), that the change of  can be realized, at least over Zg(m+1) (R  ), for some integral basis from {α, β} to { α , β}  alg extension R ⊂ K of R. Thus, we conclude that P ⊗OL ⊗W (R) Zgm (R  ) is equal to Py ⊗OL ⊗W (k) Zgm (R  ) and so G[p m ] ⊗R R  ∼ = Ay [pm ] ⊗k R  . Note that this R  point [m] [m] of W(j,j ) factors through K, hence through R. Thus, the morphism W(j,j ) → W(j,j ) is proper and quasi-finite, hence finite [EGA IV, Thm. 8.2.1]. 

Corollary 6.1.7. The RM p-divisible group associated to the universal abelian scheme over W(j,j ) is geometrically constant. In particular, the central leaf CAx at any point x of W(j,j ) coincides with W(j,j ) . Proof. Let x be a geometric point of W(j,j ) . Let Gx be the p-divisible group defined by x. The case i = j = g/2, occurring only for g even, is covered by Remark 6.1.4. The case j = 0 is the case of ordinary abelian varieties, where the result is well known. Assume now i > j > 0. Apply Proposition 6.1.5 to the OL -display over R = k(x) associated to Gx . The k(x)-algebras Rs are finite as k(x)-modules. Therefore, since k(x) is an algebraically
 closed field,
there  exist compatible sections Rs → k(x). Note that OL ⊗Z W k(x) = lim Zs k(x) . Hence, α := lim αs and β := lim βs ←− ←− ←−
 are well defined and form an OL ⊗Z W k(x) -basis of the Dieudonné module of Gx such that F (α) = T j α + T j β and F (β) = T i α. Since F  V = p, we deduce that also Verschiebung V has a canonical form with respect to the basis {α, β} independent of x. Since the category of connected p-divisible groups and the category of displays are equivalent over perfect fields, we conclude.  Corollary 6.1.8. Let 0 ≤ j ≤ g/2. The scheme W(j,j ) is quasi-affine. Proof. If j = g/2, then dim(W(j,j ) ) = 0 and W(j,j ) consists of superspecial points. The corollary is trivial in this case. Suppose j < g/2. By Corollary 6.1.6 there exists

146

Fabrizio Andreatta and Eyal Z. Goren

[1] a finite covering W(j,j ) of W(j,j ) over which the p-torsion of the universal RM abelian scheme can be trivialized. It follows from Raynaud’s trick that the pull-back of the [1] Hodge bundle to W(j,j ) is torsion [Oo3, §4]. Since the Hodge bundle is ample on M,

[1] it follows that W(j,j ) is quasi-affine, hence so is W(j,j ) .



Let αp be the group scheme over k defined as the kernel of Frobenius on the additive group Ga,k . We make OL act on it via its quotient OL /T = Fp . Proposition 6.1.9. Let 0 ≤ m ≤ j ≤ g/2. Let j  be either j or g − j . There exists a smooth connected affine scheme Um over k, of dimension m, and a finite surjective map [m] ψm : W(j,j  ) ×k Um → W(j −m,j  ) ,

such that: [m] • for every u ∈ Um (k) the image of W(j,j  × {u} is contained in the central leaf
[m] ) through any point of ψm W(j,j  ) × {u} ; [m] • for every s ∈ W(j,j  ) (k) the image of {s} × Um is the image of As via iterated αp -Hecke correspondences.

Proof. Let s ∈ W(j,j  ) . Define the schemes Un for 0 ≤ n ≤ j by induction on n as follows. Let U0 := Spec(k). Suppose that Un has been defined and it is a smooth, connected affine scheme of dimension n and that every u ∈ Un defines an iterated αp quotient As → Au of invariants (j − n, j  ). Let Un+1 be the scheme
over Un whose  fiber over any geometric point u ∈ Un is the subscheme of HomOL αp , Au [p] of those maps for which the quotient Au /αp has invariants (j − (n + 1), j  ). By [AG1, Prop. 6.6, Prop. 8.7] the morphism Un+1 → Un is an affine bundle and the fiber over u is a non-empty open subscheme of P1k(u) . It follows that Un+1 is a smooth, connected affine scheme of dimension n + 1. Fix m. Define the map [m] ψm : W(j,j  ) ×k Um −→ M

as follows. By Corollary 6.1.6 or Remark 6.1.4 we have a canonical isomorphism [m] ∼ [m] m τm : A[p m ]×M W(j,j  ) = As [p ]×W(j,j  ) . View Um as classifying suitable subgroup m schemes of As [p ]. Then, ψm is the unique map such that the pull-back of the universal [m] RM abelian scheme via ψm coincides with the quotient of A×M (W(j,j  ) ×k Um ) by the inverse image via τm of the tautological subgroup scheme of As [pm ] ×k Um defined by Um . Note that such a quotient is a RM abelian scheme by [AG1, Cor. 3.2]; in particular, the definition of ψm makes sense. By construction, the image of ψm lies in W(j −m,j  ) . To conclude it suffices to prove that ψm is finite and surjective. We proceed by induction on m. By Corollary 6.1.7, since ψ0 is the identity, the proposition is true

147

Hilbert modular varieties of low dimension

for m = 0. Suppose that ψm−1 is finite and surjective. Consider the diagram [m] W(j,j  ) ×k Um

δ



 / π −1 W(j −m,j  ) ∩ π −1 W(j −m+1,j  ) 2 1

γ

 [m−1] W(j,j  ) ×k Um−1

π2

/ W(j −m,j  )

π1 ψm−1

 / W(j −m+1,j  ) ,

[m] [m−1] where γ is the product of the natural maps γ1 : W(j,j  ) → W(j,j  ) , γ2 : Um → Um−1 and δ is the unique morphism making the diagram commute and satisfying π2 δ = ψm . [m−1] By construction of Um , for every point s = (s1 , s2 ) ∈ W(j,j  ) ×k Um−1 and any

point t in the finite scheme γ1−1 (s1 ) the map {t} ×k γ2−1 (s2 ) → π1−1 (ψm (s)) is an isomorphism. Hence, δ is quasi-finite, proper (by the valuative criterion) and surjective.

 By [AG1, Lem. 8.6], π1 π2−1 W(j −m,j  ) has invariants (j − m + 1, j  ) and, if j − m > 0, also (j − m − 1, j  ). Since
the maps π1 and π2 are proper by [AG1, Lem. 8.4] and the intersection with π1−1 W(j −m+1,j  ) of the fiber of π2 over a point of W(j −m,j  ) is non-empty and finite by [AG1, Prop. 6.6], we conclude that the  composite ψm = π2  δ is quasi-finite, proper and surjective as claimed.

Corollary 6.1.10. For every (j, n) the scheme W(j,n) is quasi-affine. Proof. By Corollary 6.1.8 the claim holds for the loci W(n,n) . The locus W(j,g−j ) is zero dimensional and, hence, quasi affine. By Proposition 6.1.9 the locus W(j,n) is the image via a finite map of a quasi-affine scheme. Hence, the conclusion. 

6.2 Connectedness of T1 and T2 Definition 6.2.1. Let 0 ≤ a ≤ g be an integer. Let Ta be the closed subscheme of M defined by   Ta := [A] ∈ M(k)|a(A) ≥ a . Remark 6.2.2. By [AG1, Lem. 4.12] we have Ta = (j,n) W(j,n) where the union is taken over all pairs of integers (j, n) such that 0 ≤ j ≤ g2 and j ≤ n ≤ g − j and a ≤ j + n. It also follows from op. cit. that Ta has dimension g − a. Theorem 6.2.3. Assume that g > 1. The intersection of T1 with any irreducible component of M is connected. The same holds for T2 if g > 2. Proof. Suppose g > 1. Then, T1 is the complement of the ordinary locus in M. Hence, it is the zero locus of the Hasse invariant h. Since h is a section of the determinant of the Hodge bundle over M, and the Hodge bundle is ample, it follows that T1 is connected (cf. [Har, Cor. III.7.9]).

148

Fabrizio Andreatta and Eyal Z. Goren

Assume now that g > 2. Let C be the set of connected components of the intersection of T2 with an irreducible component of M. Let π1 , π2 : N → M be as in § 2.2. The Hecke correspondence π2 π1−1 preserves properties such as being closed, or being irreducible, or being connected, for closed subschemes not intersecting the non-singular (j = 0) locus of M, see [AG1, Prop. 8.7]. For every (j, n), it sends an irreducible component of W(j,n) surjectively to the union of irreducible components of loci W(j  ,n ) with (j  , n ) in a given set (j, n) depending only on (j, n) [AG1, Prop. 8.10]. Moreover, for every (j  , n ) ∈ (j, n) we have j  + n ≥ j + n − 1. The Hecke correspondence has the additional property of sending each component of M into a single component of M. Fix a component C ∈ C. By Remark 6.2.2, the irreducible components of C c with j + n ≥ 2. We conclude that consist of irreducible components of strata W(j,n) 
−1 locus π2 π1 (C) is closed and connected, it lies in T1 and its irreducible components c for suitable pairs (j, n) consist of union of irreducible components of loci W(j,n) with j + n ≥ 1.
 Suppose that |C| > 1. Since π2 π1−1 (T2 ) = T1 , the irreducibility of T1 in each component of M
implies that connected components C1  there 
exist distinct and C2 such E := π2 π1−1 (C1 ) ∩ π2 π1−1 (C2 ) is non-empty. If two irreducible components of the loci W(j,n) and W(j  ,n ) intersect, then (j, n) = (j  , n ) and they must coincide, because W(j,n) is smooth [AG1, Cor. 7.4]. Hence, E is closed and consists of irreducible components of loci of type W(j,n) for suitable (j, n) with j + n ≥ 1. By Corollary 6.1.10 the loci W(j,n) do not contain any complete curve. We conclude that E contains a point [A] of type (j, n) with
j + n ≥ 3 and j ≥ 1. Note that [A∨ ] is of type (j, n) by [AG1, Lem. 8.5] and π2 π1−1 (A∨ ) lies in T2 . Hence, its
∨ image π2 π1−1 (A∨ ) via the map [A]  → [A∨ ] lies in T2 . Since j  = 0 the image I consists of a Moret–Bailly family. In particular, it is connected. We show that I connects C1 to C2 in T2 . For i = 1, 2 let [Ai ] be a point of Ci and let Hi ⊂ Ai be an OL -invariant subgroup A /H . Then, the moduli point corresponding scheme of rank p such that A ∼
−1= ∨ i i ∨ ∨ ∨ ∼ to A /Hi = Ai lie in π2 π1 (A ) . Hence, [A1 ] and [A2 ] lie in the connected ∨
subscheme π2 π1−1 (A∨ ) of T2 . This contradicts the assumption that C1 and C2 were distinct.  Remark 6.2.4. The argument in the proof of Theorem 6.2.3 shows that if a is odd and Ta is connected, then Ta+1 is connected. This is used in the proof in the claim that π2 (π1−1 (Ta+1 )) = Ta ; a claim which is false for a even, cf. Diagram B in § 3.2. It is an interesting question to know whether the loci Ta are connected for all a ≤ g − 1 or not. An affirmative answer would have strong consequences (perhaps too strong).

Hilbert modular varieties of low dimension

149

6.3 Irreducibility results The singularity strata Sj were defined in § 3.2. Lemma 6.3.1. Let g/2 ≥ s ≥ j ≥ 0 be integers. Let x ∈ Ss . The completed S ,x of Sj at x is a complete intersection, regular in codimension 2. In local ring O j S ,x is a normal domain. particular, O j

c Proof. One deduces as in [DP, §4.3], cf. § 4.3.2, that the completed local ring of W(j,j ) at x has the presentation k[[a, b, c, d]]/(ad − bc + aT s + dT g−s ) with a := aj T j + · · · + ag−s−1 T g−s−1 , b := bj T j + · · · + bs−1 T s−1 , c := cj T j + · · · + cg−s−1 T g−s−1 S ,x is defined by g − 2j equations and d := dj T j + · · · + ds−1 T s−1 . Hence, O j S ,x is g − 2j . Hence, O S ,x in 2g − 4j variables. By [DP, §4.2] the dimension of O j j S ,x is is a complete intersection and, in particular, Cohen–Macaulay. By loc. cit. O j S ,x smooth in codimension 2. Using Serre’s criterion for normality we deduce that O j is a normal domain.  c Corollary 6.3.2. For every j , the irreducible components of W(j,j ) are disjoint. c c Proof. Recall that Sj = W(j,j ) . The Lemma implies that for every x ∈ W(j,j ) the S ,x is a domain. In particular, OS ,x is a domain. Hence, there exists only one ring O j j c irreducible component of W(j,j  ) containing x.

Proposition 6.3.3. Let g > 2. Every irreducible component of M contains exactly c . The same holds one irreducible component of the non-ordinary locus T1 = W(0,1) c for the locus W(1,1) . Proof. By Theorem 6.2.3, every irreducible component of M contains exactly one c c connected component of W(0,1) . Let x ∈ W(0,1) . The completed local ring of M c at x is at x is Cohen–Macaulay of dim g. Hence, the completed local ring of W(0,1) Cohen–Macaulay of dim g − 1 by [Eis, Prop. 18.13]. c . Let {Ti } be the set of irreducible Let C be a connected component of W(0,1) components of C. Assume its cardinality is > 1. Let Z be the union of all the intersections Ti ∩ Tj for i = j . Then, C\Z is disconnected. Hence, by Hartshorne’s connectedness theorem, see [Eis, Thm. 18.12], there must exist indices i and j and an irreducible component T of Ti ∩ Tj of codimension 1 in C and, hence, of dimension g − 2. Since the locus ∪n W(0,n) is smooth, T consists of points with singularity index > 0. Since the types (j, n) define a stratification and W(j,n) is pure dimensional of dimension < g −2 for j > 0 and n > 1, T consists of a full irreducible c c . Hence, it contains a full component of the locus W(1,2) . component of the locus W(1,1) W c ,x at a closed By Lemma 6.3.4 below, the nilradical of the completed local ring O (0,1) point x of type (1, 2) is a prime ideal. This implies that the prime ideals defined by Ti

150

Fabrizio Andreatta and Eyal Z. Goren

c and Tj in the local ring of the locus W(0,1) at a closed point of W(1,2) ∩ Ti ∩ Tj are equal. Hence, Ti = Tj . Contradiction. This proves the first part of the proposition. 
c c Since π2 π1−1 (W(0,1) ) = W(1,1) , the second claim follows.  c be the Lemma 6.3.4. Let g > 2. Let x be a closed point of W(1,2) . Let D = W(0,1)  non-ordinary locus of M. Then, the nilradical of the completed local ring OD,x of D at x is a prime ideal.

M,x of M at x is isomorphic to the Proof. By § 4.3.2 the completed local ring O quotient of the ring k[[a0 , . . . , ag−2 , b0 , c0 , . . . , cg−2 , d0 ]] by the relations ad − bc + aT + dT g−1 = 0, viz., a0 d0 − b0 c0 = 0, ai d0 + ai−1 − b0 ci = 0,

1 ≤ i ≤ g − 2,

ag−2 + d0 = 0. Eliminating the variables ai , using these equations, we get M,x ∼ O = k[[b0 , c0 , . . . , cg−2 , d0 ]]/




g−2

b0 c0 − d0 b0 c1 + d02 b0 c2 − d03 b0 c3 + · · · + (−1)g−2 d0

g b0 cg−2 + (−1)g−2 d0 .

The equations of the non-ordinary locus can be deduced
as in § 9.1.1 and coincide with equations (Eq1)–(Eq4) given there with a0 := b0 c1 − d0 c2 + d02 c3 + · · · −  g−3 g−1 (−1)g−2 d0 cg−2 − (−1)g−2 d0 . If d0 = 0, then a power of b0 and c0 is zero. The c D,x [d −1 ], at x. If b0 = 0 in O reduced ring coincides with the completion of W(1,1) 0 then a0 = 0 and c0 = 0. It follows that d0 = 0 (contradiction). Let h := c1 − d0 c2 + g−3 g−1 d02 c3 + · · · − (−1)g−2 d0 cg−2 . Then, a0 = hb0 − (−1)g−2 d0 . As in §9.1.1 the lemma is reduced to proving that there exists a unique minimal prime ideal associated to the ideal I in k[[b0 , c0 , . . . , cg−2 , d0 ]][b0−1 , d0−1 ] defined by p+1

• b0

p

p+g−1

+ hb0 d0 − (−1)g−2 d0

= 0;

g

• b0 c0 − hb0 d0 + (−1)g−2 d0 = 0. Consider the ideal J in the ring k[[b0 , c0 , . . . , cg−2 , d0 ]][b0−1 , d0−1 ] defined by p

p−1

• b0 + c0 d0 = 0 (obtained dividing by b0 the sum of the first equation and the p−1 second equation multiplied by d0 );  p
p p pg • b0 c0 − d0 hp + (−1)pg d0 = 0 (obtained by raising to the p-th power the second equation).

Hilbert modular varieties of low dimension

151

Then, the minimal primes associated to I and to J in k[[b0 , c0 , . . . , cg−2 , d0 ]][b0−1 , d0−1 ] pg−p+1 are the same. We can write the second equation as c0 (c0 − d0 h)p = (−1)pg d0 . p(g−2) = 0. Define the rings Let f (X) := Xp+1 − hp X − (−1)g d0
 p p R0 := k[[c1 . . . , cg−2 , d0 ]], R1 := R0 [X]/ f (X) , R2 := R1 [b0 ]/(b0 + Xd0 ). Since R2 is (d0 X, c1 , . . . , cg−2 , d0 , b0 )-adically complete and separated, the homomorphism of k[[c1 , . . . , cg−2 , d0 ]]-algebras from k[[b0 , c0 , . . . , cg−2 , d0 ]][b0−1 , d0−1 ]/J to R2 [b0−1 , d0−1 ] given by c0  → d0 X and b0  → b0 is well defined and it is an isomorphism. It therefore suffices to prove that the nilradical of R2 is prime. Let P be a prime ideal of R1 containing 0. Then, • either X is not a p-th power in Frac(R1 /P ) and then, since R2 is a flat R1 algebra, it follows that P R2 is a prime ideal of R2 ; in particular, if P is minimal in R1 then P R2 is minimal in R2 . p

• or X is a p-th power in Frac(R1 /P ) and then Xd0 = t p for some t ∈ Frac(R1 /P ). In this case let P2 be a minimal prime ideal of R2 containing P . By the going down theorem we must have P2 ∩ R1 = P . Hence, P2 defines a prime ideal in (R2 /P )⊗R1 Frac(R1 /P ) ∼ = Frac(R1 /P )[b0 ]/(b0 +t)p . Hence, P2 must be the kernel of R2 → R2 /P → Frac(R1 /P ) the latter map being b0  → −t. Hence, P2 is unique. In any case the map Spec(R2 ) → Spec(R1 ) defines a one to one correspondence between the irreducible components of Spec(R2 ) and those of Spec(R1 ). Therefore, it suffices to prove that the nilradical of R1 is prime. We show that in fact R1 is a domain. p p(g−2) factors as the product Assume that the polynomial X p+1 − c1 X − (−1)pg d0 n of the monic polynomials f1 (X) = X 1 + · · · + α1 X + α0 and f2 (X) = Xn2 + · · · + p(g−2) β1 X + β0 over k[[c1 , d0 ]]. Then, we have α0 β0 = −(−1)pg d0 . Without loss of generality we may assume that α0 = u0 d0m for some integer p(g − 2) ≥ m > 0 and p some u0 ∈ R0 not divisible by d0 . Since f (X) ≡ Xp+1 − c1 X = (X − c1 )p X in the polynomial ring over R0 /(d0 , c2 , . . . , cg−2 ) ∼ = k[[c1 ]] (which is factorial), we must have β0 = ±c1n + v0 d0 for some integer n > 0 and v0 ∈ R0 . In particular, β0 ≡ 0 p(g−2) − mod (d0 , c1 ). Since α0 β0 = u0 d0m (±c1n +v0 d0 ), then ±u0 c1n d0m = −(−1)pg d0 m+1 u0 v0 d0 . Since u0 and c1 are not divisible by d0 , we must have m = p(g − 2). Hence, β0 u0 = −(−1)pg i. e., β0 is a unit (contradiction). This implies that the polynomial f (X) is irreducible over R0 . Since R0 is local and regular, it is also factorial and, in particular, normal. It follows from [Eis, Cor. 4.12] that R1 is an integral domain.  The following lemma shows that the situation is different if we start with a closed point x of W(1,1) .

152

Fabrizio Andreatta and Eyal Z. Goren

c Lemma 6.3.5. Let x be a closed point of D = W(0,1) of type (1, 1). The completed  local ring OD,x of D at x has exactly two minimal associated prime ideals. Each of M,x . them has height 1 in O

M,x of M at x is Proof. As in the proof of Lemma 6.3.4 the completed local ring O isomorphic to


k[[b0 , c0 , . . . , cg−2 , d0 ]]/ g−2

b0 c0 − d0 b0 c1 + d02 b0 c2 − d03 b0 c3 + · · · + (−1)g−2 d0

g b0 cg−2 + (−1)g−2 d0 .

The equations of the non-ordinary locus can be deduced as in § 9.2 and coincide with equations (Eq1)–(Eq4) given there. The reduced subscheme defined by d0 = 0 c coincides with the W(1,1) locus. Inverting d0 , we get that the non-ordinary locus in k[[b0 , c0 , . . . , cg−2 , d0 ]][d0−1 ] is defined by the ideal I :

g−2

• b0 c0 −d0 b0 c1 +d02 b0 c2 −d03 b0 c3 +· · ·+(−1)g−2 d0 p2

p2

p p 2 −p

• −b0 + d0 − c0 d0

= 0. g−3

Let h := c1 − d0 c2 + d02 c3 + · · · − (−1)g−2 d0 ring k[[b0 , c0 , . . . , cg−2 , d0 ]][d0−1 ] defined by p

p

g

b0 cg−2 +(−1)g d0 = 0;

cg−2 . Consider the ideal J in the

p−1

• −b0 + d0 − c0 d0 = 0;  p
p p pg • b0 c0 − d0 hp + (−1)pg d0 = 0. Then, the minimal primes associated to I and to J in k[[b0 , c0 , . . . , cg−2 , d0 ]][d0−1 ] are the same. pg−p+1 . We can write the second equation as (c0 − d0 )(c0 − d0 h)p = (−1)pg d0 p(g−2) = 0. Define the rings Let f (X) := X p+1 − X p − hp X + hp − (−1)g d0 R0 := k[[c1 . . . , cg−2 , d0 ]],
 R1 := R0 [X]/ f (X) , p

p

p

R2 := R1 [b0 ]/(b0 − d0 + Xd0 ). Since R2 is (d0 X, c1 , . . . , cg−2 , d0 , b0 )-adically complete and separated, the map of k[[c1 , . . . , cg−2 , d0 ]]-algebras from k[[b0 , c0 , . . . , cg−2 , d0 ]][d0−1 ]/J to R2 [d0−1 ] given by c0 → d0 X and b0  → b0 is well defined. It is easily checked that it is an isomorphism. As in the proof of Lemma 6.3.4 one concludes that the map Spec(R2 ) → Spec(R1 ) defines a one to one correspondence between the irreducible components of Spec(R2 ) and those of Spec(R1 ). It thus suffices to prove that R1 has 2 minimal prime ideals. By Hensel’s lemma, f (X) admits a unique root x ∈ R0 which is congruent to 1 modulo the maximal ideal of R0 . Write f (X) = (X−x)q(X) with q(X) prime to X−x.

Hilbert modular varieties of low dimension

153


 Let R := R0 [X]/ q(X) . We claim that R is a domain. Since k[[c1 . . . , cg−2 , d0 ]] is local and regular, it is also factorial and, in particular, normal. Therefore, by [Eis, Cor. 4.12], R is a domain if and only if the polynomial q(X) is irreducible. It suffices to check the irreducibility of the reduction s(X) of q(X) modulo (c1 , . . . , cg−2 ). Let V be a normal, local, noetherian extension of k[[d0 ]] such that s(X) admits a root z ∈ V . Let y be the image of x in V . Since (X − y)s(X) = X p+1 − Xp − p(g−2) p(g−2) −p (−1)g d0 with y a unit, the element z is not a unit and z = 1 + (−1)g d0 z . (g−2)  p   p+1  p g Hence, z = (z ) where z satisfies (z ) − (z ) − (−1) d0 = 0. Applying inductively the same trick we find that there exists a positive integer r prime to p and an element w in the maximal ideal of V such that w p+1 − w p − (−1)g d0r = 0. Hence, pvalV (w) = valV (wp+1 − w p ) = rvalV (d0 ). Hence, valV (d0 ) is a multiple of p. Hence, the degree of k[[d0 ]] ⊂ V is ≥ p and it must then be equal to p, proving that s(X) is irreducible as claimed. It follows that R1 ∼ = R0 × R is the product of two integral domains of dimension g − 1 which are flat R0 -algebras. Since minimal associated primes behave nicely under localization [Eis, Thm. 3.10(d)], the zero ideal in R2 [d0−1 ] ∼ = k[[b0 , c0 , . . . , cg−2 , d0 ]][d0−1 ]/J is contained in exactly two minimal prime ideals, each of codimension 1. 

7 Intersection theory on a singular surface We survey here intersection theory on complete surfaces with isolated normal singularities. The main references for this theory are [Arc, RT1, RT2]; see also [Mum, II (b)]. By a singular surface we mean in this section an irreducible projective normal algebraic surface over an algebraically closed field. In [Mum, RT1, RT2] the fundamentals of intersection theory on singular surfaces are presented only over the complex numbers. The reason for that is that resolution of singularities for surfaces in characteristic p was not known at those times. In fact, even the situation over the complex numbers was not yet common knowledge as one gathers from the assumptions made in [RT1, §1] and the addendum [RT2]. Since then a very strong result about resolution of singularities in arbitrary characteristic was obtained by Lipman [Lip2], building on the works of Zariski and Abhyankar [Lip2, Introd.]. Indeed, [Lip2, I §2] proves that resolution of singularities for surfaces can be achieved in arbitrary characteristic by a succession of normalizations and blow-ups. In particular, the results of [Lip2] (see also §26 of loc. cit.) show that the set-up [RT1, §1] can be achieved in arbitrary characteristic. The thesis of Archibald [Arc] contains a thorough discussion and development of intersection theory on singular surfaces (of not necessarily locally principal divisors) and comparison with other available intersection theories such as, for example, Snapper–Kleiman’s [Kle].

154

Fabrizio Andreatta and Eyal Z. Goren

7.1 Definition of the intersection number Given a singular surface V , one can find a resolution of singularities, π : V ∗ −→ V , such that V ∗ is non-singular, π is an isomorphism over the set V  := V \ V sing , π −1 (V sing ) := ϒ (the “fundamental manifold”) is of pure dimension 1, each irreducible component of it is non-singular, every two irreducible components have at most simple intersections, no three components have a common point. In fact, V ∗ can be obtained by a succession of blow-ups and normalizations. Moreover, any two such resolutions are dominated by a third one. Cf. [RT2, §1]  the strict transform, as the closure Let C ⊂ V be an irreducible curve. Define C, in V ∗ of π −1 (C ∩V  ). One says that C1 ≡Q C2 on V , and calls this relation algebraic 1 − C 2 ) is equivalence with division, if for some m > 0 and some π : V ∗ −→ V , m(C algebraically equivalent to a divisor supported on ϒ. This notion is independent of V ∗ and defines an equivalence relation. Given a resolution of singularities π : V ∗ −→ V as above, let µ1 , . . . , µs be the irreducible components of ϒ. Let d = (µi
µj )i,j =1,...,s , be the intersection matrix. It is an invertible, symmetric, negative definite matrix with no negative elements except on the diagonal. It follows that k = −d −1 is a symmetric, positive definite matrix with no negative elements.  D  Let C, D be two curves on V . One can find V ∗ as above such that in addition: C, have no common point on ϒ, neither passes through a point of µi
µj and they intersect each µi simply. The contribution to the intersection multiplicity coming from V sing is then  
µi ][D 
µj ] = (. . . , C 
µi , . . . )k t (. . . , D 
µi , . . . ). k ij [C i,j

 i , . . . ) by C ϒ . The total intersection It will be convenient to denote the vector (. . . , C
µ number is 
D  + C ϒ k t Dϒ C
D =C

(7.1)

One can prove [RT1] that this defines a symmetric bilinear pairing on divisor classes modulo ≡Q .

Hilbert modular varieties of low dimension

155

7.2 Pull-back and intersection Let µ1 , . . . , µs be the irreducible components of ϒ. We want to define for an irreducible curve C in V a divisor C ∗ in V ∗ , such that + C =C ∗

s 

γi µi ,

(7.2)

∀j.

(7.3)

i=1

and such that s

C ∗
µj = 0,


µj + Since C ∗
µj = C i=1 γi µi µj , we see that we need to solve the equat tion d (γ1 , . . . , γs ) = − t C ϒ . This has a unique solution given by t

(γ1 , . . . , γs ) = t (γ1 (C), . . . , γs (C)) = k t C ϒ .

(7.4)

The definition of C ∗ extends by linearity to any divisor. Proposition 7.2.1. The following identities hold. 1. Let C be a divisor on V , then C ∗
µj = 0 for all j = 1, . . . , s. 2. Let C, D be divisors on V , then C ∗
D ∗ = C
D. 3. Let C be a divisor on V and D a divisor on V ∗ , then C ∗
D = C
π∗ D. Proof. The first part follows from the definition and the calculation above. For part (2), on the one hand, we have 
D  + C ϒ k t Dϒ , C
D =C and on the other hand       + + C ∗
D∗ = C γi (C)µi
D γi (D)µi =

 i

−

i



i

∗

∗

γi (C)µi
D + C


 i


D − =C

 
D  γi (D)µi + C

i

   γi (C)µi
γi (D)µi

 i


D − =C





i

   γi (C)µi
γi (D)µi i

γi (C)γj (D)µi
µj

i,j


D  − (γ1 (C), . . . , γs (C)) d t (γ1 (D), . . . , γs (D)) = =C

156

Fabrizio Andreatta and Eyal Z. Goren


D  − (C ϒ k)d(k t D ϒ ) =C 
D  + C ϒ k t Dϒ . =C For part (3), we calculate that C ∗
D = C ∗
(π∗ D)∗ − C ∗
(D − (π∗ D)∗ ) = C ∗
(π∗ D)∗ = C
π∗ D.



7.3 Adjunction Let K[V ∗ ] be the canonical divisor of V ∗ and let K = π∗ K[V ∗ ]. We note that K is the unique extension of the canonical divisor on V  and hence is independent of the choice of V ∗ . We call it the canonical divisor of V . One may ask if K satisfies the adjunction formula. The answer is NO as we show by a simple example: Suppose that ϒ = µ is irreducible and µ2 = −n. This happens for example in the case of the blow-up at the origin of the cone over the curve x n + y n = zn . Then µ
(µ + K[V ∗ ]) = 2g(µ) − 2 and therefore µ
K[V ∗ ] = 2g(µ) − 2 + n. Let C be  + 1 µ. We a nonsingular curve passing simply through the point π(µ) then C ∗ = C n find that C
(C + K) = C ∗
(C ∗ + K[V ∗ ])



 + 1 µ + K[V ∗ ] + 1µ
C = C n n 
K[V ∗ ] + 1 µ
K[V ∗ ] 2 + 1 + C =C n n   1 
K[V ∗ ] + 2 + C µ2 + µ
K[V ∗ ] + n + 1 =C n 2g(µ) +n−1  −2+ = 2g(C) n 2g(µ) + n − 1 . = 2g(C) − 2 + n Since the term (2g(µ) + n − 1)/n is not zero in general, we see that adjunction does not hold in the same way.

157

Hilbert modular varieties of low dimension

Proposition 7.3.1. Define a vector κ ϒ as κ ϒ = −k t (2g(µ1 ) − 2 − µ21 , . . . , 2g(µs ) − 2 − µ2s ) = −k t (µ1
K[V ∗ ], . . . , µs
K[V ∗ ]) = −k t K[V ∗ ]ϒ . Then K[V ∗ ] = K ∗ +



κi µi ,

(7.5)

i

and C
(C + K) = 2g(C) − 2 + C ϒ k t (C ϒ + K[V ∗ ]ϒ ). Proof. Write K[V ∗ ] = K ∗ +



i κi µi ,

(7.6)

where the κi need to be calculated. We have

2g(µi ) − 2 − µ2i = K[V ∗ ]
µi  = K ∗
µi + κj µj
µi =



j

κj µj
µi .

j

We conclude that t (2g(µ1 ) − 2 − µ21 , . . . , 2g(µs ) − 2 − µ2s ) = d t (κ1 , . . . , κs ).  + Write C ∗ = C γi (C)µi and use i


(C ∗ + K ∗ ). C
(C + K) = C ∗
(C ∗ + K ∗ ) = C We get,

    
C + γi (C)µi + K[V ∗ ] − κi µi C
(C + K) = C 2 + =C



i

i


µi + C 
K[V ∗ ] − γi (C)C



i

(7.7)


µi κi C

i

2 + C ϒ k t C ϒ + C 
K[V ∗ ] − C ϒ t κ ϒ =C = 2g(C) − 2 + C ϒ k t C ϒ − C ϒ t κ ϒ = 2g(C) − 2 + C

ϒ

t

ϒ

∗ ϒ

k (C + K[V ] ).

(7.8) 

Remark 7.3.2. Observe that if C passes through none of the singular points then adjunction holds in the usual sense.

158

Fabrizio Andreatta and Eyal Z. Goren

8 Hilbert modular surfaces Let L be a real quadratic field. We let M = M(µN ) be the moduli space with µN level structure, where N ≥ 4, (N, p) = 1.

8.1 The inert case 8.1.1 Calculation of some intersection numbers. Assume p > 2 in this section. To conform with the notation in § 7 we let V be the Satake compactification of M, V ∗ be a smooth toroidal compactification of V , π : V ∗ −→ V be the projection, V  be the complement in M of the singular locus of V . We also let Di = W({i+1}) . Let C(N) be the degree of M over the coarse moduli space of abelian surfaces with RM and no level structure. Let η = 21 ζL (−1)C(N). We know [BG] that each Di is a disjoint union of η nonsingular rational curves, that D1 and D2 intersect transversely, the set of intersection points is the set of superspecial points, and that D1
D2 = η(p2 + 1).

(8.1) p−1

p−1

Let h be the total Hasse invariant [Go1, Thm. 2.1]. It is a section of L1 ⊗ L2 Over V  the Kodaira–Spencer isomorphism gives that det 1V /k ∼ = L21 ⊗ L22 , thus K∼

2 2 (h) = (D1 + D2 ), p−1 p−1

.

(8.2)

hence this also holds over V (since V is normal and V − V  is of codimension 2). p ∼  Note also that over V  we have Li L−1 i+1 = OV (Di ), as follows from the properties p of the partial Hasse invariants [Go1]. Since Di is closed in V  we conclude that Li L−1 i+1 extends to V and therefore we may define unique classes i ∈ CH (V ) ⊗ Q so that p

c1 (Li L−1 i+1 ) = pi − i+1 ,

i = 1, 2.

Now, D1
K =



C
K

C∈D1

= −2η − = −2η



C∈D1 − D12

C2

(adjunction, each C ∼ = P1 ) (D1 is a disjoint union of its components).

159

Hilbert modular varieties of low dimension

On the other hand, 2 D1
(D1 + D2 ) p−1 2 2 D12 + η(p 2 + 1) = p−1 p−1

D1
K =

(Equation (8.2)) (Equation (8.1)).

This yields D12 = −2pη,

D22 = −2pη.

Solving for 1 , 2 , one finds 21 = 0,

22 = 0,

1 2 = η.

(8.3)

8.1.2 On ampleness. The sections of the line bundle La11 La22 are Hilbert modular forms of weight (a1 , a2 ). This motivates our interest in its ampleness. Theorem 8.1.1. The class a1 1 + a2 2 is ample if and only if pa1 > a2 >

1 p a1 .

Proof. We prove the claim by using the Nakai–Moishezon criterion [Kle, III.1, Thm. 1], cf. [Har, App. A, Thm. 5.1]. Though, strictly speaking, this criterion uses Snapper– Kleinman’s intersection theory, we can use the Reeve–Tyrrell intersection theory, since the theories agrees when both are defined [Arc, Thm. 2.5.15]. We first make some preliminary calculations. Let C be a component of D1 . We have C 2 = −2 − C
K by adjunction. On the 2 2 C
(D1 + D2 ) = p−1 (C 2 + p 2 + 1), where we have used other hand, C
K = p−1 that D1 is a disjoint union of its components, one of which is C, and that C
D2 is the set of superspecial points on C, which has cardinality p 2 + 1 [BG, Thm. 6.1]. 2 (C 2 + p 2 + 1), which gives Therefore, C 2 = −2 − p−1 C 2 = −2p.

(8.4)

We conclude that C
D1 = C 2 = −2p and C
D2 = p2 + 1. Using that D1 = p1 − 2 , D2 = p2 − 1 , we solve for 1 , 2 and get C
1 = −1,

C
2 = p.

(8.5)

We conclude that if C
(a1 1 + a2 2 ) > 0 then pa2 > a1 . By symmetry, if C is a component of D2 such that C
(a1 1 + a2 2 ) > 0 then pa1 > a2 . Applying the Nakai–Moishezon criterion to the class a1 1 + a2 2 , we conclude that if a1 1 + a2 2 is ample then pa1 > a2 > p1 a1 . We now claim that the converse also holds. It is enough to prove that for every irreducible curve C we have C
(a1 1 + a2 2 ) > 0. If C is contained in D1 ∪ D2 then this follows from our calculations above. Else, write a1 1 + a2 2 = b1 D1 + b2 D2 . One checks that b1 , b2 are both positive. Since C is generically ordinary, it intersect the non-ordinary locus D1 ∪ D2 by the  “Raynaud trick” [Oo3, §4], hence has positive intersection with b1 D1 + b2 D2 .

160

Fabrizio Andreatta and Eyal Z. Goren

8.2 The split case To conform with the notation of § 7, we let V be the Satake compactification of M, the moduli space with µN -level structure, V ∗ be a smooth toroidal compactification of M, π : V ∗ → V be the projection and V  be the complement in M of the singular locus of V . One knows that the non-ordinary locus consists of two divisors D1 = W({1},∅) and D2 = W(∅,{1}) that intersect transversely; the intersection being the set of superspecial points. We also know that each Di is a disjoint union of non-singular curves. See [BG, Thm. 6.1]. However, we have very little information on the components of the Di . They are not Moret–Bailly families and one can show that they are not Shimura curves. Here by a “Shimura curve” we mean the following. Let B/Q be a quaternion algebra split at infinity. Fix a maximal order OB of B and a positive involution ∗ of B fixing OB . There is a moduli space for special polarized abelian surfaces with multiplication by OB (such that ∗ is the Rosati involution) [Dri, §4 Dfn. and Prop. 4.4]. It is easy to see that every abelian surface A with multiplication by OB over a field k is either simple or isogenous to E 2 where E is an elliptic curve. In particular, if char(k) = p > 0 then A is either ordinary or supersingular. Assume now that OL ⊂ OB and that ∗ preserves OL , then we get a forgetful morphism to the Hilbert moduli space M. We call the images of such curves, and their images under Hecke correspondences, Shimura curves. In the following, we obtain some information on the field of definition and genus of the components of the divisors D1 , D2 . 8.2.1 Fields of definition. We examine the field of definition of the superspecial points and the non-ordinary locus, under some restriction on N and p. The following lemma holds for any totally real field L of degree g > 1 and for any prime p. Lemma 8.2.1. Let N ≥ 3 be an integer such that N|(p − 1) or N|(p + 1). Every superspecial point on the moduli space M of RM abelian varieties with µN -level structure can be defined over Fp2 . Proof. We use Honda-Tate theory for which [Wat] is a good reference. Consider the Weil numbers ±p over Fp2 . There exist elliptic curves E± over Fp2 with that Weil number. The endomorphism ring of E± after tensoring with Q is “the” quaternion algebra Bp,∞ over Q ramified at p and ∞. However, one easily sees that if f ∈ EndFp (E± ) and mf ∈ EndFp2 (E± ), for some non-zero integer m, then f ∈ EndFp2 (E± ). It follows that EndFp2 (E) is a maximal order in Bp,∞ . The Frobenius endomorphism π := Fr p2 : E −→ E is equal to ±p. It follows that E± [N ] ⊆ E± (Fp2 ) iff N|(π − 1) in End(E± ). But π = ±p as an endomorphism and we conclude that E± [N ] ⊆ E± (Fp2 ) iff N|(±p − 1) as integers.

Hilbert modular varieties of low dimension

161

g

Note that End(E± ) = Mg (End(E± )) is defined over Fp2 . It follows that any OL g g structure on E± is defined over Fp2 . Note also that E± has an obvious polarization defined over Fp2 induced from the canonical identification of E with its dual, and hence (using that polarization to identify the polarizations with the symmetric positive g g elements of End(E± )) every polarization of End(E± ) is defined over Fp2 . To conclude the proof, we notice that by a theorem of P. Deligne [Shi, Thm. 3.5] every superspecial abelian variety of dimension g > 1 is isomorphic over Fp to E g and, under our assumptions, µN ∼  = Z/N Z as group schemes over Fp2 . Corollary 8.2.2. Every component of Di is defined over Fp2 . Proof. It is enough to show that if C is a component of Di then σ (C) = C if σ ∈ Gal(Fp2 /Fp2 ). We first note that Di is defined over Fp . Let x ∈ C be a superspecial point (such exists, because Di \ W(1,1) is quasi-affine by applying [Oo3, Thm. 6.5], but see also below). It is a Fp2 rational point of V and hence σ (C) is also a component of Di passing through x. However, there is a unique such component passing through x. We conclude that σ (C) = C.  8.2.2 Calculation of intersection numbers. We shall make the following assumption regarding continuity of intersection numbers (cf. Equation (8.3), Remark 8.3.3). Assumption:

21 = 0,

22 = 0,

1 2 = η.

It follows that D12 = 0,

D22 = 0,

D1
D2 = (p − 1)2 η.

(8.6)

Therefore, 0 = D12  = C2 C∈D1

=



(2g(C) − 2 − C
K)

C∈D1

=



(2g(C) − 2) − D1
K

C∈D1

=



(2g(C) − 2) − (p − 1)1
2(1 + 2 )

C∈D1

=



C∈D1

(2g(C) − 2) − 2(p − 1)η.

(8.7)

162

Fabrizio Andreatta and Eyal Z. Goren

That is, (p − 1)η =



(g(C) − 1).

(8.8)

C∈D1

This already shows that on average the genus of components of C should be greater than 1. We can do slightly better. Assume that N ≥ 3 and either N|(p − 1) or N|(p + 1). ri be the number of Let {C1 , . . . , C } be the irreducible components of D1 . Let  superspecial points on Ci . Let gi be the genus of Ci , and G = i=1 gi . Then R :=  2 i=1 ri = (p − 1) η and, together with Equation (8.8), we get,   R = (p − 1) (gi − 1) = (p − 1)(G − ).

(8.9)

i=1

We have the estimate ri > 0 (because Di \ W(1,1) is quasi-affine), but since ri =
p−1  deg L2 |Ci (existence of partial Hasse invariants and simplicity of their zeros [Go1, Thm. 2.1]) we actually have ri ≥ p − 1. Summing over the components, we get R ≥ (p − 1).

(8.10)

We obtain the following: Proposition 8.2.3. Assume that N ≥ 3 and N |(p −1) or N |(p +1). Then the average genus g of the non-ordinary locus satisfies the inequality g = G/ ≥ 2. Proposition 8.2.4. The line bundle Ln1 1 Ln2 2 is ample if and only if both n1 and n2 are positive. The proof is along the same lines as the proof of Theorem 8.1.1.

8.3 The ramified case Again, to conform with the notation of § 7, we let V be the Satake compactification of M, the moduli space with µN -level structure, V ∗ be a smooth toroidal compactification of N (sic!), π : V ∗ → V the projection. Let V  = V \ V sing . For every S ⊂ W(1,1) , let µS = π −1 (S). 8.3.1 The local structure of the moduli space. First we compute the local deformation theory at a point of M. It follows from Example 4.3.1 that the moduli space is regular at points of type (0, n), 0 ≤ n ≤ 2. By loc. cit., at a point of type (0, 2), the universal deformation ring is k[[c0 , c1 ]]. Recall Remark 6.1.4. We may take m = ∞ and c3 = 1 as in (6.3) so that the universal Frobenius is

0 T2 . F = 1 −c0σ − c1σ T

163

Hilbert modular varieties of low dimension

A deformation has type (0, 1) if and only if it is not ordinary. This is equivalent to T F 2 ≡ 0 (mod T 2 ). Equivalently,



0 0 0 0 = 0 mod T . 2 1 −c0σ 1 −c0σ This gives the condition c0 = 0. We conclude that in the local deformation space the condition for deforming into W(0,1) is given by c0 = 0 and it defines a smooth formal curve. By Example 4.3.2, at a point of type (1, 1) the universal deformation ring R is defined by k[[a0 , b0 , c0 , d0 ]]/(a0 + d0 , a0 d0 − b0 c0 ) ∼ = k[[a0 , b0 , c0 ]]/(a02 + b0 c0 ). Hence, Spec(R) is a cone. By (6.3) we may take m = ∞ and c3 = 1 so that the universal Frobenius is given by

T + a0σ −b0σ . F = T − a0σ −c0σ In order to have deformation of (0, 1) we must have T F 2 = 0 (mod T 2 ), which is equivalent to σ

 σ2 σ σ2 a a −b −b 0 0 0 0 = 0. F2 = 2 2 −a0σ −c0σ −a0σ −c0σ This gives the system of equations modulo p: p+1

b0

p+1

− a0

= 0,

p

p

a0 b0 + c0 a0 = 0,

p

p

b0 a0 + a0 c0 = 0,

p+1

−a0

p+1

+ c0

= 0.

If b0 = 0 it follows that a0 and c0 = 0 are nilpotent. The associated reduced scheme is the point we started with. Inverting b0 , the second equation can be eliminated using p p p+1 p p+1 p+1 b0 (a0 b0 + c0 a0 ) = a0 b0 − a02 a0 = a0 (b0 − a0 ). If a0 = 0, the associated p reduced scheme is the point we started with. Inverting a0 we deduce from a0 (b0 a0 + p p+1 p+1 p p a0 c0 ) = b0 (a0 − c0 ) and from the other relations that b0 a0 + a0 c0 = 0. Hence, on the complement of the point we are reduced to the equations a02 + b0 c0 = 0,

p+1

b0

p+1

− a0

= 0,

p+1

−a0

p+1

+ c0

= 0.

(8.11)

We conclude that the non-ordinary locus consists of p + 1 branches given by b0 = ζ a0 and c0 = ζ −1 a0 for ζ a p + 1-st root of unity. Finally, we compute the structure of π1 : N → M. The morphism π1 is proper
[AG1, Lem. 8.4]. Outside π1−1 W(1,1) it is one-to-one [AG1, Prop. 6.5] and so is
 an isomorphism. Since M\W(1,1) is smooth, we conclude that π1−1 M\W(1,1) is smooth. Let s ∈ W(1,1) . Let R := k[[a0 , b0 , c0 ]]/(a02 + b0 c0 ) be the completed local ring of M at s. Let A → Spec(R) be the universal abelian scheme over R. Using the theory of local models §4.3.2, we can find a R ⊗k k[T ]/(T 2 )-basis α, β 1 (A/R) such that the relative cotangent space H 0 (A,  1 of HdR A/R ) in HdR (A/R) is

164

Fabrizio Andreatta and Eyal Z. Goren

generated as R ⊗Z k[T ]/(T 2 )-module by (T + a0 )α + b0 β and c0 α + (T − a0 )β. The scheme N ×M Spec(R) can be interpreted as representing the Grassmannian of R ⊗k k[T ]/(T 2 ) rank 1 submodules of H 0 (A, A/R ), free as R-modules and killed by T . Any such module is generated by an element T Xα + T Zβ which is zero 1 (A/R)/H 0 (A,  in HdR A/R ). Hence, N ×M Spec(R) ∼ = Proj R[X, Z]/(a0 X + c0 Z, −b0 X + a0 Z),

(8.12)

Proposition 8.3.1. The following hold: 1. the singular points of M are the cusps and the points contained in W(1,1) ; 2. the variety N is smooth over k; 3. π : N → M is the blow-up along W(1,1) ; 4. for every s ∈ W(1,1) , the scheme µs is a non-singular rational curve with self intersection −2. Proof. The first assertion is a summary of part of the discussion above. Next, it follows from (8.12) that N is a smooth variety. o be the blow-up of M at W(1,1) . Since W(1,1) is reduced, we also get that the Let V inverse image of W(1,1) is a disjoint union of curves and, hence, is a divisor. By the o compatible with universal property of blow-up we get a birational map ρ : N → V the projections onto M. It is an isomorphism over M\W(1,1) . The completed local ring of M at a point of W(1,1) is isomorphic to R = k[[a0 , b0 , c0 ]]/(a02 + b0 c0 ). Since the blow-up is defined in terms of Proj of the ideal defining W(1,1) and W(1,1) is reduced, o ×M Spec(R) coincides with the blow-up of Spec(R) at its closed the fibre product V o ×M Spec(R) is point. In particular, the inverse image of the closed point of R in V 1 isomorphic to Pk and has self intersection −2. Using (8.12) one easily checks that the base change of ρ to the product of the completed local rings at the points of W(1,1) is an isomorphism. Hence, ρ is an isomorphism.  8.3.2 Calculation of some intersection numbers. Assume that p > 2 in this section. Let D be the reduced divisor that is equal to the non-ordinary locus of V .√Let h be the total Hasse invariant, h ∈ (V  , det√Ep−1 ); it admits a square root h ∈ (V  , det E(p−1)/2 ) – see [AG2]. We have ( h) = D. It follows from the Kodaira– Spencer isomorphism that (initially on V  , but then on V ) K∼

4 D. p−1

(8.13)

We know [BG, Thm. 5.3] that the number of components of D is η = 21 ζL (−1)C(N ), where C(N ) is the degree of the level structure, and that the number of points of W(1,1) is also η. We also note that Proposition 8.3.1 implies that the variety V ∗ is suitable

165

Hilbert modular varieties of low dimension

for calculating the intersections of divisors support on D. The following calculations are done using the results and notations of § 7. On the one hand, D 2 = (D ∗ )2  2  + p + 1 µW = C (1,1) 2 C∈D

=



C∈D

=



2   (p + 1)2 2  + (p + 1) 
µW µW(1,1) C C + (1,1) 4 C∈D

2 + (p + 1) C

C∈D

=







u∈W(1,1) C∈D

2 + (p + 1)2 η + (p C

+ 1)2 4

C∈D

=



C∈D

 u∈W(1,1)

µ2u

(−2)η

2 2 + (p + 1) η. C 2

On the other hand,   2 = 
K[V ∗ ]) C (−2 − C C∈D

2 
µu + (p + 1) C 4

(adjunction on V ∗ )

C∈D

=




K ∗) (−2 − C

C∈D

= −2η −



(Prop. 7.3.1 + Prop. 8.3.1)

C∗
K ∗

(Prop. 7.2.1)

C
K

(Prop. 7.2.1)

C∈D

= −2η −



C∈D

= −2η −



C∈D

= −2η −

C


4 D p−1

(Equation (8.13))

4 D2 . p−1

We conclude that D 2 = −2η −

4 2 p−1 D

+

(p+1)2 2 η,

which gives:

Proposition 8.3.2. The self intersection of D is given by D2 =

(p − 1)2 η. 2

Remark 8.3.3. Note that if we could argue by ‘continuity of intersection numbers’, (p−1)2 (p−1)2 2 we could write D = p−1 2 (1 + 2 ), whence D = 2 1
2 = 2 η.

166

Fabrizio Andreatta and Eyal Z. Goren

9 Hilbert modular threefolds Let L be a totally real cubic field. In this section we study the local structure of the moduli variety M. Given the results for g = 2 and the unramified case, we may restrict our attention to the case when p = p3 is maximally ramified. Assume that henceforth. We recall from § 3.2 the strata and their hierarchy in terms of “being in the closure” as encoded in the following diagram

g=3

(1, 2)I (0, 3) III (1, 1)I (0, 2) III (0, 1) (0, 0)

c for j = 0 To begin with, it follows from Example 4.3.1 that the locus W(j,n) and n = 0, . . . , 3, or for j = 1 and n = 1, 2 (performing a similar computation), is formally smooth at points of type (j, n ) with n ≥ n. Thus, we are interested in the c c structure of the strata W(0,1) at a point of type (1, 1) and (1, 2), and W(0,2) at a point of type (1, 2).

9.1 Points of type (1, 2) In this case j = 1, i = 2, and, since the point is superspecial, we may assume m = ∞ and c3 = 1 in Equation (6.3). The universal deformation space is of the form (cf. Example 4.3.3): k[[a0 , a1 , b0 , c0 , c1 , d0 ]]/(a0 d0 − b0 c0 , a0 + a1 d0 − b0 c1 , a1 + d0 ) ∼ = k[[a0 , b0 , c0 , c1 , d0 ]]/(a0 d0 − b0 c0 , a0 − d 2 − b0 c1 ). 0

The results of §5.6 imply that the universal “mod p” Frobenius is given over this ring by

T 2 + a0σ − d0σ T −b0σ . (9.1) F = T + d0σ −c0σ − c1σ T c . By Corollary 5.6.3, the condition that the 9.1.1 The non-ordinary locus W(0,1) deformation is non-ordinary is equivalent to the condition

 σ2 σ 2 a0σ a0σ −b0 −b0 ≡ 0 (mod T ). 2 2 d0σ −c0σ d0σ −c0σ

167

Hilbert modular varieties of low dimension

This gives the following system of equations: p+1

p

+ a0 d0 = 0

(Eq1)

b0

(Eq2)

b 0 a 0 + a0 c 0 = 0

(Eq3)

d0 b0 + c0 d0 = 0

(Eq4)

d0 a0 + c0

(Eq5)

a0 d0 − b0 c0 = 0

(Eq6)

a0 − d02 − b0 c1 = 0.

p

p

p

p

p

p+1

=0

We note that if any of the variables a0 , b0 , c0 , or d0 is zero then so is a power of all the others. In this case, the associated reduced subscheme defines a smooth 1-dimensional deformation which coincides with the j = 1 locus, generically having invariants (1, 1). Else, to find the components of the non-ordinary locus, we may invert a0 , b0 , c0 , and d0 . Using (Eq5) one checks that b0 · (Eq4) = d0 · (Eq2),

b0 · (Eq3) = d0 · (Eq1),

p

p

b0 · (Eq2) = a0 · (Eq1).

Thus, we may consider only the three equations (Eq1), (Eq5), (Eq6). Substituting using a0 = b0 c1 + d02 we reduce to the equations p+1

p

p+2

+ b0 c1 d0 + d0

b0

= 0,

b0 c1 d0 + d03 − b0 c0 = 0

in the ring k[[b0 , c0 , c1 , d0 ]][b0−1 , c0−1 , c1−1 , d0−1 ]. Multiply the second equation by p−1 d0 and subtract from the first equation to reduce to the equations p

p−1

b0 + c0 d0

= 0,

d03 − b0 c0 + b0 c1 d0 = 0.

In order to compute the components of the non-ordinary locus through the given point, one proceeds as in the proof of 6.3.4 and computes the minimal prime ideals of k[[b0 , c0 , c1 , d0 ]][b0−1 , c0−1 , c1−1 , d0−1 ] associated to the ideal defined by the equations p

p−1

b0 + c0 d0

= 0,

2p+1

d0

p+1

+ c0

p p

− c0 c1 d0 = 0.

As in loc. cit., one concludes that those prime ideals are in one to one correspondence with the minimal prime ideals associated to the ideal (0) in the ring R1 := p+1 p p 2p+1 k[[c1 , d0 ]][c0 ]/(c0 − c0 c1 d0 + d0 ) not containing d0 . Since the polynomial p+1 2p+1 in the variable c0 is irreducible over k[[d0 ]], one concludes that R1 is a c 0 + d0 domain. We conclude that the non-ordinary locus is locally irreducible at points of type (1, 2). One can also calculate that the tangent space at a point of type (1, 2) to the deformation space into non-ordinary abelian varieties (given by (Eq1)–(Eq6)) is three c . dimensional and conclude that every point of type (1, 2) is a singular point of W(0,1)

168

Fabrizio Andreatta and Eyal Z. Goren

c 9.1.2 The locus W(0,2) . We next consider the problem of deforming a point of type (1, 2) into the (0, 2) locus. The condition that the a-number is at least 2 is equivalent to the condition T F 2 ≡ 0 (mod T 3 ), where F is given by



−b0σ F = T + d0σ

T 2 + a0σ − d0σ T −c0σ − c1σ T

.

This is equivalent to the following matrix being congruent to 0 modulo T 2 :

−b0σ T + d0σ

a0σ − d0σ T −c0σ − c1σ T



−b0σ 2 T + d0σ

a0σ − d0σ T 2 2 −c0σ − c1σ T

2

2

2

.

This provides the following equations: (Eq1)

a0 d0 − b0 c0 = 0

(Eq2)

a0 − d02 − b0 c1 = 0

(Eq3)

b0

(Eq4)

a 0 − d0

p+1

p

+ a0 d0 = 0 p+1

=0

(Eq8)

p p d0 b0 + c0 d0 = 0 p p b0 + c0 + c1 d0 = 0 p p b0 a0 + a0 c0 = 0 p p p b0 d0 − a0 c1 + d0 c0

(Eq9)

d 0 a0 + c 0

(Eq10)

a 0 − d0

(Eq5) (Eq6) (Eq7)

p

p

p+1

p+1

p+1

We now substitute using (Eq4) a0 = d0 variables b0 , c0 , c1 , and d0 :

=0

=0 p

and obtain the following equations in the

p+2

− b 0 c0 = 0

p+1

− d02 − b0 c1 = 0

p+1

+ d0

(Eq1)

d0

(Eq2)

d0

(Eq3)

b0

(Eq5)

d0 b0 + c0 d0 = 0

p

p

+ c1 c0 + c0 c1 = 0.

2p+1 p

=0

169

Hilbert modular varieties of low dimension p

p

(Eq6)

b0 + c0 + c1 d0 = 0

(Eq7)

b0 d0

(Eq8)

b0 d0 − d0

(Eq9)

d0

(Eq10)

d0

p 2 +p

+ d0

p

p+1 p c1

p 2 +p+1 p 2 +p

p+1 p c0

p+1

− d0

p

+ d0 c0 = 0

p+1

+ c0

=0

=0 p

p

+ c1 c0 + c0 c1 = 0.

We distinguish two cases: Case 1: d0 = 0. This implies that a power of b0 and of c0 is zero. The associated reduced subscheme is the smooth curve given by c1 , which is the (1, 1) curve already noticed above. Case 2: we invert d0 . We now multiply each equation by a suitable power of d0 so p p that to substitute expressions of the form c0 d0 by −b0 d0 (using (Eq5)). We remark that the elimination of c0 was justified by (Eq6). We arrive at the following system of equations in b0 , c1 , and d0 : (Eq1 )

d0

(Eq2 )

p+1 d0

(Eq6 )

b0 d0

(Eq7 )

2p+1

p+1

+ b0

− d02 − b0 c1 = 0

p p−1

p 2 −1

− b0 d0

(Eq9 )

d0

2p 2 +p

d0

2p−1

=0

p2

b0 d0

(Eq8 )

(Eq10 )

p

− b0 + c1 d0

2p 2 −p−1

2p 2

=0

− b0 = 0 p2 p

p2

+ d 0 c1 + b 0 = 0 p 2 +p

− b0

p 2 +1

− d0

=0 p2

p 2 −2p+1 p p b0 c1

− b0 c1 − d0

= 0.

Note that (Eq1 ) implies that b0  = 0 and implies (Eq7 ) and (Eq9 ). We may therefore consider only the system (Eq1 )

d0

(Eq2 )

d0

(Eq6 )

b0 d0

2p+1 p+1

p+1

+ b0

− d02 − b0 c1 = 0

p p−1

p

(Eq8 )

− b0 d0

(Eq10 )

d0

2p−1

− b0 + c1 d0

p 2 −1

2p 2

=0

p

=0 p2

+ d 0 c1 + b 0 = 0

p 2 +1

− d0

p2

p2

p 2 −2p+1 p p b0 c1

− b0 c1 − d0

= 0.

170

Fabrizio Andreatta and Eyal Z. Goren

We now show that (Eq1 ) and (Eq2 ) imply (Eq6 ) and (Eq8 ), (Eq10 ). Indeed, multiplying (Eq6 ) by b0 we get b0 (Eq6 ) = d0

2p+1

p+1

+ (−d02 + d0

2p−1

)d0

2p+1 p−1 d0

− d0

= 0.

p

Multiplying (Eq8 ) by b0 , we get p

2p+1 p 2 −1 d0

b0 (Eq8 ) = d0

p+1 p p 2 ) d0

+ (−d02 + d0

2p+1 p

+ (−d0

) = 0.

Finally, 2p 2

p 2 +1

(Eq10 ) = d0 −d0

2p+1 p−1

−(−d0

)

p+1

(−d02 +d0

p 2 −2p+1

)−d0

p+1 p

(−d02 +d0

) = 0.

Hence, we are left with the system of equations (Eq1 )

d0

(Eq2 )

d0

2p+1 p+1

p+1

+ b0

=0

− d02 − b0 c1 = 0.

Recall that these equations are taken in a ring where d0 is invertible, viz. in the ring k[[b0 , c1 , d0 ]][d0−1 ]. If I is the ideal generated by the equations (Eq1 ), (Eq2 ) then the ring k[[b0 , c1 , d0 ]][d0−1 ]/I is equal to the ring k[[b0 , c1 , d0 ]][b0−1 , d0−1 ]/(d0

2p+1

p+1

+ b0

p+1

, d0

− d02 − b0 c1 ).

p−1

p+1

Hence, we can eliminate c1 , putting c1 = d02 (d0 − 1)b0−1 (note that c1 = p−1 p+1 −d0 (d0 − 1) , justifying the substitution) and conclude that the (0, 2)-locus is given locally at a point (1, 2) by the irreducible equation 2p+1

d0

p+1

+ b0

=0

in the ring k[[b0 , d0 ]] and hence is irreducible there.

9.2 Points of type (1, 1) In this case j = n = 1 and i = 1. Hence, we may assume that c3 = 1 in (6.3). The universal deformation space of [A0 ] is defined by the ring R := k[[a0 , a1 , b0 , c0 , c1 , d0 ]]/(a0 d0 − b0 c0 , a1 d0 + a0 − b0 c1 , a1 + d0 ). The matrix M of Frobenius F of the universal display is defined by

T − b0σ + d0σ T 2 + a σ − cσ T + d0σ −cσ with a := a0 + a1 T and c := c0 + c1 T . The deformations in the non-ordinary locus, c i.e., inside W(0,1) , are defined by the condition that T 2 F 2 = 0 modulo T . This is

171

Hilbert modular varieties of low dimension

equivalent to require that M · M σ = 0 mod T , i. e., to the vanishing of  p p p p

p2 p2 p2 p2 a0 − c 0 −b0 + d0 −b0 + d0 a0 − c0 p p p2 p2 d0 −c0 d −c , 0

which is equal to  2 2 p +p

b0

p p

p2 p

p 2 +p

p p2

−b0 d0 −b0 d0 +d0 p2 p

p 2 +p

−b0 d0 +d0

p p2

+a0 d0 −c0 d0

p2 p

0

p2 p

p p2

p2 p

p p2

p2 +p

−a0 b0 +a0 d0 +b0 c0 −c0 d0 −a0 c0 +c0

p p2

p2 p

−c0 d0

p2 p



p2 +p

a0 d0 −c0 d0 +c0

.

Hence, we get the following seven equations in the variables a0 , a1 , b0 , c0 , c1 and d0 : p 2 +p

p p2

p2 p

p 2 +p

− b0 d0 − b0 d0 + d0

p p2

p p2

+ a0 d0 − c0 d0 = 0

(Eq1)

b0

(Eq2)

− b0 d0 + d0

(Eq3)

− a0 b0 + a0 d0 + b0 c0 − c0 d0 − a0 c0 + c0 p2

p2 p

p 2 +p

p2 p

p2 p

p2

p

p

p p2

− c0 d0 = 0 p p2

p 2 +p

(Eq4)

a0 d0 − c0 d0 + c0

(Eq5)

a 0 d0 − b 0 c 0 = 0

(Eq6)

a1 d0 + a0 − b0 c1 = 0

(Eq7)

a1 + d0 = 0.

p2 p

p p2

p 2 +p

=0

=0

Case 1: Assume d0 = 0. Then, a power of b0 is 0 from (Eq1), a power of c0 is 0 from (Eq4), a0 = 0 from (Eq6) and a1 = 0 from (Eq7). The only free variable left is c1 . Hence, the reduced subscheme defined by d0 = 0 is 1-dimensional and c , as already known. coincides with universal deformation space inside the locus W(1,1) Case 2: Let us invert d0 = 0. Then p

p

p2

p2

p

p2

• d0 (Eq1) = (−b0 + d0 )(Eq2) + d0 (Eq5)p ; p

• c0 (Eq2) = −d0 (Eq4) + d0 (Eq5)p ; p 2 +p

• d0

p

2

p2 p2

p p2

(Eq3 − Eq4) = −b0 d p (Eq5)p − d0 c0 (Eq5)p + b0 c0 (Eq2). 2

−p

Hence, using a1 = −d0 , a0 = d02 +b0 c1 , (Eq5) and d0 (Eq2), the system of equations (Eq1)–(Eq7) becomes equivalent to the system of equations • d03 + b0 c1 d0 − b0 c0 = 0; p

p

p−1

• −b0 + d0 − c0 d0

in k[[b0 , c0 , c1 , d0 ]][d0−1 ].

= 0;

172

Fabrizio Andreatta and Eyal Z. Goren ‘close-up’ on a (1,2) point

(0,2) locus

(1,1) locus non-ordinary locus

Figure 9.1. Hilbert modular threefold – maximally ramified case.

It follows from Lemma 6.3.5 that the nilradical of the ideal defined by these equations c is not analytically irrehas exactly two minimal prime ideals. Hence, the locus W(0,1) c ducible at the points of W(1,1) . Studying the tangent space it is easily seen that W(1,1) c is singular in W(0,1) .

9.3 Summary We now come to some conclusions concerning the global structure of moduli space M for L cubic totally real field and p maximally ramified in L. Let B be any component of M. By Proposition 6.3.3 the non-ordinary locus c = W(1,1) ∪ W(1,2) is irreducible and non-singular, is irreducible. The locus W(1,1) c by loc. cit. and (3.1). The locus W(0,2) = W(0,2) ∪ W(1,2) ∪ W(0,3) is a union of Moret–Bailly families, each component is singular only at the unique point (cf. [AG1, c are disjoint, Prop. 6.6]) of W(1,2) lying on it. The components of the locus W(0,2) because intersection points can only be of type (1, 2), and by § 9.1.2 the locus is c c locus, and the W(1,1) locus are locally irreducible there. One can prove that the W(0,1) irreducible in each component of the moduli space in a different way. In fact, a similar use of the correspondences π1 π2−1 , π2 π1−1 , shows that one is irreducible if an only c c ∪ W(0,2) is connected, if the other is. We know by Theorem 6.2.3 that T2 = W(1,1) c c we know that each component of W(0,2) meets W(1,1) at a unique point, and we know that the locus S1 = W(1,1) ∪ W(1,2) is non-singular. The implies that there is a unique component of W(1,1) in every component of M. Acknowledgments. Some ideas developed in this paper arose in discussions with E. Bachmat. The authors would like to thank the Department of Mathematics and Statistics of McGill University and the Department of Mathematics, Padova University,

Hilbert modular varieties of low dimension

173

for their generous hospitality during visits on several occasions, including the Dwork semester, during which much of this paper was written.

References [AG1]

Andreatta, F., Goren, E. Z., Geometry of Hilbert modular varieties over totally ramified primes. Internat. Math. Res. Notices 33 (2003), 1785–1835.

[AG2]

Andreatta, F., Goren, E. Z., Hilbert modular forms: mod p and p-adic aspects. Mem. Amer. Math. Soc., to appear.

[AG4]

Andreatta, F., Goren, E. Z., Displays and local models. In preparation.

[Arc]

Archibald, A., Intersection theory of surfaces. Masters thesis, McGill University, March 2003.

[AtM]

Atiyah, M. F., Macdonald, I. G., Introduction to commutative algebra. AddisonWesley Publishing Co., Reading, Mass.–London–Don Mills, Ont. 1969.

[BG]

Bachmat, E., Goren, E. Z., On the non ordinary locus in Hilbert-Blumenthal surfaces. Math. Ann. 313 (1999) 3, 475–506.

[Bou]

Boutot, J.-F., Schéma de Picard local. Lecture Notes in Math. 632, Springer-Verlag, Berlin 1978.

[DP]

Deligne, P., Pappas, G., Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant. Compositio Math. 90 (1) (1994), 59–79.

[Dri]

Drinfel’d, V. G., Coverings of p-adic symmetric regions. Funct. Anal. Appl. 10 (2) (1976), 107–115.

[Eis]

Eisenbud, David, Commutative algebra. With a view toward algebraic geometry. Grad. Texts in Math. 150, Springer-Verlag, New York 1995.

[FC]

Faltings, G., Chai, C.-L., Degenerations of abelian varieties. Ergeb. Math. Grenzgeb. (3) 22, Springer-Verlag, 1990.

[Fre]

Freitag, E., Hilbert modular forms. Springer-Verlag, Berlin 1990.

[Go1]

Goren, E. Z., Hasse invariants for Hilbert modular varieties. Israel J. Math. 122 (2001), 157–174.

[Go2]

Goren, E. Z., Hilbert modular forms modulo pm : the unramified case. J. Number Theory 90 (2001), no. 2, 341–375.

[GO]

Goren, E. Z., Oort, F., Stratifications of Hilbert modular varieties. J. Algebraic Geom. 9 (2000), 111–154.

[Gro]

Grothendieck, A., Groupes de Barsotti-Tate et cristaux de Dieudonné. Séminaire de Mathématiques Supérieures 45 (Été, 1970). Les Presses de l’Université de Montréal, Montreal, Que., 1974.

[EGA IV] Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math. 28 (1966), 32 (1967).

174

Fabrizio Andreatta and Eyal Z. Goren

[SGA 2]

Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2). Augmenté d’un exposé par Michèle Raynaud. Séminaire de Géométrie Algébrique du Bois-Marie, 1962, Adv. Stud. Pure Math. 2, North-Holland Publishing Co., Amsterdam, Masson & Cie, Éditeur, Paris, 1968.

[Har]

Hartshorne, R., Algebraic geometry. Grad. Texts in Math. 52, Springer-Verlag, 1977.

[deJ]

de Jong, A. J., The moduli spaces of principally polarized abelian varieties with 0 (p)-level structure. J. Algebraic Geom. 2 (4) (1993), 667–688.

[Kle]

Kleiman, S. L., Toward a numerical theory of ampleness. Ann. of Math. (2) 84 (1966), 293–344.

[Lip1]

Lipman, J., Unique factorization in complete local rings. In Algebraic geometry (Humboldt State Univ., Arcata, Calif., 1974), Proc. Sympos. Pure Math. 29, Amer. Math. Soc., Providence, R.I., 1975, 531–546.

[Lip2]

Lipman, J., Rational singularities, with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279.

[MM]

Mazur, B., Messing, W., Universal extensions and one dimensional crystalline cohomology. Lecture Notes in Math. 370, Springer-Verlag, Berlin-New York 1974.

[Mum]

Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes Études Sci. Publ. Math. 9 (1961), 5–22.

[Oo1]

Oort, F., Subvarieties of moduli spaces. Invent. Math. 24 (1974), 95–119.

[Oo2]

Oort, F., Which abelian surfaces are products of elliptic curves? Math. Ann. 214 (1975), 35–47.

[Oo3]

Oort, F., A stratification of a moduli space of polarized abelian varieties. In Moduli of Abelian Varieties (C. Faber, G. van der Geer and F. Oort, eds.), Progr. Math. 195, Birkhäuser, 2001, 345–416.

[Oo4]

Oort, F., Foliations in moduli spaces of abelian varieties. J. Amer. Math. Soc. 17 (2004), 267–296.

[Rap]

Rapoport, M., Compactifications de l’espace de modules de Hilbert-Blumenthal. Compositio Math. 36 (1978), no. 3, 255–335.

[RZ]

Rapoport, M., Zink, Th., Period spaces for p-divisible groups. Ann. of Math. Stud. 141, Princeton University Press, Princeton, NJ, 1996.

[RT1]

Reeve, J. E., Tyrrell, J. A., Intersection theory on a singular algebraic surface. Proc. London Math. Soc. (3) 12 (1962), 29–54.

[RT2]

Reeve, J. E., Tyrrell, J. A., Intersection theory on a singular algebraic surface— Addendum. J. London Math. Soc. 41 (1966), 619–626.

[Sam]

Samuel, P., On unique factorization domains. Illinois J. Math. 5 (1961), 1–17.

[Shi]

Shioda, T., Supersingular K3 surfaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math. 732, SpringerVerlag, Berlin 1979, 564–591.

[Wat]

Waterhouse, W. C., Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 (1969), 521–560.

Hilbert modular varieties of low dimension [Zin]

175

Zink, T., The display of a formal p-divisible group, Cohomologies p-adiques et applications arithmétiques, I. Astérisque 278 (2002), 127–248.

Fabrizio Andreatta, Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, via Belzoni 7, 35131, Padova (PD), Italy E-mail: [email protected] Eyal Z. Goren, Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montreal H3A 2K6, QC, Canada E-mail: [email protected]

On Dwork cohomology for singular hypersurfaces Francesco Baldassarri and Pierre Berthelot ∗

Abstract. Let Z be a projective hypersurface over a finite field. With no smoothness assumption, we relate the p-adic cohomology spaces constructed by Dwork in his study of the zeta function of Z (cf. [29], [30], [31]), to the rigid homology spaces of Z. The key result is a general † theorem based on the Fourier transform for DX, Q -modules [40], which extends to the rigid context results proved in the algebraic one by Adolphson and Sperber [3], and Dimca, Maaref, Sabbah and Saito [27]. If V, V are dual vector bundles over a smooth p-adic formal scheme X, u : X → V a section, Z the zero locus of its reduction mod p, this theorem gives an identification between the overconvergent local cohomology of OX, Q with supports in Z and the relative rigid cohomology of V with coefficients in the Dwork isocrystal associated to u. Thanks to this result, we also give an interpretation of a canonical filtration on the Dwork complexes in terms of the rigid homology spaces of the intersections of Z with intersections of coordinate hyperplanes. 2000 Mathematics Subject Classification: 13N10, 14F30, 14F40, 14G10, 14J70, 16S32, 32C38

Contents Introduction

178

1

180

Specialization and cospecialization in rigid cohomology

2 The overconvergent Fourier transform

191

3 Applications to rigid cohomology

208

4 The algebraic and analytic Dwork complexes

214

5 The coordinate filtration

228

∗ This work has been supported by the research network Arithmetic Algebraic Geometry of the European Community (Programme IHP, contract HPRN-CT-2000-00120).

178

Francesco Baldassarri and Pierre Berthelot

Appendix: Logarithmic versus rigid cohomology of an overconvergent isocrystal with logarithmic singularities

237

References

241

Introduction Following his proof of the rationality of the zeta function of an algebraic variety over a finite field [28], Dwork wrote, between 1962 and 1969, a series of papers ([29], [30], [31], [32]) in which he developed a cohomological theory in order to express the zeta function of a projective hypersurface as an alternating product of characteristic polynomials for a suitable Frobenius action, as predicted by the Weil conjectures. Since Dwork’s theory was based on the study of complexes of differential operators, it is natural to ask for the relations between his theory and other cohomological theories based on differential calculus. For non-singular hypersurfaces, this question was answered by Katz’s thesis [41], which gave interpretations of Dwork’s algebraic and analytic cohomologies in terms of algebraic de Rham cohomology and Monsky– Washnitzer cohomology. In this article, we revisit this problem and give similar relations without the nonsingularity assumption. For algebraic Dwork cohomology, the method we use here was introduced by Adolphson and Sperber in [3], where they generalize Katz’s result to the case of smooth complete intersections in a smooth affine variety. It was then generalized to the case of singular subvarieties by Dimca, Maaref, Sabbah and Saito [27] using the techniques of algebraic D-module theory. In particular, they made explicit the role played by the Fourier transform in the Adolphson–Sperber isomorphism. They also introduced and studied a vector bundle V (m) which allowed them to relate algebraic de Rham cohomology spaces with supports in a projective hypersurface Z of degree m with certain algebraic Dwork cohomology groups. † Our main observation is that, thanks to the Fourier transform for DX, Q -modules developed by Huyghe ([36], [38]), [40]), the methods of [27] can be extended to give comparison theorems between rigid cohomology groups with supports in Z and Dwork’s analytic cohomology. We also prove that the comparison isomorphisms are compatible with Frobenius actions. This allows us to give a cohomological interpretation of some formulas of Dwork relating Fredholm determinants and zeta functions [29], and, more generally, to complete Dwork’s program by proving that the constructions developed in [29] and [30] to treat the smooth case yield the expected cohomology groups in the singular case as well. We note that our methods can also be used to obtain comparison theorems between Dwork’s dual theory, used in [31] to deal with the singular case, and de Rham and rigid cohomologies with compact supports. However, in order to keep this article to a reasonable size, we do not include these results here, and we hope to develop them subsequently.

Dwork cohomology for singular hypersurfaces

179

Let us indicate now more precisely the content of the various sections. The first section is devoted to general results underlying the relation between Dwork’s algebraic and analytic theories. We explain the construction of the specialization morphisms relating algebraic de Rham cohomology and rigid cohomology, both for ordinary cohomology and cohomology with compact supports. In the case of varieties over number fields, we prove that these specialization morphisms are isomorphisms outside a finite set of primes, as in Dwork’s theory in the case of hypersurfaces in a projective space. In the second section, we recall some general definitions and results about the † Fourier transform for coherent DV, Q -modules on a p-adic formal vector bundle V, with dual V . Our main result here is theorem 2.14, which is the analogue of [27, th. 0.2] in our context. As in the algebraic case, the core of the proof is the identification of the Fourier transform of the structural sheaf (with overconvergence conditions at infinity) of V with the overconvergent local cohomology sheaf of OV with supports in the zero section. In addition, we prove that these isomorphisms are compatible with Frobenius actions. The third section gives some consequences of theorem 2.14 in rigid cohomology. The most important is theorem 3.1, which provides a canonical isomorphism, compatible with Frobenius actions, between the rigid homology of the zero locus Z of a section u of a vector bundle V , and the rigid cohomology of the dual vector bundle V with coefficients in the Dwork isocrystal Lπ,u defined by the section u and the canonical pairing V × V → A1 . We also verify that this isomorphism is compatible under the specialization morphisms with the similar isomorphism defined in [27] for algebraic de Rham cohomology. The last two sections are devoted to the actual comparison theorems with Dwork cohomology. For simplicity, we only consider here Dwork’s original theory for hypersurfaces, although the same methods could clearly be applied to give similar comparison theorems (even in the singular case) for the complexes introduced by Adolphson and Sperber to compute the zeta function of smooth complete intersections [1] (cf. also [20], [21]). Let R be the ring of integers in a finite extension K of Qp , and f ∈ R[X1 , . . . , Xn+1 ] an homogeneous polynomial of degree d, defining a projective hypersurface Z ⊂ PnR . Let Y ⊂ PnR be the complement of the coordinate hyperplanes, and Yk , Zk the special fibers of Y , Z. In section 4, we first recall the construction of the Dwork complexes associated to f , as given in [29], and of the operator α which enters in Dwork’s computation of the zeta function of Zk ∩Yk . Dwork’s algebraic complex is un+1 built from the graded K-algebra L generated by monomials Xu = X0u0 X1u1 . . . Xn+1 such that du0 = u1 + · · · + un+1 . For the analytic complex, we use, as in rigid cohomology, the point of view of Monsky and Washnitzer, and we replace Dwork’s spaces L(b), which are Banach spaces of series in the Xu satisfying appropriate growth conditions, by the union L(0+ ) of all L(b), b > 0. This does not change Dwork’s characteristic series. Let V be the vector bundle associated to the sheaf OPn (d), D ⊂ V the union of the inverse images of the coordinate hyperplanes in Pn and of the zero section of V ,

180

Francesco Baldassarri and Pierre Berthelot

VK , DK the generic fibers, Lπ,f the algebraic module with connection constructed as above using the section of V defined by f , Lπ,f the corresponding Dwork isocrystal. We show that the Dwork complexes are isomorphic to the complexes of global algebraic differential forms (resp. analytic with overconvergence at infinity) on VK , with logarithmic poles along DK , and coefficients in Lπ,f (resp. Lπ,f ). We then use [27, theorem 0.2] and theorem 3.1 to identify the cohomology of the Dwork complexes to the algebraic de Rham cohomology and to the rigid cohomology of the generic and special fibers of Y with supports in Z, in a manner which is compatible with specialization morphisms (theorem 4.6). This provides a cohomological interpretation of Dwork’s formula [28, (21)] relating the characteristic series det(I − tα) with the zeta function of Zk ∩ Yk . In the last section, we follow Dwork’s method to relate det(I − tα) to the zeta functions of Zk and of all its intersections with intersections of coordinate hyperplanes. For that purpose, we define an increasing filtration on the Dwork complexes with the following properties. On the one hand, its Fil0 term computes the primitive algebraic de Rham cohomology and the primitive rigid cohomology of the generic and special fibers of Z (the algebraic statement was proved in [27]). On the other hand, its graded pieces of higher degree decompose as direct sums of Fil0 terms for the Dwork complexes of the intersections of Z with intersections of coordinate hyperplanes. This also provides a cohomological interpretation of a combinatorial formula of Dwork [29, (4.33)]. Most of this work was done during the special period “Dwork Trimester in Italy” (May–July 2001). It is a pleasure for the second author to thank the University of Padova for its hospitality, as well as all the colleagues in the Mathematics Department who contributed to creating a wonderful working environment. General conventions. Throughout this paper, we will adopt the following conventions: (i) If E is an abelian group, then EQ := E ⊗ Q. (ii) All schemes are assumed to be separated and quasi-compact. (iii) Notation and shift conventions for cohomological operations on D-modules are those of Bernstein and Borel [19]. (iv) In most of this article, a prime number p and a power q = p s of p will be fixed. For simplicity, we will then call “Frobenius action” an action of the s-th power of the absolute Frobenius endomorphism, and “F -isocrystal” an isocrystal endowed with such an action (cf. 1.9 for details).

1 Specialization and cospecialization in rigid cohomology One of the essential ingredients in Dwork’s study of the zeta function for a singular projective hypersurface is the fact that, when the hypersurface is defined over a number field, the analytic cohomology spaces which carry the Frobenius action are isomorphic

Dwork cohomology for singular hypersurfaces

181

for almost all prime to their algebraic analogues. We give here a general result from which this comparison theorem follows. For that purpose, we first construct, for an algebraic variety Z over the ring of integers of a local field of mixed characteristics, a specialization map which relates the algebraic de Rham homology of the generic fiber of Z with the rigid homology of its special fiber. We then prove that, when Z comes from a number field, this map is an isomorphism for almost all primes. We also give a similar result for rigid cohomology with compact supports.

1.1. In this section, we fix a complete discretely valued field K of mixed characteristics (0, p). We denote by R its valuation ring, by m its maximal ideal, and by k its residue field. Let S = Spec(R), and let X be a smooth S-scheme. We first recall how the rigid cohomology of its special fiber Xk can be computed using the analytic space XKan associated to its generic fiber XK . The scheme X defines a p-adic formal scheme X over R, and we denote by XK its generic fiber (in the sense of Raynaud), which is a quasi-compact open rigid analytic subspace of XKan . For example, if X is a closed S-subscheme of an affine space ArS , XK is the intersection of XKan with the closed unit ball in the analytic affine space ArKan , which is independent of the chosen embedding into an affine space over S. In the general case, the construction can be deduced from the affine case by a glueing argument (cf. [12, 0.2] or [14]). Thanks to results of Nagata ([45], [46]), one can find a proper S-scheme X and an open immersion X → X. Let X be the formal scheme defined by X. Note that, an and X coincide, and that X since X is proper over S, the two analytic spaces X K K K is the tube ]Xk [ X of Xk in XK . We refer to [12, 1.2] for the general notion of a strict an = X . In particular, X an is a strict neighbourhood of neighbourhood of XK in X K K K an [12, (1.2.4) (ii)]. Therefore, the strict neighbourhoods of X contained in XK in X K K XKan form a fundamental system of strict neighbourhoods of XK . Moreover, an open an if and only if one of the two subset V ⊂ XKan is a strict neighbourhood of XK in X K following equivalent conditions is satisfied: (i) The covering (V , XKan \ XK ) of XKan is admissible. (ii) For any affine open subset U ⊂ X, and any closed embedding U ⊂ ArS , there exists a real number ρ > 1 such that V ∩ UKan contains UKan ∩ B(0, ρ), where B(0, ρ) is the closed ball of radius ρ in ArKan . Since these conditions are intrinsic on X (i.e. do not depend upon the compactification X), it is thus possible to define directly on XKan the notion of a (fundamental system of) strict neighbourhood(s) of XK in XKan . an , let j If V ⊂ V is a pair of strict neighbourhoods of XK in X K V ,V : V → V be the inclusion. For any abelian sheaf E on V , we define j † E := lim jV ,V ∗ jV−1 ,V E, − → V ⊂V

182

Francesco Baldassarri and Pierre Berthelot

where the limit is taken over all strict neighbourhoods V of XK contained in V . Note that the functor j † is an exact functor [12, (2.1.3)]. The sheaf j † E is actually independent of V in the sense that, if V1 ⊂ V is a strict neighbourhood of XK , j1† the analogue of j † on V1 , and E1 = jV−1 ,V1 E, there is a canonical isomorphism ∼

j † E −−→ RjV ,V1 ∗ j1† E1

(1.1.1)

(cf. [14, 1.2 (iv)]). Since, for any j † OV -module E (resp. j1† OV1 -module E1 ), the map E → j † E (resp. E1 → j1† E1 ) is an isomorphism [12, (2.1.3)], it follows that the functors jV−1 ,V1 and jV ,V1 ∗ (resp. RjV ,V1 ∗ ) give quasi-inverse equivalences between the

categories (resp. derived categories) of j † OV -modules and j1† OV1 -modules. Moreover, for any j † OV -module E, the canonical morphism R(V , E) −→ R(V1 , jV−1 ,V1 E)

is an isomorphism. an and V = X an , and to the de In particular, we can apply this remark to V = X K 1 K † an † Rham complex of X K . If j , jX denote the corresponding functors, we obtain in this way a canonical isomorphism ∼

•

•

an † , j X an ) −−→ R(XKan , jX† X an ), Rrig (Xk /K) := R(X K K

K

(1.1.2)

which shows that the rigid cohomology of Xk can be computed directly on XKan without using a compactification of Xk . 1.2. Let Z ⊂ X be a closed subscheme, U = X \ Z, and let U = X \ Zk be the formal completion of U . We denote by jU† the analogue of jX† obtained by taking the limit on strict neighbourhoods of UK . Thanks to (1.1.2), the rigid cohomology groups of Xk with support in Zk [14, 2.3] are given by •

•

RZk , rig (Xk /K) R(XKan , (jX† X an → jU† X an ) t ), K

K

(1.2.1)

where the subscript t denotes the total complex associated to a double complex. On the other hand, we can consider the de Rham cohomology groups of XK with support in ZK . Let u denote the inclusion of U in X. If
• is an injective resolution of •XK as a complex of sheaves of K-vector spaces over XK , we obtain by definition RZK , dR (XK /K) = (XK , (
→ uK ∗ u−1 K
) t ). •

•

We now construct a canonical morphism, called the specialization morphism: ρZ : RZK , dR (XK /K) −→ RZk , rig (Xk /K).

(1.2.2)

Observe that, if J in a flasque sheaf on XKan , then, for any U ⊂ X, the sheaf jU† J is acyclic for the functor (XKan , −). Indeed, the isomorphism (1.1.1) allows to replace XKan by any strict neighbourhood of UK . Thus we can replace XKan by a quasi-compact

183

Dwork cohomology for singular hypersurfaces

strict neighbourhood V of UK (for example, the complement of an open tube ]Xk \Uk [λ of radius λ < 1). This insures that H ∗ (V , −) commutes with direct limits, and the claim is clear. Choose an injective resolution J • of •X an , and denote by  : XKan → XK the K canonical morphism. The functoriality morphism for the de Rham complex can be extended to a morphism an an −1 J )t,  −1 (
→ uK ∗ u−1 K
) t −→ (J → uK ∗ uK •

•

•

•

which can then be composed with the canonical morphism an an −1 J ) t −→ (jX† J → jU† J ) t (J → uK ∗ uK •

•

•

•

(we use here the fact that UKan is a strict neighbourhood of UK ). Taking sections on XKan and composing with the functoriality map induced by  yields the morphism ρZ . Remark. By [35], the groups RZK , dR (XK /K) are independent of the embedding of ZK into the smooth scheme XK , and define the algebraic de Rham homology of Z. Similarly, the groups RZk , rig (Xk /K) depend only upon Zk , and define the rigid homology of Zk [47]. It is easy to check that the specialization morphism ρZ depends also only upon Z. However, we will not use these facts here. 1.3. Let us change notation, and assume that K is a number field, R = OK its ring of integers, S = Spec R, S 0 its set of closed points, and X an S-scheme, with generic fiber XK . For any s ∈ S 0 , the subscript s will denote the special fiber at s. If s corresponds to p ⊂ R, let K(s) be the completion of K at p, R(s) its valuation ring, an X(s) = Spec R(s) ×Spec R X, XK(s) the generic fiber of X(s) over Spec R(s), XK(s) its associated analytic space, X(s) the formal completion of X(s) with respect to the maximal ideal of R(s), X(s)K(s) its generic fiber. Assume that X is smooth over S, and fix a closed subscheme Z ⊂ X. Together with the base change map for algebraic de Rham cohomology, the specialization homomorphism (1.2.2) provides, for each s, a canonical morphism ρZ,s : K(s) ⊗K RZK , dR (XK /K) −→ RZs , rig (Xs /K(s)),

(1.3.1)

which we call the specialization morphism at s. Theorem 1.4. Under the previous assumptions, there exists a finite subset  ⊂ S 0 such that the specialization homomorphism (1.3.1) is an isomorphism for all s  ∈ . We begin the proof with the following remarks: (i) Since the algebraic de Rham cohomology complexes RZK , dR (XK /K) commute with base field extensions, the morphism (1.3.1) is an isomorphism at a point s if and only if, on K(s), the corresponding local morphism (1.2.2) is an isomorphism. (ii) If there exists a non empty open subset S ⊂ S over which X is proper and smooth, it follows from the construction of rigid cohomology and GAGA that the

184

Francesco Baldassarri and Pierre Berthelot

morphism ρX,s is an isomorphism for all s ∈ S . In particular, the theorem then holds for the pair (X, X). (iii) Both algebraic de Rham cohomology and rigid cohomology satisfy the standard excision properties (cf. [35, (3.3)] for de Rham cohomology, and [14, 2.5] for rigid cohomology). It follows immediately from the above constructions that the specialization morphisms define a morphism between the corresponding distinguished triangles. We use an induction argument similar to the one used in [14] to prove the finiteness of rigid cohomology. We will show inductively the following assertions: (a)n : For any number field K and any smooth OK -scheme X such that dim XK ≤ n, there exists a finite subset  ⊂ (Spec OK )0 such that the morphism ρX,s : K(s) ⊗K RdR (XK /K) −→ Rrig (Xs /K(s)) is an isomorphism for s  ∈ . (b)n : For any number field K, any OK -scheme Z such that dim ZK ≤ n and any closed immersion Z → X into a smooth OK -scheme X, there exists a finite subset  ⊂ (Spec OK )0 such that the morphism ρZ,s : K(s) ⊗K RZK , dR (XK /K) −→ RZs , rig (Xs /K(s)) is an isomorphism for s  ∈ . Let us first check (a)0 . The scheme X is then étale over S, and XK is finite over K. It follows that there exists a non empty open subset in S over which the morphism X → S is finite. Thus the assertion follows from remark (ii) above. Let us now prove that (b)n−1 implies (a)n . Let X be a smooth S-scheme such that dim XK = n. Since K is of characteristic zero, we may use resolution of singularities to find an isomorphism between XK and a dense open subset of a proper and smooth K-scheme YK . By general arguments on direct limits, there exists a non empty open subset S ⊂ S, a proper and smooth S -scheme Y and an open immersion X|S → Y extending over S the previous immersion XK → YK . By remark (ii), the morphism ρY,s is an isomorphism for all s ∈ S . Let Z = Y \ X. As XK is dense in YK , we have dim ZK < n. Therefore, the induction hypothesis implies that the morphism ρZ,s is an isomorphism for all s outside a finite subset of S 0 . Shrinking S if necessary, the result for X then follows from remark (iii). We finally prove that (b)0 holds, and that (b)n−1 + (a)n implies (b)n . Let Z → X be a closed immersion into a smooth S-scheme X, with dim ZK = n. We may replace Z by Z red , since both source and target of (1.3.1) only depend upon the reduced subscheme. Then, if T ⊂ Z is the closed subset where Z → S is not smooth, we have dim TK < n. Using the excision exact sequences and the induction hypothesis, we are reduced to the case where Z is smooth over S. Let r = codim(Z, X). We have Gysin isomorphisms for algebraic de Rham cohomology [35, (3.1)] and for rigid cohomology [14, 5.2-5.5]. Moreover, the Gysin isomorphism for rigid cohomology is deduced from the Gysin morphism between algebraic de Rham complexes by taking the analytification and applying suitable j † functors. Therefore, the specialization

185

Dwork cohomology for singular hypersurfaces

morphisms fit in a commutative diagram K(s) ⊗K RdR (ZK /K)

∼

/ K(s) ⊗K RZK , dR (XK /K)[2r]

ρZ,s

ρZ,s [2r]

 Rrig (Zs /K(s))

 / RZs , rig (Xs /K(s))[2r].

∼

Since Z is smooth, and dim ZK = n, the induction hypothesis implies that the left vertical arrow is an isomorphism, and the theorem follows. Remark. In the step (b)n−1 ⇒ (a)n , we could use de Jong’s theorem on alterations instead of resolution of singularities. We would then argue as in [14, 3.5], using the fact that the specialization morphisms commute with the trace maps associated with a finite étale morphism between two affine schemes. 1.5. We now give an analogue of theorem 1.4 for rigid cohomology with compact supports. Let us first briefly explain the construction of the cospecialization morphism between rigid cohomology with compact supports and algebraic de Rham cohomology with compact supports (the reader can refer to [7, section 6] for more details). We consider again the situation of 1.1 and 1.2, where K was a complete discretely valued field of mixed characteristics (0, p), and we keep the same notation and hypotheses. Let Z be the closure of Z in X, T = Z \ Z, and let u : ]Tk [ X → ]Z k [ X be the inclusion. By construction [9], the rigid cohomology of Zk with compact supports is given by •

Rc, rig (Zk /K) := R]Zk [ (]Z k [ X , X an ) K

•

R(]Z k [ X , (]Z

•

k[

→ u ∗ ]Tk [ ) t ).

(1.5.1)

On the other hand, by [7, 1.2], the algebraic de Rham cohomology with compact supports of ZK is defined as •

•

RdR, c (ZK /K) := R(XK , ((X )/Z K → (X )/TK ) t ), K

(1.5.2)

K

where (•X )/Z K and (•X )/TK are the formal completions along Z K and TK reK K spectively. Using the functoriality of the de Rham complex and GAGA, we therefore obtain an isomorphism •

•

an , ((X an )/Z an → (X an )/TKan ) t ). RdR, c (ZK /K) R(X K K

K

(1.5.3)

K

an and T an are closed analytic subsets of the open subsets ]Z [ and ]T [ Since Z K k X k X K an , we now have a functoriality morphism of X K •

R(]Z k [ X , (]Z

•

k[

•

•

an → u ∗ ]Tk [ ) t ) −→ R(X K , ((X an )/Z an → (X an )/TKan ) t ). K

K

K

(1.5.4)

186

Francesco Baldassarri and Pierre Berthelot

The cospecialization morphism ρc,Z : Rc, rig (Zk /K) −→ RdR, c (ZK /K)

(1.5.5)

is then obtained by composing (1.5.4) with the inverse of (1.5.3). Remarks. (i) It is easy to check that ρc,Z only depends upon Z, and not upon the scheme X used to define both cohomologies with compact supports. (ii) Algebraic de Rham cohomology and rigid cohomology both satisfy Poincaré duality. Indeed, if we assume that X is of constant relative dimension n over S, and if the exponent ∨ denotes the K-linear dual, we have canonical isomorphisms ∼

RZK , dR (XK /K) −−→ RdR, c (ZK /K)∨ [−2n]

(1.5.6)

(cf. [7, 3.4]) and ∼

RZk , rig (Xk /K) −−→ Rc, rig (Zk /K)∨ [−2n]

(1.5.7)

(cf. [15, 2.4]). One verifies easily that ρZ and ρc,Z are compatible with cup-products on cohomology. On the other hand, the rigid trace map is constructed in [15] starting from Hartshorne’s algebraic trace map for projective smooth varieties [34], and this ensures the commutation of the rigid and de Rham trace maps with ρc,Z . It follows that, under the Poincaré duality pairings, ρZ and ρc,Z are dual to each other (a detailed proof can be found in [7, 6.9]). 1.6. We consider now the global situation of 1.3, and we use again the notation and hypotheses of that section. For each s ∈ S 0 , the base change map for algebraic de Rham cohomology with compact supports K(s) ⊗K RdR, c (ZK /K) −→ RdR, c (ZK(s) /K(s)) is an isomorphism, because algebraic de Rham cohomology commutes with base field extensions [35, 5.2], and this property extends to algebraic de Rham cohomology with compact supports using the standard distinguished triangle defined by a compactification. Composing the inverse of this isomorphism with (1.5.5) gives the cospecialization morphism at s ρc,Z,s : Rc, rig (Zs /K(s)) −→ K(s) ⊗K RdR, c (ZK /K).

(1.6.1)

Theorem 1.7. Under the assumptions of 1.3, there exists a finite subset  ⊂ S 0 such that the cospecialization homomorphism (1.6.1) is an isomorphism for all s ∈ . It follows from remark (ii) of 1.5 that the morphisms ρZ,s and ρc,Z,s are dual to each other under Poincaré duality over K(s). Hence ρc,Z,s is an isomorphism if and only if ρZ,s is an isomorphism, and theorem 1.4 and theorem 1.7 are equivalent. One can also give a direct proof of 1.7 as in 1.4. One observes first that, if Z ⊂ Z is an open subset, with T = Z \ Z , the cospecialization morphisms define a morphism between the corresponding distinguished triangles for cohomologies with compact

187

Dwork cohomology for singular hypersurfaces

supports. This allows to proceed by induction on the dimension of ZK . Indeed, as in 1.4 (ii), the theorem is true if there is an open subset S ⊂ S such that Z is proper and smooth over S . In particular, the theorem is true when dim ZK = 0. In general, we can remove from Z a closed subscheme T such that dim TK < dim ZK , so as to of Z := Z \ T is smooth. It is enough to prove the insure that the generic fiber ZK theorem for Z , and we can use resolution of singularities to find a compactification Z of Z which is proper and smooth over a non empty open subset of S, and in which Z is dense. By induction, the result for Z implies the result for Z . 1.8. In the local case, we will also use the specialization and cospecialization morphisms for some cohomology groups with coefficients. For simplicity, we will only consider here the case where Z = X, as this is the only case which will be needed in the present article (the reader interested in the general case will easily generalize our constructions, following the method used in 1.2 and 1.5). Our notation and hypotheses are again those of 1.1. Let us first observe that, if an , the equivalences j −1 and V1 ⊂ V are two strict neighbourhoods of XK in X K V ,V1 jV ,V1 ∗ between the categories of j † OV -modules and j1† OV1 -modules induce quasiinverse equivalences between the categories of coherent j † OV -modules endowed with an integrable and overconvergent connection and of coherent j1† OV1 -modules endowed with an integrable and overconvergent connection. The category of overconvergent isocrystals on Xk can thus be realized equivalently on V or on V1 . Therefore, for any overconvergent isocrystal L on Xk , we get as in 1.1 a canonical isomorphism ∼

•

Rrig (Xk /K, L) −−→ R(XKan , L ⊗ X an ), K

(1.8.1)

where L is viewed as a coherent jX† OXKan -module with an integrable and overconvergent connection. Let (L, ∇) be a locally free finitely generated OXK -module, endowed with an integrable connection, (L an , ∇ an ) its inverse image on XKan , and L = jX† L an , endowed with the corresponding connection. We assume that this connection on L is overconvergent, so that L can be viewed as defining an overconvergent isocrystal on Xk , still denoted by L. The specialization morphism for de Rham and rigid cohomologies with coefficients in L is then defined as the composed morphism L : R(XK , L ⊗ XK ) −→ R(XKan , L an ⊗ X an ) ρX •

•

K

•

(1.8.2)

−→ R(XKan , L ⊗ X an ) Rrig (Xk /K, L). K

To define the cospecialization morphism, we choose a compactification X of X, and a coherent OXK -module L extending L on XK . Let I be the ideal of T := X \ X in X. In general, the connection ∇ does not extend to L, but it can be extended as a n L. This allows to define the de Rham proconnection on the pro-OXK -module “ lim”I ← −n • • • n L) ⊗  . The algebraic de Rham cohomology complex I L ⊗ X := (“ lim”I XK ← −n K

188

Francesco Baldassarri and Pierre Berthelot

with compact supports and coefficients in L is then defined (cf. [4, App. D.2]) as •

•

RdR, c (XK /K, L) := R(XK , R lim I L ⊗ X ) ← − K • •

R lim R(XK , I L ⊗ X ). ← − K Note first that GAGA provides a canonical isomorphism •

•

∼

•

•

∼

•

•

(1.8.3)

an , I L an ⊗ X an ) R lim R(XK , I L ⊗ X ) −−→ R lim R(X K ← − ← − K K an −−→ R(X K , R lim I L an ⊗ X an ). ← − K

(1.8.4)

an the given open immersion. We can consider Let us now denote by j : XKan → X K an an on XK and X K the functors R ]Xk [ of local sections supported in the tube ]Xk [ X . an , the canonical morphism As (XKan , ]Tk [ X ) is an admissible covering of X K

R ]Xk [ (Rj∗ E) −→ Rj∗ (R ]Xk [ E)

(1.8.5)

is an isomorphism for any complex of abelian sheaves E on XKan . For the same reason, the canonical morphism R lim(I • L an ⊗ •X an ) → Rj∗ (L an ⊗ •X an ) induces ← − K K

an isomorphism

•

•

∼

•

R ]Xk [ (R lim(I L an ⊗ X an )) −−→ R ]Xk [ (Rj∗ (L an ⊗ X an )). ← − K K

(1.8.6)

The cospecialization morphism for de Rham and rigid cohomologies with compact supports and coefficients in L is then defined as the composed morphism L ρc,X : Rc, rig (Xk /K, L) := R]Xk [ (XKan , L an ⊗ X an ) •

K

•

an

R(X K , R ]Xk [ (Rj∗ (L an ⊗ X an ))) K

•

•

an

R(X K , R ]Xk [ (R lim(I L an ⊗ X an ))) ← − K • • an an → R(X K , R lim(I L ⊗ X an )) ← − K

RdR, c (XK /K, L) (1.8.7)

deduced from the previous isomorphisms. Remark. It is again easy to check that the specialization and cospecialization morphisms for cohomologies with coefficients in L are compatible with pairings on cohomology. Together with the compatibility of the trace maps with cospecialization, this shows that, if the exponent ∨ is used to denote O-linear duals, and if X is of pure relative dimension n, we obtain a commutative diagram RdR (XK /K, L∨ ) ∨

L ρX

 Rrig (Xk /K, L∨ )

/ RdR, c (XK /K, L)∨ [−2n] L )∨ (ρc,X

 / Rc, rig (Xk /K, L)∨ [−2n],

Dwork cohomology for singular hypersurfaces

189

and similarly exchanging the roles of cohomology and cohomology with compact ∗ (X /K, L∨ ) and H ∗ (X /K, L) are supports. In particular, if L is such that Hrig k k c, rig ∨

L and ρ L are dual to each finite dimensional and satisfy Poincaré duality, then ρX c,X other (note that these properties are not necessarily true without additional assumptions on L).

1.9. Apart from the constant coefficients case, our main interest in this article will be in cohomology groups with coefficients in Dwork’s F -isocrystal Lπ [10, (1.5)]. We recall briefly here its construction and properties. We assume now that K is a complete discretely valued field of mixed characteristics (0, p), containing Qp (ζp ), where ζp is a primitive pth root of 1. Let R be the valuation ring of K, m its maximal ideal, k its residue field, S = Spec R, S = Spf R. We recall that, for each root π of the polynomial t p−1 + p, there exists a unique primitive pth root ζ of 1 such that ζ ≡ 1 + π mod π 2 (cf. [28, p. 636]). Therefore, the choice of an element π ∈ K such that π p−1 = −p is equivalent to the choice of a non trivial additive K-valued character of Z/pZ. In the following, we fix such an element π. To deal with Frobenius actions, we will also assume that there exists an endomorphism σ : R → R lifting a power F s of the Frobenius endomorphism of k, and such that σ (π ) = π. In this article, the integer s and the endomorphism σ will be fixed, and we will work systematically with F s -isocrystals with respect to (K, σ ) rather than with F -isocrystals in the usual (absolute) sense. Therefore, we will simplify the terminology, and use the expression “F -isocrystal” to mean “F s -isocrystal with respect to (K, σ )”. Similarly, a Frobenius action will mean a σ -semi-linear action of the s-th power of the absolute Frobenius endomorphism. The datum of π defines a rank 1 bundle with connection Lπ on the affine line A1S , by endowing the sheaf OA1 with the connection ∇π such that S

 da + π a ⊗ dt, (1.9.1) ∇π (a) = dt where t is the canonical coordinate on A1S . For any S-morphism ϕ : X → A1S , we will denote by Lπ,ϕ the inverse image of Lπ , endowed with the inverse image connection. Let A1K an be the rigid analytic affine line over K,  A1S the formal affine line over S,  A1K an its generic fiber (the closed unit disk in A1K an ), and let jA† 1 be the functor defined as in 1.1 using the strict neighbourhoods of  A1K an in A1K an . We denote by Lπan the analytic vector bundle with connection associated to Lπ on A1K an , and we define Lπ = jA† 1 Lπan . Then the connection ∇πan induces an overconvergent connection on Lπ [10, (1.5)]. The natural embedding of A1k into A1S allows to realize overconvergent isocrystals on A1k as jA† 1 OA1 an -modules endowed with an integrable and overconvergent connection. K

Therefore, we can view Lπ as an overconvergent isocrystal on A1k , defined by the sheaf jA† 1 OA1 an endowed with the connection (1.9.1). Note that, if ψ is the character K of Z/pZ corresponding to π as above, then Lπ = Lψ −1 in the notation of [10].

190

Francesco Baldassarri and Pierre Berthelot

In addition, Lπ has a canonical structure of F -isocrystal: if one lifts the s-th power of the absolute Frobenius endomorphism of A1k as the σ -linear endomorphism FA1 : A1S → A1S such that FA∗1 (t) = t q , the Frobenius action φ : FA∗1 Lπ → Lπ is given by φ(1 ⊗ a) = exp(π(t q − t))a.

(1.9.2)

Since the category of overconvergent F -isocrystals is functorial with respect to morphisms of k-schemes of finite type, any such morphism ϕ : Xk → A1k defines by pull-back an overconvergent F -isocrystal on Xk , which will be denoted by Lπ,ϕ . When ϕ is the reduction mod m of a morphism of smooth S-schemes ϕ˜ : X → A1S , then Lπ,ϕ is obtained as the inverse image of (jA† 1 OA1 an , ∇πan ) by the morphism of K ringed spaces ϕ˜Kan : (XKan , jX† k OXanK ) −→ (A1Kan , jA† 1 OA1 an ), K

an is given by j † O an endowed with the inverse image connection i.e. Lπ,ϕ = jX† k Lπ, Xk XK ϕ˜ ϕ˜Kan ∗ (∇πan ) (cf. [12, 2.5.5]). Moreover, if there exists a lifting FX : X → X of the s-th power of the absolute Frobenius morphism of Xk as a σ -linear endomorphism of X, the action of Frobenius on Lπ,ϕ is given by the composite isomorphism ∼

∼

FX∗ ϕ˜ ∗ Lπ −−→ ϕ˜ ∗ FA∗1 Lπ −−→ ϕ˜ ∗ Lπ , where the first isomorphism is the identification between the two inverse images provided by the Taylor series of the connection ∇πan , and the second one is the inverse image of φ by ϕ. ˜ Let us point out that the hypotheses needed in the remark of 1.8 are satisfied by Dwork isocrystals. This is now known to be the case for any F -isocrystal, thanks to Kedlaya’s results [42], but it can also be deduced from the case of the constant isocrystal. Indeed, this is a consequence of the relation between Dwork isocrystals and Artin–Schreier coverings, which we recall now in the algebraically liftable case (cf. [10, (1.5)], [14, 3.10]). Note that the case of L∨ π follows from the case of Lπ , = L (and L

L if p = 2). Let u : C → A1S be the finite since L∨ −π −π π π covering defined by the equation y p − y − t = 0. Then u is étale outside of the closed subscheme Spec(R[y]/(py p−1 − 1)) ⊂ C, which is quasi-finite over Spec R, concentrated in the generic fiber, and whose image in A1Kan lies outside the open disk of radius p p/(p−1) > 1. Let Y = X ×A1 C, v : Y → X, and let Y be S the formal completion of Y . Then vKan is étale in a strict neighbourhood of ]Yk [ Y in YKan . The additive group Z/pZ acts on the sheaf jX† k v∗an OYKan , and Lπ,ϕ is the direct factor of jX† k v∗an OYKan on which Z/pZ acts through the character ψ −1 . As

∗ (X /K, L∨ ) v∗an jY†k OYKan jX† k v∗an OYKan , it follows that the cohomology spaces Hrig k π,ϕ ∗ ∗ (Y /K) (resp. (resp. Hc, rig (Xk /K, Lπ,ϕ )) can be identified with the subspaces of Hrig k Hc,∗ rig (Yk /K) on which Z/pZ acts through ψ (resp. ψ −1 ). Thus the finiteness of

Dwork cohomology for singular hypersurfaces

191

∗ (X /K, L∨ ) and H ∗ (X /K, L the spaces Hrig k k π,ϕ ) follows from the finiteness of π,ϕ c, rig rigid cohomology with constant coefficients. ∗ (Y /K) Moreover, the same argument shows that Poincaré duality for Hrig k ∗ (X /K, L∨ ) and induces a perfect pairing between the subspaces Hrig k π,ϕ Hc,∗ rig (Xk /K, Lπ,ϕ ). On the other hand, the transitivity of the trace map implies ∗ (Y /K) can be identified with the Poincaré pairing that the Poincaré pairing for Hrig k ∗ (X /K, v an j † O an ) (defined via the trace map on the finite étale j † O an for Hrig k ∗ Yk YK Xk XK

algebra v∗an jY†k OYKan ). Therefore, the previous pairing is equal to the pairing defined between these cohomology groups by Poincaré duality on Xk .

2 The overconvergent Fourier transform Unless otherwise specified, we assume for the rest of the paper that the base field K satisfies the hypotheses of 1.9. Our goal in this section is to prove theorem 2.14, which will be the key result to interpret the cohomology of the analytic Dwork complexes for a projective hypersurface in terms of rigid homology groups. This theorem can be viewed as an analogue for rigid cohomology of [3, th. 1.1] † and [27, th. 0.2]. Our main tool here is the theory of DX, Q -modules, and our proof follows the method of [27] based on the Fourier transform. Therefore, we begin by † briefly recalling some notions about DX, Q -modules and their Fourier transform. X the p-adic 2.1. Let X be a smooth formal S-scheme of relative dimension n, and D † completion of the standard sheaf of differential operators on X. The sheaf DX, Q is  the subsheaf of rings of DX, Q such that, if x1 , . . . , xn are local coordinates on an affine open subset U ⊂ X, and ∂i = ∂/∂xi , 1 ≤ i ≤ n, then     † ak ∂ [k]  ∃ c, η ∈ R, η < 1, such that ak  ≤ cη|k| , (U, DX, Q) = P = k∈Nn

where ∂ [k] = k!1 ∂ k , ak ∈ (U, OX, Q ), and − is a quotient norm on the Tate algebra (U, OX, Q ). It can also be written canonically as a union of p-adically complete subsheaves of rings  (m) †  = D DX, X, Q , Q m≥0

(m) being defined by D X, Q     (m) [k] (m) ) = P = X, Q )  bk → 0 for |k| → ∞ , (U, D q !b ∂ ∈ (U, D k X, Q k k

(m)

(m)

with k = pm q (m) + r k , 0 ≤ rki k

< pm for all i [13].

192

Francesco Baldassarri and Pierre Berthelot

One can also introduce overconvergence conditions along a divisor H ⊂ X, where X is the special fiber of X. Let j : Y → X be the inclusion of the complement of H in X. The sheaf OX, Q († H ) of functions with overconvergent singularities along H is the subsheaf of the usual direct image j∗ OY, Q such that, if U ⊂ X is an affine open subset, and h ∈ (U, OX ) a lifting of a local equation of H in the special fiber U of U, then     ai / hi+1  ∃ c, η ∈ R, η < 1, such that ai  ≤ cηi , (U, OX, Q († H )) = g = i∈N

where the ai ’s belong to (U, OX, Q ) and the norm is again a quotient norm. As for † † DX, Q , there is a canonical way to write OX, Q ( H ) as a union of p-adically complete sub-algebras. Indeed, if we fix m ≥ 0, there exists a p-adically complete OX -algebra (m) (H ), depending only on X and H , such that, on any affine open subset U as B X above, (m) (H )|U OU {T }/(hpm+1 T − p), B X T being an indeterminate [13]. The algebra OX, Q († H ) is then given by  (m)  OX, Q († H ) = (H ). B m≥0

X, Q

It depends only upon the support of H , and not upon the multiplicities of its components. (m) , compati(m) (H ) is endowed with a natural action of D Moreover, each B X, Q X, Q ble with its OX -algebra structure [13]. Therefore, it is possible to endow the com(m) with a ring structure extending those (m) (H )⊗  OX D pleted tensor product B X, Q X, Q (m) (m)   (H ) and D . One can then define the ring of differential operators of B X, Q

† † DX, Q ( H ) as

X, Q

† † DX, Q ( H ) :=

 m≥0

(m) (H )⊗ (m) .  OX D B X, Q X, Q

It follows easily from this definition that, for any affine open subset U ⊂ X on which there exist local coordinates, and a local equation for H in U , the sections of † † DX, Q ( H ) on U can be described as † † (U, DX, Q ( H )) =    ai,k  [k]  i+|k| g= ∃ c, η ∈ R, η < 1, such that a ∂  ≤ cη , i,k hi+1 i,k

the notation being as above, and ai,k ∈ (U, OX, Q ).

Dwork cohomology for singular hypersurfaces

193

When X is proper, and H is viewed as a divisor at infinity providing a compactification of Y := X \ H , it is often convenient to replace the notation OX, Q († H ) and † † † DX, Q ( H ) by OX, Q (∞) and DX, Q (∞), if no confusion arises.

† † Recall that OX, Q († H ) and DX, Q ( H ) are coherent sheaves of rings, and that coherent modules over these sheaves satisfy the standard A and B theorems [13, 4.3.2 and 4.3.6].

2.2. Let X be affine, with H , Y be as before, and let q : V =  ArX → X be the formal affine space of relative dimension r over X, q : V → X the dual affine space. Assume that X has local coordinates x1 , . . . , xn relative to S, and let t1 , . . . , tr (resp. t1 , . . . , tr ) denote the standard coordinates on V (resp. V ) relative to X, ∂xj , ∂ti (resp. ∂ti ) the corresponding derivations. Assume also that there exists a section h ∈ (X, OX ) lifting a local equation of H in X. We will use the results of [36], [37], [38] for the affine space V over (X, H ). While these references are written in the absolute case, i.e. X = S, it is easy to check that the proofs remain valid in our setting, requiring only obvious modifications. Let us first define the weakly complete Weyl algebra A†r (X, H ) associated to the ArY , A = (X, OX ) ⊗ K, affine space V over (X, H ). Let W = q −1 (Y) =  r (Y) can be written r (Y) = (W , D W ) ⊗ K. An element P ∈ A A  [k] [l] P = ai,j ,k,l h−(i+1) t j ∂ t ∂ x , i,j ,k,l

with coefficients ai,j ,k,l ∈ A such that ai,j ,k,l → 0 when i + |j | + |k| + |l| → ∞. r (Y) iff the ai,j ,k,l can be chosen so that there exists c, Then P ∈ A†r (X, H ) ⊂ A η ∈ R, with η < 1, such that ai,j ,k,l  ≤ cηi+|j |+|k|+|l| .

(2.2.1)

r (Y). It is easy to check that A†r (X, H ) is a sub-K-algebra of A r  If P = PX is the formal projective space of relative dimension r over X, and P the dual projective space, we will keep the notation q and q for the projections P → X and P → X. Let V , P be the special fibers of V and P , H∞ = P \ V , , H ). We will use the notation H1 = q −1 (H ) ∪ H∞ (resp. V , P , H∞ 1 OP , Q (∞) := OP , Q († H1 ),

DP† , Q (∞) := DP† , Q († H1 ),

OP , Q (∞) := OP , Q († H1 ),

DP† , Q (∞) := DP† , Q († H1 ).

The following theorem shows that coherent DP† , Q (∞)-modules are determined by their global sections: Theorem 2.3 (cf. [36], [37]). (i) The ring A†r (X, H ) is coherent.

194

Francesco Baldassarri and Pierre Berthelot

(ii) There exists a canonical isomorphism of K-algebras A†r (X, H ) (P , DP† , Q (∞)).

(2.3.1)

(iii) The functor (P , −) induces an equivalence between the category of coherent and the category of coherent A†r (X, H )-modules.

DP† , Q (∞)-modules

2.4. Under the previous hypotheses, let us describe the naive Fourier transform for coherent DP† , Q (∞)-modules. Let A r† (X, H ) be the weakly complete Weyl algebra associated to the dual affine space V over (X, H ). A basic observation is that the p-adic absolute value |π k /k!| satisfies the unequalities 1/kp ≤ |π k /k!| ≤ 1 for any k ∈ N. Comparing to the condition (2.2.1), it follows that the datum of π allows to define a continuous isomorphism ∼

φ : A r† (X, H ) −−→ A†r (X, H ), characterized by φ(ti ) = −∂ti /π,

φ(∂ti ) = π ti .

If M is a coherent DP† , Q (∞)-module, (P , M) is a coherent A†r (X, H )-module. By restriction of scalars via φ, it can be viewed as a coherent A r† (X, H )-module. The previous theorem shows that, up to canonical isomorphism, there is a unique coherent DP† , Q (∞)-module M such that (P , M ) = φ∗ (P , M). By definition, the naive Fourier transform F naive (M) of M is the DP† , Q (∞)-module M .

2.5. To define the geometric Fourier transform, we will use the standard cohomo† logical operations for DX, Q -modules. We refer to [17] and [18] for their general definitions and basic properties, and we only recall here a few facts needed for our constructions. a) Let X, X be smooth formal schemes of relative dimensions dX , dX over S, with special fibers X, X , f : X → X an S-morphism, and let H ⊂ X, H ⊂ X † † be divisors such that f −1 (H ) ⊂ H . We use the notation DX, Q (∞), DX , Q (∞) for

† † † † DX, Q ( H ), DX , Q ( H ).

† In this situation, the morphism f defines transfer bimodules DX→X , Q (∞) and

† † −1 D † DX ←X, Q (∞) (cf. [18], or [36, 1.4.1]). The first is a (DX, Q (∞), f X , Q (∞))bimodule and can be used to define an inverse image functor, which associates to a † † left DX , Q (∞)-module N the left DX, Q (∞)-module given by † f ∗ N := DX→X , Q (∞) ⊗f −1 D †

X , Q

(∞)

f −1 N .

Dwork cohomology for singular hypersurfaces

195

† Note that there is an abuse of notation here, since the definition of DX→X , Q (∞) involves completions, and therefore this functor cannot be identified in general with the inverse image for OX ,Q -modules or OX , Q (∞)-modules. b (D † For any complex N in Dcoh X , Q (∞)), the extraordinary inverse image functor ! f is then defined as usual by

f ! (N ) := Lf ∗ (N )[dX/X ], where dX/X = dX − dX . † −1 D † ∗ When f is smooth, DX→X , Q (∞) is flat over f X , Q (∞), and f preserves coherence (cf. [17], [18]). † † −1 D † b) The bimodule DX ←X, Q (∞) is a (f X , Q (∞), DX, Q (∞))-bimodule. It b (D † can be used to define a direct image functor f+ on Dcoh X, Q (∞)), which associates † b (D † to M ∈ Dcoh X, Q (∞)) the complex of left DX , Q (∞)-modules given by


† L f+ (M) := Rf∗ DX ←X, Q (∞) ⊗D †

X, Q (∞)

 M .

When f is projective, and H is the support of a relatively ample divisor, the acyclicity theorem of Huyghe [39, 5.4.1] shows that, if n ≥ 1, R n f∗ vanishes for coherent † DX, Q (∞)-modules. On the other hand, f+ does not preserve coherence in general. c) Finally, let us recall that overconvergent isocrystals may be viewed as † DX, Q (∞)-modules in the following way. If X is a smooth formal S-scheme, H ⊂ X a divisor in its special fiber, Y = X \ H , there is a specialization morphism sp : XK → X, which is a continuous map, functorial with respect to X, such that sp ∗ jY† OXK OX, Q († H ) (cf. [12] or [14, 1.1]). The functor sp∗ is exact on the category of coherent OXK -modules, and, since H is a divisor, it is also exact on the category of coherent jY† OXK -modules (cf. [14, proof of 4.2]). If L is an isocrystal on Y which is overconvergent along H , then sp ∗ L is a coherent OX, Q († H )-module, endowed with † † a canonical structure of DX, Q ( H )-module [13, 4.4]. By [13, 4.4.5 and 4.4.12], the functor sp ∗ allows to identify the category of isocrystals on Y which are overconver† † gent along H with a full subcategory of the category of coherent DX, Q ( H )-modules. Moreover, this identification is compatible with inverse images [36, 1.5.4]. Therefore, we will generally misuse notation, and simply keep the letter L to denote sp ∗ L. 2.6. We will need the geometric Fourier transform in a more general setup than the situation considered for the definition of the naive Fourier transform. We assume here that X is a smooth formal scheme of relative dimension n over S, endowed with a divisor H ⊂ X, and that q : V → X, q : V → X are dual vector bundles of rank r over X. We denote Y = X \ H , W = q −1 (Y), W = q −1 (Y). Let q : P → X and q : P → X be relative projective closures of V and V , P := P ×X P , with projections p : P → P , p : P → P , q : P → X. We write V , V , Y , W , , H , H as in W , P , P , P for the special fibers. We define the divisors H∞ , H∞ 1 1

196

Francesco Baldassarri and Pierre Berthelot

2.2, and we endow P with the divisor H2 = p−1 (H∞ ) ∪ p −1 (H∞ ) ∪ q −1 (H ),

whose support is p −1 (H1 ) ∪ p −1 (H1 ). We will use the notation OP , Q (∞) := OP , Q († H2 ),

DP† , Q (∞) := DP† , Q († H2 ).

To construct the kernel of the geometric Fourier transform, we apply 2.5 c) to P . Let µ : V ×X V −→ A1X −→ A1k be the morphism obtained by composing the canonical pairing between V and V with the projection to A1k . As seen in 1.9, µ defines by functoriality a canonical rank † 1 overconvergent F -isocrystal Lπ,µ over V ×X V . A fortiori, jW ×W Lπ,µ defines an F -isocrystal on W ×Y W , overconvergent along H2 . We will denote by LW π,µ the † † rank one OP , Q (∞)-module sp ∗ (jW ×W Lπ,µ ), endowed with its natural DP , Q (∞)module structure, and its Frobenius action. In particular, LW π,µ has a canonical basis e, and, above an open subset of X on which V has linear coordinates t1 , . . . , tr (with dual coordinates t1 , . . . , tr ), its underlying connection ∇π,µ is given by   ∇π,µ (ae) = e ⊗ da + π a ti dti + ti dti . (2.6.1) i

If M is a coherent DP† , Q (∞)-module, then LW π,µ ⊗OP , Q (∞) M , viewed as a

left DP† , Q (∞)-module through the standard tensor product structure, is still coherent. We can now define the geometric Fourier transform of a coherent DP† , Q (∞)module M by ∗ (LW F geom (M) := p+ π,µ ⊗OP , Q (∞) p (M)).

(2.6.2)

For simplicity, we do not use the standard shifts here, so that, when M consists in a ∗ single coherent DP† , Q (∞)-module placed in degree 0, LW π,µ ⊗ p (M) is a coherent DP† , Q (∞)-module placed in degree 0.

A priori, F geom (M) is only known to be a complex in D b (DP† , Q (∞)). The fol-

lowing theorem, due to Huyghe, shows that F geom transforms a coherent DP† , Q (∞)-

module into a coherent DP† , Q (∞)-module, and provides the comparison between the naive and geometric Fourier transforms:

Theorem 2.7 ([36], [40]). Let M be a coherent DP† , Q (∞)-module. (i) The complex F geom (M) is acyclic in degrees  = 0. In degree 0, its cohomology sheaf is a coherent DP† , Q (∞)-module.

Dwork cohomology for singular hypersurfaces

197

(ii) Under the assumptions of 2.4, there is a natural isomorphism of DP† , Q (∞)-modules F geom (M) F naive (M).

(2.7.1)

One of the main steps in the proof of (ii) is the computation of the geometric Fourier transform of DP† , Q (∞). We will actually use this result under the more

general assumptions of 2.6. By construction, the bimodule DP† ←P , Q (∞) is iso-

morphic to DP†, d→P ,Q (∞) ⊗OP ωP /P , where DP†, d→P ,Q (∞) is the analogue of

DP† →P , Q (∞) obtained using the right OP , Q (∞)-module structure of DP† , Q (∞) )), where rather than the left one, and ωP /P = ∧r 1P /P . If we write V = Spf(S(E E is locally free of rank r over OX , then there is a canonical isomorphism OP , Q (∞) ⊗OP 1P /P OP , Q (∞) ⊗OP q ∗ E . Since LW π,µ has a canonical section, one can use this remark to define a canonical map DP† , Q (∞) ⊗ q ∗ (∧r E ) → p∗ (DP† ←P , Q (∞) ⊗ LW π,µ ). L

† On the other hand, DP† ←P , Q (∞) ⊗ (LW π,µ ⊗ DP →P , Q (∞)) can be computed

using the Spencer resolution of DP† →P , Q (∞)

· · · → DP† , Q (∞) ⊗ TP /P → DP† , Q (∞) → DP† →P , Q (∞) → 0, which gives a canonical map † 0 W p∗ (DP† ←P , Q (∞) ⊗ LW π,µ ) → H (p+ (Lπ,µ ⊗ DP →P , Q (∞))).

Using appropriate division theorems, one can then prove that the composite map is an isomorphism ∼

DP† , Q (∞) ⊗ q ∗ (∧r E ) −−→ F geom (DP† , Q (∞))

(2.7.2)

(cf. [40], and [22] for the complex analytic case). When E is a free OX -module, the choice of a basis of E provides a trivialisation of ∧r E and the isomorphism (2.7.2) reduces to the inverse of (2.7.1) for DP† , Q (∞). Remark. One can also give a Gauss–Manin style description of F geom (M), using : the de Rham resolution of the bimodule DP† ←P , Q (∞) to compute p+ † † † r · · · → r−1 P /P ⊗ DP , Q (∞) → P /P ⊗ DP , Q (∞) → DP ←P , Q (∞) → 0.

In general, this is only a resolution in the category of (p −1 OP , Q (∞), DP† , Q (∞))bimodules. However, when V is the trivial bundle  ArX , it can be viewed as a resolution in the category of (p −1 DP† , Q (∞), DP† , Q (∞))-bimodules. Indeed, the product PrS and the fact that H2 = p −1 (H1 ) ∪ p −1 (H1 ) allow to structure P = P ×S 

198

Francesco Baldassarri and Pierre Berthelot

define a ring homomorphism DP† , Q (∞) → p∗ DP† , Q (∞) (this is a consequence

of [17, 2.3.1]). The ring DP† , Q (∞) is thus endowed with a natural structure of left (p −1 DP† , Q (∞), p−1 OP , Q (∞))-bimodule from which the claim follows easily. Therefore, one obtains for F geom (M) the OP , Q (∞)-linear presentation
W ∗ F geom (M) Coker p∗ (r−1 P /P (∞) ⊗ (Lπ,µ ⊗ p M)) (2.7.3)  ∗ → p∗ (rP /P (∞) ⊗ (LW π,µ ⊗ p M)) ,

where •P /P (∞) = •P /P ⊗ OP , Q (∞), all tensor products are taken over OP , Q (∞), and the arrows are defined by the tensor product connection on LW π,µ ⊗ p ∗ M. Over an open subset on which V is trivial, the choice of a trivialisation turns this presentation into a DP† , Q (∞)-linear presentation, which induces on the cokernel

the canonical DP† , Q (∞)-module structure of F geom (M). In particular, this induced structure is independent of the trivialisation, and can be glued on variable open subsets of X.

2.8. In view of our applications to Dwork cohomology, we want now to describe the geometric Fourier transform of the constant DP† , Q (∞)-module OP , Q (∞) as a local cohomology sheaf. So let us first recall (in the smooth and liftable case, and for an overconvergent isocrystal) the definition of the overconvergent local cohomology sheaves with supports in a closed subvariety. As before, we denote by X a smooth formal scheme, H ⊂ X a divisor in its special fiber, Y = X \ H . Let Z ⊂ X be a closed subscheme, U = Y \ Z = X \ (H ∪ Z). If L is an isocrystal on Y overconvergent along H , the overconvergent local cohomology of L with support in Z is the complex of OX, Q († H )-modules given by R †Z (L) := R sp ∗ (L → jU† (L)). † † This complex can be endowed with a natural structure of complex of DX, Q ( H )module: the case where H = ∅ is treated in [11, (4.1.5)], and one proceeds in the same way in the general case, using [13, 4.4.3]. Its cohomology sheaves will be denoted by HZ†i (L). When L is the constant isocrystal, we will use the notation R †Z (OX, Q († H )), HZ† i (OX, Q († H )).

Remark. Using the method of [17, 4.4.4], the definition of overconvergent local † † cohomology can be extended to coherent DX, Q ( H )-modules (we refer to [18] for the comparison between the two methods for overconvergent isocrystals). If i : Z → X b (D † † is a closed immersion of smooth formal S-schemes, and M ∈ Dcoh X, Q ( H )), we obtain with this definition a canonical isomorphism [17, (4.4.5.2)] ∼

i+ i ! M −−→ R †Z (M).

(2.8.1)

199

Dwork cohomology for singular hypersurfaces

For smooth subvarieties, the local structure of overconvergent local cohomology is similar to the local structure of algebraic local cohomology: Proposition 2.9. With the previous notation, assume that Z is smooth of codimension r in X. Then: (i) For all i  = r, HZ† i (OX, Q († H )) = 0. (ii) Let t1 , . . . , tn be local coordinates on X such that Z = V (t 1 , . . . , t r ), where t i is the reduction of ti mod m. Then the map sending 1 to 1/t1 . . . tr provides a † † DX, Q ( H )-linear isomorphism r 

 † † DX, Q( H )

i=1

† † DX, Q ( H )ti

+

n  i=r+1

∼ † † DX, ( H )∂ −−→ HZ† r (OX, Q († H )). t i Q (2.9.1)

We may assume that X is affine and has local coordinates t1 , . . . , tn as in (ii). For 1 ≤ i1 < · · · < ik ≤ r, let Hi1 ...ik = H ∪V (ti1 )∪· · ·∪V (tik ), and Ui1 ...ik = X\Hi1 ...ik . ˇ exact sequence Using the open covering of U given by U1 , . . . , Ur , we get a Cech [14, (1.2.2)] 0 → jU† OXK →

r  i=1

jU† i OXK → · · · → jU† 1...r OXK → 0.

Since Hi1 ...ik is the support of a divisor, the complex R sp∗ jU† i

1 ...ik

OXK is reduced

to its cohomology sheaf in degree 0, which is OX, Q († Hi1 ...ik ). Thus the complex R †Z (OX, Q († H )) is isomorphic to OX, Q († H ) →

r 

OX, Q († Hi ) → · · · → OX, Q († H1...r ) → 0 → . . . .

i=1

On the other hand, the sequence t1 , . . . , tr is regular on OX, Q († H ), hence the complex 0 → OX, Q ( H ) → †

r 

OX, Q († H )[1/ti ] → · · · → OX, Q († H )[1/t1 . . . tr ] → 0

i=1

is acyclic in degrees  = r. Note that this is a complex of OX, Q († H ) ⊗OX, Q DX, Q (m) is (m) (H )⊗  OX D modules; let DX, Q († H ) = OX, Q († H ) ⊗O DX, Q . Since B (m)

(m)

X, Q

X

X

 (H )⊗OX D for all m [13, (3.3.4)], D † († H ) is flat over DX, Q († H ). flat over B X X X, Q Hence, assertion (i) will follow if we prove that, for any sequence i1 < · · · < ik , the canonical map † † † † DX, Q ( H ) ⊗DX, Q († H ) OX, Q ( H )[1/ti1 . . . tik ] → OX, Q ( Hi1 ...ik )

(2.9.2)

200

Francesco Baldassarri and Pierre Berthelot

is an isomorphism. A standard computation shows that the map sending 1 to 1/ti1 . . . tik yields an isomorphism k  

DX, Q († H )

j =1

DX, Q († H )∂tij tij +



DX, Q († H )∂ti



i =i1 ,...,ik ∼

−−→ OX, Q († H )[1/ti1 . . . tik ]. Similarly, the map sending 1 to 1/ti1 . . . tik gives an isomorphism k  

† † DX, Q( H )

j =1

† † DX, Q ( H )∂tij tij +

 i =i1 ,...,ik

∼ † † DX, ( H )∂ −→ OX, Q († Hi1 ...ik ). ti − Q

(2.9.3) Indeed, this is proposition (4.3.2) of [11] if H = ∅; as observed in the remark of [14, 4.7], this remains true in the general case, using the method of the proof of [14, 4.6]. These presentations imply that the map (2.9.2) is an isomorphism. Finally, the presentation (2.9.3), combined with the exact sequence r  i=1

OX, Q († H1...i...r ) → OX, Q († H1...r ) → HZ† r (OX, Q († H )) → 0,

implies assertion (ii). Returning to the setting of 2.6, our next result gives the Fourier transform of OP , Q (∞): Proposition 2.10. Under the assumptions of 2.6, let i : X → V be the zero section, and let us identify X with its image in V ⊂ P . Then there exists a canonical isomorphism of DP† , Q (∞)-modules F geom (OP , Q (∞)) HX† r (OP , Q (∞)).

(2.10.1)

Let us first assume that X is affine, with local coordinates x1 , . . . , xn defining derivations ∂x1 , . . . , ∂xn and that V = ArX , with standard linear coordinates t1 , . . . , tr defining derivations ∂t1 , . . . , ∂tr . The Spencer resolution of OP , Q (∞) over DP† , Q (∞) yields an isomorphism (P , OP , Q (∞))

A†r (X, H )

r   i=1

A†r (X, H )∂ti +

 j

A†r (X, H )∂xj .

Therefore, the naive Fourier transform of OP , Q (∞) is defined by (P , F naive (OP , Q (∞))) A†r (X, H )

r   i=1

A†r (X, H )ti +

 j

A†r (X, H )∂xj .

201

Dwork cohomology for singular hypersurfaces

On the other hand, proposition 2.9 (ii) shows that (P , HX† r (OP , Q (∞))) has precisely the same presentation. Using 2.7, we obtain a DP† , Q (∞)-linear isomorphism F geom (OP , Q (∞)) F naive (OP , Q (∞)) HX† r (OP , Q (∞))

(2.10.2)

as in (2.10.1). To complete the construction of (2.10.1) in the general case, we need to glue the previous isomorphisms (2.10.2) on variable open subsets where V is trivial, and therefore to prove that they are independent of the choice of coordinates. r , with basis t , . . . , t , defining the dual basis t , . . . , t . Thus Let E = OX 1 r r 1 )), and t , . . . , t are a regular sequence t1 , . . . , tr are coordinates on ArX = Spf(S(E r 1 of generators for the ideal of the zero section in V . The DP† , Q (∞)-linear surjective map DP† , Q (∞) → OP , Q (∞) provides the following diagram F geom (OP , Q (∞)) OO

∼

F geom (DP† , Q (∞)) O

∼

∼

DP† , Q (∞) ⊗ q ∗ (∧r E )

/ F naive (OP , Q (∞)) OO

∼

/ H † r (O

P , Q (∞))

X

/ F naive (D †

OO

†

2 D (∞) ffffff P , Q f f f f f ffff ffff∼ff f f f f f ffff P , Q (∞))

.

In this diagram, the upper composite arrow is the isomorphism (2.10.2), and the right vertical arrow is the map corresponding to (2.9.1). The left square is commutative by functoriality, and the right one because of the definition of the isomorphism (2.10.2). The oblique arrow is the trivialisation given by the basis t1 ∧· · ·∧tr of ∧r E , and yields a commutative triangle as explained in 2.7. Since the left composite arrow is canonical, it suffices to check that the composite map DP† , Q (∞)⊗q ∗ (∧r E ) → HX† r (OP , Q (∞)) is independent of the choice of coordinates. Equivalently, it suffices to check that the image of 1 under the corresponding map DP† , Q (∞) → HX† r (OP , Q (∞)) ⊗ q ∗ (∧r (E ∨ )) is independent of the choice of coordinates. Since it is the section 1 ⊗ t1 ∧ · · · ∧ tr , this is clear. t ...t 1

r

Remarks. (i) It follows from this local calculation that, if one computes the Fourier transform of OP , Q (∞) using the isomorphism F geom (OP , Q (∞)) H 0 (p∗ (P /P (∞) ⊗ LW π,µ [r])) •

(2.10.3)

as in (2.7.3), the isomorphism ∼

ε : H r (p∗ (P /P (∞) ⊗ LW −→ HX† r (OP , Q (∞)) π,µ )) − •

(2.10.4)

202

Francesco Baldassarri and Pierre Berthelot

defined by (2.10.1) is the unique DP† , Q (∞)-linear isomorphism such that, for any basis t1 , . . . , tr of E , ε((dt1 ∧ · · · ∧ dtr ) ⊗ e) =

1 t1 . . . tr

,

where e is the canonical section of LW π,µ . (ii) Following the method of [8], one could also give a more conceptual proof of proposition 2.10. However, our local computation will be useful to check the compatibility of (2.10.1) with Frobenius actions. 2.11. To define Frobenius actions, we will use the fact that the inverse image func† tor for DX, Q (∞)-modules can be defined with respect to non necessarily liftable morphisms of schemes between the special fibers, as explained in [16, 2.1.6]. In par† ticular, the inverse image F ∗ M of a left DX, Q (∞)-module M by the s-th power of the absolute Frobenius endomorphism of X can be defined without assuming that it can be lifted to X. When such a lifting F exists, F ∗ M is the usual inverse image by F , and, up to canonical isomorphism, it is independent of the choice of F . Applying † this remark to DX, Q (∞), one can associate to the s-th power of the absolute Frobe† nius endomorphism of X a transfer bimodule DX→X, Q (∞) which can be locally

† identified to F ∗ DX, Q (∞), for any local lifting F . This allows to extend globally

† the definition of the functor F ∗ to the derived category D b (DX, Q (∞)) by the usual L

† formula F ∗ M = DX→X, Q (∞) ⊗D †

X, Q (∞)

M. A Frobenius action on a complex ∼

† ∗ M ∈ D b (DX, −→ M in Q (∞)) can then be defined as an isomorphism  : F M −

† D b (DX, Q (∞)). The existence of Frobenius actions will generally follow from the functoriality properties of rigid cohomology. Thus, the isomorphism W (LW F geom (OP , Q (∞)) = p+ π,µ ) p∗ (P /P (∞) ⊗ Lπ,µ )[r] •

provides a Frobenius action on F geom (OP , Q (∞)) coming from the F -isocrystal struc ture of LW π,µ and the functoriality properties of rigid cohomomology for V relatively to V . Similarly, the canonical F -isocrystal structure of OP , Q (∞) and the fact that F ∗ commutes with the j † functors provides a Frobenius action on HX† r (OP , Q (∞)). Proposition 2.12. The canonical isomorphism (2.10.1) commutes with the Frobenius actions defined above on F geom (OP , Q (∞)) = H r (p∗ (•P /P (∞) ⊗ LW π,µ )) and HX† r (OP , Q (∞)).

This is a local property, hence we may assume that X is affine and that V = ArX ; let X = Spf A . We may also assume that there exists h ∈ A lifting a local equation of H in X. We fix a lifting F : X → X of the s-th power of the absolute Frobenius q endomorphism of X, and we extend it to V and V by setting F ∗ (ti ) = ti , F ∗ (ti ) = ti q .

203

Dwork cohomology for singular hypersurfaces

The Frobenius action on H r (p∗ (•P /P (∞) ⊗ LW π,µ )) is deduced by functoriality from the chosen liftings of Frobenius, and the given action on Lπ,µ . The latter is the composite isomorphism ∼

∼

F ∗ Lπ,µ = F ∗ µ ∗ Lπ −−→ µ ∗ F ∗ Lπ −−→ µ ∗ Lπ = Lπ,µ , where the first isomorphism is the Taylor isomorphism comparing the two inverse images (since F  µ  = µ  F ), and the second is the pull-back of the Frobenius action on Lπ given by (1.9.2). It follows that the Frobenius action on Lπ,µ is given by multiplication by     q    q q q q exp π ti ti )q − ti ti exp π ti ti − ti ti exp π(ti ti −ti ti ). = i

i

i

i

Thus we want to compare the action induced by (2.12.1) on

i

(2.12.1) ⊗

H r (p∗ (•P /P (∞)

†r LW π,µ )) with the canonical action of Frobenius on HX (OP , Q (∞)). In order to follow the action of Frobenius, we will use the following description of (2.10.1). Let B be the weak completion of A[h−1 , t1 , . . . , tr , t1 , . . . , tr ]. Then the • complex (P , p∗ (•P /P (∞) ⊗ LW π,µ )) can be identified with the total complex K associated to the r-uple complex such that K j1 ,...,jr = BQ if (j1 , . . . , jr ) ∈ {0, 1}r and 0 otherwise, the differentials being defined by

∇i = ∂ti + π ti : K j1 ,...,ji−1 ,0,ji+1 ,...,jr −→ K j1 ,...,ji−1 ,1,ji+1 ,...,jr . On the other hand, the covering of V × V \ X × V by the open subsets D(ti ) provides ˇ (thanks to [14, (1.2.3)]) a Cech exact sequence 0 → BQ →

  −1 †  −1 −1 † B ti Q → · · · → B t1 , . . . , tr Q i

  −1  −1   −1 †  −1 , . . . , t −1 † → 0, → B t1 , . . . , tr B t , . . . , t r 1 i Q Q i

in which the last term is equal to (P , p∗ (HP† r (OP , Q (∞)))); here P is embedded into P thanks to the closed immersion i = i × IdP . Since the arrows commute with the ∇i ’s, we can build out of this sequence a similar exact sequence of r-uple complexes. Thus, the total complexes associated to these r-uple complexes sit in a similar exact sequence of complexes  • • • • Ki → · · · → K1,...,r → K → 0, (2.12.2) 0→K → i

in which the complexes K • and K • are respectively (P , p∗ (P /P (∞) ⊗ LW π,µ )) •

204

Francesco Baldassarri and Pierre Berthelot

and (P , p∗ (P /P (∞) ⊗ HP† r (LW π,µ ))). •

It is clear from its construction that this is an exact sequence of complexes of (P , DP† , Q (∞))-modules (if we view here (P , DP† , Q (∞)) as a subring of

(P , p∗ DP† , Q (∞)) thanks to the choice of coordinates as in the remark of 2.7), and that it is compatible with the action of Frobenius. In the next lemma, we will show that ∇i is an isomorphism on any term of the form B[ti 1−1 , . . . , ti k−1 ]†Q when i is one of the ij ’s. Therefore, all complexes Ki•1 ,...,ik are b (D † acyclic, and the exact sequence (2.12.2) provides in Dcoh P , Q (∞)) an isomorphism ∼

K • [r] −−→ K • . Both complexes are actually reduced to a single cohomology sheaf in degree 0, and we obtain a DP† , Q (∞)-linear isomorphism ∼

H r (p∗ (P /P (∞) ⊗ LW −→ H 0 (p∗ (P /P (∞) ⊗ HP† r (LW π,µ )) − π,µ ))), (2.12.3) •

•

compatible with the natural Frobenius actions. Using 2.9 (i) and (2.8.1), we obtain † W ! W ∗ W HP† r (LW π,µ ) R P (Lπ,µ )[r] i+ i (Lπ,µ )[r] i+ i (Lπ,µ ).

But i ∗ (LW π,µ ) is the trivial F -isocrystal OP , Q (∞), as follows from (2.6.1) and (2.12.1) (or from the fact that the restriction of µ to V → V ×X V factors through the zero section of A1k ). Thus (2.12.3) can be written as a Frobenius compatible isomorphism ∼

H r (p∗ (P /P (∞) ⊗ LW −→ H 0 (p∗ (P /P (∞) ⊗ HP† r (OP , Q (∞)))). π,µ )) − •

•

The target can be computed using the canonical isomorphisms ∼

H 0 (p∗ (P /P (∞) ⊗ HP† r (OP , Q (∞)))) −−→ H 0 (p+ i+ (OP , Q (∞))[−r]) •

∼

−−→ H 0 (i+ q+ (OP , Q (∞))[−r]) (cf. [17, 4.3.6, 4.3.7]). The complex q+ (OP , Q (∞))[−r] is given by the relative de Rham cohomology of an overconvergent power series algebra over OX, Q (∞). Therefore, it is isomorphic to OX, Q (∞), and we obtain ∼

H 0 (p∗ (P /P (∞) ⊗ HP† r (OP , Q (∞)))) −−→ HX† r (OP , Q (∞)). •

It is easy to check that this isomorphism is compatible with the functoriality actions of Frobenius. By composition, we finally obtain an isomorphism ∼

H r (p∗ (P /P (∞) ⊗ LW −→ HX† r (OP , Q (∞)) π,µ )) − •

(2.12.4)

which is compatible with the Frobenius actions. To end the proof, we only have to check that this isomorphism is equal to (2.10.4). Remark 2.10 (i) shows that it suffices to check that (2.12.4) maps (dt1 ∧ · · · ∧ dtr ) ⊗ e to 1/t1 . . . tr . If r = 1 (which will be the case in our application), the sequence (2.12.2) is a short exact sequence of length 1 complexes, and the claim follows from

Dwork cohomology for singular hypersurfaces

205

an easy computation based on the snake lemma. In the general case, one can first observe that it is enough to prove the analogous claim in the algebraic situation, where each B[ti 1−1 , . . . , ti k−1 ]† is replaced by A[h−1 , t1 , . . . , tr , t1 , . . . , tr , ti 1−1 , . . . , ti k−1 ], because it provides a complex similar to (2.12.2), mapping to (2.12.2). Thus one can define for algebraic de Rham cohomology a morphism similar to (2.12.4) and mapping to it. It is then enough to observe that, in the algebraic situation, the rank r case can be reduced to the rank 1 case by a multiplicativity argument. We now check the acyclicity lemma used in the above proof. Lemma 2.13. For any sequence 1 ≤ i1 < · · · < is ≤ r, and any i ∈ {i1 , . . . , is }, the map  −1  −1 −1 † −1 † → B ti 1 , . . . , ti s ∇i = ∂ti + π ti : B ti 1 , . . . , ti s Q Q is an isomorphism. We may assume that i = i1 = 1, and write t, t , ∂, ∇ for t1 , t1 , ∂t1 , ∇1 . Let C =
−1 ] A[h−1 , t , . . . , t , t2 , . . . , tr , t −1 , . . . , t −1 ]† . We endow the Tate algebra A[h r

Q −1 = ti  = 1 for all i, and ∈ B[t −1 , . . . , ti s−1 ]†Q can be

is

ti 

with any Banach norm, extend it by setting ti  = take the induced norm on CQ . Then any element ϕ  written uniquely as a series ϕ = k≥0 αk t k , where the coefficients αk ∈ CQ are such that αk  ≤ cηk for some constants c, η ∈ R, η < 1. If ∇(ϕ) = 0, then (k + 1)αk+1 + π t αk = 0 for all k ≥ 0. Then the coefficient αk is given by αk = (−1)k α0

π k k t , k!

and αk  = α0 |π k /k!|. As lim|π k /k!|1/k = 1, the αk cannot be the coefficients of an element of B[t −1 , . . . , ti s−1 ]†Q if α0  = 0. Therefore, ϕ = 0.  k To check the surjectivity of ∇, let ψ = k≥0 βk t be a given element in † −1 −1 B[t , . . . , tis ]Q . We must find a sequence of elements αk ∈ CQ such that (k + 1)αk+1 + π t αk = βk for all k ≥ 0. Because there exists c, η ∈ R such that βk  ≤ cηk , with η < 1, we can define αk as the sum of the series 1 (−π t )k  j! αk := βj , πt k! (−π t )j j ≥k

which converges in CQ . The coefficients αk satisfy the previous relation, and it is easy to check that, for any η such that η < η < 1, there exists c ∈ R such that αk  ≤ c η k . Thus they define a series ϕ ∈ B[t −1 , . . . , ti s−1 ]†Q such that ∇(ϕ) = ψ.

206

Francesco Baldassarri and Pierre Berthelot

Remark. Similar computations show that the algebraic analogue of lemma 2.13, where B[ti 1−1 , . . . , ti s−1 ]† is replaced by A[h−1 , t1 , . . . , tr , t1 , . . . , tr , ti 1−1 , . . . , ti k−1 ], is also true. We can now deduce from 2.10 the main result of this section. Our proof follows the method of [27]. Theorem 2.14. Under the assumptions of 2.6, let u : X → V be a section, LW π,u = ∗ W (u×Id) Lπ,µ the overconvergent F -isocrystal on W obtained by functoriality, Z ⊂ X the zero locus of u, Z the special fiber of Z. Assume that Z is locally a complete b (D † intersection of codimension r in X. Then there exists in Dcoh X, Q (∞)) a canonical isomorphism † q+ (LW π,u ) R Z (OX, Q (∞))[r],

(2.14.1)

compatible with the Frobenius actions on both sides. As in 2.11, the Frobenius actions are defined by functoriality using comparison with rigid cohomology. Let u = u × Id : P → P . We consider the cartesian square  P 

u

/ P

u

 / P ,

q

  X

p

and we apply the functor u! to the isomorphism (2.10.1). In view of 2.9, we obtain an isomorphism †r † ! ! u! (p+ (LW π,µ )) u (HX (OP , Q (∞))) u (R X (OP , Q (∞))[r])

(2.14.2)

† in D b (DX, Q (∞)). Thus it is enough to check that there exists canonical isomorphisms W (LW u! (p+ π,µ )) q+ (Lπ,u )[−r],

u

!

(R †X (OP , Q (∞)))



(2.14.3)

R †Z (OX, Q (∞))[−r].

(2.14.4)

We only give a rough sketch here, referring to [18] for more details. Using the techniques of [17] to handle direct and inverse limits, one can reduce to proving (m) the analogs of (2.14.3) and (2.14.4) in D b (DXi ), where the subscript i denotes the reduction mod p i , and m is any positive integer. The first isomorphism is a base change result, which follows from the following two facts: (m) (m) (m) (m) (m) (m) = (m) = D ⊗ B (H ), D a) If D = D ⊗ B (H2 ) and D (m) DP ←P i i

Pi

Pi (m) BP (H2 ), i

Pi

1

Pi

Pi

Pi

Pi ←Pi

(m) (m) ⊗ then D is a flat D -module, whose formation comPi ←Pi Pi mutes with base changes;

Dwork cohomology for singular hypersurfaces

207

(m) (m) (m) (m) = D (m) ⊗ B (m) (H1 ), D b) If D Pi Pi Pi Xi ←Pi = DXi ←Pi ⊗ BPi (H1 ) and M is a (m) flat quasi-coherent D -module, then the canonical base change morphism Pi

(m) (m) ⊗ (m) u ∗ (M)) ⊗ (m) M)) → Rq∗ (D Lu ∗ (Rp∗ (D Xi ←Pi P ←P D D i

i

Pi

Pi

is an isomorphism. The proof of the second isomorphism is more delicate, and uses the description of overconvergent local cohomology with support in a closed subscheme defined by an n (
) of
(cf. [17, ideal
in terms of the RHom of the divided power envelopes P(m) 4.4.4]). This allows to reduce the assertion to the following claim: c) If J and
are the ideals of Xi and Zi in Pi and Xi respectively, the canonical morphism n n Lu ∗ (RHomOP (P(m) (J), OPi )) → RHomOXi (P(m) (
), OXi ) i

is an isomorphism. The key point here is that, thanks to our complete intersection hypothesis for Z in X, the two copies of OXi viewed as OPi -modules via the section u and the zero section, are Tor-independent over OPi . Using known results on the structure of divided power envelopes in the case of complete intersections [13, 1.5.3], it follows that the canonical map n n (J)) → P(m) (
) Lu ∗ (P(m)

is an isomorphism, which implies our claim. † This provides the construction of (2.14.1) in D b (DX, Q (∞)). However, the right hand side of (2.14.1) is known to have coherent cohomology (thanks to a straightforward generalization of [17, 4.4.9] adding overconvergent poles along some divisor). b (D † Thus, (2.14.1) is an isomorphism in Dcoh X, Q (∞)). Since (2.10.1) is compatible with Frobenius actions, the isomorphism (2.14.2) defined by applying u! to (2.10.1) is compatible with the Frobenius actions obtained by inverse image (thanks to [17, 4.3.4]). Using the construction of functoriality maps in rigid cohomology [14, 1.5] to define Frobenius actions, it is easy to check that the isomorphism (2.14.3) identifies the inverse image of the Frobenius action on • W Rp∗ (•P /P (∞) ⊗ LW π,µ ) with the Frobenius action on Rq∗ (P /X (∞) ⊗ Lπ,u ). On the other hand, using the rigid analytic construction of overconvergent local cohomology, it is also immediate to check that the isomorphism (2.14.4) identifies the inverse image of the Frobenius action on R †X (OP , Q (∞)) with the Frobenius action on R †Z (OX, Q (∞)). It follows that the isomorphism (2.14.1) commutes with Frobenius actions. Remark. The complete intersection hypothesis on Z has only been used to give a simple proof of the isomorphism (2.14.4). While (2.14.4) has not yet been checked in

208

Francesco Baldassarri and Pierre Berthelot

the general case, there is no doubt that it should be true in full generality, and therefore that 2.14 should remain valid without the complete intersection hypothesis. For example, it is worth noting that the theorem is true when the section u reduces to the zero section X → V , hence Z = X. Indeed, the functors u+ and u! only depend upon the reduction of u over Spec(k) [16], so we may assume that u itself is the zero section. Then, thanks to [17, (4.4.5.2)], there is a canonical isomorphism ∼

R †X (OP , Q (∞)) −−→ u+ (u! (OP , Q (∞))). Moreover, the functors u+ and u! are quasi-inverse equivalences between coherent † † DX, Q (∞)-modules and coherent DP , Q (∞)-modules with support in X [17, 5.3.3]. Therefore, the previous isomorphism gives an isomorphism ∼

u! (R †X (OP , Q (∞))) −−→ u! (OP , Q (∞)) = OX, Q (∞)[−r], and (2.14.4) is an isomorphism. As LW π,u = OP , Q (∞) in this case, the isomorphism (2.14.1) is simply the isomorphism ∼

q+ (OP , Q (∞)) −−→ OX, Q (∞)[r] resulting from the triviality of the relative de Rham cohomology of a vector bundle.

3 Applications to rigid cohomology We now derive consequences of 2.14 for rigid cohomology, including rigid cohomology with compact supports. We will also check the compatibility between our isomorphism and its algebraic analog, constructed in [27]. Theorem 3.1. With the notation and hypotheses of 2.14, assume in addition that X is proper over Spf R. Then there exists a canonical isomorphism Rrig (W/K, Lπ,u ) RZ∩Y, rig (Y /K)

(3.1.1)

which commutes with the natural Frobenius actions F ∗ on both cohomology spaces. Let f : X → S be the structural morphism, and n the relative dimension of X over † S. Since (2.14.1) is an isomorphism in D b (DX, Q (∞)), it defines an isomorphism † f+ (q+ (LW π,u )[−r])[−n] f+ (R Z (OX, Q (∞)))[−n]. † b As q+ (LW π,u ) belongs to Dcoh (DX, Q (∞)), we obtain W f+ (q+ (LW π,u )[−r])[−n] (f  q)+ (Lπ,u )[−r − n]

R(P , P ⊗ LW π,u ) •

Dwork cohomology for singular hypersurfaces

209

•

† an Lπ,µ(u×Id) ))

R(P , R sp∗ (PK ⊗ jW •

† an

R(PK , PK ⊗ jW Lπ,µ(u×Id) )

= Rrig (W/K, Lπ,u ), the latter isomorphism being due to the fact that (P , H2 ) is a smooth compactification of W . On the other hand, if U = X \ Z, we obtain •

f+ (R †Z (OX, Q (∞)))[−n] R(X, X ⊗ R †Z (OX, Q (∞))) •

R(X, R sp∗ ((XK ⊗ (jY† OXK → jU† ∩Y OXK )) t )) = RZ∩Y, rig (Y/K). Therefore, we obtain the isomorphism (3.1.1). As (2.14.1) is compatible with Frobenius actions, the same holds for (3.1.1). Remark. As for theorem 2.14, theorem 3.1 remains valid when the reduction of u over Spec(k) is the zero section. Corollary 3.2. Under the assumptions of 3.1, there exists a canonical isomorphism Rc, rig (W/K, Lπ,u ) Rc, rig (Z ∩ Y/K)[−2r],

(3.2.1)

which commutes with the Frobenius actions F ∗ on Rc, rig (W/K, Lπ,u ) and q r F ∗ on Rc, rig (Z ∩ Y/K). Replacing Lπ by L−π in (3.1.1) and taking K-linear duals yields an isomorphism Rrig (W/K, L−π,u )∨ RZ∩Y, rig (Y /K)∨ which commutes with the dual actions of Frobenius F∗ = F ∗ ∨ on both sides. Poincaré duality is compatible with F ∗ , and provides isomorphisms Rrig (W/K, L−π,u )∨ Rc, rig (W/K, Lπ,u )[2n + 2r], RZ∩Y, rig (Y/K)∨ Rc, rig (Z ∩ Y/K)[2n]. Since F ∗ = q n+r σ on Hc, rig (W/K) (resp. q n σ on Hc,2nrig (Y /K)), these isomorphisms identify F∗ on Rrig (W/K, L−π,u )∨ to q n+r (F ∗ )−1 on Rc, rig (W/K, Lπ,u ), and F∗ on RZ∩Y, rig (Y/K)∨ to q n (F ∗ )−1 on Rc, rig (Z ∩ Y/K). The corollary follows. 2(n+r)

3.3. We now want to check that, when the previous situation is algebraizable, the isomorphism (3.1.1) is compatible with specialization. As we are returning to a situation similar to 1.1, we change notation. For any S-scheme X, we denote by XK and Xk the generic and special fibers of X, αX : XK → X the inclusion of the generic fiber, and X the (p-adic) formal completion of X. Let f : X → S be a proper and smooth morphism of relative dimension n, q : V → X a vector bundle of rank r over X, q : V → X the dual vector bundle,

210

Francesco Baldassarri and Pierre Berthelot

P and P their projective closures over X, u : X → V a section, Z → X its zero locus. We also assume that H ⊂ X is a relative divisor over S, and we set Y = X \ H , W = q −1 (Y ), W = q −1 (Y ). Let Lπ,µ be the rank 1 module with integrable connection on VK × VK defined as the inverse image of Lπ by the canonical pairing VK ×XK VK → A1K , and Lπ,u its inverse image by the section uK × Id : VK → VK ×XK VK . Note that, on A1K , Lπ is the inverse image of the usual exponential module under the automorphism defined by multiplication by π. Therefore, we can deduce from [27, th. 0.2] canonical isomorphisms qK, + (Lπ,u )[−r] R ZK (OXK ),

(3.3.1)

RdR (WK /K, Lπ,u ) RZK ∩YK , dR (YK /K).

(3.3.2)

Proposition 3.4. Under the previous assumptions, the square RdR (WK /K, Lπ,u )

(3.3.2) ∼

/ RZK ∩YK , dR (YK /K)

L

ρWπ,u

 Rrig (Wk /K, Lπ,u )

(3.1.1) ∼



(3.4.1)

ρZ∩Y

/ RZk ∩Yk , rig (Yk /K),

where the vertical arrows are the specialization morphisms defined in 1.8 and 1.2, is commutative. To check this compatibility, we will first give an interpretation of the specialization morphisms in terms of D-modules. 3.5. Let X be a smooth S-scheme, H ⊂ X a relative divisor, j : Y → X the inclusion of Y := X \ H in X, i : X → X the canonical morphism, and i an : XK → XKan (m) the inclusion. We consistently regard OX -modules as OX -modules via i∗ . If DX (resp. DXK ) is the sheaf of differential operators of level m on X (resp. the sheaf (m) (m) of differential operators on XK ), we will use the notation DX (∞) = j∗ (DY ), DXK (∞) = jK ∗ (DYK ), DX, Q (∞) = j∗ (DY ) ⊗ Q αX ∗ (DXK (∞)), as well as OXK (∞) = jK ∗ (OYK ), OX, Q (∞) = j∗ (OY ) ⊗ Q αX ∗ (OXK (∞)). For any m, there is a canonical ring isomorphism (m)

DX, Q (∞) DX (∞) ⊗ Q. † On the other hand, the construction of DX, Q (∞) provides a ring homomorphism (0) (0) (Hk )⊗ (0) ⊗ Q → D † (∞). D DX (∞) → B X X X, Q

Thus we obtain a canonical ring homomorphism † DX, Q (∞) → DX, Q (∞).

(3.5.1)

Dwork cohomology for singular hypersurfaces

211

† If M ∈ D b (DXK (∞)), and M ∈ D b (DX, Q (∞)), a specialization morphism from M to M is by definition a morphism

RαX ∗ (M) → M,

(3.5.2)

in D b (DX, Q (∞)); note that RαX ∗ (M) = αX ∗ (M) if M is a quasi-coherent DXK (∞)module. For example, the morphism (3.5.1) itself, as well as the canonical morphism OX, Q (∞) → OX, Q (∞) defined similarly, are specialization morphisms. More generally, let M be a DXK (∞)-module, M an the associated analytic sheaf, which is a (DXK (∞)) an -module. Note that, for any open subset U ⊂ X, with formal completion U, we have UK ⊂ XK ∩UKan . It follows that there is a natural DX, Q (∞)linear morphism αX ∗ (M) → sp∗ (i an ∗ (M an )). Therefore, if M an is a (DXK (∞)) an -module such that M := sp∗ (i an ∗ (M an )) is en† dowed with a structure of DX, Q (∞)-module inducing its natural DX, Q (∞)-module structure, the datum of a (DXK (∞)) an -linear morphism M an → M an defines a specialization morphism from M to M. In particular, we will use this remark in the following situations: a) Let L be a coherent OYK -module endowed with an integrable connection, such that the induced connection on L = jY†k (L an ) is overconvergent along Hk . If M = jK ∗ (L) and M = sp∗ (i an ∗ (L)), then there is a canonical specialization morphism from M to M. b) Let Z ⊂ X be a closed subscheme, U = Y \ Z = X \ H ∪ Z, M = R ZK (OXK (∞)), M = R †Zk (OX, Q (∞)). If J • is an injective resolution of OXK (∞) over DXK (∞), and J • an injective resolution of (OXK (∞)) an over (DXK (∞)) an , one can choose a (DXK (∞)) an -morphism ϕ : J • an → J • inducing the identity on (OXK (∞)) an . As XKan \ ZKan is a strict neighbourhood of ]Uk [ X , ϕ induces a morphism ( ZK (J • )) an → (J • → jU† k (J • )) t . One obtains in this way a canonical specialization morphism from M to M. Specialization morphisms are functorial in X in the following sense. Let f : X → X be an S-morphism, H ⊂ X, H ⊂ X relative divisors such that f −1 (H ) ⊂ H , Y = X \ H , Y = X \ H . As for (3.5.1), there are natural specialization morphisms on X † αX ∗ (DXK →XK (∞)) → DX→X , Q (∞) † αX ∗ (DXK ←XK (∞)) → DX ←X, Q (∞),

where DXK →XK (∞) = jK ∗ (DYK →YK ), DXK ←XK (∞) = jK ∗ (DYK ←YK ). Moreover, these morphisms are semi-linear with respect to the ring homomorphism † f −1 (DX , Q (∞)) → f −1 (DX , Q (∞)) deduced from (3.5.1) on X . It follows that, for

212

Francesco Baldassarri and Pierre Berthelot

† any complexes M ∈ D b (DXK (∞)), M ∈ D b (DX , Q (∞)) (resp. M

D b (D

XK (∞)),

† D b (DX, Q (∞))),

M ∈ a specialization morphism from (resp. from M to M) defines canonically specialization morphisms

M

∈

to M

αX ∗ (fK∗ (M )) → f ∗ (M ), αX ∗ (fK + (M)) → f+ (M). Finally, a specialization morphism defines a morphism between de Rham cohomologies. This is the particular case of the previous situation where X = S, and it can be described in the following way. A specialization morphism from † M ∈ D b (DXK (∞)) to M ∈ D b (DX, Q (∞)) defines a morphism ωX ⊗L (0) RαX ∗ (M) → ωX ⊗L (0) M. DX

DX

(0)

Using the de Rham resolution of ωX over DX and taking global sections, we obtain a morphism RdR (XK /K, M) → RdR (X/S, M).

(3.5.3)

When X is proper, and M, M come from an OYK -module with connection L as in a) above, the morphism (3.5.3) can be written as RdR (YK /K, L) → Rrig (Yk /K, L). The computation based on de Rham resolutions shows that this morphism is the specialization morphism ρYL defined in (1.8.2). Similarly, if X is proper, M = R ZK (OXK (∞)) and M = R †Zk (OX, Q (∞)) as in b) above, the morphism (3.5.3) can be written as RZK ∩YK , dR (YK /K) → RZk ∩Yk , rig (Yk /K), and this morphism is the morphism ρZ∩Y defined in (1.2.2). 3.6. We now return to the proof of 3.4. We endow P , P and P = P ×X P with the divisors defined by H and the hyperplanes at infinity as in 2.6. Using the natural specialisation morphism for Lπ,µ , and applying the previous remarks, we obtain a specialization morphism W αP ∗ (p+ (LW π,µ )) → p+ (Lπ,µ ), W where LW π,µ is defined as in 2.6, Lπ,µ denotes the direct image of (Lπ,µ )|WK ×WK by the inclusion WK × WK → PK × PK , and we keep the notation p for the projections PK → PK and P → P . On the other hand, we also obtain a specialization morphism

αP ∗ (HXr K (OPK (∞))) → HX† kr (OP , Q (∞)).

Dwork cohomology for singular hypersurfaces

213

These morphisms fit in a commutative square ∼

/ αP ∗ (HXr (OP (∞))) K K

(2.10.1) ∼

 / H † r (OP , Q (∞)),

(LW )) αP ∗ (p+ π,µ



(LW ) p+ π,µ

(3.6.1)

Xk

where the upper isomorphism is the algebraic analogue of (2.10.1) (cf. [27, 2.3]). Indeed, this commutativity is a local property on X, hence one may assume that V = ArX , with coordinates t1 , . . . , tr , and then it follows from the fact that both isomorphisms send the section (dt1 ∧ · · · ∧ dtr ) ⊗ e, where e is the basis of LW π,µ (resp. . . . t of the corresponding local cohomology sheaf. LW ), to the section 1/t π,µ r 1 Using the isomorphisms (2.14.3) and (2.14.4), and their algebraic analogues, it follows by functoriality that the specialization morphisms defined in 3.5 fit in a commutative square RαX ∗ (q+ (LW π,u ))

∼

/ RαX ∗ (R Z (OXK (∞)))[r] K

 q+ (LW π,u )

(2.14.1) ∼

 / R † (OX, Q (∞))[r], Zk

(3.6.2)

where the upper isomorphism is the image by RαX ∗ of the isomorphism defined in [27, 0.2]. Taking de Rham cohomology, the proposition follows as explained in 3.5. 3.7. Let us now assume that we are in the situation considered in 1.3, where K is a number field, with ring of integers R, and S = Spec R. Consider a proper and smooth S-scheme X, endowed with a divisor H , Y = X \ H , a vector bundle V of rank r over X, and a section u : X → V of the dual vector bundle, such that the zero locus Z of u is flat over S, and locally a complete intersection of codimension r in X. For each closed point s ∈ S 0 , let K(s) be the completion of K at s, k(s) its residue field, ps the characteristic of k(s). We choose for each s a root πs of the polynomial Xps −1 + ps in a finite extension K (s) of K(s), with residue field k (s). If R is an R-algebra, we denote by the subscript R objects deduced from S-objects by base change from Spec(R) to Spec(R ). Then, combining 1.4 with the previous proposition, and using Poincaré duality, we obtain the following corollary: Corollary 3.8. Under the previous assumptions, there exists a finite subset  ⊂ S 0 such that the morphisms L

ρWπs ,u : RdR (WK (s) /K (s), Lπs ,u ) → Rrig (Wk (s) /K (s), Lπs ,u ),

(3.8.1)

πs ,u : Rc, rig (Wk (s) /K (s), Lπs ,u ) → RdR, c (WK (s) /K (s), Lπs ,u ) ρc,W

(3.8.2)

L

are isomorphisms for all s ∈ / .

214

Francesco Baldassarri and Pierre Berthelot

4 The algebraic and analytic Dwork complexes We will now use the results of the previous sections to explain the geometric interpretation of the algebraic and analytic complexes constructed by Dwork to obtain a rationality formula for the zeta function of a projective hypersurface over a finite field. In this section, K will be a finite extension of Qp , R its ring of integers, k its residue field, of cardinality q = ps . We assume that K contains the primitive p-th roots of 1, and we fix an element π ∈ K such that π p−1 = −p. Let X = PSn be the projective space of relative dimension n over S = Spec(R), X1 , . . . , Xn+1 the standard projective coordinates on X, H1 , . . . , Hn+1 the corresponding coordinate hyperplanes, Ui = X \ Hi , H = H1 ∪ · · · ∪ Hn+1 , Y = X \ H . We fix an homogeneous polynomial f ∈ R[x1 , . . . , xn+1 ] of degree d ≥ 1, and we denote by Z ⊂ X the projective hypersurface defined by f . As before, the subscripts K and k will denote the generic and special fibers. In [28, (21)], Dwork introduces a characteristic series χF (t) defined by a Frobenius operator, such that the zeta function of the affine hypersurface Zk ∩ Yk can be expressed by the formula n

ζ (Zk ∩ Yk , qt) = (1 − t)−(−δ) χF (t)−(−δ)

n+1

,

(4.0.1)

where the operator δ on the multiplicative group K[[t]]× is defined by A(t)δ = A(t)/A(qt). Although the proof given in [28] is non-cohomological, Dwork gave in subsequent articles a cohomological interpretation of this formula when Zk is non singular ([29], [30]). We will show here that, using Dwork’s computations and our previous results, this formula has an interpretation in terms of rigid cohomology which holds also in the singular case. 4.1. We first recall the construction of the algebraic and analytic Dwork complexes associated to f (cf. [29, §3]). Let T be the set of multi-indexes u = (u0 , u1 , . . . , un+1 ) ∈ Nd+2 such that du0 = u1 + · · · + un+1 . a) We denote by L the graded sub-algebra of K[X0 , X1 , . . . , Xn+1 ] whose elements are polynomials of the form  P (X0 , X1 , . . . , Xn+1 ) = au Xu . u∈T

For any b ∈ R, b > 0, we denote by L(b) the sub-algebra of the power series algebra K[[X0 , X1 , . . . , Xn+1 ]] defined by     L(b) = ξ = au Xu  ∃ c ∈ R such that ord(au ) ≥ bu0 + c . u∈T

The algebra L(b) can be endowed with the norm ξ  = supu |au |pbu0 , for which it is a p-adic Banach algebra. If b < b , then L(b ) ⊂ L(b), and the inclusion is a

215

Dwork cohomology for singular hypersurfaces

completely continuous map [48]. We define  L(0+ ) = L(b). b>0

b) For any i ≥ 1, the differential operator Di = Xi

∂ ∂f + π X0 Xi ∂Xi ∂Xi

(4.1.1)

acts on L and L(b), for all b, hence also on L(0+ ). Moreover, we have Di  Dj = Dj  Di for all i, j . Therefore, we can form “Koszul complexes” K (L; D) ⊂ K (L(b); D) ⊂ K (L(0+ ); D). •

•

•

using the sequence D = (D1 , . . . , Dn+1 ). For example, K • (L; D) is defined as L→

n+1 

L · ei →

i=1



L · ei ∧ ej → · · · → L · e1 ∧ · · · ∧ en+1 ,

i<j

the differential being defined by d(P ei1 ∧ · · · ∧ eik ) =

n+1  (Di P )ei ∧ ei1 ∧ · · · ∧ eik . i=1

The definition of K (L(b); D) and K (L(0+ ); D) is similar. We will consider these complexes as being concentrated in degrees in [0, n + 1]. •

•

c) Let F (X0 , . . . , Xn+1 ) ∈ K[[X0 , . . . Xn+1 ]] be the formal power series q

q

q

F (X0 , . . . , Xn+1 ) = exp(π(X0 f (X1 , . . . , Xn+1 ) − X0 f (X1 , . . . , Xn+1 ))). If b0 is small enough, then F ∈ L(b0 ), and multiplication by F in K[[X]] induces hand, one can a continuous endomorphism of L(b ) for any
b ≤ bu0. On the other u , and ψ induces define an endomorphism ψ of K[[X]] by ψ = a X a X u u u qu a continuous homomorphism ψ : L(b/q) → L(b) for any b ≤ b0 . Finally, one can define an endomorphism α : K[[X]] → K[[X]] by α(ξ ) = ψ(F ξ ). If b ≤ qb0 , then α induces a completely continuous endomorphism of L(b), which is the composite F

ψ

α : L(b) → L(b/q) − → L(b). → L(b/q) − Dwork defines the characteristic series χF (t) of α as the limit (for the topology of pointwise convergence of coefficients in K) of the characteristic polynomials det(I − tαN ), where αN is the endomorphism of the vector space of polynomials of degree ≤ N defined by replacing F by its truncation in degrees ≤ (q − 1)N [29, §2]. In particular, χF does not depend upon b, and will not change if K is replaced by an extension which is complete under a valuation inducing the p-adic valuation on K. On the other hand, we can view α as a completely continuous endomorphism of L(b)

216

Francesco Baldassarri and Pierre Berthelot

for small b, which allows to define its Fredholm determinant det(I − tα). For any such b, we have det(I − tα) = χF (t), thanks to [48, §9]. Since α is a completely continuous endomorphism of L(b), it endows L(b) with the structure of a nuclear space in Monsky’s sense [44, 1.3], and det(I − tα) is equal to Monsky’s characteristic series for nuclear operators. Moreover, since det(I − tα) is independent of b, the space L(0+ ) = ∪b L(b) endowed with α is still nuclear, and the characteristic series of α on L(0+ ) is still equal to det(I − tα) = χF (t) [44, 1.6]. For any i, we have α  Di = qDi  α. Thus, one can define an endomorphism (again be denoted by α) of the complex K • (L(0+ ); D), by setting α(P ei1 ∧ · · · ∧ eik ) = q n+1−k α(P )ei1 ∧ · · · ∧ eik

(4.1.2)

in degree k. The characteristic series of α on K (L(0+ ); D) is then defined by •

det(I − tα|K (L(0+ ); D)) := •

n+1 

det(I − tα|K i (L(0+ ); D))(−1)

i

i=0

= χF (t)(−δ)

n+1

.

(4.1.3)

On the other hand, α acts on the cohomology spaces H i (K • (L(0+ ); D)). By [44, 1.4], they are still nuclear spaces, and we have det(I − tα|K (L(0+ ); D)) = •

n+1 

i

det(I − tα|H i (K (L(0+ ); D)))(−1) . •

i=0

hence n+1 

i

det(I − tα|H i (K (L(0+ ); D)))(−1) = χF (t)(−δ) •

n+1

.

(4.1.4)

i=0

4.2. To give a geometric interpretation of Dwork’s complexes, we follow [27], and we introduce the vector bundle V = Spec(S(OX (d))) over X. Let q : V → X be the projection, Vi = q −1 (Ui ), Hi = q −1 (Hi ), H = ∪i Hi = q −1 (H ). We view X as a closed subscheme of V via the zero section, and we define D = H ∪ X, W = V \ H = q −1 (Y ), W ∗ = V \ D = W \ Y . We observe that D is a relative normal crossings divisor in V above S = Spec(R). For all r ≥ 0, we will denote by rV (log D) the sheaf of differential forms of degree r over V with logarithmic poles along D. Note also that these definitions, as well as the definition of L given above, make sense over any base ring R. Lemma 4.3. With the above notation, we have: (i) Over any base ring R, there is a natural isomorphism of R-algebras ∼

L −−→ (V , OV ),

(4.3.1)

217

Dwork cohomology for singular hypersurfaces

and H i (V , OV ) = 0 for all i ≥ 1. (ii) If d is invertible in R, the sheaf 1V (log D) is a free OV -module. Since V = Spec(S(OX (d))), we have Rq∗ (OV ) = q∗ (OV ) =



OX (md),

m≥0

hence R(V , OV ) R(X,



OX (md))

m≥0



(X, OX (md)) (V , OV ).

m≥0

If we denote by k[X1 , . . . , Xn+1 ]i the R-submodule of homogeneous polynomials of degree i, we can compose the standard isomorphism  ∼  k[X1 , . . . , Xn+1 ]md −−→ (X, OX (md)) m≥0

m≥0

with the obvious isomorphism ∼

L −−→



k[X1 , . . . , Xn+1 ]md

m≥0

defined by the substitution X0  → 1, and the first assertion follows. To construct a basis for 1V (log D), one can use the following local coordinates on the open subsets Vj , 1 ≤ j ≤ n + 1. For i = j , 1 ≤ i ≤ n + 1, let xi,j = Xi /Xj , so that the xi,j are local coordinates on Uj . On the other hand, the section tj = Xjd ∈ (X, OX (d)) ⊂ (V , OV ) defines a relative coordinate on V above Uj . Keeping the notation xi,j for the inverse images of the xi,j on Vj , we obtain a system of local coordinates (xi,j , tj )i=j on Vj . It will sometimes be convenient to use also the notation xj,j = Xj /Xj = 1. Thus, the image in (Vj , OV ) of a monomial X u ∈ L is the section u

u1 n+1 . . . xn+1,j . tju0 x1,j

(4.3.2)

If d is invertible in R, we can now define a family (ωi )1≤i≤n+1 of global differential forms on V by setting dxi,j 1 dtj + if j  = i, d tj xi,j 1 dti 1 dti dxi,i = (= + ). d ti d ti xi,i

ωi |Vj = ωi |Vi

(4.3.3)

When j varies, the ωi |Vj glue to define global sections ωi = d1 dtti i ∈ (V , 1V (log D)), and it is clear that, on any Vj , they form a basis of 1V (log D). 4.4. Returning to our initial context where R is the ring of integers of K, we denote by X, V, Uj , Vj the formal p-adic completions of X, V , Uj , Vj . Over Uj,K , the

218

Francesco Baldassarri and Pierre Berthelot

generic fiber VK of V is the compact open subset of VKan defined by the condition |tj (y)| ≤ 1, so that the map VK → XK is a locally trivial fibration whose fibers are closed unit discs. Let P be the projective closure of V and P be the formal completion of P along its special fiber. We recall that the tube ]Vk [ P of Vk in PK is equal to VK . Let jV† be the functor of overconvergent sections around VK , as defined in 1.1. The first assertion of the previous lemma has the following analytic counterpart: Lemma 4.5. With the previous notation, there is a natural isomorphism of K-algebras ∼

L(0+ ) −−→ (VKan , jV† (OVKan ))

(4.5.1)

extending the isomorphism (4.3.1). Furthermore, H i (VKan , jV† (OVKan )) = 0 for all i ≥ 1. For any ρ ∈ R, ρ ≥ 1, and any j ∈ {1, . . . , n + 1}, let Vj,ρ ⊂ (q an )−1 (Uj,K ) be the affinoid open subset defined as Vj,ρ = {y | |tj (y)| ≤ ρ}. If j, j ∈ {1, . . . , n + 1}, then tj = xjd ,j tj . As |xj ,j (x)| = 1 for all x ∈ Uj ,K ∩ Uj,K , it follows that Vj,ρ ∩ (q an )−1 (Uj,K ∩ Uj ,K ) = Vj ,ρ ∩ (q an )−1 (Uj,K ∩ Uj ,K ). Hence the Vj,ρ for variable j glue together to define an open subset Vρ ⊂ VKan . Note that V1 = VK . > It is clear that, for ρ → 1, the Vρ are a fundamental system of strict neighbourhoods of VK . Thus, if jρ : Vρ → VKan denotes the inclusion morphism, there is a natural ∼ isomorphism jV† (OVKan ) −−→ limρ →1 > jρ ∗ (OVρ ). − →  For b > 0, let ξ = u∈T au Xu ∈ L(b), and let ρ be such that 1 < ρ < pb . Since xi,j  ≤ 1 in (Uj,K , OXKan ), the series   u  u    u u au tj 0 xi,ji = au xi,ji tj 0 u∈T

u0

i

u1 +···+un+1 =du0

i

converges towards an element ξj,ρ ∈ Vj,ρ . Moreover, since tj = xjd ,j tj and xi,j = xi,j xj,j above Uj ∩ Uj , these series glue for variable j to define an element ξρ ∈ (Vρ , OVKan ). Then the homomorphism (4.5.1) is obtained by sending ξ ∈ L(b) ⊂ L(0+ ) to the image of ξρ in (VKan , jV† (OVKan )), for any ρ such that 1 < ρ < pb . If ξ  = 0, then ξj,ρ  = 0, hence (4.5.1) is injective. To prove it is surjective, we define, for b ≥ 0,     au Xu  ord(au ) − bu0 → +∞ if u0 → +∞ . L (b) = ξ = u∈T

Thus L (b) ⊂ L(b) for all b > 0, and L(0+ ) = tion provides a natural homomorphism



b>0 L

(b).

L (logp (ρ)) −→ (Vρ , OVρ )

The previous construc(4.5.2)

for any ρ ≥ 1. Then it suffices to construct a decreasing sequence of real numbers ρm , with limit 1, such that the following holds when ρ is one of the ρm ’s:

Dwork cohomology for singular hypersurfaces

219

a) The homomorphism (4.5.2) is an isomorphism. On the other hand, it follows from 1.1 that, for any fixed ρ0 > 1, we have H i (VKan , jV† (OVKan )) H i (Vρ0 , jV† (OVKan ))

lim H i (Vρ0 , jρ ∗ (OVρ )) − → > ρ →1

lim H i (Vρ , OVρ ), − → > ρ →1

where the second isomorphism is due to the fact that Vρ0 is quasi-compact and separated, and the third one to the fact that, for any affinoid A in Vρ0 , A∩Vρ is affinoid. As above, the vanishing of H i (VKan , jV† (OVKan )) will follow if we construct a decreasing sequence of real numbers ρm , with limit 1, such that the following holds when ρ is one of the ρm ’s: b) For any i > 0, H i (Vρ , OVρ ) = 0. Let us prove that assertions a) and b) hold when ρ belongs to the sequence ρm = p 1/m . As it suffices to prove these properties after a finite extension of the base field K, we may assume that there exists an element λ ∈ K such that |λ| = ρ. Then multiplication by λ in the vector bundle VKan induces an isomorphism of rigid spaces ∼ hλ : V1 −−→ Vρ . Moreover, the substitution X0  → λX0 defines an isomorphism ∼ h λ : L (logp (ρ)) −−→ L (0), and the homomorphisms (4.5.2) are compatible with h λ and h∗λ . Therefore, it suffices to prove a) and b) when ρ = 1. In this case, we have V1 = VK , and H i (V1 , OV1 ) = H i (V, OV ) ⊗ K for all   i ≥ 0. On the other hand, L (0) = L R ⊗ K, where LR is the p-adic completion of the algebra L constructed over the base ring R. Denoting by Rj , Vj the reductions modulo p j of R, V, and applying lemma 4.3 over Rj , we obtain that ∼

H i (Vj , OVj ) = 0

LRj −−→ (Vj , OVj ),

if i ≥ 1.

In particular, the cohomology groups H i (Vj , OVj ) satisfy the Mittag-Leffler condition for all i ≥ 0, and therefore this gives an isomorphism ∼

H i (V, OV ) −−→ lim H i (Vj , OVj ) ← − j

 for all i. Since L R = lim j LRj , assertions a) and b) follow. ← − Theorem 4.6. Under the assumptions of 4.2 and 4.4, let q : V → X be the dual vector bundle of V , u : X → V the section defined by the homogeneous polynomial f ∈ (X, OX (d)), Lπ,f the rank one module with connection on V obtained as the inverse image of Lπ by the morphism V → V × V → A1S defined by u, Lπ,f = an ) the corresponding overconvergent F -isocrystal on V . We denote again by jV† (Lπ,f k the subscripts K and k the generic fiber and the special fiber of an S-scheme.

220

Francesco Baldassarri and Pierre Berthelot

(i) There exists an isomorphism of complexes ∼

•

•

θ : K (L; D) −−→ (VK , VK (log DK ) ⊗ Lπ,f ),

(4.6.1)

which can be identified in degree 0 to the isomorphism (4.3.1) (using the canonical basis of Lπ,f ). In the derived category of K-vector spaces, θ defines an isomorphism ∼

K (L; D) −−→ RdR (WK∗ /K, Lπ,f ). •

(4.6.2)

(ii) There exists an isomorphism of complexes ∼

an θ † : K (L(0+ ); D) −−→ (VKan , jV† (V an (log DKan ) ⊗ Lπ,f )) •

•

K

(4.6.3)

which can be identified in degree 0 to the isomorphism (4.5.1). In the derived category of K-vector spaces, θ † defines an isomorphism ∼

K (L(0+ ); D) −−→ Rrig (Wk∗ /K, Lπ,f ), •

(4.6.4)

in which the endomorphism α of K • (L(0+ ); D) corresponds to the endomorphism F∗ = q n+1 (F ∗ )−1 on rigid cohomology. Let us recall that, by construction, Lπ,f is a rank 1 OV -module endowed with a natural basis e. If we use this basis to identify Lπ,f to OV , then the isomorphisms θ and θ † have been defined in degree 0 by the previous lemmas. Thus, if θj is the composed homomorphism ∼

θj : L −−→ (VK , OVK ) → (Vj,K , OVK ), and u ∈ T, we obtain in (Vj,K , Lπ,f ) u

u1 n+1 θ(Xu ) = θj (Xu ) ⊗ e = tju0 x1,j . . . xn+1,j ⊗ e,

(4.6.5)

thanks to (4.3.2). In higher degrees, we define θ (resp. θ † ) as the unique isomorphism which is semi-linear with respect to (4.3.1) (resp. (4.5.1)), and sends any product ei1 ∧ · · · ∧ eik to ωi1 ∧ · · · ∧ ωik ⊗ e, where (ωi )i is the basis defined in (4.3.3). We obtain in this way isomorphisms of graded modules θ and θ † . For each j , let tj be the dual coordinate associated to tj on Vj = q −1 (Uj ). Under the composed morphism ϕj : Vj → V × V → A1S , the inverse image of the coordinate t ∈ (A1S , OA1 ) is S

ϕj∗ (t)

= u ∗ (tj )tj = tj f (x1,j , . . . , xn+1,j ) = θj (f ).

It follows that, viewing f as an element of (V , OV ) through (4.3.1), the connection ∇π,f of Lπ,f is given by ∇π,f (g e) = (d(g) + πg d(f )) ⊗ e for any section g of OV .

(4.6.6)

Dwork cohomology for singular hypersurfaces

221

Since the ωi are a basis of 1VK (log DK ) over OVK , we can define derivations ∂i of OVK by setting d(g) =

n+1 

∂i (g) ωi ,

i=1

so that ∇π,f is given by ∇π,f (g e) =

n+1  (∂i (g) + πg∂i (f )) ωi ⊗ e. i=1

To prove the commutation of θ and θ † with the differentials, it is then enough to prove that, when g ∈ K[X1 , . . . , Xn+1 ] is homogeneous of degree dk, the isomorphism (4.3.1) maps X0k Xi ∂g/∂Xi to ∂i (g) for all i. We can compute in (Vn+1,K , OVK ), k g(x and use the coordinates tn+1 , x1,n+1 , . . . , xn,n+1 to write g = tn+1 n+1 , 1), where x n+1 stands for x1,n+1 , . . . , xn,n+1 . Then we obtain k d(g) = ktn+1 g(x n+1 , 1)

n

∂g dtn+1  k dxi,n+1 + tn+1 xi,n+1 (x n+1 , 1) tn+1 ∂Xi xi,n+1 i=1 n 

k = dktn+1 g(x n+1 , 1) ωn+1 + k (dkg(x n+1 , 1) − = tn+1

+

n  i=1

k tn+1 xi,n+1

n  i=1

i=1

k tn+1 xi,n+1

xi,n+1

∂g (x , 1)(ωi − ωn+1 ) ∂Xi n+1

∂g (x , 1)) ωn+1 ∂Xi n+1

∂g (x , 1) ωi , ∂Xi n+1

from which the claim follows. The acyclicity property of lemma 4.3 implies that •

•

(VK , VK (log DK ) ⊗ Lπ,f ) = R(VK , VK (log DK ) ⊗ Lπ,f ) in the derived category. As Lπ,f has no singularities along D, Deligne’s theorem [25, II 3.14] shows that the canonical morphism R(VK , VK (log DK ) ⊗ Lπ,f ) −→ R(WK∗ , W ∗ ⊗ Lπ,f ) •

•

K

is an isomorphism. Combined with (4.6.1), it provides the isomorphism (4.6.2). Similarly, lemma 4.5 implies that •

•

an an )) = R(VKan , jV† (V an (log DKan ) ⊗ Lπ,f )). (VKan , jV† (V an (log DKan ) ⊗ Lπ,f K

K

222

Francesco Baldassarri and Pierre Berthelot

in the derived category. On the other hand, corollary A.4 of the Appendix provides an isomorphism ∼

an R(VKan , jV† (V an (log DKan ) ⊗ Lπ,f )) −−→ Rrig (Wk∗ /K, Lπ,f ), •

K

so that this gives (4.6.4) by composition as in the algebraic case. Indeed, let T be the infinity divisor in the special fiber of P , and D the closure of D in P . As Lπ,f has an ) satisfies the hypothesis of A.1, and we can apply no singularities along D, jV† (Lπ,f an ) on the strict corollary A.4 to P endowed with the divisors T , D, and to jV† (Lπ,f neighbourhood VKan of VK . Then (A.4.1) gives the above isomorphism. To compare the Frobenius actions on K • (L(0+ ); D) and Rrig (Wk∗ /K, Lπ,f ), we must describe explicitly the F -isocrystal structure of Lπ,f . We first observe that V can be endowed with a global lifting FV of the Frobenius morphism of Vk by q q q setting FV∗ (Xi ) = Xi for all i, hence FV∗ (xi,j ) = xi,j , FV∗ (tj ) = tj for all i, j . Let ∼

u = u × IdV : V → V × V . Then the Frobenius action φπ,f : FV∗ (Lπ,f ) −−→ Lπ,f is given by the composed isomorphism ∼

∼

FV∗ (Lπ,f ) = FV∗ u ∗ µ∗ (Lπ ) −−→ u ∗ µ∗ FA∗1 (Lπ ) −−→ u ∗ µ∗ (Lπ ) = Lπ,f , where the first isomorphism is the Taylor isomorphism relating the two inverse images of Lπ under the morphisms µ  u  FV and FA1  µ  u , and the second one is the ∼ pull-back of φ : FA∗1 (Lπ ) −−→ Lπ . Over Vj , the inverse images of the coordinate t on q q A1 under the morphisms µ  u  FV and FA1  µ  u are equal respectively to tj f (xi,j ) q and tj f (xi,j )q . Thus, the restriction of the first isomorphism over (q an )−1 (Uj,K ) is q q given in the canonical basis by multiplication by exp(π tj (f (xi,j ) − f (xi,j )q )). From (1.9.2) we deduce that the restriction of the second one is given by multiplication by q exp(π(tj f (xi,j )q − tj f (xi,j ))). Therefore, φπ,f is given by q

q

φπ,f (1 ⊗ e) = exp(π(tj f (xi,j ) − tj f (xi,j ))) e.

(4.6.7)

The inverse image morphism F ∗ on Rrig (Wk∗ , Lπ,f ) is obtained by applying the functor R(VKan , −) to the morphism of complexes π,f : † • jW ∗ (V an ) ⊗j † (O K

V

VKan )

Lπ,f

F ∗ ⊗Id

† • / F∗ (jW ∗ (V an ) ⊗j † (O K

V

VKan )

F ∗ (Lπ,f ))

∼ Id ⊗φπ,f



† • F∗ (jW ∗ (V an ) ⊗j † (O an ) V K V K

Lπ,f ).

The direct image morphism F∗ on Rrig (Wk∗ , Lπ,f ) is defined as the Poincaré dual of n+1 (F ∗ )−1 the morphism F ∗ on Rc, rig (Wk∗ , L∨ π,f ), and it is easy to check that F∗ = q

∗ ∗ (since F ∗ = q n+1 on Hc,2n+2 rig (Wk /K) [47, 6.5], and F is an isomorphism compatible with pairings). On the other hand, FV is finite étale of rank q n+1 over WK∗ , and the

223

Dwork cohomology for singular hypersurfaces

corresponding trace morphism extends to a morphism of complexes •

•

† † Tr F : F∗ (jW ∗ (V an )) → jW ∗ (V an ) K

K

such that the composed morphism F∗

•

Tr F

•

•

† † † → F∗ (jW −→ jW jW ∗ (V an ) − ∗ (V an )) − ∗ (V an ) K

is multiplication by

q n+1 .

† • F∗ (jW ∗ (V an ) ⊗j † (O K

V

K

K

Let π,f be the composed morphism

F ∗ (Lπ,f )) o an )

−1 Id ⊗φπ,f

VK

∼

† • F∗ (jW ∗ (V an ) ⊗j † (O K

V

VKan )

Lπ,f )

∼

 † • F∗ (jW ( )) ∗ V an ⊗j † (O K

V

VKan )

Lπ,f

Tr F ⊗ Id

† • / jW ∗ (V an ) ⊗j † (O K

V

VKan )

Lπ,f .

It is clear that π,f  π,f = q n+1 , hence π,f induces F∗ on Rrig (Wk∗ /K, Lπ,f ). As F ∗ (dtj /tj ) = q dtj /tj and F ∗ (dxi,j /xi,j ) = q dxi,j /xi,j , the morphism Tr F can also be defined on jV† (•V an (log DKan )), and the canonical morphism K

• jV† (V an (log DKan )) K

•

† −→ jW ∗ (V an ) K

commutes with the morphisms Tr F on both complexes. Repeating the definition of π,f , we obtain an endomorphism F∗ of the complex (VKan , jV† (•V an (log DKan )) ⊗ K Lπ,f ) such that the canonical isomorphism ∼

(VKan , jV† (V an (log DKan )) ⊗ Lπ,f ) −−→ Rrig (Wk∗ /K, Lπ,f ) •

K

commutes with F∗ . Therefore, it suffices to show that the isomorphism θ † identifies α to F∗ . But on the one hand θ † identifies q n+1 ψ on L(0+ ) with the trace of F on the algebra (VKan , OVKan ). On the other hand we have on (q an )−1 (Uj,K ) q

q

θ † (F (X0 , . . . , Xn+1 )) = exp(π(tj f (xi,j ) − tj f (xi,j ))), −1 which by (4.6.7) is the series defining φπ,f . The claim then follows easily from (4.1.2).

As a consequence, general results known for rigid cohomology also apply to Dwork cohomology: Corollary 4.7. Without assumption on f , the Dwork cohomology spaces H i (K • (L(0+ ); D)) are finite dimensional K-vector spaces, and α induces an automorphism on these spaces. Thanks to (4.6.4), this follows from 1.9, or from [42].

224

Francesco Baldassarri and Pierre Berthelot

Corollary 4.8. Let K be a number field, R its ring of integers, S = Spec(R), S 0 the set of closed points in S. For each s ∈ S 0 , let K(s) be the completion of K at s, ps its residue characteristic, K (s) a finite extension of K(s) containing a root πs of the polynomial X ps −1 + ps . Assume that f ∈ R[X1 , . . . , Xn+1 ] is a homogeneous polynomial of degree d ≥ 1, and denote by K • (Ls ; D), K • (Ls (0+ ); D) the Dwork complexes built with f on K (s). Then there exists a finite subset  ⊂ S 0 such that, for all s ∈ S 0 \ , the inclusion K (Ls ; D) ⊂ K (Ls (0+ ); D) •

•

induces an isomorphism on the cohomology spaces. Using again (4.6.4), this is a consequence of 3.7. Our next theorem relates Dwork’s cohomology with the rigid homology of the affine hypersurface Zk ∩ Yk . Theorem 4.9. Under the assumptions of 4.2 and 4.4, there exists distinguished triangles +1

•

RZK ∩YK , dR (YK /K) −→ K (L; D) −→ RdR (YK /K)[−1] −−−→

(4.9.1)

+1

RZk ∩Yk , rig (Yk /K) −→ K (L(0+ ); D) −→ Rrig (Yk /K)[−1] −−−→ (4.9.2) •

and a canonical morphism of distinguished triangles RZK ∩YK , dR (YK /K) ρZ∩Y



RZk ∩Yk , rig (Yk /K)

/ K • (L; D)

/ RdR (YK /K)[−1]

 / K • (L(0+ ); D)

 / Rrig (Yk /K)[−1]

+1

/

+1

/

ρY [−1]

(4.9.3) where the left and right vertical maps are the specialisation morphisms (1.2.2) and the middle one is the canonical inclusion. Moreover, the arrows in the triangle (4.9.2) commute with the following endomorphisms: a) q n+1 F ∗ −1 on RZk ∩Yk , rig (Yk /K), b) α on K • (L(0+ ); D), c) q n F ∗ −1 on Rrig (Yk /K)[−1]. With the notation of 4.2, the decomposition D = H ∪ X gives rise to an exact sequence of logarithmic de Rham complexes Res

0 −→ V (log H ) −→ V (log D) −−→ X (log H )(−1)[−1] −→ 0, •

•

•

(4.9.4)

where Res is the residue map [25, II 3.7]. This sequence is compatible with the action of FV by functoriality, provided that the functoriality map on •X (log H ) is multiplied

225

Dwork cohomology for singular hypersurfaces

by q, as indicated by the −1 twist: this is due to the fact that, on each Vj , we have FV∗ (dtj /tj ) = q dtj /tj . Tensoring with Lπ,f , which is trivial on the zero section X ⊂ V , and taking cohomology on VK , we obtain a distinguished triangle R(XK , •XK (log HK ))[−1] gNNN pp NNN +1 ppp NNN p p p NN w pp p • / R(VK , •VK (log DK ) ⊗ Lπ,f ). R(VK , VK (log HK ) ⊗ Lπ,f )

(4.9.5)

Using the Grothendieck–Deligne theorem, we obtain the following isomorphisms: ∼

•

∼

•

R(VK , VK (log HK ) ⊗ Lπ,f ) −−→ R(WK , WK ⊗ Lπ,f ), •

R(VK , VK (log DK ) ⊗ Lπ,f ) −−→ R(WK∗ , W ∗ ⊗ Lπ,f ), •

K

∼

•

•

R(XK , XK (log HK )) −−→ R(YK , YK ). On the other hand, (3.3.2) gives a canonical isomorphism ∼

•

R(WK , WK ⊗ Lπ,f ) −−→ RZK ∩YK , dR (YK /K). Thus the triangle (4.9.5) can be written as +1

RZK ∩YK , dR (YK /K) −→ RdR (WK∗ /K, Lπ,f ) −→ RdR (YK /K)[−1] −→ . (4.9.6) The exact sequence (4.9.4) defines a similar sequence on VKan . Applying jV† , tensoring with Lπ,f and taking cohomology on VKan provides a distinguished triangle R(XKan , •X an (log HKan ))(−1)[−1] (4.9.7) K g O O o OOO o +1 ooo OOO o o OOO o o o wo / R(V an , j † (• an (log D an )) ⊗ Lπ,f ), R(VKan , jV† (•V an (log HK an )) ⊗ Lπ,f ) V K K V K

K

in which the arrows are compatible with the action of FV by functoriality. Using A.4, we obtain as in the algebraic case isomorphisms ∼

† (V an ) ⊗ Lπ,f ), R(VKan , jV† (V an (log HK an )) ⊗ Lπ,f ) −−→ R(VKan , jW •

•

K

∼ • R(VKan , jV† (V an (log DKan )) ⊗ Lπ,f ) −−→ K ∼

•

K

• † (VKan , jW ∗ (V an ) ⊗ Lπ,f ) K •

R(XKan , X an (log HKan )) −−→ R(XKan , jY† (X an )), K

K

and the targets are respectively equal by construction to Rrig (Wk /K, Lπ,f ), Rrig (Wk∗ /K, Lπ,f ), Rrig (Yk /K). Note that these isomorphisms identify FV∗ to

226

Francesco Baldassarri and Pierre Berthelot

F ∗ . Thanks to (3.1.1), we also have an isomorphism Rrig (Wk /K, Lπ,f ) RZk ∩Yk , rig (Yk /K) which is compatible to F ∗ . Taking the Frobenius actions into account, the triangle (4.4) can therefore be rewritten as +1

RZk ∩Yk , rig (Yk /K) −→ Rrig (Wk∗ /K, Lπ,f ) −→ Rrig (Yk /K)(−1)[−1] −→ . (4.9.8) Because of the functoriality of the constructions used to build the triangles (4.9.5) and (4.4) from the exact sequence (4.9.4), it is clear from the definition of the specialization morphisms given in 1.2 and 1.8 that they fit in a morphism of triangles RdR (WK /K, Lπ,f )

/ RdR (W ∗ /K, Lπ,f ) K Lπ,f

Lπ,f

ρW

 Rrig (Wk /K, Lπ,f )

+1

/

+1

/.

/ RdR (YK /K)[−1] ρY [−1]

ρW ∗

 / Rrig (W ∗ /K, Lπ,f ) k

 / Rrig (Yk /K)[−1]

On the other hand, proposition 3.4 shows that the isomorphisms (3.3.2) and (3.1.1) L identify ρZ∩Y with ρWπ,f . Therefore, the specialization morphisms define a morphism of triangles from (4.9.6) to (4.9.8). Finally, we can use the isomorphisms (4.6.2) and (4.6.4) to rewrite triangle (4.9.6) as (4.9.1), and triangle (4.9.8) as (4.9.2). Under these isomorphisms, the inclusion K • (L; D) ⊂ K • (L(0+ ); D) corresponds to the inclusion •

•

(VK , VK (log DK ) ⊗ Lπ,f ) ⊂ (VKan , jV† (V an (log DKan ) ⊗ Lπ,f )), K

L hence to the specialization morphism ρWπ,f ∗ . K

Therefore, we obtain the morphism of tri-

angles (4.9.3). As the endomorphism α of K (L(0+ ); D) corresponds to q n+1 (FV∗ )−1 in the derived category, the morphisms of the triangle (4.9.2) satisfy the announced compatibilities with Frobenius actions. •

Using the triangle (4.9.2), one can give an interpretation of Dwork’s formula (4.0.1) in terms of rigid cohomology: Corollary 4.10. The characteristic series χF (t) of α satisfies the relation χF (t)

(−δ)n+1

= (1 − t)

−(−δ)n

= (1 − t)

−(−δ)n

2n−2 

det(I − t qF ∗ |Hc,i rig (Zk ∩ Yk /K))(−1)

i

i=n−1

ζ (Zk ∩ Yk , qt)−1 .

(4.10.1)

227

Dwork cohomology for singular hypersurfaces

Since all complexes in the triangle (4.9.2) have finite dimensional cohomology groups, we obtain the relation det(I − t α|K (L(0+ ); D)) = det(I − t q n+1 F ∗ −1 |RZk ∩Yk , rig (Yk /K)) •

× det(I − t q n F ∗ −1 |Rrig (Yk /K)[−1]), and, by (4.1.4), we have det(I − t α|K (L(0+ ); D)) = χF (t)(−δ) •

n+1

.

The affine variety Yk is an n-dimensional torus, hence we have for any i n

i Hrig (Yk /K) = K ( i ) , i (Y /K). It follows that and F ∗ is multiplication by q i on Hrig k n

det(I − t q n F ∗ −1 |Rrig (Yk /K)[−1]) = (1 − t)−(−δ) . Finally, Poincaré duality provides a perfect pairing RZk ∩Yk , rig (Yk /K) ⊗K Rc, rig (Zk ∩ Yk /K) −→ K(−n)[−2n], which is compatible to F ∗ . Therefore the automorphism q n+1 F ∗ −1 is dual to qF ∗ on Rc, rig (Zk /K), and we obtain det(I − t q n+1 F ∗ −1 |RZk ∩Yk , rig (Yk /K))  i det(I − t qF ∗ |Hc,i rig (Zk ∩ Yk /K))(−1) . = i

Thanks to [33, 6.3 I], the second relation of the corollary follows. To complete the proof of the first one, we observe first that Hc,i rig (Zk ∩ Yk /K) = 0 for i > 2n − 2 because dim(Zk ∩ Yk ) = n − 1. On the other hand, Hc,i rig (Yk \ Zk /K) = Hc,i rig (Yk /K) = 0 for i < n, because Yk \ Zk and Yk are affine and smooth of dimension n, so that their rigid cohomology with compact supports is Poincaré dual to their Monsky–Washnitzer cohomology, which is concentrated in degrees ≤ n. Hence Hc,i rig (Zk ∩ Yk /K) = 0 for i < n − 1. Corollary 4.11. For any i, the eigenvalues of α acting on the Dwork cohomology space H i (K • (L(0+ ); D)) are Weil numbers, whose weights belong to the interval [2n − 2i + 2, 2n − i + 2]. The first assertion results from the fact that the eigenvalues of F ∗ on RZk ∩Yk , rig (Yk /K) are Weil numbers, a result proved by Chiarellotto using comparison with -adic cohomology [24, I 2.3], and more recently by direct p-adic methods by Kedlaya [43]. To get estimates on the weights, we use the exact sequences i−1 HZi k ∩Yk , rig (Yk /K) −→ H i (K (L(0+ ); D)) −→ Hrig (Yk /K). •

228

Francesco Baldassarri and Pierre Berthelot

From the above discussion, it follows that q n F ∗ −1 is pure of weight 2n − 2(i − 1) i−1 (Yk /K). From [24, I 2.3], we get that q n+1 F ∗ −1 is mixed of weights in on Hrig [2n − 2i + 4, 2n − i + 2] on HZi k ∩Yk , rig (Yk /K). The above estimate follows.

5 The coordinate filtration While theorems 4.6 and 4.9 relate the cohomology of Dwork complexes to the de Rham and rigid cohomologies of the open subsets ZK ∩ YK and Zk ∩ Yk , the cohomologies of the projective hypersurfaces ZK and Zk themselves can also be computed using the Dwork complexes, and the same holds for the cohomologies of all their intersections with any intersection of coordinate hyperplanes. The method developed by Dwork relies on the construction of certain subcomplexes of L and L(0+ ), and we want now to give a cohomological interpretation of these subcomplexes, and of the following formula (5.0.1) of Dwork, which is closely related. We denote by N the subset {1, . . . , n+1} ⊂ N. For each non empty subset A ⊂ N , we define  HA = Hi , ZA = Z ∩ HA . i ∈A /

In particular, ZN = Z. Let m(A) = #(A) − 1, so that HA is a linear projective subspace of dimension m(A) of PnS , and let PA (t) be the rational function defined by ζ (ZA,k , t) = PA (t)(−1)

m(A)

(−1)m(A)

= PA (t)

  m(A) (1 − q i t)

(1 − q m(A) t) (1 − q

m(A)

i=0 m(A) t)ζ (Pk , t).

Then a combinatorial argument based on (4.0.1) shows that  n+1 χF (t)δ = (1 − t) PA (qt)

(5.0.1)

A⊂N A =∅

(cf. [29, (4.33)]). 5.1. For all A ⊂ N , let A : L → L, †A : L(0+ ) → L(0+ ) be the homomorphisms / A, and Xi to Xi otherwise. We define defined by sending Xi to 0 if i ∈ LA = Im(A : L → L),

LA (0+ ) = Im(†A : L(0+ ) → L(0+ )).

Thus we obtain L∅ = K, LN = L. We denote respectively by fA and DA,i , for i ∈ A, the polynomial and the differential operator deduced from f and Di by substituting 0 to Xj if j ∈ / A. Thus fA is the

Dwork cohomology for singular hypersurfaces

229

equation defining ZA as an hypersurface in HA , and DA,i is the differential operator defined by fA as in (4.1.1). We can introduce the algebras LA and LA (0+ ) built on the variables X0 and Xi for i ∈ A, and define the Koszul complexes K • (LA ; D A ), K • (LA (0+ ); D A ) associated to the differential operators DA,i , i ∈ A. We denote by αA the endomorphism of K • (LA (0+ ); D A ) defined by the polynomial fA . For all B ⊂ A ⊂ N, let MB = i∈B Xi , let (MB ) ⊂ K[[X0 , . . . , Xn+1 ]] be the ideal generated by MB , and let LB A = LA ∩ (MB ),

+ + LB A (0 ) = LA (0 ) ∩ (MB ).

In particular, L∅A = LA . We want to define an increasing filtration on the complexes K • (L; D) and • K (L(0+ ); D). We first define for each subset A ⊂ N an increasing filtration FilA r L + )) on L (resp. L(0+ )) by L(0 (resp. FilA r   0 if r < 0,     LB if 0 ≤ r ≤ #A, B⊂A FilA N r L= #(A\B)=r     L if r > #A A A A + (resp. L(0+ ), LB N (0 )). Note that Fil0 L = LN , and Fila L = L for a = #A. • + We can now define Filr K (L; D) (resp. L(0 )) by  Filr K j (L; D) = FilA (5.1.1) r L · eA #A=j

(resp. L(0+ )), where eA = ei1 ∧ · · · ∧ eij if A = {i1 , . . . , ij } with i1 < · · · < ij . To check that this is indeed a sub-complex, we observe that  d(P · eA ) = Di (P ) ei ∧ eA . i ∈A /

Assume i ∈ / A, and write Ai = A ∪ {i}. If P is divisible by MB for some B ⊂ A with #(A \ B) = r, then i ∈ / B, and Di (P ) is divisible by Xi , hence by Xi MB = MB∪{i} . i Therefore Di (P ) ∈ FilA r , hence Filr is a subcomplex. A For all A ⊂ N , LN (0+ ) is stable under the endomorphism α of L(0+ ). Indeed, this is clearly the case for both multiplication by F (X0 , . . . , Xn+1 ) and ψ. It follows that, for all r, the subcomplex Filr K • (L(0+ ); D) is stable under the endomorphism α of K • (L(0+ ); D). Finally, we observe that, since LA and LA (0+ ) are built from fA as L and L(0+ ) from f , the complexes K • (LA ; D A ) and K • (LA (0+ ); D A ) can be endowed with an analogous filtration. To describe the associated graded complex, we first prove a lemma.

230

Francesco Baldassarri and Pierre Berthelot

Lemma 5.2. Let A ⊂ N be a subset, and A = N \ A. For all r, there exists a canonical isomorphism  ∼ LB −→ gr A (5.2.1) r L A ∪B − B⊂A #(A\B)=r

(resp. L(0+ )). The proofs for L and L(0+ ) are completely parallel, so we limit ourselves here to A B the case of L. For B ⊂ A, with #(A \ B) = r, we have LB A ∪B ⊂ LN ⊂ Filr L, which defines the map. u Let a = #A. We first observe that FilA r L has a K-basis formed by monomials X , A u ∈ T, such that ui ≥ 1 for at least a − r indices in A. Therefore gr r L has a basis formed by the classes of monomials X u such that ui ≥ 1 for exactly (a − r) indices in A. This set of monomials is the disjoint union, for all B ⊂ A with #(A \ B) = r, of the subsets of monomials divisible by MB and not by any Xi for i ∈ A \ B. It follows immediately that the map (5.2.1) is an isomorphism. Proposition 5.3. With the notation of 5.1, there exists for all r a canonical isomorphism of complexes  ∼ • • Fil0 K (LA ; D A )[−r] −−→ gr r K (L; D) (5.3.1) #A=n+1−r

(resp. LA (0+ ), L(0+ )). In the case of L(0+ ), this isomorphism is compatible with the actions of the endomorphisms α. We first describe Fil0 K • (LA ; D A ), with A ⊂ N . For all B ⊂ A, we have • B FilB 0 LA = LA , hence Fil0 K (LA ; D A ) is the complex   A LA −→ LB LB A · eB −→ . . . −→ A · eB −→ LA · eA −→ 0. B⊂A #B=1

B⊂A #B=#A−1

Using lemma 5.2, we obtain for each r and each j  gr A gr r K j (L; D) = r L · eA #A=j







#A=j

B⊂A #B=j −r

LB A ∪B · eA ,

with A = N \ A. Let A = A ∪ B. Then #A = n + 1 − r, and the datum of (A, B) is equivalent to the datum of (A , B). Let C = A \ B. If we map eB to (−1)rj ε(B, C)eA , where ε(B, C) is the signature of the permutation {B, C} of A, we

Dwork cohomology for singular hypersurfaces

obtain an isomorphism gr r K j (L; D)







#A =n+1−r B⊂A #B=j −r



#A =n+1−r

231

LB A · eB

Fil0 K j −r (LA ; D A ).

It is then easy to check that, for variable j , these isomorphisms define an isomorphism of complexes as claimed in (5.3.1). The same proof applies to gr r K • (L(0+ ); D), and the compatibility with the endomorphisms α is straightforward. Corollary 5.4. The characteristic series χF (t) satisfies the relation  n+1 #A • = (1 − t) det(I − tα| Fil0 K (LA (0+ ); D A ))(−1) . χF (t)δ

(5.4.1)

A=∅

n+1

= det(I − tα|K • (L(0+ ); D)). Since By definition (cf. (4.1.3)), χF (t)(−δ) • the filtration Fil• K (L(0+ ); D) is stable under α, the corollary follows from the proposition by observing that gr n+1 K (L(0+ ); D) Fil0 K (L∅ (0+ ); 0)[−n − 1], •

•

and that Fil0 K • (L∅ (0+ ); 0) = K • (L∅ (0+ ); 0) = K, with α∅ = Id. We now want to give a geometric interpretation of the complexes Fil0 K • (LA ; D A ) and Fil0 K • (L(0+ )A ; D A ), providing a computation of det(I − tα) in terms of zeta functions. It is clearly sufficient to treat the case where A = N. 5.5. We first introduce a notation used in our next theorem. Let C be an abelian category, and C(C) the category of complexes of objects of C. For any n ∈ Z, the truncation functor τ≥n associates to a complex E • ∈ C(C) the complex τ≥n E

•

: · · · −→ 0 −→ E n /d(E n−1 ) −→ E n+1 −→ · · · . •,•

The category C(C) is itself an abelian category. If E is a double complex of objects of C, we can view it as a complex of objects of C(C) indexed by the second II E •,• the double complex obtained by applying index. We will then denote by τ≥n •,• •,• the truncation functor in C(C(C)). If g : E → E is a morphism of double II g : τ II E •,• → τ II E •,• . It is easy to complexes, then g induces a morphism τ≥n ≥n ≥n •,• •,• are homotopic in the sense of [23, IV 4], then the check that, if g, g : E → E II g and τ II g are also homotopic. In particular, the morphisms τ II g morphisms τ≥n ≥n ≥n t II g induced between the associated simple complexes are homotopic (in the and τ≥n t usual sense).

232

Francesco Baldassarri and Pierre Berthelot

Let C be another abelian category, and F : C → C a left exact functor. Assume that C has enough injectives. Then any complex E • ∈ C + (C) has an injective •,• Cartan–Eilenberg resolution I for which there exists an index i0 such that I i,j = 0 II to the double complex F (I •,• ) defined for i < i0 . We can apply the functor τ≥n II F (I •,• ) . The above by such a resolution, and take the associated total complex τ≥n t remarks show that, viewed as an object in the derived category D(C ), this complex •,• is independent of the choice of the resolution I , and depends functorially upon • + E ∈ C (C). We will denote II τ≥n RF t : C + (C) −→ D + (C ).

the functor defined in this way. For n = 1 and any E • ∈ C + (C), this construction gives rise to a distinguished triangle •

•

+1

•

II RF t (E ) −−→ F (E ) −→ RF (E ) −→ τ≥1

(5.5.1)

in D + (C ). This triangle is functorial with respect to E • when E • varies in C + (C) (but of course not in D + (C)). •,• When E • ∈ C ≥0 (C), one can find a Cartan–Eilenberg resolution I of E • such • i,j II = 0 for i < 0. Therefore, τ≥n RF t (E ) is concentrated in degree ≥ n. that I Applying the usual truncation functor τ≥n to the second morphism in (5.5.1), one obtains a canonical morphism •

•

II τ≥n RF (E ) −→ τ≥n RF t (E )

(5.5.2)

in D ≥n (C).

Theorem 5.6. (i) There exists natural isomorphisms of complexes ∼

•

•

Fil0 K (L; D) −−→ (VK , VK ⊗ Lπ,f ), ∼

(5.6.1)

Fil0 K (L(0+ ); D) −−→ (VKan , jV† (V an ) ⊗ Lπ,f ), •

•

K

(5.6.2)

the latter being compatible with the Frobenius actions α on Fil0 K • (L(0+ ); D) and F∗ = π,f on (VKan , jV† (•V an ) ⊗ Lπ,f ) (cf. 4.6 for the definition of π,f ). K (ii) Let s : X → V be the zero section. With the notation of 5.5, the functoriality morphisms •

•

•

q∗ (V ⊗ Lπ,f ) −→ q∗ s∗ (X ) Pn , S

•

•

•

q∗an (jV† (V an ) ⊗ Lπ,f ) −→ q∗an s∗an (X an ) Pn an , K

K

K

233

Dwork cohomology for singular hypersurfaces

induced by s give rise to a commutative diagram of isomorphisms

∼

/ τ≥1 R(PnK , •Pn )

∼

II R (V , • ⊗ L τ≥1 t K π,f ) VK

K



II R (V an , j † (• ) ⊗ L τ≥1 t K π,f ) V V an

∼



(5.6.3)

∼

/ τ≥1 R(PnKan , •Pn an ), K

K

in which the lower horizontal isomorphism is compatible with the Frobenius actions F ∗ . The algebraic part of this theorem is essentially [27, 3.3]. We give here a proof which extends to the rigid context. To prove assertion (i), we consider the isomorphisms of complexes ∼

K • (L; D) _  K (L(0+ ); D) •

∼

/ (VK , •V (log DK ) ⊗ Lπ,f ) K _  (log DKan )) ⊗ Lπ,f ) V an

/ (V an , j † (• K

V

K

provided by (4.6.1) and (4.6.3), and we look at the images of Fil0 under these isomorphisms. For all A ⊂ N, the element eA is mapped to    1 dtj dxi,j ωA = ωi = + , xi,j d tj i∈A

i∈A

where the last equality takes place in (Vj,K , •VK (log DK )) for any j . Expanding this form, one obtains a sum of terms whose denominator is of degree ≤ 1 with respect A to any of the coordinates tj , xi,j , i ∈ A. As any monomial X u ∈ FilA 0 L = LN is divisible by the X i ’s for i ∈ A, its image in (Vj,K , OVK ) is divisible by the xi,j , i ∈ A. If A  = ∅, i∈A ui > 0, hence u0 > 0, and the image of Xu is also divisible by tj . It follows that the image of X u eA is a differential form which has no poles along DK . Therefore the isomorphisms (4.6.1) and (4.6.3) map respectively Fil0 K • (L; D) and Fil0 K • (L(0+ ); D) to (VK , •VK ⊗ Lπ,f ) and (VKan , jV† (•V an ) ⊗ Lπ,f ). K Conversely, let X u ∈ L be a monomial such that the image of X u eA is in un+1 u1 . . . xn+1,j ωA belongs to (VK , •VK ⊗ Lπ,f ). Then, for each j , the form tju0 x1,j • (Vj,K , VK ). Using (4.3.3), it follows that, for each j , ui ≥ 1 for all i ∈ A, i  = j , hence that ui ≥ 1 for all i ∈ A. Therefore Xu eA belongs to Fil0 K • (L; D), and (5.6.1) is an isomorphism. Applying the same argument with a series in L(0+ ), we see that (5.6.2) is also an isomorphism. The compatibility with Frobenius actions results from the proof of 4.6. Using a Cartan–Eilenberg resolution of •VK ⊗ Lπ,f , we obtain a canonical isomorphism ∼

II II R t (VK , VK ⊗ Lπ,f ) −−→ τ≥1 R t (PnK , q∗ (VK ⊗ Lπ,f )) τ≥1 •

•

234

Francesco Baldassarri and Pierre Berthelot

which defines by functoriality the upper horizontal morphism in (5.6.3). Using the first cohomology spectral sequences, the proof that it is an isomorphism is reduced to proving that the maps H i (PnK , q∗ (rVK )) → H i (PnK , rPn ) are isomorphisms for all K

i ≥ 1 and all r. As 1V /Pn q ∗ (OPnS (d)), there are exact sequences S

0 −→ q ∗ (rPn ) −→ rV −→ q ∗ (r−1 Pn (d)) −→ 0 S

(5.6.4)

S

for all r. They give rise to exact sequences   rPn (kd) −→ q∗ (rV ) −→ r−1 0 −→ Pn (kd) −→ 0, S

k≥0

k>0

S

in which the map rPn → q∗ (rV ) is a section of the map q∗ (rV ) → rPn defined by S

S

s. Since H i (PnS , rPn (m)) = 0 for all r and i, m > 0 (and any basis S) [26, 1.1], the S claim follows. Moreover, since •Pn is concentrated in positive degrees, (5.5.2) gives K a canonical morphism II R t (PnK , Pn ). τ≥1 R(PnK , Pn ) −→ τ≥1 •

•

K

It is an isomorphism since II R τ≥1 t

K

(PnK , rPn ) K

= 0 for r ≥ 1, and this allows to replace

by τ≥1 R in the right column of (5.6.3). The lower horizontal morphism in (5.6.3) is defined similarly. To prove that it is an isomorphism, one takes the exact sequence of analytic sheaves corresponding to (5.6.4) and one applies the exact functor jV† . Taking direct images, one obtains an exact sequence 0 → q∗an (jV† (q an ∗ (rPn an ))) → q∗an (jV† (rV an )) → q∗an (jV† (q an ∗ (r−1 Pn an (d)))) → 0. K

K

K

The surjectivity here results from the fact that R i q∗an (jV† (E)) = 0 for i ≥ 1 and any coherent OVKan -module E. Indeed, if we denote by Vρ the strict neighbourhoods of V introduced in the proof of lemma 4.5, and if U ⊂ PnKan is any affinoid open subset, then q an −1 (U ) ∩ Vρ is affinoid, which implies that H i (q an −1 (U ), jV† (E)) = 0 for i ≥ 1. In this exact sequence, we have isomorphisms ∼

q∗an (jV† (OVKan )) ⊗ rPn an −−→ q∗an (jV† (q an ∗ (rPn an ))), K

∼ q∗an (jV† (OVKan )) ⊗ r−1 −→ PnKan (d) −

K

q∗an (jV† (q an ∗ (r−1 PnKan (d)))).

Furthermore, the canonical morphism OPnKan (d) → q∗an (jV† (OVKan )) induces an isomorphism ∼

q∗an (jV† (OVKan ))(d) −−→ q∗an (jV† (OVKan ))/OPnKan .

235

Dwork cohomology for singular hypersurfaces

Therefore, proving that the lower horizontal line of (5.6.3) is an isomorphism is reduced as above to proving that H i (PnKan , q∗an (jV† (OVKan )) ⊗ rPn an (d)) = 0 K

for all i ≥ 1 and all r. One can then proceed as in the proof of lemma 4.5 to reduce this claim to a similar vanishing statement for the (algebraic) projective space over a Z/pn Z-scheme, which results again from the vanishing of the spaces H i (PnS , rPn (m)) S for all i, m > 0, all r, and any basis S. Finally, one obtains the commutative square (5.6.3) by functoriality. As the right vertical arrow is an isomorphism, the same holds for the left one. It is clear that the lower isomorphism commutes with the actions of F by functoriality.

Corollary 5.7. There exists distinguished triangles, related by a morphism of triangles defined by the specialization morphisms, Fil0 K • (L; D)

/ RZK , dR (Pn /K) K

 Fil0 K • (L(0+ ); D)



/ RdR (Pn /K) K

ρZ

/ RZk , rig (Pn /K) k



+1

/

+1

/.

∼

/ τ≥1 Rrig (Pn /K) k

(5.7.1) Moreover, the morphisms in the lower triangle commute with the following Frobenius actions: a) α on Fil0 K • (L(0+ ); D), b) q n+1 (F ∗ )−1 on RZ, rig (Pnk /K), c) q n+1 (F ∗ )−1 on τ≥1 Rrig (Pnk /K). We only explain the construction of the analytic triangle, since the argument is similar in the algebraic case. Applying (5.5.1) to the functor (PnKan , −) and to the complex q∗an (jV† (•V an ) ⊗ K Lπ,f ), we obtain a distinguished triangle II R (Pn an , q an (j † (• ) ⊗ L τ≥1 t K π,f )) ∗ V VKan g P o P o PPP +1oooo PPP o o PPP o o o wo / R(Pn an , q∗an (j † (• an ) ⊗ Lπ,f )) (PnKan , q∗an (jV† (•V an ) ⊗ Lπ,f )) K V V K

K

(5.7.2) in which all arrows are compatible with F ∗ . Thanks to (5.6.2), we have an isomorphism ∼

Fil0 K (L(0+ ); D) −−→ (PnKan , q∗an (jV† (V an ) ⊗ Lπ,f )) •

•

K

236

Francesco Baldassarri and Pierre Berthelot

which identifies α on Fil0 K • (L(0+ ); D) to the endomorphism π,f defined by Tr F . Therefore the composed morphism Fil0 K (L(0+ ); D) −→ R(PnKan , q∗an (jV† (V an ) ⊗ Lπ,f )) •

•

K



• R(VKan , jV† (V an ) ⊗ Lπ,f ) K

= Rrig (Vk /K, Lπ,f )

commutes with α and q n+1 (F ∗ )−1 . On the other hand, theorem 3.1 (applied for H = ∅) provides an isomorphism ∼

Rrig (Vk /K, Lπ,f ) −−→ RZk , rig (Pnk /K) which commutes with F ∗ . This gives the definition of the first arrow in (5.7.1) and shows its compatibility with Frobenius actions. To complete the construction of (5.7.1), we use the bottom line of (5.6.3), which gives an isomorphism ∼

II R t (PnKan , q∗an (jV† (V an ) ⊗ Lπ,f )) −−→ τ≥1 R(PnKan , Pn an ) τ≥1 •

K

•

= τ≥1 Rrig (Pnk /K)

K

compatible with F ∗ . The corollary follows.

5.8. To conclude, we explain how corollary 5.7 implies Dwork’s formula (5.0.1). Thanks to (5.4.1), it suffices to prove that, for any A ⊂ N , A  = ∅, det(I − tα| Fil0 K (LA (0+ ); D A )) = PA (qt)(−1) . #A

•

Recall that m(A) = #A − 1. The triangle (5.7.1) relative to fA gives det(I − tα| Fil0 K (LA (0+ ); D A )) = •

m(A)

det(I − qt (q m(A) (F ∗ )−1 )|RZA,k , rig (Pk · det(I − qt (q

m(A)

∗ −1

(F )

/K))

m(A) )|τ≥1 Rrig (Pk /K))−1 .

Since q m(A) (F ∗ )−1 = F∗ , the first factor is equal to ζ (ZA,k , qt)−1 , while the second m(A) one is equal to ζ (Pk , qt)(1−q m(A)+1 t). The claim then follows from the definition of PA (t). Remark. When A = N , the previous equation can be written as det(I − tα| Fil0 K (L(0+ ); D))−1 = ζ (Zk , qt)(1 − qt) . . . (1 − q n t), •

(5.8.1)

a formula which does not seem to appear explicitly in Dwork’s papers, but was pointed out by Adolphson and Sperber (cf. [1, p. 289]).

237

Dwork cohomology for singular hypersurfaces

Appendix: Logarithmic versus rigid cohomology of an overconvergent isocrystal with logarithmic singularities In the body of this paper, we made use of the following strengthened version of the comparison theorems proved in [5] and [14, 4.2]. We place ourselves in the local situation of 1.1 and consider a smooth formal scheme X over S = Spf R. We fix a divisor T of the special fiber X of X and let V = X \T denote the complementary open subscheme of X. We also choose a divisor Z in X with relative normal crossings over S, union of smooth irreducible components Z1 , . . . , Zn , and let U = X \ Z. We have the functor jU† (resp. jV† ) defined for abelian sheaves on XK , and more generally for abelian sheaves on any strict neighborhood of ]U [X (resp. of ]V [X ) in XK , by using the full system of strict neighborhoods of ]U [X (resp. of ]V [X ) in XK . We let sp : XK −→ X be the specialization morphism. Theorem A.1. Let L be a coherent jV† OXK -module, such that sp ∗ L is a locally free OX, Q († T )-module (of finite type). Assume L is endowed with an integrable connexion ∇ : L −→ jV† (1XK (log ZK )) ⊗j † O V

XK

L

(A.1.1)

with logarithmic singularities along ZK and that jU† (L, ∇) = jU† ∩V (L, ∇) is overconvergent along Z ∪ T . Consider the following assumption: (NL)G The additive subgroup  of K alg generated by the exponents of monodromy of (L, ∇) around the components of ZK , consists of p-adically non-Liouville numbers. Then, if the assumption (NL)G is satisfied, and none of the exponents of monodromy of (L, ∇) around the components of ZK is a negative integer, the canonical inclusion of complexes of K-vector spaces on X •

ιX : sp ∗ (jV† (XK (log ZK )) ⊗j † O V

•

XK

L) −→ sp ∗ (jU† ∩V (XK ) ⊗j † O V

XK

L) (A.1.2)

is a quasi-isomorphism. We have the following general result [14, p. 355]. Proposition A.2. Let E be a coherent jV† OXK -module. For any open affine Y ⊂ X, and any q ≥ 1, H q (YK , E) = 0.

(A.2.1)

Therefore: (i) R q sp ∗ E = 0, ∀q ≥ 1; (ii) H q (Y, sp ∗ E) = 0, for any open affine Y ⊂ X, and q ≥ 1. Let h ∈ O(Y) be a lifting of an equation of the divisor T ∩ Y of Y , and let 0 ⊂ R× be the multiplicative group of absolute values of non zero elements of K,

238

Francesco Baldassarri and Pierre Berthelot

 = 0 ⊗ Q ⊂ R× . For λ ∈ , let jλ : Uλ → YK be the inclusion of the open affinoid Uλ = {x ∈ YK | |h(x)| ≥ λ}. Then there exists a λ0 ∈  and a coherent OUλ0 -module E such that E |YK = lim − →

λ0 ≤λ<1 λ∈

−1 jλ∗ jλ,λ E, 0

(A.2.2)

where jλ,λ0 : Uλ → Uλ0 denotes the immersion. So, YK being quasi-compact and separated, and by Kiehl’s acyclicity theorem applied to the affinoid Uλ and the coherent −1 E, sheaf jλ,λ 0 −1 −1 H q (YK , E) = lim H q (YK , jλ∗ jλ,λ E ) = lim H q (Uλ , jλ,λ E ) = 0. 0 0 − → − → λ0 ≤λ<1 λ∈

λ0 ≤λ<1 λ∈

This proves (A.2.1). Then (i) follows because R q sp ∗ E is the sheaf associated to the presheaf U  → H q (UK , E) on X. Finally, H q (Y, sp ∗ E) = H q (R(Y, −)  Rsp ∗ (E)) = H q (R(YK , E)) = H q (YK , E). A.3. We now prove theorem A.1. Let u : Y → X be an open formal subscheme of X. The inverse image on YK of the morphism ιX of (A.1.2) via uK is the morphism ιY for the data (uK )−1 (L), T ∩ Y, Z ∩ Y (cf. [12, 2.1.4.2]). So, it will suffice to prove that (Y, ιY ) is a quasi isomorphism for all Y in a basis of open affine formal subschemes of X. When X is affine, the previous proposition shows that (X, ιX ) identifies with the natural morphism of hypercohomology •

R(XK , jV† (XK (log ZK )) ⊗j † O

•

L) −→ R(XK , jU† ∩V (XK ) ⊗j † O

L). (A.3.1) We will therefore prove that, for any q ≥ 0, the natural morphism of hypercohomology groups V

XK

H q (XK , jV† (XK (log ZK )) ⊗j † O •

V

XK

L) −→ H q (XK , jU† ∩V (XK ) ⊗j † O •

L) (A.3.2) is an isomorphism of K-vector spaces, when X is replaced by any element of a basis of open affine formal subschemes of the original formal scheme X. We may then assume that L is a free jV† OXK -module of finite type and that there exists a system of coordinates t1 , . . . , td on X such that Z is defined by t1 . . . tr = 0. We choose a lifting h ∈ O(X) of an equation of T . We now follow the arguments of [5], with minor modifications. For λ, σ ∈ (0, 1)∩  we define strict neighborhoods {W λ,σ } (resp. {V σ }) of ]U ∩ V [X (resp. ]V [X ) in XK , by V

XK

W λ,σ = {x ∈ XK | |h(x)| ≥ σ, |t1 (x)| > λ, . . . , |tr (x)| > λ},

V

XK

239

Dwork cohomology for singular hypersurfaces

while V σ = {x ∈ XK | |h(x)| ≥ σ }, and we denote by jλ,σ : W λ,σ → V σ , the open embedding. For σ sufficiently close to 1, say σ ≥ σ0 , there is a free OV σ -module of finite type Lσ , equipped with a logarithmic connection ∇σ : Lσ −→ 1V σ (log ZK ∩ V σ ) ⊗OV σ Lσ ,

(A.3.3)

such that jV† (Lσ , ∇σ ) = (L, ∇). The condition of overconvergence of (L, ∇) along T ∪ Z means that: (SC) There exist functions σ, λ : (0, 1) ∩  −→ (0, 1) ∩  such that for any η ∈ (0, 1) ∩ , σ (η) ≥ σ0 , and for any open affinoid C ⊂ W λ(η),σ (η) and any section e ∈ (C, Lσ (η) ) lim ||(α!)−1 ∇(∂/∂t)α (e)||C η|α| = 0,

|α|→∞

(A.3.4)

 where ∇(∂/∂t)α = di=1 ∇(∂/∂ti )αi and || ||C denotes any Banach norm on (C, Lσ (η) ). For any α ∈ Nd and e ∈ (V σ , Lσ ), t α ∇(∂/∂t)α (e) has no poles along ZK ∩ V σ . On the other hand, if C is a strict affinoid neighborhood of ]V [X of the form C = {x ∈ XK | |h(x)| ≥ }, for any  ∈ , it is easy to check that, for any holomorphic function f ∈ (C, OC ), ||f ||C = ||f ||C∩]U [X , where the Banach norms || ||C and || ||C∩]U [X are related by the presentation (C∩]U [X , OC ) (C, OC ){T }/(T t1 . . . tr − 1). So, the condition of overconvergence of (L, ∇) along T ∪ Z may be reformulated as follows: (SC) There exists a function σ : (0, 1) ∩  −→ (0, 1) ∩  such that for any η ∈ (0, 1) ∩ , any open affinoid C ⊂ V σ (η) and any section e ∈ (C, Lσ (η) ) lim ||(α!)−1 t α ∇(∂/∂t)α (e)||C η|α| = 0

|α|→∞

(A.3.5)

where || ||C denotes any Banach norm on (C, Lσ (η) ). Let η0 , η1 : (0, 1) ∩  −→ (0, 1) ∩  be non-decreasing functions such that λ < η0 (λ) < η1 (λ), for all λ. We parametrize a fundamental system of strict neighborhoods {Wλ } (resp. {Vλ }) of ]U ∩ V [X (resp. ]V [X ) in XK , by λ ∈ (0, 1) ∩ , as follows: Wλ = W λ,σ (η1 (λ)) = {x ∈ XK | |h(x)| ≥ σ (η1 (λ)), |t1 (x)| > λ, . . . , |tr (x)| > λ}, while Vλ = V σ (η1 (λ)) = {x ∈ XK | |h(x)| ≥ σ (η1 (λ))},

240

Francesco Baldassarri and Pierre Berthelot

where σ is a function satisfying property (SC) such that σ (η) > η for any η ∈ (0, 1) ∩ , and we denote by jλ : Wλ → Vλ the open embedding. We abusively write (Lλ , ∇λ ) for (Lσ (η1 (λ)) , ∇σ (η1 (λ)) ). It will be enough to compare, for any λ ∈ (0, 1)∩, the effect in hypercohomology of the morphism of complexes of abelian sheaves on Vλ •

•

Vλ (log(ZK ∩ Vλ )) ⊗ Lλ −→ jλ∗ (Wλ ⊗ Lλ|Wλ )

(A.3.6)

We slightly modify the construction of [5, §4], and, for any fixed λ ∈ (0, 1) ∩ , we construct an admissible covering {U S,λ } of XK parametrized by the subsets S of T = {1, . . . , r}, where U S,λ = {x ∈ XK | |ti (x)| < η1 (λ), for i ∈ S, |tj (x)| ≥ η0 (λ), for j ∈ T \ S}. (A.3.7) As shown in [5, 4.2], U S,λ is a trivial bundle in open polydisks of radius η1 (λ) of relative dimension s = card S over the affinoid space V S,λ = {x ∈ XK | |ti (x)| = 0, for i ∈ S, |tj (x)| ≥ η0 (λ), for j ∈ T \ S}. (A.3.8) Let h0 be the restriction of the function h to the closed analytic subspace V S,λ . Under the previous description of V S,λ , U S,λ ∩ Vλ identifies with W S,λ × D s (0, η1 (λ)− ), the trivial bundle in open polydisks of radius η1 (λ) of relative dimension s over the affinoid space W S,λ = {x ∈ V S,λ | |h0 (x)| ≥ σ (η1 (λ))}.

(A.3.9)

Under the previous identification, U S,λ ∩ Wλ identifies with W S,λ × C s (0, (λ, η1 (λ))), the trivial bundle in open polyannuli of interior radius λ and exterior radius η1 (λ) of relative dimension s over W S,λ . Let now A := O(W S,λ ) be the affinoid algebra corresponding to W S,λ and || ||A be any Banach norm on A. We set Aη1 (λ) {{x}} = Aη1 (λ) {{x1 , . . . , xs }} := O(W S,λ × D s (0, η1 (λ)− )), where x = (x1 , . . . , xs ) denotes an ordered arrangement of {ti , i ∈ S}, a topological K-algebra with the locally convex topology defined by the family of seminorms    α a x   = sup aα A  |α| , α α∈Ns



α

indexed by  ∈ (0, η1 (λ)). Let LA be the K-Lie-algebra of continuous derivations of A/K, trivially extended to Aη1 (λ) {{x}}, and LA,x be the K-Lie-subalgebra of the K-Lie-algebra of continuous derivations of Aη1 (λ) {{x}}/K generated by LA and the xi ∂/∂xi , for i = 1, . . . , s. So, the free Aη1 (λ) {{x}}-module of finite type M := Lλ (W S,λ × C s (0, (λ, η1 (λ)))), carries an action of LA,x . By a trivial dilatation, we may assume that η1 (λ) = 1, so that Aη1 (λ) {{x}} = A{{x}} in the notation of [6, 4.1.1]. It readily follows from condition (SC) that M satisfies the local overconvergence condition [6, 5.1]

241

Dwork cohomology for singular hypersurfaces

(SCL ) For all m ∈ M and , η ∈ (0, 1) lim ||(α!)−1 x α ∇(∂/∂x)α (m)|| η|α| = 0,

(A.3.10)

|α|→∞

f

So, in the notation of [6, §5], M is an object of SCModA, (A{{x}}, LA,x ), where  :=  + Z is an additive subgroup of K alg which, by the assumption (NL)G , does not contain any p-adically Liouville number. So, the classification result [6, 6.5.2] f applies and we may assume that M is a λ-simple object of SCModA, (A{{x}}, LA,x ) [6, §4.2], for λ = (λ1 , . . . , λs ) ∈  s . As in the proof of [5, 6.4], we can assume, by localisation on an admissible affinoid covering of W S,λ , that M = A{{x}} with trivial action of LA , while xi ∂/∂xi , for i = 1, . . . , s, acts via ∇(xi ∂/∂xi ) = xi ∂/∂xi + λi . Our result then follows from [5, 6.6, (i)]. Corollary A.4. Under the hypothesis of A.1, let us assume in addition that X is proper over S. Then, for any strict neighborhoood W of ]V [ X , there exists a canonical isomorphism ∼

•

R(W, jV† (XK (log ZK )) ⊗j † O V

XK

L) −−→ Rrig (U ∩ V , L),

(A.4.1)

where we still denote by L the overconvergent isocrystal on U ∩V defined by jU† ∩V (L). From (1.1.1), we deduce that the canonical morphism •

R(XK , jV† (XK (log ZK ))⊗j † O V

•

XK

L) −→ R(W, jV† (XK (log ZK ))⊗j † O V

XK

L)

is an isomorphism. On the other hand, applying the functor R(X, −) to the isomorphism (A.1.2) and using A.2 (i), we get an isomorphism ∼

•

R(XK , jV† (XK (log ZK )) ⊗j † O V

XK

•

L) −−→ R(XK , jU† ∩V (XK ) ⊗j † O V

XK

L).

Since X is proper, the target space is equal to Rrig (U ∩ V , L), and the corollary follows.

References [1]

A. Adolphson and S. Sperber, On the zeta function of a complete intersection. Ann. Sci. École Norm. Sup. 29 (1996), 287–328.

[2]

A. Adolphson and S. Sperber, A remark on local cohomology. J. Algebra 206 (1998), 555–567.

[3]

A. Adolphson and S. Sperber, Dwork cohomology, de Rham cohomology, and hypergeometric functions. Amer. J. Math. 122 (2000), 319–348.

242

Francesco Baldassarri and Pierre Berthelot

[4]

Y. André and F. Baldassarri, De Rham cohomology of differential modules on algebraic varieties. Prog. Math. 189, Birkhäuser, 2001.

[5]

F. Baldassarri and B. Chiarellotto, Algebraic versus rigid cohomology with logarithmic coefficients. In Barsotti Symposium in Algebraic Geometry, (V. Cristante, W. Messing, eds.), Perspect. Math. 15, Academic Press, 1994, 11–50.

[6]

F. Baldassarri and B. Chiarellotto, Formal and p-adic theory of differential systems with logarithmic singularities depending upon parameters. Duke Math. J. 72 (1993), 1, 241–300.

[7]

F. Baldassarri, M. Cailotto and L. Fiorot, Poincaré Duality for Algebraic De Rham Cohomology. Preprint, Università di Padova (2002).

[8]

F. Baldassarri and A. D’Agnolo, On Dwork cohomology and algebraic D-modules. In Geometric Aspects of Dwork Theory (A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz and F. Loeser, eds.), Volume 1, Walter de Gruyter, Berlin 2004, 245–253.

[9]

P. Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristique p. In Introduction aux cohomologies p-adiques (Journées d’analyse p-adique, 1982), Mém. Soc. Math. France (N.S.) 23 (1986), 7–32.

[10] P. Berthelot, Cohomologie rigide et théorie de Dwork : Le cas des sommes exponentielles. Astérisque 119-120 (1984), 17–49. [11] P. Berthelot, Cohomologie rigide et théorie des D-modules. In Proc. Conference p-adic analysis (Trento 1989), Lecture Notes in Math. 1454, 78–124, Springer-Verlag, 1990. [12] P. Berthelot, Cohomologie rigide et cohomologie rigide à supports propres, première partie. Prépublication IRMAR 96-03, Université de Rennes (1996). [13] P. Berthelot, D-modules arithmétiques I. Opérateurs différentiels de niveau fini. Ann. Sci. École Norm. Sup. 29 (1996), 185–272. [14] P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide, avec un appendice par A. J. de Jong. Invent. Math. 128 (1997), 329–377. [15] P. Berthelot, Dualité de Poincaré et formule de Künneth en cohomologie rigide. C. R. Acad. Sci. Paris 325 (1997), 493–498. [16] P. Berthelot, D-modules arithmétiques II. Descente par Frobenius. Mém. Soc. Math. France 81 (2000). [17] P. Berthelot, Introduction à la théorie arithmétique des D-modules. Astérisque 279 (2002), 1–80. [18] P. Berthelot, D-modules arithmétiques III. Images directes et inverses. In preparation. [19] A. Borel et al., Algebraic D-modules, Perspect. Math. 2, Academic Press, 1987. [20] P. Bourgeois, Annulation et pureté des groupes de cohomologie rigide associés à des sommes exponentielles. C. R. Acad. Sci. Paris 328 (1999), 681–686. [21] P. Bourgeois, Comparaison entre la cohomologie rigide et la théorie de Dwork pour les intersections complètes affines. Application à la conjecture de Katz sur la cohomologie rigide. Prépublication IRMAR 00-10 (2000).

Dwork cohomology for singular hypersurfaces

243

[22] J.-L. Brylinski, B. Malgrange and J.-L. Verdier, Transformation de Fourier géométrique II. C. R. Acad. Sci. Paris 303 (1986), 193–198. [23] H. Cartan and S. Eilenberg, Homological Algebra. Princeton Math. Series 19, Princeton University Press, 1956. [24] B. Chiarellotto, Weights in rigid cohomology. Applications to unipotent F -isocrystals. Ann. Sci. École Norm. Sup. 31 (1998), 683–715. [25] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer-Verlag, 1970. [26] P. Deligne, Cohomologie des intersections complètes. In Groupes de Monodromie en Géométrie Algébrique (SGA 7 II), Lecture Notes in Math. 340, Springer-Verlag, 1973. [27] A. Dimca, F. Maaref, C. Sabbah and M. Saito, Dwork cohomology and algebraic Dmodules. Math. Ann. 318 (2000), 107–125. [28] B. Dwork, On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82 (1960), 631–648. [29] B. Dwork, On the zeta function of a hypersurface. Inst. Hautes Études Sci. Publ. Math. 12 (1962), 5–68. [30] B. Dwork, On the zeta function of a hypersurface II. Ann. of Math. 80 (1964), 227–299. [31] B. Dwork, On the zeta function of a hypersurface III. Ann. of Math. 83 (1966), 457–519. [32] B. Dwork, On the zeta function of a hypersurface IV. A deformation theory for singular hypersurfaces. Ann. of Math. 90 (1969), 335–352. [33] J.-Y. Etesse and B. Le Stum, Fonctions L associées aux F -isocristaux surconvergents. I. Interprétation cohomologique. Math. Ann. 296 (1993), 557–576. [34] R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20, Springer-Verlag, 1966. [35] R. Hartshorne, On the de Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. 45 (1975), 5–99. [36] C. Huyghe, Construction et étude de la transformation de Fourier des D-modules arithmétiques. Thèse Univ. Rennes 1 (1995). [37] C. Huyghe, Interprétation géométrique sur l’espace projectif des AN (K)† -modules cohérents. C. R. Acad. Sci. Paris 321 (1995), 587–590. † [38] C. Huyghe, Transformation de Fourier des DX, Q (∞)-modules. C. R. Acad. Sci. Paris 321 (1995), 759–762.

[39] C. Huyghe, D † (∞)-affinité des schémas projectifs. Ann. Inst. Fourier (Grenoble) 48 (1998), 913–956. [40] C. Noot-Huyghe, Transformation de Fourier des D-modules arithmétiques I. In Geometric Aspects of Dwork Theory (A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz and F. Loeser, eds.), Volume II, Walter de Gruyter, Berlin 2004, 855–905. [41] N. Katz, On the differential equations satisfied by period matrices. Inst. Hautes Études Sci. Publ. Math. 35 (1968), 71–106.

244

Francesco Baldassarri and Pierre Berthelot

[42] K. Kedlaya, Finiteness of rigid cohomology with coefficients. Preprint, Berkeley (December 2002). [43] K. Kedlaya, Fourier Transforms and p-adic Weill II. Preprint, Berkeley (January 2003). [44] P. Monsky, Formal cohomology: III. Fixed point theorems. Ann. of Math. 93 (1971), 315–343. [45] M. Nagata, Imbedding of an abstract variety in a complete variety. J. Math. Kyoto Univ. 2 (1962), 1–10. [46] M. Nagata, A generalization of the embedding problem of an abstract variety in a complete variety. J. Math. Kyoto Univ. 3 (1963), 89–102. [47] D. Pétrequin, Classes de Chern et classes de cycles en cohomologie rigide. Thèse Univ. Rennes 1 (2000). [48] J.-P. Serre, Endomorphismes complètement continus des espaces de Banach p-adiques. Inst. Hautes Études Sci. Publ. Math. 12 (1962), 69–85. Francesco Baldassarri, Dipartimento di Matematica, Università di Padova, Via Belzoni 7, 35131 Padova, Italy E-mail: [email protected] Pierre Berthelot, IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France E-mail: [email protected]

On Dwork cohomology and algebraic D-modules Francesco Baldassarri and Andrea D’Agnolo

Abstract. After works by Katz, Monsky, and Adolphson–Sperber, a comparison theorem between relative de Rham cohomology and Dwork cohomology is established in a paper by Dimca–Maaref–Sabbah–Saito in the framework of algebraic D-modules. We propose here an alternative proof of this result. The use of Fourier transform techniques makes our approach more functorial. 2000 Mathematics Subject Classification: 32S40, 14F10

1 Review of algebraic D-modules For the reader’s convenience, we recall here the notions and results from the theory of algebraic D-modules that we need. Our references were [4, 3, 6, 7].

1.1 Basic operations Let X be a smooth algebraic variety over a field of characteristic zero, and let OX and DX be its structure sheaf and the sheaf of differential operators, respectively. Let Mod(DX ) be the abelian category of left DX -modules, Db (DX ) its bounded derived category, and Dbqc (DX ) the full triangulated subcategory of Db (DX ) whose objects have quasi-coherent cohomologies. Let f : X − → Y be a morphism of smooth algebraic varieties, and denote by DX→Y and DY ←X the transfer bimodules. We use the following notation for the operations of tensor product, inverse image, and direct image for D-modules1 ⊗ : Dbqc (DX ) × Dbqc (DX ) − → Dbqc (DX ), (M, M )  → M ⊗OL M , X

∗

f :

Dbqc (DY )

f+ :

Dbqc (DX )

− →

Dbqc (DX ),

N  → DX→Y ⊗fL−1 D f −1 N ,

− →

Dbqc (DY ),

L M  → Rf ∗ (DY ←X ⊗D M).

Y

X

L L 1About the tensor product, note that M ⊗L M  (M ⊗ OX DX ) ⊗DX M  M ⊗DX (DX ⊗OX M ), OX where M ⊗OX DX (resp. DX ⊗OX M ) is given the natural structure of left-right (resp. left-left) DX L always uses up the “trivial” D -module structure. bimodule, and ⊗D X X

246

Francesco Baldassarri and Andrea D’Agnolo

If f : X − → Y and g : Y − → Z are morphisms of smooth algebraic varieties, then there are natural functorial isomorphisms f ∗g∗ g+ f+ For

N,N

∈

Dbqc (DY ),

(g f )∗ , (g f )+ .

 

(1.1) (1.2)

there is a natural isomorphism f ∗ (N ⊗ N )  f ∗ N ⊗ f ∗ N .

(1.3)

For M ∈ Dbqc (DX ) and N ∈ Dbqc (DY ), there is a projection formula2 f+ (M ⊗ f ∗ N )  f+ M ⊗ N .

(1.4)

Consider a Cartesian square of smooth algebraic varieties X f

 Y

h

/X

(1.5)


f h

 / Y.

For M ∈ Dbqc (DX ), there is a base change formula2 f+ h ∗ M[dX − dX ]  h∗ f+ M[dY − dY ],

(1.6)

where dX denotes the dimension of X.

1.2 Relative cohomology Let S be a closed reduced subscheme of X, and denote by IS ⊂ OX the corresponding ideal of OX . For F ∈ Mod(OX ) one sets3 Γ[S] (F ) = lim Hom OX (OX /ISm , F ). − → m If M ∈ Mod(DX ) one checks that Γ[S] M has a natural left DX -module structure, and one considers the right derived functor RΓ[S] : Dbqc (DX ) − → Dbqc (DX ). Let i : X \ S − → X be the open embedding, and M ∈ Dbqc (DX ). There is a distinguished triangle in Dbqc (DX ) +1

→M− → i+ i ∗ M −→ . RΓ[S] M − 2 In the appendix we recall the proofs of base change and projection formulae.

(1.7)

3 In other words, for any open subset V ⊂ X, Γ F (V ) = {s ∈ F (V ) : (I | )m s = 0, m  0}. [S] S V Note that if F is quasi-coherent, Hilbert’s Nullstellensatz implies that Γ[S] F  ΓS F , the subsheaf of F whose sections are supported in S.

On Dwork cohomology and algebraic D-modules

247

For S, S ⊂ X reduced closed subschemes, and M ∈ Dbqc (DX ), one has RΓ[S] M RΓ[S] RΓ[S ] M

 M ⊗ RΓ[S] OX ,  RΓ[S∩S ] M.

(1.8) (1.9)

Let f : X − → Y be a morphism of smooth varieties, Z ⊂ Y a reduced closed subscheme, and set S = f −1 (Z) ⊂ X. Then there is an isomorphism f+ RΓ[S] M  RΓ[Z] f+ M.

(1.10)

Let Y be a closed smooth subvariety of X of codimension cY , and denote by j : Y − →X the embedding. Recall that Kashiwara’s equivalence states that the functors M  → j ∗ M[−cY ] and N → j+ N establish an equivalence between the category Modqc (DY ) of quasi-coherent DY -modules, and the full abelian subcategory of Modqc (DX ) whose ∼ → M. This extends to derived categories. In particular, the objects M satisfy Γ[Y ] M − functor j+ : Dbqc (DY ) − → Dbqc (DX )

is fully faithful,

(1.11)

and for M ∈ Dbqc (DX ) one has RΓ[Y ] M  j+ j ∗ M[−cY ].

(1.12)

1.3 Fourier–Laplace transform To ϕ ∈ Γ (X; OX ) one associates the DX -module4 DX eϕ = DX /Iϕ ,

Iϕ (V ) = {P ∈ DX (V ) : P eϕ = 0},

∀V ⊂ X open.

For f : X − → Y a morphism of smooth algebraic varieties, and ψ ∈ Γ (Y ; OY ), one has f ∗ DY eψ  DX eψ f .

(1.13)

Let us denote by A1X the trivial line bundle on X, and by t ∈ Γ (A1X ; OA1 ) its fiber X coordinate. Let π : V − → X be a vector bundle of finite rank, πˇ : Vˇ − → X be the dual p1 p2 bundle, γV : V ×X Vˇ − → A1X be the natural pairing, and V ←− V ×X Vˇ −→ Vˇ be the natural projections. The Fourier–Laplace transform for D-modules is the functor FV : Dbqc (DV ) − → Dbqc (DVˇ )

N  → p2+ (p1∗ N ⊗ γV∗ DA1 et ). X

4 Equivalently, D eϕ is the sheaf O with the D -module structure given by the flat connection X X X 1  → dϕ.

248

Francesco Baldassarri and Andrea D’Agnolo

The Fourier–Laplace transform is involutive, in the sense that (cf. [10, Lemma 7.1 and Appendix 7.5]) FVˇ FV  (− idV )∗ .

(1.14)

Let f : V − → W be a morphism of vector bundles over X, and denote by tf : Wˇ − → Vˇ the transpose of f . Then for any N ∈ Dbqc (DV ) and P ∈ Dbqc (DW ) there are natural isomorphisms5 FW f+ N  (tf )∗ FV N , FV f ∗ P  (tf )+ FW P .

(1.15) (1.16)

If X is viewed as a zero-dimensional vector bundle over itself, the projection π : V − → ˇ X and the zero-section ιˇ : X − → V are transpose to each other. Hence (1.16) gives for M ∈ Dbqc (DX ) and Q ∈ Dbqc (DVˇ ) the isomorphisms6 ιˇ+ M  FV π ∗ M, ιˇ∗ Q  π+ FVˇ Q.

(1.17) (1.18)

2 Dwork cohomology Let π : V − → X be a vector bundle of rank r, and let s : X − → Vˇ be a section of the dual bundle πˇ : Vˇ − → X, and set s˜ = idV ×X s : V − → V ×X Vˇ . Recall that → A1X denotes the pairing, and let F ∈ Γ (V ; OV ) be the function γV : V ×X Vˇ − F = t γV s˜ . Let us denote by S the reduced zero locus of s in X, and by j : S − → X the embedding. The geometric framework is thus summarized in the following commutative diagram whose squares are Cartesian in the category of reduced schemes XO j

S

ιˇ

/ Vˇ o O

p2




s




j

/Xo

π

γV

/ A1 V ×O X VHˇ X HH p1 HH s˜ HH HH # idV V o V.

Generalizing previous results of [9, 13, 2], Theorem 0.2 of [5] gives the following link between relative cohomology and Dwork cohomology 5 See the appendix for a proof. 6 Note that isomorphism (1.17) is the content of [5, Lemma 2.3], of which we have thus provided a more natural proof.

On Dwork cohomology and algebraic D-modules

249

Theorem 2.1. For M ∈ Dbqc (DX ) there is an isomorphism RΓ[S] M[r]  π+ (π ∗ M ⊗ DV eF ). Our aim here is to provide a more natural proof of this result. Proof. For M = OX , the statement reads RΓ[S] OX [r]  π+ DV eF .

(2.1)

For a general M ∈ Dbqc (DX ), there are isomorphisms RΓ[S] M  M ⊗ RΓ[S] OX

by (1.8),

and π+ (π ∗ M ⊗ DV eF )  M ⊗ π+ DV eF

by (1.4).

It is thus sufficient to prove (2.1). Setting L = γV∗ DA1 et , there is a chain of isomorX phisms DV eF  s˜ ∗ L  s˜ ∗ L ⊗ OV  p1+ s˜+ (˜s ∗ L ⊗ OV )  p1+ (L ⊗ s˜+ OV )  p1+ (L ⊗ s˜+ π ∗ OX )  p1+ (L ⊗ p2∗ s+ OX ) = FVˇ s+ OX .

by (1.13) by (1.2) by (1.4) by (1.6)

Hence we have π + DV eF  π + FVˇ s+ OX  ιˇ∗ s+ OX

by (1.18),

and to prove (2.1) we are left to establish an isomorphism ιˇ∗ s+ OX  RΓ[S] OX [r].

(2.2)

By (1.11), this follows from the chain of isomorphisms ιˇ+ ιˇ∗ s+ OX  ιˇ+ ιˇ∗ s+ s ∗ OVˇ  RΓ[ˇι(X)] RΓ[s(X)] OVˇ [2r]

by (1.12)

 RΓ[ˇι(S)] RΓ[ˇι(X)] OVˇ [2r]

by (1.9)

 RΓ[ˇι(S)] ιˇ+ OX [r]

by (1.12)

 ιˇ+ RΓ[S] OX [r]

by (1.10).

 

250

Francesco Baldassarri and Andrea D’Agnolo

Remark 2.2. Kashiwara’s equivalence allows one7 to develop the theory of algebraic D-modules on possibly singular varieties, so that the formulae stated in the previous section still hold. In this framework, (2.2) is obtained by ιˇ∗ s+ OX  j+ j ∗ OX [r − cS ]  RΓ[S] OX [r]

by (1.6) by (1.12),

where cS denotes the codimension of S in X.

A Appendix A.1 Base change and projection formulae The base change formula (1.6) is proved in [4, Theorem VI.8.4] for h a locally closed embedding8 . Let us recall how to deal with the general case. Proof of (1.6). The Cartesian square (1.5) splits into the two Cartesian squares X f

 Y

(f ,h )

/ Y × X


(idY ,h)

idY ×f

 / Y × Y

p2
p2

/X f

 / Y,

where p2 and p2 are the natural projections. Since (idY , h) is a closed embedding, by [4] the base change formula holds for the Cartesian square on the left hand side. We are thus left to prove the base change formula for the Cartesian square on the right hand side. For M ∈ Dbqc (DX ), one has the chain of isomorphisms (idY ×f )+ p2 ∗ M  (idY ×f )+ (OY
M)  OY
f+ M  p2∗ f+ M, where
denotes the exterior tensor product.

 

Let us also recall, following [3], how projection formula is deduced from base change formula. 7 This is done for example in [3]. For S a reduced closed subschemes of a smooth variety X, the idea is ∼ → M. to define Mod(DS ) as the full abelian subcategory of Mod(DX ) whose objects M satisfy Γ[S] M − 8 In the language of Gauss–Manin connections, the base change formula is stated in [1, § 3.2.6] for h

flat.

On Dwork cohomology and algebraic D-modules

251

Proof of (1.4). Consider the diagram with commutative square X tt δX ttt δf t tt ytt f  / X×Y X×X

f

/Y


f

δY

 / Y × Y,

where δX and δY are the diagonal embeddings, δf is the graph embedding, f = f × idY , and f = idX ×f . Then there is a chain of isomorphisms ∗ (M
f ∗ N ) f+ (M ⊗ f ∗ N )  f+ δX ∗ ∗ f (M
N )  f + δX ∗  δY f+ (M
N )  δY∗ (f+ M
N )  f+ M ⊗ N .

by (1.6)  

A.2 Fourier–Laplace transform The formulae stated in section 1.3 for the Fourier–Laplace transform of algebraic D-modules have their analogues for the Fourier–Deligne transform of -adic sheaves (see [12] or [11, §III.13]), and for the Fourier–Sato transform of conic abelian sheaves (see [8]). Apart from [10], we do not have specific references for the algebraic Dmodule case. We thus provide here some proofs.

Proof of (1.15) and (1.16). The following arguments are parallel to those in the proof of [12, Théorème 1.2.2.4] or [8, Proposition 3.7.14]. Consider the diagram with Cartesian squares VˇO o p2

tf


α

WˇO eLLL LLL q2 LLL r2 LL β / W × Wˇ V ×X Wˇ X

V ×X VˇL o LLL LL r1 p1 LLL LL  % V


f

q1

 / W,

where the morphisms pi , qi , ri , for i = 1, 2 are the natural projections. Note that γV α = γW β. The isomorphism (1.15) is obtained via the following chain of

252

Francesco Baldassarri and Andrea D’Agnolo

isomorphisms9 , where we set L1 = DA1 et . X

(tf )∗ FV N = (tf )∗ p2+ (p1∗ N ⊗ γV∗ L1 )  r2+ α ∗ (p1∗ N ⊗ γV∗ L1 )  r2+ (α ∗ p1∗ N ⊗ α ∗ γV∗ L1 )  r2+ (r1∗ N ⊗ α ∗ γV∗ L1 )  r2+ (r1∗ N ⊗ β ∗ γW∗ L1 )  q2+ β+ (r1∗ N ⊗ β ∗ γW∗ L1 )  q2+ (β+ r1∗ N ⊗ γW∗ L1 )  q2+ (q1∗ f+ N ⊗ γW∗ L1 ) = FW f+ N .

by (1.6) by (1.3) by (1.1) by (1.1) by (1.2) by (1.4) by (1.6)

Applying the functor FWˇ to the isomorphism (1.15) with N = FVˇ Q, Q ∈ Dbqc (DVˇ ), and using (1.14), we get f+ FVˇ Q  FWˇ (tf )∗ Q. The isomorphism (1.16) is obtained from the one above by interchanging the roles of f and tf .  

References [1]

Y. André and F. Baldassarri, De Rham cohomology of differential modules on algebraic varieties. Progr. Math. 189, Birkhäuser, 2001.

[2]

A. Adolphson and S. Sperber, Dwork cohomology, de Rham cohomology, and hypergeometric functions. Amer. J. Math. 122 (2) (2000), 319–348.

[3]

J. Bernstein, Lectures on algebraic D-modules at Berkeley. 2001 (unpublished).

[4]

A. Borel, Algebraic D-modules. Perspect. Math. 2, Academic Press, 1987.

[5]

A. Dimca, F. Maaref, C. Sabbah, M. Saito, Dwork cohomology and algebraic D-modules. Math. Ann. 318 (1) (2000), 107–125.

[6]

M. Kashiwara, Algebraic study of systems of partial differential equations. Mém. Soc. Math. France (N.S.) 63 (1995), xiv+72, Kashiwara’s Master’s Thesis, Tokyo University 1970, translated from the Japanese by A. D’Agnolo and J.-P. Schneiders.

[7]

—, D-modules and microlocal calculus. Transl. Math. Monogr. 217, Amer. Math. Soc., 2003, translated from the 2000 Japanese original by M. Saito.

9 Note that these arguments still apply if one replaces L = D et with an arbitrary quasi-coherent 1 A1X DA1 -module. On the other hand, in order to prove (1.16) we will use the fact that the Fourier transform is X

involutive.

On Dwork cohomology and algebraic D-modules

253

[8]

M. Kashiwara and P. Schapira, Sheaves on manifolds. Grundlehren Math. Wiss. 292, Springer-Verlag, 1990.

[9]

N. M. Katz, On the differential equations satisfied by period matrices. Inst. Hautes Études Sci. Publ. Math. 35 (1968) 223–258.

[10] N. M. Katz and G. Laumon, Transformation de Fourier et majoration de sommes exponentielles. Inst. Hautes Études Sci. Publ. Math. 62 (1985), 361–418. [11] R. Kiehl and R. Weissauer, Weil conjectures, preverse sheaves and l’adic Fourier transform. Ergeb. Math. Grenzgeb. 42, Springer-Verlag, 2001. [12] G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil. Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131–210. [13] P. Monsky, Finiteness of de Rham cohomology. Amer. J. Math. 94 (1972), 237–245. Francesco Baldassarri, Dipartimento di Matematica Pura ed Applicata, Università di Padova, via G. Belzoni, 7, 35131 Padova, Italy E-mail: [email protected] Andrea D’Agnolo, Dipartimento di Matematica Pura ed Applicata, Università di Padova, via G. Belzoni, 7, 35131 Padova, Italy E-mail: [email protected]

An introduction to the theory of p-adic representations Laurent Berger

Abstract. This informal article is an expanded version of two lectures given in Padova during the “Dwork Trimester” in June 2001. Their goal was to explain the proof of the p-adic monodromy theorem for de Rham representations and to give some background on p-adic representations. Résumé. Cet article informel est une version longue de deux exposés donnés à Padoue en Juin 2001 au “Trimestre Dwork”. Leur objet était d’expliquer la démonstration du théorème de monodromie p-adique pour les représentations de de Rham et de donner des rappels sur les représentations p-adiques. 2000 Mathematics Subject Classification: 11F80, 11R23, 11S25, 12H25, 13K05, 14F30

Contents I

Introduction I.1 Prolegomenon . . . . . . . . . . . I.1.1 Motivation . . . . . . . . I.1.2 Organization of the article I.1.3 Acknowledgments . . . . I.2 p-adic representations . . . . . . . I.2.1 Some notations . . . . . . I.2.2 Definitions . . . . . . . . I.2.3 Fontaine’s strategy . . . . I.3 Fontaine’s classification . . . . . .

II p-adic Hodge theory II.1 The field C and the theory of Sen II.1.1 The action of GK on C . II.1.2 Sen’s theory . . . . . . . II.2 The field BdR . . . . . . . . . . II.2.1 Reminder: Witt vectors . II.2.2 The universal cover of C II.2.3 Construction of BdR . . II.2.4 Sen’s theory for B+ dR . . II.3 The rings Bcris and Bst . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

257 257 257 257 258 258 258 258 259 260

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

261 261 261 262 263 263 264 265 266 266

256

Laurent Berger

II.3.1 Construction of Bcris . . . . . . . . . . . II.3.2 Example: elliptic curves . . . . . . . . . II.3.3 Semi-stable representations . . . . . . . II.3.4 Frobenius and filtration . . . . . . . . . . II.3.5 Some remarks on topology . . . . . . . . II.4 Application: Tate’s elliptic curve . . . . . . . . . II.4.1 Tate’s elliptic curve . . . . . . . . . . . . II.4.2 The p-adic representation attached to Eq II.4.3 p-adic periods of Eq . . . . . . . . . . . II.4.4 Remark: Kummer theory . . . . . . . . . II.5 p-adic representations and Arithmetic Geometry . II.5.1 Comparison theorems . . . . . . . . . . II.5.2 Weil–Deligne representations . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

266 267 268 269 269 270 270 271 271 272 273 273 274

III Fontaine’s (ϕ, )-modules III.1 The characteristic p theory . . . . . . . . . . . . . . . . . III.1.1 Local fields of characteristic p . . . . . . . . . . . III.1.2 Representations of GEK and differential equations III.2 The characteristic zero theory . . . . . . . . . . . . . . . . III.2.1 The field of norms . . . . . . . . . . . . . . . . . III.2.2 (ϕ, )-modules . . . . . . . . . . . . . . . . . . . III.2.3 Computation of Galois cohomology . . . . . . . . III.3 Overconvergent (ϕ, )-modules . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

274 274 274 275 276 276 276 277 277

IV Reciprocity formulas for p-adic representations IV.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . IV.1.1 Reciprocity laws in class field theory . . . . . . IV.2 A differential operator on (ϕ, )-modules . . . . . . . . IV.3 Crystalline and semi-stable representations . . . . . . . . IV.3.1 Construction of Dcris (V ) and of Dst (V ) . . . . . IV.3.2 Rings of periods and limits of algebraic functions IV.3.3 Regularization and decompletion . . . . . . . . IV.4 De Rham representations . . . . . . . . . . . . . . . . . IV.4.1 Construction of NdR (V ) . . . . . . . . . . . . . IV.4.2 Example: C-admissible representations . . . . . IV.5 The monodromy theorem . . . . . . . . . . . . . . . . . IV.5.1 -adic monodromy and p-adic monodromy . . . IV.5.2 p-adic differential equations . . . . . . . . . . . IV.5.3 The monodromy theorem . . . . . . . . . . . . . IV.5.4 Example: Tate’s elliptic curve . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

279 279 279 279 280 280 281 281 282 282 283 283 283 284 284 284

. . . . . . . . . . . . . . .

V Appendix 285 V.1 Diagram of the rings of periods . . . . . . . . . . . . . . . . . . . . . . . . . . 285 V.2 List of the rings of power series . . . . . . . . . . . . . . . . . . . . . . . . . . 286 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

An introduction to the theory of p-adic representations

257

I Introduction I.1 Prolegomenon I.1.1 Motivation. One of the aims of arithmetic geometry is to understand the structure of the Galois group Gal(Q/Q), or at least to understand its action on representations coming from geometry. A good example is provided by the Tate module T E of an elliptic curve E defined over Q. The action of Gal(Q/Q) on T E carries a lot of arithmetical information, including the nature of the reduction of E at various primes and the number of points in E(Fq ). Let Dp ⊂ Gal(Q/Q) be the decomposition group of a place above p; it is naturally isomorphic to Gal(Qp /Qp ). The aim of the theory of p-adic representations is to extract information from the action of Dp , on Qp -vector spaces. This is in stark contrast to the theory of -adic representations, which endeavors to understand the action of Dp on Q -vector spaces with   = p. In this latter situation, the topology of Dp is mostly incompatible with that of an -adic vector space (essentially because the wild inertia of Dp is a pro-p-group), and the result is that the theory of -adic representations is of an algebraic nature. On the other hand, in the p-adic case, the topologies are compatible and as a result there are far too many representations. The first step is therefore to single out the interesting objects, and to come up with significant invariants attached to them. Unlike the -adic situation, the study of p-adic representations is therefore of a rather (p-adic) analytic nature. For example, there exists a p-adically continuous family of characters of the group Gal(Qp /Qp ), given by χ s where χ is the cyclotomic character and s varies in weight space (essentially p −1 copies of Zp ). Out of those characters, only those corresponding to integer values of s “come from geometry”. This kind of phenomenon does not arise in the -adic case, where every character is “good”. The aim of this article is to introduce some of the objects and techniques which are used to study p-adic representations, and to provide explanations of recent developments.

I.1.2 Organization of the article. This article is subdivided in chapters, each of which is subdivided in sections made up of paragraphs. At the end of most paragraphs, I have added references to the literature. This article’s goal is to be a quick survey of some topics and a point of entry for the literature on those subjects. In general, I have tried to give my point of view on the material rather than complete detailed explanations. References are indicated at the end of paragraphs. For each topic, I have tried to indicate a sufficient number of places where the reader can find all the necessary details. I have not always tried to give references to original articles, but rather to more recent (and sometimes more readable) accounts.

258

Laurent Berger

I.1.3 Acknowledgments. The basis for this article are the two lectures which I gave at the “Dwork Trimester” in Padova, and I thank the organizers, especially F. Baldassarri, P. Berthelot and B. Chiarellotto for the time and effort they spent to make this conference a success. After I wrote a first version of this article, M. Çiperiani, J.-M. Fontaine and H. J. Zhu took the time to read it, pointed out several inaccuracies and made many suggestions for improvement. Any remaining inaccuracies are entirely my fault.

I.2 p-adic representations I.2.1 Some notations. The results described in this article are true in a rather general setting. Let k be any perfect field of characteristic p (perfect means that the map x  → x p is an automorphism), and let F = W (k)[1/p] be the fraction field of OF = W (k), the ring of Witt vectors over k (for reminders on Witt vectors, see  =K paragraph II.2.1). Let K be a finite totally ramified extension of F , and let C = F be the p-adic completion of the algebraic closure of F (not to be confused with the field C of complex numbers). If k is contained in the algebraic closure of Fp , then C = C p , the field of so-called p-adic complex numbers. An important special case is when k is a finite extension of Fp , so that K is a finite extension of Qp , and F is then the maximal unramified extension of Qp contained in K. The reader can safely assume that we’re in this situation throughout the article. Another important special case though is when k is algebraically closed. Let µm denote the subset of K defined by µm = {x ∈ K, x m = 1}. We’ll choose once and for all a compatible sequence of primitive p n -th roots of unity, ε(0) = 1, and ε (n) ∈ µpn ⊂ K, such that ε (1)  = 1 and (ε(n+1) )p = ε(n) . Let Kn = K(ε(n) ) (n) is like choosing an orientation and K∞ = +∞ n=0 Kn . Making such a choice of ε in p-adic Hodge theory, in the same way that choosing one of ±i is like choosing an orientation in classical geometry. Here are the various fields that we are considering: GK

   F ⊂ K ⊂ Kn ⊂ K∞ = K∞ ⊂ F = K ⊂ C       K

HK

Let GK be the Galois group Gal(K/K). The cyclotomic character χ : GK → Z∗p is defined by σ (ζ ) = ζ χ(σ ) for every σ ∈ GK and ζ ∈ µp∞ . The kernel of the cyclotomic character is HK = Gal(K/K∞ ), and χ therefore identifies K = Gal(K∞ /K) = GK /HK with an open subgroup of Z∗p . I.2.2 Definitions. A p-adic representation V of GK is a finite dimensional Qp -vector space with a continuous linear action of GK . The dimension of V as a Qp -vector space will always be denoted by d. Here are some examples of p-adic representations:

An introduction to the theory of p-adic representations

259

1. If r ∈ Z, then Qp (r) = Qp · er where GK acts on er by σ (er ) = χ (σ )r er . This is the r-th Tate twist of Qp ; 2. if E is an elliptic curve, then the Tate module of E, V = Qp ⊗Zp Tp E is a p-adic representation of dimension d = 2; 3. more generally, if X is a proper and smooth variety over K, then the étale cohomology Héti (XK , Qp ) is a p-adic representation of GK . This last example is really the most interesting (the first two being special cases), and it was the motivation for the systematic study of p-adic representations. Grothendieck had suggested in 1970 the existence of a “mysterious functor” (le foncteur mystérieux) directly linking the étale and crystalline cohomologies of a p-divisible group. Fontaine gave an algebraic construction of that functor which conjecturally allowed one to recover, for any i and any proper and smooth X/K, the de Rham cohomology of X/K (which is a filtered K-vector space) from the data of Héti (XK , Qp ) as a p-adic representation. His construction was shown to be the right one in general by Tsuji; we’ll come back to that in II.5.1. The above result is a p-adic analogue of the well-known fact that if X is a proper smooth variety over a number field L, then over the complex numbers C one has an isomorphism i (X/L) C ⊗Z H i (X, Z) C ⊗L HdR

given by integrating differential forms on cycles. I.2.3 Fontaine’s strategy. Fontaine’s strategy for studying p-adic representations was to construct rings of periods, which are topological Qp -algebras B, with a continuous and linear action of GK and some additional structures which are compatible with the action of GK (for example: a Frobenius ϕ, a filtration Fil, a monodromy map N , a differential operator ∂), such that the B GK -module DB (V ) = (B ⊗Qp V )GK , which inherits the additional structures, is an interesting invariant of V . For Fontaine’s constructions to work, one needs to assume that B is GK -regular, which means that if b ∈ B is such that the line Qp · b is stable by GK , then b ∈ B ∗ . In particular, B GK has to be a field. In general, a simple computation shows that dimB GK DB (V ) ≤ d = dimQp V , and we say that V is B-admissible if equality holds, which is equivalent to having B ⊗Qp V B d as B[GK ]-modules. In this case, B ⊗B GK DB (V ) B ⊗Qp V , and the coefficients of a matrix of this isomorphism in two bases of DB (V ) and V are called the periods of V . Let us briefly mention a cohomological version of this: a p-adic representation V determines a class [V ] in H 1 (GK , GL(d, Qp )), and therefore a class [V ]B in H 1 (GK , GL(d, B)). The representation V is B-admissible if and only if [V ]B is trivial. In this case, [V ]B is a coboundary, given explicitly by writing down a GK invariant basis of B ⊗Qp V .

260

Laurent Berger

Here are some examples of rings of periods: 1. If B = K, then B GK = K and V is K-admissible if and only if the action of GK on V factors through a finite quotient. This is essentially Hilbert 90; 2. If B = C, then B GK = K (the so-called theorem of Ax–Sen–Tate, first shown by Tate) and V is C-admissible if and only if the action of the inertia IK on V factors through a finite quotient. This was conjectured by Serre and proved by Sen. We will return to this in IV.4.2; 3. Let B = BdR be Fontaine’s ring of p-adic periods (defined below in II.2.3). It is a field, equipped with a filtration, and B GK = K. If V = Héti (XK , Qp ), for a proper smooth X/K, then V is BdR -admissible, and DdR (V ) = DBdR (V )

i (X/K) as filtered K-vector spaces. This is one of the most important HdR theorems of p-adic Hodge theory. For rings of periods and Tannakian categories in a general setting, see Fontaine’s [Fo94b].

I.3 Fontaine’s classification By constructing many rings of periods, Fontaine has defined several subcategories of the category of all p-adic representations, and in this paragraph, we shall list a number of them along with categories of invariants which one can attach to them. Many of the words used here will be defined later in the text, but the table below should serve as a guide to the world of p-adic representations. p-adic representations

Some invariants attached to those representations

References in the text

-adic analogue

all of them

(ϕ, )-modules

III.2.2

–

Hodge–Tate

Hodge–Tate weights

de Rham

1. p-adic differential equations 2. filtered K-vector spaces

II.2.3, IV.4.1

potentially semi-stable

1. quasi-unipotent differential equations 2. admissible filtered (ϕ, N, GL/K )-modules

II.3.3, IV.3.3

semi-stable

1. unipotent differential equations 2. admissible filtered (ϕ, N )-modules

II.3.3, IV.3.3

unipotent representations

crystalline

1. trivial differential equations 2. admissible filtered ϕ-modules

II.3.1, IV.3.3

representations with good reduction

II.1.2

– all -adic representations quasi-unipotent representations

Each category of representations is a subcategory of the one above it. One can associate to every p-adic representation a (ϕ, )-module, which is an object defined

An introduction to the theory of p-adic representations

261

on the boundary of the open unit disk. This object extends to a small annulus, and if V is de Rham, the action of the Lie algebra of  gives a p-adic differential equation. This equation is unipotent exactly when V (restricted to GKn for some n) is semi-stable. In this case, the kernel of the connection is a (ϕ, N )-module which coincides with the (ϕ, N )-module attached to V by p-adic Hodge theory (one loses the filtration, however). All of this will be explained later in the body of the text.

II p-adic Hodge theory In this chapter, we’ll define various rings of periods which are used in p-adic Hodge theory, and give some simple examples of Fontaine’s construction for an explicit geometric object (an elliptic curve).

II.1 The field C and the theory of Sen Before we define the rings of periods which are used in p-adic Hodge theory, we’ll review some simple properties of the field C of p-adic complex numbers. As we have seen above, C is not a great ring of periods (since C-admissible representations are potentially unramified while representations coming from arithmetic geometry are much more complicated than that), but one can still extract a lot of arithmetic information from the data of C ⊗Qp V : this is the content of Sen’s theory. II.1.1 The action of GK on C. An important property of C that we will need is that we can explicitly describe C H where H is a closed subgroup of GK . Clearly,  H H K ⊂ C H and therefore K ⊂ C H . The Ax–Sen–Tate theorem says that the latter  H inclusion is actually an equality: K = C H . This was first shown by Tate, and the proof was later improved and generalized by Sen and Ax. Following Sen, Ax gave a natural proof of that result, by showing that if an element of K is “almost invariant” H by H , then it is “almost” in K . The first indication that C was not a good choice for a ring of periods was given by a theorem of Tate, which asserts that C does not contain periods for characters which are too ramified (for example: the cyclotomic character). More precisely, he showed that if ψ : GK → Z∗p is a character which is trivial on HK but which does not factor through a finite quotient of K , then H 0 (K, C(ψ −1 )) = {x ∈ C, g(x) = ψ(g)x for all g ∈ GK } = {0}. In particular, there is no period in C for the cyclotomic character (a non-zero element of the above set is a period for ψ −1 ). Let us explain the proof of Tate’s result: by the Ax–

262

Laurent Berger

∞ . Sen–Tate theorem, the invariants of C under the action of HK are given by C HK = K The main argument in Tate’s proof is the construction of generalized trace maps ∞ → Kn . The map pr K is a continuous, Kn -linear, and K -equivariant secpr Kn : K n ∞ . In addition, if x ∈ K ∞ , then x = limn→∞ pr K (x). tion of the inclusion Kn ⊂ K n We see that we can and should set pr Kn (x) = limm→+∞ [Kn+m : Kn ]−1 Tr Kn+m /Kn (x). The proof of the convergence of the above limit depends essentially on a good understanding of the ramification of K∞ /K. Using these maps, one can prove Tate’s theorem. Let x be a period of ψ. Since ∞ . We therefore have x = limn→∞ xn where ψ|HK = 1, one has x ∈ C HK = K xn = pr Kn (x), and since g(x) − ψ(g)x = 0 for all g ∈ GK , and pr Kn is Galoisequivariant, one also has g(xn ) − ψ(g)xn = 0 for all g ∈ GK . If xn  = 0, this would imply that ψ factors through Gal(Kn /K), a contradiction, so that xn = 0 for every n. Since x = limn→∞ xn , we also have x = 0. General facts on C can be found in Koblitz’s [Kob84], which is a good introduction to p-adic numbers. The beginning of [DGS94] is a wonderful introduction too. The proof of Ax–Sen–Tate’s theorem that we referred to is in Ax’s [Ax70], see also Colmez’s [Col02, §4]. Tate’s theorems on the cohomology of C are in [Tat66] or in Fontaine’s [Fon00, §1]. II.1.2 Sen’s theory. The point of Sen’s theory is to study the residual action of K ∞ -vector space (C ⊗Qp V )HK , where V is a p-adic representation of GK . on the K We can summarize his main result as follows. If d ≥ 1, then H 1 (HK , GL(d, C)) is ∞ )) induced trivial and the natural map: H 1 (K , GL(d, K∞ )) → H 1 (K , GL(d, K ∞ is a bijection. by the inclusion K∞ ⊂ K One can show that this implies the following: given a p-adic representation V , ∞ -vector space (C ⊗Qp V )HK has dimension d = dimQp (V ), and the union the K of the finite dimensional K∞ -subspaces of (C ⊗Qp V )HK stable under the action of K is a K∞ -vector space of dimension d. We shall call it DSen (V ), and the natural ∞ ⊗K∞ DSen (V ) → (C ⊗Qp V )HK is then an isomorphism. The K∞ -vector map K space DSen (V ) is endowed with an action of K , and Sen’s invariant is the linear map giving the action of Lie(K ) on DSen (V ). It is the operator defined in End(DSen (V )) by V = log(γ )/ logp (χ(γ )), where γ ∈ K is close enough to 1 (the definition of V doesn’t depend on the choice of γ ). More precisely, for any k ≥ 1, (1 − γ )k is a K-linear operator on DSen (V ) and one can show that if γ ∈ K is close enough to 1, then the series of operators: −

 (1 − γ )k 1 logp (χ(γ )) k k≥1

converges (in End(DSen (V ))) to an operator V which is K∞ -linear and does not depend on the choice of γ . The operator V is then an invariant canonically attached to V . Let us give a few examples: we say that V is Hodge–Tate, with Hodge–Tate weights h1 , . . . , hd ∈ Z,

An introduction to the theory of p-adic representations

263

if there is a decomposition of C[GK ]-modules: C ⊗Qp V = ⊕dj =1 C(hj ). This is equivalent to V being diagonalizable with integer eigenvalues. For this reason, the eigenvalues of V are usually called the generalized Hodge–Tate weights of V . All representations coming from a proper smooth variety X/K (the subquotients of its étale cohomology groups) are Hodge–Tate, and the integers −hj are the jumps of the filtration on the de Rham cohomology of X. For example, the Hodge–Tate weights of V = Qp ⊗Zp Tp E, where E is an elliptic curve, are 0 and 1. Here is a representation which is not Hodge–Tate: let V be a two dimensional Qp -vector space on which GK acts by     1 logp (χ(g)) 0 1   so that V =  . 0 1 0 0 Relevant papers of Sen are [Sen72, Sen73, Sen80] and [Sen93] which deals with families of representations. Colmez has given a different construction more in the spirit of the “ring of periods” approach (by constructing a ring BSen ), see [Col94]. For an interesting discussion of all this, see Fontaine’s course [Fon00, §2].

II.2 The field BdR II.2.1 Reminder: Witt vectors. Before we go any further, we’ll briefly review the theory of Witt vectors. Let R be a perfect ring of characteristic p. For example, R could be a finite field or an algebraically closed field, or the ring of integers of an algebraically closed field (in characteristic p, of course). The aim of the theory of Witt vectors is to construct a ring A, in which p is not nilpotent, and such that A is separated and complete for the topology defined by the ideals p n A. We say that A is a strict p-ring with residual ring R. The main result is that if R is a perfect ring of characteristic p, then there exists a unique (up to unique isomorphism) strict p-ring A = W (R) with residual ring R. It is called the ring of Witt vectors over R. Furthermore, because of the unicity, if one has a map ξ : R → S, then it lifts to a map ξ : W (R) → W (S). In particular, the map x → x p lifts to a Frobenius automorphism ϕ : W (R) → W (R). Let us give a few simple examples: if R = Fp , then W (R) = Zp and more generally, if R is a finite field, then W (R) is the ring of integers of the unique unramified extension of Qp whose residue field is R. If R = Fp , then W (R) = OQ unr . In the p following paragraphs, we will see more interesting examples. xn in A whose image in If x = x0 ∈ R, then for every n ≥ 0, choose an element  −n pn xn then converges in A to an element [x] which depends R is x p . The sequence  only on x. This defines a multiplicative map x  → [x] from R → A, which is a section of the projection x → x, called the Teichmüller map. The Teichmüller elements (the elements in the image of the Teichmüller map) are a distinguished set of representatives of the elements of R. One can write every element x ∈ A in a unique

264 way as x = write

Laurent Berger

+∞

n=0 p

x+y =

n [x

n]

+∞ 

with xn ∈ R. Given two elements x, y ∈ A, one can then

pn [Sn (xi , yi )]

and

xy =

n=0

+∞ 

pn [Pn (xi , yi )]

n=0 p −n

p −n

where Sn and Pn ∈ Z[Xi , Yi ]i=0,...,n are universal homogeneous polynomials of degree one (if one decides that the degrees of the Xi and Yi are 1). For example, 1/p 1/p S0 (X0 , Y0 ) = X0 + Y0 and S1 (X0 , X1 , Y0 , Y1 ) = X1 + Y1 + p−1 ((X0 + Y0 )p − +∞ X0 − Y0 ). The simplest way to construct W (R) is then by setting W (R) = n=0 R and by making it into a ring using the addition and multiplication defined by the Pn and Sn , which are given by (not so) simple functional equations. Finally, let us mention that if R is not perfect, then there still exist strict p-rings A such that A/pA = R, but A is not unique anymore. Such a ring is called a Cohen ring. For example, if R = Fp [[X]], then one can take A = Zp [[X]], but for all α ∈ pZp , the map X  → X + α is a non-trivial isomorphism of A which induces the identity on R. The above summary is inspired by a course given by P. Colmez. The best place to start further reading is Harder’s survey [Har97]. The construction of Witt vectors is also explained by Serre in [Ser68] (or in English in [Ser79]).

II.2.2 The universal cover of C. Let  E+ be the set defined by  E+ = lim OC = {(x (0) , x (1) , . . . ) | (x (i+1) )p = x (i) } ← −p x →x

which we make into a ring by deciding that if x = (x (i) ) and y = (y (i) ) are two elements of  E+ , then their sum and their product are defined by: j

(x + y)(i) = lim (x (i+j ) + y (i+j ) )p and (xy)(i) = x (i) y (i) . j →∞

This makes  E+ into a perfect local ring of characteristic p. Let ε = (ε(i) ) where (i) the ε are the elements which have been chosen in I.2.1. It is easy to see that E= E+ [(ε − 1)−1 ] and one can show that  E is a field which is the Fp ((ε − 1)) ⊂  completion of the algebraic (non-separable!) closure of Fp ((ε − 1)), so it is really a familiar object. E by vE (x) = vp (x (0) ) so that  E+ is the integer ring We define a valuation vE on  n (n) p of  E for vE . For example, vE (ε − 1) = limn→∞ vp (ε − 1) = p/(p − 1). Finally, let us point out that there is a natural map  E+ → limx→x p OC /p and it’s ← − not hard to show that this map is an isomorphism.

An introduction to the theory of p-adic representations

265

There is a natural map θ from  E+ to OC , which sends x = (x (i) ) to x (0) , and the +  A+ = W ( E+ ) and map θ : E → OC /p is a homomorphism. Let    B+ =  A+ [1/p] = { pk [xk ], xk ∈  E+ } k−∞

E+ . The map θ then extends where [x] ∈  A+ denotes the Teichmüller lift of x ∈ 

k

k (0) +  p [xk ] to p xk . to a surjective homomorphism θ : B → C, which sends p Let [ε1 ] = [(ε (1) , . . . )] so that ε1 = ε, and let ω = ([ε] − 1)/([ε1 ] − 1). Then θ (ω) = 1 + ε (1) + · · · + (ε (1) )p−1 = 0, and actually, the kernel of θ is the ideal generated by ω. Here is a simple proof: obviously, the kernel of θ :  E+ → OC /p is the ideal + of x ∈  E such that vE (x) ≥ 1. Let y be any element of  A+ killed by θ whose + A → ker(θ ) is then injective, reduction modulo p satisfies vE (y) = 1. The map y  and surjective modulo p; since both sides are complete for the p-adic topology, it is an isomorphism. Now, we just need to observe that the element ω is killed by θ and that vE (ω) = 1. These constructions are given in Fontaine’s [Fo94a], but the reader should be warned that the notation is rather different; for example,  E+ is Fontaine’s R and +  A is his Ainf . In [Fo94a], the title of this paragraph is also explained (the pair B+ → C) is the solution of a universal problem). The most up-to-date place ( B+ , θ :  to read about these rings is Colmez’ [Col02, §8]. II.2.3 Construction of BdR . Using this we can finally define BdR ; let B+ dR be the ring obtained by completing  B+ for the ker(θ)-adic topology, so that  B+ B+ /(ker(θ ))n . dR = lim ← − n In particular, since ker(θ) = (ω), every element x ∈ B+ dR can be written (in many

+∞ n +  ways) as a sum x = n=0 xn ω with xn ∈ B . The ring B+ dR is then naturally an F -algebra. Let us construct an interesting element of this ring; since θ (1 − [ε]) = 0, the element 1 − [ε] is “small” for the topology of B+ dR and the following series −

+∞  (1 − [ε])n n=1

n

will converge in B+ dR , to an element which we call t. Of course, one should think of t as t = log([ε]). For instance, if g ∈ GF , then g(t) = g(log([ε])) = log([g(ε(0) , ε(1) , . . . )]) = log([εχ(g) ]) = χ (g)t so that t is a period for the cyclotomic character. We now set BdR = B+ dR [1/t], which is a field that we endow with the filtration defined by Fili BdR = t i B+ dR . This is the natural filtration on BdR coming from the

266

Laurent Berger

fact that B+ dR is a complete discrete valuation ring. By functoriality, all the rings we have defined are equipped with a continuous linear action of GK . One can show that K GK is BG dR = K, so that if V is a p-adic representation, then DdR (V ) = (BdR ⊗Qp V ) naturally a filtered K-vector space. We say that V is de Rham if dimK DdR (V ) = d. We see that Gr BdR ⊕i∈Z C(i), and therefore, if V is a de Rham representation (a BdR -admissible representation), then there exist d integers h1 , . . . , hd such that C ⊗Qp V ⊕dj =1 C(hj ). A de Rham representation is therefore Hodge–Tate. Furthermore, one sees easily that the jumps of the filtration on DdR (V ) are precisely the opposites of Hodge–Tate weights of V (that is, Fil−hj (D)  = Fil−hj +1 (D)). For example, if V = Qp ⊗Zp Tp E, then the Hodge–Tate weights of V are 0 and 1. References for this paragraph are Fontaine’s [Fo94a] for the original construction of BdR , and Colmez’s [Col02, §8] for a more general presentation. For the behavior of BdR under the action of some closed subgroups of GK , one can see Iovita–Zaharescu’s [IZ99a, IZ99b]. II.2.4 Sen’s theory for B+ dR . Fontaine has done the analogue of Sen’s theory for + BdR , that is, he defined a K∞ [[t]]-module D+ dif (V ) which is the union of the finite HK which are stable by  . He ⊗ dimensional K∞ [[t]]-submodules of (B+ K Q p V) dR + then proved that Ddif (V ) is a d-dimensional K∞ [[t]]-module endowed with a residual action of K . The action of Lie(K ) gives rise to a differential operator ∇V . The representation V is de Rham if and only if ∇V is trivial on K∞ ((t)) ⊗K∞ [[t]] D+ dif (V ). (V ), ∇ ) simply by applying Furthermore, one recovers (DSen (V ), V ) from (D+ V dif the map θ : B+ dR → C. This construction is carried out in Fontaine’s course [Fon00, §3,4], where BdR representations are classified.

II.3 The rings Bcris and Bst II.3.1 Construction of Bcris . One unfortunate feature of B+ dR is that it is too coarse a ring: there is no natural extension of the natural Frobenius ϕ :  B+ →  B+ to a + + 1/p 1/p p ] − p)  = 0, so that [ p ]−p continuous map ϕ : BdR → BdR . For example, θ ([ 1/p ] − p) ∈ B+ . But if ϕ is a natural extension of is invertible in B+ , and so 1/([ p dR dR B+ , then one should have ϕ(1/([ p1/p ] − p)) = 1/([ p ] − p), and since ϕ : B+ →  . θ ([ p ] − p) = 0, 1/([ p ] − p) ∈ / B+ dR L Another way to see this is that since BG dR = L for every finite extension L/K, the existence of a canonical Frobenius map ϕ : BdR → BdR would imply the existence of a Frobenius map ϕ : K → K, which is of course not the case. One would still like to have a Frobenius map, and there is a natural way to complete  B+ (where one avoids 1/p inverting elements like [ p ] − p) such that the completion is still endowed with a Frobenius map.

An introduction to the theory of p-adic representations

267

+ The ring B+ cris is a subring of BdR , consisting of the limits of sequences of elements

+∞ −n2 n t converges of B+ n=0 p dR which satisfy some growth condition. For example, + + but not in B . The ring B is then equipped with a continuous Frobenius in B+ dR cris cris can be written (in many ways) ϕ. More precisely, recall that every element x ∈ B+ dR

n with x ∈  + . One then has: as x = +∞ x ω B n n=0 n +∞    ωn + +  = x ∈ B , x = x → 0 in B , where x B+ n n cris dR n! n=0

Let Bcris = B+ cris [1/t] (note that Bcris is not a field. For example, ω − p is in Bcris but not its inverse); one can show that (Bcris )GK = F , the maximal absolutely unramified subfield of K. Those representations V of GK which are Bcris -admissible are called crystalline, and using Fontaine’s construction one can therefore associate to every such V a filtered ϕ-module Dcris (V ) = (Bcris ⊗Qp V )GK (a filtered ϕ-module D is an F -vector space with a decreasing, exhaustive and separated filtration indexed by Z on K ⊗F D, and a σF -semi-linear map ϕ : D → D. We do not impose any compatibility condition between ϕ and Fil). One can associate to a filtered ϕ-module D two polygons: its Hodge polygon PH (D), coming from the filtration, and its Newton polygon PN (D), coming from the slopes of ϕ. We say that D is admissible if for every subobject D  of D, the Hodge polygon of D  lies below the Newton polygon of D  , and the endpoints of the Hodge and Newton polygons of D are the same. One can show that Dcris (V ) is always admissible. Furthermore, a theorem of Colmez and Fontaine shows that the functor V  → Dcris (V ) is an equivalence of categories between the category of crystalline representations and the category of admissible filtered ϕ-modules 1 . The construction of Bcris can be found in Fontaine’s [Fo94a] or Colmez’s [Col02, §8]. One should also look at Fontaine’s [Fo94b] for information on filtered ϕ-modules. The theorem of Colmez–Fontaine is proved in Colmez–Fontaine’s [CF00], as well as in Colmez’s [Col02, §10] and it is reviewed in Fontaine’s [Fon00, §5]. The ring B+ cris has an interpretation in crystalline cohomology, see Fontaine’s [Fon83] and FontaineMessing’s [FM87]. II.3.2 Example: elliptic curves. If V = Qp ⊗Zp Tp E, where E is an elliptic curve over F with good ordinary reduction, then Dcris (V ) is a 2-dimensional F -vector space with a basis x, y, and there exists λ ∈ F and α0 , β0 ∈ OF∗ depending on E such that: 
 if i ≤ −1 Dcris (V ) −1 ϕ(x) = α0 p x i and Fil Dcris (V ) = (y + λx)F if i = 0  ϕ(y) = β0 y  {0} if i ≥ 1 1 admissible modules were previously called weakly admissible, but since Colmez and Fontaine showed that being weakly admissible is the same as being admissible (previously, D was said to be admissible if there exists some V such that D = Dcris (V )), we can drop the “weakly” altogether.

268

Laurent Berger

The Newton and Hodge polygons of Dcris (V ) are then as follows: t @ @ @ @ @t

t @

@

@

@

@t

t

Newton polygon

t

Hodge polygon

If on the other hand an elliptic curve E has good supersingular reduction, then the operator ϕ : Dcris (V ) → Dcris (V ) is irreducible and the Newton and Hodge polygons are as follows: t H HH HH HH HH HH t HH HH HH HH Ht

t @ @ @ @ @t

Newton polygon

t

Hodge polygon

In both cases, it is clear that Dcris (V ) is admissible. For basic facts about elliptic curves, see for example Silverman’s [Sil86, Sil96]. For basic facts on Newton polygons, see the first chapter of [DGS94] and for isocrystals, see [Fon79, Kz79]. II.3.3 Semi-stable representations. If an elliptic curve E has bad semi-stable reduction, then V is not crystalline but it is semi-stable, that is, it is Bst -admissible where Bst = Bcris [Y ], where we have decided that ϕ(Y ) = Y p and g(Y ) = Y + c(g)t, where n n c(g) is defined by the formula g(p1/p ) = p1/p (ε (n) )c(g) . Of course, the definition of Y depends on a number of choices, but two such Bst ’s are isomorphic. In addition to a Frobenius, Bst is equipped with the monodromy map N = −d/dY . (0) = p, and let log[ ∈  p ] ∈ B+ Let p E+ be an element such that p dR be the element defined by log[ p ] = logp (p) −

+∞  (1 − [ p ]/p)n−1 n=1

n

.

One can define a Galois equivariant and Bcris -linear embedding of Bst into BdR , by mapping Y to log[ p] ∈ B+ dR , but doing so requires us to make a choice of logp (p). As a consequence, there is no canonically defined filtration on Dst (V ), only on DdR (V ): one has to be a little careful about this. This in contrast to the fact that the inclusion K ⊗F Dcris (V ) ⊂ DdR (V ) is canonical. It is customary to choose logp (p) = 0 which is what we’ll always assume from now on.

An introduction to the theory of p-adic representations

269

One can then associate to every semi-stable representation V a filtered (ϕ, N )module and Colmez and Fontaine showed that the functor V  → Dst (V ) is an equivalence of categories between the category of semi-stable representations and the category of admissible filtered (ϕ, N)-modules. See the references for paragraph II.3.1 on Bcris . See II.4.3 for Fontaine’s original definition of Bst .

II.3.4 Frobenius and filtration. Although the ring Bcris is endowed with both a Frobenius map and the filtration induced by the inclusion Bcris ⊂ BdR , these two structures have little compatibility. For example, here is an exercise: let r = {rn }n≥0 be a sequence with rn ∈ Z. Show that there exists an element xr ∈ k≥0 ϕ k Bcris such that for every n ≥ 0, one has ϕ −n (xr ) ∈ Filrn BdR \ Filrn +1 BdR (for a solution, 0 see paragraph IV.3.2). The reader should also be warned that B+ cris ⊂ Fil Bcris = + Bcris ∩ BdR but that the latter space is much larger. It is true however that if Bcris is the set of elements x ∈ Bcris such that for every n ≥ 0, one has ϕ n (x) ∈ Fil0 Bcris , then  2  ϕ(Bcris ) ⊂ B+ cris ⊂ Bcris (ϕ (Bcris ) if p = 2). Given the above facts, it is rather surprising that there is a relation of some sort ϕ=1 between ϕ and Fil. One can show that the natural map Bcris → BdR /B+ dR is surjective, and that its kernel is Qp . This gives rise to an exact sequence ϕ=1

0 → Qp → Bcris → BdR /B+ dR → 0 called the fundamental exact sequence. It is used to define Bloch–Kato’s exponential. See Fontaine’s [Fo94a] and [Fo94b] or Colmez’s [Col02, §8]. For Bloch–Kato’s exponential, see Bloch–Kato’s [BK91, §3] and Kato’s [Kat93].

II.3.5 Some remarks on topology. We’ll end this section with a few remarks on the topologies of the rings we just introduced. Although B+ dR is a discrete valuation ring, + complete for that valuation, the natural topology on BdR is weaker than the topology coming from that valuation. It is actually the topology of the projective limit on  B+ =  A+ [1/p] combines the p-adic B+ B+ / ker(θ)n , and the topology of  dR = lim ← −n E+ which is a valued ring. topology and the topology of the residue ring  A+ /p =  + n In particular, BdR / ker(θ) is p-adic Banach space, which makes B+ dR into a p-adic Fréchet space. The topology on Bcris is quite unpleasant, as Colmez points out: “By the construcp n −1 /(p n − 1)! does not converge to 0 in B+ , but tion of B+ cris , the sequence xn = ω cris the sequence ωxn does; we deduce from this the fact that the sequence txn converges + to 0 in B+ cris , and therefore that xn → 0 in Bcris , so that the topology of Bcris induced by that of Bcris is not the natural topology of B+ cris .” The reason for this is that the sequence n! converges to 0 in a pretty chaotic manner, and it is more convenient to

270

Laurent Berger

use the ring +∞    ωn + +  , = x ∈ B , x = x , where x → 0 in B B+ n n max dR pn n=0

which is also endowed with a Frobenius map. In any case, the periods of crystalline representations live in  B+ rig [1/t] =

+∞

n=0

ϕ n B+ cris [1/t] =

+∞

n=0

ϕ n B+ max [1/t]

because they live in finite dimensional F -vector subspaces of Bcris stable by ϕ. Finally, let us mention an interesting result of Colmez, that has yet to be applied: + K is naturally a subring of B+ dR , and he showed that BdR is the completion of K for the induced topology, which is finer than the p-adic topology. This generalizes an earlier 2 result of Fontaine, who showed that K is dense B+ dR /t . The topology of K induced + by BdR is a bit like the “uniform convergence of a function and all its derivatives”, if

n (n) one views x ∈ K as an algebraic function of p. For example, the series +∞ n=0 p ε + + is not convergent in BdR . A series which converges in BdR does so in C, so we get a map θ : B+ dR → C, which coincides with the one previously defined. The remark on the topology of Bcris can be found in Colmez’s [Col98a, §III], and Colmez’s theorem is proved in the appendix to Fontaine’s [Fo94a]. Fontaine’s earlier result was used by Fontaine and Messing in [FM87]. The ring  B+ rig has an interpretation in rigid cohomology, as was explained to me by Berthelot in [Blt01].

II.4 Application: Tate’s elliptic curve We will now explicitly show that if E is an elliptic curve with bad semi-stable reduction, then V = Qp ⊗Zp Tp E is BdR -admissible. After that, we will show that V is actually semi-stable. We’ll assume throughout this section that K = F , ie that K is absolutely unramified. II.4.1 Tate’s elliptic curve. Let q be a formal parameter and define sk (q) =

+∞  nk q n , 1 − qn

a4 (q) = −s3 (q),

a6 (q) = −

n=1

x(q, v) =

+∞ 

q nv − 2s1 (q), (1 − q n v)2 n=−∞

y(q, v) =

5s3 (q) + 7s5 (q) , 12

+∞ 

(q n v)2 + s1 (q). (1 − q n v)3 n=−∞

/ q Z = q (the multiplicative All those series are convergent if q ∈ pOF and v ∈ ∗ subgroup of F generated by q). For such q  = 0, let Eq be the elliptic curve defined by the equation y 2 + xy = x 3 + a4 (q)x + a6 (q). The theorem of Tate is then:

An introduction to the theory of p-adic representations

271

the elliptic curve Eq is defined over F , it has bad semi-stable reduction, and Eq is ∗ ∗ uniformized by F , that is, there exists a map α : F → Eq (F ), given by
(x(q, v), y(q, v)) if v ∈ / qZ v → 0 if v ∈ q Z ∗

which induces an isomorphism of groups with GF -action F /q → Eq (F ). Furthermore, if E is an elliptic curve over F with bad semi-stable reduction, then there exists q such that E is isomorphic to Eq over F . For basic facts about Tate’s elliptic curve, see Silverman’s [Sil96, V.3] for example. II.4.2 The p-adic representation attached to Eq . Using Tate’s theorem, we can give an explicit description of Tp (Eq ). Let ε(i) be the pi -th roots of unity chosen in I.2.1 and let q (i) be elements defined by q (0) = q and the requirement that (q (i+1) )p = q (i) . Then α induces isomorphisms ∗

F /q ∗

−−−−→

Eq (F )

n

{x ∈ F /q, x p ∈ q} −−−−→ Eq (F )[pn ] ∗

n

and one sees that {x ∈ F /q, x p ∈ q} = {(ε (n) )i (q (n) )j , 0 ≤ i, j < pn −1}. The elements ε (n) and q (n) therefore form a basis of Eq (F )[p n ], so that a basis of Tp (Eq ) is given by e = limn ε(n) and f = limn q (n) . This makes it possible to compute ← − ← − explicitly the Galois action on Tp (Eq ). We have g(e) = limn g(ε(n) ) = χ (g)e and ← − g(f ) = limn g(q (n) ) = limn q (n) (ε (n) )c(g) = f + c(g)e where c(g) is some p-adic ← − ← − integer, determined by the fact that g(q (n) ) = q (n) (ε (n) )c(g) . Note that [g  → c(g)] ∈ H 1 (F, Zp (1)). The matrix of g in the basis (e, f ) is therefore given by   χ(g) c(g)  . 0 1 II.4.3 p-adic periods of Eq . We are looking for p-adic periods of V = Qp ⊗Zp Tp (Eq ) which live in BdR , that is for elements of (BdR ⊗Qp V )GF . An obvious candidate is t −1 ⊗ e since g(t) = χ(g)t and g(e) = χ (g)e. Let us look for a second element of (BdR ⊗Qp V )GF , of the form a ⊗ e + 1 ⊗ f . We see that this element will be fixed by GF if and only if g(a)χ (g) + c(g) = a. q = (q (0) , q (1) , . . . ). Observe that we Let  q be the element of  E+ defined by  (0) (1) c(g) have g( q ) = (g(q ), g(q ), . . . ) =  q ε , and that θ ([ q ]/q (0) − 1) = 0, so that (0) [ q ]/q − 1 is small in the ker(θ)-adic topology. The series logp (q

(0)

)−

+∞  (1 − [ q ]/q (0) )n n=1

n

272

Laurent Berger

therefore converges in B+ dR to an element which we call u. One should think of u as being u = log([ q ]). In particular, g(u) = g(log([ q ])) = log([g( q )]) = log([ q ]) + c(g) log([ε]) = u + c(g)t, and we readily see that a = −u/t satisfies the equation g(a)χ (g) + c(g) = a. A basis of DdR (V ) = (BdR ⊗Qp V )GF is therefore given by
x = t −1 ⊗ e y = −ut −1 ⊗ e + 1 ⊗ f and this shows that Tp (Eq ) is BdR -admissible. Furthermore, one sees that θ (u − logp (q (0) )) = 0, so that u − logp (q (0) ) is divisible by t and   if i ≤ −1 DdR (V ) i (0) Fil DdR (V ) = (y + logp (q )x)F if i = 0   {0} if i ≥ 1 This gives us a description of DdR (V ). We shall now prove that V is semi-stable. It’s clearly enough to show that t, u ∈ B+ st . The series which defines t converges (that is, the cyclotomic character is crystalline), and the series which dein B+ cris fines log[ q / p vp (q) ] also does. As a consequence, one can write u = vp (q)Y + log[ q / p vp (q) ] ∈ B+ st . This implies that V is semi-stable. Actually, Fontaine defined Bst so that it would contain Bcris and a period of Eq , so that the computation of this paragraph is a little circular. Let us compute the action of Frobenius in the case of Tate’s elliptic curve. On a ring of characteristic p, one expects Frobenius to be x  → x p , and therefore ϕ([x]) should be [x p ] so that ϕ(log[x]) = p log[x]. In particular, one has ϕ(t) = pt and ϕ(u) = pu and the action of Frobenius on Dst (V ) is therefore given by ϕ(x) = p−1 x and ϕ(y) = y. Let us point out one more time that the filtration is defined on DdR (V ), and that the identification Dst (V ) DdR (V ) depends on a choice of logp (p). The

p-adic number logp (q (0) /p vp (q L-invariant of V .

(0) )

) is canonically attached to V and is called the

II.4.4 Remark: Kummer theory. What we have done for Tate’s elliptic curve is really a consequence of the fact that V = Qp ⊗Zp Tp Eq is an extension of Qp by Qp (1), namely that there is an exact sequence 0 → Qp (1) → V → Qp → 0. All of these extensions are classified by the cohomology group H 1 (K, Qp (1)), which is described by Kummer theory. Recall that for every n ≥ 1, there is an isomorphism n δn : K ∗ /(K ∗ )p → H 1 (K, µpn ). By taking the projective limit over n, we get a ∗ → H 1 (K, Zp (1)) because lim µpn Zp (1) once we have chosen map δ : K ← −n a compatible sequence of ε (n) . By tensoring with Qp , we get an isomorphism δ : ∗ → H 1 (K, Qp (1)) which is defined in the following way: if q = q (0) ∈ Qp ⊗Zp K ∗ , choose a sequence q (n) such that (q (n) )p = q (n−1) , and define c = δ(q) Qp ⊗Zp K by (ε(n) )c(g) = g(q (n) )/(q (n) ). Of course, this depends on the choice of q (n) , but two different choices give cohomologous cocycles.

An introduction to the theory of p-adic representations

273

It is now easy to show that every extension of Qp by Qp (1) is semi-stable. This is because tc(g) = g(log[ q ]) − log[ q ] with notations similar to those above, and ∗ then the series which defines log[   q ] converges q = (q (n) ). If q ∈ Qp ⊗Zp O K + ∗ , then in Bcris and the extension V is crystalline. In general, if q ∈ Qp ⊗Zp K + + log[ q ] will be in Bcris + vp (q)Y ⊂ Bst . The F -vector space Dst (V ) will then have q ]t −1 ⊗ e + 1 ⊗ f so that ϕ(x) = p−1 x and a basis x = t −1 ⊗ e and y = − log[ ϕ(y) = y. If one chooses logp (p) = 0, then the filtration on DdR (V ) is given by Fil0 DdR (V ) = (y + logp (q)x)F .

II.5 p-adic representations and Arithmetic Geometry II.5.1 Comparison theorems. If X/K is a proper smooth variety over K, then by a i (X/K). comparison theorem, we mean a theorem relating Héti (XK , Qp ) and HdR It was shown early on by Fontaine that the Tate modules V = Qp ⊗Zp Tp A of all abelian varieties A are de Rham (he actually showed in a letter to Jannsen that they were potentially semi-stable), and that DdR (V ) is isomorphic to the dual of the de Rham cohomology of A. Fontaine and Messing then found another proof, in which they explicitly construct a pairing between V (interpreted as a quotient of 1 (A/K) (interpreted as the group of isomorphism classes of the étale π1 (A)) and HdR vectorial extensions of A). One should remember that for an abelian variety A, we have Hom(Tp A, Zp (1)) Hét1 (AK , Zp ). After that, Fontaine and Messing proved the comparison theorem for the Héti (XK , Qp ) of proper smooth X for i ≤ p − 1 and K/Qp finite unramified. These results were then extended by Kato and his school (Hyodo, Tsuji). Finally, the geni (X/K) from the data of eral statement that for a variety X/K, one can recover HdR V = Héti (XK , Qp ) as a p-adic representation was shown by Tsuji. He showed that if X has semi-stable reduction, then V = Héti (XK , Qp ) is Bst -admissible. A different proof was given by Niziol (in the good reduction case) and also by Faltings (who proved that V is crystalline if X has good reduction and that V is de Rham otherwise). In the case of an abelian variety, the rings Bcris and Bst are exactly what it takes to decide, from the data of V alone, whether A has good or semi-stable reduction. Indeed, Coleman–Iovita and Breuil showed that A has good reduction if and only if V is crystalline, and that A has semi-stable reduction if and only if V is semi-stable. This can be seen as a p-adic analogue of the (-adic) Néron–Ogg–Shafarevich criterion. In another direction, Fontaine and Mazur have conjectured the following: let V be a p-adic representation of Gal(Q/L) where L is a finite extension of Q. Then, V “comes from geometry” if and only if it is unramified at all but finitely many primes , and if its restriction to all decomposition groups above p are potentially semi-stable. Note that among all potentially semi-stable representations V of GK , where K is a p-adic field, there are many which do not come from geometry: indeed, if V = Héti (XK , Qp ) then the eigenvalues of ϕ on Dst (V ) should at least be Weil numbers.

274

Laurent Berger

There were many partial results before Tsuji’s theorem was proved in [Tsu99] (see [Tsu02] for a survey), and we refer the reader to the bibliography of that article. For a different approach (integrating forms on cycles), see Colmez’s [Col98b]. The conjecture of Fontaine and Mazur was proposed by them in [FM93]. There is little known in that direction, but there are some partial results: see Taylor’s [Tay01] and Kisin’s [Kis03] for example. Regarding the criteria for good or semi-stable reduction, see Coleman–Iovita’s [CI99] and Breuil’s [Bre00]. II.5.2 Weil–Deligne representations. Let V be a potentially semi-stable representation of GK , so that there exists L, a finite extension of K such that the restriction of V to GL GL is semi-stable. One can then consider the F -vector space DL st (V ) = (Bst ⊗Qp V ) . It is a finite dimensional (ϕ, N)-module with an action of Gal(L/K). One can attach to such an object several interesting invariants: L-factors, -factors, and a representation of the Weil–Deligne group. In particular, if E is an elliptic curve, one can recover from the p-adic representation Tp E pretty much the same information as from the -adic representation T E. The action of the Weil–Deligne group on DL st (V ) was defined by Fontaine in [Fo94c].

III Fontaine’s (ϕ, )-modules III.1 The characteristic p theory A powerful tool for studying p-adic representations is Fontaine’s theory of (ϕ, )modules. We will first define ϕ-modules for representations of the Galois group of a local field of characteristic p (namely k((π))) and then apply this to the characteristic zero case, making use of Fontaine–Wintenberger’s theory of the field of norms. III.1.1 Local fields of characteristic p. Let πK be a formal variable (for now), let F  be the maximal unramified extension of F in K∞ and 2 let AK be the ring AK =

∞   k=−∞

 ak πKk , ak ∈ OF  , a−k → 0 ,

so that AK /p = kF  ((πK )). The ring AK (which is an example of a Cohen ring, as p in II.2.1) is endowed with actions of ϕ and K , such that ϕ(πK ) = πK mod p. The exact formulas depend on K, but if K = F then ϕ(πK ) = (1 + πK )p − 1 and if 2 it is incorrectly assumed throughout [Ber02] that F = F  . The problem is that even if K/F is totally ramified, K∞ /F∞ does not have to be. In general in [Ber02] one should take eK = e(K∞ /F∞ ) and not [K∞ : F∞ ].

An introduction to the theory of p-adic representations

275

γ ∈ K , then γ (πK ) = (1 + πK )χ(γ ) − 1. We won’t use the action of K in the “characteristic p case”. Let EK = kF  ((πK )) = AK /p, and let E be the separable closure of EK . Let GEK be the Galois group of E/EK . In this paragraph, we will look at p-adic representations of GEK , that is, finite dimensional Qp -vector spaces V , endowed with a continuous linear action of GEK . Let BK be the fraction field of AK (one only needs to invert p). A ϕ-module M is a finite dimensional BK -vector space with a semi-linear action of ϕ. We say that M is étale (or slope 0 or also unit-root) if M admits an AK -lattice MA which is stable by ϕ and such that ϕ ∗ MA = MA (which means that ϕ(MA ) generates MA over AK ). This follows for example from ϕ(MA ) ⊂ MA and p
det(ϕ). The first result is that there is an equivalence of categories {p-adic representations of GEK } ←→ {étale ϕ-modules} Let us explain where this comes from. The correspondence T  → (E ⊗Fp T )GEK is (by Hilbert 90) an equivalence of categories between the category of Fp representations of GEK , and étale EK -modules. Let A be a Cohen ring over E (we will give a more precise definition of A below. Suffice it to say that A should be the ring of Witt vectors over E, but E is not perfect, so that there are several possible choices for A). The ring A is endowed with an action of GEK and AGEK = AK . Then by lifting things to characteristic 0 and inverting p, we get an equivalence of categories between the category of Qp -representations of GEK , and étale BK -modules with a Frobenius (these constructions were previously used, for example, by Bloch and Katz). We will now give a construction of the ring of periods A. Let  E+ be the ring +   E. introduced in II.2.2 and let E be the field of fractions of E . Then EK embeds in  E. Let E be the completion of the For example, if K = F , then EF = k((ε − 1)) ⊂  E. separable closure of EK in  One can show that  E is the completion of the algebraic closure of EF so that  E is the completion of the perfection of E. By a theorem of Ax,  E is also the completion of E. Let  A = W ( E) and  B= A[1/p]. It is easy to see (at least when K = F ) that  BK is a subfield of B, with πF = [ε] − 1. If K  = F , then one should take for πK an element of A whose image modulo p is a uniformiser of EK = EHK . Let B be the completion of the maximal unramified extension of BK in  B, and A = B ∩  A. The field B is endowed with an action of GEK , and one indeed has BGEK = BK . The field B is also naturally endowed with a Frobenius map ϕ. These ideas appear for example in Katz’ [Kz73, chap 4]. We gave their local version, which is in Fontaine’s [Fon91, A1]. III.1.2 Representations of GEK and differential equations. Let us mention an application of the theory we just sketched. Let δ be the differential operator defined by δ(f (π )) = (1 + π )df/dπ on the field BF . This operator extends to B because it extends to the maximal unramified extension of BF , and then to its completion by continuity. One can use it to associate to every p-adic representation of GEK a BF -vector

276

Laurent Berger

space with a Frobenius ϕ and a differential operator δ which satisfy δ  ϕ = pϕ  δ. When the action of the inertia of GEK factors through a finite quotient on a representation V , then there exists a basis of D(E) in which δ is “overconvergent” (in the sense of III.3 below). One can use this fact to associate to every potentially unramified representation of GEK an overconvergent differential equation. This condition (ϕ and δ overconvergent) is much stronger than merely requiring ϕ to be overconvergent (which happens very often, see III.3). There are many interesting parallels between the theory of finite Galois representations in characteristic p and differential equations: see Crew’s [Cre85, Cre00] and Matsuda’s [Mat95] for a starting point.

III.2 The characteristic zero theory III.2.1 The field of norms. The next step of the construction is the theory of the field of norms (of Fontaine and Wintenberger) which gives a canonical isomorphism between GEK and HK . Let NK be the set limn Kn where the transition maps are given ← − by NKn /Kn−1 , so that NK is the set of sequences (x (0) , x (1) , . . . ) with x (n) ∈ Kn and NKn /Kn−1 (x (n) ) = x (n−1) . If we define a ring structure on NK by (xy)(n) = x (n) y (n)

and

(x + y)(n) = lim NKn+m /Kn (x (n+m) + y (n+m) ), m→+∞

then NK is actually a field, called the field of norms of K∞ /K. It is naturally endowed with an action of HK . Furthermore, for every finite Galois extension L/K, NL /NK is a finite Galois extension whose Galois group is Gal(L∞ /K∞ ), and every finite Galois extension of NK is of this kind so that the absolute Galois group of NK is naturally isomorphic to HK . On the other hand, one can prove that NK is a local field of characteristic p isomorphic to EK kF  ((πK )). More precisely, by ramification theory, the map NKn /Kn−1 is “close” to the p-th power map and there is therefore a well-defined ring E given by sending (x (n) ) ∈ NK to (y (n) ) ∈  E where homomorphism from NK to  m (n) (n+m) p ) . This map then realizes an isomorphism between NK y = limm→+∞ (x and EK , so that the two Galois groups HK and GEK are naturally isomorphic. For the theory of the field of norms in a much more general setting, see Fontaine and Wintenberger’s [FW79] and Wintenberger’s [Win83]. For the construction of the isomorphism NK → EK and its relation to Coleman series, see Fontaine’s appendix to [Per94] and Cherbonnier–Colmez’s [CC99]. III.2.2 (ϕ, )-modules. By combining the construction of III.1.1 and the theory of the field of norms, we see that we have an equivalence of categories: {p-adic representations of HK } ←→ {étale ϕ-modules}. We immediately deduce from this the equivalence of categories we were looking for: {p-adic representations of GK } ←→ {étale (ϕ, K )-modules}.

An introduction to the theory of p-adic representations

277

One associates to V the étale ϕ-module D(V ) = (B ⊗Qp V )HK , which is an étale ϕmodule endowed with the residual action of K : it is a (ϕ, K )-module. The inverse functor is then given by D → (B ⊗BK D)ϕ=1 . In general, it is rather hard to write down the (ϕ, )-module associated to a representation V . We can therefore only give a few trivial examples, such as D(Qp (r)) = BF (r). See also the examples in IV.5.4. The original theory of (ϕ, )-modules is the subject of Fontaine’s [Fon91]. It has been modified a bit by Cherbonnier and Colmez in [CC99], whose constructions we have followed. For explicit families of (ϕ, )-modules, see [BLZ03].

III.2.3 Computation of Galois cohomology. Since the category of étale (i.e. slope 0) (ϕ, )-modules is equivalent to that of p-adic representations, it should be possible to recover all properties of p-adic representations in terms of (ϕ, )-modules. For example, Herr showed in his thesis how one could compute the Galois cohomology of V from D(V ). Let H i (K, V ) denote the groups of continuous cohomology of V . Herr’s main result is that one can recover the H i (K, V )’s from D(V ). Let K be the torsion subgroup of K ; since K is an open subgroup of Z∗p , K is a finite subgroup whose

order divides p − 1 (or 2 if p=2). Let p be the idempotent defined by p = |1K | δ∈K δ so that if M is a Zp [[K ]]-module, then p is a projection map from M to M K (at least if p = 2). Let γ be a topological generator of K /K . Let D (V ) = D(V )K . If α is a map α : D (V ) → D (V ) which commutes with K , let Cα,γ (K, V ) be the following complex : f

g

0 → D (V ) → D (V ) ⊕ D (V ) → D (V ) → 0 where f (x) = ((α − 1)x, (γ − 1)x) and g(x, y) = (γ − 1)x − (α − 1)y. The cohomology of the complex Cϕ,γ (K, V ) is then naturally isomorphic to the Galois cohomology of V . For example, we see immediately that H i (K, V ) = 0 if i ≥ 3. This was proved by Herr in [Her98]. For various applications, see Herr’s [Her98, Her01, Her00], Benois’[Ben00], [Ber01, chapVI] and [Ber03a, Ber03c], Cherbonnier– Colmez’s [CC99], and Colmez’s [Col99].

III.3 Overconvergent (ϕ, )-modules Since the theory of (ϕ, )-modules is so good at dealing with p-adic representations, we would like to be able to recover from D(V ) the invariants associated to V by p-adic Hodge theory. This is the subject of the next chapter, on reciprocity formulas, but in this paragraph we will introduce the main technical tool, the ring of overconvergent elements.

278

Laurent Berger

By construction, the field B is a subfield of  B = W ( E)[1/p] =

 

 p k [xk ], xk ∈  E .

k−∞

Let B†,r be the subring of B defined as follows:    p−1 pk [xk ], k + vE (xk ) → +∞ . B†,r = x ∈ B, x = pr k−∞

If rn = pn−1 (p − 1) for some n ≥ 0, then the definition of B†,rn boils down to

(n) requiring that k−∞ pk xk converge in C, which in turn is equivalent to requiring

p −n that k−∞ p k [xk ] converge in B+ dR . If eK denotes the ramification index of K∞ /F∞ , and F  is the maximal unramified extension of F contained in K∞ , then one can show that if πK ∈ AK is the “variable” introduced previously, (see the end of III.1.1) and r  0, then the invariants of B†,r under the action of HK are given by

†,r HK

(B

)

=

B†,r K

=

+∞   k=−∞

ak πKk ,



where ak ∈ F and

+∞ 

ak X k

k=−∞

 is convergent and bounded on p−1/eK r ≤ |X| < 1 .

If K = F (so that eK = 1), then one can take πF = π , and the above description is valid for all r ≥ p − 1. A p-adic representation is said to be overconvergent if, for some r  0, D(V ) has a basis consisting of elements of D†,r (V ) = (B†,r ⊗Qp V )HK . This is equivalent to requiring that there exist a basis of D(V ) in which Mat(ϕ) ∈ M(d, B†,r K ) for some r  0. The main result on the (ϕ, )-modules of p-adic representations (or, equivalently, on étale (ϕ, )-modules) is a theorem of Colmez and Cherbonnier which shows that every p-adic representation of GK (equivalently, every étale (ϕ, )-module) is overconvergent. It is not true that every étale ϕ-module is overconvergent, and their proof uses the action of K in a crucial way. For instance, there is no such result in the characteristic p theory. The above result is the main theorem of Cherbonnier–Colmez’s [CC98]. Most applications of (ϕ, )-modules to p-adic Hodge theory make use of it. If V is absolutely crystalline, then one can say more about the periods of D(V ), see Colmez’s [Col99], [Ber02, 3.3] and [Ber03b]. See also the next chapter.

An introduction to the theory of p-adic representations

279

IV Reciprocity formulas for p-adic representations IV.1 Overview IV.1.1 Reciprocity laws in class field theory. The aim of this chapter is to give constructions relating the theory of (ϕ, )-modules to p-adic Hodge theory. The first thing we’ll do is explain why we (and others) have chosen to call such constructions reciprocity formulas. Recall that, in its simplest form, the aim of class field theory is to provide a description of Gal(K ab /K), where K is a field. For example, if K is a local field, then one has for every finite extension L/K the norm residue symbol (·, L/K) : K ∗ → Gal(L/K)ab , which is a surjective map whose kernel is NL/K (L∗ ). This is a form of the local reciprocity law, and the aim of explicit reciprocity laws is to describe (explicitly!) the map (·, L/K) (more precisely, the Hilbert symbol). For example, a theorem of Dwork shows that if ζ is a p n -th root of unity, then one has (u−1 , Qp (ζ )/Qp ) · ζ = ζ u . Let V = Qp (1), which is the Tate module of the multiplicative group Gm . The classical reciprocity map relates the tangent space DdR (V ) of Gm to the Galois cohomology H 1 (GK , V ). This is why we call a reciprocity map those maps which relate Galois cohomology and p-adic Hodge theory. Since the Galois cohomology of V naturally occurs in (ϕ, )-modules, it is natural to call “reciprocity map” those maps which relate (ϕ, )-modules and p-adic Hodge theory. This is the aim of this chapter: we will show how to recover Dcris (V ) or Dst (V ) from D(V ) and how to characterize de Rham representations. As an application, we will explain the proof of Fontaine’s monodromy conjecture. The first important constructions relating (ϕ, )-modules and p-adic Hodge theory were carried out in Cherbonnier–Colmez’s [CC99], and are closely related to PerrinRiou’s exponential, as in her [Per94] and Colmez’s [Col98a]. See also [Ber03a] for “explicit formulas” for Bloch–Kato’s maps.

IV.2 A differential operator on (ϕ, )-modules In order to further relate the theory of (ϕ, )-modules to p-adic Hodge theory, we will need to look at the action of the Lie algebra of K on D† (V ). On B†K it acts through a differential operator ∇, given by ∇ = log(γ )/ logp (χ (γ )), and one can easily show that ∇(f (π )) = log(1 + π )(1 + π )df/dπ . We see in particular that ∇(f (π )) ∈ / B†K , and so it is necessary to extend the scalars to +∞   = f (π ) = ak πKk , where ak ∈ F  B†,r K rig,K k=−∞

 and f (X) is convergent on p−1/eK r ≤ |X| < 1 .

280

Laurent Berger

The definition is almost the same as that of B†,r K , but we have dropped the bound†,r edness condition. A typical element of Brig,K is t = log(1 + π ). We see that B†,r rig,K is a Fréchet space, with all the norms given by the sup norms on “closed” annuli, and † †,r that it contains B†,r r0 Brig,K is the K as a dense subspace. The union Brig,K =

Robba ring RK of p-adic differential equations, and EK† = B†K is the subring of RK consisting of those functions which are bounded. The p-adic completion of EK† = B†K is EK = BK . This being done, we see that the formula ∇V = log(γ )/ logp (χ (γ )) (this operator is defined in the same way as in paragraph II.1.2) gives the action of Lie(K ) on D†rig (V ) = B†rig,K ⊗B† D† (V ). Unfortunately, the action of Lie(K ) on B†rig,K is not K very nice, because ∇(f (π)) = log(1 + π )(1 + π )df/dπ and this operator has zeroes at all the ζ − 1 with ζ ∈ µp∞ . In particular, it is not a basis of 1 † and it is Brig,K

not the kind of differential operator that fits in the framework of p-adic differential 1 equations. The “right” differential operator is ∂V = log(1+π) ∇V , but this operator acting on D†rig (V ) has poles at all the ζ − 1. In the following paragraphs, we will see that one can “remove” these poles exactly when V is de Rham. See [Ber01, Ber02] or for detailed constructions and the basic properties of those rings and operators.

IV.3 Crystalline and semi-stable representations IV.3.1 Construction of Dcris (V ) and of Dst (V ). We will start by studying the action of K on D†rig (V ), and our main result is that Dcris (V ) = (D†rig (V )[1/t])K , in a sense which will be made precise below. In addition, one can define B†log = B†rig [log(π )] with the obvious actions of ϕ and K , and we shall also see that Dst (V ) = (D†log (V )[1/t])K . If the Hodge–Tate weights of V are negative (if V is positive), then

Dcris (V ) = D†rig (V )K and Dst (V ) = D†log (V )K . Recall that Bst is a subring of BdR equipped with a Frobenius. The periods of V are the elements of Bst which “occur” in the coefficients of Dst (V ), they form a finite dimensional subspace of Bst , stable by Frobenius. Therefore, these periods

+∞ F n-vector + live in n=0 ϕ (Bst )[1/t]. The main strategy for comparing the theory of (ϕ, )-modules and p-adic Hodge theory is to construct a rather large ring  B†rig , which contains B† , B†rig,K and

+∞ n + ϕ (B ) so that B† ⊗ † D† (V ) ⊂  B† ⊗Q V and Dcris (V ) ⊂ ( B† ⊗Q n=0 V )G K .

cris

rig,K

BK

rig

p

rig

p

The result alluded to above, for positive crystalline representations, is that the two F -vector subspaces of  B†rig ⊗Qp V , Dcris (V ) and D†rig (V )K , actually coincide. This means that if V is crystalline, then the Frobenius ϕ on D† (V ) has a rather special form. We’ll give an informal justification for the above result in the next paragraph.

An introduction to the theory of p-adic representations

281

IV.3.2 Rings of periods and limits of algebraic functions. First of all, one should think of most rings of periods as rings of “limits of algebraic functions” on certain subsets of C. For example, the formula B =  Bunr F tells us that B is the ring of limits of (separable) algebraic functions on the boundary of the open unit disk. The ring  B is then the ring of all limits of algebraic functions on the boundary of the open unit disk. Heuristically, one should view other rings in the same fashion: the ring B+ cris “is” the ring of limits of algebraic functions on the disk D(0, |ε(1) − 1|p ), and B+ max “is” the ring of limits of algebraic functions on a slightly smaller disk D(0, r). One should therefore think of ϕ n (B+ cris ) as the ring of limits of algebraic functions on the disk D(0, |ε (n) − 1|p ), and finally  B+ rig “is” the ring of limits of algebraic functions on the open unit disk D(0, 1). Similarly,  B†,r rig “is” the ring of limits of algebraic functions on an annulus C[s, 1[, B†,r where s depends on r, and ϕ −n ( rig ) “is” the ring of limits of algebraic functions on

−n †,r an annulus C[sn , 1[, where sn → 0, so that +∞ n=0 ϕ (Brig ) “is” the ring of limits of algebraic functions on the open unit disk D(0, 1) minus the origin; furthermore, if an element of that ring satisfies some simple growth properties near the origin, then it “extends” to the origin (remember that in complex analysis, a holomorphic function on D(0, 1− ) − {0} which is bounded near 0 extends to a holomorphic function on D(0, 1− )). As for the ring B+ dR , it behaves like a ring of local functions around a circle (in n particular, there is no Frobenius map defined on it). Via the map ϕ −n : B†,r rig → †,rn B+ dR , we have for n ≥ 1 a filtration on Brig , which corresponds to the order of vanishing at ε(n) − 1. For instance, we can now give a short solution to the exercise in paragraph II.3.4: given a sequence rn of integers, let q = ϕ(π)/π and set xr = +∞ n−1 r 0 (q/p)rn . This infinite product converges to a “function” whose order π n=1 ϕ of vanishing at ε (n) − 1 is exactly rn .

IV.3.3 Regularization and decompletion. We shall now justify the above results on Dcris (V ). The analogous results on Dst (V ) follow by adding log(π ) everywhere. We’ve already seen that the periods of positive crystalline representations live in  B+ rig (if we don’t assume that V is positive, then they live in  B+ [1/t]). rig The elements of ( B† ⊗Q V )GK form a finite dimensional F -vector space, so rig

p

GK , and furthermore this B†,r that there is an r such that ( B†rig ⊗Qp V )GK = ( rig ⊗Qp V ) F -vector space is stable by Frobenius, so that the periods of V (in this setting) not

+∞ −n †,r  only live in  B†,r n=0 ϕ (Brig ) and they also satisfy some simple rig but actually in growth conditions (depending, say, on the size of det(ϕ)), which ensure that they too can be seen as limits of algebraic functions on the open unit disk D(0, 1− ), that is as GK = ( GK . This is † B+ elements of  B+ rig . In particular, we have (Brig ⊗Qp V ) rig ⊗Qp V ) what we get by regularization (of the periods).

282

Laurent Berger

It’s easy to show that ( B†rig ⊗Qp V )HK =  B†rig,K ⊗B† step is to show that ( B†rig,K ⊗B†

rig,K

rig,K

D†rig (V ), and the last

D†rig (V ))GK = D†rig (V )GK . This is akin to a

B†rig,K /B†rig,K decompletion process, going from  B†rig,K to B†rig,K . The ring extension  ∞ /K, so that by using Colmez’s decompletion maps, which looks very much like K are analogous to Tate’s pr Kn maps from paragraph II.1.1, one can finally show that in fact, Dcris (V ) = (B†rig,K ⊗B† D† (V ))GK . In particular, V is crystalline if and only if K

(B†rig,K ⊗B† D† (V ))GK is a d-dimensional F -vector space. K See [Ber01, Ber02]. For decompletion maps and the “Tate–Sen” conditions, see [BC03] and Colmez’ Bourbaki talk [Col01].

IV.4 De Rham representations In the previous paragraph, we have shown how to recognize crystalline and semi-stable representations in terms of their (ϕ, )-modules. We shall now do the same for de Rham representations, and show that a representation V is positive de Rham if and only if there exists a free B†rig,K -submodule of rank d of D†rig (V ), called NdR (V ), which is stable by the operator ∂V (when V is not positive, then NdR (V ) ⊂ D†rig (V )[1/t]). Of course, when V is crystalline or semi-stable, one can simply take NdR (V ) = B†rig,K ⊗F Dcris (V ) or NdR (V ) = (B†log,K ⊗F Dst (V ))N =0 . IV.4.1 Construction of NdR (V ). In general, let us give an idea of how one can construct NdR (V ). In the paragraph II.2.4, we recalled Fontaine’s construction of −n sends D†,rn (V ) into (B+ ⊗ HK , which “Sen’s theory for B+ rig dR ”. The map ϕ dR Qp V ) should be thought of as “localizing at ε(n) − 1” in geometrical terms. The module †,rn −n D+ dif (V ) of Fontaine is then equal to K∞ [[t]] ⊗ϕ −n (B†,rn ) ϕ (Drig (V )). Recall that rig,K

Fontaine has shown that a positive V is de Rham and if and only if the connection ∇V has a full set of sections on D+ dif (V ) (in which case the kernel of the connection is K∞ ⊗K DdR (V )). In geometrical terms, this means that if V is positive and de Rham, then ∇V has some “local” solutions around the ε(n) − 1. In that case, one can glue all of those solutions together to obtain NdR (V ). More precisely, there exists n0  0 and r  0 such that we have NdR (V ) = B†rig,K ⊗B†,r Nr (V ) where Nr (V ) is the set rig,K

−n of x ∈ D†,r rig (V ) such that for every n ≥ n0 , one has ϕ (x) ∈ Kn [[t]] ⊗K DdR (V ).

It’s easy to see that Nr (V )[1/t] = D†,r rig (V )[1/t] and that Nr (V ) is a closed (for the †,r Fréchet topology) B†,r rig,K -submodule of Drig (V ). The fact that Nr (V ) is free of rank

d d then follows form the following fact: if M ⊂ (B†,r rig,K ) is a closed submodule, such †,r d that Frac B†,r rig,K ⊗B†,r M = (Frac Brig,K ) , then M is free of rank d. rig,K

An introduction to the theory of p-adic representations

283

One can then show that NdR (V ) is uniquely determined by the requirement that it be free of rank d and stable by ∂V , so that in particular ϕ ∗ NdR (V ) = NdR (V ). We therefore have the following theorem: if V is a de Rham representation, then there exists NdR (V ) ⊂ D†rig (V )[1/t], a B†rig,K -module free of rank d, stable by ∂V and ϕ, such that ϕ ∗ NdR (V ) = NdR (V ). Such an object is by definition a p-adic differential equation with Frobenius structure (see IV.5.2 below). Using this theorem, one can construct a faithful and essentially surjective exact ⊗-functor from the category of de Rham representations to the category of p-adic differential equations with a Frobenius structure. The above theorem is the main result of [Ber02]. For applications, see [Ber02, d Ber03c]. The result on closed submodules of (B†,r rig,K ) is proved in [Ber02, 4.2], see also [For67]. IV.4.2 Example: C-admissible representations. Let us give an example for which it is easy to characterize NdR (V ). We’ve already seen that when V is crystalline or semi-stable, one can take NdR (V ) = B†rig,K ⊗F Dcris (V ) or NdR (V ) = (B†log,K ⊗F Dst (V ))N=0 . Another easy case is when V is C-admissible. This was one of the examples in I.2.3 where we mentioned Sen’s result: a representation V is C-admissible if and only if it is potentially unramified. We’ll give a proof of that result which relies on a theorem of Tsuzuki on differential equations. Let V be a C-admissible representation. This means that C ⊗Qp V = C ⊗K (C ⊗Qp V )GK , so that V is Hodge–Tate and all its weights are 0. In particular, Sen’s map V is zero. Since we recovered Sen’s map from ∇V by localizing at ε(n) − 1, this implies that the coefficients of a matrix of ∇V are holomorphic functions which are 0 at ε (n) − 1 for all n  0. These functions are therefore multiples of t = log(1 + π ) in B†rig,K and so ∇V (D†rig (V )) ⊂ log(1 + π )D†rig (V ) so that we have NdR (V ) = D†rig (V ). The RK -module NdR (V ) is then endowed with a differential operator ∂V and a unit-root Frobenius map ϕ which is overconvergent. One can show that if ϕ is overconvergent, then so is ∂V (because ϕ regularizes functions). The module NdR (V ) is therefore an overconvergent unit-root isocrystal, and Tsuzuki proved that these are potentially trivial (that is, they become trivial after extending the scalars to RL /RK for a finite extension L/K). This implies easily enough that the restriction of V to IK is potentially trivial. See [Ber02, 5.6]. For Tsuzuki’s theorem, see his [Tsk99] and Christol’s [Chr01]. Sen’s theorem was first proved in Sen’s [Sen73].

IV.5 The monodromy theorem IV.5.1 -adic monodromy and p-adic monodromy. As was pointed out in the introduction, -adic representations are forced to be well-behaved, while the group GK has far too many p-adic representations. Over the years it became apparent that the

284

Laurent Berger

only representations related to arithmetic geometry were the de Rham representations (see II.5.1). In particular it was conjectured (and later proved) that all representations coming from geometry were de Rham. Among these, some are more pleasant, they are the semi-stable ones, which are the analogue of the -adic unipotent representations. Grothendieck has shown that all -adic representations are quasi-unipotent, and after looking at many examples, Fontaine was led to conjecture the following p-adic analogue of Grothendieck’s -adic monodromy theorem: every de Rham representation is potentially semi-stable. We shall now explain the proof of that statement. An excellent reference throughout this section is Colmez’ Bourbaki talk [Col01]. IV.5.2 p-adic differential equations. A p-adic differential equation is a module M, free of finite rank over the Robba ring RK , equipped with a connection ∂M : M → M. We say that M has a Frobenius structure if there is a semi-linear Frobenius ϕM : M → M which commutes with ∂M . A p-adic differential equation is said to be quasi-unipotent if there exists a finite extension L/K such that ∂M has a full set of solutions on RL [log(π )] ⊗RK M. Christol and Mebkhout extensively studied p-adic differential equations. Crew and Tsuzuki conjectured that every p-adic differential equation with a Frobenius structure is quasi-unipotent. Three independent proofs were given in the summer of 2001. One by André, using Christol–Mebkhout’s results and a Tannakian argument. One by Kedlaya, who proved a “Dieudonné–Manin” theorem for ϕ-modules over RK . And one by Mebkhout, relying on Christol–Mebkhout’s results. We refer the reader to Christol and Mebkhout’s surveys [CM00, CM02] and Colmez’s Bourbaki talk [Col01] for enlightening discussions of p-adic differential equations. The above theorem is proved independently in André’s [And02b], Mebkhout’s [Meb02] and Kedlaya’s [Ked00]. See also André’s [And02a] for a beautiful discussion of a special case. IV.5.3 The monodromy theorem. Using the previous results, one can give a proof of Fontaine’s monodromy conjecture. Let V be a de Rham representation, then one can associate to V a p-adic differential equation NdR (V ). By André, Kedlaya, and Mebkhout’s theorem, this differential equation is quasi-unipotent. Therefore, there exists a finite extension L/K such that (RL [log(π )] ⊗RK NdR (V ))GL is an F -vector space of dimension d and by the results of paragraph IV.3.3, V is potentially semistable. See [Ber02, 5.5] for further discussion of the above result. IV.5.4 Example: Tate’s elliptic curve. To finish this chapter, we will sketch this for Tate’s elliptic curve (or indeed for all ordinary elliptic curves). For simplicity, assume that k is algebraically closed. If q = q0 is the parameter associated to Eq , then there exists qn ∈ Fn = F (ε (n) ) such that NFn+1 /Fn (qn+1 ) = qn (this is the

An introduction to the theory of p-adic representations

285

only place where we use the fact that k is algebraically closed), and by a result of σ −n

Coleman, there is a power series Colq (π) such that qn = ColqF (ε(n) −1). If Fq (π ) = (1+π) dlog Colq (π), then Fq (π) ∈ π −1 OF [[π]] and one can show that there is a basis (a, b) of the (ϕ, )-module D(V ) associated to V such that the action of F = γ  is given by:   1−η Fq (π ) χ(η) 1−γ  Mat(η) =  0 1 Let ∇ be the differential operator giving the action of the Lie algebra of F on power series, so that we have (∇f )(π) = (1+π ) log(1+π )f  (π ) (recall that t = log(1+π )). The Lie algebra of F then acts on D†rig (V ) by an operator ∇V given by   ∇ 1 1−γ Fq (π )  Mat(∇V ) =  0 0 One then sees that ∂V (t −1 a) = 0 and that ∂V (b) belongs to B†rig,F (t −1 a), so that the p-adic differential equation t −1 a, b is unipotent. This shows that V is indeed semi-stable. The extensions of Qp by Qp (1) are important and also a source of explicit examples. They are related to Kummer theory as in paragraph II.4.4, and Coleman series as above, among other topics. Some interesting computations can be found in Cherbonnier–Colmez’s [CC99, V].

V Appendix V.1 Diagram of the rings of periods The following diagram summarizes the relationships between the different rings of / is an / / are surjective, the dotted arrow periods. The arrows ending with †,r + n inductive limit of maps defined on subrings (ιn :  Blog → BdR ), and all the other ones are injective. All the rings with tildes (  ) also have versions without a tilde: one goes from the latter to the former by making Frobenius invertible and completing. For example,  E is the completion of the perfection of E. The three rings in the leftmost column (at least their tilde-free versions) are related to the theory of (ϕ, K )-modules. The three rings in the rightmost column are related to p-adic Hodge theory. To go from one theory to the other, one goes from one side to the other through all the intermediate rings. The best case is when one can work in the middle column. For example, from top to bottom: semi-stable, crystalline, or finite height representations. The ring that binds them all is  B†log .

286

Laurent Berger

4 B+ dR O  B†log o

 B+ log

 B†rig o

 B+ rig

 BO o

 BO † o

 B+ O

θ

 Ao

 A† o

 A+

θ

O

O



 Eo

/ B+ st O

O

/ B+

max

O



 E+

θ

θ

//Crr O / / OC  / / OC /p

V.2 List of the rings of power series Let us review the different rings of power series which occur in this article; let C[r; 1[ be the annulus{z ∈ C, p−1/r ≤ |z|p < 1}. We then have: E+ F

A+ F B+ F EF

k[[T ]] OF [[T ]] F ⊗OF OF [[T ]]

BF

k((T ))  ]][T −1 ] OF [[T  F ⊗OF OF [[T ]][T −1 ]

A†,r F

Laurent series f (T ), convergent on C[r; 1[, and bounded by 1

B†,r F †,r Brig,F B†,r log,F

Laurent series f (T ), convergent on C[r; 1[, and bounded

B+ rig,F

f (T ) ∈ F [[T ]], f (T ) converges on the open unit disk D[0; 1[

B+ log,F

B+ rig,F [log(T )]

AF

Laurent series f (T ), convergent on C[r; 1[ B†,r rig,F [log(T )]

An introduction to the theory of p-adic representations

287

References [And02a]

André, Y., Représentations galoisiennes et opérateurs de Bessel p-adiques. Ann. Inst. Fourier (Grenoble) 52 (3) (2002), 779–808.

[And02b]

André, Y., Filtrations de Hasse-Arf et monodromie p-adique. Invent. Math. 148 (2002), 285–317.

[Ax70]

Ax, J., Zeros of polynomials over local fields—The Galois action. J. Algebra 15 (1970), 417–428.

[Ben00]

Benois, D., On Iwasawa theory of crystalline representations. Duke Math. J. 104 (2000), 211–267.

[Ber01]

Berger, L., Représentations p-adiques et équations différentielles. Thèse de l’Université de Paris VI, Mai 2001.

[Ber02]

Berger, L., Représentations p-adiques et équations différentielles. Invent. Math. 148 (2002), 219–284.

[Ber03a]

Berger, L., Bloch and Kato’s exponential map: three explicit formulas. Doc. Math., Extra Volume: Kazuya Kato’s Fiftieth Birthday (2003), 99–129.

[Ber03b]

Berger, L., Limites de représentations cristallines. To appear in Compositio Math.

[Ber03c]

Berger, L., Représentations de de Rham et normes universelles. To appear in Bull. Soc. Math. France. .

[BLZ03]

Berger, L., Li, H., Zhu, H., Construction of some families of 2-dimensional crystalline representations. To appear in Math. Ann.

[BC03]

Berger, L., Colmez, P., Familles de représentations p-adiques. In preparation.

[Blt01]

Berthelot, P., Lettre du 29 Mars 2001.

[BK91]

Bloch, S., Kato, K., L-functions and Tamagawa numbers of motives. In The Grothendieck Festschrift, Vol. I, Progr. Math. 86, Birkhäuser, Boston, MA, 1990, 333–400.

[Bre00]

Breuil, C., Groupes p-divisibles, groupes finis et modules filtrés. Ann. of Math. 152 (2000), 489–549.

[Bre02]

Breuil, C., Integral p-adic Hodge theory. Adv. Stud. Pure Math. 36, Math. Soc. Japan, Tokyo 2002.

[BM02]

Breuil, C., Messing, W., Torsion étale and crystalline cohomology. Astérisque 279 (2002), 81–124.

[CC98]

Cherbonnier, F., Colmez, P., Représentations p-adiques surconvergentes. Invent. Math. 133 (1998), 581–611.

[CC99]

Cherbonnier, F., Colmez, P., Théorie d’Iwasawa des représentations p-adiques d’un corps local. J. Amer. Math. Soc. 12 (1999), 241–268.

[Chr01]

Christol, G., About a Tsuzuki theorem. In p-adic functional analysis (Ioannina, 2000), Lecture Notes in Pure and Appl. Math. 222, Dekker, NewYork 2001, 63–74.

288

Laurent Berger

[CM00]

Christol, G., Mebkhout, Z., p-adic differential equations. In Algebra and number theory (Fez), Lecture Notes in Pure and Appl. Math. 208, Dekker, New York 2000, 105–116.

[CM02]

Christol, G., Mebkhout, Z., Équations différentielles p-adiques et coefficients padiques sur les courbes. Astérisque 279 (2002), 125–183.

[CR94]

Christol, G., Robba, P., Équations différentielles p-adiques. Applications aux sommes exponentielles. Actualités Math., Hermann, Paris 1994.

[CI99]

Coleman, R., Iovita, A., The Frobenius and monodromy operators for curves and abelian varieties. Duke Math. J. 97 (1) (1999), 171–215.

[Col94]

Colmez, P., Sur un résultat de Shankar Sen. C. R. Acad. Sci. Paris Sér. I Math. 318 (11) (1994), 983–985.

[Col98a]

Colmez, P., Théorie d’Iwasawa des représentations de de Rham d’un corps local. Ann. of Math. 148 (1998), 485–571.

[Col98b]

Colmez, P., Intégration sur les variétés p-adiques. Astérisque 248 (1998), viii+155 pp.

[Col99]

Colmez, P., Représentations cristallines et représentations de hauteur finie. J. Reine Angew. Math. 514 (1999), 119–143.

[Col00]

Colmez, P., Fonctions L p-adiques. Séminaire Bourbaki, 1998/99, Astérisque 266 (2000), Exp. 851.

[Col02]

Colmez, P., Espaces de Banach de dimension finie. Journal de l’Inst. Math. de Jussieu 1 (3) (2002), 331–439.

[Col01]

Colmez, P., Les conjectures de monodromie p-adiques. Séminaire Bourbaki, 2001/02, Astérisque, Exp. 897.

[CF00]

Colmez, P., Fontaine J-M., Constructions des représentations p-adiques semistables. Invent. Math. 140 (2000), 1–43.

[Cre85]

Crew, R., F -isocrystals and p-adic representations. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Part 2, Amer. Math. Soc., Providence, RI, 1987, 111–138.

[Cre98]

Crew, R., Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve. Ann. Sci. École Norm. Sup. 31 (1998), 717–763.

[Cre00]

Crew, R., Canonical extensions, irregularities, and the Swan conductor. Math. Ann. 316 (1) (2000), 19–37.

[DGS94]

Dwork, B., Gerotto G., Sullivan F., An introduction to G-functions. Ann. of Math. Stud. 133. Princeton University Press, Princeton, NJ, 1994, xxii+323 pp.

[Fal98]

Faltings, G., Crystalline cohomology and p-adic Galois representations. In Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, 25–80.

[Fon79]

Fontaine, J.-M., Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate. In Journées de Géomètrie Algébrique de Rennes (Rennes, 1978), Vol. III, Astérisque 65 (1979), 3–80.

An introduction to the theory of p-adic representations

289

[Fon82]

Fontaine, J.-M., Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate. Ann. of Math. (2) 115 (3) (1982), 529–577.

[Fon83]

Fontaine, J.-M., Cohomologie de de Rham, cohomologie cristalline et représentations p-adiques. In Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, Springer-Verlag, Berlin 1983, 86–108.

[Fo94a]

Fontaine, J.-M., Le corps des périodes p-adiques. Périodes p-adiques (Bures-surYvette, 1988), Astérisque 223 (1994), 59–111.

[Fo94b]

Fontaine, J.-M., Représentations p-adiques semi-stables. Périodes p-adiques, (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 113–184.

[Fo94c]

Fontaine, J.-M., Représentations -adiques potentiellement semi-stables. Périodes p-adiques, (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 321–347.

[Fon91]

Fontaine, J.-M., Représentations p-adiques des corps locaux I. In The Grothendieck Festschrift, Vol. II, Progr. Math. 87, Birkhäuser, Boston, MA, 1990, 249–309.

[Fon96]

Fontaine, J.-M., Deforming semistable Galois representations. Elliptic curves and modular forms (Washington, DC, 1996). Proc. Nat. Acad. Sci. U.S.A. 94 (21) (1997), 11138–11141.

[Fon00]

Fontaine, J.-M., Arithmétique des représentations galoisiennes p-adiques. À paraître dans Astérisque.

[Fon92]

Fontaine, J.-M., Valeurs spéciales des fonctions L des motifs. Séminaire Bourbaki, Vol. 1991/92. Astérisque 206 (1992), Exp. No. 751, 4, 205–249.

[FI93]

Fontaine, J.-M., Illusie, L., p-adic periods: a survey. In Proceedings of the IndoFrench Conference on Geometry (Bombay, 1989), Hindustan Book Agency, Delhi 1993, 57–93.

[FL82]

Fontaine, J.-M., Laffaille, G., Construction de représentations p-adiques. Ann. Sci. École Norm. Sup. (4) 15 (4) (1982), 547–608 (1983).

[FM93]

Fontaine, J.-M., Mazur, B., Geometric Galois representations. In Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, Internat. Press, Cambridge, MA, 1995, 41–78.

[FM87]

Fontaine, J.-M., Messing, W., p-adic periods and p-adic étale cohomology. In Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math. 67, Amer. Math. Soc., Providence, RI, 1987, 179–207.

[FW79]

Fontaine, J.-M., Wintenberger, J.-P., Le “corps des normes” de certaines extensions algébriques de corps locaux. C. R. Acad. Sci. Paris Sér. A-B 288 (6) (1979), A367–A370.

[For67]

Forster, O., Zur Theorie der Steinschen Algebren und Moduln. Math. Z. 97 (1967), 376–405.

[Har97]

Harder, G., Wittvektoren. Jahresber. Deutsch. Math.-Verein. 99 (1) (1997), 18–48.

[Her98]

Herr, L., Sur la cohomologie galoisienne des corps p-adiques. Bull. Soc. Math. France 126 (1998), 563–600.

290

Laurent Berger

[Her01]

Herr, L., Une approche nouvelle de la dualité locale de Tate. Math. Ann. 320 (2) (2001), 307–337.

[Her00]

Herr, L., --modules and Galois cohomology. In Invitation to higher local fields (Münster, 1999), Geom. Topol. Monogr. 3, Geom. Topol. Publ., Coventry, 2000, 263–272 (electronic).

[IZ99a]

Iovita, A., Zaharescu, A., Generating elements for B+ dR . J. Math. Kyoto Univ. 39 (2) (1999), 233–248.

[IZ99b]

Iovita, A., Zaharescu, A., Galois theory of B+ dR . Compositio Math. 117 (1) (1999), 1–31.

[Kat93]

Kato, K., Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR . In Arithmetic algebraic geometry, Lecture Notes in Math. 1553, SpringerVerlag, Berlin 1993, 50–63.

[Kz73]

Katz, N., p-adic properties of modular schemes and modular forms. In Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math. 350, Springer-Verlag, Berlin 1973, 69–190.

[Kz79]

Katz N., Slope filtration of F -crystals. In Journées de Géomètrie Algébrique de Rennes (Rennes, 1978), Vol. I, Astérisque 63 (1979), 113–163.

[Ked00]

Kedlaya, K., A p-adic local monodromy theorem. To appear in Ann. of Math.

[Kis03]

Kisin, M., Overconvergent modular forms and the Fontaine-Mazur conjecture. Invent. Math. 153 (2) (2003), 373–454.

[Kob84]

Koblitz, N., p-adic numbers, p-adic analysis, and zeta-functions. Second edition. Grad. Texts in Math. 58, Springer-Verlag, New York 1984.

[Laz62]

Lazard, M., Les zéros des fonctions analytiques d’une variable sur un corps valué complet. Inst. Hautes Études Sci. Publ. Math. 14 (1962), 47–75.

[Mat95]

Matsuda, S., Local indices of p-adic differential operators corresponding to ArtinSchreier-Witt coverings. Duke Math. J. 77 (3) (1995), 607–625.

[Meb02]

Mebkhout, Z., Analogue p-adique du théorème de Turrittin et le théorème de la monodromie p-adique. Invent. Math. 148 (2002), 319–351.

[Nek93]

Nekováˇr, J., On p-adic height pairings. In Séminaire de Théorie des Nombres, Paris, 1990–91, Progr. Math. 108, Birkhäuser, Boston, MA, 1993, 127–202.

[Niz98]

Niziol, W., Crystalline conjecture via K-theory. Ann. Sci. École Norm. Sup. (4) 31 (5) (1998), 659–681.

[Per88]

Perrin-Riou, B., Représentations p-adiques ordinaires. In Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 185–220.

[Per94]

Perrin-Riou, B., Théorie d’Iwasawa des représentations p-adiques sur un corps local. Invent. Math. 115 (1994), 81–161.

[Per95]

Perrin-Riou, B., Fonctions L p-adiques des représentations p-adiques. Astérisque 229 (1995), 198 pp.

An introduction to the theory of p-adic representations

291

[Per99]

Perrin-Riou, B., Théorie d’Iwasawa et loi explicite de réciprocité. Doc. Math. 4 (1999), 219–273 (electronic).

[Per01]

Perrin-Riou, B., Théorie d’Iwasawa des représentations p-adiques semi-stables. Mém. Soc. Math. Fr. (N.S.) 84 (2001), vi+111 pp.

[Sen72]

Sen, S., Ramification in p-adic Lie extensions. Invent. Math. 17 (1972), 44–50.

[Sen73]

Sen, S., Lie algebras of Galois groups arising from Hodge–Tate modules. Ann. of Math. 97 (1973), 160–170.

[Sen80]

Sen, S., Continuous cohomology and p-adic Galois representations. Invent. Math. 62 (1980/81), 89–116.

[Sen93]

Sen, S., An infinite-dimensional Hodge–Tate theory. Bull. Soc. Math. France 121 (1) (1993), 13–34.

[Ser92]

Serre, J.-P., Lie Algebras and Lie Groups. Second edition. Lecture Notes in Math. 1500. Springer-Verlag, Berlin 1992.

[Ser89]

Serre, J.-P., Abelian -adic representations and elliptic curves. With the collaboration of Willem Kuyk and John Labute. Second edition. Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989, xxiv+184 pp.

[Ser68]

Serre, J.-P. Corps locaux. Deuxième édition. Publications de l’Université de Nancago, No. VIII. Hermann, Paris 1968, 245 pp.

[Ser79]

Serre, J.-P. Local fields. Translated from the French by Marvin Jay Greenberg. Grad. Texts in Math. 67, Springer-Verlag, New York–Berlin 1979, viii+241 pp.

[Sil86]

Silverman, J., The arithmetic of elliptic curves. Grad. Texts in Math. 106, SpringerVerlag, New York 1986, xii+400 pp.

[Sil96]

Silverman, J., Advanced topics in the arithmetic of elliptic curves. Grad. Texts in Math. 151, Springer-Verlag, New York, 1994, xiv+525 pp.

[Tat66]

Tate, J., p-divisible groups. In Proc. Conf. Local Fields (Driebergen, 1966), Springer-Verlag, Berlin, 158–183.

[Tay01]

Taylor, R., Remarks on a conjecture of Fontaine and Mazur. Journal de l’Inst. Math. de Jussieu 1 (2001), 1–19.

[Tay02]

Taylor, R., Galois representations. To appear in Ann. Fac. Sci. Toulouse Math.

[Tsu99]

Tsuji, T., p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137 (1999), 233–411.

[Tsu02]

Tsuji, T., Semi-stable conjecture of Fontaine-Jannsen: a survey. Astérisque 279 (2002), 323–370.

[Tsk98]

Tsuzuki, N., Slope filtration of quasi-unipotent overconvergent F -isocrystals. Ann. Inst. Fourier (Grenoble) 48 (1998), 379–412.

[Tsk99]

Tsuzuki, N., Finite local monodromy of overconvergent unit-root F -isocrystals on a curve. Amer. J. Math. 120 (1998), 1165–1190.

[Wa96]

Wach, N., Représentations p-adiques potentiellement cristallines. Bull. Soc. Math. France 124 (1996), 375–400.

292

Laurent Berger

[Wa97]

Wach, N., Représentations cristallines de torsion. Compositio Math. 108 (1997), 185–240.

[Win83]

Wintenberger, J.-P., Le corps des normes de certaines extensions infinies des corps locaux; applications. Ann. Sci. École Norm. Sup. 16 (1983), 59–89.

[Win88]

Wintenberger, J.-P., Théorème de comparaison p-adique pour les schémas abéliens. I: Construction de l’accouplement de périodes. Périodes p-adiques (Bures-surYvette, 1988), Astérisque 223 (1994), 349–397.

Laurent Berger, Harvard University, Department of Mathematics, One Oxford Street, Cambridge, MA 02138-2901, U.S.A. E-mail: [email protected]

Smooth p-adic analytic spaces are locally contractible. II Vladimir G. Berkovich∗

Contents 0

Introduction

293

1

Piecewise RS -linear spaces

298

2

R-colored polysimplicial sets

308

3

R-colored polysimplicial sets of length l

313

4 The skeleton of a nondegenerate pluri-stable formal scheme

327

5 A colored polysimplicial set associated with a nondegenerate poly-stable fibration

336

6

p-Adic analytic and piecewise linear spaces

346

7

Strong local contractibility of smooth analytic spaces

355

8

Cohomology with coefficients in the sheaf of constant functions

362

0 Introduction Let k be a field complete with respect to a non-Archimedean valuation, k  its ring of integers, and  k its residue field. Every formal scheme X locally finitely presented over k, and a generic k  has a closed fiber Xs , which is a scheme of locally finite type over  fiber Xη , which is a strictly k-analytic space (in the sense of [Ber2]) whose underlying ∗ This research was supported by Minerva Foundation, Germany, and US-Israel Binational Science Foundation

294

Vladimir G. Berkovich

topological space is a paracompact locally compact space of dimension dim(Xη ), and there is a reduction map π : Xη → Xs . Given a formal scheme X for which there is a sequence of morphisms from a certain fl−1

f1

f0

class X = (X = Xl → · · · → X1 → X0 = Spf(k  )), in [Ber7] we constructed a strong deformation retraction of the generic fiber Xη to a closed subset S(X) called the skeleton of X. (The morphisms from that class are called poly-stable, such a sequence X is called a poly-stable fibration, and such a formal scheme X is called pluri-stable.) We also constructed a canonical homeomorphism between the skeleton S(X) and the geometric realization of a simplicial set associated with the closed fiber of X. This homotopy description of the spaces Xη together with the results of J. de Jong from [deJ] were used in [Ber7] to prove that in the case, when the valuation on k is nontrivial, any strictly analytic subdomain of a smooth k-analytic space is locally contractible. In our work in progress on integration on p-adic analytic spaces, the following stronger property turns out to play an important role. Assume that the valuation on k is nontrivial, and let X be a strictly analytic domain in a smooth k-analytic space. Then each point x ∈ X has a fundamental system of open neighborhoods V such that: (a) there is a contraction  of V to a point x0 ∈ V ; (b) there is an increasing sequence of compact strictly analytic domains X1 ⊂ X2 ⊂ · · · ⊂ V which exhaust V and are K has a finite preserved under ; (b) for any bigger non-Archimedean field K, V ⊗ number of connected components and  lifts to a contraction of each of them to a point over x0 ; and (d) there is a finite separable extension L of k such that, if K from K → V ⊗ L induces a bijection between the sets of (c) contains L, then the map V ⊗ connected components. One of the main purposes of this paper is to prove the above property. The proof is based on a further study of the skeleton S(X) for those poly-stable fibrations X in which all of the poly-stable morphisms fi : Xi+1 → Xi are so called nondegenerate. This study has an independent interest. It turns out that S(X) depends only on X = Xl (it is therefore denoted by S(X)), and that it is provided with a canonical piecewise linear structure of a special type. This piecewise linear structure on the skeleton S(X) is closely related to the analytic structure on the generic fiber Xη , and is in fact reflected in many familiar properties and objects related to analytic functions (such as the growth and Newton polygon of an analytic function). We now give a summary of the material which follows. In §1, we introduce and study a subcategory of the category of piecewise linear spaces. The exposition is slightly non-traditional in the sense that the model vector space for us is the multiplicative group (R∗+ )n provided with the following action of R: (s, (t1 , . . . , tn )) → (t1s , . . . , tns ). Similarly, linear functions considered are maps to R∗+ of the form (t1 , . . . , tn ) → rt1s1 . . . tnsn . The subcategory introduced consists of the piecewise linear spaces which are built from the polytopes defined by linear inequalities with certain restrictions on their coefficients. Namely, the coefficients at the linear terms are required to belong to a sub-semiring S ⊂ R, and the constant terms are required to belong to a submonoid R ⊂ R∗+ such that for any r ∈ R and s ∈ S

Smooth p-adic analytic spaces are locally contractible. II

295

one has r s ∈ R. The polytopes defined in such a way are called RS -polytopes, and the spaces obtained are called piecewise RS -linear. If S = R and R = R∗+ , one gets the whole category of piecewise linear spaces. The skeleton S(X) of a nondegenerate pluri-stable formal scheme over k  is provided (in §5) with a piecewise RZ+ -linear structure for R = |k ∗ | ∩ [0, 1]. There is at least a formal similarity between piecewise linear and k-analytic spaces. Namely, both are provided with a Grothendieck topology formed by piecewise linear subspaces in the former and by analytic subdomains in the latter. Coverings are defined in the same way: a family {Yi }i∈I of subspaces of Y is a covering if every point y ∈ Y has a neighborhood of the form Yi1 ∪ · · · ∪ Yin with y ∈ Yi1 ∩ · · · ∩ Yin . In §6, a direct relation between the Grothendieck topologies on S(X) and Xη is established, and it is very important for applications in §7 and §8. To describe the constructions of §2 and §3, recall that in [Ber7] we associated with the closed fiber of a poly-stable fibration X over k  of length l a polysimplicial set, i.e., an object of the category  E ns of contravariant functors from a certain category  to the category of sets E ns. (The simplicial set mentioned at the beginning of the introduction was in fact derived from the latter.) If l = 1, we associated with the formal scheme X = X1 itself a more refined object, an R-colored polysimplicial set, i.e., an object of the category R E ns, where the category R was associated with a submonoid R ⊂ [0, 1]. (In the case considered, R = |k| ∩ [0, 1].) The geometric realization of an R-colored polysimplicial set was provided with an extra structure, a monoid of continuous functions to [0, 1] (which were eventually related to the absolute values of the functions from the monoid O(X) ∩ O(Xη )∗ ). Let R be a category provided with a geometric realization functor that takes an object A to a pair (|A|, MA ), where |A| is a topological space and MA is a semiring of continuous functions on |A| with values in [0, 1]. (The semirings are considered ˙ = max(f, g).) In §2, with the usual multiplication and the following addition: f +g we construct a category R provided with a similar geometric realization functor. It gives rise to a category of R-colored polysimplicial sets R E ns and a similar geometric realization functor on it. If R is a one point category with the geometric realization functor that takes the only object of R to a one point space with a submonoid R ⊂ [0, 1], one gets the category R introduced in [Ber7, §4]. The only difference is that the monoids, considered in loc. cit., are submonoids of the semirings considered here, but the former can be characterized inside the latter. In §3, we study the category obtained by iteration of the latter construction. Namely, given a submonoid R ⊂ [0, 1], we set R,1 = R and R,l = R,l−1 for l ≥ 2. In this way we get the category R,l E ns of R-colored polysimplicial sets of length l. The main facts established here are as follows. The geometric realization of an R-colored polysimplicial set of length l is always Hausdorff and, if the set is locally finite and 0 ∈ R, the geometric realization is provided with a canonical piecewise RZ+ -linear structure so that the semiring associated with it consists of certain piecewise RZ+ -linear functions.

296

Vladimir G. Berkovich

In §4, we recall the notion of a poly-stable morphism and introduce an additional property of nondegenerateness. (A pluri-stable formal scheme over k  is nondegenerate if and only if its generic fiber is a normal strictly k-analytic space.) We introduce a partial ordering on the generic fiber Xη of a formal scheme X locally finitely presented over k  , and prove that the skeleton S(X) of a nondegenerate poly-stable fibration X of length l coincides with the set of maximal points with respect to the ordering on Xl,η . This implies that S(X) depends only on Xl , and so the skeleton S(X) of a nondegenerate pluri-stable formal scheme X is well defined. We also recall the construction of the retraction map τ : Xη → S(X), which in general depends on the choice of X with Xl = X, and introduce a class of so called strongly nondegenerate pluri-stable formal schemes for which τ does not depend on the choice of X. In §5, we associate with every nondegenerate poly-stable fibration X over k  of length l a locally finite R-colored polysimplicial set D(X) of length l, where R = ∼ |k ∗ | ∩ [0, 1], and construct a canonical homeomorphism |D(X)| → S(X) such that, for any f ∈ O (Xl ), the function x  → |f (x)| on S(X) is contained in the semiring MX associated with the geometric realization of D(X). (Here O (X) is the set of all f ∈ O(X) whose restriction to every connected component of X is not zero.) This provides the skeleton S(X) with a piecewise RZ+ -linear structure and a semiring of piecewise RZ+ -linear functions MX . In §6.1, we prove that the latter depend only on Xl , i.e., given a nondegenerate pluri-stable formal scheme X over k  , a piecewise RZ+ -linear structure on S(X) and a semiring of piecewise RZ+ -linear functions MX on it are well defined and, for any f ∈ O (X), the function x → |f (x)| on S(X) is contained in MX . We also prove that any pluri-stable morphism ϕ : X → X from a similar formal scheme X gives rise to a piecewise RZ+ -linear map S(X ) → S(X) and it takes functions from MX to functions from MX . In §6.2, we get a first application of the above results whose elementary particular case tells the following. Given a compact strictly analytic domain X in the analytification of a separated scheme of finite type over k and invertible analytic functions f1 , . . . , fn on X, the image of the mapping X → (R∗+ )n : x → (|f1 (x)|, . . . , |fn (x)|) is a finite union of RZ+ -polytopes of dimension at most dim(X). (This result was recently extended by A. Ducros to arbitrary compact strictly k-analytic spaces.) Moreover, if such X is connected, the quotient group O(X)∗ /(k ∗ O(X)1 ) is finitely generated, where O(X)1 = {f ∈ O(X)∗ | |f (x)| = 1 for all x ∈ X}. Let X be a nondegenerate pluri-stable formal scheme over k  . In §6.3, we prove that, for any strictly analytic subdomain V ⊂ Xη , the intersection V ∩ S(X) is a piecewise RZ+ -linear subspace of S(X) and, for any analytic function f ∈ O (V ), the function x  → |f (x)| on V ∩ S(X) is piecewise |k ∗ |Z+ -linear. In particular, the canonical embedding S(X) → Xη is continuous with respect to the Grothendieck topologies of S(X) and Xη formed by piecewise RZ+ -linear subspaces and strictly analytic subdomains, respectively. In §6.4, we prove that the retraction map τ : Xη → S(X) is continuous with respect to the same Grothendieck topologies on S(X) and Xη . (This result is used in §7 and §8.) We also prove that, given an arbitrary

Smooth p-adic analytic spaces are locally contractible. II

297

morphism ϕ : X → X from a similar formal scheme X over k  , the composition √ √ ϕ τ map S(X ) → Xη → S(X) is piecewise ( |k ∗ |)Q+ -linear, where |k ∗ | = {α ∈ R∗+ | α n ∈ |k ∗ | for some n ≥ 1}. In §7, we prove the property mentioned at the beginning of the introduction. In §8, we prove results which have a direct relation to p-adic integration. Assume that the characteristic of k is zero. The sheaf of constant functions cX on a reduced d strictly k-analytic space X is the étale sheaf of k-vector spaces Ker(OX → 1X ). If k is algebraically closed, it is the constant sheaf kX associated with k, but in general it is much bigger. Assume X is smooth. It is well known that the de Rham complex d

d

d

OX → 1X → 2X → · · · is not exact. On the other hand, the similar de Rham complex for the sheaf of naive analytic functions (i.e., the functions analytic in an open neighborhood of each point from the dense subset X0 = {x ∈ X | [H(x) : k] < ∞}) is exact, but the kernel of the first differential is too large. One of the purposes of a p-adic integration theory is to find an intermediate class of functions between the analytic and naive analytic ones such that the corresponding de Rham complex is an exact resolution of the sheaf of constant functions cX . It is what was essentially done by R. Coleman in [Col] and [CoSh] for smooth k-analytic curves. In our generalization of his work, the following two facts are of crucial importance. The first one (Theorem 8.2.1) tells that each point of X has a fundamental system of open neighborhoods V such that H n (V , cX ) = 0 for all n ≥ 1. The second one (Corollary 8.3.3) tells that, given a nondegenerate strictly pluri-stable formal scheme X over k  , an irreducible component Y ⊂ Xs , and a Zariski closed subset Z ⊂ Xη , then for X = π −1 (Y)\Z one has H n (X, cX ) = 0 for all n ≥ 1. To give some idea on how these two facts are used (in our work in progress), notice that, if the above integration theory exists and X is a smooth k-analytic space with H 1 (X, cX ) = 0, then every closed analytic one-form on X has a primitive (of course, in a bigger class of functions) which is defined uniquely up to an element of c(X). The second of the above facts provides a class of spaces (of the form X = π −1 (Y)) where one constructs such a primitive. The construction depends on X and Y (and not only on X), and the first fact is used to show that the primitive constructed actually depends only on X. In another work in progress, we generalize many of the results of this paper to the whole class of pluri-stable formal schemes. In particular, we show that the skeleton S(X) always depends only on X = Xl , but in the general case S(X) is provided with a so called piecewise monomial structure which is more general than the piecewise linear structure considered here (see Remark 1.3.2(ii)). It is for that reason certain constructions in §2, §3 and §5 are considered in a more general setting. I am very grateful to the referee for many corrections, suggestions and remarks that significantly improved the paper.

298

Vladimir G. Berkovich

1 Piecewise RS -linear spaces 1.1 RS -polytopes Recall that a (compact) polytope in a vector space is the convex hull of a finite set of points. This object is a building block of the classical notion of a piecewise linear space. A basic fact is that a compact subset of a vector space is a polytope if and only if it can be defined by a finite number of linear inequalities (see [Zie, Theorem 1.1]). We say that a set is a semiring if it is a commutative monoid by multiplication and addition related by the identity a(b + c) = ab + ac and which contains 1. An example of a semiring is the set of all continuous non-negative real valued functions on a topological space provided with the usual multiplication and the following addition: ˙ = max(f, g). In this section we consider only sub-semirings of the field of real f +g numbers R. Let S be a sub-semiring of R that contains 0, and let R be a nonempty S-submonoid of R∗+ , i.e., it is a nonempty submonoid of R∗+ such that for any r ∈ R and s ∈ S one has r s ∈ R. The simplest example is S = R and R = R∗+ , and the main examples considered in the paper are provided by a non-Archimedean field √ k and are as follows: S = Z+ and R = |k ∗ | ∩ [0, 1] or |k ∗ |, and S = Q+ and R = |k ∗ | = {α ∈ R∗+ | α n ∈ |k ∗ | for some n ≥ 1}. If R = {1} (e.g., if the valuation on k is trivial), everything we are going to consider is trivial, but has a meaning. We denote by  S (resp. S) the subring (resp. subfield) of R generated by S and by   R (resp. R) the S-submodule (resp. S-vector subspace) of R∗+ generated by R, and we denote by R the convex hull of R in R∗+ , which is also an S-submonoid of R∗+ . (Here are all possible values of R: {1}, [1, ∞[, ]0, 1] and R∗+ .) For n ≥ 0, we denote by An (RS ) the S-monoid of functions on (R∗+ )n of the form (t1 , . . . , tn )  → rt1s1 . . . tnsn , where r ∈ R and s1 , . . . , sn ∈ S, and, for a subset V ⊂ (R∗+ )n , we denote by AV (RS ) the set of the restrictions to V of the functions from An (RS ). An RS -polytope in (R∗+ )n is a compact subset of Rn which is defined by a finite system of inequalities of the form f (t) ≤ g(t) with f, g ∈ An (RS ). Of course, any RS -polytope is also an R S -polytope. An easy criterion for the latter is as follows. A point of (R∗+ )n is said to be an R-point if all of its coordinates are contained in R, and a line in (R∗+ )n is said to be S-rational if there exist s1 , . . . , sn ∈ S such that, for some (and therefore every) pair of distinct points x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) of the line, one has xyii = t si with t ∈ R∗+ , 1 ≤ i ≤ n. Notice that, if the above x and y are R-points, then t ∈ R and, in fact,

f (y) f (x)

∈ R for all f ∈ An (RS ).

1.1.1 Lemma. The following properties of a polytope V ⊂ (R∗+ )n are equivalent: (a) V is an R S -polytope; (b) V is defined by a finite system of inequalities of the form f (t) ≤ g(t) with f, g ∈ An (RS );

Smooth p-adic analytic spaces are locally contractible. II

299

(c) all vertices of V are R-points and all edges of V are S-rational. Notice that if dimS (R) = 1 then the second property in (c) follows from the first one. Proof. The equivalence of (a) and (b) is trivial, and the equivalence of (b) and (c) is a simple linear algebra.   1.1.2 Corollary. Let V be an RS -polytope in (R∗+ )n . Then any subset of V , which is defined by a finite system of inequalities of the form t1s1 . . . tnsn ≤ r with s1 , . . . , sn ∈ S and r ∈ R, is an RS -polytope. In particular, all faces of V and the intersection of two   RS -polytopes are RS -polytopes. An (abstract) RS -polytope is a topological space X provided with a set of continu∼ ous functions AX for which there exists a homeomorphism ϕ : X → V , where V is an ∼ RS -polytope in (R∗+ )n , such that ϕ ∗ induces a bijection AV (RS ) → AX . For example, a subset V ⊂ (R∗+ )n provided with the set of functions AV (RS ) is an (abstract) RS polytope if and only if V is an RS -polytope in (R∗+ )n . A morphism of RS -polytopes ψ : X → X is a continuous map that takes functions from AX to functions from AX . In this way we get a category of RS -polytopes. For example, there is an evident anti-equivalence between the category of zero dimensional RS -polytopes and the category of S-monoids R ⊂ R ⊂ R ∩ R, which are generated over S by R and a finite number of elements, and with inclusions as morphisms. In particular, if the S-monoid R is not divisible, the minimal dimension of an affine space which contains a zero dimensional RS -polytope isomorphic to a given one may be sufficiently large. A subset Y of an RS -polytope X is said to be an RS -polytope in X if one of the above maps ϕ takes it to an RS -polytope in V . Such a subset is provided with the evident RS -polytope structure. 1.1.3 Corollary. Let ϕ : X → X be a morphism of RS -polytopes. Then (i) the image ϕ(X ) is an RS -polytope in X; ∼

(ii) ϕ induces an isomorphism X → ϕ(X ) if and only if the map AX → AX is surjective; (iii) for any RS -polytope Y in X, the preimage ϕ −1 (Y ) is an RS -polytope in X , and   the induced map ϕ −1 (Y ) → Y is a morphism of RS -polytopes. A morphism of RS -polytopes ϕ : X → X is said to be an immersion if it satisfies the equivalent properties of Corollary 1.1.3(ii).

300

Vladimir G. Berkovich

1.2 RS -polyhedra An RS -polyhedron in (R∗+ )n is a finite union of RS -polytopes. Let V be an RS polyhedron. A continuous function f : V → R∗+ is said to be piecewise RS -linear if V can be represented as a union of RS -polytopes V = V1 ∪ · · · ∪ Vk such that f |Vi ∈ AVi (RS ) for all 1 ≤ i ≤ k. Let PV (RS ) denote the set of all piecewise RS -linear functions on V . From Corollary 1.1.2 it follows that, given f1 , . . . , fm ∈ PV (RS ), one can find RS -polytopes V1 , . . . , Vk ⊂ V such that V = V1 ∪ · · · ∪ Vk and fi |Vj ∈ AVj (RS ) for all 1 ≤ i ≤ m and 1 ≤ j ≤ k. In particular, PV (RS ) is an S-monoid, and it contains the functions max{f1 , . . . , fn } and min{f1 , . . . , fn }. 1.2.1 Lemma. Let V ⊂ (R∗+ )n and U ⊂ (R∗+ )m be RS -polyhedra. Then the following properties of a continuous map ϕ : V → U are equivalent: (a) there exist RS -polytopes V1 , . . . , Vk ⊂ V and U1 , . . . , Uk ⊂ U such that V = V1 ∪ · · · ∪ Vk and ϕ induces morphisms of RS -polytopes Vi → Ui , 1 ≤ i ≤ k; (b) ϕ ∗ takes functions from PU (RS ) to PV (RS ). Proof. The implication (a)⇒(b) easily follows from Corollary 1.1.3(iii). Assume that ϕ ∗ takes functions from PU (RS ) to PV (RS ), and let f1 , . . . , fm be the preimages of the coordinate functions on (R∗+ )m in PV (RS ). We can find RS -polytopes V1 , . . . , Vk ⊂ V such that V = V1 ∪ · · · ∪ Vk and fi |Vj ∈ AVj (RS ) for all 1 ≤ i ≤ m and 1 ≤ j ≤ k. Then the image Ui of each Vi under ϕ is an RS -polytope in (R∗+ )m , which is contained in U , and the induced maps Vi → Ui are morphisms of RS polytopes.   A continuous map between RS -polyhedra ϕ : V → V is said to be piecewise RS -linear if it possesses the equivalent properties of Lemma 1.2.1. An (abstract) RS -polyhedron is a topological space X provided with a set of con∼ tinuous functions PX for which there exists a homeomorphism ϕ : X → V , where ∼ V is an RS -polyhedron in (R∗+ )n , such that ϕ ∗ induces a bijection PV (RS ) → PX . A morphism of RS -polyhedra ϕ : X → X is a continuous map that takes functions from PX to functions from PX . A subset Y of an RS -polyhedron X is said to be an RS -polyhedron in X if the above map ϕ takes it to an RS -polyhedron in V . This property of Y does not depend on the choice of ϕ, and in this case Y is provided with the evident RS -polyhedron structure. 1.2.2 Lemma. Let ϕ : X → X be a morphism of RS -polyhedra. Then (i) the image ϕ(X ) is an RS -polyhedron in X; (ii) for any RS -polyhedron Y in X, ϕ −1 (Y ) is an RS -polyhedron in X , and the induced map ϕ −1 (Y ) → Y is a morphism of RS -polyhedra.  

Smooth p-adic analytic spaces are locally contractible. II

301

We say that a morphism of RS -polyhedra ϕ : X → X is an immersion if it induces ∼ an isomorphism X → ϕ(X ). 1.2.3 Lemma. The following properties of a morphism of RS -polyhedra ϕ : X → X are equivalent: (a) ϕ is an isomorphism (resp. an immersion); (b) for every RS -polyhedron Y in X, the induced morphism ϕ −1 (Y ) → Y is an isomorphism (resp. an immersion); (c) there exists a finite covering of X by RS -polyhedra {Yi } such that the induced morphisms ϕ −1 (Yi ) → Yi are isomorphisms (resp. immersions).   Notice that, if a morphism of RS -polytopes is an isomorphism (resp. immersion) as a morphism of RS -polyhedra, then it is an isomorphism (resp. immersion) as a morphism of RS -polytopes.

1.3 Piecewise RS -linear spaces Let X be a locally compact space. (All locally compact spaces are assumed to be Hausdorff.) An RS -polyhedron chart on X is a compact subset V ⊂ X provided with an RS -polyhedron structure. Two charts U and V are said to be compatible if U ∩ V is an RS -polyhedron in U as well as in V , and the RS -polyhedron structures on it induced from U and V are the same. An piecewise RS -linear atlas on X is a family τ of compatible RS -polyhedron charts with the property that every point x ∈ X has a neighborhood of the form V1 ∪ · · · ∪ Vn with V1 , . . . , Vn ∈ τ . Given a piecewise RS -linear atlas τ on X, we say that an RS -polyhedron chart on X is compatible with τ if it is compatible with every chart from τ . Two piecewise RS linear atlases on X are said to be compatible if every chart of one atlas is compatible with the other atlas. From Lemma 1.2.3 it follows that, if two RS -polyhedron charts are compatible with a piecewise RS -linear atlas, then they are compatible. It follows that compatibility is an equivalence relation on the set of piecewise RS -linear atlases on X. A piecewise RS -linear space is a locally compact space X provided with an equivalence class of piecewise RS -linear atlases. Notice that each equivalence class has a unique maximal atlas. It consists of all RS -polyhedron charts which are compatible with some (and, therefore, with any) piecewise RS -linear atlas from the equivalence class. The charts from the maximal atlas will be called RS -polyhedra in X. A function f : X → R∗+ is said to be piecewise RS -linear if its restriction to every RS -polyhedron Y in X is contained in PY . The set of such functions on X will be denoted by PX . A morphism of piecewise RS -linear spaces is a continuous map ϕ : X → X with the following property. There exist piecewise RS -linear atlases τ on X and τ on X

302

Vladimir G. Berkovich

that define the piecewise RS -linear structures on X and X and such that for every V ∈ τ there exists V ∈ τ for which ϕ(V ) ⊂ V and the induced map V → V is a morphism of RS -polyhedra. Notice that in this case, for every pair of RS -polyhedra V ⊂ X and V ⊂ X with ϕ(V ) ⊂ V , the induced map V → V is a morphism of RS -polyhedra. It follows that one can compose piecewise RS -linear morphisms, and so we get a category of piecewise RS -linear spaces PLR S . This category admits finite direct products. A subset Y of a piecewise RS -linear space X is said to be a piecewise RS -linear subspace if every point y ∈ Y has a neighborhood in Y of the form V1 ∪· · ·∪Vn , where V1 , . . . , Vn are RS -polyhedra in X. Such a subset Y is locally closed in X, and has a canonical structure of a piecewise RS -linear space. Given a morphism of piecewise RS -linear spaces ϕ : X → X, the preimage of any piecewise RS -linear subspace of X is a piecewise RS -linear subspace of X . If ϕ is proper, then the image ϕ(X ) is a piecewise RS -linear subspace of X. The morphism ϕ is said to be an immersion if it induces an isomorphism between X and a piecewise RS -linear subspace of X. Let X be a piecewise RS -linear space. The family of its piecewise RS -linear subspaces can be considered as a category, and it gives rise to a Grothendieck topology XG generated by the pretopology in which the set of coverings of a piecewise RS linear subspace Y consists of families {Yi }i∈I of piecewise RS -linear subspaces of Y such that every point y ∈ Y has a neighborhood of the form Yi1 ∪ · · · ∪ Yin with y ∈ Yi1 ∩· · ·∩Yin . Since all open subsets of X are piecewise RS -linear subspaces, there is a morphism of sites XG → X. Moreover, every morphism of piecewise RS -linear → X . The correspondence spaces ϕ : X → X gives rise to a morphisms of sites XG G Y  → PY is a sheaf in the Grothendieck topology XG , denoted by PXG . Its restriction to the usual topology of X will be denoted by PX . More generally, for any piecewise RS -linear space X , the correspondence Y  → Hom(Y, X ) is a sheaf of sets on XG . A morphism of piecewise RS -linear spaces ϕ : Y → X is said to be a G-local immersion (G stands for Grothendieck topology) if for every point y ∈ Y there exist RS -polyhedra V1 , . . . , Vn ⊂ Y such that V1 ∪ · · · ∪ Vn is a neighborhood of y in Y and all of the induced morphisms Vi → X are immersions. Notice that a G-local immersion ϕ : Y → X, which induces a homeomorphism of Y with its image in X, is an immersion. If S is a sub-semiring of R that contains S and R is an S -submonoid of R∗+ that R contains R, then there is the evident functor PLR S → PLS . Of course, this functor does not change the underlying topological spaces, but it can change their Grothendieck topology. From Corollary 1.1.2 it follows that the Grothendieck topology is not changed if S ⊂ S and R ⊂ R. Let {Xi }i∈I be a family of piecewise RS -linear spaces, and suppose that, for each pair i, j ∈ I , we are given a piecewise RS -linear subspace Xij ⊂ Xi and an isomor∼ phism νij : Xij → Xj i so that Xii = Xi , νij (Xij ∩Xil ) = Xj i ∩Xj l and νil = νj l νij on Xij ∩ Xil . In this case one can construct a topological space X obtained by gluing    is the disjoint union where X of Xi along Xij . (It is the quotient space X/E, i Xi

Smooth p-adic analytic spaces are locally contractible. II

303

 defined by the system {νij }.) Let µi denote the and E is the equivalence relation on X induced map Xi → X. 1.3.1 Lemma. In each of the following cases, there exists a unique piecewise RS -linear structure on X such that all µi are immersions: (a) all Xij are open in Xi and X is Hausdorff; (b) for any i ∈ I , all Xij are closed in Xi and the number of j ∈ I with Xij  = ∅ is finite. Furthermore, in the case (a), all µi (Xi ) are open in X and, in the case (b), all µi (Xi ) are closed in X. In the situation of the lemma, X is said to be obtained by gluing of Xi along Xij . Proof. In the case (a), the equivalence relation E is open (see [Bou, Ch. I, §9, n 6]) and, therefore, all µi (Xi ) are open in X. In the case (b), the equivalence relation E is closed (see loc. cit., n 7) and, therefore, all µi (Xi ) are closed in X, µi induce ∼ homeomorphisms Xi → µi (Xi ), and X is Hausdorff. Let τ denote the family of all subsets V ⊂ X for which there exists i ∈ I such −1 that V ⊂ µi (Xi ) and µ−1 i (V ) is an RS -polyhedron in Xi (in this case µi (V ) is an RS -polyhedron in Xj for every j with V ⊂ µj (Xj )). The family τ is a piecewise RS -linear atlas on X and, for the piecewise RS -linear space structure on X it defines, all µi are immersions. That the piecewise RS -linear structure on X with the latter property is unique is trivial.   1.3.2 Remarks. (i) The definition of a piecewise linear space given in this subsection is an easy version of the definition of a non-Archimedean analytic space in [Ber2]. Both are examples of a global object defined by gluing local objects (affinoid spaces in the former and polyhedra in the latter) which are closed subsets. The main difference between our definition and that in [Hud] is in the freeing of the requirement that every point has a neighborhood isomorphic to a polyhedron. The latter property (appropriately adjusted) is established in the following subsection and used in §7 (see also Remark 1.4.5). (ii) If R = {1}, then any RS -polyhedron is a point and any piecewise RS -linear space is a discrete topological space with the only one piecewise RS -linear function which takes value 1. (iii) The piecewise monomial spaces introduced in our work in progress and mentioned in the introduction are glued from certain compact subsets of Rn+ which are defined by a finite number of inequalities f (t) ≤ g(t) with f and g of the form rt1s1 . . . tnsn , where si are elements of a sub-semiring S ⊂ R and r are elements of an S-submonoid R ⊂ R+ such that if 0 ∈ R then S ⊂ R+ .

304

Vladimir G. Berkovich

1.4 An embedding property 1.4.1 Proposition. Every point of a piecewise RS -linear space has a compact piecewise RS -linear neighborhood which admits a piecewise R S -linear isomorphism with an R S -polyhedron. The statement is trivial if R = {1}, and so we assume that R  = {1}. Let X be an RS -polyhedron in (R∗+ )n . An RS -polytopal subdivision of X is a finite family τ of RS -polytopes that cover X and are such that (1) if V ∈ τ , then all faces of V are contained in τ , and (2) if U, V ∈ τ , then U ∩ V is a face in U and in V . The subdivision τ is a refinement of a similar subdivision τ if each V ∈ τ is contained in some V ∈ τ . If τ is a family of subsets of a set and U is a subset of the same set, then τ |U denotes the family {V ∈ τ | V ⊂ U }. 1.4.2 Lemma. Let X be an RS -polyhedron in (R∗+ )n , and let σ be a finite family of RS -polyhedra in X. Then there exists an RS -polytopal subdivision τ of X such that for every U ∈ σ the following is true: (a) τ |U is an RS -polytopal subdivision of U ; (b) if V ∈ τ , then U ∩ V is a face in V . Proof. Step 1. There exists τ that satisfies (a). Indeed, replacing each polyhedron U ∈ σ by a finite set of RS -polytopes whose union is U , we may assume that σ consists of RS -polytopes. We may also assume that σ contains a finite set of RS -polytopes whose union is X. For each U ∈ σ , we fix a finite set F (U ) of pairs (f, g) of functions from An (RS ) such that U = {x ∈ (R∗+ )n | f (x) ≤ g(x) for all (f, g) ∈ F (U )}. Let F be the union of F (U ) for all U ∈ σ . Then the required RS -polytopal subdivision τ consists of the polytopes W for which there exist subsets T ⊂ σ and F≤ , F≥ ⊂ F with F≤ ∩ F≥ = ∅ such that W is the set of all points x ∈ U ∈T U satisfying the inequalities f (x) ≤ g(x) for (f, g) ∈ F≤ and f (x) ≥ g(x) for (f, g) ∈ F≥ and the equalities f (x) = g(x) for (f, g) ∈ F \(F≤ ∪ F≥ ). Indeed, let W (T , F≤ , F≥ ) denote the above polytope. Since W (T , F≤ , F≥ ) ∩ W (T , F≤ , F≥ ) = W (T ∪ T , F≤ ∩ F≤ , F≥ ∩ F≥ ), it suffices to check that, if W = W (T , F≤ , F≥ ) is contained in W = W (T , F≤ , F≥ ), then W is a face of W . For this we can replace T by T ∪ T , F≤ by F≤ ∩ F≤ and F≥ by F≥ ∩ F≥ and, therefore, we may assume that T ⊃ T , F≤ ⊂ F≤ and F≥ ⊂ F≥ . Since W (T , F≤ , F≥ ) is evidently a face of W , we may assume that F≤ = F≤ and F≥ = F≥ . It remains, therefore, to consider the case when T = T ∪ {U } for some U ∈ σ . In this case, one has W = W (T , F≤ , F≥ \F (U )), and the latter is evidently a face of W . Step 2. If τ satisfies (a), there exists a refinement of τ that satisfies (b). If U ∈ σ and V ∈ τ , U ∩ V is a union of faces of V . Let M(V , U ) denote the set of the faces of V in V ∩ U which are maximal by inclusion. For each pair of distinct faces W1 , W2 ∈ M(V , U ) of V , we fix a hyperplane L ⊂ (R∗+ )n defined by an equation f (x) = g(x) with f, g ∈ An (RS ) and such that L ∩ W1 = L ∩ W2 = W1 ∩ W2

Smooth p-adic analytic spaces are locally contractible. II

305

and, for every pair of points x1 ∈ W1 \W2 and x2 ∈ W2 \W1 , the interval connecting them intersects L. Let σ be the union of σ , τ and of {L ∩ X} for all quadruples (U, V , W1 , W2 ) as above. By Step 1, there exists an RS -polytopal subdivision τ of X with the property (a) for σ . We claim that τ satisfies the property (b) for σ . Indeed, suppose there exist U ∈ σ and V ∈ τ for which there exist two distinct faces W1 , W2 ∈ M(V , U ), and let V ∈ τ contain V . Then W1 and W2 cannot lie in one face of V in V ∩ U because they are maximal among the faces of V in V ∩ U . Thus, there exist two distinct faces W1 , W2 ∈ M(V , U ) such that W1 ⊂ W1 , W2 ⊂ W2 , W1 ⊂ W1 ∩ W2 and W2  ⊂ W1 ∩ W2 . Let x1 and x2 be points from the interiors of W1 and W2 , respectively, which do not lie in W1 ∩ W2 , and let L be the hyperplane associated with (U, V , W1 , W2 ). Then L contains a point from the interval connecting x1 and x2 . Such a point lies in the interior of a face of V that contains W1 and W2 . Since τ |L∩X is a subdivision of L ∩ X, it follows that W1 , W2 ⊂ L. This contradicts   the equalities L ∩ W1 = L ∩ W2 = W1 ∩ W2 . An RS -polytopal subdivision τ is said to be simplicial if all polytopes from τ are simplices. 1.4.3 Lemma. If dimS (R) = 1, then any RS -polytopal subdivision of an RS -polyhedron X ⊂ (R∗+ )n has an RS -simplicial refinement with the same set of vertices. Proof. The assumption implies that the convex hull of any subset of the set of vertices of an RS -polytope is an RS -polytope and, therefore, the proof of the corresponding classical fact (see [RoSa, Proposition 2.9]) is applicable. (The same reasoning will be used in the proof of Lemma 1.4.4 below.)   Proof of Proposition 1.4.1. First of all, we may assume that S is a field and, therefore, R is a vector space over S. It suffices to show that every point x of a piecewise RS -linear space X, which is a union of two RS -polyhedra X and X , has an RS polyhedron neighborhood. Of course, we may assume that x ∈ X ∩ X . Let R be a fixed one-dimensional S-vector subspace of R. We claim that there exists a compact piecewise RS -linear neighborhood of x, which is isomorphic to a piecewise RS -linear space. (1) By Lemma 1.4.2, there exists an RS -polytopal subdivision τ of X with the properties (a) and (b) for σ = {X ∩ X }. Furthermore, we can find an RS -polytopal subdivision τ of X with the properties (a) and (b) for σ = τ |X ∩X . (2) Let W be the minimal polytope from τ that contains the point x, and let τ be the family of all polytopes from τ ∪ τ that contain W . (Notice that τ is preserved under intersections.) Then V ∈τ V is a neighborhood of x in X. The point x lies in the interior W˚ of W . Let x0 be a fixed R-point in W˚ . We say that a point y from the above union is marked if for some (and therefore any) V ∈ τ with y ∈ V one (y) has ff(x ∈ R for all f ∈ AV (RS ). A polytope in V ∈ τ is said to be special if 0) it contains the point x0 and all its vertices are marked points, and a polyhedron in V is said to be special if it is a finite union of special polytopes. Notice that a line in

306

Vladimir G. Berkovich

V passing through two different marked points is S-rational and, by Lemma 1.1.1, special polytopes are RS -polytopes, and special polyhedra are RS -polyhedra. Notice also that a polytope, which is special as a polyhedron, is special as a polytope.  ⊂ V, (3) We are going to construct for every V ∈ τ a special polyhedron V which is a neighborhood of the point x in V and such that if U ∈ τ and U ⊂ V = V  ∩ U . The construction is made inductively and, at the beginning, for then U polytopes from τ ∩ τ . First of all, since the set of marked points is dense in W˚ , we  ⊂ W , which is a neighborhood of the point x in W and can find a special polytope W  are ˚ is contained in W . Let V be a bigger polytope from τ ∩ τ , and assume that U already constructed for all U ∈ τ ∩ τ with U ⊂ V˙ , where V˙ = V \V˚ is the boundary , where the union is taken over all U ∈ τ ∩ τ of V . Then the polyhedron V1 = ∪U with U ⊂ V˙ , is a neighborhood of the point x in V˙ . We take an arbitrary marked  as the join of y and V1 in V (i.e., the set {λy + µz}, where point y ∈ V˚ and define V  are constructed for z ∈ V1 , λ, µ ≥ 0 and λ + µ = 1). After the special polyhedra V all V ∈ τ ∩ τ , we continue the same construction for polytopes V ∈ τ ∩ τ . Namely, assume first that V ⊂ X ∩ X . Then V is a union of some U ∈ τ , and we define  as the union ∪U , taken over all U ∈ τ ∩ τ with U ⊂ V . Assume now that V is V minimal among those polytopes from τ ∩ τ that contain a point from X\X . Then the intersection V = V ∩ X is a face of V of smaller dimension. It follows that the  is a neighborhood of the point x in the boundary V˙ of V . We special polyhedron V  as the join of y and V  in V . If a take an arbitrary special point y ∈ V˚ and define V polytope V ∈ τ ∩ τ is not minimal among those, that contain a point from X\X , and  are constructed for all U ∈ τ ∩ τ with U ⊂ V˙ , we denote by the special polyhedra U ’s and define V  as the join of some special point V1 the union of the corresponding U y ∈ V˚ and V1 .  is a compact piecewise RS -linear neighborhood of the (4) The union Y = V ∈τ V point x in X. We claim that Y is isomorphic to a piecewise RS -linear space. Indeed, assume that V ∈ τ is an RS -polytope in (R∗+ )n , and let the coordinates of the point x0 be (α1 , . . . , αn ). Then the automorphism ϕ of (R∗+ )n : (y1 , . . . , yn ) → ( αy11 , . . . , αynn ) takes marked points to R -points and, therefore, it takes every special polytope U in V to an RS -polytope ϕ(U ) in ϕ(V ). Moreover, ϕ induces a bijection between Aϕ(U ) (RS ) (y) and the subspace of AU (RS ) consisting of functions of the form y → r ff(x with 0) r ∈ R and f ∈ AV (RS ). It follows that this RS -polytope structure on U does not depend on the embedding of V in a vector space, and it gives rise to RS -polyhedron structures on special polyhedra in V . Moreover, if V , V ∈ τ , then the RS -polyhedron structures on special polyhedra in V ∩ V , induced from V and V , are compatible. In this way we get a piecewise RS -linear structure on Y . The proposition now follows from the following lemma, which is a straightforward generalization of the classical result for S = R and R = R∗+ . 1.4.4 Lemma. If S is a field and dimS (R) = 1, then any compact piecewise RS linear space is isomorphic to an RS -polyhedron.

Smooth p-adic analytic spaces are locally contractible. II

307

Proof. It suffices to show that a compact piecewise RS -linear space X, which is a union of two RS -polyhedra X and X , is isomorphic to an RS -polyhedron. (A) An RS -polytope chart on X is a compact subset V ⊂ X provided with an RS -polytope structure which gives rise to an RS -polyhedron in X. Two RS -polytope charts U and V are said to be compatible if U ∩ V is an RS -polytope in U as well as in V , and the RS -polytope structures on it induced from U and V are the same. We claim that X can be covered by a finite family τ of RS -simplex charts such that (1) if V ∈ τ , then all faces of V are contained in τ , and (2) if U, V ∈ τ , then U ∩ V is a face in U and in V . Indeed, by Lemma 1.4.2, there exists an RS -polytopal subdivision τ of X with the properties (a) and (b) for σ = {X ∩ X }, and we can find an RS -polytopal subdivision τ of X with the properties (a) and (b) for σ = τ |X ∩X . Since dimS (R) = 1, we may apply Lemma 1.4.3 and assume that τ is simplicial. Let V1 , . . . , Vm be all of the polytopes from τ , which are not contained in X ∩ X and such that if Vi is a face of Vj then i ≤ j . We set Y1 = X and Yi+1 = Yi ∪ Vi , and provide as follows each Yi with a family of RS -simplex charts τi possessing the properties (1) and (2) and such that τ1 = τ and τi+1 |Yi = τi for all 1 ≤ i ≤ m. For this we fix an ordering of the set of the vertices in τ outside X ∩ X , and assume that, for some 1 ≤ i ≤ m, τi is already constructed. If x is the first vertex of Vi outside X ∩ X , we define τi+1 as consisting of all simplices from τi and the joins of x and U ∈ τi with U ⊂ V˙i . (The latter are RS -simplices since dimS (R) = 1.) The family τ = τm+1 on Ym+1 = X is the required one. (B) Let {x1 , . . . , xn+1 } be the set of all vertices in τ , and let {y1 , . . . , yn+1 } be a set of independent R-points in (R∗+ )n . For a simplex V ∈ τ , let ϕ(V ) be the RS simplex in (R∗+ )n , which is the convex hull of those points from {y1 , . . . , yn+1 } which corresponds to the vertices of V . Then the correspondence xi  → yi gives rise to an isomorphism between X and the RS -polyhedron which is the union of all ϕ(V ) with V ∈ τ.   1.4.5 Remarks. (i) It is not true in general that every point of a piecewise RS linear space has a compact piecewise RS -linear neighborhood isomorphic to an RS polyhedron. For example, assume that S = Z+ and R is an arbitrary submonoid of R∗+ that contains a number 0 < r < 1, and let W be the triangle in (R∗+ )2 defined by the inequalities t1 ≤ 1 and r ≤ t2 ≤ t1 . If U1 and U2 are the edges of W defined ∼ by the equalities t2 = r and t1 = t2 , respectively, there is an isomorphism U1 → U2 that takes a point (t1 , r) to the point (t1 , t1 ), and it defines an involutive automorphism ϕ of V = U1 ∪ U2 . Let X be the piecewise RZ+ -linear space obtained by gluing of two copies of W along the isomorphism ϕ of V (see Lemma 1.3.1). Then the point x = (r, r) has no an RZ+ -polyhedron neighborhood in X. Indeed, let f be a piecewise RZ+ -linear function in a neighborhood of x in X. The preimage of the neighborhood in W contains a triangle W defined in W by the inequality t1 ≤ r for some r ∈ R with r < r < 1, and one has f (y) = f (ϕ(y)) for all y ∈ W ∩ U1 . But the restriction of f to a vertical interval in W (defined by the equality t1 = α for r < α ≤ r ) is nondecreasing as a function on t2 . It follows that the restriction of

308

Vladimir G. Berkovich

f to each vertical interval is constant. Since piecewise RZ+ -linear functions separate points of an RZ+ -polyhedron, the point x has no an RZ+ -polyhedron neighborhood. (ii) Although Proposition 1.4.1 is enough for an application in §7, it would be  (instead of S and R), and if interesting to know if its statement is true for  S and R Lemma 1.4.4 is true without the assumption dimS (R) = 1.

2 R-colored polysimplicial sets 2.1 Categories with a geometric realization functor Given a topological space X, the set of all non-negative real valued functions on X forms a semiring with respect to the usual multiplication and the following addition: ˙ = max(f, g). We denote by Topsr the category of the pairs (X, M) consisting f +g of a topological space X and a semiring M of continuous functions on X with values in [0, 1] such that 1 ∈ M. The set of morphisms Hom((X , M ), (X, M)) consists of the continuous maps X → X that take functions from M to M . The category Topsr admits direct limits. Let R be a small category provided with a functor R → Topsr : A  → (|A|, MA ) (which will be called a geometric realization functor). In this section we introduce certain categories which are related to R and also provided with a geometric realization functor. The first example is the category R E ns of contravariant functors from R to the category of sets E ns. The category R can be considered as its full subcategory under the fully faithful functor R → R E ns : A → RA that takes an object to the contravariant functor represented by it. The geometric realization functor R E ns → Top sr : C → (|C|, MC ) is the one that extends R → Topsr to the functor which commutes with direct limits. For an object A ∈ Ob(R) and an element c ∈ CA , where CA is the value of C at A, we denote by σc the corresponding map |A| → |C|.

2.2 The category R Recall the definition of the category  from [Ber7, §3]. First of all, for a tuple n = (n0 , . . . ,np ) with either p = n0 = 0 or p ≥ 0 and ni ≥ 1 for all 0 ≤ i ≤ p, let [n] denote the set [n0 ] × · · · × [np ], where [n] = {0, 1, . . . , n}. The set [n] ∈ Ob() is endowed with a metric as follows. The distance between two elements i and j of [n] is the number of distinct coordinates of i and j . Objects of the category  are the sets [n] for the tuples n as above, and morphisms are isometric maps. By [Ber7, Lemma 3.1], each isometric map γ : [n ] → [n] can be described as follows. First of all, we set ω(n) = [p], if [n]  = [0], and ω(n) = ∅, otherwise. Then there is a pair (f, α) consisting of an injective map f : ω(n ) → [p] and α = {αi }0≤i≤p , where αi is an injective map [n f −1 (i) ] → [ni ] for i ∈ Im(f ), and is a map [0] → [ni ]

Smooth p-adic analytic spaces are locally contractible. II

309

for i  ∈ Im(f ). The map γ takes an element i = (i0 , . . . ,ip ) ∈ [n ] to the element i = (i0 , . . . ,ip ) ∈ [n] with ij = αj (if −1 (j ) ) for j ∈ Im(f ), and ij = αj (0) for j  ∈ Im(f ). It follows that, for every subset J ⊂ ω(n), the morphism γ : [n ] → [n] gives rise to a morphism [n f −1 (J ) ] → [nJ ], where nJ denotes the tuple (nj0 , . . . , njt ) if J = {j0 , . . . , jt } is non-empty and j0 < · · · < jt , and the zero tuple 0, otherwise. Assume we are given a category R and a functor R → Topsr : A  → (|A|, MA ) (as in §2.1). We introduce as follows a category R , whose objects are denoted by [n]A,r , n , M n ). First of all, the objects [n] and a functor R → Top sr : [n]A,r  → (A,r A,r A,r correspond to the following data: [n] = [n0 ] × · · · × [np ] ∈ Ob(), A ∈ Ob(R) p+1 and r = (r0 , . . . , rp ) ∈ MA , which satisfy the condition that r0 = 1, if [n] = [0], and ri = 1 for all 0 ≤ i ≤ p, if [n]  = [0]. Given an object [n]A,r and a morphism ψ : A → A, let J (ψ, r) denote the set of all j ∈ ω(n) with rj (x) < 1 for some x ∈ Im(|ψ|), where |ψ| is the map |A | → |A|. A morphism [n ]A ,r → [n]A,r is a pair consisting of a morphism ψ : A → A in R and a morphism γ : [n ] → [nJ ] in , where J = J (ψ, r), which satisfy the following condition: if γ is associated with a pair (f, α) as above, then rj = |ψ|∗ (rf (j ) ) for all j ∈ ω(n ). Furthermore, we set n = {(x, t) ∈ |A| × [0, 1][n] | ti0 . . . tini = ri (x), 0 ≤ i ≤ p} A,r n the semiring of continuous functions on  n generated by all and denote by MA,r A,r functions from MA and the coordinate functions t → tij . Given a morphism (γ , ψ) : n n [n ]A ,r → [n]A,r as above, the corresponding map A ,r → A,r takes a point (x , t ) to the point (x, t), where x = |ψ|(x ) and (a) if i  ∈ J (ψ, r), then tij = 1 for all 0 ≤ j ≤ ni , (b) if i ∈ J (ψ, r)\Im(f ), then tij = ri (x) for j = αi (0) and tij = 1 for j  = αi (0), and (c) if i ∈ Im(f ), then tij = t −1 for j ∈ Im(αi ) and tij = 1 −1 f

(i),αi (j )

for j  ∈ Im(αi ). In this way we get a geometric realization functor R → Topsr .

2.3 Connections between the categories R and R First of all, there is a fully faithful functor R → R : A  → [0]A,1 and a functor R → R : [n]A,r  → A. The latter makes R a fibered category over the category R in the sense of [SGA1, Exp. VI] and can be seen using the following general construction. Let R be another small category provided with a functor R → Top sr : A  → (|A |, MA ), and assume we are given a functor R → R : A  → A and a morphism of hA

functors from R to Top sr : (|A |, MA ) → (|A|, MA ). Then one can define a functor R ×R R → R : ([n]A,r , A )  → [n ]A ,r , where n = nJ , r = h∗A (r J ) and J = {j ∈ ω(n) | rj (x) < 1 for some x ∈ Im(hA )}. (The truncation r J has the same meaning as nJ .) Notice that there is an isomorphism ∼ n n of functors from R ×R R to Top sr : A ,r → A,r ×|A| |A |.

310

Vladimir G. Berkovich

2.3.1 Examples. (i) Given an object A ∈ Ob(R), let {A} denote the category consisting of one object A (with only identity morphism) provided with the following functor to Topsr : A  → (|A|, MA ). The above construction, applied to the canonical ∼ functor {A} → R, gives an equivalence of categories R ×R {A} → {A} . (ii) Given a morphism ψ : A → A in R, the above construction, applied to the |ψ|

functor {A } → R : A  → A and the morphism (|A |, MA ) → (|A|, MA ), gives the inverse image functor ψ ∗ : R ×R {A} → R ×R {A } that makes R a fibered category over R. (iii) Given an object A ∈ Ob(R) and a point x ∈ |A|, let {x} denote the category consisting of one object x (with only identity morphism) provided with the functor to Top sr : x  → (x, Mx ), where Mx = {f (x) | f ∈ MA }. The above construction, applies to the functor {x} → R : x → A and the canonical morphism (x, Mx ) → (|A|, MA ), gives a functor R ×R {A} → {x} . Recall that one can associate with each small category L a partially ordered set O(L) (see [GaZi, Ch. II, §5.1]). Namely, it is the partially ordered set associated with the set Ob(L) provided with the following partial preorder structure: C ≤ D if there is a morphism C → D. As a set, O(L) is the set of equivalence classes in Ob(L) with respect to the following equivalence relation: C ∼ D if there are morphisms C → D and D → C. The partially ordered set O(L) can be considered as a category so that the map Ob(L) → O(L) is the underlying map of the evident functor L → O(L). A functor O(L) → L, whose composition with the latter is the identity functor on O(L), will be said to be a section of L → O(L). The following simple lemma describes the partially ordered set O([n]A,r ), associated with the category R /[n]A,r , in terms of the partially ordered set O(A), associated with the category R/A. First of all, we notice that, given [n]A,r and two ϕ

ψ

morphisms A → A → A, one has J (ψ  ϕ, r) ⊂ J (ψ, r) and, in particular, the subset J (ψ, r) depends only on the equivalence class of ψ in Ob(R/A). We also say that a non-empty subset C ⊂ [n] = [n0 ] × · · · × [np ] is of the direct product type if C = C0 × · · · × Cp , where Ci is the image of C under the canonical projection [n] → [ni ]. 2.3.2 Lemma. (i) There is a one-to-one correspondence between O([n]A,r ) and the set of pairs (ψ, C) consisting of an element ψ ∈ O(A) and a subset C ⊂ [nJ ] of the direct product type, where J = J (ψ, r); (ii) (ψ , C ) ≤ (ψ , C ) if and only if ψ ≤ ψ and C is contained in the image of C under the canonical projection [nJ ] → [nJ ], where J = J (ψ , r) and J = J (ψ , r); (iii) any section O(A) → R/A of the functor R/A → O(A) can be lifted to a section O([n]A,r ) → R /[n]A,r of the functor R /[n]A,r → O([n]A,r ).

Smooth p-adic analytic spaces are locally contractible. II

311

Proof. Given a morphism ψ : A → A in R, let J = J (ψ, r) = {0 ≤ j0 < · · · < jq ≤ p}. For a non-empty subset C = C0 × · · · × Cq ⊂ [nJ ], let nC be the tuple consisting of the numbers #Ci − 1 that are greater than zero, and let r C be the corresponding subtuple of r. Then the canonical injective maps Ci → [nji ] define a morphism µψ,C : [nC ]A ,r C → [n]A,r in R . It is easy to see that, when ψ runs through a system of representatives of O(A) in Ob(R/A), the morphisms µψ,C run through a system of representatives of O([n]A,r ) in Ob(/[n]A,r ), i.e., (i) is true. The statements (ii) and (iii) also easily follow from the construction.   Assume that R has a structure of a symmetric strict monoidal category, i.e., there is a multiplication bifunctor R × R → R : (A , A )  → A A which satisfies certain conditions (see [Mac, Ch. VII]). Assume also that the canonical morphisms of partially ordered sets O(A ) × O(A ) → O(A A ) are isomorphisms, and that there ∼ is an isomorphism of functors from R × R to Top sr : (|A |, MA ) × (|A |, MA ) → (|A A |, MA A ). Then this structure is naturally extended to the category R and the same properties also hold. Namely, the multiplication bifunctor R × R → R : ([n ]A ,r , [n ]A ,r ) → [n]A,r = [n ]A ,r [n ]A ,r is defined as follows: A = A A and (a) n = n and r = r , if [n ] = [0], (b) n = n and r = r , if [n ] = [0], and (c) n = (n 0 , . . . , n p , n 0 , . . . , n p ) and r = (r0 , . . . , rp , r0 , . . . , rp ), otherwise. The first property follows from Lemma 2.3.2, and the second one follows n , M n )). from the definition (of (A,r A,r

2.4 R-colored polysimplicial sets The category of R-colored polysimplicial sets is the category R E ns. By §2.1, there is a geometric realization functor R E ns → Top sr : D  → (|D|, MD ) which n , M n ). commutes with direct limits and extends the functor [n]A,r  → (A,r A,r The functor representable by an object [n]A,r ∈ Ob(R ) is denoted by [n]A,r r . One and, for D ∈ Ob(R E ns), the image of [n]A,r under D is denoted by DA,n ∼

r evidently has Hom([n]  r A,r , D) → DA,n and, therefore, there is a canonical bijection between the set DA,n of polysimplices of D and the set of objects of the category R /D. In particular, there is an equivalence relation on the set of polysimplices of D, and the set of equivalence classes is provided with a partial ordering. It is denoted by O(D). Notice that O([n]A,r ) coincides with the partially ordered set O([n]A,r ) considered in §2.3. The correspondence D  → O(D) is a functor from R E ns to the category of partially ordered sets Or, and this functor commutes with direct limits (cf. [Ber7, 3.3]). There is a fully faithful functor R E ns → R E ns : [n]C,r  → [n]C,r which commutes with direct limits and extends the functor R → R E ns : [n]A,r  → [n]A,r . Namely, [n]C,r is the polysimplicial set D with the property that, for s [m]A,s ∈ Ob(R ), DA,m is the set of pairs consisting of an element c ∈ CA and a

312

Vladimir G. Berkovich

morphism γ = (f, α) : [m] → [nI ] in  with I = I (c, r) = {i ∈ ω(n) | ri (x) < 1 for some x ∈ Im(σc )} such that sj = σc∗ (rf (j ) ) for all j ∈ ω(m). 2.4.1 Lemma. There is a canonical isomorphism of functors from R E ns to Top sr : ∼

n n (|[n]C,r |, M[n]C,r ) → (C,r , MC,r ).

Proof. If n = (n0 , . . . , np ) and r = (r0 , . . . , rp ), then n = {(x, t) ∈ |C| × [0, 1][n] | ti0 . . . tini = ri (x), 0 ≤ i ≤ p} C,r n is the semiring generated by M and the coordinate functions t . On the and MC,r C ij ∼

∼

other hand, there are canonical isomorphisms lim RA → C and lim [nI ]A,r I → D, −→ −→ where both limits are taken over the category R/C (whose objects are morphisms c RA → C) and I = I (c, r). The required isomorphism is defined by the canonical nI n that take a point (x, t ) to the point (σ (x), t) with t = t and → C,r maps A,r c ij ij I   tij = 1 for all 0 ≤ j ≤ ni , if i ∈ I and i  ∈ I , respectively. The canonical functor R → R E ns : [n]A,r  → RA can be extended to a functor R E ns → R E ns : D  → D which commutes with direct limits. (It is left adjoint to the functor R E ns → R E ns induced by the functor [n]A,r  → A.) One can describe D as follows. Given A denote the set of the polysimplices of D over A, i.e., the union A ∈ Ob(R), let D r taken over all [n]A,r ∈ Ob(R ). Since R is a fibered category over R, ∪DA,n A is an object of R E ns. We provide the set D A with the correspondence A → D A the minimal equivalence relation with respect to which any two elements d, d ∈ D r , d ∈ Dr with the following property are equivalent: d ∈ DA,n and there exists a A,n morphism γ : [n ]A,r → [n]A,r over the identity morphism of A with d = D(γ )(d). A with respect to the above equivalence relation (i.e., D A Then D A is the quotient of D A ). The following properties of the functor is the set of connected components of D D → D easily follow from the construction. 2.4.2 Lemma. ∼

(i) For every C ∈ Ob(R E ns), there is a canonical isomorphism [n]C,r → C; (ii) the functor D  → D makes R E ns a fibered category over R E ns, namely, given a morphism ψ : C → D in R E ns, the inverse image ψ ∗ D is as follows: r ×D A CA ; (ψ ∗ D)rA,n = DA,n (iii) the structure of a fibered category R E ns over R E ns extends that on R over R.  

Smooth p-adic analytic spaces are locally contractible. II

313

n → |A| give rise to functorial Notice that the canonical surjective projections A,r surjective projections |D| → |D|.

3 R-colored polysimplicial sets of length l 3.1 The category R,l Let R be a nontrivial submonoid of [0, 1] that contains 1. (In §3.5, it will be assumed that 0 ∈ R.) We can consider R as a semiring of continuous functions on a one point space. If R is a one point category and  is the functor that associates with the only object of R the above space, then R is the category R introduced in [Ber7, §4]. We iterate this construction by setting R,1 = R and R,l = R,l−1 for l ≥ 2. We also denote by R,l the corresponding functor R,l → Topsr . We represent objects of the category R,l as pairs [n]r of the following form, and n n we denote the image of [n]r under the functor R,l by (r , Mr ). First of all, n is a (i) (i) tuple (n(1) , . . . , n(l) ) with [n(i) ] = [n0 ] × · · · × [npi ] ∈ Ob(). Furthermore, r is a (i) (i) (1) (1) tuple (r (1) , . . . , r (l) ) with r (i) = (r0 , . . . , rpi ) of the following type: r0 , . . . , rp1 ∈ (i)

(i)

n≤i−1

R and, for i ≥ 2, r0 , . . . , rpi ∈ Mr ≤i−1 , where n≤i = (n(1) , . . . , n(i) ) and r ≤i = (i)

(r (1) , . . . , r (i) ) for 1 ≤ i ≤ l. Finally, the tuples r (i) satisfy the condition that r0 = 1, (i) if [n(i) ] = [0], and rj  = 1 for all 0 ≤ j ≤ pi , otherwise. The object with [n(i) ] = [0] for all 1 ≤ i ≤ l will be denoted by [0]1,l . One has n

r = {t = (t (1) , . . . , t (l) ) ∈ [0, 1][n

(1) ]

×· · ·×[0, 1][n

(l) ]

(i)

(i) (i) j nj

| tj 0 . . . t

(i)

= rj (t ≤i−1 )},

n

where t ≤i−1 = (t (1) , . . . , t (i−1) ), and Mr is the semiring of continuous functions (i) generated by R and the coordinate functions t  → tj k . Notice that for any morphism n

n

[n ]r → [n]r the corresponding map r → r is injective. By Lemma 2.3.2(iii), the canonical functor R,l /[n]r → O([n]r ) has a section O([n]r ) → R,l /[n]r .

pi (i) ˚ rn denote nj . Furthermore, let  We set |n| = li=1 |n(i) |, where |n(i) | = j =0 n the open subset of r that consists of the points as above with the additional conditions (i) (i) tj 0 < 1, . . . , t (i) < 1 for all 1 ≤ i ≤ l and 0 ≤ j ≤ pi with [n(i) ]  = [0]. It is called j nj n of r .

˚ rn . The proof of ˙ rn of rn is the complement of  the interior The boundary  the following lemma is trivial. n ˚ rn . Furthermore,  ˚ rn 3.1.1 Lemma. If r ∈ Mr and r = 1, then r(x) < 1 for all x ∈  n is dense in r , and it coincides with the set of points that have an open neighborhood   homeomorphic to an open ball (of dimension |n|).

314

Vladimir G. Berkovich

n ˚ n with respect to the injective A subset of r , which is the image of the interior  r n

n

n

map r → r that corresponds to a morphism [n ]r → [n]r , is called a cell of r . n

The closure of a cell will be called a cell closure. (It coincides with the image of r under the above map.) Notice that a cell depends only on the equivalence class of the n morphism [n ]r → [n]r in the partially ordered set O([n]r ). Let O(r ) denote the n set of cells of r provided with the following partial ordering: A ≤ B if A ⊂ B. 3.1.2 Lemma. (i) A cell closure is a disjoint union of cells; n

(ii) two distinct cells are disjoint (and, therefore, O(r ) can be also viewed as the set of all cell closures partially ordered by inclusion); ∼

n

(iii) there is an isomorphism of partially ordered sets O([n]r ) → O(r ). Proof. Assume that the statements are true for l − 1. By Lemma 3.1.1(ii), to prove (i), n it suffices to verify that r is a disjoint union of cells. First of all, if [n(l) ] = [0], then n ∼

n≤l−1

r → r ≤l−1 , and the required fact for [n]r easily follows from that for [n≤l−1 ]r ≤l−1 . n

Assume therefore that [n(l) ]  = [0], and let t ∈ r . To show that the point t is ≤l−1 ˚ n≤l−1 . contained in a cell, we may assume, by the induction hypothesis, that t ≤l−1 ∈  (l)

r

(l)

For 0 ≤ i ≤ pl , let Ci denote the subset of all j ∈ [ni ] with tij < 1. (The subset Ci is (l)

non-empty since ri (t ≤l−1 ) < 1.) Furthermore, let J be the subset of all j ∈ ω(n(l) ) with #Cj > 1, and let m be the tuple of the numbers #Cj − 1 for j ∈ J , if J  = ∅, and m = (0), if J = ∅. Then the sets Cj define a morphism [m] → [n(l) ] in . Let s be (l) the tuple of the functions rj for j ∈ J , if J  = ∅, and s = (1), if J = ∅. Then there is a well defined morphism [n ]r → [n]r in R,l , where [n ≤l−1 ]r ≤l−1 = [n≤l−1 ]r ≤l−1 ,

n (l) = m and r (l) = s, and the point t is contained in the cell that corresponds to this ≤l−1 n ˚ n≤l−1 , and all morphism. Notice that in this way we described all cells of r over  r

of them are pair-wise disjoint, i.e., (i) and (ii) are true. The statement (iii) now easily follows from the induction hypothesis and Lemma 2.3.2.   Notice that the symmetric strict monoidal category structure on the category R in the sense of [Mac, Ch. VII], defined in [Ber7, §3], extends naturally to the category R,l . Namely, the multiplication bifunctor R,l × R,l → R,l : ([n ]r , [n ]r )  → [n]r = [n ]r [n ]r is defined as follows: (a) [n(i) ] = [n (i) ] and r (i) = r (i) , if [n (i) ] = [0], (b) [n(i) ] = [n (i) ] and r (i) = r (i) , if [n (i) ] = [0], and (c) (i) (i) (i) (i) (i) (i) (i) (i) n(i) = (n 0 , . . . , n p , n 0 , . . . , n p ) and r (i) = (r 0 , . . . , r p , r 0 , . . . , r p ), i i i i otherwise. Notice also that there is a canonical isomorphism of partially ordered ∼ n n sets O([n ]r ) × O([n ]r ) → O([n]r ) and of objects of Top sr : (r , Mr ) × n

n

∼

n

n

(r , Mr ) → (r , Mr ).

Smooth p-adic analytic spaces are locally contractible. II

315

3.2 R-colored polysimplicial sets of length l The category of R-colored polysimplicial sets of length l is the category R,l E ns of contravariant functors from R,l to the category of sets E ns. If R is a bigger submonoid of [0, 1], there are fully faithful functors R,l → R ,l and R,l E ns → R ,l E ns. The standard r-colored n-polysimplex [n]r is the functor representable r by [n]r . If D ∈ Ob(R,l E ns), the image of [n]r under D is denoted by Dn (the set ∼

r

of r-colored n-polysimplices of D). One evidently has Hom([n]r , D) → Dn and,  r therefore, there is a canonical bijection between the set Dn of all polysimplices of D and the set of objects of the category R,l /D. In particular, there is an equivalence relation on the set of polysimplices of D, and the set of equivalence classes is provided with a partial ordering. It is denoted by O(D). Notice that O([n]r ) coincides with the partially ordered set O([n]r ). The correspondence D → O(D) is a functor from R,l E ns to the category of partially ordered sets Or, and this functor commutes with direct limits. A polysimplicial set is said to be finite if it has a finite number of polysimplices. It is said to be locally finite if each polysimplex is contained in a finite number of other polysimplices (i.e., the corresponding element of O(D) is smaller than at most a finite number of other elements of O(D)). r The dimension of a polysimplex d ∈ Dn is |n|. Notice that it is equal to the n topological dimension of r . Let m ≥ 0. The m-skeleton Skm (D) of a polysimplicial set D is the polysimplicial subset of D which is formed by the polysimplices of dimension at most m. We also set Sk−1 (C) = ∅. For example, [n]r = Sk m ([n]r ), ˙ r = Sk m−1 ([n]r ) (the boundary of [n]r ). For where m = |n|, and we set [n] r d ∈ Dn , let Gd denote the stabilizer of d in the automorphism group Aut([n]r ). 3.2.1 Lemma. Let P m be a set of representatives of the equivalence classes of polysimplices of D of dimension m. Then the following diagram is cocartesian:  / Sk m−1 (D) ˙ d∈P m Gd \[nd ]r d



d∈P m

 Gd \[nd ]r d

 / Sk m (D).

Proof. Let E be the cocartesian product, and let N and S denote the polysimplicial sets at the north-west and the south-west of the diagram, respectively. Given [n]r , if r ∼ r r ∼ r |n| < m, one evidently has Nn → Sn and Sk m−1 (D)n → Sk m (D)n and, therefore, r ∼

r

r

r

En → Sk m (D)n . On the other hand, if |n| = m, then Nn = Sk m−1 (D)n = ∅ and

r ∼ Sn →

r Sk m (D)n

∼

and, therefore, E → Sk m (D).

 

316

Vladimir G. Berkovich

The canonical functor R,l → R,l−1 E ns : [n]r  → [n≤l−1 ]r ≤l−1 can be extended to a functor R,l E ns → R,l−1 E ns : D  → D ≤l−1 which commutes with direct limits. (It is the functor D  → D from §2.4.) By Lemma 2.4.2, the latter functor makes R,l E ns a fibered category over R,l−1 E ns which is compatible with the fibered category structure of R,l over R,l−1 . The symmetric strict monoidal structure on the category R,l is naturally extended to the category R,l E ns, i.e., there is a bifunctor R,l E ns × R,l E ns → R,l E ns : (D , D )  → D D that commutes with direct limits and extends the functor ([n ]r , [n ]r ) → [n ]r [n ]r . One easily sees that the canonical morphism D D → D × D is injective and that there is an isomorphism of partially ordered ∼ sets O(D ) × O(D ) → O(D D ). n n The functor R,l → Top sr : [n]r  → (r , Mr ) can be extended to a geometric realization functor R,l E ns → Topsr : D → (|D|, MD ) which commutes with direct limits. Notice that there are functorial projections (|D|, MD ) → (|D ≤l−1 |, MD ≤l−1 ), which are surjective on the underlying topological spaces, and that there are functorial bijective continuous maps |D D | → |D | × |D |.

3.3 Elementary functions Given a semiring M of continuous non-negative real valued functions on a topological space X, we say that a nonzero function f ∈ M is elementary if it possesses the ˙ for some nonzero g, h ∈ M, then following property: if f = max(g, h) (= g +h) either f = g or f = h. The subset of elementary functions in M will be denoted by e(M). 3.3.1 Proposition. n

n

(i) Given f, g ∈ e(Mr ), if f |U = g|U for a non-empty open subset U ⊂ r , then f = g; n

n

(ii) given a nonzero f ∈ Mr , there exists a unique finite subset {fi }i∈I ⊂ e(Mr ) such that f = maxi∈I {fi }, but f = maxi∈J {fi } for strictly smaller subsets J ⊂ I. 3.3.2 Lemma. n n (r , Mr )

For every [n]r ∈ Ob(R,l ) different from [0]1,l , the object n

n

of is isomorphic to an object (r , Mr ) with the tuple n of the form ((1), . . . , (1)) (of length |n|). Topsr

Proof. We may assume that [n(i) ] = [0] for all 1 ≤ i ≤ l, and we notice that n n ∼ n n there is an evident isomorphism (r , Mr ) → (r , Mr ), where n and r are the

Smooth p-adic analytic spaces are locally contractible. II (1)

(1)

(2)

(l)

(1)

(1)

317 (2)

(l)

tuples ((n0 ), . . . , (np1 ), (n0 ), . . . , (npl )) and ((r0 ), . . . , (rp1 ), (r0 ), . . . , (rpl ))

of length li=1 (pi + 1). Thus, we may assume that all pi ’s are zero, i.e., n = ((n(1) ), . . . , (n(l) )) and r = ((r (1) ), . . . , (r (l) )). We now notice that the equation t0 . . . tn = r is equivalent to the system of two equations t0 . . . tn−2 ·tn−1 = r and n

n

∼

. Thus, if n(i) > 1 for some 1 ≤ i ≤ l, then (r , Mr ) → tn−1 · tn = tn−1 n

n

(r , Mr ), where n and r are the tuples (. . . , (n(i−1) ), (n(i) − 1), (1), (n(i+1) ), . . . )

and (. . . , (r (i−1) ), (r (i) ), (tn (i) −1 ), (r (i+1) ), . . . ). Repeating this procedure, we construct the required isomorphism.  

Proof of Proposition 3.3.1. Lemmas 3.1.1 and 3.3.2 reduce the proposition to the verification of the following fact. Assume we are given an object (X, M) of Top sr , which possesses the properties (i) and (ii). Given a function r ∈ M such that the open set V = {x ∈ X | r(x) < 1} is dense in X, we set X = {(x, t0 , t1 ) ∈ X × [0, 1]2 | t0 · t1 = r(x)}. Let M denote the monoid of continuous functions on X generated by M and the coordinate functions t0 and t1 , and let M denote the semiring of continuous functions generated by M . (Notice that e(M ) ⊂ M .) Then (1) every nonzero function from M has a unique representation in either the form f t0m or the form f t1n with f ∈ M\{0}, m ≥ 0 and n ≥ 1; (2) the elementary functions among them are precisely those with f ∈ e(M);

(3) the semiring M possesses the properties (i) and (ii). Notice that the statements (1)–(3) hold when X is a one point space and that the canonical projection π : X → X is an open map. That any nonzero function F ∈ M is of the form considered is trivial. The form of the restriction of F to the fiber π −1 (x) of a point x ∈ V is unique and, since V is dense in X, we see that the form of F is unique, i.e., (1) is true. If F is of the form from (1), let us call the function f ∈ M the base of F . Assume the restrictions of two nonzero functions F, G ∈ M to a non-empty open subset U ⊂ X coincide. Then for every point x from the non-empty open set U = π(U ) ∩ V the restrictions of F and G to π −1 (x) coincide. It follows that F and G have similar forms and for their bases f and g one has f |U = g|U . It follows that f = g, i.e., (i) is true for the nonzero functions from M with an elementary base. Let E denote the latter class of functions. It is clear that any nonzero function from M is the maximum of a finite set of functions from E and, in particular, e(M ) ⊂ E. Assume that for F ∈ E one has F = max{F1 , . . . , Fn } with F1 , . . . , Fn ∈ E and that the family F1 , . . . , Fn is minimal. Then there exists a non-empty set U ⊂ X such that F1 (x ) > Fi (x ) for all x ∈ U and 2 ≤ i ≤ n. It follows that F |U = F1 |U , and the validity of the property (i) for functions from E implies that F = F1 , i.e., E = e(M ) and (2) is true.

318

Vladimir G. Berkovich

Assume now that max{F1 , . . . , Fm } = max{G1 , . . . , Gn } for Fi , Gj ∈ E and that the families of functions on both sides are minimal. Given 1 ≤ i ≤ m, there exists a non-empty open subset U ⊂ M such that Fi (x ) > Fk (x ) for all x ∈ U and k  = i. Furthermore, we can find 1 ≤ j ≤ n and a non-empty open subset U ⊂ U such that Gj (x ) > Gl (x ) for all x ∈ U and l  = j . It follows that Fi |U = Gj |U and therefore Fi = Gj . Hence, {F1 , . . . , Fm } ⊂ {G1 , . . . , Gn }. By symmetry, the converse inclusion also holds, i.e., (3) is true.   n

The set e(Mr ) consists of the functions which can be uniquely  (i) (i) represented in the form of a product λ (tj k )aj k taken over all 1 ≤ i ≤ l with

3.3.3 Corollary.

(i)

(i)

[n(i) ]  = [0], 0 ≤ j ≤ pi and 0 ≤ k ≤ nj , where λ ∈ R\{0} and aj k ∈ Z+ are such (i)

that for every i and j there is k with aj k = 0.

  n

3.3.4 Corollary. The family of cell closures in r coincides with the family of all n n non-empty subsets of the form {x ∈ r | f (x) = 1} with f ∈ e(Mr ). In particular, n

n

n

∼

n

any isomorphism (r , Mr ) → (r , Mr ) in Topsr gives rise to an isomorphism of n

n

∼

partially ordered sets O([n]r ) = O(r ) → O(r ) = O([n ]r ). Proof. Assume that the statement is true for l − 1. To prove the direct implication it n≤l−1

n

suffices to consider the cells of r over the interior of r ≤l−1 . Such a cell corresponds (l)

(l)

to a subset C ⊂ [n(l) ] = [n0 ] × · · · × [npl ] of the form C0 × · · · × Cpl with n (l) Ci ⊂ [ni ], and its closure coincides with the set {x ∈ r | f (x) = 1} for the  pl  (l) elementary function f = i=0 j ∈Ci tij . To prove the converse implication, it suffices to consider an elementary function f represented in the form of Corollary (i) 3.3.3 with λ = 1 and aj k = 0 for all 1 ≤ i ≤ l − 1. For 0 ≤ i ≤ pl , we set (l)

n

(l)

Ci = {j ∈ [ni ] | aij = 0}. Then the set {x ∈ r | f (x) = 1} coincides with the closure of the cell that corresponds to the subset C = C0 × · · · × Cpl ⊂ [n(l) ].  

3.4 Hausdorffness of the geometric realization 3.4.1 Proposition. Hausdorff.

For every D ∈ Ob(R,l E ns), the topological space |D| is

˙ r → [n]r induces a homeomorphism 3.4.2 Lemma. The morphism [n] ∼

n

˙r . ˙ r| →  |[n]

Smooth p-adic analytic spaces are locally contractible. II

319 n

Proof. Step 1. For i ≥ 0, we define as follows a subset Pi of the set of cells of r : ˚ rn } and, for i ≥ 1, Pi is the set of maximal cells in the complement of the P0 = { union of all cells from i−1 j =0 Pj . Let P i denote the set of the closures of cells from Pi . (Recall that the map Pi → P i : A  → A is a bijection.) We claim that (a) every cell from P2 is contained in exactly two cell closures from P 1 ; (b) if a cell A is contained in B ∩ C for B, C ∈ P 1 with B = C, then there exist B1 = B, B2 , . . . , Bk = C ∈ P 1 and D1 , . . . , Dk−1 ∈ P 2 such that A ⊂ D1 ∩· · ·∩Dk−1 and Di ⊂ Bi ∩Bi+1 with Bi  = Bi+1 for all 1 ≤ i ≤ k −1. Indeed, assume the claim is true for l − 1. By Lemma 3.3.2 and Corollary 3.3.4, m (l) we may assume that n(l) = (1). Let m = n≤l−1 , s = r ≤l−1 , r = r0 and S = s . n One has r = {(x, t0 , t1 ) ∈ S×]0, 1]2 | t0 · t1 = r(x)}. Let π denote the canonical n m projection r → s , and let Qi and Qi denote the sets of cells and cell closures m in s similar to Pi and P i . For X ∈ Qi , the preimage π −1 (X) is a disjoint union of three cells X ∈ Pi , X0 ∈ Pi+1 (defined by t0 = 1) and X1 ∈ Pi+1 (defined by  ∈ Pi+1 , if r|X = 1. For Y = X, we denote by Y , t1 = 1), if r|X = 1, and is a cell X 0 1  the closures of X , X 0 , X 1 and X,  respectively. For example, S = rn . Y , Y and Y We now verify (a) and (b) case by case.  for X ∈ Q1 with r|X = 1, then A is contained only in (a) Let A ∈ P2 . If A = X 0 1 i S and S . If A = X for i = 0, 1 and X ∈ Q1 with r|X  = 1, then A is contained only in X and S i . If A = X for X ∈ Q2 with r|X  = 1, then A is contained only in Y and Z , where Y and Z are the cell closures from Q1 that contain X.  for a cell X in S with (b) Assume first that B = S 0 and C = S 1 . Then A = X  ∈ P 2 and r|X = 1. Let Y be a cell closure from Q1 that contains X. If r|Y = 1, then Y  ⊂ S 0 ∩ S 1 . If r|Y  = 1, then Y 0 , Y 1 ∈ P 2 , Y ∈ P 1 , and one has A ⊂ Y 0 ∩ Y 1 , A⊂Y Y 0 ⊂ S 0 ∩ Y and Y 1 ⊂ Y ∩ S 1 . Assume now that B = S 0 and C = Y , where Y ∈ Q1 . Then Y 0 ∈ P 2 , and one has A ⊂ Y 0 ⊂ S 0 ∩ Y . Assume finally that B = Y and C = Z with Y, Z ∈ Q1 , and let X be the image of A in S. If r|X = 1 (and,  then Y 0 , Z 0 ∈ P 2 , and one has A ⊂ Y 0 ∩ Z 0 , Y 0 ⊂ Y ∩ S 0 and therefore, A = X), 0 0 Z ⊂ S ∩Z . If r|X  = 1, we apply induction and find Y1 = Y, Y2 , . . . , Yk = Z ∈ Q1 and V1 , . . . , Vk−1 ∈ Q2 such that X ⊂ V1 ∩ · · · ∩ Vk−1 and Vi ⊂ Yi ∩ Yi+1 with Yi = Yi+1 for all 1 ≤ i ≤ k − 1. Since r|Vi  = 1, then Vi ∈ P 2 and Yi ∈ P 1 , and one has A ⊂ V1 ∩ · · · ∩ Vk and Vi ⊂ Yi ∩ Yi+1 for all 1 ≤ i ≤ k − 1. n

Step 2. Let us fix a section O(r ) = O([n]r ) → /[n]r : A  → ([nA ]r A → [n]r ) of the canonical functor R,l /[n]r (see Lemma 2.3.2(iii)). By Step 1, for every cell B ∈ P2 there are exactly two cells B1 , B2 ∈ P1 with B ≤ B1 and B ≤ B2 . We claim that there is a canonical isomorphism of polysimplicial sets    ∼ →  ˙ r, Coker [nB ]r B → [nA ]r A → [n] B∈P2

A∈P1

320

Vladimir G. Berkovich

where the upper and lower morphisms are induced by the canonical morphisms [nB ]r B → [nB1 ]r B and [nB ]r B  → [nB2 ]r B , respectively. Indeed, let C denote 1 2 ˙ r induces an isothe cokernel. From Step 1 it follows that the morphism C → [n] ∼ ˙ r ). The claim now follows from morphism of partially ordered sets O(C) → O([n] the following simple observation. Given a morphism of polysimplicial sets C → D ∼ which induces an isomorphism of partially ordered sets O(C) → O(D), assume that ∼ n the stabilizer of every polysimplex d ∈ Dr in Aut([n]r ) is trivial. Then C → D. The statement of the lemma now follows from the fact that the geometric realization functor commutes with cokernels.   3.4.3 Corollary. In the situation of Lemma 3.2.1, the following diagram of topological spaces is cocartesian:  / |Sk m−1 (D)| ˙ nd d∈P m Gd \r d  nd G d∈P m d \r d



 / |Sk m (D)|.

Proof. The statement follows from Lemmas 3.2.1 and 3.4.2 and the fact that the geometric realization functor commutes with direct limits.   Proof of Proposition 3.4.1. By Corollary 3.4.3, the canonical map |Sk m−1 (D)| → |Sk m (D)| identifies the first space with a closed subspace of the second one. It follows also that a subset U ⊂ |Sk m (D)| is open in |Sk m (D)| if and only if the intersection U ∩ |Sk m−1 (D)| is open in |Sk m−1 (D)| and the preimages of U under all maps n n r dd → |Sk m (D)| that correspond to the polysimplices d ∈ P m are open in r dd . Given a polysimplicial set C, let us say that two subsets U, V ⊂ |C| are strongly n disjoint if the closures of their preimages in r cc are disjoint for every c ∈ C. We claim that (a) given strongly disjoint open subsets U, V ⊂ |Sk m−1 (D)|, there exist strongly disjoint open subsets U , V ⊂ |Sk m (D)| with U ∩ |Sk m−1 (D)| = U and V ∩ |Sk m−1 (D)| = V; (b) given an open subset U ⊂ |Sk m−1 (D)|, a polysimplex d ∈ P m , and a set X in ˚ rnd under the corresponding map rnd → |Sk m (D)| the image of the interior  d d ˚ rnd is relatively compact, there exists an open such that the preimage of X in  d subset U ⊂ |Sk m (D)| with U ∩ |Sk m−1 (D)| = U which is strongly disjoint from X. ˙ rnd . (a) For a polysimplex d ∈ P m , let U(d) denote the preimage of U in Gd \ d nd Since the closures of U(d) and V(d) are disjoint and Gd \r d is a compact space,

Smooth p-adic analytic spaces are locally contractible. II

321

it contains open subsets U(d) and V (d) whose closures are disjoint and such that ˙ rnd ) = U(d) and V (d) ∩ (Gd \ ˙ rnd ) = V(d) . The required sets U U(d) ∩ (Gd \ d d and V are constructed as the unions of the images of U(d) and V (d) in |Sk m (D)|, respectively, taken over all d ∈ P m . n (b) For the given polysimplex d, we can find an open subset U(d) ⊂ Gd \r dd with ˙ rnd ) = U(d) and such that its closure does not intersect with the closure U(d) ∩ (Gd \ d ˚ rnd . If e is a polysimplex from P m different from d, of the preimage of X in Gd \ d n ˙ rnd ) = U(e) . we take for U(e) an arbitrary open subset of Gd \r dd with U(e) ∩ (Gd \ d The required set U is the union of the images of U(d) and U(e) in |Sk m (D)| taken over e ∈ P m different from d. Step 2. |D| is a Hausdorff space. Let x and y be two distinct points of |D|. They ˚ rn under the maps sm → |D| and rn → |D| ˚ sm and  are contained in the images of  that corresponds to (unique) polysimplices d ∈ P m and e ∈ P n . Assume that m ≤ n. First of all, to construct disjoint open neighborhoods U of x and V of y in |D|, it suffices to construct strongly disjoint open neighborhoods U of x and V of y in |Sk n (D)|. Indeed, if U and V are already constructed then, by Step 1(a), there exist increasing sequences of subsets Un = U ⊂ Un+1 ⊂ · · · and Vn = V ⊂ Vn+1 ⊂ · · · such that Ui and Vi are open and strongly disjoint in |Sk i (D)|, Ui+1 ∩|Sk i (D)| = Ui and Vi+1 ∩|Sk i (D)| = Vi . Since |D| is a direct limit of the spaces |Sk i (D)|, it follows that the unions U and V of all Ui and Vi , respectively, are open and disjoint in |D|. Assume first that m = n. By Corollary 3.4.3, |Sk n (D)|\|Sk n−1 (D)| is a disjoint union of open subsets of |Sk n (D)|, which are evidently Hausdorff and locally compact, and therefore any two open neighborhoods of x and y with disjoint closures are also open and strongly disjoint in |Sk n (D)|. Assume now that m < n. Let U be an arbitrary open neighborhood of the point x in |Sk n−1 (D)|, and let V be an open neighborhood ˚ rn in |Sk n (D)| such that the preimage of V in  ˚ rn is of the point y in the image of  relatively compact. By Step 1(b), there exists an open subset U ⊂ |Sk n (D)| with   U ∩ |Sk n−1 (D)| = U which is strongly disjoint from V, and we are done. n

n

˚ r under the map r → |D| that A subset of |D|, which is the image of the interior  r corresponds to a polysimplex d ∈ Dn , is called a cell of |D|. Corollary 3.4.3 implies ˚ rnd | and that |Sk m (D)|\|Sk m−1 (D)| is a that such a cell is homeomorphic to Gd \| d disjoint union of the cells that correspond to polysimplices from P m . Proposition 3.4.1 n implies that the closure of the above cell in |D| coincides with the image of r in |D|. Such a compact subset of |D| is called a cell closure. Let O(|D|) denote the set of cells of |D| provided with the following partial ordering: A ≤ B if A ⊂ B. 3.4.4 Corollary. (i) A cell closure is a disjoint union of cells; (ii) two distinct cells are disjoint (and, therefore, O(|D|) can be also viewed as the set of all cell closures partially ordered by inclusion); ∼

(iii) there is an isomorphism of partially ordered sets O(D) → O(|D|).

 

322

Vladimir G. Berkovich

3.4.5 Corollary. (i) Given an injective morphism of polysimplicial sets D → D, the corresponding map |D | → |D| identifies |D | with a closed subset of |D|; (ii) for any polysimplicial set D, there is a one-to-one correspondence between polysimplicial subsets of D and the closed subsets of |D|, which are unions of cells.   r

A polysimplicial set D is said to be free if for every polysimplex d ∈ Dn the corresponding morphism [n]r → D is injective. Notice that every polysimplicial set that admits a morphism to free polysimplicial set is also free. 3.4.6 Lemma. If D is a free polysimplicial set, the following properties of a morphism D → D are equivalent: (a) the morphism D → D is injective; (b) the map |D | → |D| identifies |D | with a closed subset of |D|; (c) the map of partially ordered sets O(D ) → O(D) is injective. Proof. The implications (a)⇒(b) and (b)⇒(c) follow from Corollaries 3.4.5(i) and 3.4.4(iii) (and do not require the assumption on D). Assume (c) is true, and let r two polysimplices d1 , d2 ∈ D nr have the same image d in Dn . Then there is an automorphism γ of [n]r with D (γ )(d1 ) = d2 and, therefore, D(γ )(d) = d. Since D   is free, γ is the identity automorphism and, therefore, d1 = d2 . Recall that a Kelley space is a Hausdorff topological space X possessing the property that a subset of X is closed whenever its intersection with each compact subset of X is closed. For example, every locally compact space is Kelley. Proposition 3.4.1 implies that the geometric realization |D| of any polysimplicial set D is a Kelley space. It is locally compact if and only if D is locally finite. Given polysimplicial sets D ∼ and D , there is a homeomorphism |D D | → |D | × |D |, where the latter direct product is taken in the category of Kelley spaces.

3.5 A piecewise RZ+ -linear structure on the geometric realization In this subsection we assume that the monoid R does not contain zero. In this case, (1) (l) n n r is evidently an RZ+ -polyhedron in (R∗+ )[n ] × · · · × (R∗+ )[n ] . The semiring Mr is generated by R and the coordinate functions and, in particular, all functions from n Mr are piecewise RZ+ -linear. We remark that one can easily see, by induction on l, n Z+ -linear. that the inverse of any coordinate function on r is piecewise R

Smooth p-adic analytic spaces are locally contractible. II

323

n

Given a function f ∈ Mr , let {fi }i∈I be the finite set of elementary function from Proposition 3.3.1(ii) that are associated with f . For i ∈ I , we set Vi (f ) = n {x ∈ r | fi (x) ≥ fj (x) for all j ∈ I }. (Notice that each Vi (f ) contains a point x with fi (x) > fj (x) for all j  = i.) We set σ (f ) = {Vi (f )}i∈I and, for a subset n F = {f1 , . . . , fm } ⊂ Mr , we denote by σ (F ) the family of all sets of the form V1 ∩ · · · ∩ Vm with Vi ∈ σ (fi ). Notice that the union of all V ∈ σ (F ) coincides with n n n n (i) r . Finally, we set Fr = {rj }1≤i≤l,0≤j ≤pi and σr = σ (Fr ). n

n

3.5.1 Lemma. Let F be a finite subset of Mr that contains Fr . Then (i) every V ∈ σ (F ) is an RZ+ -polytope, and the restriction to V of each function from the monoid generated by F and the coordinate functions is RZ+ -linear on V ; (ii) if U, V ∈ σ (F ), then U ∩ V is a face in U and in V ; n

(iii) if  is a cell closure in r and V ∈ σ (F ), then  ∩ V is a face of V . (1)

(l)

Proof. (i) The set V is defined in [0, 1][n ] × · · · × [0, 1][n ] by the following equalities and inequalities for all 1 ≤ i ≤ l, 0 ≤ j ≤ pi and f ∈ F : (1) (i) (i) (i) tj 0 (x) . . . t (i) (x) = rj (x), and (2) fk (x) ≥ fk (x) for some k and all k , where j nj

(i)

{fk } is the finite set of elementary functions associated with f . Since rj ∈ F and is the maximum of the corresponding fk ’s, (2) implies that (1) is equivalent to the (i) (i) equality tj 0 (x) . . . t (i) (x) = fk (x), and the statement follows. j nj

(ii) The polytopes U and V are defined by the same equalities (1) and similar inequalities (2) with different k’s, and their intersection U ∩ V is defined by the additional equalities of the corresponding elementary functions fk ’s. It follows that U ∩ V is a face in U and in V . n (i) (iii) Since  is defined in r by the equalities tj k = 1 for some i, j and k (see Corollary 3.3.4), it follows that  ∩ V is a face of V .   From Lemma 3.5.1 it follows that the family τ (F ) of all of the faces of the polytopes n from σ (F ) is an RZ+ -polytopal subdivision of r . It follows also that every cell n closure  in r is an RZ+ -polyhedron and τ (F )| is an RZ+ -polytopal subdivision n n of . The subdivision τ (Fr ) will be denoted by τr . 3.5.2 Corollary.

Every morphism γ : [n ]r → [n]r in R,l gives rise to an n

n

n

immersion of RZ+ -polyhedra r → r , and the restriction of τr to the image of the n

n

latter gives rise to an RZ+ -polytopal subdivision of r , which is a refinement of τr . If γ is an isomorphism, both subdivisions coincide.   n

Thus, the correspondence [n]r  → r gives rise to a functor from R,l to the category of RZ+ -polyhedra in which morphisms are immersions.

324

Vladimir G. Berkovich

3.5.3 Proposition. One can provide the geometric realization |D| of every locally finite R-colored polysimplicial set D of length l with a unique piecewise RZ+ -linear structure so that n

(a) if D = [n]r , it is the canonical RZ+ -polyhedron structure on r ; (b) for any morphism D → D between locally finite R-colored polysimplicial sets of length l, the induced map |D | → |D| is a G-local immersion of piecewise RZ+ -linear spaces. 3.5.4 Lemma. Assume we are given a piecewise RZ+ -linear space X and an equivalence relation E on X which is a piecewise RZ+ -linear subspace of X × X and satisfies the following two properties: (1) both projections p1 , p2 : E → X are proper G-local immersions of piecewise RZ+ -linear spaces; (2) for every point x ∈ X, there exist RZ+ -polyhedra X1 , . . . , Xn in X with the property that any two equivalent points of Xi are equal and such that X1 ∪ · · · ∪ Xn is a neighborhood of x in X. Then the quotient space Y = X/E can be provided with a unique piecewise RZ+ -linear structure such that the canonical map X → Y is a G-local immersion. Proof. First of all, the space Y is locally compact since both projections p1 , p2 : E → X are proper. Let σ be the family of RZ+ -polyhedrons U in X such that any two equivalent points of U are equal. By (2), σ is a piecewise RZ+ -linear atlas on X. Furthermore, let τ be the family of the compact subsets V of Y for which there ∼ exists U ∈ σ with U → V . Since the fibers of both projections p1 , p2 : E → X are finite, it follows that for every point y ∈ Y there exist V1 , . . . , Vn ∈ τ such that V1 ∪ · · · ∪ Vn is a neighborhood of y in Y . Finally, let V , V ∈ τ , and let ∼ ∼ U , U ∈ σ be such that U → V and U → V . The set W = (U × U ) ∩ E is an RZ+ -polyhedron and, by the assumptions, the projections p1 : W → U and p2 : W → U are injective G-local immersions, i.e., they are immersions. It follows that the RZ+ -polyhedron structures on V and V , provided by the homeomorphisms with U and U , respectively, are compatible on the intersection V ∩ V . Thus, τ is a piecewise RZ+ -linear atlas on Y , and the canonical map X → Y is a G-local immersion. That the piecewise RZ+ -linear structure on Y with the latter property is unique is already clear.   Proposition 3.5.3 is established using the construction of Lemma 3.5.4 and the following two simple facts which are proved without the assumption 0  ∈ R. n r Given a polysimplex d ∈ Dn , let Ed denote the equivalence relation on r induced n n n by the canonical map λd : r → |D|. We consider Ed as a subset of r × r . Furthermore, for a morphism γ : [n ]r → [n]r in R,l , let γ denote the induced map n

n

λD(γ )(d) = λd  (γ ) : r → r → |D|.

Smooth p-adic analytic spaces are locally contractible. II

325

n

3.5.5 Lemma. Ed coincides with the union ∪(γ 1 , γ 2 )(r ) taken over all pairs of morphisms γ1 , γ2 : [n ]r → [n]r with D(γ1 )(d) = D(γ2 )(d). Proof. It is clear that the union is contained in Ed . On the other hand, assume that n the images of two points x1 , x2 ∈ r coincide in |D|. We have to verify that there exist two morphisms γ1 , γ2 : [n ]r → [n]r with D(γ1 )(d) = D(γ2 )(d) and a point n

x ∈ r such that x1 = (γ1 )(x ) and x2 = (γ2 )(x ). First of all, let x1 and n

x2 lie in the cells of r associated with morphisms γ1 : [n ]r → [n]r and γ2 : ˚ n and  ˚ n in |D| coincide, from [n ]r → [n]r , respectively. Since the images of  r

r

∼

Corollary 3.4.4(iii) it follows that there exists an isomorphism α : [n ]r → [n ]r with D(γ1 )(d) = D(γ2  α)(d). Thus, replacing γ2 by γ2  α, we may assume that [n ]r = ˚ n . [n ]r . Furthermore, let x1 = (γ1 )(x ) and x2 = (γ )(x ) for some x , x ∈  r

2

Since the images of the points x and x in |D| coincide, from Corollary 3.4.3 it follows that there is an automorphism α of [n ]r with D(α)(d ) = d , where d = D(γ1 )(d) = D(γ2 )(d), such that x is the image of x under the corresponding automorphism of n

r . Hence, we get the required fact for the morphisms γ1 and γ2 = γ2  α and the   point x .

Assume that for every 1 ≤ i ≤ l we are given an ordering on the set [n(i) ] = (i) (i) [n0 ] × · · · × [npi ]. Let us represent elements of [n(i) ] as pairs (j, µ), where 0 ≤ n (i) j ≤ pi and 0 ≤ µ ≤ nj , and consider the following subset of r (i)

n

(i)

(i)

X = {x = (xj k ) ∈ r | xj µ ≤ xkν for (j, µ) ≤ (k, ν) in [n(i) ], 1 ≤ i ≤ l} . n

Notice that the sets of this form cover r , but some of them may be empty. 3.5.6 Lemma. If the images of two points x, y ∈ X in |D| coincide, then x = y. Proof. (A) Given [n ] ∈ Ob(), the set X has a non-empty intersection with at most one cell which corresponds to an equivalence class of [n ]-polysimplices of [n]r . (An [n ]-polysimplex is an r -colored [n ]-polysimplex for some [n ]r ∈ Ob(R,l ).) Assume that the statement is true for l − 1. We set [m]s = [n≤l−1 ]r ≤l−1 and [m ] = n

m

[n ≤l−1 ]. The image of X under the canonical projection r → s is contained in a set of the same type and, therefore, it has a non-empty intersection with at most one cell which corresponds to an equivalence class of [m ]-polysimplices of [m]s . If the latter cell exists, we may assume that [m ] = [m]. By Lemma 2.3.2, the equivalences classes of [n ]-polysimplices of [n]r correspond bijectively to non-empty subsets of (l) (l) (l) [n(l) ] = [n0 ] × · · · × [npl ] of the form C = C0 × · · · × Cpi with |Cjk | = n k + 1 for 0 ≤ k ≤ pl and |Cj | = 1 for j  ∈ {j0 , . . . , jpl }. Given such a subset C, the n ˚ sm with x (l) < 1 for µ ∈ Cj corresponding cell of r consists of the points x over  jµ

326

Vladimir G. Berkovich (l)

(l)

and xj µ = 1 for µ  ∈ Cj . After a permutation of the coordinate functions {tj µ }µ , we may assume that the ordering on the set [n(l) ] satisfies the property (j, µ) < (j, ν) for µ < ν. Hence, if the above cell has a non-empty intersection with X, then (l) Cj = {0, . . . , cj }, where cj = n k , if j = jk , and cj = 0, if j  ∈ {j0 , . . . , jpl }. Moreover, in this case one has (j, cj ) < (k, ck + 1) for all 0 ≤ j, k ≤ pl with (l) ck < nk . These inequalities uniquely determine the sequence c0 , c1 , . . . , cpl among those obtained from it by a permutation, and this implies the required fact. n (B) By (A), the points x and y are contained in one cell of r and, therefore, the claim follows from Corollary 3.4.3 and the following elementary fact. If, for a non-decreasing sequence of numbers x1 ≤ · · · ≤ xn and a permutation σ ∈ Sn , one has xσ (1) ≤ · · · ≤ xσ (n) , then xσ (i) = xi for all 1 ≤ i ≤ n.   Proof of Proposition  n 3.5.3. We apply the construction of Lemma 3.5.4 to the disjoint union X = r dd , taken over all polysimplices d of D, and the equivalence relation E ⊂ X×X induced on X by the canonical surjective map X → |D|. Since the validity of the properties (1) and (2) follows from Lemmas 3.5.5 and 3.5.6, respectively, we are done.   3.5.7 Corollary. Let D be a locally finite R-colored polysimplicial set of length l. Then (i) all cells and cell closures are piecewise RZ+ -linear subspaces of |D|; (ii) all functions from MD are piecewise RZ+ -linear.

 

Thus, if D is a locally finite R-colored polysimplicial set of length l, its geometric realization |D| is a piecewise RZ+ -linear space provided with a semiring MD of piecewise RZ+ -linear functions and a locally finite stratification by relatively compact piecewise RZ+ -linear subspaces, cells, with the property that the closure of a cell, a cell closure, is also a piecewise RZ+ -linear subspace and a (finite) union of cells. Furthermore, given a morphism D → D between R-colored polysimplicial sets of length l, the corresponding map |D | → |D| is a G-local immersion of piecewise RZ+ -linear spaces that takes functions from MD to functions from MD and induces a surjective open map from every cell of |D | to a cell of |D|. 3.5.8 Remarks. (i) It is very likely that the property (2) in Lemma 3.5.4 always follows from (1). (ii) It follows from the remark at the beginning of this subsection that, given a piecewise RZ+ -linear subspace X of the geometric realization |D| of a locally finite Z -linear (resp. R Q -linear) function R-colored polysimplicial set D, every piecewise R  on X is in fact piecewise RZ+ -linear (resp. R Q+ -linear).

Smooth p-adic analytic spaces are locally contractible. II

327

4 The skeleton of a nondegenerate pluri-stable formal scheme 4.1 Poly-stable fibrations and pluri-stable morphisms Let k be a non-Archimedean field whose valuation is not assumed to be nontrivial. For a strictly k-analytic space X, we denote by O (X) the multiplicative monoid of all analytic functions f ∈ O(X) for which the Zariski closed set {x ∈ X | f (x) = 0} is nowhere dense in X. If X is normal (i.e., all strictly affinoid subdomains of X are normal), then O (X) coincides with the set of all f ∈ O(X) whose restriction to every connected component of X is not zero. For a formal scheme X locally finitely presented over k  , we denote by O (X) the multiplicative monoid of all f ∈ O(X) whose image in O(Xη ) is contained in O (Xη ). For an affine formal scheme X = Spf(A) finitely presented over k  , an element a ∈ A and an integer n ≥ 0, we set X(n, a) = Spf(A{T0 , . . . , Tn }/(T0 . . . Tn − a)) , and for m ≥ 0 we set X(m) = X(m, 1). (If n = 0, we assume that a = 1 and set X(0, 1) = X.) Furthermore, given tuples n = (n0 , . . . , np ) ∈ Zp+1 and a = (a0 , . . . , ap ) ∈ Ap+1 such that ni ≥ 1 and each ai is not invertible in A, or p = n0 = 0 and a0 = 1, we set X(n, a) = X(n0 , a0 ) ×X · · · ×X X(np , ap ) . (If X = Spf(k  ), the latter is the formal scheme which was denoted in [Ber7] by T(n, a).) If, in addition, a non-negative integer m is given, we set X(n, a, m) = X(n, a) ×X X(m). Recall ([Ber7, §1]) that a morphism ϕ : Y → X of formal schemes locally finitely presented over k  is said to be strictly poly-stable if, for every point y ∈ Y, there exist an open affine neighborhood X = Spf(A) of ϕ(y) and an open neighborhood Y ⊂ ϕ −1 (X ) of y such that the induced morphism Y → X goes through an étale morphism Y → X (n, a, m) for some triple (n, a, m) as above. If the latter morphisms can be found in such a way that ai ∈ O (X ) ⊂ A for all 0 ≤ i ≤ p, then ϕ will be said to be nondegenerate. Furthermore, ϕ is said to be (nondegenerate) poly-stable if there exists a surjective étale morphism Y → Y for which the induced morphism Y → X is (nondegenerate) strictly poly-stable. A (nondegenerate, strictly) poly-stable fibration over k  of length l ≥ 0 is a sequence of (nondegenerate, strictly) poly-stable morphisms fl−1

f1

f0

X = (Xl → · · · → X1 → X0 = Spf(k  )) . fl−2

f1

For the above X, we denote by X≤l−1 the poly-stable fibration (Xl−1 → · · · → X1 ) of length l − 1. (We omit f0 and X0 = Spf(k  ) if their presence is evident.) Recall that in [Ber7] we denoted by k  -P stl the category of poly-stable fibrations over k 

328

Vladimir G. Berkovich

of length l, and we considered the category k  -P stlét with the same family of objects as k  -P stl but with étale morphisms between them. We denote by k  -P stnd,l and ét the full subcategories of the latter consisting of the objects for which all of k  -P stnd,l the morphisms fi , 0 ≤ i ≤ l − 1 are nondegenerate. A morphism ϕ : Y → X of formal schemes locally finitely presented over k  is said to be (nondegenerate, strictly) pluri-stable if it is a composition of (nondegenerate, strictly) poly-stable morphisms. For example, a formal scheme X over k  is (nondegenerate, strictly) pluri-stable (i.e., the morphism X → Spf(k  ) is a such one) if there exists a (nondegenerate, strictly) poly-stable fibration X over k  of some length l with Xl = X. The category of pluri-stable formal schemes over k  will be denoted by k  -Pst, and k  -Pst ét and k  -Pst pl will denote the categories with the same family of objects but with étale and pluri-stable morphisms between them, respectively. The full subcategories of the latter consisting of the nondegenerate pluri-stable ét and k  -Pst pl . formal schemes will be denoted by k  -Pstnd , k  -Pstnd nd Pluri-stable morphisms and formal schemes are examples of pluri-nodal morphisms and formal schemes introduced in [Ber7, §1] (see Remark 4.1.5). Recall that a morphism ϕ : Y → X between formal schemes locally finitely presented over k  is called strictly pluri-nodal if locally in the Zariski topology it is a composition of étale morphisms and morphisms of the form Spf(A{u, v}/(uv −a)) → Spf(A), a ∈ A, and it is called pluri-nodal if there exists a surjective étale morphism Y → Y such that the induced morphism Y → X is strictly pluri-nodal. We also say that such a morphism is nondegenerate if the above morphisms Spf(A{u, v}/(uv − a)) → Spf(A) can be found in such a way that a ∈ O (Spf(A)) ⊂ A. (Notice that this is consistent with the notion of a nondegenerate pluri-stable morphism.) Recall that for any pluri-nodal formal scheme X over k  the reduction map π : Xη → Xs is surjective (see [Ber7, Corollary 1.7]). 4.1.1 Lemma. Every pluri-nodal morphism is flat. Proof. Since étale morphisms are evidently flat, it suffices to consider morphisms of the form Spf(B) → Spf(A) with B = A{u, v}/(uv − a), a ∈ A. Let α be an on k is element of the maximal ideal k  which is not equal to zero if the valuation

nontrivial. Each element of B has a unique representation in the form ∞ i=−∞ fi wi , where fi → 0 in the α-adic topology of A, and wi = u−i for i < 0 and wi = v i for i ≥ 0. It follows that, if the valuation on k is trivial, B is a free A-module. If the valuation on k is nontrivial, it follows that, for every n ≥ 1, B/(α n B) is a free module   over A/(α n A) and, by [BoLü1, Lemma 1.6], B is flat over A. 4.1.2 Corollary. π −1 (ϕs (Ys )).

Given a pluri-nodal morphism ϕ : Y → X, one has ϕη (Yη ) =

Proof. First of all, increasing the field k, we may assume that its valuation is nontrivial. It suffices to show that, given a faithfully flat morphism Y = Spf(B) → X = Spf(A),

Smooth p-adic analytic spaces are locally contractible. II

329

the induces map Y = Yη → X = Xη is surjective. For this we notice that the set X0 = {x ∈ X | [H(x) : k] < ∞} coincides with the set of prime ideals ℘ ⊂ A with dim(A/℘) = 1 and for which the canonical homomorphism k  → A/℘ is injective. It follows that X0 is contained in the image of Y . Since X0 is dense in X and both spaces X and Y are compact, the map Y → X is surjective.   4.1.3 Lemma. A pluri-nodal formal scheme X over k  is nondegenerate if and only if its generic fiber Xη is a normal strictly k-analytic space. Proof. The direct implication follows straightforwardly from [Ber7, Lemma 1.5]. To prove the direct implication (and the corollary that follows), it suffices to verify the following fact. Let ϕ : Z = Spf(C) → X = Spf(A) be a morphism of pluri-nodal formal schemes that goes through an étale morphism ψ : Z → Y = Spf(B) with B = A{u, v}/(uv), and assume that Zη is normal. Then (a) ϕs (Zs ) is an open subscheme of Xs ; (b) the strictly k-analytic space π −1 (ϕs (Zs )) is normal; ∼

∼

(c) ψ(Z) ⊂ U ∪ V, where U = Spf(B{u} ) → Spf(A{u, u1 }) and V = Spf(B{v} ) → Spf(A{v, v1 }). Indeed, (a) is true since the morphism of schemes ϕs : Zs → Xs is flat and of finite type. Furthermore, (b) is true since Zη is normal, C is flat over A and π −1 (ϕs (Zs )) = ϕη (Zη ), by Corollary 4.1.2. Finally, since the reduction map π : Zη → Zs is surjective, to prove (c) it suffices to show that for every point y ∈ ψη (Zη ) either |u(y)| = 1 or |v(y)| = 1. Assume this is not true, i.e., there exists a point y ∈ ψη (Zη ) with |u(y)| < 1 and |v(y)| < 1 (since uv = 0 then in fact either u(y) = 0 or v(y) = 0). Then for the point y = π(y) ∈ Ys one has u(y) = v(y) = 0. Consider the closed immersion χ : X → Y defined by the surjection B → A that takes u and v to zero. Since the reduction map π : Xη → Xs is surjective, it follows that there exists a point y ∈ π −1 (y) with u(y ) = v(y ) = 0. Since π −1 (y) ⊂ π −1 (ψs (Zs )) = ψη (Zη ), the latter contradicts [Ber7, Lemma 1.5].   4.1.4 Corollary. Any pluri-nodal morphism from a nondegenerate pluri-nodal to a   pluri-nodal formal scheme over k  is always nondegenerate. The closed fiber Xs of a pluri-stable formal scheme X over k  is provided with a stratification, i.e., a partition of Xs by locally closed irreducible normal subschemes with the property that the closure of a stratum is a union of strata (see [Ber7, §2]). The set of the generic points of the strata is denoted by str(Xs ). It is a partially ordered set with respect to the following ordering: x ≤ y if y is contained in the closure of x. A pluri-stable (and, in particular, an étale) morphism ϕ : Y → X induces a map of partially ordered sets str(Ys ) → str(Xs ) and, in fact, str(Ys ) = ∪str(Ys,x ), where the union is taken over all x ∈ str(Xs ). If k is a bigger non-Archimedean field, then

330

Vladimir G. Berkovich

Spf(k  ) Spf(k  ) → X induces a surjective map of partially the morphism X = X⊗ ordered sets str(Xs ) → str(Xs ). If all of the strata of Xs are geometrically irreducible, the latter map is an isomorphism. If the valuation on k is trivial then, by [Ber7, 1.5], the closed fiber Xs of a nondegenerate pluri-nodal formal scheme X is normal and, therefore, str(Xs ) coincides with the set gen(Xs ) of generic points of the irreducible components of Xs . As in [Ber7], we introduce categories P stnd,l and Pstnd whose objects are pairs (k, X) and (k, X), where k is a non-Archimedean field and X is from k  -P stnd,l and X is from k  -Pstnd , and morphisms (K, Y) → (k, X) and (K, Y) → (k, X) are k K  pairs consisting of an isometric embedding k → K and morphisms Y → X⊗ ét , k  K  in K  -Pstnd , respectively. Similarly, P stnd,l in K  -P stnd,l and Y → X⊗ pl

ét and Pst denote the categories with the same families of objects but with Pstnd nd k K  k  K  and Y → X⊗ the above morphisms for which the morphisms Y → X⊗ are étale and pluri-stable, respectively. ét are full subcategories of the categories P st and Notice that P stnd,l and Pstnd l ét P stl from [Ber7], respectively, and all of these categories are fibered ones over the category dual to the category of non-Archimedean fields. Notice also that the correspondence X  → X≤l−1 gives rise to a functor P stl → P stl−1 . For brevity, the pairs (k, X) and (k, X) will be denoted by X and X, respectively.

4.1.5 Remarks. Assume that the valuation on k is nontrivial, and let a ∈ k  \{0}, A = k  {u, v}/(uv − a) and B = A{x, y}/(xy − (u + v)). The localization B{u} is canonically isomorphic to k  {u, x, u1 , x1 }. Let X1 and X2 be two copies of Spf(B), X12 and X21 two copies of Spf(B{u} ) considered as open subschemes of X1 and X2 , respectively, and X the separated formal scheme constructed by gluing X1 and X2 ∼ along the isomorphism X12 → X21 that takes u to x1 and x to u1 . We believe that the strictly pluri-nodal formal scheme X is not pluri-stable.

4.2 The skeleton of a poly-stable fibration Recall that in [Ber7] we constructed for every poly-stable fibration X over k  of length l a closed subset S(X) ⊂ Xl,η , the skeleton of X, and a proper strong deformation retraction  : Xl,η ×[0, l] → Xl,η of Xl,η to S(X). The retraction map Xl,η → S(X) : x  → xτ = (x, l) is denoted by τ . In this subsection we briefly recall a part of the construction and some basic facts from [Ber7]. (The construction of the retraction map τ will be recalled in §4.4.) First of all, if X = T(n, a, m) with T = Spf(k  ), then Xη = M(B), where B = C/b, C = A{T00 , . . . , Tpnp }, A = k{T1 , . . . , Tm , T11 , . . . , T1m }, and b is the ideal of C generated by the elements Ti0 . . . Tini − ai , 0 ≤ i ≤ p. If we provide A and C with the canonical

norms and B with the quotient norm, then the set D, consisting of the elements µ aµ T µ such that aµ = 0 for all µ = {µij }0≤i≤p,0≤j ≤ni

Smooth p-adic analytic spaces are locally contractible. II

331

with min0≤j ≤ni {µij } ≥ 1 for some 0 ≤ i ≤ p, is a Banach A-submodule of C, and ∼ the canonical surjection C → B induces an isometric isomorphism D → B. The skeleton S(X) is the image of the set S = {t ∈ [0, 1][n] | ti0 . . . tini = |ai |, 0 ≤ i ≤ p} under the following injective mapping S → Xη . It takes a point t ∈ S to the bounded multiplicative seminorm on B which is induced by the function D → R+ : f =

µ µ µ aµ T  → maxµ {||aµ ||t }. If X is a formal scheme over k  that admits an étale morphism to some Y = T(n, a, m), then the skeleton S(X) is the preimage of S(Y) under the induced map Xη → Yη . One show that this subset of Xη is well defined. If the closed fiber Xs has a unique maximal stratum, then the map S(X) → S(Y) is injective, and if, in addition, ∼ this maximal stratum goes to the unique maximal stratum of Ys , then S(X) → S(Y). If X is an arbitrary strictly poly-stable formal scheme over k  , one defines the skeleton S(X) as the union i∈I S(Xi ), where {Xi }i∈I is a covering of X by open subschemes that admit an étale morphism to a formal scheme of the form T(n, a, m). If X is an arbitrary poly-stable formal scheme over k  , one takes a surjective étale morphism X → X from a strictly poly-stable formal scheme X and defines the skeleton S(X) as the image of S(X ) under the induced map X η → Xη . Furthermore, one defines the skeleton S(Y/X) of a poly-stable morphism ϕ : Y → X as follows. Given a point x ∈ Xη , Yx = Y ×X Spf(H(x) ) is a poly-stable ∼ formal scheme over H(x) , and there are canonical isomorphisms Yx,η → Yη,x ∼
where x is the image of x under the reduction map and Yx,s → Ys,x ⊗k(x) H(x), π : Xη → Xs . The skeleton of ϕ is the closed set  S(Y/X) = S(Yx ) . x∈Xη One also constructs a strong deformation retraction ϕ : Yη × [0, 1] → Yη of Yη to S(Y/X). fl−1

f1

Finally, let X = (Xl → · · · → X1 ) be a poly-stable fibration over k  of length l ≥ 0. If l = 0, then S(X) = X0,η = M(k). If l = 1, then S(X) = S(X1 ) and, if l ≥ 2, then the skeleton S(X) is the closed set −1 S(X) = S(Xl /Xl−1 ) ∩ fl−1 (S(X≤l−1 )) .

The correspondence X  → S(X) is a subfunctor of the following functor from P stlét to the category of paracompact locally compact spaces: X  → Xl,η . This functor is denoted by S l . Notice that there is a canonical morphism of functors S(X) → S(X≤l−1 ). One constructs the strong deformation retraction  : Xl,η × [0, l] → Xl,η of Xl,η to S(X) inductively as a composition of the strong deformation retraction fl−1 of Xl,η to S(Xl /Xl−1 ) with a strong deformation retraction of S(Xl /Xl−1 ) to S(X), which is a certain lifting of the strong deformation retraction  : Xl−1,η ×[0, l −1] → Xl−1,η . One has (xt )t = xmax(t,t ) for all points x ∈ Xl,η and all t, t ∈ [0, l], where xt = (x, t).

332

Vladimir G. Berkovich

Recall that the image of every point from S(X) under the reduction map π : Xl,η → Xl,s is contained in str(Xl,s ). The preimage of a point from str(Xl,s ) in S(X) is a locally closed subset called a cell of S(X). The cells form a partially ordered set O(S(X)) with respect to the following ordering: A ≤ B if A ⊂ B. The reduction ∼ map induces an isomorphism of partially ordered sets O(S(X)) → str(Xl,s ). For example, if the valuation on k is trivial and X is nondegenerate, then str(Xl,s ) coincides with the set gen(Xl,s ) of generic points of the irreducible components of Xl,s . By [Ber7, Corollary 1.7], for any point x ∈ gen(Xl,s ), there exists a unique point x ∈ Xl,η with π(x) = x. It follows that S(X) is a discrete set which is the preimage of gen(Xl,s ) in Xl,η . In particular, if Xl is connected, Xl,η is contractible. Given a formal scheme X locally finitely presented over k  , one provides its generic fiber Xη with a partial ordering as follows (see [Ber7, §5]). If X = Spf(A) is affine, then x ≤ y if |f (x)| ≤ |f (y)| for all f ∈ A. If X is arbitrary, the partial orderings on the generic fibers of open affine subschemes of X are compatible, and they define a partial ordering on Xη . Given a poly-stable fibration X over k  of length l, one has x ≤ xt for all x ∈ Xl,η and all t ∈ [0, t] and, in particular, x ≤ xτ , where xτ = τ (x). One of the key ingredients of the above constructions is the following fact, which will be also used here. Recall (see [Ber7, §7) that a strictly poly-stable morphism ϕ : Y → X is said to be geometrically elementary if, for every point x ∈ Xs , the partially ordered set str(Ys,x ) has a unique maximal element and all of the strata of Ys,x are geometrically irreducible. Notice that if ϕ : Y → X is another strictly poly-stable morphism, which is also geometrically elementary, and we are given an étale morphism Y → Y over X, then the induced map S(Y /X) → S(Y/X) is injective. The fact is as follows (see [Ber7, Corollary 7.4]). Given a strictly polystable morphism ϕ : Y → X, for every point y ∈ Ys there exists an étale morphism X → X and an open subscheme Y ⊂ Y ×X X such that the image of Y s in Ys contains the point y and the induced morphism Y → X is geometrically elementary.

4.3 The dependence of S(X) on Xl Given a formal scheme X locally finitely presented over k  , we introduce as follows a partial ordering  on the generic fiber Xη , which is stronger than the partial ordering ≤ considered in [Ber7]: x  y if, for every étale morphism X → X and every point x ∈ X η over x, there exists a point y ∈ X η over y such that |f (x )| ≤ |f (y )| for all f ∈ O(X η ). Notice that, given a morphism ϕ : Y → X, for any pair of points x, y ∈ Yη with x  y one has ϕη (x)  ϕη (y).

4.3.1 Theorem. Then

fl−1

f1

Let X = (Xl → · · · → X1 ) be a poly-stable fibration over k  .

(i) for all points x ∈ Xl,η and all t ∈ [0, l], one has x  xt ;

Smooth p-adic analytic spaces are locally contractible. II

333

(ii) if X is nondegenerate, the skeleton S(X) coincides with the set of the points of Xl,η which are maximal with respect to the partial ordering . Proof. (i) Given an étale morphism ϕ : X → Xl and a point x ∈ X η over x, let fl−1 ϕ

fl−2

f1

X be the poly-stable fibration (X −−−→ Xl−1 → · · · → X1 ). By [Ber7, Theorem 8.1(viii)], one has ϕη (xt ) = xt . Since x ≤ xt , it follows that x  xt . (ii) By (i), the skeleton S(X) contains the set of maximal points and, therefore, it remains to show that for any pair of distinct points x, y ∈ S(X) none of the inequalities x  y or y  x is true. Since this property is local in the étale topology, we may assume that all formal schemes Xi are affine, i.e., Xi = Spf(Ai ), and every morphism fi : Xi+1 → Xi goes through an étale Xi+1 → Xi (ni , a i , mi ) and is geometrically elementary. It suffices to show that there exist two functions f, g ∈ Al with |f (x)| < |f (y)| and |g(x)| > |g(y)|. This is trivially true for l = 0, and assume that l ≥ 1 and that this is true for l − 1. We may assume that the images of x and y in S(X≤l−1 ) coincide. Let z be this image. If nl = (n0 , . . . , np ) and a l = (a0 , . . . , ap ), then for every 0 ≤ i ≤ p one has |(Ti0 . . . Tini )(x)| = |(Ti0 . . . Tini )(y)| = |ai (z)|. Notice that |ai (z)|  = 0 because X is nondegenerate. Since the morphisms Xl → Xl−1 and Y = Xl−1 (nl , a l , ml ) → Xl−1 are geometrically elementary, it follows that the canonical map S(Xl /Xl−1 ) → S(Y/Xl−1 ) is injective and, therefore, there exist 0 ≤ i ≤ p and 0 ≤ j ≤ ni with |Tij (x)|  = |Tij (y)|. Assume that |Tij (x)| < |Tij (y)|. Then for g = Ti0 . . . Ti,j −1 Ti,j +1 . . . Tini one has |g(x)| > |g(y)|.   Thus, the skeleton S(X) is well defined for any nondegenerate pluri-stable formal scheme X. 4.3.2 Corollary. Let ϕ : Y → X be a pluri-stable morphism between nondegenerate pluri-stable formal schemes over k  . Then (i) ϕη (S(Y)) ⊂ S(X); (ii) if ϕ is étale, then S(Y) = ϕη−1 (S(X)). Proof. (i) By Corollary 4.1.4, it suffices to consider the case when the morphism ϕ is nondegenerate poly-stable. Assume that X = Xl−1 for a nondegenerate poly-stable fl−2

f1

ϕ

fl−2

fibration (Xl−1 → · · · → X1 ) of length l − 1, and we set X = (Y → Xl−1 → f1

· · · → X1 ). Then the morphism ϕ takes S(X) to S(X≤l−1 ). Since S(Y) = S(X) and S(X) = S(X≤l−1 ), the required fact follows. (ii) By (i), one has S(Y) ⊂ ϕη−1 (S(X)). Let x ∈ S(X) and y ∈ ϕη−1 (x). By Theorem 4.3.1(i), one has y  yt and, therefore, yt ∈ ϕη−1 (x) for all t ∈ [0, l]. Since ϕη−1 (x) is a discrete topological space, it follows that y = yl ∈ S(Y).   Corollary 4.3.2 implies that the correspondence X → S(X) is a subfunctor of the pl functor X  → Xη on the category Pstnd .

334

Vladimir G. Berkovich

4.3.3 Remarks. (i) For any nondegenerate pluri-nodal formal scheme X, there exists a surjective étale morphism ϕ : Y → X from a nondegenerate strictly pluri-stable formal scheme Y. From Corollary 4.3.2 it follows that the image of S(Y) in Xη does not depend on the choice of ϕ and coincides with the set of maximal points with respect to the partial ordering  on Xη . It can be called the skeleton S(X) of X, and both statements of Corollary 4.3.2 hold for any pluri-nodal morphism ϕ. (ii) In our work in progress, we give a similar description of the skeleton S(X) of an arbitrary poly-stable fibration X of length l as the set of maximal points with respect to a partial ordering on Xl,η which is stronger than the above one (but coincides with it if X is nondegenerate).

4.4 The retraction map τ : Xl,η → S(Xl ) fl−1

f1

Let X = (Xl → · · · → X1 ) be a poly-stable fibration over k  of length l. In this subsection we recall the construction of the retraction map τ = τX : Xl,η → S(X), and we introduce a class of nondegenerate poly-stable fibrations X for which the retraction map τ depends only on Xl . Assume that l ≥ 1 and that the retraction map is already constructed for polystable fibrations of length l − 1. Consider first the case when Xl−1 is affine and Xl = Xl−1 (n, a, m) with n = (n0 , . . . , np ) and a = (a0 , . . . , ap ). The continuous mapping Xl,η → Xl−1,η × [0, 1][n] : y  → (fl−1 (y); |Ti0 (y)|, . . . , |Ti,ni (y)|)0≤i≤p induces a homeomorphism between S(Xl /Xl−1 ) and the closed set S = {(x; t) ∈ Xl−1,η × [0, 1][n] | t0i . . . tini = |ai (x)|, 0 ≤ i ≤ p} , and it gives rise to a retraction map ρ : Xl,η → S. If l = 1, then S(X) = S and τ = ρ. Assume therefore that l ≥ 2. In this case the retraction map τ is a composition of the above map ρ with a retraction map γ : S → S(X) constructed as follows (see [Ber7, p. 62]). First of all, one defines for each n ≥ 0 a strong deformation retraction ψn : [0, 1][n] × [0, 1] → [0, 1][n] to the point (1, . . . , 1). The map ψn is required to possess the property that ψn (σ (t), s) = σ (ψn (t, s)) for all permutations σ of degree n + 1, and so it suffices to define ψn (t, s) only for the points t ∈ [0, 1][n] with t0 ≤ t1 ≤ · · · ≤ tn . First, if s ≤ t0 . . . tn , then ψn (t, s) = t. Furthermore, if i+2 ti+2 . . . tn for some 0 ≤ i ≤ n − 1, then tii+1 ti+1 . . . tn ≤ s < ti+1  ψn (t, s) =

s ti+1 . . . tn



1 i+1

 ,...,

s ti+1 . . . tn



1 i+1

 , ti+1 , . . . ,tn .

Smooth p-adic analytic spaces are locally contractible. II

335


1 1  Finally, if s ≥ tnn+1 , then ψn (t, s) = s n+1 , . . . ,s n+1 . The retraction map γ : S → S(X) is as follows
 γ (x, t 1 , . . . , t p ) = xτ ; ψn0 (t 0 , |a0 (xτ )|), . . . , ψnp (t p , |ap (xτ )|) . If X is such that Xl−1 is affine, the morphism fl−1 : Xl → Xl−1 is geometrically elementary and goes through an étale morphism ϕ : Xl → Y = Xl−1 (n, a, m), then fl−2

f1

the morphism X → Y = (Y → Xl−1 → · · · → X1 ) gives rise to embeddings S(Xl /Xl−1 ) → S(Y/Xl−1 ) and S(X) → S(Y), and the above retraction maps ρ : Yη → S(Y/Xl−1 ) and γ : S(Y/Xl−1 ) → S(Y) give rise to retractions maps ρ : Xl,η → S(Xl /Xl−1 ) and γ : S(Xl /Xl−1 ) → S(X). The latter do not depend on the choice of ϕ, and one has τ = γ  ρ. If X is arbitrary, one can find surjective étale morphisms X → X and X → : X → X : X l → X l−1 and fl−1 X ×X X such that the morphisms fl−1 l l−1 are disjoint unions of morphisms satisfying the assumptions of the previous paragraph. Since the retraction maps τ : X l,η → S(X ) and τ : X l,η → S(X ) are compatible, they give rise to a retraction map τ : Xl,η → S(X). We say that a strictly poly-stable morphism ϕ : Y → X is strongly nondegenerate if, for every point y ∈ Y, there exist an open affine neighborhood X = Spf(A) of ϕ(y) and an open neighborhood Y ⊂ ϕ −1 (X ) of y such that the induced morphism Y → X goes through an étale morphism Y → X (n, a, m), where all ai are invertible in A = A ⊗k  k (i.e., ai (x)  = 0 for all x ∈ X η ). A poly-stable morphism ϕ : Y → X is said to be strongly nondegenerate if there exists a surjective étale morphism Y → Y for which the induced morphism Y → X is strongly nondegenerate strictly polystable. For example, a poly-stable formal scheme X over k  is strongly nondegenerate if and only if it is nondegenerate. One can easily see that a poly-stable morphism ϕ : Y → X is strongly nondegenerate if and only if the induced morphism of strictly k-analytic spaces ϕη : Yη → Xη is smooth in the sense of rigid geometry (or rig-smooth). (A morphism of strictly k-analytic spaces ϕ : Y → X is said to be rig-smooth if every point y ∈ Y has a neighborhood of the form V1 ∪ · · · ∪ Vn , where each Vi is a strictly affinoid subdomain of Y such that the induced morphism Vi → X goes through a quasi-étale morphism (see [Ber5, §3]) to an affine space Am X over X. A morphism between good strictly k-analytic spaces is smooth in the sense of [Ber2] if and only if it is rig-smooth and has no boundary.) A pluri-stable formal scheme X over k  is said to be strongly nondegenerate if the canonical morphism X → Spf(k  ) is a sequence of strongly nondegenerate polystable morphisms. Similarly, a poly-stable fibration X of length l is said to be strongly nondegenerate if all the morphisms fi : Xi+1 → Xi are strongly nondegenerate. 4.4.1 Theorem. Let X be a strongly nondegenerate poly-stable fibration of formal schemes. Then, for every point x ∈ Xl,η , xτ is a unique point of S(X) = S(Xl ) which is greater than or equal to x (with respect to the partial ordering  on Xl,η ).

336

Vladimir G. Berkovich

Proof. As in the proof of Theorem 4.3.1(ii), the property considered is local in the étale topology and, therefore, we may assume that all formal schemes Xi are affine, and every morphism fi : Xi+1 → Xi goes through an étale Xi+1 → Xi (ni , a i , mi ) and is geometrically elementary. For every 0 ≤ i ≤ p, one has Ti0 . . . Tini = ai . Since ai are invertible on Xl−1,η , it follows that all of the coordinate functions Tij are invertible on Xl,η . Since x ≤ xτ , the latter implies that |Tij (x)| = |Tij (xτ )|. But, by the proof of Theorem 4.3.1(ii), we know that for any pair of distinct points y, z ∈ S(X) there exist functions f, g ∈ Al which are representable in the form of products of coordinate functions and such that |f (y)| < |g(z)| and |f (y)| > |g(z)|. The required fact follows.   From Theorem 4.4.1 it follows that, for any strongly nondegenerate poly-stable formal scheme X, there is a well defined retraction map τ : Xη → S(X). 4.4.2 Corollary. Let X be a strongly nondegenerate pluri-stable formal scheme. Given a poly-stable fibration X of length l and a morphism of formal schemes ϕ : X l → X, the following diagram is commutative: S(X ) O

τ ϕη

τ

τ

X l,η

/ S(X) O

ϕη

/ Xη .

where τ is the retraction map associated with X . Proof. For every point x ∈ Xl,η , one has x  xτ and, therefore, ϕη (x )  ϕη (xτ ).   Theorem 4.4.1 implies that ϕη (x )τ = ϕη (xτ )τ .

5 A colored polysimplicial set associated with a nondegenerate poly-stable fibration 5.1 Formulation of the result In this section we construct for every non-Archimedean field k and every l ≥ 0 a ét → ,lf E ns, where R k = |k ∗ |∩[0, 1]. This family of functors for functor k  -P stnd,l R k ,l different k’s forms a functor between fibered categories over the category dual to that ét , and the second one is of non-Archimedean fields. The first one is the category P stnd,l ,lc  E ns introduced as follows. Its objects are pairs (k, D) consisting of the category  l

a non-Archimedean field k and a locally finite polysimplicial set D ∈ Ob(,lf E ns), R k ,l and morphisms (k , D ) → (k, D) are pairs consisting of an isometric embedding

Smooth p-adic analytic spaces are locally contractible. II

337

k → k and a morphism D → D in ,lfk E ns. For brevity, a pair (k, D) is denoted R ,l by D. 5.1.1 Theorem-Construction. One can construct for every l ≥ 0: ét →  ,lc E ns (it takes X to D(X)), (a) a functor of fibered categories Dl : P stnd,l l ∼

(b) an isomorphism of functors θl : |D(X)| → S(X), and (c) a morphism of functors D(X)≤l−1 → D(X≤l−1 ) compatible with θl and θl−1 , which possess the following properties: (1) if X is strictly poly-stable, the polysimplicial set D(X) is free; (2) given a surjective étale morphism X → X, there is an isomorphism of polysim→ ∼ plicial sets Coker(D(X ×X X ) → D(X )) → D(X); ∼

(3) the homeomorphism θl : |D(X)| → S(X) induces an isomorphism of partially ∼ ordered sets O(|D(X)|) → O(S(X)); (4) for every g ∈ O (Xl ), one has θl∗ (|g|) ∈ MD(X) , where |g| is the function x  → |g(x)|; (5) if X is strictly poly-stable, then each point of Xl has an open affine neighborhood fl−1

fl−2

f1

X = Spf(A) such that, for X = (X → Xl−2 → · · · → X1 ), D(X ) is a n standard polysimplex [n]r and the map A\{0} → Mr : g  → θl∗ (|g|) is surjective. The construction is done by induction in §§5.2–5.5. If l = 0, then X = (X0 = Spf(k  )), S(X) = X0,η = M(k) and D(X) = [0]1 . Assume that l ≥ 1 and that the r above objects are already constructed for l − 1. For a polysimplex d ∈ D(X)n , we n σd

θl

shall denote by σ d the map r → |D(X)| → S(X). 5.1.2 Remark. In our work in progress, we extend the above construction to the whole class of poly-stable fibrations. Namely, we construct a functor X  → D(X) from the category of all poly-stable fibrations of length l over k  to the category of ∼ |k  |-colored polysimplicial sets of length l, an isomorphism of functors θl : |D(X)| → ≤l−1 ≤l−1 S(X), and a morphism of functors D(X) → D(X ). They possess the same ∗ properties (1)–(5) with the only difference that, in (4), θl (|g|) ∈ MD(X) for all g ∈ n O(Xl ) and, in (5), the map A → Mr : g → θl∗ (|g|) is surjective. The combinatorial part of the proof of Theorem 5.1.1 in §§5.2–5.4 works also in the general case. The assumption on nondegenerateness of X is used here only for the verification of the property (4) in §5.5 since, in the general case, its verification is more involved.

338

Vladimir G. Berkovich

5.2 Construction of D(X) for strictly poly-stable X Before starting the construction, we recall some facts from [Ber7, §3]. Let X be a strictly poly-stable scheme over a field K. For a point x ∈ str(X), the set irr(X, x) of the irreducible components of X passing through x is provided with a metric as follows: the distance between two components X, X ∈ irr(X, x) is the codimension of the intersection X ∩ X at the point x. Given an étale morphism ϕ : Y → X and a point y ∈ str(Y), for any point x ∈ str(X) with ϕ(y) ≤ x the canonical map irr(Y, y) → irr(X, x) is isometric. For example, if T = T0 × · · · × Tp × S, where Ti = Spec(K[Ti0 , . . . , Tini ]/(Ti0 . . . Tini )) with ni ≥ 1 and ∼ −1 ]), then there is an isometric bijection [n] → S = Spec(K[S1 , . . . , Sm , S1−1 , . . . , Sm irr(T , t) that takes j = (j0 , . . . , jp ) ∈ [n] to the irreducible component defined by the equations T0j0 = · · · = Tpjp = 0, where t is the maximal point in str(T ). Thus, any étale morphism ϕ : X → T from an open neighborhood X of the point x to the above scheme T , that takes x to the above point t, gives rise to an isometric bijection ∼ µϕ : [n] → irr(X, x). The latter property of ϕ is equivalent to the fact that all of the coordinate functions Tij vanish at the point x. fl−1

f1

Let X = (Xl → · · · → X1 ) be a nondegenerate strictly poly-stable fibration over k  . We set X = Xl−1 , Y = Xl and ϕ = fl−1 . By induction, there is a free locally finite polysimplicial set C = D(X≤l−1 ) and a continuous map |C| → Xη that ∼ ∼ identifies |C| with S(X≤l−1 ). Since O(C) → O(|C|) and O(S(X≤l−1 )) → str(Xs ), ∼ the latter map induces an isomorphism of partially ordered sets O(C) → str(Xs ) : c ([n]r → C)  → c. We construct as follows an R k -colored polysimplicial set D of length l. r Given [n]r ∈ Ob(R k ,l ), let Dn be the set of all triples d = (y, c, µ) consisting r ≤l−1

of a point y ∈ str(Ys ), a polysimplex c ∈ Cn≤l−1 with c = x, where x = ϕs (y) ∈ ∼

str(Xs ), and an isometric bijection µ : [n(l) ] → irr(Ys,x , y) such that there exists an open affine neighborhood X ⊂ X of x and an open neighborhood Y ⊂ ϕ −1 (X ) of y for which the induced morphism Y → X goes through an étale morphism ψ : Y → X (n(l) , a, m) such that all of the coordinate functions Tij of X (n(l) , a, m) vanish at y, µψ = µ and σ ∗c (|a|) = r (l) . From [Ber7, Proposition 4.3] it follows that the object [n]r is uniquely defined by the triple d = (y, c, µ). Furthermore, let γ : [n ]r → [n]r be a morphism in R k ,l . It gives rise to a morphism γ ≤l−1 : [n ≤l−1 ]r ≤l−1 → [n≤l−1 ]r ≤l−1 , and we set c = C(γ ≤l−1 )(c) ∈ C

r ≤l−1 n ≤l−1

and x = c ∈ str(Xs ). One has x ≤ x and, by [Ber7, Proposition 2.9],

the set of points y ∈ str(Ys ) with ϕs (y ) = x and y ≤ y is non-empty and has a unique maximal point. Let y be this point. By [Ber7, Lemma 6.1], there exists a unique pair (J, µ ) consisting of a subset J ⊂ ω(n(l) ) and an isometric bijection

Smooth p-adic analytic spaces are locally contractible. II (l)

339

∼

µ : [nJ ] → irr(Ys,x , y ) for which the following diagram is commutative [n(l) ] 

(l) [nJ ]

∼ µ

/ irr(Ys,x , y)

∼

 / irr(Ys,x , y ).

µ

(5.1)

(Here the left vertical arrow is the canonical projection, and the right one is from [Ber7, Proposition2.9].) By the proof of loc. cit., one has J = {j ∈ ω(n(l) ) | aj (x ) = 0 in  k(x )}, i.e., J = {j ∈ ω(n(l) ) | |aj (x)| < 1 for some (and therefore all) x ∈ −1 π (x )}. It follows that J is precisely the set of all j ∈ ω(n(l) ) with rj (x) < 1 for some x ∈ Im((γ ≤l−1 )) and, therefore, the morphism γ gives rise to a morphism (l) γ (l) : [n (l) ] → [nJ ] in  such that rj = rf (j )  (γ ≤l−1 ) for all j ∈ ω(n (l) ), where f is the map ω(n (l) ) → J defined by γ ≤l−1 (see §2.1). By [Ber7, Lemma 3.13], there exists a unique pair (y , µ ) consisting of a point y ∈ str(Ys,x ) with y ≤ y and ∼ an isometric bijection µ : [n (l) ] → irr(Ys,x , y ) for which the following diagram is commutative (l)

[nJ ] O

∼ µ

/ irr(Ys,x , y ) O

(5.2)

γ (l)

[n (l) ]

∼ µ

/ irr(Ys,x , y ).

Let now Y be the open subscheme of Y where all of the coordinate functions of X (n(l) , a, m), which do not vanish at the point y , are invertible. We also set aj =

af (j ) for j ∈ ω(n (l) ). Then the morphism Y → X goes through an étale morphism ψ : Z → X (n (l) , a , m ) (for some m ≥ 0). Thus, the triple d = (y , c , µ ) is an r element of Dn , and we get an R k -colored polysimplicial set D of length l. We claim that the following is true: (i) the polysimplicial set D is free and locally finite; (ii) the correspondence d = (y, c, µ)  → y defines an isomorphism of partially ∼ ∼ ordered sets O(D) → str(Ys ) over the isomorphism O(C) → str(Xs );

(iii) the morphism D ≤l−1 → C : d = (y, c, µ) → c (see § 2.4) is surjective (resp. injective) if and only if the map str(Ys ) → str(Xs ) is surjective (resp. for every x ∈ str(Xs ), Ys,x is connected). (i) That D is locally finite is trivial. To show that it is free, we have to verify r that, given d = (y, c, µ) ∈ Dn and two morphisms γ1 , γ2 : [n ]r → [n]r with D(γ1 )(d) = D(γ2 )(d), then γ1 = γ2 . Let d = (y , c , µ ) = D(γ1 )(d). Since

340

Vladimir G. Berkovich

c = C(γ1≤l−1 )(c) = C(γ2≤l−1 )(c) and C is free, it follows that γ1≤l−1 = γ2≤l−1 . The (l) (l) equality γ1 = γ2 now follows from the fact that both morphisms appear as left vertical arrows in the corresponding diagrams (5.2) with the same sets and three other arrows. s (ii) Given a point y ∈ str(Ys ), let c ∈ Cm be a polysimplex with c = x = ϕs (y). One can find an open affine neighborhood X of x and an open neighborhood Y ⊂ ϕ −1 (X ) of y for which the induced morphism Y → X goes through an étale morphism ψ : Y → X (n, a, m) such that all of the coordinate functions Tij on X (n, a, m) vanish at y. Then the étale morphism gives rise to an isometric ∼ bijection µ : [n] → irr(Ys,x , y). Let a = (a0 , . . . , ap ). By the property (4), m ri = σ ∗c (|ai |) ∈ Ms for all 0 ≤ i ≤ p. Thus, if n = (m, n) and r = (s, r), where r r = (r0 , . . . , rp ), then the triple d = (y, c, µ) gives rise to an element of Dn , i.e., the canonical map O(D) → str(Ys ) : d = (y, c, µ)  → y is surjective. r Assume now that there are two polysimplices d = (y, c, µ) ∈ Dn and d = r

(y , c , µ ) ∈ Dn with y ≤ y. Then for x = ϕs (y) and x = ϕs (y ) one has x ≤ x. ∼

Since c = x, c = x and O(C) → str(Xs ), there is a morphism α : [n ≤l−1 ]r ≤l−1 → [n≤l−1 ]r ≤l−1 with c = C(α)(c). Let y ∈ str(Ys,x ) be the unique maximal point with the property y ≤ y. As above, there exists a unique pair (J, µ ) consisting of a (l) ∼ subset J ⊂ ω(n(l) ) and an isometric bijection µ : [nJ ] → irr(Ys,x , y ) for which the diagram (5.1) is commutative, and we know that J = {j ∈ ω(n(l) ) | |aj (x)| < 1 for some x ∈ Im((α))}. Let β denote the isometric map

[n

(l)

µ −1 ∼

µ ∼

(l)

] → irr(Ys,x , y ) → irr(Ys,x , y ) → [nJ ] .

It induces an injective map f : ω(n (l) ) → J . From [Ber7, Proposition 4.3] it follows (l) (l) that r j = (α)∗ (rf (j ) ) for all j ∈ ω(n (l) ) and, therefore, the pair (α, β) induces a morphism γ : [n ]r → [n]r for which d = D(γ )(d). It follows that the map O(D) → str(Ys ) is an isomorphism of partially ordered sets. (iii) The direct implications follows straightforwardly from the description of D ≤l−1 in terms of D. Assume first that the map str(Ys ) → str(Xs ) is surjective. s r We have to show that for every c ∈ Cm there exists d = (y, c, µ) ∈ Dn with r

[n≤l−1 ]r ≤l−1 = [m]s . By (ii), there exists d = (y, c , µ ) ∈ Dn with c = c. ∼

∼

Since O(C) → str(Xs ), there exists an isomorphism γ : [m]s → [n ≤l−1 ]r ≤l−1 with c = C(γ )(c ). If [n]r is the inverse image of [n ]r under γ (in the sense of Example ∼

2.3.1(ii)) and µ is the composition of the isometric bijection [n(l) ] → [n (l) ] with µ , r then the triple d = (y, c, µ) is an element of Dn . Assume now that, for every x ∈ Xs , r Ys,x is connected. We have to show that any two polysimplices d = (y, c, µ) ∈ Dn r

and d = (y , c, µ ) ∈ Dn (over the same c) are equivalent. By the assumption, it

Smooth p-adic analytic spaces are locally contractible. II

341

suffices to consider the case when y ≤ y, but in this case the required fact is obtained from the construction in the proof of (ii) (with the identity morphism α). We set D(X) = D. It is easy to see that the correspondence X  → D(X) is functorial ét that consists of strictly poly-stable fibrations. on the full subcategory of P stnd,l

5.3 Construction of D(X) for arbitrary X 5.3.1 Lemma. Assume we are given a surjective étale morphisms X → X between nondegenerate strictly poly-stable fibrations of length l. Then there is an isomorphism → ∼ of polysimplicial sets Coker(D(X ×X X ) → D(X )) → D(X). Proof. We set X = X ×X X , X = Xl−1 , C = D(X≤l−1 ), Y = Xl , D = D(X) and so on. By the induction hypothesis, there is an isomorphism of polysimplicial sets → ∼ Coker(C → C ) → C. The morphism of polysimplicial sets D → D is surjective. Indeed, let d = r (y, c, µ) ∈ Dn . By [Ber7, Corollary 2.8], there exists a point y ∈ str(Y s ) over the point y ∈ Ys . Let x and x be the images of y and y in Xs and X s , respectively. One has c = x. To prove the claim, it suffices to show that there exists a polysimplex r ≤l−1

c ∈ C n≤l−1 over c with c = x (since the triple d = (y , c , µ ) will then represent

an element of D nr over d, where µ is the composition of µ with the inverse of the ∼

∼

canonical isometric bijection irr(Y s,x , y ) → irr(Ys,x , y)). Since O(C) → str(Xs ) ∼

and O(C ) → str(X s ), the necessary fact is a consequence of the following simple observation. Given a morphism E → E in R,l E ns, the canonical map E nr → r En ×O(E) O(E ) is surjective for every [n]r ∈ Ob(R,l ). To see the latter, let us r

r

consider a pair of polysimplices e ∈ En and e ∈ E n such that the class of d, r

the image of e in En , coincides with that of e in O(E). It follows that there is an ∼

isomorphism γ : [n]r → [n ]r with e = E(γ )(d). Then the image of the polysimplex r E (γ )(e ) in En is e and its class in O(E ) coincides with that of e . → The morphism Coker(D → D ) → D is an isomorphism. Assume there are two r polysimplices d1 = (y 1 , c1 , µ1 ) and d2 = (y 2 , c2 , µ2 ) in D nr whose images in Dn coincide. Then c1 = x 1 and c2 = x 2 are the images of the points y 1 and y 2 in X s , respectively. By [Ber7, Corollary 2.8], we can find a point y ∈ str(Y s ) over the pair of points (y 1 , y 2 ). Let x be the image of y in X s . It suffices to show that there exists a polysimplex c ∈ C nr over the pair of polysimplices (c1 , c2 ) with c = x . Since ∼

∼

O(C ) → str(X s ) and O(C ) → str(X s ), this follows from the above observation   applied to the morphism C → C ×C C . We fix for each nondegenerated poly-stable fibrations X of length l a surjective étale morphism X → X so that, if X is strictly poly-stable, then X = X = X,

342

Vladimir G. Berkovich →

and define D(X) as the cokernel Coker(D(X ×X X ) → D(X )). We get a functor ét ,lf E ns that possesses the properties (1) and (2). We also get Dl : P stnd,l →  l a morphism of functors D(X)≤l−1 → D(X≤l−1 ) and functorial isomorphisms of ∼ partially ordered sets O(D(X)) → str(Xl,s ). ∼

5.4 Construction of an isomorphism of functors θl : |Dl | → S l 5.4.1 Lemma. Given an open immersion Y → X, the induced morphism D(Y) → D(X) is injective (and, therefore, it identifies D(Y) with the polysimplicial subset of D(X) which corresponds in O(D(X)) = str(Xl,s ) to the subset str(Yl,s )). Proof. If X is strictly poly-stable, the statement follows for l − 1 (resp. l) from the induction hypothesis and Lemma 3.4.6 (resp. the explicit construction of D(X)). In the general case, assume that two polysimplices d1 and d2 of D(Y) go to the same polysimplex of D(X). Let X → X be a surjective étale morphism with strictly poly-stable X , and let Y be the preimage of Y in X . We can find polysimplices d1 = (y 1 , c1 , µ 1 ) and d2 = (y 2 , c2 , µ 2 ) in D(Y ) over d1 and d2 , respectively. The assumption implies that there exist polysimplices di = (x i , ci , µ i ) of D(X ×X X ), ) for 1 ≤ i ≤ n−1 and p (d ) = d . 1 ≤ i ≤ n, with p1 (d1 ) = d1 , p2 (di ) = p1 (di+1 2 n 2 It follows that p1 (x 1 ) = y 1 and, therefore, p2 (x 1 ) ∈ Y l,s , i.e., x 1 ∈ str(Y l,s ×Y l,s

Y l,s ). For the same reason, the same is true for all points x i and, therefore, all of the   polysimplices di come from D(Y ×Y Y ), i.e., d1 = d2 . ∼

Notice that it suffices to construct an isomorphism of functors |Dl | → S l on a full ét with the property that any object of the whole category is the subcategory of P stnd,l image of an object of the subcategory under a surjective étale morphism. It suffices ∼ therefore to construct functorial homeomorphisms |D(X)| → S(X) for X which are strictly poly-stable and such that Xl−1 is affine, and the morphism fl−1 : Xl → Xl−1 is geometrically elementary and goes through an étale morphism Xl → Xl−1 (n, a, m). (Notice that in this case the formal scheme Xl is quasi-compact.) We set X = Xl−1 = Spf(A), Y = Xl , ϕ = fl−1 , C = D(X≤l−1 ) and D = D(X). The first example of a geometrically elementary morphism is a morphism of the form X(n, a, m) → X. ∼

5.4.2 Lemma. If Y = X(n, a, m), then D → [n]C,|a| (see §2.4). s

Proof. Given a polysimplex d = (y, c, µ) ∈ Dm , the set I = {i ∈ ω(n) | |ai (x)| < 1 for some x ∈ Im(σ c )} coincides with the set I (c, |a|) defined in §2.3. If y is the maximal point in Ys,x , where x = ϕs (y) = c, there is a canonical isometric bijection ∼ ∼ [nI ] → irr(Ys,x ) and, therefore, the isometric bijection µ : [m(l) ] → irr(Ys,x , y) defines a morphism γ = (f, α) : [m(l) ] → [nI ] in  such that sj = σ ∗c (|af (j ) |) for all

Smooth p-adic analytic spaces are locally contractible. II

343

s ≤l−1

j ∈ ω(m(l) ). The pair, consisting of c ∈ Cm≤l−1 and the morphism γ , represents an scolored m-polysimplex of [n]C,|a| , and the correspondence d = (y, c, µ) → (c, γ ) gives rise to the required isomorphism.   Assume that Y = X(n, a, m), where n = (n1 , . . . , np ) and a = (a0 , . . . , ap ). Recall (see Step 1 from [Ber7, §5]) that the continuous mapping Yη → Xη × [0, 1][n] : y  → (ϕ(y); |T00 (y)|, . . . , |Tpp (y)|) induces a homeomorphism between S(Y/X) and the closed set S = {(x; t) ∈ Xη × [0, 1][n] | ti0 . . . tini = |ai (x)|, 1 ≤ i ≤ p} . Since S(X) = S(Y/X) ∩ ϕ −1 (S(X≤l−1 )), the isomorphism of Lemma 2.4.1 defines ∼ a homeomorphism |D| → S(X) which possesses the property (3). Indeed, it suffices s to verify that, given a function g ∈ O (Y) and a polysimplex d ∈ Dm , one has m σ ∗d (|g|) ∈ Ms . This easily follows from [Ber7, Lemma 5.6]. Consider now a geometrically elementary morphism ϕ : Y → X that goes through fl−2

f1

an étale morphism Y → Z = X(n, a, m). We set Z = (Z → Xl−1 → · · · → X1 ) and E = D(Z). By the claim (iii) from §5.2, the morphisms of polysimplicial sets D ≤l−1 → C and E ≤l−1 → C are injective and bijective, respectively, and, by the ∼ above construction, there is a homeomorphism |E| → S(Z) that possesses the property (3). Since for every point x ∈ str(Xs ) the induced map of partially ordered sets str(Ys,x ) → str(Zs,x ) is injective, from Lemma 3.4.6 it follows that the morphism of polysimplicial sets D → E is injective. On the other hand, let x be a point of |C| = S(X≤l−1 ) and x its image in Xs . Notice that x ∈ str(Xs ) (see [Ber7, Theorem 8.1(v)]). Since Ys,x is geometrically irreducible, the maps str(Yx,s ) → str(Zx,s ) and D(Yx ) → D(Zx ) are injective and, by [Ber7, Theorem 5.4], the map S(Yx ) → S(Zx ) is injective, and its image is the union of the cells of S(Zx ) that are the preimages of the points coming from str(Yx,s ). It follows that the map S(X) → S(Z) is injective, and its image is the union of the cells of S(Z) that are the preimages of the points coming ∼

∼

from str(Ys ). Since O(D) → str(Ys ), we get a homeomorphism |D| → S(X). The restriction of the latter to the fibers at the point x gives rise to a homeomorphism ∼ D(Yx ) → S(Yx ) which coincides with that of [Ber7, Theorem 5.4]. It follows that ∼ the homeomorphism |D| → S(X) is well defined and, in fact, functorial. ∼ Thus, an isomorphism of functors θl : |Dl | → S l that possesses the property (3) is constructed. It follows from the construction that the morphism D(X)≤l−1 → D(X≤l−1 ) is compatible with θl and θl−1 .

344

Vladimir G. Berkovich

5.5 Verification of the properties (4) and (5) In this subsection we use the assumption that the poly-stable fibrations considered are nondegenerate. If the valuation on k is trivial, both properties are evidently true, and so we assume that the valuation on k is nontrivial. It is clear that it suffices to verify the property (4) only for strictly poly-stable X. Let y 0 ∈ Xl,s , y the generic point of the stratum of Xl,s that contains the point y 0 , and x the image of y in Xl−1,s . First of all, we can shrink X so that Xl−1 = Spf(A) is affine, the point x is a unique maximal one in the partially ordered set str(Xl−1,s ), m ∗ (|f |) is surjective. D(X≤l−1 ) = [m]s , and the map A\{0} → Ms : f → θl−1 Furthermore, we can shrink Xl so that Xl = Spf(B) is affine, the point y is a unique maximal one in str(Xl,s ), and the canonical morphism Xl → Xl−1 goes through an étale morphism ϕ : Xl → Z = Xl−1 (n, a, m) such that the image z of y in Zs is a ∼ unique maximal point in str(Zs ). It follows that D(X) → [n]r , where n = (m, n) ∼

fl−1

f1

and r = (s, |a|), and that S(X) → S(Z), where Z = (Z → Xl−1 → · · · → X1 ). Since the retraction maps Yη → S(X) and Zη → S(Z) commute with ϕ, it follows n that S(X) = ϕ −1 (S(Z)). From [Ber7, Lemma 5.6] it follows that θl∗ (|h|) ∈ Mr for n all h ∈ C\{0}, where Z = Spf(C), and that the map C\{0} → Mr : h  → θl∗ (|h|) n is surjective. Thus, to prove the claim, it suffices to show that θl∗ (|g|) ∈ Mr for all g ∈ B\{0}. For this we need, first of all, the following criterion for a real valued n n continuous function on r to be contained in Mr . n r denote the set of all continuous functions α : rn → R∗+ with the property Let M ˚ rn , there exists a function f ∈ Mrn that, for every relatively compact open subset U ⊂  n n r . with α|U = f |U . One evidently has Mr ⊂ M rn there exists β ∈ M rn with α · β ∈ Mrn . Then 5.5.1 Lemma. Assume that for α ∈ M n α ∈ Mr . n

Proof. Given a function f ∈ Mr , let {fi }i∈I be the finite set of elementary functions n from Proposition 3.3.1(ii) that are associated with f . For i ∈ I , Ui (f ) = {x ∈ r | n fi (x) > fj (x) for all j ∈ I , j  = i} is a nonempty open subset of r , and the n union i∈I Ui (f ) is dense in r . Furthermore, we set A(f ) = {Ui (f )}i∈I and, for a n subset F = {f1 , . . . , fm } ⊂ Mr , we denote by A(F ) the family of all sets of the form n U1 ∩· · ·∩Um with Ui ∈ A(fi ). (Notice that the union of all U ∈ A(F ) is dense in r .) n n n (i) Finally, for f ∈ Mr we set B(f ) = A({f } ∪ Fr ), where Fr = {rj }1≤i≤l,0≤j ≤pi . n Each set U ∈ B(f ) is contained in an RZk + -subpolytope of r and is convex in it, and the restriction f |U is a linear function on U (see Lemma 3.5.1(i)). n Let α and β be from the formulation, and set h = α · β ∈ Mr . We claim for n every U ∈ B(h) there exists a unique f (U ) ∈ e(Mr ) with α|U = f (U ) |U . Indeed, the uniqueness of f (U ) follows from Proposition 3.3.1(i). Let U be a relatively compact

Smooth p-adic analytic spaces are locally contractible. II

345

˚ rn , and let f and g be functions from Mrn with α|U = f |U convex open subset of U ∩  and β|U = g|U . Then h|U = f |U · g|U . The function on the left hand side is linear. On the other hand, both functions f |U and g|U are maxima of a finite number of linear functions. It follows that they are in fact linear. This easily implies that U is a subset of some set from A(f ), and if f (U ) is the corresponding elementary component of f then α|U = f (U ) |U . From Proposition 3.3.1(i) it follows that f (U ) does not depend on the choice of the set U and the function f and, by the continuity of α, α|U = f (U ) |U . Thus, α = maxU ∈B(h) {f (U ) } since this equality is true for the restrictions of both ˚ rn . It follows that α ∈ Mrn .   sides to every relatively compact open subset of  Let g ∈ B\{0}. By [Ber7, Theorem 8.1(vi)], the local ring of every point from S(X) is a field. It follows that ε = min{|g(y)| | y ∈ S(X)} > 0. rn . First of all, we recall that the interior A. The function θl∗ (|g|) is contained in M ˚ rn is the preimage of S(X) ∩ π −1 (y) under θl , and that the morphism Xl → Xl−1  goes through an étale morphism ϕ : Xl → Z = Xl−1 (n, a, m). 1. We may assume that m = 0. Indeed, consider first the case l = 1. If t is the maximal point of X0 (m)η (it corresponds to the supremum norm of the alge∼ bra k{T1 , . . . , Tm , T1−1 , . . . , Tm−1 }), then D(X 1 ) → D(X1 ), where X 1 = (X1 )t = ∼ X1 ×X0 (m) Spf(H(t) ), and S(X 1 ) → S(X1 ). Since |H (t)| = |k|, the situation is ∼

reduced to X 1 (for which m = 0). In the case l ≥ 2, one has D(X ) → D(X) and fl−1

∼

fl−2

fl−3

f1

is the S(X ) → S(X), where X = (Xl → Xl−1 (m) → Xl−2 → · · · → X1 ), fl−1 composition of ϕ with the canonical projection Xl−1 (n, a, m) → Xl−1 (m), and fl−2 is the composition of the canonical projection Xl−1 (m) → Xl−1 with fl−2 . 2. We may assume that [n]  = [0]. Indeed, if [n] = [0], then the morphism fl−1 is étale. If l = 1, the whole statement of this subsection is trivial. If l ≥ 2, there is ∼

fl−2 fl−1

fl−3

an isomorphism D(X) → [0]D(X ),1 (see §2.4), where X = (Xl −−−→ Xl−2 → f1

· · · → X1 ) is of length l − 1. 3. We may assume that the étale morphism from the maximal stratum Y of Xl,s to the maximal stratum Z of Zs , induced by ϕ, is an open immersion. Indeed, let n = (n0 , . . . , np ) and a = (a0 , . . . , ap ). The reductions of the functions a0 , . . . , ap  vanish at the maximal stratum X of Xl−1,s . (Notice that X is closed in Xl−1,s .) in A The maximal stratum Z of Zs , which is defined in the preimage of X by vanishing of all coordinate functions Tij for 0 ≤ i ≤ p and 0 ≤ j ≤ ni , maps isomorphically onto X, and the maximal stratum Y of Xl,s is the preimage of Z in Xl,s . The induced morphism Y → X is étale, and we can find an étale morphism X l−1 = Spf(A ) → Xl−1 such that X l−1,s contains a closed subset X provided with an open immersion X → Y compatible with the étale morphisms Y → X and X → X. Shrinking Xl , we ∼ may assume that X → Y . Let X l be the connected component Xl ×Xl−1 X l−1 that contains the image of Y under the evident morphism to the closed fiber of the latter.

346

Vladimir G. Berkovich

Then the required property is true for X = (X l → X l−1 → Xl−2 → . . . X1 ) and ∼

D(X ) → D(X). 4. Shrinking Xl , we may assume that ϕ identifies Y with a closed subset Z of Zs , where Z is an open subset of Z of the form Spf(C ) with C = C{c} , and we may also assume that the image of Xl is contained in Z . By [Ber7, Lemma 4.4], there is ∼ ∼  →  B an isomorphism of analytic spaces π −1 (Y ) → π −1 (Z ) and of completions C with respect to the ideals J C and J B, respectively, where J = (α, h1 , . . . , hm ) ⊂ C,  generate the ideal of Z, and α h1 , . . . , hm are elements of C, whose reductions in C ˚ rn is contained is a fixed non-zero element of k  . Any relatively compact subset of  in θl−1 (Vδ ) for some δ > 0, where Vδ = {y ∈ S(X) | |hi (y)| < 1 − δ, 1 ≤ i ≤ m}. Let n be a sufficiently large integer with (1 − δ)j |α|n−j < ε for all 0 ≤ j ≤ n. Then |h(y)| < ε for all y ∈ Vδ and all h ∈ J n B. Finally, we can find an element h ∈ C and an integer ν ≥ 0 such that g − chν ∈ J n B. Since |c(y)| = 1 for all y ∈ Xl,η , it follows that |g(y)| = |h(y)| for all y ∈ Vδ . n

B. The function θl∗ (|g|) is contained in Mr . We can shrink Xl so that B = B{f } with B = C[T ]/(P ) and f ∈ B , where P (T ) is a monic polynomial in C[T ] such that the image of its derivative in B is invertible. Furthermore, we can find g ∈ B and m ≥ 0 such that |(g − fgm )(y)| < ε for all y ∈ Xl,η . Since |f (y)| = 1 for all y ∈ Xl,η , it follows that |g(y)| = |g (y)| for all y ∈ S(X). Thus, we may assume that g ∈ B . Since the strictly k-affinoid algebra C = C ⊗k  k is normal, the coefficients of the minimal polynomial T n + h1 T n−1 + · · · + hn of g over its fraction field are in fact elements of C. From [BGR, Proposition 3.8.1/7(a)] it follows that hi ∈ C  , and since C  = C, by [Ber7, Proposition 1.4], it follows that hi ∈ C for all 1 ≤ i ≤ n. One has hn = 0 and hn = −g(g n−1 + h1 g n−2 + · · · + hn−1 ), and the required fact follows from Lemma 5.5.1  

6 p-Adic analytic and piecewise linear spaces 6.1 A piecewise linear structure on the skeleton of a pluri-stable formal scheme fl−1

f1

Let X = (Xl → · · · → X1 ) be a nondegenerate poly-stable fibration over k  of length l. By Theorem 5.1.1, there is a canonical homeomorphism between the geometric realization of the R k -colored polysimplicial set D(X) of length l and the skeleton S(X). This homeomorphism provides S(X) with a piecewise RZk + -linear structure and a semiring MX of piecewise RZk + -linear functions on S(X). Recall that the skeleton fl −1

f1

S(X), as a subset of Xl,η , depends only on Xl (see §4.3). Let X = (X l → · · · → X 1 ) be another nondegenerate poly-stable fibration of length l over k  .

Smooth p-adic analytic spaces are locally contractible. II

347

ét , the induced map 6.1.1 Theorem. For any morphism ϕ : X l → Xl in Pstnd S(X ) → S(X) is a G-local immersion of piecewise RZk + -linear spaces, and it takes functions from MX to functions from MX .

k  k  → Xl , we can replace Proof. Since the statement is true for the morphism Xl ⊗  k  k so that we may assume that k = k and ϕ is an étale k  -morphism. X by X⊗ fl−1 ϕ

fl−2

f1

Furthermore, if Y = (Xl → Xl−1 → · · · → X1 ), then S(Y) = S(X l ) = S(X ) and, by Theorem 5.1.1, applied to the canonical morphism Y → X, we can replace X by Y so that we may assume that ϕ is an isomorphism. Finally, given a surjective étale morphism ψ : Y → Xl , we denote by ψ the surjective étale morphism Y = X l ×Xl Y → X l , and we set Y = (Y fl −1 ψ

fl −2

fl−1 ψ

→

fl−2

f1

Xl−1 → · · · → X1 ) and Y =

f1

(Y → X l −1 → · · · → X 1 ). Since the canonical maps S(Y) → S(X) and S(Y ) → S(X ) are surjective G-local immersions of piecewise RZk + -linear spaces, we may always replace X by Y and X by Y . This reduces the situation to the case when X is strictly poly-stable, Xl = Spf(A) is affine, D(X) is a standard polysimplex n [n]r , and the map A\{0} → Mr : g  → θl∗ (|g|) is surjective. It follows that the n homeomorphism S(X ) → S(X) takes functions from MX = Mr to functions from n MX . Since S(X) is isomorphic to the RZk + -polyhedron r , the map S(X ) → S(X)

is piecewise RZk + -linear. Applying the latter to the inverse morphism ϕ −1 : Xl → X l , we deduce that the map S(X ) → S(X) is in fact a piecewise RZk + -linear isomorphism.  

Thus, for any nondegenerate pluri-stable formal scheme X over k  , the skeleton S(X) is provided with a well defined piecewise RZk + -linear structure and a semiring MX of piecewise RZk + -linear functions. 6.1.2 Corollary. Let ϕ : X → X be a pluri-stable morphism between nondegenerate pluri-stable formal schemes over k  . Then the induced map S(X ) → S(X) is piecewise RZk + -linear and it takes functions from MX to functions from MX . Proof. The statement is deduced from Theorem 6.1.1 in the same way as Corollary 4.3.2(i) is deduced from Theorem 4.3.1.   6.1.3 Corollary. Let ϕ : X → X be a morphism between nondegenerate pluristable formal schemes, and assume that X is strongly nondegenerate. Then the induced map τ  ϕη : S(X ) → S(X) is piecewise RZk + -linear and it takes functions from MX functions from MX . Proof. Let X be a strongly nondegenerate poly-stable fibration of length l with Xl = X. As in the proof of Theorem 6.1.1, the situation is reduced to the case when all

348

Vladimir G. Berkovich

formal schemes Xi = Spf(Ai ) are affine and every morphism fi : Xi+1 → Xi goes through an étale Xi+1 → Xi (ni , a i , mi ) and is geometrically elementary. In this case, |f (x)| = |f (xτ )| for all coordinate function f from Al and all points x ∈ Xη . It follows that for every point x ∈ S(X ) one has |f (τ (ϕη (x )))| = |ϕη∗ (f )(x )|. Since the restriction of the function |ϕη∗ (f )| to S(X ) is contained in MX , it follows that the map τ  ϕη takes functions from MX to functions from MX and, in particular, it is piecewise RZk + -linear.    denote the Given a nondegenerate pluri-stable formal scheme X over k  , let M X semiring of real valued functions f on S(X) with the following property: for every quasi-compact open subscheme Y ⊂ X, there exists α ∈ |k ∗ | such that (αf )|S(Y) ∈  ∗ denote  consists of piecewise |k ∗ |Z+ -linear functions. Let M MY . Notice that M X X  (i.e., such that there exists g ∈ M  the subset of the functions f invertible in M X X ∗ with f g = 1). It is a group by multiplication that contains |k |. 6.1.4 Corollary.  ; (i) If f ∈ O (Xη ), the restriction of |f | to S(X) is contained in M X  ∗ , and it gives (ii) if f ∈ O(Xη )∗ , the restriction of |f | to S(X) is contained in M X ∗ . rise to an embedding O(Xη )∗ /O(X)∗ → M X Proof. (i) If f ∈ O (Xη ), one can find for every quasi-compact open subscheme Y ⊂ X an element α ∈ k ∗ with (αf )|Y ∈ O (Y). It follows that |αf ||S(Y) ∈ MY , η  . i.e., |f ||S(X) ∈ M X (ii) If f ∈ O(Xη )∗ , there exists g ∈ O(Xη )∗ with f g = 1, and the inclusion  ∗ follows from (i). Furthermore, since x ≤ xτ for all points x ∈ Xη , |f ||S(X) ∈ M X it follows that |f (x)| = |f (xτ )| for all f ∈ O(Xη )∗ and, therefore, the kernel of the  ∗ : f → |f || homomorphism O(Xη ) → M S(X) coincides with the set of the functions X f ∈ O(Xη ) with |f (x)| = 1 for all x ∈ Xη . But from [Ber4, Proposition 1.4] it follows that the latter set coincides with O(X)∗ .   6.1.5 Corollary. If X is quasi-compact and connected, O(Xη )∗ /(k ∗ O(X)∗ ) is a finitely generated torsion free group.  ∗ /|k ∗ |. If {Yj }j ∈J Proof. By Corollary 6.1.4, the group considered is embedded in M X  ∗ /|k ∗ | is embedded in the is a finite étale covering of X with connected Yj ’s, then M X  ∗ /|k ∗ |. We may therefore assume that X = Xl for a strictly direct product of M Yj pluri-stable fibration X over k  of length l with affine Xi ’s and for which D(X) is a  ∗ is generated standard polysimplex [n]r . In this case one can easily show that M X (l) (l)  ∗ . Since M  ∗ = |k ∗ |, the ∗ and the coordinate functions tj ν with rj ∈ M by M Xl−1 Xl−1 X0 required statement easily follows.  

Smooth p-adic analytic spaces are locally contractible. II

349

6.1.6 Remark. To represent the above results in a functorial form, let us intro sr over the category dual to the category of duce as follows a fibered category PL non-Archimedean fields. Its objects are triples (k, X, MX ) consisting of a nonArchimedean field k, a piecewise RZk + -linear space X, and a semiring MX of piecewise RZk + -linear functions on X. Morphisms (k , X , MX ) → (k, X, MX ) are pairs con

sisting of an isometric embedding k → k and a piecewise RZk + -linear map X → X  sr that takes functions from MX to functions from MX . Let also PL G be the category with the same family of objects but with those of the above morphisms for which the map X → X is a G-local immersion of piecewise RZk + -linear spaces. Then the correspondence X  → (S(X), MX ) gives rise to functors between fibered categories pl ét → PL  sr  sr  sr Pstnd G , Pstnd → PL and Pstsnd → PL .

6.2 The image of an analytic space in the skeleton Recall that a strictly k-analytic space X is said to be quasi-algebraic if every point of X has a neighborhood of the form V1 ∪ · · · ∪ Vn , where each Vi is a strictly affinoid subdomain of X isomorphic to an affinoid domain in the analytification of a scheme of finite type over k. Recall also that a morphism of k-analytic spaces is said to be compact if it induces a proper map between the underlying topological spaces. 6.2.1 Theorem. Let X be a strongly nondegenerate pluri-stable formal scheme over k  , τ the retraction map Xη → S(X), and Y a quasi-algebraic strictly k-analytic space. Then for any compact morphism ϕ : Y → Xη the image τ (ϕ(Y )) is a piecewise RZk + -linear closed subspace of S(X) of dimension at most dim(Y ). Proof. It suffices to consider the case when the formal scheme X is affine and Y is a strictly affinoid domain in Zan , where Z is an integral affine scheme of finite type over k. Replacing k by the separable closure of k in O(Z), we may assume that Z is geometrically irreducible. By [Ber7, Lemma 9.4], there is an open embedding of Z in Yη , where Y is an integral scheme proper finitely presented and flat over k  , /W )η , where π is and an open subscheme W of Ys such that Y = π −1 (W ) = (Y an  the reduction map Yη = Yη → Ys . Since Z is geometrically irreducible, then so is Yη . By de Jong’s results [deJ] (in the form of [Ber7, Lemma 9.2]), there exist a fl−1

f1

finite normal extension k of k and a poly-stable fibration Y = (Yl → · · · → Y1 ) over k  , where all morphisms fi are projective of dimension one and have smooth geometrically irreducible generic fibers, and a dominant morphism Yl → Y that induces a proper generically finite morphism Yl,η → Yη . Notice that, since the morphisms fi have smooth geometrically irreducible generic fibers, the poly-stable  is nondegenerate. fibration Y

350

Vladimir G. Berkovich

, Y the formal completion of Y along Let W be the preimage of W in Yl,s l and Y = Yη . The morphism ϕ gives rise to a surjective generically finite morphism of strictly k-analytic spaces Y → Y . We claim that the induced morphism Y → Xη comes from a unique morphism of formal schemes ϕ : Y → X. Indeed, let X = Spf(A), and let Y = Spf(B) be an open affine subscheme of Y . The morphism of strictly k-affinoid spaces Y η → Xη is defined by a homomorphism of strictly k-affinoid algebras A = A ⊗k  k → B = B ⊗k  k . By [Ber7, Proposition ∼ ∼ 1.4], one has A → A and B → B  . It follows that the homomorphism A → B defines a unique homomorphism A → B which, in its turn, defines a morphism of affine formal schemes Y → X that induces the morphism Y η → Xη we started from. Thus, we have ϕ(Y ) = ϕη (Y η ). By Corollary 4.4.2, the image of ϕη (Y η ) under the retraction map τ : Xη → S(X) coincides with the image of the skeleton S(Y ) under the map Sϕ : S(Y ) → S(X). But, by Corollary 6.1.3, the latter map is piecewise RZk + -linear. Hence, the image of S(Y ) under Sϕ is a piecewise RZk + -linear closed   subspace of S(X) of dimension at most dim(Y ) = dim(Y ).

W ,

6.2.2 Corollary. Let Y be a compact quasi-algebraic strictly k-analytic space, and f1 , . . . , fn invertible analytic functions on Y . Then the image of Y under the map Y  → (R∗+ )n : y  → (|f1 (y)|, . . . , |fn (y)|) is a |k ∗ |Z+ -polyhedron in (R∗+ )n of dimension at most dim(Y ). Proof. Since Y is compact, we can multiply all of the functions by an element of k ∗ so that the image is contained in the set S = {t ∈ (R∗+ )n | |a| ≤ |ti | ≤ 1 for all 1 ≤ i ≤ n} with a ∈ k ∗ . Let X be the direct product of n copies of the affine formal scheme Spf(k  {u, v}/(uv − a)). It is a strongly nondegenerate poly-stable formal scheme. The projection of Xη to the coordinate v of each of the affine formal schemes identifies Xη with the poly-annulus {x ∈ An | |a| ≤ |Ti (x)| ≤ 1 for all 1 ≤ i ≤ n}, and the functions f1 , . . . , fn give rise to a morphism of strictly k-analytic spaces ϕ : Y → Xη . Furthermore, the continuous map (A1 \{0})n → (R∗+ )n : x  → (|T1 (x)|, . . . , |Tn (x)|) identifies the skeleton S(X) with the set S, and gives rise to the retraction map τ : Xη → S(X) = S. Thus, the map from the statement of the corollary coincides the composition τ  ϕ : Y → S(X) = S and, by Theorem 6.2.1,   its image is a RZk + -polyhedron in S. The following is a consequence of Corollary 6.1.5 and the proof of Theorem 6.2.1. For an analytic space Y , we set O(Y )1 = {f ∈ O(Y ) | |f (y)| = 1 for all y ∈ Y }. 6.2.3 Corollary. If a quasi-algebraic strictly k-analytic space Y is compact and connected, then the group O(Y )∗ /(k ∗ O(Y )1 ) is finitely generated.

Smooth p-adic analytic spaces are locally contractible. II

351

Proof. As in the proof of Theorem 6.2.1, one can apply de Jong’s results to show that there is a finite surjective family of morphisms Yi → Y , where each Yi is the generic fiber Xiη of a connected pluri-stable formal scheme Xi over ki , where ki is a finite extension of k. Then the group considered is embedded in the direct product of the groups O(Yi )∗ /(k ∗ O(Yi )1 ). Since the groups ki∗ /(k ∗ ki1 ) are finite, the required statement follows from Corollary 6.1.5.  

6.3 Continuity of the embedding S(X) → Xη in the Grothendieck topology Let X be a nondegenerate pluri-stable formal scheme over k  . The piecewise RZk + linear structure on the skeleton S(X) provides it with a Grothendieck topology formed by piecewise RZk + -linear subspaces. Recall (see [Ber2, §1.3]) that Xη is also provided with a Grothendieck topology formed by strictly analytic subdomains. 6.3.1 Theorem. For any strictly analytic subdomain V ⊂ Xη , the intersection V ∩ S(X) is a piecewise RZk + -linear subspace of S(X) and, for any f ∈ O (V ), the restriction of the function |f | to V ∩ S(X) is piecewise |k ∗ |Z+ -linear. In particular, the canonical embedding S(X) → Xη is continuous with respect to the Grothendieck topologies of S(X) and Xη . Proof. It suffices to consider the case when X = Spf(A) is affine and connected. By Gerritzen–Grauert Theorem ([BGR, 7.3.5/2]), a basis of the Grothendieck topology on a strictly k-affinoid space is formed by rational strictly affinoid domains, and so we may assume that V is such a domain. This means that there are functions f1 , . . . , fn , g ∈ A = A⊗k  k without common zeros on Xη such that V = {x ∈ Xl,η | |fi (x)| ≤ |g(x)| for all 1 ≤ i ≤ n}. Multiplying all of the above functions by an element of k ∗ , we may assume that f1 , . . . , fn , g ∈ A. Since any function on S(X) of the form x  → |f (x)| with f ∈ A\{0} is piecewise RZ+ -linear, it follows that V ∩ S(X) is a piecewise RZk + -linear subspace of S(X). Furthermore, let f ∈ O (V ). Then ε = min{|f (x)| | x ∈ V ∩ S(X)} > 0,  and one can find an element h ∈ A and an integer n ≥ 0 such that (f − ghn )(x) < ε |h| for all x ∈ V and, therefore, the restrictions of the functions |f | and |g| n to V ∩ S(X) ∗ coincide. The latter function is evidently piecewise |k |Z -linear. That it is in fact |k ∗ |Z+ -linear follows from Remark 3.5.8(ii).  

352

Vladimir G. Berkovich

6.4 Continuity of the retraction map τ : Xη → S(X) in the Grothendieck topology Let X be a nondegenerate pluri-stable formal scheme over k  . We choose a nondegenfl−1

f1

erate poly-stable fibration X = (Xl → · · · → X1 ) over k  of length l with Xl = X and denote by τ = τX the corresponding retraction map Xη → S(X). 6.4.1 Theorem. For any piecewise RZk + -linear subspace E ⊂ S(X), τ −1 (E) is a strictly analytic subdomain of Xη . In particular, the retraction map τ is continuous with respect to the Grothendieck topologies of S(X) and Xη . Assume that the above X possesses the following properties: (1) for every 1 ≤ i ≤ l, Xi = Spf(Ai ) is affine; (2) D(X) is a standard polysimplex [n]r , and D(X≤i ) are the standard polysimplices [n≤i ]r ≤i for all 1 ≤ i ≤ l; n≤i

(3) the maps Ai \{0} → Mr ≤i : f  → θi∗ (|f |), are surjective for all 1 ≤ i ≤ l; (4) for every 1 ≤ i ≤ l, the morphism fi−1 : Xi → Xi−1 goes through an étale morphism Xi → Xi−1 (n(i) , a (i) , mi ). n≤i

In what follows we identify S(X≤i ) = S(Xi ) with r ≤i . Furthermore, we introduce as follows a positive integer ν(n). If l = 1, then ν(n) = 1. If l ≥ 2, then ν(n) = ν(n≤l−1 ) · µ(n(l) ) where, for n = (n0 , . . . , np ), µ(n) is the least common multiple of the integers 1, 2, . . . , max0≤i≤p {ni } + 1. 6.4.2 Lemma. following data:

n

In the above situation, for every element α ∈ Mr there exist the n

(a) a finite covering of S(X) = r by RZk + -polyhedra {Ei }i∈I ; (b) for every i ∈ I , a finite covering of the preimage τ −1 (Ei ) by strictly analytic domains {Vij }j ∈Ji with τ (Vij ) = Vij ∩ S(X); (c) for every i ∈ I and j ∈ Ji , functions fij , gij ∈ Al such that for all x ∈ Vij one has |fij (xτ )| = |fij (x)|, |gij (xτ )| = |gij (x)| and   1  fij (x)  ν(n)   . α(xτ ) =  gij (x)  n

Proof. First of all, we notice that if an element α ∈ Mr possesses the properties of the lemma then, for any function f ∈ Al , the sets {x ∈ Xη | |f (x)| ≤ α(xτ )} and {x ∈ Xη | |f (x)| ≥ α(xτ )} are strictly analytic subdomains of Xη .

Smooth p-adic analytic spaces are locally contractible. II

353

We prove the lemma by induction on l. Since it is evidently true for l = 0, we assume that l ≥ 1 and that the statement is true for X≤l−1 . The morphism fl−1 : X → Xl−1 goes through an étale morphism X → X = Xl−1 (n, a, m) with ∼ n = n(l) , a = a (l) and m = ml . Since D(X) → D(X ) and the map B\{0} → fl−2

n

f1

Mr : g → θl∗ (|g|) is surjective, where X = (X → Xl−1 → · · · → X1 ) and X = Spf(B), we may assume that X = X . Of course, we assume that [n]  = [0]. n

Step 1. We may assume that the element α is a coordinate function on r . Indeed, it n suffices to show that if the lemma is true for two elements α, α ∈ Mr , then it is also true for their product α ·α and their maximum max{α, α }. Let us take the data provided by the assumption for the functions α and α , and mark the data for α with the prime sign. Then the data for the product α · α consist of the RZk + -polyhedra Ei ∩ Ei , the strictly analytic domains Vij ∩ Vi j , and the functions fij · fi j and gij · gi j . The data for the maximum max{α, α } consist of the same RZk + -polyhedra Ei ∩Ei , the strictly analytic      fi j (x)   fij (x)   subdomains of Vij ∩ Vi j , defined in it by the inequalities  gij (x)  ≥  g (x)  and i j      fij (x)   fi j (x)   gij (x)  ≤  g (x) , respectively, and the functions fij · gi j , fi j · gij and {gij · gi j }. i j

n

Step 2. By Step 1, we may assume that the element α ∈ Mr is one of the coordinate functions t0j = θl∗ (|T0j |). We denote n0 , a0 and T0j by n, a and Tj , respectively. For a point y ∈ Xη , we denote by x its image in Xl−1,η , and we denote by yτ and xτ the images of y and x in S(X) = S(X) and S(X≤l−1 ), respectively. First of all, we define the following covering of S(X) by RZk + -polyhedra which correspond to permutations σ ∈ Sn+1 : Eσ = {y ∈ S(X) | |Tσ (0) (y)| ≤ |Tσ (1) (y)| ≤ · · · ≤ |Tσ (n) (y)|} . It suffices to consider the restrictions of the coordinate functions to E, which correspond to the trivial permutation. From the description of τ , recalled in §4.4, it n −1 follows that τ (E) = i=0 Vi , where Vi consists of all points y ∈ Xη that satisfy the following three inequalities: i+2 |a(xτ )| ≤ |(Ti+1 Ti+2 . . . Tn )(y)|} ,

max {|(Tji+1 Ti+1 . . . Tn )(y)|} ≤ |a(xτ )| ,

0≤j ≤i

max {|Tj (y)|} ≤ |Ti+1 (y)| ≤ · · · ≤ |Tn (y)| .

0≤j ≤i

∗ (|a|), the first two inequalities Applying the induction hypothesis to the function θl−1 define a finite union of rational strictly affinoid subdomains of Xη , and the functions Ti+1 , . . . , Tn are invertible on each of them. It follows that the third inequality also defines a rational strictly affinoid subdomain in each of them and, therefore, Vi is a

354

Vladimir G. Berkovich

finite union of rational strictly affinoid domains. Furthermore, the description of τ implies that for y ∈ Vi one has   1   i+1 a(xτ )  |Tj (yτ )| =  (Ti+1 . . . Tn )(y)  for 0 ≤ j ≤ i, and |Tj (yτ )| = |Tj (y)| for i + 1 ≤ j ≤ n. It follows that τ (Vi ) = Vi ∩S(X) = {y ∈ S(X) | |T0 (y)| = · · · = |Ti (y)| ≤ |Ti+1 (y)| ≤ · · · ≤ |Tn (y)|}. ∗ (|a|), we get the required Applying again the induction hypothesis to the function θl−1 fact.  

Proof of Theorem 6.4.1. First of all, since the retraction map τ is proper, the statement is local in the Zariski topology. Furthermore, by Raynaud’s theorem (see [BoLü2, Corollary 5.11), given a flat morphism of strictly k-affinoid spaces ϕ : Y → X, for any strictly affinoid domain V ⊂ Y the image ϕ(V ) is a finite union of strictly affinoid subdomains of X, i.e., is a compact strictly analytic subdomain of X. It follows that the statement of the theorem is local in the étale topology and, in particular, we may assume that X is strictly poly-stable. Of course, we may assume that all Xi = Spf(Ai ) are affine. After that we can shrink X so that it satisfies the assumptions of Lemma 6.4.2. n It suffices to show that, given two elements α, α ∈ Mr , the preimage τ −1 (D) of D = {x ∈ S(X) | α(x) ≤ α (x)} is a strictly analytic subdomain of Xη . Let us take the data provided by Lemma 6.4.2 for the functions α and α , and mark the data for α with the prime sign. It suffices to show that, for every quadruple i ∈ I , j ∈ Ji , i ∈ I and j ∈ Ji , the intersection τ −1 (D) ∩ Vij ∩ Vi j is a strictly analytic subdomains of Vij ∩ Vi j . We have       f (x)   (x) f ij j i   ≤  τ −1 (D) ∩ Vij ∩ Vi j = x ∈ Vij ∩ Vi j |   gij (x)   gi j (x)  Since all of the functions in the inequality are invertible on Vij ∩Vi j , the set considered   is a strictly analytic subdomain of Vij ∩ Vi j . 6.4.3 Corollary. The following properties of a subset E ⊂ S(X) are equivalent: (a) E is a piecewise RZk + -linear subspace of S(X); (b) τ −1 (E) is a strictly analytic subdomain of Xη .

 

The following result is a consequence of Lemma 6.4.2. Let X and X be nondegenerate pluri-stable formal schemes over k  and k  , respectively, and let ϕ : X → X be a morphism in Pstnd . We fix a nondegenerate poly-stable fibration of length l over k  with Xl = X which gives rise to a retraction map τ : Xη → S(X).

Smooth p-adic analytic spaces are locally contractible. II

6.4.4 Theorem. The map τ  ϕη : S(X ) → S(X) is piecewise If l = 1, this map is in fact |k ∗ |Z+ -linear.




355 |k ∗ |

 Q+

-linear.

k  k  , we may assume that k = k and that ϕ is a kProof. Replacing X by X⊗ morphism. Furthermore, since the statement is local in the étale topology of X and X , we may assume that X satisfies the assumptions of Lemma 6.4.2 and that X is strictly pluri-stable and small enough so that S(X ) is an RZk + -polyhedron. To prove the n

statement, it suffices to show √that if l ≥ 2 (resp. l = 1) then, for every α ∈ MX = Mr , ϕη∗ (τ ∗ (α)) is a piecewise ( |k ∗ |)Q+ -linear (resp. |k ∗ |Z+ -linear) function on S(X ). By Lemma 6.4.2, there exists a finite covering of Xη by strictly analytic domains {Vij }i∈I,j ∈Ji and, for each i ∈ I and j ∈ Ji , functions fij , gij ∈ Al such that for all x ∈ Vij one has |fij (xτ )| = |fij (x)|, |gij (xτ )| = |gij (x)| and   1  fij (x)  ν(n)   . α(xτ ) =  gij (x) 

By Theorem 6.3.1, each Eij = S(X ) ∩ ϕη−1 (Vij ) is a piecewise RZk + -linear subspace of the RZk + -polyhedron S(X ) and, by the above formula, the restriction of ϕη∗ (τ ∗ (α)) √ to Eij coincides with the restriction of the piecewise ( |k ∗ |)Q -linear function  1  ∗  (ϕ fij )(x )  ν(n)  x  →  ∗ . (ϕ gij )(x )  √ The latter function is piecewise ( |k ∗ |)Q+ -linear, by Remark 3.5.8(ii). If l = 1, then ν(n) = 1 and, therefore, it is even piecewise |k ∗ |Z+ -linear. Since S(X ) is a union of   all Eij , the required fact follows.

7 Strong local contractibility of smooth analytic spaces 7.1 Formulation of the result Let k be a non-Archimedean field with a non-trivial valuation. Recall (see [Ber7, §9]) that a k-analytic space is said to be locally embeddable to a smooth space if each point x ∈ X has an open neighborhood isomorphic to a strictly analytic domain in a smooth k-analytic space. This class includes the class of spaces smooth in the sense of [Ber2], their strictly analytic subdomains, and is contained in the class of spaces smooth in the sense of rigid geometry (i.e., rig-smooth spaces). Notice also that any rig-smooth affinoid space is locally embeddable in a smooth space. Recall also that a strong deformation retraction of a topological space X to a subset S ⊂ X is a continuous mapping  : X × [0, 1] → X such that (x, 0) = x and (x, 1) ∈ S for all x ∈ X, and (x, t) = x for all x ∈ S and t ∈ [0, 1]. We say that

356

Vladimir G. Berkovich

a subspace Y ⊂ X is preserved under  if (Y × [0, 1]) ⊂ Y . If S is a point,  is said to be a contraction of X to the point. 7.1.1 Theorem. Let X be a k-analytic space locally embeddable in a smooth space. Each point x ∈ X has a fundamental system of open neighborhoods V which possess the following properties: (a) there is a contraction  of V to a point x0 ∈ V ; (b) there is an increasing sequence of compact strictly analytic ∞domains X1 ⊂ X2 ⊂ · · · which are preserved under  and such that V = n=1 Xn ;  K has a finite number of connected (c) given a non-Archimedean field K over k, V ⊗ components, and  lifts to a contraction of each of the connected components to a point over x0 ; (d) there is a finite separable extension L of k such that, if K from (c) contains L, K → V ⊗  L induces a bijection between the sets of connected then the map V ⊗ components. Recall that [Ber7, Theorem 9.1] states that each point x ∈ X has a fundamental system of contractible open neighborhoods V . In §7.2, we recall the main construction from the proof of loc. cit.. After that, instead of using [Ber7, Theorem 8.2], we use results from §1 and §6. But before doing this, we establish a simple fact which will be used in the last step of the proof and is true without the assumption that the valuation on k is nontrivial. Let k be a finite extension of k. Then every strictly k -affinoid algebra A is evidently a strictly k-affinoid algebra, and so the strictly k -affinoid space X = M(A) can be considered as a strictly k-affinoid space, i.e., there is a canonical functor from the category of strictly k -affinoid spaces to that of strictly k-affinoid ones. From the following proposition it follows that the latter can be extended to a functor stk -An → st-k-An from the category of strictly k -analytic spaces to that of strictly k-analytic ones, and it takes strictly k -analytic domains to strictly k-analytic ones. Notice that the above functor is left adjoint to the ground field extension functor  k . st-k-An → st-k -An : X  → X ⊗ 7.1.2 Proposition. Let X be a strictly k -affinoid space. Then any strictly k -affinoid subdomain V ⊂ X is a strictly k-affinoid subdomain of X, considered as a strictly k-affinoid space. 7.1.3 Lemma. Assume that the valuation on k is trivial, and let ϕ : Y = M(B) → X = M(A) be a morphism of strictly k-affinoid spaces. Then the following are equivalent: (a) ϕ identifies Y with a strictly affinoid subdomain of X;

Smooth p-adic analytic spaces are locally contractible. II

357

(b) the induced morphism of affine schemes Y = Spec(B) → X = Spec(A) is an open immersion. ∼

Proof. (a)⇒(b) For any point y ∈ Y with [H (y) : k] < ∞, one has OX,x → OY,y , where x is the image of y in X. But the images x of x in X corresponds to a maximal X,x of OX,x by the maximal ideal of A, and OX,x coincides with the completion O ideal (see [Ber1, Theorem 3.5.1]), and the same is true for the image y of y in Y. It ∼ Y,y and, X,x → O follows that the morphism of schemes induces an isomorphism O therefore, it is an étale morphism. On the other hand, since for any bigger field K (also provided with the trivial valuation) the map Y(K) = Y (K) → X(K) = X(K) is injective, the morphism of schemes is radicial. It remains to use the fact that any étale and radicial morphism between affine schemes of finite type over a field is an open immersion. (b)⇒(a) If Y is identified with a principal open subset {x ∈ X | f (x)  = 0}, then Y is identified with the rational subdomain {x ∈ X | |f (x)| = 1}. In the general case, Y is a finite union of principal open subsets, and so Y = ni=1 Yi , where each Yi is identified with a rational subdomain of X of the above forms. From [Ber2, Remark 1.2.1] it follows that ϕ identifies Y with a strictly affinoid subdomain of X.   7.1.4 Corollary. If the valuation on k is trivial, then any strictly k-analytic space is Hausdorff. Proof. By [Ber2, Lemma 1.1.1(ii)], it suffices to show that any strictly analytic subdomain Y of a strictly k-analytic space X = M(A) is compact. From Lemma 7.1.3 it follows that Y corresponds to an open subscheme of X = Spec(A). Since the ring A is Noetherian, any open subscheme of X is quasicompact and, therefore, Y is compact.   Proof of Proposition 7.1.2. If the valuation on k is trivial, the statement follows from Lemma 7.1.3. Thus, assume that the valuation on k is nontrivial, and let X = M(A) and V = M(AV ). The statement is trivial if V is a rational domain since it is defined by the inequalities |fi (x)| ≤ |g(x)|, where f1 , . . . , fn , g are elements of A that generate the unit ideal. Assume V is arbitrary. By Gerritzen–Grauert Theorem ([BGR, 6.3.5/2]), it is a finite union ni=1 Vi of rational strictly affinoid subdomains of X. By Tate’s Acyclicity Theorem, there an isomorphism of commutative Banach  ∼ →  k-algebras AV → Ker( i AVi → i,j AVi ∩Vj ). Since AV is strictly k-affinoid and the canonical map V → M(AV ) is a bijection, V is a strictly k-affinoid subdomain of X (see [Ber2, Remark 1.2.1]).  

7.2 Proof: Step 1 We follow the proof of [Ber7, Theorem 9.1]. It is done by induction on the dimension of X at x. First of all, we may assume that X is a strictly analytic domain in Xan ,

358

Vladimir G. Berkovich

where X = Spec(A) is a smooth irreducible affine scheme over k. Let x be the image of the point x in X. There are the following two cases: (α) x is not the generic point of X; (β) x is the generic point of X. Case (α). As in loc. cit., Steps 1 and 2 of Case (a), one reduces the situation to the case when the field k(x) is separable over k and, after that, one shows that there is a sufficiently small open neighborhood of x isomorphic to Y × D(0; r) with x = (y, 0), where Y is a strictly analytic domain in the analytification of a smooth scheme over k and D(0; r) is the open disc with center at zero and of radius r > 0. Thus, we may assume that X = Y × D(0; r), and it suffices to show that the point x = (y, 0) has an open neighborhood with the properties (a)–(d). In loc. cit., Step 3, one constructs a continuous mapping X × [0, 1] → X : (x , t) → xt , which is a retraction of X to a closed subset homeomorphic to Y ×[0, r[ and such that |T (xt )| = |T (x )| for all x ∈ X and t ∈ [0, 1]. Thus, if V is an open neighborhood of the point y and Y1 ⊂ Y2 ⊂ · · · is an increasing sequence of compact strictly analytic domains in V possessing the properties (a)–(d), then the open neighborhood V × D(0, r) of the point x and the sequence of compact strictly analytic domains Y1 × E(0; r1 ) ⊂ Y2 × E(0, r2 ) ⊂ · · · possess √ the same properties, where r1 < r2 < · · · is an increasing sequence of numbers from |k ∗ | with ri → r as i → ∞, and E(0; r) is the closed disc of radius r. Case (β). As in loc. cit., Case (b), we may assume that X is compact and X is geometrically irreducible, and it suffices to show that, given a rational strictly affinoid neighborhood W of x in Xan , there exists an open neighborhood of x in X which possesses the properties (a)–(d) and is contained in W ∩ X. By loc. cit., Lemma 9.4, there is an open embedding of X in Yη , where Y is an integral scheme proper finitely presented and flat over k  , open subschemes Z and W of Ys , and a closed subscheme V of Ys such that (1) X = π −1 (Z), W = π −1 (W ) and π(x) ∈ V; (2) V ⊂ W ; (3) V and Ys \Z are unions of irreducible components of Ys . By J. de Jong results [deJ] (in the form of [Ber7, Lemma 9.2]), there exist a finite normal extension k of k, a poly-stable fibration Y of length l over k  such that all morphisms fi : Yi+1 → Yi are projective of dimension one and have smooth geometrically irreducible generic fibers, an action of a finite group G on Y over k  , and a dominant G-equivariant morphism ϕ : Yl → Y that induces a proper generically )G is purely inseparable → Yη and such that the field R(Yl,η finite morphism Yl,η  over k  is nondegenerate. over R(Y). Notice that the poly-stable fibration Y , respectively. Then Let Z , W and V be the preimages of Z, W and V in Yl,s \Z are unions of irreducible components of Y and V ⊂ W . For V and Yl,s l,s

Smooth p-adic analytic spaces are locally contractible. II

359

X = π −1 (Z ) and W = π −1 (W ), one has X = ϕη−1 (X) and W = ϕη−1 (W ). Moreover, π −1 (V ) ∩ X is an open subset of X contained in W ∩ X . By the construction, we can find a nonempty open affine subscheme U ⊂ X such that the morphism U := ϕη−1 (U) → U is finite and the finite morphism G\U → U is radicial. By the assumption (β), the point x is contained in Uan . It follows that the set U := π −1 (V) ∩ X ∩ Uan is an open neighborhood of x in X contained in W ∩ X. The set U := π −1 (V ) ∩ X ∩ U an is open in X and dense Zariski open in π −1 (V ) ∩ X = π −1 (V ∩ Z ), and the radicial morphism G\U → U induces ∼ \Z are unions of irreducible a homeomorphism G\U → U . Since V and Yl,s . components of Yl,s , it follows that V ∩ Z is a strata subset of Yl,s By [Ber7, Theorems 8.1], there is a G-equivariant strong deformation retraction an   : Y an l,η × [0, 1] → Y l,η : (y , t)  → yt to the skeleton S = S(Y ) of the formal  completion of Y along the closed fiber. (Notice that Y an l,η = Y .) Furthermore, l,η

 induces a G-equivariant strong deformation retraction of the set π −1 (V ∩ Z ) to S is contained in the its intersection  S with the skeleton S of Y . This intersection  Zariski open subset U of π −1 (V ∩ Z ), and U is preserved under  . Thus,  induces a strong deformation retraction  : Uan × [0, 1] → Uan to the closed subset S = G\ S . S = G\S , as well as a strong deformation retraction of U to 

7.3 Proof: Step 2 We can shrink U so that the finite morphisms U → G\U and G\U → U are flat. In this case, the induced morphisms between the analytifications are also flat (see [Ber2, Proposition 3.2.10), and M. Raynaud’s theorem (see [BoLü2, Corollary 5.11]) implies that the image of any strictly analytic subdomain of U an and G\U an is a strictly analytic domain in G\U an and Uan , respectively. In particular, we can , and so we may assume that there are isomorphisms replace Y by the quotient G\Yl,η ∼

∼

of schemes G\U → U and of analytic spaces G\U → U , and we may assume that k G = k and, in particular, that k is a Galois extension of k. We now claim that there exists a sequence of compact strictly analytic domains Y1 ⊂ Y2 ⊂ · · · in Yan which are preserved under  and such that Uan = ∞ n=1 Yn . of Y along its closed  is the generic fiber of the formal completion Y Indeed, Y an l,η l l fiber. The latter formal scheme is a finite union of G-invariant open affine subschemes Yi . If we can find an exhausting sequence of G-invariant compact strictly analytic domains Y1 i ⊂ Y2 i ⊂ · · · in Yiη ∩ U an which are preserved under  and for which

i i the quotients i Yn = G\Yn exist, then the sequence of the compact analytic domains Yn = i Yn possesses the required properties. It suffices therefore to consider an  . open affine formal subscheme Y of Y l Let Y = Spf(A ). The generic fiber Y η is the strictly k-affinoid space M(A ), where A = A ⊗k  k . The complement of U an in Y η is defined by a finite number

360

Vladimir G. Berkovich

of equations fi (x ) = 0, 1 ≤ i ≤ m, with fi ∈ A G . We take a decreasing sequence of positive numbers r1 > r2 > · · · in R k with limn→∞ rn = 0, and consider the G-invariant strictly affinoid domains Yni = {x ∈ Y η | |fi (y )| ≥ ri }. Since |f (yt )| ≥ |f (y )| for all elements f ∈ A and all t ∈ [0, 1] (see [Ber7, Theorem 8.1(iii)]), Yni are preserved under  . It follows that the compact strictly analytic domains i and G-invariant, and the quotients G\Y exist. Yn = m n i=1 Yn are preserved under  ∞ One also has Y1 ⊂ Y2 ⊂ · · · and n=1 Yn = Y η ∩ U an .

7.4 Proof: Step 3  = D(Z ), where Z  ) and D Consider the R k -colored polysimplicial sets D = D(Y   is the poly-stable fibration (Z → Y l−1 → · · · → Y1 ) over k and Z is the formal  along the open subset Z of Y . By §4.3, there are canonical completion of Y l l,s ∼ ∼  | →  ) and |D S(Z ). Setting D = G\D and homeomorphisms |D | → S = S(Y  (Notice  = G\D  , we can identify S with |D| and  S with an open subset of |D|. D  is a closed subset of |D|.) that |D| Let x0 be the image of the point x under the retraction map τ : Uan → S induced by . By Proposition 1.4.1, one can find a compact RZk + -piecewise linear neighborhood √  which is isomorphic to an ( |k ∗ |)Q -polyhedron in an affine E of the point x0 in |D|, space (R∗+ )d . For 0 < r < 1, let B(x0 , r) denote the open box {y ∈ (R∗+ )d | (y) r < | ttii(x | < r −1 , 1 ≤ i ≤ d}. One can find 0 < r0 < 1 such that, for every 0)  r0 ≤ r < 1, the open set E(r) = E ∩ B(x0 , r) is contained in  S and possesses the  property that, for each point y ∈ E(r), the interval {x0t · y 1−t }t∈[0,1] , connecting the  points x0 and y, is contained in E(r). Let us fix such r, and let  be the contraction    to the point x0 . Furthermore, let E(r) × [0, 1] → E(r) : (y, t) → x0t · y 1−t of E(r) √ 1 ≥ r1 > r2 > · · · > r be a sequence of numbers from |k ∗ | with limn→∞ rn = r. (y)  Then the RZk + -polyhedrons En = {y ∈ E(r) | rn ≤ | ttii(x | ≤ rn−1 , 1 ≤ i ≤ d} 0) ∞   ⊂  S, the set V (r) = are preserved under  and E(r) = n=1 En . Since E(r)  ∩ Uan is an open neighborhood of the point x in U . τ −1 (E(r)) We claim that, for every r0 ≤ r < 1, V (r) possesses the properties (a) and (b), and that one can find r0 ≤ r0 ≤ 1 such that, for every r0 ≤ r < 1, V (r) also possesses the properties (c) and (d).   to E(r), (a) The composition of the strong deformation retraction of τ −1 (E(r))  induced by , and of the contraction  of E(r) to x0 , gives rise to a contraction of V (r) to the point x0 . (b) We claim that Zn = τ −1 (En ) is a strictly analytic subdomain of Uan . Indeed, let En be the preimage of En in S . By Theorem 6.4.1, Zn = τ −1 (En ) is a strictly k -analytic subdomain of U an , where τ is the retraction map U an → S induced by  . Proposition 7.1.2 implies that Zn is a strictly k-analytic subdomain of U an considered as a strictly k-analytic space. Since Zn is the image of Zn under the flat

Smooth p-adic analytic spaces are locally contractible. II

361

morphism U an → Uan , the claim follows from M. Raynaud’s theorem. Thus, the intersection Xn = Yn ∩ Zn , where Yn is constructed in §7.2, is a compact strictly analytic subdomain of V (r), preserved under the contraction from ∞ ∞ and it is evidently an =  X since U Y and E(r) = E (a). One also has V (r) = ∞ n=1 n n=1 n n=1 n . To establish the properties (c) and (d), we need the following additional fact. Given a G-local immersion of compact piecewise R k -linear spaces g : T → S, one can find r0 ≤ r0 < 1 such that, for every r0 ≤ r < 1, the contraction  of the  to x0 lifts to a contraction of each of the connected component of g −1 (E(r))  set E(r) to a point above x0 . Indeed, let y1 , . . . , yn be the preimages of the point x0 in T . We can find pairwise disjoint neighborhoods D1 , . . . , Dn of the points y 1 , . . . , yn , i respectively, with the following property: for every 1 ≤ i ≤ n, Di = m j =1 Dij ,

where each Dij is a compact piecewise R k -linear subspace of T that contains the point yi and such that g induces an isomorphism of Dij with an R k -polyhedron Eij in E. One can find r0 ≤ r0 < 1 such that, for every 1 ≤ i ≤ n, 1 ≤ j ≤ mi and every  ), the interval, connecting the points x0 and y, is contained in point y ∈ Eij ∩ E(r 0   and  for all Eij ∩ E(r0 ). This construction guarantees the required property of E(r) r0 ≤ r < 1. K is a strictly analytic (c) and (d). For a non-Archimedean field K over k, V (r)⊗  ⊗ K. The latter is a quotient of Y  K under the action of the group G. domain in Yηan ⊗ l,η Since k is a finite Galois extension of k, the tensor product k ⊗k K is isomorphic to a direct product of m copies of a finite Galois extension K of K with m·[K : K] = [k : k]. This isomorphism gives rise to an action of G on the direct product and, therefore, to an action of G on the corresponding disjoint union YK i of m copies of each of the     formal schemes Yi ⊗k K , 1 ≤ i ≤ l. Thus, we have a nondegenerate poly-stable K  fibration YK = (YK l → · · · → Y1 ) over K provided with an action of the group ∼ an  G over K  , and an isomorphism of strictly K-analytic spaces G\YK l,η → Yη ⊗K.

be the R K -colored polysimplicial set associated with YK , and S the Let DK K ∼

| → S . It gives rise skeleton of YK . There is a G-equivariant homeomorphism |DK K ∼

, where D = G\D . Furthermore, the to a homeomorphism |DK | → SK = G\SK K K gives rise to a strong G-equivariant strong deformation retraction  K of YK to S l,η K K to SK compatible with the strong deformation deformation retraction K of Yηan ⊗ retraction  of Yηan to S. If gK denotes the canonical G-local immersion of compact piecewise R K -linear spaces SK → S, then K induces a strong deformation retrac−1  K to gK (E(r)). It follows that the number of connected components of tion of V ⊗ K is finite. V⊗ Furthermore, we can find finite unramified extensions L1 , . . . , Ln of k such that for any K, as above, there is an embedding of some Li into K which induces an ∼ ⊗ K ⊗ L  ) →  str(Yl,s isomorphism of partially ordered sets str(Yl,s   k k i ) and, there

∼

→ D and fore, it induces isomorphisms of R K -colored polysimplicial sets DK Li

362

Vladimir G. Berkovich ∼

DK → DLi . The composition of the morphism inverse to the latter with the canonical surjection DK → DK gives rise to a surjective morphism of R K -colored polysimplicial sets DLi → DK . Since the polysimplicial sets DLi are finite, it follows that there are only finite many possible polysimplicial sets DK and all of them are R k -colored (because |L∗i | = |k ∗ |). Let D1 , . . . , Dµ be these R k -colored polysimplicial sets. We apply the above µadditional fact to the G-local immersion of compact piecewise R k -linear spaces i=1 |Di | → S. It follows that there is a number r0 ≤ r0 < 1 such  to the point x0 lifts that for any K and any r0 ≤ r < 1 the contraction  of E(r) −1  to a contraction of each of the connected component of gK (E(r)) to a point above x0 . The composition of K with such a lifting gives rise to a contraction of each K to a point above x0 , i.e., (c) is true. connected component of V ⊗ Finally, let L be a finite unramified extension of k such that all of the strata of the ⊗ L  are geometrically irreducible over L.  Then for any K as above scheme Yl,s  k ∼

→ D with L ⊂ K there are isomorphisms of R K -colored polysimplicial sets DK L ∼ and DK → DL . (Notice that in this case K = K since k ⊂ K.) It follows that the canonical map SK → SL is a homeomorphism and, therefore, it induces a ∼ −1   (E(r)) → gL−1 (E(r)). This implies (d).   homeomorphism gK

8 Cohomology with coefficients in the sheaf of constant functions 8.1 The sheaf of constant functions Let k be a non-Archimedean field with a non-trivial valuation. Recall that in every strictly k-analytic space X the subset X0 = {x ∈ X | [H (X) : k] < ∞} is dense. For a reduced strictly k-analytic space X, we denote by c(X) the set of all analytic functions f ∈ O(X) such that the image of each connected component of X under the morphism f : X → A1 is a point. (Since such a point should lie in (A1 )0 , a function f ∈ O(X) is contained in c(X) if and only if the restriction of f to each connected component of X is algebraic over k.) The correspondence U → c(U ) is a sheaf of k-algebras in the étale topology of X (as well as in the G-topology of X), denoted by cX . Of course, if k is algebraically closed, it is the constant sheaf kX associated with k. 8.1.1 Lemma. Assume that X is connected. Then (i) c(X) is a field finite over k; (ii) assume that the algebra of any connected strictly affinoid subdomain of X has no zero divisors (e.x., X is normal ); if the restriction of a function f ∈ O(X) to a non-empty open subset U is in c(U), then f ∈ c(X).

Smooth p-adic analytic spaces are locally contractible. II

363

Proof. (i) Let f be a nonzero element of c(X). Then the image of X under the morphism f : X → A1 is a nonzero point from (A1 )0 and, therefore, P (f ) = 0 for a monic polynomial P (T ) = T n + a1 T n−1 + · · · + an ∈ k[T ] with an  = 0. It follows that f is invertible in c(X), i.e., c(X) is a field. It is embedded in the field H (x) of every point x ∈ X. Since there is a point x with [H (x) : k] < ∞, c(X) is finite over k. (ii) We may assume that X = M(A) is strictly k-affinoid, and we can find a nonzero polynomial P (T ) over k with P (f |U ) = 0, i.e., for the element g = P (f ) ∈ A one has g|U = 0. It follows that the image of g in the local ring OX,x of any point x ∈ U is zero. This local ring is faithfully flat over the local ring OX,x of the affine scheme X = Spec(A) at the image x of x in X (see [Ber2, 2.1.4]). It follows that the image of g in the localization of A with respect to the prime ideal of the point x is zero and, therefore, g is a zero divisor in A. The assumption implies that g = 0.   A strictly k-analytic space X is said to be geometrically reduced (resp. geometri cally normal) if the strictly  k a -analytic space X = X ⊗ k a is reduced (resp. normal). For example, the generic fiber of Xη of a nondegenerate pluri-stable formal scheme X over k  is geometrically normal. 8.1.2 Lemma. Let X be a geometrically reduced strictly k-analytic space. Then (i) the set of points x ∈ X0 such that X is smooth at x and the field H(x) is separable over k is dense in X; (ii) if x is a point from X0 with the properties (i), then there is an open neighborhood of x isomorphic to an open polydisc in an affine space over H(x). Proof. (i) We can replace X by an open neighborhood of any point from X0 in the interior of X so that it may be assumed to be closed. Since the field  k a is algebraically closed and the regular locus of X is non-empty, from [Ber5, Theorem 5.2] it follows that the smooth locus of X is dense in X. Replacing X by the smooth locus, we may assume that X is smooth. We then can shrink it and assume that there is an étale morphism ϕ : X → An . For each point x ∈ X, H(x) is a finite separable extension of H (ϕ(x)). We may therefore assume that X = An . In this case the statement follows from the well known fact that the set of all elements of an algebraic closure k a of k, which are separable over k, is dense in k a (see [BGR, 3.4.1/6]). (ii) As in (i), we can shrink X and assume that there is an étale morphism X → H (x) → AnH (x) . The point x An : x → y. It induces an étale morphism X = X ⊗ has an H(x)-rational preimage x in X and, therefore, the étale morphism X → X is a local isomorphism at the point x . Thus, shrinking X, we get an étale morphism ∼ X → AnH (x) : x → y with H(y ) → H (x). It follows that the latter morphism is a local isomorphism at the point x.   8.1.3 Corollary. Let X be a geometrically reduced strictly k-analytic space. Then (i) if X is connected, c(X) is a finite separable extension of k;

364

Vladimir G. Berkovich

(ii) the stalk cX,x of cX at a point x ∈ X coincides with the algebraic separable closure of k in H(x); s associated with (iii) the pullback of the étale sheaf cX to X is the constant sheaf kX s a the separable closure k of k in k .

Proof. (i) trivially follows from Lemma 8.1.2, and it implies that the image of cX,x in H(x) is contained in the algebraic separable closure. Let k be a finite separable k → subextension of k in H(x), and consider the canonical étale morphism X = X ⊗ X. The canonical character H(x) ⊗ k → H(x) defines a point x over x with ∼ H (x) → H(x ). From [Ber2, Proposition 3.4.2] it follows that the above étale morphism is a local isomorphism at the point x and, therefore, k is contained in the image of cX,x in H(x). The statement (iii) is already trivial.   8.1.4 Lemma. The following properties of a geometrically reduced strictly kanalytic space X are equivalent: (a) c(X) = k; k is connected for every finite extension k of k; (b) X⊗ (c) X is connected. k is not connected. If k is the Proof. (a)⇒(b) Assume that there is k such that X ⊗ k → X⊗ k maximal subextension of k separable over k, then the canonical map X ⊗ is a homeomorphism and, therefore, we may assume that k = k . We may also k . assume that k is a Galois extension of k. Let X be a connected component of X ⊗ The morphism X → X is a finite étale Galois covering of X of degree less than [k : k]. If G is the Galois group of this covering, then c(X) = c(X )G ⊃ k G . The latter field is bigger than k, and this contradicts the assumption (a). (b)⇒(c) Assume that X is a disjoint union of non-empty open subsets U1 and U2 . k Since for every compact analytic subdomain Y ⊂ X the canonical map Y → lim Y ⊗ ←− is a homeomorphism, where the inverse limit is taken over finite separable extensions k are open and k of k in k a , it follows that the images of U1 and U2 in every X ⊗ k are surjective. But we can find k such closed and, therefore, the maps Ui → X⊗ k has a k -rational point. Since the preimage of the latter in X is a one point that X⊗ subset, we get a contradiction. (c)⇒(a) From (c) it follows that X is connected and, in particular, c(X) is a finite ∼ c(X) (c(X) ⊗k ka ). The latter tensor ka → X⊗ separable extension of k. One has X ⊗ product is a direct product of [c(X) : k] copies of ka , and so the connectedness of ka implies that c(X) = k.   X⊗ 8.1.5 Corollary. Let Y be a strictly analytic domain in a geometrically reduced strictly k-analytic space X. Then the sheaf cY is canonically isomorphic to the pullback of the sheaf cX on Y .

Smooth p-adic analytic spaces are locally contractible. II

365

Proof. It suffices to show that, given a compact strictly analytic domain Y in X, there ∼ exists a compact neighborhood U of Y with c(U ) → c(Y ). For this we may assume ∼ that Y and X are connected. Furthermore, we may shrink X so that c(X) → c(U ) for any connected compact neighborhood U of Y in X. Finally, we may assume that c(X) = k (see Remark 8.1.7). We claim that in this case c(Y ) = k. Indeed, if k is not this is not true, we can find a finite separable extension k of k such that Y ⊗  connected. Let {Yi }1≤i≤n be the connected components of Y ⊗k , and let {Ui }1≤i≤n be their pairwise disjoint compact neighborhoods. Then there exists aconnected compact k is contained in ni=1 Ui . It follows neighborhood U of Y whose preimage in X ⊗ k is not connected. Since c(U ) = k, this contradicts Lemma 8.1.4. that U ⊗   8.1.6 Lemma. Assume that the characteristic of k is zero, and let X be a reduced strictly k-analytic space that satisfies the assumption of Lemma 8.1.1 (ii). Then cX = d

Ker(OX → 1X ). Proof. We may assume that X is connected. Let f be a function from O(X) with df = 0. Any strictly affinoid subdomain V ⊂ X is regular at a dense open subset V ⊂ V and, therefore, V is smooth at each point from V ∩ X0 (see [Ber5, 5.2]). By Lemma 8.1.2, there exists a non-empty open subset W ⊂ V isomorphic to an open polydisc in an affine space over k , a finite extension of k. It follows that f |W ∈ c(W ), and Lemma 8.1.1 (ii) implies that f ∈ c(X).   8.1.7 Remark. Let X = M(A) be a strictly k-affinoid space, and V a strictly k-affinoid subdomain of X. Assume that A contains a finite extension k of k. Then X and V can be considered as strictly k -affinoid spaces, and it is easy to see (in comparison to Proposition 7.1.2) that V is a strictly k -affinoid subdomain of X.

8.2 Local cohomological triviality of the sheaf cX 8.2.1 Theorem. Assume that the characteristic of k is zero, and let X be a k-analytic space locally embeddable in a smooth space. Then each point of X has a fundamental system of open neighborhoods V with H n (V , cX ) = 0 for all n ≥ 1. Since the characteristic of k is zero, the stalks of cX are uniquely divisible abelian groups, and since the Galois cohomology of such a group is trivial, [Ber2, Proposition 4.2.4] implies that, for any reduced strictly k-analytic space X, the étale cohomology groups H n (X, cX ) of X coincide with the cohomology groups H n (|X|, cX ) of the underlying topological space |X|. Proof. By Theorem 7.1.1, each point of X has a fundamental system of open neighborhoods V with the properties (a)–(d). We claim that, for such V , one has H n (V , cX ) = 0, n ≥ 1.

366

Vladimir G. Berkovich

Let X1 ⊂ X2 ⊂ · · · be the increasing sequence of compact strictly analytic subdomains of V from the property (b). By Corollary 8.1.5, the pullback of the étale sheaf cX to Xm coincides with cXm . From [Ber2, Lemma 6.3.12] it follows that to prove the claim it suffices to show that H n (Xm , cXm ) = 0 for all n ≥ 1. By Corollary 8.1.3(ii), the pullback of the étale sheaf cXm to Xm is the constant s . Since X is compact, there is a Hochschield–Serre spectral sequence sheaf kX m m

p,q

E2

= H p (G, H q (Xm , k s )) ⇒ H p+q (Xm , cXm ) ,

where G is the Galois group of k s over k. The étale cohomology groups H q (Xm , k s ) coincide with H q (|Xm |, k s ). Since all of the connected components of Xm are conp,q tractible, it follows that H q (Xm , k s ) = 0 and, therefore, E2 = 0 for all q ≥ 1. Furthermore, since H p (G, k s ) = 0 for all p ≥ 1, the spectral sequence implies that   H n (Xm , cXm ) = 0 for all n ≥ 1.

8.3 Cohomology of certain analytic spaces In this subsection, k is assumed to be of characteristic zero. Let X be a nondegenerate pluri-stable formal scheme over k  , and let Y be a quasi-compact locally closed strata subset of the closed fiber Xs (i.e., Y is a locally closed subset which is a finite union of strata of Xs ). The set S(X/Y ) = S(X) ∩ π −1 (Y) is a piecewise RZk + -linear subspace of S(X). It is a union of strata and contained in each dense Zariski open subset of k  ( k a ) . It is a nondegenerate pluri-stable formal π −1 (Y). We also set X = X⊗ a  scheme over ( k ) with the closed fiber Xs = Xs ⊗k  k a , Y = Y ⊗k  k a is a subscheme a of the latter, and so a piecewise RZk + -linear subspace S(X/Y ) of S(X) is defined. Let G be the Galois group of k s over k. 8.3.1 Theorem. Let X = π −1 (Y)\Z, where Z is a nowhere dense Zariski closed subset of Xη . Then the canonical maps S(X/Y ) → X → X induce isomorphisms of finitely dimensional vector spaces over k ∼

∼

H n (X, cX ) → H n (X, k s )G → H n (S(X/Y ), k s )G ,

n≥0.

Since the characteristic of k is zero, the first two groups can be considered in the étale as well as in the usual topology. The third group H n (S(X/Y ), k s ) is of course considered in the usual topology, it coincides with the singular cohomology group and is evidently finitely dimensional over k s . From [Ber7, Theorem 8.1] it follows that S(X/Y ) is a strong deformation retraction of X, and this implies the second isomorphism. Furthermore, if Y is open in Xs and X coincides with π −1 (Y), then X is compact and, therefore, the first isomorphism follows from the Hochschield–Serre spectral sequence. The non-triviality of the first isomorphism is in the fact that such a spectral sequence does not hold if X is not compact.

Smooth p-adic analytic spaces are locally contractible. II

367

Proof. By [Ber7, Theorem 8.1], there is a proper strong deformation retraction  of Xη to the skeleton S(X), and it lifts to a strong deformation retraction  of X to S(X). Let τ and τ denote the corresponding retraction maps Xη → S(X) and Xη → S(X). We set S = S(X/Y ) and S = S(XY ). From loc. cit. it follows that π −1 (Y) = τ −1 (S), π −1 (Y) = τ −1 (S), and that X and X contain S and S and are preserved under  and , respectively. 8.3.2 Lemma. There is an increasing sequence X1 ⊂ X2 ⊂ · · · of compact strictly analytic subdomains of π −1 (Y) with the following properties: (a) X = ∞ n=1 Xn ; (b) all Xn are preserved under ; (c) all τ (Xn ) are compact piecewise RZk + -linear subspaces of S. Proof. First of all, shrinking X we may assume that it is quasi-compact and Y is closed in Xs . We claim that it suffices to consider the case when X is affine. Indeed, assume the lemma is true in this case, and let {Xi }i∈I be a finite covering of X by open affine subschemes. By the assumption, we can find, for every i ∈ I , an increasing sequence X1i ⊂ X2i ⊂ · · · of compact strictly analytic domains of π −1 (Yi ) with the properties (a)–(c) for X ∩ π −1 (Yi ), where Yi = Y ∩ X i,s . Then the properties (a)–(b) hold for the compact strictly analytic domains Xn = i∈I Xni . Thus, let X = Spf(A). Let f1 , . . . , fm be nonzero elements of A with Z = {x ∈ Xη | fi (x) = 0 for all 1 ≤ i ≤ m}. Let ε be a positive integer which is smaller than all of the minima of the functions x  → |fi (x)| on the skeleton S(X), and let ε ≥ r1 > r2 > · · · be a decreasing sequence of numbers from |k ∗ | tending to zero. By [Ber7, Theorem 8.1(iii)], for every 1 ≤ i ≤ m and n ≥ 1, the strictly affinoid domain Yni = {x ∈ Xη | |fi (x)| ≥ rn } is preserved under . Then the same is true for the compact strictly analytic domain i Y Yn = m i=1 n . Thus, we have an increasing sequence Y1 ⊂ Y2 ⊂ · · · of compact strictly analytic domains in Xη which contain S(X), are preserved under  and such that π −1 (Y)\Z = ∞ Y n=1 n . Let E1 ⊂ E2 ⊂ · · · be an increasing sequence of compact piecewise RZk + -linear −1 (E ) is a compact subspaces of S with S = ∞ n n=1 En . By Theorem 6.4.1, each τ −1 −1 strictly analytic domain in π (Y). Then Xn = τ (En ) ∩ Yn is a compact strictly analytic domain in X = π −1 (Y)\Z, it is preserved under  and its image under τ is   En , i.e., the sequence X1 ⊂ X2 ⊂ · · · possesses the properties (a)–(c). Lemma 8.3.2 implies that the compact strictly analytic domains Xn of X are prea served under  and τ (Xn ) are piecewise RZk + -linear subspaces of S. In particular, H q (Xn , k s ) are of finite dimension over k s . Since X = ∞ n=1 X n , there is an isomorphism of finitely dimensional vector spaces over k s ∼

H q (X, k s ) → lim H q (Xn , k s ), q ≥ 0 . ←−

n

368

Vladimir G. Berkovich q

Let hn denote the dimension over k s of the image of H q (Xm , k s ) in H q (Xn , k s ) for q q q q sufficiently large m. One has h1 ≤ h2 ≤ . . . and hn = h for sufficiently large n, q where h is the dimension of H q (X, k s ) over k s . Recall that, by the Hochschield–Serre ∼ spectral sequence, one has H q (Xn , cXn ) → H q (Xn , k s )G . Let K be a finite unramified Galois extension of k such that all of the strata of the k  K  are geometrically irreducible. Then the action of G on the closed fiber of X⊗ skeleton S(X) goes through an action of its finite quotient Gal(K/k). It follows that s H q (Xn , k s )Gal(k /K) = H q (Xn , K), and we get ∼

H q (Xn , cXn ) → H q (Xn , k s )G = H q (Xn , K)Gal(K/k) . The latter space has finite dimension over k and, in particular, there is an isomorphism ∼

H q (X, cX ) → lim H q (Xn , cXn ), q ≥ 0 . ←−

n

It follows also that the image of H q (Xm , cXm ) in H q (Xn , cXn ) for sufficiently large m is q of dimension at most [K : k]·hn over k. Hence, the dimension of H q (X, cX ) over k is at ∼ q  most [K : k] · h , and there is a canonical isomorphism H q (X, cX ) → H q (X, k s )G .  8.3.3 Corollary. Let X be a nondegenerate strictly pluri-stable formal scheme over k  , Y an irreducible component of Xs , and X = π −1 (Y)\Z, where Z is a Zariski closed subset of Xη . Then H n (X, cX ) = 0 for all n ≥ 1. Proof. By Theorem 8.3.1, we may assume that k is algebraically closed, and it suffices to show that S(X/Y ) is contractible. (Of course, at this point the assumption on the characteristic of k is already not important.) To prove the contractibility, it is more convenient to use [Ber7, Theorem 8.2] instead of Theorem 5.1.1 of this paper. Let X be a strictly poly-stable fibration over k  with Xl = X. Recall that [Ber7, Theorem 8.2] identifies the skeleton S(X) = S(X) with the geometric realization |C| of a polysimplicial set C = C(X) associated with X. The polysimplicial set C here is an object of the category  E ns, where  is a category with the same family of objects as  but with larger sets of morphisms, and the geometric realization functor extends the functor that takes [n] ∈ Ob() with n = (n0 , . . . , np ) to  n = {(uij )0≤i≤p,0≤j ≤ni ∈ [0, 1][n] | ui0 + · · · + uini = 1, 0 ≤ i ≤ p} . Since X is strictly poly-stable, the polysimplicial set C is interiorly free, i.e., the stabilizer of any nondegenerate n-polysimplex of C in Aut([n]) is trivial. It follows ˚ n of  n . Let y be that the corresponding map  n → |C| is injective on the interior  the vertex of |C| that corresponds to the generic point of Y. Then S(X/Y ) is identified with the union S of all cells of |C| whose closure contains the vertex y. We define a map  : S × [0, 1] → S as follows (x, t) = ty + (1 − t)x. (Notice that the latter makes sense in S.) The map  is evidently continuous and defines a contraction of S to the point y.  

Smooth p-adic analytic spaces are locally contractible. II

369

8.3.4 Corollary. Let X be a reduced strictly k-analytic space isomorphic to W \V an , where W is a compact strictly analytic domain in the analytification Xan of a separated scheme X of finite type over k and V is a Zariski closed subset of X. Then there are canonical isomorphisms of finitely dimensional vector spaces over k ∼

H n (X, cX ) → H n (X, k s )G , n ≥ 0 .

Proof. By [Ber7, Theorem 10.1], the abelian group H n (X, Z) is of finite rank and G acts on it through a finite quotient. Since H n (X, k s ) = H n (X, Z) ⊗Z k s , it follows that the action of G on H n (X, k s ) is discrete. It follows that, if there exists a proper hypercovering X• → X such that the statement is true for all Xn ’s, then it is also true for X. Using this remark and de Jong’s results [deJ] (as in the proof of loc. cit.), the situation is reduced to the case when X is of the form considered in Theorem 8.3.1.   8.3.5 Remark. Assume that k is a finite extension of Qp , and let X be a separated reduced scheme of finite type over k. By [Ber8, Theorem 1.1(a )], there are ∼ an an canonical isomorphism H n (|X |, Qp ) → H n (X, Qp )sm , where H n (|X |, Qp ) are an k a )an , the cohomology groups of the underlying topological space of X = (X ⊗  n a H (X, Qp ) are the p-adic étale cohomology groups of X = X ⊗ k and, for a p-adic representation V , V sm denotes the subspace of V consisting of the elements with open stabilizer in G. Together with Corollary 8.3.4, this implies that there are canonical isomorphisms ∼

H n (Xan , cXan ) → (H n (X, Qp )sm ⊗Qp k s )G = (H n (X, Qp ) ⊗Qp k s )G . It follows that dimk H n (Xan , cXan ) = dimQp H n (X, Qp )sm .

References [Ber1]

Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields. Math. Surveys Monogr. 33, Amer. Math. Soc., Providence, R.I., 1990.

[Ber2]

Berkovich, V. G., Étale cohomology for non-Archimedean analytic spaces. Inst. Hautes Études Sci. Publ. Math. 78 (1993), 5–161.

[Ber3]

Berkovich, V. G., Vanishing cycles for formal schemes. Invent. Math. 115 (1994), 539–571.

[Ber4]

Berkovich, V. G., The automorphism group of the Drinfeld half-plane. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 1127–1132.

[Ber5]

Berkovich, V. G., Vanishing cycles for non-Archimedean analytic spaces. J. Amer. Math. Soc. 9 (1996), 1187–1209.

370

Vladimir G. Berkovich

[Ber6]

Berkovich,V. G.,Vanishing cycles for formal schemes. II. Invent. Math. 125 (1996), 367–390.

[Ber7]

Berkovich, V. G., Smooth p-adic analytic spaces are locally contractible. Invent. Math. 137 (1999), 1–84.

[Ber8]

Berkovich, V. G., An analog of Tate’s conjecture over local and finitely generated fields. Internat. Math. Res. Notices 13 (2000), 665–680.

[BGR]

Bosch, S., Güntzer, U., Remmert, R., Non-Archimedean analysis. A systematic approach to rigid analytic geometry. Grundlehren Math. Wiss. 261, SpringerVerlag, Berlin–Heidelberg–New York 1984.

[BoLü1]

Bosch, S., Lütkebohmert, W., Formal and Rigid geometry I. Rigid spaces. Math. Ann. 295 (1993), 291–317.

[BoLü2]

Bosch, S., Lütkebohmert, W., Formal and Rigid geometry II. Flattening techniques, Math. Ann. 296 (1993), 403–429.

[Bou]

Bourbaki, N., Topologie générale. Hermann, Paris 1951.

[Col]

Coleman, R., Dilogarithms, regulators, and p-adic L-functions. Invent. Math. 69 (1982), 171–208.

[CoSh]

Coleman, R., de Shalit, E., p-Adic regulators on curves and special values of p-adic L-functions. Invent. Math. 93 (1988), 239–266.

[GaZi]

Gabriel, P., Zisman, M., Calculus of fractions and homotopy theory. Ergebnisse Math. Grenzgeb. 35, Springer-Verlag, Berlin–Heidelberg–New York 1967.

[SGA1]

Grothendieck, A., Séminaire de Géométrie Algébrique. I. Revêtements étales et groupe fondamental. Lecture Notes in Math. 224, Springer-Verlag, Berlin– Heidelberg–New York 1971.

[Hud]

Hudson, J. F. P., Piecewise linear topology. University of Chicago Lecture Notes, W. A. Benjamin, Inc., New York–Amsterdam 1969.

[deJ]

de Jong, A. J., Families of curves and alterations. Ann. Inst. Fourier (Grenoble) 47 (1997) 599–621.

[Mac]

MacLane, S., Categories for the Working Mathematician. Grad. Texts in Math. 5, Springer-Verlag, Berlin–Heidelberg–New York 1971.

[RoSa]

Rourke, C. P., Sanderson, B.J., Introduction to piecewise-linear topology. Ergebnisse Math. Grenzgeb. 69, Springer-Verlag, Berlin–Heidelberg–New York 1972.

[Zie]

Ziegler, G. M., Lectures on polytopes. Grad. Texts in Math. 152, Springer-Verlag, Berlin–Heidelberg–New York 1995.

Vladimir G. Berkovich, Department of Mathematics, The Weizmann Institute of Science, P.O.B. 26, 76100 Rehovot, Israel E-mail: [email protected]

Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems Jean-Benoît Bost

Contents 1

Introduction

372

2 Algebraicity of smooth formal germs in algebraic varieties and auxiliary polynomials 2.1 Algebraic formal germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Evaluation maps and an algebraicity criterion . . . . . . . . . . . . . . . . . . 2.3 An algebraicity criterion for smooth formal germs in varieties over function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Application: positivity properties of Lie algebras of group schemes . . . . . . .

376 376 377 381 385

3 The canonical semi-norm attached to a germ of analytic curve in a complex algebraic variety 3.1 The basic construction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Birational invariance of the canonical semi-norm . . . . . . . . . . . 3.3 Functorial properties of the canonical semi-norm . . . . . . . . . . . 3.4 Canonical semi-norm and capacity . . . . . . . . . . . . . . . . . . . 3.4.1 Green functions and Riemann surfaces . . . . . . . . . . . . . 3.4.2 An upper bound on canonical semi-norms . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

388 388 390 393 394 394 396

4 Algebraicity criteria for smooth formal germs of subvarieties in algebraic varieties over number fields 4.1 Sizes of formal subschemes over p-adic fields . . . . . . . . . . . . . . . . 4.1.1 Groups of formal and analytic automorphisms . . . . . . . . . . . . 4.1.2 The size R(Vˆ ) of a formal germ Vˆ . . . . . . . . . . . . . . . . . . 4.1.3 Sizes of solutions of algebraic differential equations . . . . . . . . 4.2 Normed and semi-normed lines over number fields . . . . . . . . . . . . . 4.3 An arithmetic algebraization theorem . . . . . . . . . . . . . . . . . . . . 4.4 Proof of the algebraization theorem . . . . . . . . . . . . . . . . . . . . . 4.4.1 Auxiliary hermitian vector bundles and linear maps . . . . . . . . . 4.4.2 Application of the slope inequalities . . . . . . . . . . . . . . . . . 4.5 Analytic germs with positive canonical semi-norms . . . . . . . . . . . . . 4.6 Application to differential equations . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

398 399 399 400 401 402 403 405 405 407 409 411

. . . . . .

. . . . . .

372

Jean-Benoît Bost

A Appendix: extensions of sections of large powers of ample line bundles

412

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

Conventions. The following notation and terminology are used throughout this paper. The open (resp. closed) disk of center a and radius r in C is will be denoted D(a, r) (resp. D(a, r)). The rank of a vector bundle E (resp. of a linear map ϕ) will be denoted rk E (resp. rk ϕ). By an algebraic variety over some field k, we mean an integral scheme of finite type over k. Integral subschemes of such an algebraic variety X over k will be called algebraic subvarieties of X. On a complex analytic manifold, we write as usual d = ∂ + ∂ and we let d c := (i/4π )(∂ − ∂); consequently: dd c = (i/2π )∂∂.

1 Introduction 1.1. Consider a number field K, a quasi-projective variety X over K, a point P in X(K), and a germ Vˆ of formal subvariety of X through P , namely, a smooth formal subscheme of the formal completion Xˆ P of the K-scheme X at the closed point P . We shall say that such a formal scheme is algebraic when it is a branch (i.e.a component of the formal completion at P ) of an algebraic subvariety Y of X containing P (see section 2.1, infra, for a more complete discussion of the concept of algebraic formal germ). Various questions in arithmetic geometry may be rephrased in terms of the algebraicity of such formal germs Vˆ : one would like to know natural arithmetic conditions on Vˆ implying its algebraicity. The main examples we have in mind are the following ones: A. Formal series. Let f ∈ K[[t1 , . . . , tN ]] be a formal series in N variables which has a positive radius of convergence at every place of K, finite or infinite. In other words, for any non-zero prime ideal p in OK (resp. for any field embedding σ : K → C), the series f seen as an element of Kp [[t1 , . . . , tN ]] (resp. of C[[t1 , . . . , tN ]]) by means of the embedding K → Kp of K in its p-adic completion Kp (resp. by means of σ : K → C) has a positive p-adic (resp. complex) radius of convergence. Then the graph of f defines a smooth formal germ of dimension N , Vˆ := Graph(f )

Germs of analytic varieties in algebraic varieties

373

through (0, f (0)) – formally, it is defined by the principal ideal generated in AN+1 K by z − f (t1 , . . . , tN ) in K[[t1 , . . . , tN , z − f (0)]]) – and the algebraicity of Gr(f ) is equivalent to the algebraicity of f over the subfield K(t1 , . . . , tN ) of the field of fraction of K[[t1 , . . . , tN ]] (or to the fact that f belongs to the integral closure of the local ring OAN ,0 in its completion Oˆ AN ,0  K[[t1 , . . . , tN ]]). B. Formal subgroups of algebraic groups. Assume that X is a K-algebraic group G and P = e, the unit element of G(K), and let h be a Lie subalgebra (over K) of ˆ h of the formal g := Lie G. We may consider the formal Lie subgroup Vˆ := Exp ˆ e over K attached to G, namely the smooth formal subgroup of G ˆ e which group G 1 ˆ admits h as Lie algebra . Then the formal germ V is algebraic iff h is an algebraic Lie algebra, i.e., is the Lie algebra of some algebraic K-subgroup H of G. For instance, if G is the product G1 × G2 of two K-algebraic groups G1 and G2 with Lie algebras g 1 and g 2 , a K-Lie algebra isomorphism ϕ : g 1 → g 2 is the ˆ h is algebraic, differential of a K-isogeny from G1 to G2 iff the formal germ Vˆ := Exp where h denotes the Lie subalgebra of g = g 1 ⊕ g 2 defined by the graph of ϕ. C. Ordinary differential equations. Consider an algebraic ordinary differential equation over a number field K, and define Vˆ as its formal solution for some initial conditions defined over K. For instance, if Q = (Q1 , . . . , Qn ) is an element in K(X, Y1 , . . . , Yn )n and y0 a point in K n such that (0, y0 ) does not lie on the polar divisor of any component Qi of Q, we may consider the formal solution f in K[[t]]n of the differential equation f (t) = Q(t, f (t))

(1.1)

f (0) = y0 .

(1.2)

satisfying the initial condition +1 This solution f is an “algebraic function” iff the graph Vˆ of f in AN (0,y0 ) is algebraic. More generally, we may consider a smooth variety X over K, a point P in X(K) and a sub-vector bundle F of rank one of the tangent bundle TX/K , and consider the smooth formal germ of curve Vˆ defined by “integrating” the line bundle F . Formally, it is defined as the unique smooth formal germ of curve in X through P such that, if i : Vˆ → X denotes the inclusion morphism, the differential Di, which a priori is an element of (Vˆ , i ∗ TX ), indeed belongs to (Vˆ , i ∗ F ). We recover the previous situation by defining X as the complement of the polar divisors of the Qi ’s in An+1 , P as (0, y0 ), and F as the line bundle generated by the vector field n

∂ ∂ − Qi . ∂X ∂Yi i=1

1 It may be constructed as follows: if Exp ˆ denotes the “formal exponential map” of G – that is, the ˆe isomorphism of K-formal schemes from the completion at 0 of the K-affine space defined by g onto G ˆ h is the image by Exp ˆ of the formal completion at 0 defined by the Campbell-Hausdorff series – then Exp of the K-affine subspace h of g.

374

Jean-Benoît Bost

(Actually, both constructions B and C are special cases of the construction of formal germs of leaves of algebraic foliations over number fields; cf. [ESBT99] and [Bos01].) In the three situations A, B, and C above, the formal germ Vˆ satisfies the following analyticity conditions: For any non-zero prime ideal p in OK (resp. for any field embedding σ : K → C), the formal germ VˆKp (resp. Vσ ) in the formal completion at P of XKp (resp. Xσ ) deduced from Vˆ by the base field extension K → Kp (resp. σ : K → C) is analytic. Namely, it is the formal germ attached to some (necessarily smooth) germ of Kp analytic (resp. C-analytic) subvariety through P of the Kp -analytic (resp. C-analytic space) X(Kp ) (resp. X(C)). This is tautological in caseA; in cases B and C, this follows from the well known analyticity properties of the Campbell-Hausdorff series and from the classical Cauchy’s theorem and its p-adic versions (see for instance [Ser92], section V.4, and [DGS94], Appendix III). These analyticity conditions are easily seen to be necessary for the algebraicity of Vˆ . Actually, the latter imposes much stronger conditions. For instance, as early as 1852, Eisenstein  discovered the following fact, now known as Eisenstein’s theorem: k if a formal series +∞ k=0 ak t in Q[[t]] is algebraic, then there exists integers A, B ≥ 1 k such that AB ak ∈ Z for every k ∈ N. Concerning solutions of differential equations considered in B above, it was pointed out by Grothendieck and Katz around 1970 ([Kat72]) that, if the differential system defined by a line bundle F in the tangent bundle TX of a smooth variety X over a number field K is algebraically integrable, then the following arithmetic condition – which we shall call condition GK – necessarily holds: For almost every non-zero prime ideal p in OK , the sub-line bundle FFp of TXFp on the variety XFp obtained by reduction modulo p from some smooth model X of X over some open dense subscheme S of Spec OK and from a line bundle F → TX/S extending F is closed under the p-th power map (where p denotes the characteristic of the residue field Fp := OK /p). Actually, Grothendieck and Katz were considering linear differential systems only; the case of general differential systems explicitly appears in [Miy87], [SB92] and [ESBT99]. 1.2. In this paper, we are interested in sufficient conditions implying the algebraicity of Vˆ in the context of examples A, B, and C above. The investigation of such conditions has a long and rich history, about which we shall give only a few indications. The first result concerning sufficient conditions for algebraicity appears to be a theorem established by E. Borel in 1892 asserting that, if a formal series f ∈ Z[[t]] is the Taylor expansion at 0 of some function meromorphic on a disk D(0; R) of radius R > 1, then f is the expansion of a rational function. Concerning linear differential equations, Grothendieck and Katz conjectured that condition GK is indeed a sufficient condition for algebraic integrability. This con-

Germs of analytic varieties in algebraic varieties

375

jecture is formulated in the seminal paper [Kat72] of Katz, where he proves it in the significant special case of linear differential systems “of geometric origin” (see also [Kat82], [Kat96], [And99] and [And02] for more recent developments in this direction). Besides, in their famous works [Ser68] and [Fal83], Serre and Faltings obtained deep results concerning isogenies between elliptic curves and abelian varieties, which may be used to handle non-trivial cases of the algebraicity problem in the situation B (see for instance [ESBT99], sections 3–5). Finally, around 1984, D.V. and G.V. Chudnovsky discovered how to apply “transcendence techniques” to establish algebraicity statements in the situations A, B, and C ([CC85a] and [CC85b]). Their work was subsequently extended by André ([And89], [And99] and [And02]), Graftieaux ([Gra01a] and [Gra01b]), and the author ([Bos01]). We refer the reader to [CL02] for a synthetic view of these results. Bernard Dwork himself played a distinguished role in contributing to various aspects of the algebraicity problem in the situations A and C. It is barely necessary to recall that, in his famous rationality proof ([Dwo60]), he established a generalized version of Borel’s rationality criterion discussed above – the Borel–Dwork criterion. Let us also mention its investigations of Eisenstein’s theorem ([DR79], [DvdP92]) and his papers ([BD79], [Dwo81], [Dwo99]) devoted to the “arithmetic theory of differential equations”. The latter also constitutes one of the main themes of the beautiful book [DGS94] by B. Dwork, G. Gerotto and F. Sullivan. 1.3. This paper is devoted to some algebraicity criterions, implying the algebraicity of formal germs of curves over number fields in the situations A, B, and C considered above. These criterions, which are refined versions of the main results of [Bos01] in the special case of germs of formal curves, are expressed in terms of positivity properties – defined in terms of its Arakelov degree – of the tangent line TP Vˆ equipped with some natural p-adic and archimedean semi-norms. As our previous results in [Bos01], they are established by a geometric version of “transcendence techniques”, which avoids the traditional constructions of “auxiliary polynomials” but is based instead on some geometric version of these, namely the study of evaluation maps on the spaces of global sections of ample line bundles on a projective variety, defined by restricting them to formal subschemes or to subschemes of finite lengths. Dealing with formal germs of curves only – instead of formal germs of arbitrary dimension as in [Bos01] – allows various technical simplifications and leads to an algebraization theorem (Theorem 4.2, infra ) whose statement and proof are particularly simple. However, Theorem 4.2 admits higher dimensional generalizations on which we plan to return in the future. This paper is organized as follows. In section 2, we discuss the notion of algebraicity of formal germs in algebraic varieties, and we provide an introduction to the use of auxiliary polynomials, in the geometric guise of evaluation maps, by showing how simply they lead to non-trivial algebraicity results in some purely geometric situations. In particular, we establish an algebraicity criterion for formal germs over functions fields, which we use to

376

Jean-Benoît Bost

investigate the positivity properties of the Lie algebras of group schemes over a field of characteristic zero. Section 3 is of a more analytic nature: we assume that Vˆ is a germ of analytic curve in a complex algebraic variety X, and we explain how the consideration of the metric properties of the evaluation maps involved in the method of auxiliary polynomials leads to the construction of some remarkable semi-norm on the complex line TP Vˆ . We also study some “naturality” and “functoriality” properties of this semi-norm, and we establish some upper-bound on it in terms of potential theoretic invariants. In section 4, we present an algebraicity theorem concerning formal germs of curves in algebraic varieties over number fields, which may be seen as an arithmetic counterpart of the criterion over function fields discussed in section 2. This criterion involves the canonical complex semi-norms investigated in section 3. Actually, it may be used to formulate a conjecture about complex linear algebraic differential systems, whose solution would provide a proof of the conjecture of Grothendieck–Katz asserting that condition GK is a sufficient condition of algebraic integrability for algebraic linear differential systems over number fields.

2 Algebraicity of smooth formal germs in algebraic varieties and auxiliary polynomials 2.1 Algebraic formal germs Let X be a variety over a field K, P a point of X(K), Xˆ P the formal completion of X at P , and Vˆ → Xˆ P a smooth formal subscheme. For any non-negative integer i, we shall denote Vi the i-th infinitesimal neighborhood of P in Vˆ . Thus, V0 = {P } ⊂ V1 ⊂ V2 ⊂ · · · and Vˆ = lim Vi . →

It will be convenient to let: V−1 = ∅. We may consider the Zariski closure of Vˆ in X, namely, the smallest closed subscheme Z of X which contain Vi for every i ≥ 0, or equivalently, such that Zˆ P contain Vˆ . Observe that it is a subvariety (i.e.an integral subscheme) of X containing P . The ideal in OX,P defining its germ at P is the intersection of OX,P and of the ideal in its completion Oˆ X,P = OXˆ P that defines Vˆ . Since Zˆ P contains Vˆ , the dimension of Z is greater or equal to the dimension of Vˆ .

Germs of analytic varieties in algebraic varieties

377

The following proposition is an easy application of the basic properties of dimension and normalization: Proposition 2.1. The following three conditions are equivalent: (i) There exists an algebraic variety Y over K, a point 0 of Y (K) and a K-morphism which maps 0 to P and such that the induced morphism on formal completions fˆ0 : Yˆ0 −→ Xˆ P factorizes through Vˆ → Xˆ P and defines a formal isomorphism from Yˆ0 to Vˆ . (ii) There exists a closed subvariety Z of X such that P belongs to Z(K) and Vˆ is a branch of Z through P . (iii) The dimension of the Zariski closure Z of Vˆ in X coincides with the dimension of the formal scheme Vˆ . We shall say that the formal germ Vˆ is algebraic when the above conditions are satisfied.

2.2 Evaluation maps and an algebraicity criterion Let us keep the notation of the preceding paragraph. Let us moreover assume that X is projective and consider an ample line bundle L on X. Let us introduce the following K-vector spaces and K-linear maps: ED := (X, L⊗D ), ηD : ED −→ (Vˆ , L⊗D ) s  −→ s|Vˆ , i ηD : ED −→ (Vi , L⊗D ) s  −→ s|Vi ,

and i−1 i := {s ∈ ED | sVi−1 = 0} = ker ηD . ED

Observe that we have a canonical isomorphism (Vˆ , L⊗D )  lim (Vi , L⊗D ), ←−

i

by means of which the map ηD gets identified with i . lim ηD ←−

i

378

Jean-Benoît Bost

i define a decreasing filtration of the finite dimensional K-vector The subspaces ED space ED : i+1 i 1 0 · · · ⊂ ED ⊂ ED ⊂ · · · ⊂ ED ⊂ ED = ED .

Moreover the very definition of Z as the Zariski closure of Vˆ shows that, if
Z denotes the ideal sheaf in OX defining Z, we have  i ED = ker ηD = (X,
Z .L⊗D ). (2.1) i≥0

Finally, if TVˆ denotes the tangent space of Vˆ , then, for any non-negative integer i, the kernel of the restriction map from (Vi , L⊗D ) to (Vi−1 , L⊗D ) may be identified i i with S i TˇVˆ ⊗LD P , and the restriction of the evaluation map ηD to ED defines a K-linear map: i γDi : ED −→ S i TˇVˆ ⊗ L⊗D P .

Roughly speaking, it is the map which sends a section of L⊗D vanishing up to order i−1 at P along Vˆ to the i-th “Taylor coefficient” of its restriction to Vˆ . By construction, i+1 ker γDi = ED .

(2.2)

Proposition 2.2. The following two conditions are equivalent: (i) The formal germ Vˆ is algebraic. (ii) There exists c > 0 such that, for any (D, i) ∈ N>0 × N satisfying i/D > c, the map γDi vanishes. Condition (ii) may be also expressed by saying that, for every positive integer D, i ) i the filtration (ED i≥0 becomes stationary – or equivalently that ηD vanishes on ED – when i > cD. The direct implication (i) ⇒ (ii) will be a consequence of the following lemma, which belongs to the basic theory of ample line bundles (see for instance [Laz01], Chapter 5, notably Proposition 5.1.9). Lemma 2.3. Let M be a projective variety of dimension d over a field K, H an ample line bundle over M, and 0 a point in M(k). Let ε(H, 0) denote the Seshadri constant of H at 0 and degH M := c1 (H )d ∩ [M] be the degree of M with respect to H . Then, for any positive integer D and any regular section s of H ⊗D over M which does not vanishes identically, the order of vanishing mult0 s of s at 0 satisfies the following upper bound: mult 0 s ≤

degH M D. ε(H, 0)d−1

(2.3)

Germs of analytic varieties in algebraic varieties

379

Recall that ε(H, 0) is the positive real number defined as follows: let ν : M˜ −→ M be the blow-up of 0 in M and let E := ν −1 (0) be its exceptional divisor; then σ0 (H ) is the supremum of the rational numbers q such that the Q-line bundle ν ∗ H ⊗ O(−qE) is ample. To prove (2.3), one observes that the Cartier divisor on M˜ ν ∗ div s − mult0 s.E is effective; therefore, for any q as above, the intersection number c1 (ν ∗ H ⊗ O(−qE))d−1 ∩ (ν ∗ div s − mult0 s.E) is non-negative. Since this intersection number is easily seen to be D.degH M − mult 0 s.q d−1 , we get (2.3) by letting q go to ε(H, 0). Proof of Proposition 2.2 To prove the implication (i) ⇒ (ii), let us assume that Vˆ is algebraic and let us consider the normalization n : Zn → Z of the Zariski closure Z of Vˆ in X. Like Z, it is a projective variety of dimension d := dim Vˆ . Indeed, the line bundle n∗ L on Zn is ample and, since n is birational, degn∗ L Zn = degL Z. Let 0 ∈ Zn (K) be the preimage of P by n corresponding to the branch Vˆ of Zˆ P . In other words, the completion of n at 0 induces a formal isomorphism: n 0 −→ Vˆ . nˆ 0 : Z i . Pulling back s by n, we get a regular section n∗ s of Let s be an element of ED over Zn which vanishes at order at least i at the point 0. Lemma (2.3) shows that n∗ s vanishes on Zn if

n∗ L⊗N

i>

degL Z D. ε(n∗ L, 0)d−1

i vanishes on Vˆ when i > cD, where This proves that any s ∈ ED

c :=

degL Z . ε(n∗ L, 0)d−1

Conversely, let assume that condition (ii) holds, and let d still denote dim Vˆ . Then, for any (D, i) ∈ N2 , the quotient vector space i+1 i i ED /ED = ED / ker γDi  im γDi

vanishes if i > cD and its rank is always at most  d +i−1 i ˇ D rk (S TVˆ ⊗ LP ) = . i

380

Jean-Benoît Bost

This implies that 

rk ED /

 i≥0

i ED



=

i≥0

i+1 i rk (ED /ED )

≤

[cD]

 i=0

d +i−1 i

.

d

Moreover the last sum is equivalent to cd! D d when D goes to infinity. Besides, according to (2.1):  i ED = (X, L⊗D )/ (X,
Z .L⊗D ). ED / i≥0

For D large enough, this space may be identified with (Z, L⊗D ) and its rank is deg Z equivalent to (dimLZ)! D dim Z when D goes to infinity. This shows that dim Z is less or equal – hence equal – to d and that deg L Z ≤ cd .   The implication (ii) ⇒ (i) in Proposition 2.2 asserts that, when Vˆ is not algebraic, there exists non-vanishing maps γDi with arbitrary large values of the ratio i/D. Actually, it is possible to establish a strengthened version of this implication, which will turn out to be useful in the sequel: Lemma 2.4. If Vˆ is not algebraic, then  i+1 i i≥0 (i/D)rk (ED /ED ) = +∞. lim  i+1 i D→+∞ i≥0 rk (ED /ED ) Observe that, if (2.4) holds, then, for any λ > 0,  i+1 i i≥λD (i/D)rk (ED /ED ) = +∞. lim  i+1 i D→+∞ i≥0 rk (ED /ED )

(2.4)

(2.5)

Indeed, 

i+1 i i<λD (i/D)rk (ED /ED )  i+1 i i≥0 rk (ED /ED )

≤ λ.

Proof. As observed in the previous proof, the sum   

i+1 i i rk (ED /ED ) = rk ED / ED i≥0

i≥0

is equal to rk ((Z, L⊗D )) when D is large enough, and therefore grows like D dim Z when D goes to infinity.

381

Germs of analytic varieties in algebraic varieties

Moreover, for any λ ≥ 0 and any D ∈ N,  



i λD i+1 i+1 i i i /ED )≥λ rk (ED /ED ) = λrk ED / ED rk (ED , D i≥0

i≥λD

(2.6)

i≥0

where λD denotes the smallest integer ≥ λD. To derive a lower bound on this quantity, observe that       λD λD i i ED ED rk ED / = rk ED / − rk (ED /ED ) i≥0

and that

i≥0



λD  λD−1  = rk im γD rk ED /ED

is bounded from above by the length lg(VλD−1 ) of VλD−1 . This shows that   i+1 i lg(VλD−1 ) i≥0 (i/D)rk (ED /ED ) ≥λ 1− .  i+1 i rk ((Z, L⊗D )) i≥0 rk (ED /ED ) Finally, if Vˆ is not algebraic, then dim Z > d := dim Vˆ and, when D goes to infinity,  λD + d − 1 = O(D d ) = o(D dim Z ) = o(rk ((Z, L⊗D )), lg(VλD−1 ) = d and therefore

 lim inf

D→+∞

i+1 i i≥0 (i/D)rk (ED /ED )  i+1 i i≥0 rk (ED /ED )

≥ λ.

As λ is arbitrary, this completes the proof.

 

2.3 An algebraicity criterion for smooth formal germs in varieties over function fields Let C be a smooth projective connected curve over some field k, and let K := k(C) be its function field. Consider X a variety over K, P a point in X(K), and Vˆ ⊂ Xˆ P a smooth formal germ of subvariety through P of X. In this section, we discuss a criterion for the algebraicity of Vˆ , involving a model X of X over C and the positivity properties of the thickenings of the closure P of P in X attached to Vˆ . This algebraicity criterion will appear as a geometric model for the arithmetic algebraicity criterion presented in section 4.3 below. Moreover its proof demonstrates how simply the use of “auxiliary polynomials” leads to non-trivial results, even in a purely geometric framework (see for instance Theorem 2.6 infra). The reader is referred to [BM01] and to [Bos01], section 3.3, for related geometric

382

Jean-Benoît Bost

results and discussions of their relations with the classical works of Andreotti on pseudo-concavity, and of Hartshorne–Hironaka–Matsumura on the G2 condition. After possibly shrinking X, we may assume that it is quasi-projective and choose a quasi-projective model2 π : X −→ C such that P extends to a section P of π. As in the preceding section, we denote Vi the i-th infinitesimal neighbourhood of P in X. We may consider the subschemes Vi of X defined as the closures of these subschemes Vi of XK . For any i ∈ N, the support of Vi is exactly the image of the section P . In particular, the subschemes Vi are finite over C. Moreover their ideal sheaves
Vi satisfy the relations
Vi .
Vj ⊂
Vi+j +1 ,

for any (i, j ) ∈ N2 ;

(2.7)

indeed, the restriction to the generic fiber of any local section of
Vi .
Vj is a section of
Vi .
Vj =
Vi+j +1 . In particular
V0 .
Vi ⊂
Vi+1 , and the coherent sheaf
Vi /
Vi+1 may be identified with a coherent sheaf on V0 , or ∼ equivalently, thanks to the isomorphism P : C −→ V0 , with a coherent sheaf Ji+1 := π∗
Vi /
Vi+1 over C. Actually, the sheaves Ji+1 are easily checked to be torsion free, and therefore may be identified with the sheaves of sections of some vector bundles Ji+1 over C. Recall that, if E is a vector bundle of positive rank on C, its slope is defined as the quotient deg E , rk E and its maximal slope µmax (E) is the maximum of the slopes µ(F ) of sub-vector bundles of positive rank in E. Observe that, if L is any line bundle on C, µ(E) :=

µmax (E ⊗ L) = µmax (E) + deg L. Moreover, if E1 and E2 are vector bundles over C, with E2 of positive rank, and if there exists some (generically) injective morphism of vector bundles ϕ : E1 −→ E2 , then the following slope inequality holds: deg E1 ≤ rk E1 · µmax (E2 ).

(2.8)

We are now in position to formulate our algebraicity criterion:

2 namely, a quasi-projective k-variety X, equipped with a flat k-morphism π : X → C and an isomorphism of its generic fiber XK with X.

Germs of analytic varieties in algebraic varieties

383

Theorem 2.5. With the notation above, if 1 lim sup µmax (Jj ) < 0, j j →+∞

(2.9)

then Vˆ is algebraic. ˆ P that is smooth over C, Observe that, if Vˆ extends to a formal subscheme Vˆ of X then for any j ∈ N, we have ˆ Jj  S j (NP Vˇ), and the numerical condition (2.9) is equivalent to the ampleness3 of the vector bundle P ∗ NP Vˆ over C. In general, we still have natural maps of vector bundles over C S j J1 −→ Jj , which are isomorphisms at the generic point Spec K of C. However, they are not always isomorphisms over C, and in general condition (2.9) is stronger than the ampleness of J1ˇ. Proof. One easily checks that one may find a projective compactification of X to which the morphism π extends. Therefore, we may assume that X is indeed projective, and choose an ample line bundle L on it. Let L be its restriction LK to X, and let ED , i , ηi and γ i be as in the previous section 2.2. ED D D By replacing X by the Zariski closure Z of Vˆ in X and X by the closure Z of Z in X (which leaves the subschemes Vi and the sheaves Ji unchanged), we may also i is the assume that Vˆ is Zariski dense in X, and therefore that, for any integer D, ED zero subspace for i large enough. We are going to show that, when condition (2.9) is satisfied, the “average value” of i/D, namely   i+1 i+1 i i i≥0 (i/D)rk (ED /ED ) i≥0 (i/D)rk (ED /ED ) = , (2.10)  i+1 i rk ED i≥0 rk (ED /ED ) stays bounded when D goes to infinity. According to Lemma 2.4, this will prove that Vˆ is algebraic. To achieve this, let us consider the direct images ED := π∗ L⊗D and π|Vi ∗ L⊗D . These are torsion free coherent sheaves, or equivalently vector bundles, on C, which at the generic point Spec K of C coincide with the K-vector spaces ED and (Vi , L⊗D ). i : E −→ (V , L⊗D ) extends to a morphism of Moreover, the restriction map ηD D i 3 See for instance [Laz01], part II, and its references for the basic results of the theory of ample vector bundles; see also [Bar71] in the positive characteristic case.

384

Jean-Benoît Bost

vector bundles: ηiD : ED −→ π|Vi ∗ L⊗D s  −→ s|Vi .

i ) The filtration (ED i≥0 of ED also extends to the filtration of ED by the sub-vector i ⊗D to bundles ED := ker ηi−1 D . Finally, the kernel of the restriction map from π|Vi ∗ L π|Vi−1 ∗ L⊗D may be identified with Ji ⊗ P ∗ L⊗D and the restriction of the evaluation i defines a morphism of vector bundles map ηiD to ED i γ iD : ED −→ Ji ⊗ P ∗ L⊗D , i+1 and which coincides with γDi at the generic point of C. The kernel of γ iD is ED i therefore γ D factorizes through a (generically) injective morphism of vector bundles: i+1 i /ED −→ Ji ⊗ P ∗ L⊗D . γ˜Di : ED

Since L is ample, for D large enough, the sheaf ED is generated by its global sections, and consequently: deg ED ≥ 0.

(2.11)

i = {0} when i >> 0, we may write: Moreover, as ED

i+1 i deg ED = deg (ED /ED ).

(2.12)

i≥0

Combined with the slope inequality 2.8 applied to the morphisms γ˜Di , the relations (2.11) and (2.12) and the identity µmax (Ji ⊗ P ∗ L⊗D ) = µmax (Ji ) + D deg (P ∗ L) show that: D rk ED deg P ∗ L +

i≥0

i+1 i rk (ED /ED )µmax (Ji ) ≥ 0.

(2.13)

If we now assume that condition (2.9) is satisfied, then there exists i0 ∈ N and c > 0 such that, for any integer i ≥ i0 , µmax (Ji ) ≤ −c i. Therefore, from (2.13), we deduce that

i+1 i+1 i i D rk ED deg P ∗ L+ rk (ED /ED )(µmax (Ji )+ci)−c irk (ED /ED ) ≥ 0. 0≤i
Since the second sum is bounded by

rk (S i TˇVˆ )(µmax (Ji ) + ci), 0≤i
i≥0

Germs of analytic varieties in algebraic varieties

385

which does not depend on D, this establishes that the ratio (2.10) is indeed bounded.  

2.4 Application: positivity properties of Lie algebras of group schemes In spite of the simplicity of its proof, the algebraization criterion in Theorem 2.5 has significant applications, for instance to algebraic foliations, as demonstrated by the recent work of Bogomolov and McQuillan [BM01]. In this section, we briefly describe how easily it leads to some basic positivity properties of Lie algebras of group schemes over a field of characteristic 0. Let C be a smooth projective connected curve over a field k, and let π : G → C be a smooth quasi-projective4 group scheme over C. Let us denote by ε its zero-section, and by Lie G := ε ∗ Tπ its Lie algebra. It is a vector bundle over C, equipped with a OC -bilinear Lie bracket: [., .] : Lie G ⊗OC Lie G −→ Lie G v1 ⊗ v2  −→ [v1 , v2 ]. Observe that, by restricting to the generic point Spec K of C, one defines a bijection between the set of sub-vector bundles of Lie G and the set of K-vector subspaces of (Lie G)K . Moreover, (Lie G)K may be identified with the K-Lie algebra Lie G of the K-algebraic group G := GK , and a sub-vector bundle F of Lie G is a bundle of Lie subalgebras (in other words, its sheaf of sections is closed under the above Lie bracket) iff FK is a Lie sub-algebra of Lie G. Theorem 2.6. Assume that k is a field of characteristic zero, and consider a Lie subalgebra FK of Lie G. If the corresponding sub-vector bundle F of Lie G is ample, then FK is an algebraic Lie subalgebra of Lie G. Actually, it is the Lie algebra of a connected linear unipotent5 K-algebraic subgroup H of G. Since an abelian variety over a field of characteristic zero contains no non-trivial unipotent subgroup, this implies in particular:

4Actually, according to Raynaud [Ray70], Corollaire VI 2.6 et Théorème VIII 2, any smooth group scheme (of finite type) over a smooth curve is quasi-projective. Besides, a classical theorem of Cartier shows that, over a field of characteristic zero, any flat group scheme of finite type is smooth. 5 Recall that, over a perfect field K, a (smooth) K-algebraic group G is called unipotent if, for some n, it is isomorphic to a closed subgroup of the algebraic subgroup of the subgroup U (n)K of GL(n)K of matrices (gij )1≤i,j ≤n such that gii = 1 and gij = 0 if i < j . A connected K-algebraic group is unipotent iff there exists a composition series G = G0 ⊃ G1 ⊃ · · · ⊃ Gs = {e} by connected K-algebraic subgroups such that the quotients Gi /Gi+1 are isomorphic to additive groups Ga r over K. See for instance [Bor91], I.4 and V.15 for proofs and references.

386

Jean-Benoît Bost

Corollary 2.7. Assume that k is a field of characteristic zero. If G is an abelian variety over K, then the vector bundle 1G/C := (Lie G)ˇ is semi-positive (in other words, any quotient of 1G/C has non-negative degree). This corollary is indeed a classical result of Griffiths, which he established by transcendental techniques. (When k = C, and G is an abelian scheme, this follows from the computation of the curvature of Hodge bundles; cf. [Gri70], Theorem 5.2. When G is semi-abelian, the singularities on the metric on 1G/C , seen as a Hodge bundle, at points of bad reduction of G are logarithmic, and consequently, the positivity properties of its curvature outside the divisor of bad reduction still allows to derive the semi-positivity of 1G/C . The general case follows by semi-stable reduction.) It seems plausible that Theorem 2.6 can be recovered by combining structure theorems on algebraic groups, the rigidity properties of reductive Lie groups, and Corollary 2.7. (The author confesses that he did not check it in detail.) Observe that Corollary 2.7, and a fortiori Theorem 2.6, do not hold in general over a base field k of characteristic p > 0. As demonstrated by Moret-Bailly in [MB81], Proposition 3.1, counter-examples may be obtained from two supersingular elliptic curves E1 and E2 by considering the abelian surface G over C obtained by quotienting the abelian surface A := (E1 × E2 )C over C by a subgroup scheme of (αp ⊕ αp )C (which lies in A[p] ) that is locally isomorphic to αp,C , but not constant. Proof of Theorem 2.6 To prove the first assertion, we need to show that, if F is ample, ˆ e is algebraic. This is a ˆ G FK of G then the germ of formal subvariety Vˆ := Exp straightforward consequence of Theorem 2.5 and of the subsequent observation, since Vˆ extends to a formal subscheme of the completion Gˆ ε of G along its zero section ε ˆ pulled back to C by ε, may which is smooth over C, and the normal bundle to ε in V, be identified with F . (Indeed, we may consider the relative formal exponential map ˆ G/C . It maps the completion of the total space V(Lie Gˇ ) of the vector bundle Exp Lie G along its zero section isomorphically onto the completion Gˆ ε of G along its unit ˆ G/C of the completion along its zero section ε. Consequently, the image under Exp section of the total space V(Fˇ ) of the subbundle F of Lie G provides the required extension.) For any vector bundle E over C, we shall denote E the commutative algebraic ˇ of this vector bundle equipped with the group over C defined by the total space V(E) group law deduced from the additive structure of E. We now turn to the special case of Theorem 2.6 when FK is an abelian Lie subalgebra of Lie G, namely: Lemma 2.8. With the notation of Theorem 2.6, if the vector bundle is ample and if FK is an abelian Lie subalgebra of Lie G (i.e., if [., .] vanishes on F ⊗ F ), then there exists an embedding of algebraic group j : FK −→ G, the differential of which along ε coincides with the inclusion map iK : FK → Lie G.

Germs of analytic varieties in algebraic varieties

387

Observe that Lemma 2.8 already implies Corollary 2.7. Proof of Lemma 2.8 Consider the group scheme F ×C G over C. Its Lie algebra F ⊕ Lie G contains the graph of the inclusion i : F → Lie G as a Lie subalgebra, which moreover is ample. Therefore the graph of iK is the Lie algebra of a connected algebraic subgroup H of FK × G, which is easily shown to be the graph of a group embedding j : FK → G.   In the sequel of the proof, we freely use the basic properties of the Harder– Narasimhan filtration and of the slopes of vector bundles on a curve over a field of characteristic zero. To prove the second assertion in Theorem 2.6, we may assume that the vector bundle Lie G has positive rank and consider its Harder–Narasimhan filtration: E0 = {0} ⊂ E1 ⊂ · · · ⊂ EN = Lie G. By definition, the quotient bundles Ei /Ei−1 , 1 ≤ i ≤ N, are semi-stable and their slopes µi := µ(Ei /Ei−1 ) satisfy µ1 > · · · > µN . We also define i+ := max{i ∈ {1, . . . , n} | µi > 0} := 0

if µ1 > 0 if µ1 = 0,

and E+ := Ei+ . Any sub-vector bundle of Lie G which is ample is contained in E+ . Consequently, to complete the proof of Theorem 2.6, it is sufficient to show that E+,K is the Lie algebra of a unipotent algebraic subgroup of G. This will follow from the following Lemma, inspired by a similar observation by Shepherd-Barron ([SB92], Lemma 9.1.3.1): Lemma 2.9. (i) For any i ∈ {1, . . . , N} such that µi ≥ 0, Ei,K is a Lie subalgebra of Lie G. Moreover, for any element j ∈ {1, . . . , i}, Ej,K is a Lie ideal in Ei,K . (ii) For any i ∈ {1, . . . , N} such that µi > 0, the quotient Lie algebra Ei,K /Ei−1,K is abelian. Indeed, combined with the first assertion of Theorem 2.6, Lemma 2.9 (i) shows that the K-vector spaces E1,K ⊂ · · · ⊂ Ei+ ,K are the Lie algebras of connected algebraic subgroups H1 ⊂ · · · ⊂ Hi+ in G, and that H1 , . . . , Hi+ −1 are normal subgroups of Hi+ . Moreover, Lemma 2.9 (ii) and Lemma 2.8 show that the algebraic groups H1 , H2 /H1 ,…,Hi+ /Hi+ −1 are additive groups.

388

Jean-Benoît Bost

Proof of Lemma 2.9 Observe that, for any i ∈ {1, . . . , N}, the maximal slope of E/Ei−1 is µi . Moreover, for any (i, j ) ∈ {1, . . . , N}2 , Ei /Ei−1 ⊗ Ej /Ej −1 is semistable of slope µi + µj , and consequently the minimal slope of Ei ⊗ Ej is µi + µj . For any i ∈ {1, . . . , N − 1}, we may consider the following morphism of vector bundles over C: [.,.]

αi : Ei ⊗ Ei → E ⊗ E −−−→ E  E/Ei . If µi ≥ 0, the minimal slope 2µi of its source Ei ⊗Ei is larger than the maximal slope µi+1 of E/Ei , and therefore αi is the zero morphism. This shows that, if µi ≥ 0, then Ei,K is a Lie subalgebra of Lie G. The other assertions of Lemma 2.9 are similarly established by considering the morphisms: [.,.]

βij : Ei ⊗ Ej → E ⊗ E −−−→ E  E/Ej . and [.,.]

γi : Ei ⊗ Ei −−−→ Ei  Ei /Ei−1 .


Lemma 2.9 and Theorem 2.6

3 The canonical semi-norm attached to a germ of analytic curve in a complex algebraic variety 3.1 The basic construction Consider a complex algebraic variety X, a point P in X, and a germ C of smooth analytic curve through P in X. In this section, we describe a construction which attaches – in a canonical way – a semi-norm .(X,P ,C) on the tangent line TP C to any such data (X, P , C). This construction focuses on the metric behavior of i and γ i already considered in the proof of the algebraicity the evaluation maps ηD D criterion Proposition 2.2, and turns out to play a key role in the arithmetic algebraization theorem, Theorem 4.2 infra. For a while, let us assume that X is complete and consider a line bundle L on X. Let us also choose a norm .0 on the complex line TP C and a continuous hermitian metric . on L. Then, for any non-negative integer D, we may consider the D-th tensor power of this hermitian metric on L⊗D and the L∞ -norm .L∞ it induces on the finite dimensional complex vector space ED := (X, L⊗D ). For any non-negative integer i, we may also consider the norm .i,D on the complex line TˇP C ⊗i ⊗ L⊗D P

Germs of analytic varieties in algebraic varieties

389

deduced by duality and tensor product from the norm .0 on TP C and the norm . on LP . By applying the construction of “auxiliary functions” in section 2.2 to the formal germ of curve Vˆ through P defined by the germ of analytic curve C, we define subspaces i := {s ∈ E D | s|C has a zero of order ≥ i at P }, ED

and evaluation maps i ˇ ⊗i ⊗ L⊗D γDi : ED −→ S i TˇP C ⊗ L⊗D P  TP C P

which send a section of L⊗D vanishing up to order i − 1 at P along C to the i-th “Taylor coefficient” of its restriction to C. Finally we may consider the operator norm γDi  :=

max

i ,s ∞ ≤1 s∈ED L

γDi (s)i,D

i and Tˇ C ⊗i ⊗ L⊗D considered above. of γDi with respect to the norms on ED P P A straightforward application of Cauchy’s inequalities establishes the existence of positive real numbers r and C such that, for any non-negative integers i and D,

γDi  ≤ r −i C D .

(3.1)

Equivalently, if we let a := log r −1 and b := log C, we have: log γDi  ≤ ai + bD,

(3.2)

and consequently the upper limit ρ(X, P , C, L) := lim sup

i D →+∞

1 1 log γDi (= lim sup log γDi ) x→+∞ i i i ≥x

(3.3)

D

belongs to [−∞, +∞[. Moreover one easily checks that it does not depend on the choice of the metric . on L and that, if .0 is replaced by eλ .0 , then ρ(X, P , C, L) is replaced by ρ(X, P , C, L) − λ. This shows that .(X,P ,C,L) := eρ(X,P ,C,L) .0

(3.4)

is a semi-norm on the complex line TP C independent of the choices of the auxiliary metrics .0 and .. It vanishes iff ρ(X, P , C, L) = −∞. The following properties of the semi-norm .(X,P ,C,L) are simple consequences of its definition: Lemma 3.1. 1) For any two line bundles L1 and L2 over X such that there exists a regular section of Lˇ1 ⊗ L2 which does not vanish at P , we have: .(X,P ,C,L1 ) ≤ .(X,P ,C,L2 ) .

(3.5)

390

Jean-Benoît Bost

2) For any line bundle L over X and any positive integer k, .(X,P ,C,L⊗k ) = .(X,P ,C,L) .

(3.6)

Lemma 3.1 shows that, when X is projective, the set of semi-norms .(X,P ,C,L) on TP C obtained by varying the line bundle L possesses one greatest element, namely the semi-norm .(X,P ,C,L) where L is any ample line bundle on X. This greatest semi-norm will be called the canonical semi-norm on TP C and denoted .(X,P ,C) .

3.2 Birational invariance of the canonical semi-norm It turns out that the construction of the canonical semi-norm may be extended to the situation where X is an arbitrary complex variety (not necessarily projective) and that it satisfies remarkable “functorial” properties. This will be a consequence of the following proposition. Proposition 3.2. Let f : X → X be a morphism of complete complex algebraic varieties, and let P be a point in X and C a germ of smooth complex analytic curve through P in X such that the restriction of f to C defines an analytic isomorphism from C onto a germ of smooth complex analytic curve C through P := f (P )6 . 1) For any line bundle L on X, the isomorphism of complex lines Df|C (P ) = Df (P )|TP C : TP C −→ TP C satisfies, for any v ∈ TP C : Df (P )v(X,P ,C,L) ≤ v(X ,P ,C ,f ∗ L) .

(3.7)

2) Moreover, equality holds in (3.7) if one of the following conditions holds: i) the canonical morphism of sheaves OX −→ f∗ OX is an isomorphism; ii) the line bundle L is ample and the canonical morphism of sheaves OX −→ f∗ OX is an isomorphism on some open neighborhood of P in X. Observe that condition i) in 2) holds for instance when f is dominant (or equivalently surjective) with geometrically connected generic fiber and X is normal. Proof of Proposition 3.2 For any non-negative integer i, we shall denote Ci (resp. Ci ) the i-th infinitesimal neighbourhood of p (resp. P ) in C (resp. C ). Let us choose a 6 This condition is satisfied iff the tangent space T C is not contained in the kernel of the differential P Df (P ) (which is a linear map between Zariski tangent spaces, from TP X to TP X).

391

Germs of analytic varieties in algebraic varieties

continuous hermitian metric on L and let us endow f ∗ L with this metric pulled back by f . Let us also choose some norms on the complex lines TP C and TP C such that Df|C (P ) : TP C −→ TP C is an isometry. The inequality (3.7) will be obtained by examining the following commutative diagrams: ED := (X, L⊗D )

ϕD

/ E := (X , f ∗ L⊗D ) D η iD

i ηD

 (Ci , L⊗D )

 / (C , f ∗ L⊗D ) i

∼

where the horizontal maps are defined by pulling back sections of L⊗D by f , and i and η i denotes the restriction maps. where ηD D Indeed, these diagrams induce the following ones: i := ker ηi−1 ED D

ϕD

/ E i := ker ηi−1 D D γ iD

γDi

 TˇP C ⊗i ⊗ L⊗D P

(3.8)

i ID

/ TˇP C ⊗i

 ⊗ f ∗ L⊗D P .

As in the construction of .(X,P ,C,L) described in the previous section, the metrics introduced above may be used to define the norms of γDi and of γ iD . Moreover, in the commutative diagram (3.8), the map ϕD decreases the L∞ -norms, while the map

ˇ IDi : TˇP C ⊗i ⊗ L⊗D P −→ TP C

⊗i

ˇ ⊗ f ∗ L⊗D P  TP C

⊗i

⊗ L⊗D P

may be identified with t Df|C (P )⊗i ⊗ I dL⊗D , which is an isometry. This shows that P

i

γDi  ≤ γ D .

(3.9)

Using the definition of .(X,P ,C,L) and .(X ,P ,C ,f ∗ L) (see (3.3) and (3.4)), this yields (3.7). To prove that equality holds in (3.7) when condition i) or ii) holds, first observe that these conditions imply that f is surjective, and therefore that the maps ϕD preserve the L∞ -norms. Moreover, when condition i) is satisfied, the linear maps ϕD define isomorphisms i

i ϕD : ED −→ E D ,

and therefore equality holds in (3.9), hence in (3.7). To prove the equality in case ii), one may consider the Stein factorization of f and write it as the composition of a morphism satisfying i) and of a finite morphism. Thus one is reduced to handle the case where L is ample and where f is a finite morphism that defines an isomorphism between open neighbourhoods of P in X and of P in

392

Jean-Benoît Bost

X . In this case, as L and f ∗ L are ample, the metrics which appear in (3.7) are the canonical metrics .(X,P ,C) and .(X ,P ,C ) , and we may replace L by any ample line bundle on X. In particular, if
denotes the coherent ideal sheaf in OX defined as the annihilator of the cokernel of the canonical morphism OX −→ f∗ OX , we may assume that there exists a section s0 in (X,
.L) which does not vanishes at P . Then for any non-negative integers i and D, and any s ∈ E iD , the product f ∗ s0 ⊗ s may be i and its image s0 (P ) ⊗ γ iD (s ) by γ iD+1 coincides with written ϕD (s) with s ∈ ED+1 i (s0 (P ) ⊗ γDi (s)). This shows that ID+1 i

γ D  ≤

s0 L∞ (X) i γD+1 . s0 (P )

These estimates lead to the inequality opposite to (3.7).

 

Corollary 3.3. Let X and X be two projective complex varieties and i : U → U an isomorphism between Zariski open subsets U and U of X and X respectively. If P is a point of X and C a germ of smooth analytic curve through P in X, and if P := f (P ) and C := f (C ), then the isomorphism Di(P ) : TP C −→ TP C satisfies: Di(P )v(X,P ,C) = v(X ,P ,C ) ,

for any v ∈ TP C .

(3.10)

Proof. By considering the closure of the graph of i in X × X and its projections to X and X , we see that to prove (3.10) we may assume that i is the restriction of some (birational) morphism i : X → X. Let us also choose ample line bundles L and L on X and X respectively. By the equality case (ii) in Proposition 3.2, we have: Di(P )v(X,P ,C) = Di(P )v(X,P ,C,L) = v(X ,P ,C ,i ∗ L) .

(3.11)

Besides, if k is a large enough positive integer, the line bundle Lˇ ⊗ i ∗ L⊗k admits a regular section on X which does not vanish at P . (Indeed, (X , Lˇ ⊗ i ∗ L⊗k ) may be identified with (X, i∗ Lˇ ⊗ L⊗k ) and i∗ Lˇ ⊗ L⊗k is generated by its global sections for k >> 0.) Therefore, by applying Lemma 3.1, 1) and 2), we get: .(X ,P ,C ,L ) ≤ .(X ,P ,C ,i ∗ L⊗k ) = .(X ,P ,C ,i ∗ L) . This shows that .(X ,P ,C ,i ∗ L) = .(X ,P ,C ) . Finally, (3.10) follows from (3.11) and (3.12).

(3.12)  

Germs of analytic varieties in algebraic varieties

393

Let us now assume that the variety X is arbitrary, and consider some quasiprojective open neighbourhood U of P in X and some projective variety U containing U as an open subvariety. Corollary 3.3 shows that the canonical metric .(U ,P ,C) on TP C is independent of the choices of U and U , and we shall extend the previous definition of the canonical metric by letting: .(X,P ,C) := .(U ,P ,C) .

3.3 Functorial properties of the canonical semi-norm We may now generalize the “functoriality properties” established in Proposition 3.2 and Corollary 3.3 when the ambient varieties are projective. Indeed, from these properties and the definition of the canonical semi-norm, it is straightforward to deduce assertions 1) and 2-i) in the following proposition: Proposition 3.4. Let X (resp. X ) a complex algebraic variety, P (resp. P ) a point in X (resp. X ), and C (resp. C ) a germ of smooth analytic curve through P (resp. P ) in X (resp. X ). Let also f : X → X be a morphism of complex algebraic varieties such that f (P ) = P and f|C is an analytic isomorphism from C to C. 1) The isomorphism of complex lines Df|C (P ) : TP C −→ TP C satisfies, for any v ∈ TP C : Df (P )v(X,P ,C) ≤ v(X ,P ,C ) .

(3.13)

2) Moreover, equality holds in (3.13) if one of the following conditions holds: i) the morphism f defines an isomorphism from some open neighborhood of P in X onto some open neighborhood of P in X; ii) the morphism f is an embedding. To prove that equality holds in (3.13) when f is an embedding, we may assume that X and Y are projective. Then it is a consequence of the following proposition of independent interest, a stronger form of which is established in Appendix A at the end of this article. Proposition 3.5. Let X be a complex projective variety, Y a closed subvariety of X, L an ample line bundle over X, and . an arbitrary continuous metric on L. There exists C ∈ R∗+ satisfying the following condition: for any positive large enough integer D and any s ∈ (Y, L⊗D ), there exists s˜ ∈ (X, L⊗D ) such that s˜|Y = s

394

Jean-Benoît Bost

and ˜s L∞ ≤ C D sL∞ .

3.4 Canonical semi-norm and capacity Observe that, if a germ C of smooth analytic curve through a point P in a complex algebraic variety is algebraic, then the canonical semi-norm .(X,P ,C) on TP C vanishes. Indeed the direct implication in the algebraicity criterion Proposition 2.2 shows that, if we assume – as we can – the variety X projective and if we denote by L an ample line bundle on X, then the evaluation maps considered in 3.1 i ˇ ⊗i ⊗ L⊗D γDi : ED −→ S i TˇP C ⊗ L⊗D P  TP C P

vanish if i/D is large enough; accordingly, 1 log γDi  = −∞. i i →+∞

ρ(X, P , C, L) := lim sup D

In this section, we derive an upper bound on the canonical metric .(X,P ,C) in terms of classical potential theoretic invariants of a Riemann surface “extending” C, which implies its vanishing when C is algebraic as a very special case. As will be clear in the proof, this lower bound is a geometric version of the classical Schwarz lemma, which plays a prominent role in transcendence and Diophantine approximation proofs. We defer examples of analytic germs with non-trivial canonical semi-norms to section 4.5 infra. 3.4.1 Green functions and Riemann surfaces. Let us briefly recall some basic facts and introduce some notation concerning Green functions on Riemann surfaces. We refer the reader to the monographs [Tsu59], [Rum89], and [Ran95] and to [Bos99], 3.1 and Appendix, for proofs and additional information. Let M be a connected Riemann surface and O a point of M. Consider a relatively compact domain  in M containing O with a non-empty and regular enough boundary ∂. Precisely, we assume that  has only regular boundary points in the sense of potential theory. (This condition is satisfied for instance if the non-empty compact set ∂ is locally connected without isolated points. Actually, it would be enough for the sequel to consider the case where  is the interior of some compact submanifold of codimension 0 with C ∞ boundary.) Then we may consider the Green function, or equilibrium potential, of P in . It is the unique continuous function gP , on M \{O} satisfying the following three conditions: EP1. It vanishes identically on M \ ; EP2. It is harmonic on  \ {O};

Germs of analytic varieties in algebraic varieties

395

EP3. It possesses a logarithmic singularity at O; namely, if z denotes a local holomorphic coordinates on some open neighborhood U of P , we have: gP , = log |z − z(O)|−1 + h

on U \ {O},

where h is an harmonic function on U . The Green function gO, represents the electric field of a unit charge placed at the point O in the two-dimensional world modeled by M, when  (resp. M \ ) is made of an insulating material (resp. of a conducting material wired to the earth). It is positive on  \ {O}, and conditions EP2 and EP3 may be expressed as the equality of currents: 1 (3.14) dd c gO, = − δO on . 2 The value h(P ) of the function h in condition EP3 may be interpreted as the capacity of M \  with respect to P . Of course, this value depends on the choice of cap the local coordinate z. Intrinsically, we may define a “capacitary norm” .P , on the ∂ complex line TO M = C ∂z by the equation: |P 

∂ e−gO, (Q) cap P , := e−h(P ) = lim . Q→O |z(Q) − z(O)| ∂z |P

(3.15)

Let us now assume that M is not compact and consider an increasing sequence (n )n∈N of relatively compact domains of M containing O, with “regular” boundaries, such that M = n∈N n . Then the sequence of Green functions (gO,n )n∈N is noncap decreasing, and consequently the sequence of norms (.O,n )n∈N is non-increasing. Their limit behavior turns out to depend on the “type” of the Riemann surface M in the sense of the classical works of Myrberg–Nevanlinna–Ahlfors (see for instance [Ahl52] and [AS60], chapter IV). Recall that a connected Riemann surface S is said to be “parabolic” in the sense of Myrberg, or equivalently, to have “null boundary” in the sense of R. Nevanlinna, or to belong to the class OG , when any negative subharmonic function on S is constant. This arises for instance when S is a complex (smooth connected) algebraic curve. Otherwise, S is said to be “hyperbolic”, or to have “positive boundary”. Using this terminology, the following alternative holds: (1) If the Riemann surface M is hyperbolic, then the pointwise limit gO,M of (gO,n )n∈N is everywhere finite on M \ {O}. Moreover it is a positive harmonic function on M \ {O}, with a logarithmic singularity at O – indeed, gO,M is minimal amongst the functions satisfying these conditions, and, by definition, is the Green cap function of O in M. We may also define a capacitary norm .O,M on TO M by the equality 

e−gO,M (Q) ∂ cap O, := lim . Q→O |z(Q) − z(O)| ∂z |P cap

This norm coincides with the limit limn→∞ .O,n .

(3.16)

396

Jean-Benoît Bost

(2) If the Riemann surface M is “parabolic”, then the pointwise limit of (gO,n )n∈N is everywhere +∞ and cap

lim .O,n = 0.

n→∞

Then we let: cap

.O,M = 0. cap

To sum up, the “capacitary semi-norm” .O,M on TO M always coincide with the cap limit limn→∞ .O,n , and vanishes iff M is parabolic. It is natural to extend the preceding discussion to the situation where M is compact cap (hence parabolic) by letting .O,M = 0 in that case also. Observe finally that, if F is any closed polar subset of M (e.g., a closed discrete cap cap subset) not containing O, then the semi-norms .O,M and .O,M\F coincides. Indeed, the Riemann surfaces M and M \ F have the same type, and, when they are hyperbolic, gO,M\F is the restriction of gO,M to M \ F . 3.4.2 An upper bound on canonical semi-norms. As before, we consider a complex algebraic variety X, a point P in X, and a germ C of smooth analytic curve through P in X. Let also M be a connected Riemann surface, O a point in M and f : M −→ X an analytic map which sends O to P and maps the germ of M at O to the germ C. (Thus f defines an analytic isomorphism from the germ of M at O onto the germ C, unless Df (O) vanishes.) Proposition 3.6. For any v in TO M, we have: cap

Df (O)v(X,P ,C) ≤ vO,M .

(3.17)

In particular, this proves: Corollary 3.7. If f maps the germ of M at O isomorphically onto the germ C at P and if the Riemann surface M is parabolic, then the canonical semi-norm .(X,P ,C) vanishes. Applied to the normalization of the Zariski closure of C, this corollary shows again that the canonical semi-norm .(X,P ,C) vanishes when the germ C is algebraic. Observe that the capacitary norm at the origin on the open disk D(0, 1) is the “standard norm”: ∂ cap  = 1.  ∂z |0 0,D(0,1)

Germs of analytic varieties in algebraic varieties

397

Indeed, the disk D(0, 1) is hyperbolic and g0,D(0,1) (z) = log |z|−1 for any z ∈ C. Therefore the special case of Proposition 3.6 where (M, O) = (D(0, 1), 0) reads: Corollary 3.8. For any analytic map f : D(0, 1) −→ X which sends 0 to P and maps the germ of C at 0 to the germ C, we have: Df (0)(

∂ )(X,P ,C) ≤ 1. ∂z

(3.18)

In more geometric terms, this estimate asserts that the canonical semi-norm .(X,P ,C) on TP C is bounded from above by the Poincaré metric at P on any Riemann surface which“extends C and maps to X”. Proof of Proposition 3.6. To establish (3.17), we may assume that M is not compact (by deleting one point if necessary) and then it is enough to prove that, for any relatively compact domain  in M with regular boundary containing O, the following inequality holds for any v in TO M: cap

Df (O)v(X,P ,C) ≤ vO, .

(3.19)

Clearly, we may also assume that Df (O) is not zero (hence an isomorphism) and that X is projective. To derive (3.19), we choose an ample line bundle L on X, a C ∞ hermitian metric . on L, an holomorphic coordinate z on some open neighborhood of O in M that vanishes at O, and we define a norm .0 on TP C by letting      Df (O) ∂  = 1.  ∂z |O 0 Finally, we choose a real valued C ∞ function ψ, defined on some open neighborhood of  in M such that ψ(O) = 0 and dd c ψ ≥ f ∗ c1 (L) on . (If h1 and h2 are two holomorphic functions vanishing at O defined on some open neighborhood of  with disjoint ramification divisors, then we can take ψ := C(|h1 |2 + |h2 |2 ) for any large enough C in R∗+ .) i , γ i , γ i  and ρ(X, P , C, L) as in section 3.1, Using these data, we may define ED D D and the inequality (3.19) may be rewritten as:    ∂ cap  . (3.20) ρ(X, P , C, L) ≤   ∂z |O  O,

398

Jean-Benoît Bost

i , we have: To prove (3.20), observe that, for any section s in ED

γDi (s) = lim

Q→O

s  f (Q) . |z(Q)|i

Therefore, provided s does not vanish identically on f (M), log γDi (s) is the value at O of log s  f  − i log |z|, which defines a locally integrable continuous function with values in [−∞, +∞[ on a neighborhood of O in M. This is also the value at O of the function     ∂ cap D   (3.21) + ψ log s  f  + i gO, + log  ∂z |O O, 2 from M to [−∞, +∞[, which indeed is subharmonic on . This follows from the equality of currents on M dd c log s  f 2 = δf ∗ div s − f ∗ c1 (L), from which we derive: dd c (log s  f 2 + Dψ) ≥ iδO = −2idd c gO, on . By the maximum principle, log γDi (s) is therefore not greater than the supremum of (3.21) on ∂. Since gP , vanishes on ∂, we finally get:    ∂ cap D i   ∞ log γD (s) ≤ log sL + i log  + max ψ. ∂z |O O, 2 ∂ This shows that

   ∂ cap D 1 i  + log γD  ≤ log  max ψ   i ∂z |O O, 2i ∂

and yields (3.19).

 

4 Algebraicity criteria for smooth formal germs of subvarieties in algebraic varieties over number fields In this section, we discuss some algebraization theorems concerning formal germs of subvarieties in algebraic varieties over number fields, which involve the canonical semi-norm studied in the previous paragraphs. These theorems are improvements of the main result of [Bos01] applied to formal germs of curves. When dealing with number fields and p-adic fields, we will use the following notation and terminology.

399

Germs of analytic varieties in algebraic varieties

If K is a number field, its ring of integers will be denoted OK and the set of its finite places (or, equivalently, the set of non-zero prime ideals of OK , or of closed points of Spec OK ) will be denoted Vf (K). For any p in Vf (K), we let Fp be the finite field OK /p, Np := |Fp | the norm of p, Kp (resp. Op ) the p-adic completion of K (resp. of OK ), and | |p the p-adic absolute value on Kp normalized in such a way that, for any uniformizing element  in Op , we have: | |p = Np−1 ; equivalently, if p denotes the residue characteristic of p and e the absolute ramification index of Kp , then: |p|p = Np−e = p−[Kp :Qp ] . If  is an Op -lattice in some finite dimensional Kp -vector space E, the p-adic norm   on E attached to  is defined by the equality n     xi ei  := max |xi |p ,  1≤i≤n

i=1

for any Op -basis (e1 , . . . , en ) of  and any (x1 , . . . , xn ) ∈ K n .

4.1 Sizes of formal subschemes over p-adic fields We now recall some constructions from [Bos01], to which we refer for details and proofs. Let k be a p-adic field (i.e., a finite extension of Qp ), O its subring of integers (i.e., the integral closure of Zp in k), | | : k → R+ its absolute value, and F its residue field. (Actually we might assume more generally that k is any field equipped with a complete non-Archimedean absolute value | | : k → R+ and let O := {t ∈ k | |t| ≤ 1} be its valuation ring.) 4.1.1 Groups of formal and analytic automorphisms. If g := formal power series in k[[X1 , · · · , Xd ]] and if r ∈ R∗+ , we define



I ∈Nd

aI XI is a

gr := sup |aI |r |I | ∈ R+ ∪ {+∞}. I

The “norm” gr is finite iff the series g is convergent and bounded on the open d-dimensional ball of radius r. ˆ d , the formal completion at the origin ˆ d of automorphisms of A The group Aut A k k of the d-dimensional affine space over k, may be identified with the space of dtuples f = (fi )1≤i≤d of formal series fi ∈ k[[x1 , · · · , xd ]] such that f (0) = 0
∂fi  and Df (0) := ∂x (0) belongs to GLn (k). We shall consider the following 1≤i,j ≤d j ˆ d: subgroups of Aut A k

400

Jean-Benoît Bost

• the subgroup Gfor formed by the formal automorphisms f such that Df (0) belongs to GLn (O); • the subgroup Gω formed by the elements f := (fi )1≤i≤d of Gfor such that the series fi have positive radii of convergence; • for any r ∈ R∗+ , the subgroup Gω (r) of Gω formed by the elements f := (fi )1≤i≤d of Gfor such that the series (fi )1≤i≤d satisfy the bounds fi r ≤ r. This group may be seen as the group of analytic automorphisms, preserving the origin, of the open d-dimensional ball of radius r. Moreover,  Gω (r) = Gω . r > r > 0 ⇒ Gω (r ) ⊂ Gω (r) and r>0

4.1.2 The size R(Vˆ ) of a formal germ Vˆ . The filtration (Gω (r))r>0 of the group Gω will now be used to attach a number R(Vˆ ) in [0, 1] to any smooth formal germ Vˆ in an algebraic variety over k, which will provide some quantitative measure of its analyticity. ˆ d , we may consider ˆ d . For any ϕ in Aut A Let Vˆ be a formal subscheme of A k k ˆ d . Moreover, the its inverse image ϕ ∗ (Vˆ ), which is again a formal subscheme of A k following conditions are equivalent: 1. Vˆ is a smooth formal scheme of dimension v. ˆ v × {0} ˆ d such that ϕ ∗ (Vˆ ) is the formal subscheme A 2. There exists ϕ in Aut A k k ˆ d. of A k ˆ v × {0} of A ˆ d. 3. There exists ϕ in Gfor such that ϕ ∗ (Vˆ ) is the formal subscheme A k k Similarly, the following two conditions are equivalent: 1. Vˆ is the formal scheme attached to some germ at 0 of smooth analytic subspace of dimension v of the d-dimensional affine space over k. ˆ v × {0} of A ˆ d. 2. There exists ϕ in Gω such that ϕ ∗ (Vˆ ) is the formal subscheme A k k When they are satisfied, we shall say that the formal germ Vˆ is analytic and smooth. These observations lead to define the size of a smooth formal subscheme Vˆ of ˆ d as the supremum R(Vˆ ) in [0, 1] of the real numbers r ∈]0, 1] for dimension v of A k ˆ v × {0} of which there exists ϕ in Gω (r) such that ϕ ∗ (Vˆ ) is the formal subscheme A k ˆ d . It is positive iff Vˆ is analytic. A k More generally, if X is an O-scheme of finite type equipped with a section P ∈ X(O) and if Vˆ is a smooth formal subscheme of the formal completion Xˆ PK of X := XK at PK , then the size RX (Vˆ ) of Vˆ with respect to the model X of X may be defined as the size of i(Vˆ ), where i : U → AdO is an embedding of some open neighbourhood U in X of the section P into an affine space of large enough dimension d, which moreover maps P to the origin 0 ∈ AdO (O). This definition is independent of the choices of U , d, and i, and extends the previous one.

Germs of analytic varieties in algebraic varieties

401

When Vˆ is a smooth germ of analytic curve, we shall define a p-adic norm on the tangent line TP Vˆ by letting: .(X,P ,Vˆ ) := RX (Vˆ )−1 .0 , where .0 denotes the p-adic norm on TP Vˆ which makes the differential Di(P ) : TP Vˆ → T0 Adk  k d isometric when k d is equipped with the “standard” p-adic norm, the unit ball of which is O d . Observe that, if Vˆ extends to a formal subscheme Vˆ of the formal completion of X along P which is smooth along P , then RX (Vˆ ) = 1. If moreover Vˆ is a formal germ of curve, the norm .(X,P ,Vˆ ) on TP Vˆ is therefore the p-adic norm attached to ˆ its O-lattice defined by the normal bundle of P in V. 4.1.3 Sizes of solutions of algebraic differential equations. It is possible to establish lower bounds on the sizes of formal germs of solutions of algebraic ordinary differential equations. These play a key role in the application of our arithmetic algebraization criterion to the solutions of algebraic differential equations over number fields (see [Bos01], 2.2 and 3.4.3, and infra, 4.6). Proposition 4.1. Let X be a smooth scheme over Spec O, P a section in X(O), and F a sub-vector bundle of rank 1 in TX/O . Let X := Xk , P := Pk , and F := Fk , and let Vˆ be the formal germ of curve in Xˆ P defined by integration of the (involutive) line bundle F in TX . 1) The size R(Vˆ ) of Vˆ with respect to X satisfies the lower bound: 1

R(Vˆ ) ≥ |π| := |p| p−1 .

(4.1)

2) If moreover k is absolutely unramified and if the reduction FF → TXF of F to the closed fiber XF of X is closed under p-th power, then 1

R(Vˆ ) ≥ |p| p(p−1) .

(4.2)

This is proved in [Bos01], Proposition 3.9, with the exponent 3/p2 instead of 1/p(p − 1) in (4.2); however, a closer inspection of the proof shows that indeed it holds with the exponent 1/p(p − 1). Observe that the lower bound (4.1) is basically optimal, as demonstrated by the differential system  ∂ ∂ 2 + (y + 1) OA 2 . X := AO , P := (0, 0), and F := ∂x ∂y Indeed, then Vˆ is the formal germ Graph(x → exp x − 1),

402

Jean-Benoît Bost

the size of which is not larger than the radius of convergence |π| of the exponential series. Observe also that, after exchanging the two coordinates, Vˆ may also be seen as the graph of the series log(1 + t) :=

∞

(−1)n+1 n=1

n

t n,

whose radius of convergence is 1. This shows that the size R(Graph(φ)) of the graph of some formal series φ may be strictly smaller than its radius of convergence.

4.2 Normed and semi-normed lines over number fields We define a normed line L := (LK , (.p ), (.σ )) over a number field K as the data of a rank one K-vector space LK , of a family (.p ) of p-adic norms on the Kp -lines LK ⊗K Kp indexed by the non-zero prime ideals p of OK , and of a family (.σ ) of hermitian norms on the complex lines LK ⊗K,σ C, indexed by the fields embeddings σ : K → C. Moreover the family (.σ ) is required to be stable under complex conjugation. (The data of these families of norms is equivalent to the data of a family (.v )v , indexed by the set of all places v of K, of v-adic norms on the rank one vector spaces Lv := LK ⊗K Kv over the v-adic completions Kv of K.) ˇ (resp. by L ⊗ M) If L and M are normed lines over K, then we will denote by L the normed line over K defines by the K-line LˇK := HomK (LK , K) (resp. the K-line LK ⊗K MK ) equipped with the p-adic and hermitian norms deduced by duality (resp. by tensor product) from the ones defining L (resp. L and M). We shall say that a normed K-line is summable if, for some (or equivalently, for any), non-zero element l of LK , the family of real numbers (log lp )p is summable. Then we may define its Arakelov degree as the real number

 L := deg log l−1 + log l−1 (4.3) p σ . p

σ

Indeed, by the product formula, the right-hand side of (4.3) does not depend on the choice of l. ˇ If L and M are summable normed lines over K, then the normed K-lines L and L ⊗ M also are summable. Moreover, as a straightforward consequence of the definition of the Arakelov degree, we have: ˇ = −deg L L deg

(4.4)

Germs of analytic varieties in algebraic varieties

403

 L + deg  M.  L ⊗ M = deg deg

(4.5)

and Observe that hermitian line bundles over Spec OK , as usually defined in Arakelov geometry (see for instance [Bos01], 4.1.1) provide examples of normed lines over K. Namely, if L = (L, (.σ )σ :K→C )) is such an hermitian line bundle – so L is a projective OK -module of rank 1, and (.σ )σ :K→C ) is a family, invariant under complex conjugation, of norms on the complex lines Lσ := L ⊗σ :OK →C C – the corresponding normed K-line is LK equipped with the p-adic norms defined by the Op -lattices L⊗OK Op in L⊗OK Kp  L⊗K Kp and with the hermitian norms (.σ ). The normed lines so-defined are summable, and their Arakelov degree, as defined by (4.3), coincide with the usual Arakelov degree of hermitian line bundles. It is convenient to extend the definitions of normed lines and Arakelov degree as follows: we shall define a semi-normed K-line L as a rank one K-vector space LK equipped with families of semi-norms (.p ) and (.σ ), where the latter is assumed to be stable under complex conjugation. (In other words, we allow some of the .p or .σ to vanish.) We shall say that the Arakelov degree of a semi-normed K-line L is defined if, for some (or equivalently, for any), non-zero element l of LK , the family of real numbers (log+ lp )p is summable. Then we may again define its Arakelov degree by means of (4.3), where we follow the usual convention log 0−1 = +∞. It is a well defined element of ] − ∞, +∞]. The definition of the tensor product of normed K-lines immediately extends to semi-normed K-lines. Moreover, if two semi-normed K-lines have well defined Arakelov degrees, then their tensor product also and the additivity relation (4.5) still holds. (Observe however that the duality relation (4.4) makes sense only for summable normed K-lines.)

4.3 An arithmetic algebraization theorem We are now in position to state an arithmetic analogue of the algebraization criterion of section 2.3, which concerns germs of formal curves in algebraic varieties over number fields: Theorem 4.2. Let X be a quasi-projective variety over a number field K, P a point in X(K) and Vˆ a germ of smooth formal curve in X through P that is analytic at every place7 . Let X be a model of X, quasi-projective over Spec OK , such that P extends to a section P in X(OK ), and let t be the semi-normed K-line defined by the tangent line 7 Recall that this means that Vˆ is a one-dimensional smooth formal subscheme of X ˆ P such that, for any

non-zero prime ideal p in OK (resp. any field embedding σ : K → C), the smooth formal curve VˆKp (resp. Vˆσ ) in XK (resp. Xσ ) is indeed Kp -analytic (resp. C-analytic). p

404

Jean-Benoît Bost

TP Vˆ equipped with the p-adic norms .(XO ,PO ,VˆO ) and the canonical hermitian p p p semi-norms .(Xσ ,Pσ ,Vˆσ ) . If the Arakelov degree of t is well-defined and if  t > 0, deg

(4.6)

then the formal germ Vˆ is algebraic. Observe that, conversely, if Vˆ is any algebraic smooth formal germ through a rational point in an algebraic variety over a number field, then it is analytic at every place, almost all its p-adic sizes are equal to 1 8 , and its complex canonical seminorms vanish. In particular, the Arakelov degree of t is well defined, and assumes the value +∞. Corollary 4.3. Let Vˆ be a smooth formal germ of curve through a rational point P in an algebraic variety X over a number field K, analytic at every place. Let us denote by (Rp ) the family of p-adic sizes of Vˆ , defined with respect to some model U over OK of an open neighborhood U of P in X such that P extends to an integral point P ∈ U(OK ), and suppose that the following conditions are satisfied: 1) the product  Rp , p∈Spec OK \{(0)}

which is a well-defined number in [0, 1], is positive; 2) for at least one embedding σ : K → C, the canonical semi-norm .(Xσ ,Pσ ,Vˆσ ) vanishes. Then the formal germ Vˆ is algebraic. Observe that condition 1) is actually independent of the choice of U and U. Moreover, condition 2) is satisfied if, for some embedding σ , there exists a parabolic Riemann surface M, a point O in M and an analytic map f : M −→ Xσ (C) which defines an isomorphism from the formal germ of M at O to Vˆσ . In this way, we recover the main result of [Bos01] (Theorem 3.4) for one-dimensional formal germs as a special instance of Theorem 4.2. Corollary 4.3 is a straightforward consequence of Theorem 4.2: after possibly shrinking U and changing U, we may assume that U is quasi-projective over OK and  t is well defined, apply Theorem 4.2 to X = U; indeed, condition 1) shows that deg and condition 2) that its value is +∞. Theorem 4.2 and Corollary 4.3 are in the same spirit as the algebraization theorems for formal germs of D.V. and G.V. Chudnovsky ([CC85a], Section 5, and [CC85b], Theorem 1.2) and André ([And89], Chapter VIII, especially Theorem 1.2, [And99], 8Actually, for any model X of X over O , there is a non-empty subscheme Spec O [1/N ] and a section K K P ∈ X(OK [1/N ]) such Vˆ extends to a formal subscheme of X along P that is smooth over OK [1/N ].

Germs of analytic varieties in algebraic varieties

405

Theorem 2.3.1, and [And02], Theorem 5.4.3), which however are technically somewhat different.

4.4 Proof of the algebraization theorem The proof of Theorem 4.2 is similar to the proof of the algebraization criterion over function fields, Theorem 2.5. It constitutes a refined variant of the proof of the main result (Theorem 3.4) in [Bos01], and, like the latter, it relies on some simple inequalities relating slopes of hermitian vector bundles and heights of linear maps, for which we refer to [Bos01], 4.1. In the sequel, we freely use the basic definitions and results concerning hermitian vector bundles, slopes and height of linear maps which are recalled in loc. cit. 4.4.1 Auxiliary hermitian vector bundles and linear maps. Observe that X may be imbedded, as a scheme over Spec OK , into some projective space PN OK . By replacing N X by its closure in POK , we may assume that it is projective. We may also assume that Vˆ is Zariski dense in X, by replacing X by the closure in PN of the Zariski closure OK

Z of Vˆ in X considered in section 2.1. Observe that these reductions leave unchanged the (semi-)norms defining the generalized hermitian line bundle t. (For the p-adic norms, this follows from the independence of the size of a formal germ with respect to the imbedding i used to define it; for the archimedean canonical semi-norms, this follows from Proposition 3.4. Actually, we could avoid to rely on this non-trivial Proposition by not assuming that Vˆ is Zariski dense. This would only require more complicated notation and minor modifications in the proof below.) Let us also choose the following additional data: – an hermitian line bundle L := (L, .L ) on X such that L := LK is ample on X := XK ; – a positive Lebesgue measure µ on X(C), invariant under complex conjugation (see [Bos01], 4.1.3); – a family (.0,σ )σ :K→C , invariant under complex conjugation, of norms on the complex lines (TP Vˆσ )σ :K→C . Using these data, we may define: – for any positive integer D, the direct image ED := π∗ L⊗D of L⊗D by the structural morphism π : X → Spec OK . (In other words, ED is the locally free coherent sheaf on Spec OK associated to the OK -module (X, L⊗D ).) – the L2 -norms (.L2 ,σ )σ :K→C on the finite dimensional complex vector spaces ED,σ  (Xσ (C), L⊗D σ ) associated to the measure µXσ (C) and the D-th tensor power of the given metric .L on Lσ . By endowing ED with these hermitian norms, we obtain an hermitian vector bundle E D .

406

Jean-Benoît Bost

– a normed K-line t 0 , associated to an hermitian line bundle over Spec OK , by endowing the K-line TP Vˆ with the archimedean norms (.0,σ )σ :K→C and with its naive OK -structure9 defined from the model X of X. Theorem 4.2 will be established by applying the algebraicity criterion involving evaluation maps established in section 2.2 (see Proposition 2.2 and Lemma 2.4), and i , ηi , and γ i as in this section. Observe that E := (X, L⊗D ) we define ED , ED D D D may be identified with ED,K . Moreover, since Vˆ is Zariski dense in X, for any given D, the evaluation map i : ED := (X, L⊗D ) −→ (Vi , L⊗D ) ηD i+1 vanishes – provided i is large enough. In particular, is injective – and therefore ED

i+1 i rk (ED /ED ) = rk ED . i≥0

For any p ∈ Spec OK \ {0}, the size of the formal germ VˆKp with respect to the model XOp will be denoted Rp . Since the Arakelov degree of t is well defined, the series with positive terms

log Rp−1 p∈Spec OK \{0}

has a finite sum. By definition, the canonical semi-norm on TP Vˆσ is given by   1 .(Xσ ,Pσ ,Vˆσ ) = exp lim sup log γDi σ .0,σ , i/D→+∞ i where γDi σ denotes the operator norm of i i : ED,σ −→ TˇP Vˆσ⊗i ⊗ L⊗D γD,σ Pσ

when the source space is equipped with the L∞ -norm and the range space with the norm deduced by tensor product from the norms .0,σ on TP Vˆσ and .L on LPσ . As a matter of fact, we could – and, in the sequel, we shall – use the L2 -metric on i (namely, the restriction of the one on ED,σ = ED,σ considered above), and still ED,σ define the same canonical semi-norm. Indeed, the logarithm of the ratio of the L∞ and L2 norms on ED,σ is O(D) when D goes to infinity (see for instance [Bos01], 4.1.3). 9 In other words, for any p ∈ Spec O \ {0}, the p-adic norm on T Vˆ ⊗ K defining t is the norm 0 K P K p

.0 considered at the end of section 4.1.2. Equivalently, the OK -submodule of TˇP Vˆ defining the integral structure of the dual hermitian vector bundle tˇ0 is given by the image of the composite map P ∗ 1X/O

K

→ (P ∗ 1X/O )K  1X/K,P → TˇP Vˆ . K

407

Germs of analytic varieties in algebraic varieties

The very definitions of the normed lines t and t 0 and of their Arakelov degree show that the latter satisfy the following relation:  t = deg  t0 + deg



log Rp −

p∈Spec OK \{0}

1 log γDi σ . i/D→+∞ i lim sup

σ :K→C

(4.7)

 t belongs to ]0, +∞], there exists positive real numbers λ and Consequently, as deg d such that, for any (D, i) ∈ N>0 × N satisfying i > λD,  t0 + deg



log Rp −

p∈Spec OK \{0}

σ :K→C

1 log γDi σ ≥ d. i

(4.8)

4.4.2 Application of the slope inequalities. We are going to show that the ratio  i+1 i i≥λD (i/D)rk (ED /ED ) (4.9) rk ED stays bounded when D goes to infinity. According to Lemma 2.4 and (2.5), this will prove that Vˆ is algebraic. As in [Bos01], our main tool will be the slope inequalities applied to the evaluation morphisms n ηD : ED := ED,K −→ (Vn , L⊗D ). n is injective, the slope inequalities of loc. cit., Specifically, if n is so large that ηD n , and Proposition 4.6, applied to the hermitian vector bundle E D , the linear map ηD ⊗D the filtration of (Vn , L ) by the order of vanishing read as the following estimates (compare [Bos01], (4.18)):

 µ(E D ) ≤

1 i+1 i rk ((ED /ED ) rk ED i≥0 (4.10)   ⊗i ⊗i ⊗D i ⊗D  (tˇ0 ⊗ P ∗ L ) + [K : Q]h(E D , tˇ0 ⊗ P ∗ L , γ i ) . deg D

The left hand side of (4.10) is the slope of E D :  µ(E D ) :=

 (E D ) deg . rk ED

⊗i ⊗D Recall also that h(E D , tˇ0 ⊗ P ∗ L , γDi ) denotes the height of the linear map γDi . By definition, it is given by the sum of the “local norms” of γDi : ⊗i i ⊗D [K : Q]h(E D , tˇ0 ⊗ P ∗ L , γDi ) =

p∈Spec OK \{0}

log γDi p +

σ :K→C

log γDi σ ,

408

Jean-Benoît Bost

where the archimedean norm γDi σ has the same meaning as above, and where the p-adic norm γDi p is defined as the operator norm of i i γD,K : ED,K −→ TˇP VˆK⊗ip ⊗ L⊗D P ,Kp , p p i (resp. on TˇP VˆK⊗ip ⊗ L⊗D defined by using the p-adic norm on ED,K P ,Kp ) defined by p

i (resp. by P ∗ (tˇ0⊗i ⊗ L⊗D )Op ). the lattice ED,O p As shown in [Bos01], Proposition 4.4, the left hand side of (4.10) satisfies the following lower bound, where c denotes some positive constant, and D any natural integer:

 µ(E D ) ≥ −cD.

(4.11)

To derive an upper bound on the right hand side of (4.10), first observe that ⊗i  t 0 + D deg  L.  (tˇ0 ⊗ P ∗ L⊗D ) = −i deg deg

(4.12)

γDi ,

To estimate the height of recall that, from the definition of the p-adic sizes Rp , it follows that, for any p ∈ Spec OK \ {0}, γDi p ≤ Rp−1 (see [Bos01], Lemma 3.3 and 4.9). Consequently,

⊗i i ⊗D [K : Q] h(E D , tˇ0 ⊗ P ∗ L , γDi ) ≤ i

p∈Spec OK \{0}

log Rp−1 +

σ :K→C

log γDi σ

(4.13) Moreover, the archimedean norms γDi σ satisfy the Cauchy type estimates (3.1) and (3.2). Therefore, there exist constants α and β, such that, for any non-negative integers D and i:

log γDi σ ≤ αi + βD. (4.14) σ :K→C

From (4.13) and (4.14), we already derive the existence of some constant c(λ) such that, for any natural integer D,

 ⊗i 1 i+1  i  t 0 + [K : Q]h(E iD , tˇ0 ⊗ P ∗ L⊗D , γ i ) rk ((ED /ED ) − i deg D rk ED 0≤i≤λD

≤ c(λ)D.

(4.15)

The slope inequality (4.10), combined with the lower bound (4.11) on its left hand side and with (4.12) and (4.15), leads to the estimates:

i+1 i  P ∗ L + c(λ)D + 1 − cD ≤ D deg rk ((ED /ED ) rk ED i>λD (4.16)   i ˇ⊗i ⊗D ∗ i  t 0 + [K : Q]h(E , t ⊗ P L , γ ) . − i deg D

0

D

Germs of analytic varieties in algebraic varieties

409

Moreover, (4.8) and (4.13) show that, if i > λD, then ⊗i  t 0 + [K : Q]h(E D , tˇ0 ⊗ P ∗ L⊗D , γ i ) ≤ −id. −i deg D

Together with (4.16), this leads to the upper bound  i+1 i  P ∗L c + c(λ) + deg i≥λD (i/D)rk (ED /ED ) ≤ rk ED d and concludes the proof.

4.5 Analytic germs with positive canonical semi-norms In this section, we apply our algebraization theorem to investigate the canonical seminorm associated to a germ of smooth analytic curve in the affine plane A2 (C). We may restrict to analytic germs C through the origin (0, 0) in A2 (C), the restriction to which of the first projection A2 −→ A1 (z1 , z2 )  −→ z1 is étale. These germs are exactly the germs of the form Cϕ := Graph(ϕ), where ϕ(z) =

+∞

an z n

n=1

is a complex formal series with positive radius of convergence. For any such germ, we let ∂ ∂ vϕ := + ϕ (0) . ∂z1 ∂z2 It a basis vector of the complex line T(0,0) Cϕ . Observe that, according to Corollary 3.8, for any such series ϕ of radius of convergence at least 1, we have: vϕ (A2 ,(0,0),Cϕ ) ≤ 1. Moreover, as observed in section 3.4, vϕ (A2 ,(0,0),Cϕ ) = 0 when Cϕ , or equivalently ϕ, is algebraic. Besides, if the coefficients an of the series ϕ are integers, then the formal germ Vˆ through the origin in A2Q defined as the graph of ϕ seen as a formal series is analytic at

410

Jean-Benoît Bost

every place. Actually it is straightforward to check that, for any prime number p, the p-adic size of this germ, computed with respect to the model A2Zp is 1 and that, with

the notation of Theorem 4.2 applied with K = Q, X = A2Q , and X = A2Z , we have:  t = − log vϕ (A2 ,(0,0),C ) . deg ϕ According to Theorem 4.2, the germ Cϕ is therefore algebraic if vϕ (A2 ,(0,0),Cϕ ) < 1. These observations establish the following proposition:

 n Proposition 4.4. If ϕ(z) = +∞ n=1 an z is an element, vanishing at 0, of the ring R of formal series with integer coefficients whose complex radius of convergence is ≥ 1, then either (i) ϕ is algebraic and vϕ (A2 ,(0,0),Cϕ ) = 0, or (ii) ϕ is not algebraic and vϕ (A2 ,(0,0),Cϕ ) = 1. It is not difficult to prove that, in case (i), the series ϕ is actually the expansion of a function in Q(z)10 . We shall not use this fact in the sequel. Observe that the set of algebraic elements of R, vanishing at 0, is infinite countable (indeed the Zariski-closure in A2C of a germ Cϕ with ϕ ∈ Q[[z]] is defined over Q). Therefore the set of series of type (ii) in Proposition 4.4 constitute a set with the power of the continuum. Explicit elements of this set are provided by lacunary series such as ϕ(z) :=

+∞

k

z2 ,

k=0

or, more generally, by the series ϕn (z) :=

+∞

z nk ,

k=0

where n = (nk )k∈N is a sequence of positive integers such that nk+1 inf > 1. k∈N nk

(4.17)

Indeed, according to a classical theorem of Hadamard, the holomorphic functions on the unit disc D(0, 1) defined by such series admit the full circle ∂D(0, 1) as natural boundary, and therefore cannot be algebraic. 10 This a special case of Proposition 2.1 and Corollary 2.2 in [Har88], which may be established as follows. For any holomorphic function ϕ over the open unit disk D(0, 1) that is algebraic over C[t], there is a nonzero polynomial Q in C[t] such that Qϕ is integral over C[t], and therefore extends to a continuous function on the closed disk D(0, 1). In particular the coefficients of the Taylor expansion of Qϕ at 0 converge to 0. If moreover f belongs to R, then Q may be chosen in Z[t], and consequently these coefficients belong to Z, and only a finite number of them does not vanish.

Germs of analytic varieties in algebraic varieties

411

Observe also that, for any polynomial P in C[z], vanishing at 0, the automorphism TP :

A2C −→ A2C (z1 , z2 )  −→ (z1 , z2 + P (z1 ))

of A2C transforms the germ Cϕ into the germ CP +ϕ , and its differential DTP (0, 0) maps vϕ to vP +ϕ . In particular, vP +ϕ (A2 ,(0,0),CP +ϕ ) = vϕ (A2 ,(0,0),Cϕ ) . In particular, for any P ∈ C[z] and any non-algebraic element of R vanishing at 0, vP +ϕ (A2 ,(0,0),CP +ϕ ) = 1. This construction shows in particular that, amongst the series ϕ holomorphic on the unit disk, the ones such that vϕ (A2 ,(0,0),Cϕ ) = 1 are dense in the topology of uniform convergence on compact subsets of D(0, 1).

4.6 Application to differential equations We finally discuss how our algebraicity criterion Theorem 4.2 may be applied to ordinary differential equations. As in the situation C described in the introduction, we consider a smooth variety X over a number field K, a point P in X(K) and a sub-vector bundle F of rank 1 of the tangent bundle TX/K , and we are interested in the algebraicity of the formal germ of integral curve Vˆ through P . The conjecture of Grothendieck–Katz has been generalized to possibly non-linear differential systems by Ekedahl, Shepherd-Barron, and Taylor ([ESBT99]) as the following question: With the notation above, does the condition GK – which asserts that almost all the reductions of F modulo a prime ideal p of OK are closed under the p-th power map – imply the algebraicity of Vˆ ? Observe that the formal germ Vˆ is analytic at every place and that, when moreover the condition GK is satisfied, we may apply the lower bound (4.2) to VˆKp for almost every non-zero prime ideal p in OK . Therefore, under this assumption, the sizes Rp defined as in Corollary 4.3 satisfy the following lower bounds: Rp > 0

for every p ∈ Spec OK \ {0}

and Rp ≥ p

−

[Kp :Qp ] p(p−1)

for almost every p ∈ Spec OK \ {0}.

(As usual, p denotes the residue characteristic of p.) In particular, the product  Rp p∈Spec OK \{(0)}

412

Jean-Benoît Bost

is positive. Together with Corollary 4.3, this establishes the following: Proposition 4.5. If a sub-line bundle F of the tangent bundle TX of smooth variety X over a number field K satisfies the condition GK, then its formal germ of integral curve through a point P in X(K) is algebraic if (and only if ), for at least one embedding σ : K → C, the canonical semi-norm .(Xσ ,Pσ ,Vˆσ ) vanishes. Consequently, the conjecture of Grothendieck–Katz and its non-linear generalization leads us to wonder wether the canonical semi-norm attached to a germ of integral curve of a complex algebraic differential equation always vanishes. It seems quite sensible to expect that this is true for linear differential equations. According to Proposition 4.5, this would establish the original conjecture of Grothendieck–Katz.11

A Appendix: extensions of sections of large powers of ample line bundles A.1. Recall that a continuous metric . on a line bundle L over an analytic space X is called positive if, for any trivializing section s of L over an open subset U of X, the function log s−1 is strongly plurisubharmonic on U . In this Appendix, we prove the following sharp version of Proposition 3.5, concerning line bundles equipped with positive metrics : Theorem A.1. Let X be a complex projective variety, Y a closed subvariety of X, L an ample line bundle over X, and . a positive metric on L. There exist an integer D0 ≥ 0 and, for any ε > 0, a positive real number Cε satisfying the following condition: for any integer D ≥ D0 and any s ∈ (Y, L⊗D ), there exists s˜ ∈ (X, L⊗D ) such that s˜|Y = s and ˜s L∞ (X) ≤ Cε eεD sL∞ (Y ) .

(A.1)

Since the validity of Proposition 3.5 does not depend on the choice of the metric on L, and since any ample line bundle on a projective variety admits a positive metric, Theorem A.1 implies Proposition 3.5. Observe that, besides the proof of Proposition 3.4, Theorem A.1 also possesses applications to Arakelov geometry, in the study of heights of cycles and subschemes (cf. [Zha95], [Ran02]). Actually, similar results have been established in the literature by means of L2 estimates à la Hörmander. However, they are often less precise, and 11 Added in proof (April 2004). This expectation does not look sensible anymore, since the author has established that some variant of the canonical semi-norm does not vanish in general for germs of integral curves of hypergeometric differential equations.

Germs of analytic varieties in algebraic varieties

413

require some smoothness hypothesis on X and Y (see for instance [Man93], [Zha95] Theorem 2.2, [Dem00], [Ran02] section 3.1.1). The proof that we present in this Appendix is based instead on the classical finiteness results of Grauert on strictly pseudo-convex domains (in the spirit of the proof of Satz 2 in [Gra62], p. 343) and the Banach open mapping theorem, and allows us to handle singular varieties as well. A.2. Specifically, we shall use the following theorem of Grauert, which he established in course of his famous solution of the Levi problem ([Gra58], Proposition 4, p. 466; in this paper, Grauert considers only analytic manifolds, however, as observed in [Gra62], p. 344, the proof immediately extends to analytic spaces): Theorem A.2. Let M be a reduced complex analytic space and  a relatively compact open subset of M, with strictly pseudo-convex boundary. For any coherent analytic sheaf F on , the cohomology group H 1 (; F ) is finite dimensional. Besides, we shall use the following version of the open mapping theorem (see for instance [Bou81], I.28 exercice 4), and I.19 Corollaire 3): Theorem A.3. Let E and F be two Fréchet spaces and u : E −→ F a continuous linear map. If coker u := F /u(E) is finite dimensional, then u(E) is closed in F and the map u : E −→ u(E) is open. In particular, for any continuous semi-norm p on E, there exists a continuous semi-norm q on F satisfying the following condition: for any y in u(E), there exists x in E such that u(x) = y and p(x) ≤ q(y). A.3. In the sequel, the algebra of analytic functions on some complex analytic space M will be denoted O an (M). Let X, Y, L, and   be as in the statement of Theorem A.1, and let   also denote the metric on Lˇ dual to the metric   on L. We may consider the total spaces V(X, L) and V(Y, L) of the line bundle Lˇ over X and Y , and, for any r ∈ R∗+ , the disk bundles D(X, r) → V(X, L)(C ) and D(Y, r) → V(Y, L)(C ), formed by the elements v in the fibers of Lˇ such that v < r. These are relatively compact open subsets of the analytic spaces V(X, L)(C ) and V(Y, L)(C ), and their boundary is strongly pseudo-convex, as a consequence of the positivity of the metric   on L.

414

Jean-Benoît Bost

We shall also denote by D(X) (resp. D(Y )) the unit disk bundle D(X, 1) (resp. D(Y, 1)). Observe that the closed embedding of complex algebraic varieties i : V(Y, L) → V(X, L) restricts to a closed embedding of analytic spaces j : D(Y ) → D(X). For any r ∈]0, 1[, we denote  X,r (resp.  Y,r ) the norm  L∞ (D(X,r)) (resp. the norm  L∞ (D(Y,r)) ) on O an (D(X)) (resp. on O an (D(Y ))). The family of norms ( X,r )r∈]0,1[ (resp. ( Y,r )r∈]0,1[ ) defines the natural Fréchet space structure on O an (D(X)) (resp. on O an (D(Y ))). The spaces V(X, L) and V(Y, L) are equipped with natural Gm -actions, defined ˇ and the imbedding i is Gm -equivariant. by the action of homotheties on fibers of L, These actions restrict to analytic actions of U (1) := {u ∈ C | |u| = 1} on D(X) and D(Y ), and, for any integer k, we shall define O an (D(X))k as the subspace of O an (D(X)) consisting of the analytic functions f on D(X) such that, for any u ∈ U (1) and any z ∈ D(X), f (uz) = uk f (z). One defines a projection pX,k : O an (D(X)) −→ O an (D(X))k by letting  pX,k (f )(z) :=

1

e−2πikt f (e2πit z)dt.

0

It is continuous; indeed, for any r ∈]0, 1[ and any f ∈ O an (D(X)), pX,k (f )X,r ≤ f X,r .

(A.2)

Observe also that O an (D(X))k may be identified with the vector space (X, L⊗k ) of algebraic regular – or equivalently, of analytic – sections of L⊗k , by means of the map which sends s ∈ (X, L⊗k ) to the analytic function f on D(X) defined by f (z) := s(π(z)), z⊗k 

for any z ∈ D(X).

⊗k to the dual line L ˇ ⊗k .) (Observe that s(π(z)) belongs to the complex line L⊗k π(z) , and z Moreover, with the above notation, the norms of f and s are related by:

f X,r := r k sL∞ (X) .

(A.3)

Similarly, we may define a subspace O an (D(Y ))k of O an (D(Y )), and a projection pY,k : O an (D(Y )) −→ O an (D(Y ))k .

Germs of analytic varieties in algebraic varieties

415

The subspace O an (D(Y ))k may be identified with (Y, L⊗k ), and (A.3) still holds (with Y instead of X). Moreover, for any f ∈ O an (D(X)), pY,k (f|D(Y ) ) = pX,k (f )|D(Y ) .

(A.4)

A.4. Consider the ideal sheaf
V(Y,L) of V(Y, L) in V(X, L) and the associated short exact sequence of sheaves of OV(X,L) -modules: 0 −→
V(Y,L) −→ OV(X,L) −→ i∗ OV(Y,L) −→ 0. This sequence induces a short exact sequence of analytic coherent sheaves on D(X): an an an 0 −→
D(Y ) −→ OD(X) −→ j∗ OD(Y ) −→ 0,

and, consequently, by taking the cohomology on D(X), an exact sequence of complex vector spaces ρ

an O an (D(X)) −→ O an (D(Y )) −→ H 1 (D(X);
D(Y ) ),

where ρ denotes the restriction map from functions on D(X) to functions on D(Y ). an ) is finite According to Theorem A.2, the cohomology group H 1 (D(X);
D(Y ) dimensional. Therefore, by Theorem A.3, ρ(O an (D(X))) is a closed subspace of O an (D(Y )), and, for any positive ε, there exists Cε and r(ε) ∈]0, 1[ such that, for any f ∈ ρ(O an (D(Y ))), there exists f˜ ∈ O an (D(X)) mapped to f by ρ such that f˜X,e−ε ≤ Cε f Y,r(ε) .

(A.5)

The restriction morphism ρ is clearly equivariant with respect to the U (1)-action on O an (D(X)) and O an (D(Y )). Therefore its cokernel – which is a finite dimensional separated locally convex complex vector space – is naturally endowed with a continuous action of U (1), and consequently, may be decomposed as a finite direct sum

(coker ρ)k , coker ρ = k∈I

where (coker ρ)k denotes the subspace of coker ρ on which U (1) acts by the character (u → uk ). Let D0 be any non-negative integer larger that all the integers in I . Then, for any integer D ≥ D0 and any s ∈ (Y, L⊗D ), the class in coker ρ of the function f ∈ O an (D(Y ))D associated to s vanishes, and therefore f may be written ρ(f˜), where f˜ is an element of O an (D(X)) satisfying (A.5). Moreover, (A.4) and (A.2) show that, by replacing f˜ by pX,D (f˜), we may also assume that f˜ belongs to O an (D(X))D . Then the corresponding section s˜ in (X, L⊗D ) satisfies s˜|Y = s

416

Jean-Benoît Bost

and, according to (A.5) and (A.3), e−εD ˜s L∞ (X) ≤ Cε r(ε)D sL∞ (Y ) . Since r(ε) < 1, this establishes the required estimate (A.1).

References [Ahl52]

L. V. Ahlfors, On the characterization of hyperbolic Riemann surfaces. Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys. 125 (1952).

[And89]

Y. André, G-functions and geometry. Friedr. Vieweg & Sohn, Braunschweig 1989.

[And99]

Y. André, Sur la conjecture des p-courbures de Grothendieck et Katz. Prépublication, Institut de Mathématiques de Jussieu, 1999.

[And02]

Y.André, Sur la conjecture des p-courbures de Grothendieck et Katz et un problème de Dwork. Prépublication, Ecole Normale Supérieure, 2002.

[AS60]

L. V. Ahlfors and L. Sario, Riemann surfaces. Princeton University Press, Princeton, NJ, 1960.

[Bar71]

C. M. Barton, Tensor products of ample vector bundles in characteristic p. Amer. J. Math. 93 (1971), 429–438.

[BD79]

F. Baldassarri and B. Dwork, On second order linear differential equations with algebraic solutions. Amer. J. Math. 101 (1) (1979), 42–76.

[BM01]

F. Bogomolov and M. L. McQuillan, Rational curves on foliated varieties. Preprint, Inst. Hautes Études Sci., 2001.

[Bor91]

A. Borel, Linear Algebraic Groups. Second enlarged edition, Springer-Verlag, Berlin 1991.

[Bos99]

J.-B. Bost, Potential theory and Lefschetz theorems for arithmetic surfaces. Ann. Sci. École Norm. Sup. 32 (1999), 241–312, 1999.

[Bos01]

J.-B. Bost, Algebraic leaves of algebraic foliations over number fields. Inst. Hautes Études Sci. Publ. Math. 93 (2001), 161–221.

[Bou81]

N. Bourbaki, Espaces vectoriels topologiques. Chapitres 1 à 5, new edition, Masson, Paris 1981.

[CL02]

A. Chambert-Loir, Théorèmes d’algébricité en géométrie diophantienne. Séminaire Bourbaki, Vol. 2000/2001, Exposé 886, Astérisque 282 (2002), 175–209.

[CC85a]

D. V. Chudnovsky and G. V. Chudnovsky, Applications of Padé approximations to the Grothendieck conjecture on linear differential equations. In Number theory (NewYork, 1983–84), Lecture Notes in Math. 1135, Springer-Verlag, Berlin 1985, 52–100.

[CC85b]

D. V. Chudnovsky and G. V. Chudnovsky, Padé approximations and Diophantine geometry. Proc. Nat. Acad. Sci. U.S.A. 82 (8) (1985), 2212–2216.

Germs of analytic varieties in algebraic varieties

417

[Dem00]

J.-P. Demailly, On the Ohsawa-Takegoshi-Manivel L2 extension theorem. In Complex analysis and geometry (Paris, 1997), Progr. Math. 188, Birkhäuser, Basel 2000, 47–82.

[Dwo60]

B. Dwork, On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82 (1960), 631–648.

[Dwo81]

B. Dwork, Arithmetic theory of differential equations. In Symposia Mathematica XXIV (Sympos., INDAM, Rome, 1979), Academic Press, London 1981, 225–243.

[Dwo99]

B. M. Dwork, On the size of differential modules. Duke Math. J. 96 (2) (1999), 225–239.

[DGS94]

B. Dwork, G. Gerotto, and F. J. Sullivan, An introduction to G-functions. Princeton University Press, Princeton, N.J., 1994.

[DR79]

B. Dwork and P. Robba, On natural radii of p-adic convergence. Trans. Amer. Math. Soc. 256 (1979), 199–213.

[DvdP92]

B. M. Dwork and A. J. van der Poorten, The Eisenstein constant. Duke Math. J. 65 (1) (1992), 23–43.

[ESBT99]

T. Ekedahl, N.I. Shepherd-Barron, and R. Taylor, A conjecture on the existence of compact leaves of algebraic foliations. Available at http://www. dpmms.cam.ac.uk/∼nisb/fruit/WWW/foliation.dvi, 1999.

[Fal83]

G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (3) (1983), 349–366.

[Gra58]

H. Grauert, On Levi’s problem and the imbedding of real-analytic manifolds. Ann. of Math. 68 (1958), 460–472.

[Gra62]

H. Grauert, Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146 (1962), 331–368.

[Gra01a]

P. Graftieaux, Formal groups and the isogeny theorem. Duke Math. J. 106 (1) (2001), 81–121.

[Gra01b]

P. Graftieaux, Formal subgroups of abelian varieties. Invent. Math. 145 (1) (2001), 1–17.

[Gri70]

P. Griffiths, Periods of integrals on algebraic manifolds, III (some global differential properties of the period mapping). Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125–180.

[Har88]

D. Harbater, Galois covers of an arithmetic surface. Amer. J. Math. 110 (1988), 849–885.

[Kat72]

N. M. Katz, Algebraic solutions of differential equations (p-curvature and the Hodge filtration). Invent. Math. 18 (1972), 1–118.

[Kat82]

N. Katz, A conjecture in the arithmetic theory of differential equations. Bull. Soc. Math. France 110 (2) (1982), 203–239 and 347–348.

[Kat96]

N. M. Katz, Rigid local systems, Ann. of Math. Stud. 139, Princeton University Press, Princeton, NJ, 1996.

418

Jean-Benoît Bost

[Laz01]

R. Lazarsfeld, Positivity in Algebraic Geometry. Available at http://www. math.lsa.umich.edu/ ∼rlaz/, 2001.

[Man93]

L. Manivel, Un théorème de prolongement L2 de sections holomorphes d’un fibré hermitien. Math. Z. 212 (1) (1993), 107–122.

[Miy87]

Y. Miyaoka, Deformations of a morphism along a foliation and applications. In Algebraic geometry—Bowdoin 1985. Proc. Symp. Pure Math. 46-1, Amer. Math. Soc., Providence, RI, 1987, 245–268.

[MB81]

L. Moret-Bailly, Familles de courbes et de variétés abéliennes sur P1 II. Exemples. Astérisque 86 (1981), 125–140.

[Ran02]

H. Randriambololona, Hauteurs pour les sous-schémas et utilisation de méthodes arakeloviennes pour l’approximation diophantienne. Mémoire de Thèse, Université Paris XI-Orsay 2002.

[Ran95]

T. Ransford, Potential theory in the complex plane. Cambridge University Press, Cambridge 1995.

[Ray70]

M. Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Math. 119, Springer-Verlag, Berlin 1970.

[Rum89]

R. Rumely, Capacity theory on algebraic curves, Lecture Notes in Math. 1378, Springer-Verlag, Berlin 1989.

[Ser68]

J.-P. Serre, Abelian l-adic representations and elliptic curves. W. A. Benjamin, Inc., New York–Amsterdam 1968.

[Ser92]

J.-P. Serre, Lie algebras and Lie groups (1964 lectures given at Harvard University). Lecture Notes in Math. 1500 (second edition), Springer-Verlag, Berlin 1992.

[SB92]

N. Shepherd-Barron, Miyaoka’s theorems on the generic seminegativity of TX and on the Kodaira dimension of minimal regular threefolds. Astérisque 211 (1992), 103–114.

[Tsu59]

M. Tsuji, Potential theory in modern function theory. Maruzen, Tokyo 1959.

[Zha95]

S. Zhang, Positive line bundles on arithmetic varieties. J. Amer. Math. Soc. 8 (1) (1995), 187–221.

Jean-Benoît Bost, Département de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France E-mail: [email protected]

Thirty years later Gilles Christol

Abstract. We illustrate by recent results accurateness of concepts Dwork introduced when initiating p-adic differential equation theory thirty years ago. 2000 Mathematics Subject Classification: 12B40

1 Introduction Dwork set up p-adic differential equation theory thirty years ago in three papers ([9], [10], [11]). The few earlier works on this topic were attempts to translate the complex theory and led to poor results. On the basis of his experience with differential equations “coming from geometry”, Dwork introduced three basic notions: • generic points, • the transfer principle, • weak and strong Frobenius structures. The aim of this paper is to show that these notions were so relevant that they are still useful thirty years later. The quest for a general transfer principle has been both a stimulation and an indication for working out the structures underlying p-adic differential equations. In the end, that principle turns out to be true near regular singularities [3] but false near irregular singularities [14]. Nowadays transfer theorems remain a useful tool when computing radii of convergence of solutions even though one must usually supplement them by invoking further properties of the solutions and their radii of convergence (continuity, convexity,…). When working out the p-adic theory of differential equations, generic points became an increasingly fundamental tool. It is striking to point out that Berkovich, about ten years after Dwork, independently of him and in a quite distinct context, brought a new light onto this concept: under his definition of p-adic analytic spaces Dwork’s generic points become actual geometric points. Berkovich’s original aim was to endow analytic spaces with “a nice topology which makes geometrical considerations

420

Gilles Christol

relevant and useful over p-adic fields” [1]. However, although Dwork’s generic points and Berkovich “non standard” points are essentially the same notion, they underscore distinct aspects of the problematic. Usually Berkovich’s point of view is better for geometric considerations while Dwork’s is advantageous for computations. In this article, dedicated to Dwork, we will of course use Dwork’s point of view but it should be an interesting exercise to translate proofs into Berkovich’s language. To illustrate the enduring usefulness of generic points we choose to present a complete proof of a result given in [8]. It is less technical than it might appear. Roughly speaking it says that there is at most one morphism between two irreducible differential modules. It is fundamental for defining a category of coefficients for padic cohomology for curves over a finite field. Indeed it enables one to get good properties for morphisms from conditions placed only on objects. The interest of the proof itself lies in the use of a two step tower of generic points. Dwork’s notion of Weak Frobenius structure has been used at almost all stages of the theory. It is the natural way to get information about differential modules lying outside the “Young domain”, namely the set on which the exact generic radius of convergence cannot be read directly from coefficients. Strong Frobenius structures first came to the fore for differential equations coming from geometry. It was soon suspected that the only obstruction for the weak Frobenius structure, when it exists, to be strong (namely periodic) should be irrationality of exponents. Such a conjecture is now well understood and seems to be reachable. The material of this paper is based on the original work of Dwork and Robba but its modern form is mainly due to Mebkhout. However, the author remains solely responsible for any errors, and he acknowledges the assistance of the referee in removing several of them from an earlier draft.

2 The general setting 2.1 Differential modules In this paragraph we recall some well known results 2.1.1. A differential ring A is a commutative ring endowed with a derivation D. The ring of constants K = {x ∈ A ; D(x) = 0} will be supposed to be a field. We will denote by A[D] the noncommutative ring of differential polynomials with coefficients in A (and the rule D a = a D + D(a)), and by MC(A) the full subcategory of the category of left A[D]-modules whose objects are free of finite rank as A-modules. Objects of MC(A) will be called A-differential modules. For instance, for any µ × µ matrix G with entries in A, we will denote by (Aµ , G) the A-differential module Aµ endowed with the action of D given by D X = D(X) − G X. The notation Aµ (instead of (Aµ , 0)) will be reserved for the special case when D acts “naturally”.

Thirty years later

421

2.1.2. The category MC(A) has internal Hom. Namely, for any A-differential modules M and N, the A-module HomA (M, N) is free of finite rank and endowed with
an A[D]-module structure by means of the classical formula (Df )(m) =  D · f (m) − f (D · m). Hence it is an A-differential module. In particular, the dual M ∗ = HomA (M, A) is an A-differential module. 2.1.3. Let now suppose A is a differential subring of a differential ring B with field of constants . For any A-differential module M, the dimension of the -vector space HomA[D] (M, B) of solutions of M in B is at most dimA M. Choosing an A-basis of M amounts to giving a horizontal isomorphism (namely, an A[D]-isomorphism) of M with the A-differential module (Aµ , G) for some square matrix G. Then ν = dim HomA[D] (M, B) is the maximal rank of a matrix X with entries in B such that D X = G X. 2.1.4. When B⊗A M becomes trivial, namely isomorphic to the “trivial” B-differential module B µ with µ = dimA M, one says that M is solvable in B. For B a field, this happens when dim HomA[D] (M, B) = dimA M. An A-basis of M being chosen, M is solvable in B if and only if there exists an invertible matrix X with entries in B such that D X = G X. Then the invertible matrix Y = tX−1 satisfies the relation D Y = − tG Y and gives an (horizontal) isomorphism of MC(B) between B ⊗A M and B µ . The existence of such a matrix X, shows also that the dual module M ∗ = HomA (M, A) is solvable in B. 2.1.5. An A-differential module M is said to have an index in B if Ext1A[D] (M, B) is finite dimensional. Then one sets χ (M, B) = dim HomA[D] (M, B) − dim Ext1A[D] (M, B). Choosing an A-basis of M, we get also: χ (M, B) = dim ker(D − G, B µ ) − dim coker(D − G, B µ ). 2.1.6. In particular, when D is onto in B, the method of variation of parameters shows that Ext1A[D] (M, B) = 0 for any A-differential module M that is solvable in B. Amongst the rings defined below, the derivation d/dx is onto for rings of type A but not for rings of types H or R. 2.1.7. Let N and M be A-differential modules. The -vector space of solutions of HomA (N, M) in B is then HomA[D] (HomA (N, M), B) = HomA[D] (M, B ⊗A N) = HomB[D] (B ⊗A M, B ⊗A N) In particular, the identity being a horizontal endomorphism of M (i.e. D · id = 0), the -vector space of solutions of HomA (M, M) in B is of dimension at least one.

422

Gilles Christol

Proposition 2.1. If the A-differential module M remains irreducible in the category

of  then the -vector space

⊗K A) for any finite extension  MC( HomA[D] (HomA (M, M), B) is one dimensional. Proof. An element u of HomA[D] (HomA (M, M), B) can be viewed as a horizontal endomorphism of B ⊗A M. Then its powers are also horizontal and, by finite dimensionality over , there exists a non zero polynomial P in [X] such that P (u) = 0.

be a finite extension of  that contains the roots λi of P . By irreducibility if Let  none of the u − λi id were zero, they would all be onto and so would be their product   P (u). Hence u = λi id for some i. When M is a direct sum it is obvious that dim HomA[D] (M, B ⊗A M) ≥ 2. However, of Proposition 2.1 is not true in general. For instance M =  theconverse   0 0 is reducible in MC(A) but HomA[D] (M, B ⊗A A) is one dimensional A2 , α β when β is not in D(B) and α not in (D − β)(B); in other words when M is a non split over B extension of two A-differential modules which are not isomorphic in MC(B).

2.2 Rings of analytic functions 2.2.1. Let K be a field complete for a non archimedean valuation of mixed characteristics (0, p). 2.2.2. For (α, ρ) in K×]0, ∞[, let | · |α,ρ be the unique multiplicative norm on K(x) extending that of K and such that |x − α|α,ρ = ρ. Following Dwork, we define a generic point associated with the disk D(α, ρ) to be any number tα,ρ belonging to some sufficiently large extension α,ρ of K and such that |f (tα,ρ )| = |f |α,ρ for every f in K(x). Actually, such numbers tα,ρ do indeed exist and are unique up to continuous K-isomorphisms. In Berkovich’s theory, the multiplicative semi-norm | · |α,ρ “is” a point of the p-analytic space associated with the K-affine line and tα,ρ is just a way to talk about it. 2.2.3. Let EK,α,ρ be the field of analytic elements in the generic disk D(tα,ρ , ρ) with coefficients in K, namely the completion of K(x) for the norm | · |α,ρ , let  as x s ; as ∈ K, lim |as |ρ s = 0 HK,α,ρ = s∈N

s→∞

be the ring of analytic functions in the “closed” disk D(α, ρ) = {x ; |x − α| ≤ ρ} with coefficients in K, let  HK,α,r = as x s ; as ∈ K, (∀r < ρ) lim |as |r s = 0 AK,α,ρ = r<ρ

s∈Z

s→±∞

423

Thirty years later

be the ring of analytic functions in the “open” disk D(α, ρ − ) = {x ; |x − α| < ρ} with coefficients in K, let  A(η,ρ) = as x s ; as ∈ K, (∀r ∈]η, ρ[) lim |as |r s = 0 s∈Z

s→±∞

be the ring of analytic functions in the annulus C(α, ]η, ρ[) = {x ; η < |x − α| < ρ} with coefficients in K and let   A(η,ρ) = as x s ; as ∈ K, (∃η < ρ)(∀r ∈]η, ρ[) lim |as |r s = 0 RK,α,ρ = η<ρ

s∈Z

s→±∞

be the ring of analytic functions in the annulus C(α, ]η, ρ[) for some η < ρ with coefficients in K. All these rings are differential rings for the derivation D = d/dx and have K as field of constants. 2.2.4. In order to lighten notations, having fixed K, α and ρ once and for all, we will drop them and write t, E, H, A and R instead respectively of tα,ρ , EK,α,ρ , HK,α,ρ , AK,α,ρ and RK,α,ρ . For instance, for 0 < r < ρ, the rings Aα,ρ ,tα,ρ ,r and Aα,r ,tα,r ,r will be denoted respectively by A,t,r and Ar ,tr ,r . Among other the following inclusions will be useful: ⊂ R,t,ρ ⊂ E ⊂ H,t,ρ ⊂ A,t,ρ H ⊂ H,t,r ⊂ A,t,r ⊂ R,t,r ⊂ A ⊂ Er ⊂ Ar ,tr ,r Let us remark that there is no inclusion, for instance, between E and A or between R,t,ρ and A,t,r .

2.3 Dwork–Robba statements 2.3.1. Given a differential equation L in H[D], Dwork’s aim was to study its solutions near “the” generic point t and then to specialize information to the point α. This specialization is the easiest case of the transfer principle. The basic remark, already evident in [11], says that L can be split in E[D] according to the (growth and) radius of convergence of its solutions near the generic point t. Theorem 2.2 (Dwork). Let M be an E-differential module. Then there exists an exact sequence 0 −→ Minj −→ M −→ M sol −→ 0 of MC(E) such that: a) HomE[D] (Minj , A,t,ρ ) = 0, b) dim HomE[D] (M sol , A,t,ρ ) = dimE M sol .

424

Gilles Christol

Actually, for any r ≤ ρ there is an analogous splitting of M with A,t,ρ replaced by A,t,r or even by the subring of functions in A,t,r with a given growth. 2.3.2. Robba went deeper connecting this result with index problems [15]. First, using, among other things, Theorem 2.2, he showed that, as an operator on A,t,ρ , any differential polynomial of E[D] is onto. That basic result can be stated in the following way: Theorem 2.3. Let M be an E-differential module and let 0 < r ≤ ρ, then Ext 1E[D] (M, A,t,r ) = 0. In other words the functor solution HomE[D] (?, A,t,r ), from MC(E) to Vec() is exact. 2.3.3. Secondly, Robba proved that monic differential operators L of H[D] (i.e. without singularities in the disk D(α, ρ)) that are injective on A,t,ρ do have an index in A (namely coker L is finitely dimensional) and are one–to–one as operators on R. We sum up these properties in the following theorem. Theorem 2.4. Let M be an H-differential module. If HomH [D] (M, A,t,ρ ) = 0, then a) HomH [D] (M, Ar ,tr ,r ) = 0 for r ≤ ρ big enough, b) dimK Ext1H[D] (M, A) < ∞, c) HomH [D] (M, R) = Ext1H[D] (M, R) = 0 Remark 2.5. Result a) means that injectivity is an “open” condition. On the other hand, the injectivity hypothesis of Theorem 2.4, means that solutions of the differential module near the generic point t have a radius of convergence strictly less than ρ. So we obviously have a ) HomH [D] (M, A,t,r ) = 0 for r ≤ ρ big enough. Conditions a) and a ) should not be confused.

2.4 Duality 2.4.1. In [16], Robba conjectured that, when it exists, the index of an R-differential operator is 0. In [7], this conjecture is proved in a general setting, namely under a non Liouvilleness hypothesis that ensures existence of the index. Unfortunately this proof relies heavily on the invariance of index by compact perturbation and hence is only

425

Thirty years later

available when the field K is spherically complete (for instance discretely valued). We will only need a particular case of Robba’s conjecture (see Lemma 2.9)) which can be proved directly. 2.4.2. We also need to understand Robba’s conjecture using duality. Lemma 2.6. Let M be an A-differential module. If M has an index both in A and R then so has M ∗ = HomA (M, A) and χ(M, R) = χ (M, A) − χ (M ∗ , A). Proof. When endowed with its natural topology, A (resp. R) is a Frechet space (resp. ∗ the strong a direct limit of Frechet spaces). Moreover R = A ⊕A∗ where  A is topological dual of A. This decomposition is given by s∈Z = s≥0 + s<0 . We γ

then have an exact sequence 0 −→ A −→ R −→ A∗ −→ 0 where γ is the truncation of positive power terms. Now by choosing a basis of M, we get a square matrix G with entries in A such that χ (M, R) = χ(D − G, Rµ )

χ(M, A) = χ (D − G, Aµ )

where µ is the rank of M over A. The commutative diagram 0

/ Aµ D−G

0

/ Rµ D−G

 / Aµ

 / Rµ

γ

/ A∗µ

/0

D−γ G γ

 / A∗µ

/0

shows that, under the hypotheses of the theorem, D − γ  G has an index as operator on A∗µ and: χ (D − G, Rµ ) = χ(D − G, Aµ ) + χ (D − γ  G, A∗µ ). Now using the open mapping theorem, which is available both for Frechet spaces and for direct limits of Frechet spaces, one proves [5] that a map u on A has an index if and only if its transposed map tu has an index on A∗ . Moreover transposition interchanges the dimensions of the kernel and the cokernel so that χ(tu, A∗ ) = −χ (u, A). A straightforward computation gives t(D − γ  G) = −D − tG. Hence D + tG has an index on Aµ and 
χ D + tG, A∗µ ) = −χ(D − γ  G, A∗µ ) so that χ (M, R) = χ(D − G, Rµ ) = χ(D − G, Aµ ) − χ (D + tG, Aµ ) = χ(M, A) − χ (M ∗ , A).

 

426

Gilles Christol

Remark 2.7. In view of this lemma, Robba’s conjecture means that M and its dual M ∗ have the same index on A. Remark 2.8. Under the hypothesis of Lemma 2.6 one can go a little further. By the same argument, D + γ  tG is shown to have an index on A∗µ and using the same commutative diagram with tG instead of −G, one gets that D + tG has an index on Rµ and that
 χ (D + tG, Rµ ) = χ(D + tG, Aµ ) + χ D + γ  tG, A∗µ )
 = −χ D − γ  G, A∗µ ) − χ (D − G, Aµ ) = −χ(D − G, Rµ ) In particular, if M satisfies Robba ’s conjecture then so does M ∗ . 2.4.3. In the solvable case, Robba’s conjecture is easy to check even if the derivation on the ring R is not surjective (x −1 is not a derivative), and so even though the method of paragraph 2.1.6 is not available. Lemma 2.9. Let M be an R-differential module which is solvable in R. Then M has an index on R and χ(M, R) = 0. Proof. By definition, M is isomorphic to Rµ in MC(R). But both kernel and cokernel of D acting on R are one dimensional (generated respectively by 1 and x −1 ). Thus   χ (D, Rµ ) = µ χ (D, R) = 0.

2.5 Entirely irreducible R-differential modules In this paragraph we sum up the main structure results for the category of “solvable R-differential modules” up to ramification. Complete proofs are available in [7] and [8]. 2.5.1. Let M be an R-differential module. One cannot ask that M be solvable in A,t,ρ because this ring does not contain R. However there is a substitute for A,t,ρ solvability. Almost by definition, there exist η < ρ and an A(η,ρ) -differential module Mη such that M = R ⊗A(η,ρ) Mη . For η < r < ρ let ray(M, r) be the greatest R < r such that Mr is solvable in Ar ,tr ,R . Motivated by the continuity of the function ray(Mη , ?), we will say that M is solvable if limr→ρ ray(Mη , r) = ρ. When M is a solvable R-differential module of rank µ, for r large enough, one has ray(Mη , r) = (r/ρ)γ +1 ρ where µγ is a non negative integer. The rational number γ is called the slope of M.

Thirty years later

427

2.5.2. In the local 1 theory of p-adic differential equations, the good category happens to be MCS(R) the full subcategory of MC(R) whose objects are the solvable ones of MC(R). The category MCS(R) is easily seen to be stable by internal Hom. When K is spherically complete (for instance discretely valued), it is abelian. A much deeper result is to attach to each solvable R-differential module M its maximal injective submodule M>0 and its zero slope quotient M ≤0 . Theorem 2.10. Let suppose K to be spherically complete and let M be a solvable R-differential module. Then there exists an exact sequence 0 −→ M>0 −→ M −→ M ≤0 −→ 0 of MCS(R) such that, for r ∈]η, ρ[ with η big enough a) M>0 = R ⊗A(η,ρ) M>0,η with HomA(r,ρ) [D] (M>0,η , Ar ,tr ,r ) = 0, b) ray(M ≤0 , r) = r. This theorem may be seen as a “relative” version of Theorem 2.2. Actually the exact sequence is split. 2.5.3. Proposition 2.1 has a more sophisticated counterpart in the category MCS(R). Definition 2.11. A solvable R-differential module is said to be entirely irreducible if HomR (M, M)≤0 is one dimensional. Proposition 2.12. Any solvable R-differential module that remains irreducible after any finite extension of the field K and any ramification y d = x of the variable with order d prime to p is entirely irreducible. 2.5.4. When finite extensions of the field of constants and ramifications of order prime to p are allowed, any soluble R-differential module can be split into entirely irreducible components. More precisely one has: Theorem 2.13 (Weak p-adic Turrittin theorem). Let K be spherically complete and let M be a solvable R-differential module such that both differences of exponents of M and exponents of HomR (M, M) are non Liouville. There exists an extension R = RK [x 1/d ] of R where d prime to p and K is a finite extension of K such that the differential module R ⊗R M can be obtained by successive extensions from entirely irreducible R -differential modules. 1All this theory is built to study (affine) varieties over a field of characteristic p, namely the residue field k of K. However, following Grothendieck, to get a good de Rham type cohomology, one has to work with characteristic 0 coefficients. In that sense, disks of radius 1 in K are viewed as lifting of points of the k-affine line. One of the basic Dwork’s ideas was to consider “overconvergent” analytic functions. It leads, ultimately, to use R as the right substitute for the local ring.

428

Gilles Christol

Remark 2.14. It is expected that any difference of two exponents of M should be an exponent of HomR (M, M). Indeed, by definition, the exponents of M are the exponents of M ≤0 . Their differences should be the exponents of HomR (M ≤0 , M ≤0 ) (by the decomposition theorem of [6], such is the case when these differences are non Liouville) but HomR (M ≤0 , M ≤0 ) is a subquotient of HomR (M, M) hence its exponents are amongst those of HomR (M, M). Remark 2.15. In the zero slope case, there is no problem. Namely, when the exponents of M have non Liouville differences, M ≤0 can be obtained by successive extensions of one dimensional (solvable) R-differential modules [6]. Mebkhout proved in [13] that in Theorem 2.13 it is enough to consider entirely irreducible R -differential modules whose rank is one or is divisible by p. Actually, it is suspected that the rank of entirely irreducible differential modules is a power of p.2 Moreover, and this is the main difference between p-adic and formal theory, there are entirely irreducible differential modules of rank strictly greater than one. To construct such differential modules, the basic remark is that ramification of order d (prime to p) multiplies the slope of a solvable differential module by d. Hence a solvable R-differential module of slope k/p cannot be reduced to one dimensional differential modules because the latter have integer slopes.

3 Morphisms between entirely irreducible differential modules Our aim is to prove the following. Theorem 3.1. Suppose that K is spherically complete and let M and N be entirely irreducible solvable R-differential modules. Then HomR (M, N )≤0 is at most one dimensional.

for some η < ρ Any R-differential module M can be written M = R ⊗A(η,ρ) M

Then the dimension of M ≤0 is the diand some A(η,ρ) -differential module M.

Ar ,tr ,r ) = HomEr [D] (Er ⊗A

mension of HomA(η,ρ) [D] (M, (η,ρ) M, Ar ,tr ,r ), for r > η near enough ρ. By applying this to the R-differential modules HomR (M, N ), HomR (M, M) and HomR (N, N), and by slightly changing notations and letting ρ = r vary in a small interval, one sees that Theorem 3.1 is a straightforward consequence of the next proposition. Proposition 3.2. Assume that M and N are two E-differential modules such that HomE[D] (M, A,t,ρ ⊗E M) and HomE[D] (N, A,t,ρ ⊗E N) are one dimensional. Then HomE[D] (M, A,t,ρ ⊗E N ) is at most one dimensional. If it is not zero, then A,t,ρ ⊗E M and A,t,ρ ⊗E N are isomorphic in MC(A,t,ρ ). 2 Added in proof. This assertion has been proved by Y. André (personal communication).

Thirty years later

429

Remark 3.3. The first statement of Proposition 3.2 is not a purely formal result about indecomposable objects in MC(A,t,ρ ). Indeed an indecomposable E-differential module can become reducible when tensorising by A,t,ρ . More precisely, the proposition, and in particular its second assertion, is deeply based on the Robba’s theorem (Theorem 2.3). For instance, let N = (E, α) be some   rank-one solvable 0 β of N by the trivial E-differential module. Then any extension M = E 2 , 0 α rank one differential module E becomes trivial after tensorising by A,t,ρ . Actually, by Robba’s there exist h in A,t,ρ such that D(h) + αh = −β and the  theorem,  1 h matrix gives an isomorphism of M with the split A,t,ρ -differential module 0 1    0 0 2 . In particular, HomE[D] (M, A,t,ρ ⊗E M) is two dimensional. A,t,ρ , 0 α To prove Proposition 3.2 we need a lemma.

3.1 A two step tower of generic points The generic disk is quite regular. So its E-differential modules have nice properties and their H,t,r -differential submodules inherit a large part of this good behavior. Lemma 3.4. Let M be a E-differential module, let r < ρ be big enough to ensure that HomE[D] (Minj , A,t,r ) = 0 and let N → H,t,r ⊗E M be a H,t,r -differential submodule. Then a) Ext 1H,t,r [D] (N, A,t,r ) = 0, b) N has an index in R,t,r and χ(N, R,t,r ) = 0.

= H,t,r ⊗E M. By Theorem 2.3 Proof. Let M

A,t,r ) = Ext1 (M, A,t,r ) = 0. Ext 1H,t,r [D] (M, E[D] Hence property a) is a consequence of the long exact sequence:

A,t,r ). 0 ←− Ext1H,t,r [D] (N, A,t,r ) ←− Ext1H,t,r [D] (M, To prove b) let us observe that, for r < ρ, E ⊂ A,t,ρ ⊂ H,t,r ⊂ R,t,r . Let 0 −→ Minj −→ M −→ M sol −→ 0 be the Dwork splitting from Theorem 2.2 and let

inj = H,t,r ⊗E Minj and M

sol = H,t,r ⊗E M sol . We get the two exact sequences3 M i

inj −→ M

−→

sol −→ 0 0 −→ M M i

inj −→ N −→ i(N) −→ 0 0 −→ N ∩ M 3After tensorising by E ,t,r , they correspond to be the Dwork’s spliting of E,t,r ⊗E M and E,t,r ⊗H,t,r N in MC(E,t,r ).

430

Gilles Christol

By definition, M sol is solvable in A,t,ρ hence, for r < ρ, also in H,t,r and

sol is solvable in H,t,r . Its H,t,r -differential submodule i(N) is solvable in so M H,t,r hence in R,t,r and, by Lemma 2.9, χ(i(N), R,t,r ) = 0. On the other hand, we apply Theorem 2.4 to the disk D(t, r) instead of D(α, ρ).

= t,r are then A second step generic point T = tt,r and the corresponding field  needed. So analytic elements of E,t,r are defined in the disk D(T , r − ). The key point is that T is also a generic point for K. Thus the radius of convergence of solutions of M near t and T are the same, namely ray(M, ρ). If r is bigger than the radius of convergence of all solutions of M near t, namely big enough for HomE[D] (Minj , A,t,r ) = 0, then it is bigger than the radius of convergence of all solutions of M near T and

inj , A ) = 0. By exactHomE[D] (Minj , A,T

,r ) = 0. Thus Hom H,t,r [D] (M ,T ,r ness of the solution functor in A,T

,r (Theorem 2.3), we have Hom H,t,r [D] (N ∩

inj , A ) = 0. Hence by Theorem 2.4 c) we have χ (N ∩ M

inj , R,t,r ) = 0. M ,T ,r Property b) is now a direct consequence of the additive property of the index in exact sequences 



inj , R,t,r + χ i(N), R,t,r = 0.   χ (N, R,t,r ) = χ N ∩ M

3.2 Proof of Proposition 3.2 Let M and N be E-differential modules such that HomE[D] (M, A,t,ρ ⊗E M) and

= H,t,r ⊗E M HomE[D] (N, A,t,ρ ⊗E N ) are one dimensional. For r < ρ, let M

and N = H,t,r ⊗E N and let u = 0 be in HomE[D] (M, A,t,ρ ⊗E N). Since for r < ρ, H,t,r contains A,t,ρ , u can be extended to a horizontal mor

u

. We set

−→ N phism M

, im

u), P = HomH,t,r (N

M)

= H,t,r ⊗E HomE (N, M) Q = HomH,t,r (N,

so that the dual H,t,r -differential modules of P and Q are

) u, N P ∗ = HomH,t,r (im

N)

= H,t,r ⊗E HomE (M, N ) Q∗ = HomH,t,r (M,

Moreover

= HomH,t,r [D] (P , A,t,r ), u, A,t,r ⊗H,t,r N) id ∈ HomH,t,r [D] (im

N

) = HomH,t,r [D] (Q, A,t,r ).

u ∈ HomH,t,r [D] (M,

u

−→ im

u −→ 0 gives rise to the injective map On the other hand the surjective map M

N)

← P ∗ H,t,r ⊗E HomE (M, N) = HomH,t,r (M, i

gives rise to the injective map and the injective map 0 −→ im

u −→ N

, N

) = H,t,r ⊗E HomE (N, N ). P → HomH,t,r (N

431

Thirty years later

Hence, if r < ρ is big enough, Lemma 3.4 can be applied to Q, Q∗ , P and P ∗ giving Ext 1H,t,r [D] (Q, A,t,r ) = Ext1H,t,r [D] (Q∗ , A,t,r ) = 0,

χ (Q, R,t,r ) = 0.

Ext 1H,t,r [D] (P , A,t,r ) = Ext1H,t,r [D] (P ∗ , A,t,r ) = 0,

χ (P , R,t,r ) = 0.

By applying Lemma 2.6 to P and Q we get 0 < dim HomH,t,r [D] (P , A,t,r ) = χ (P , A,t,r ) = χ (P ∗ , A,t,r ) = dim HomH,t,r [D] (P ∗ , A,t,r ) 0 < dim HomH,t,r [D] (Q, A,t,r ) = χ(Q, A,t,r ) = χ (Q∗ , A,t,r )

= dim HomH,t,r [D] (Q∗ , A,t,r ).

To prove that dim HomE[D] (M, A,t,ρ ⊗E N) ≤ 1, it suffices to prove that, for some r < ρ, dim HomH,t,r [D] (Q, A,t,r ) = dim HomE[D] (M, A,t,r ⊗E N) ≤ 1. By the above relations this amounts to dim HomH,t,r [D] (Q∗ , A,t,r ) ≤ 1. The

−→ coker

short exact sequence 0 −→ im

u −→ N u −→ 0 gives rise to the long exact sequence: u) 0 −→ HomH,t,r [D] (P ∗ , A,t,r ) = HomE[D] (N, A,t,r ⊗H,t,r im

, A,t,r ⊗E N ) −→ HomH,t,r [D] (N

, A,t,r ⊗H,t,r coker

u) −→ HomH,t,r [D] (N 1

−→ Ext (N , A,t,r ⊗H,t,r im

u) = Ext1 H,t,r [D]

H,t,r [D] (P

∗

, A,t,r ) = 0

in which we just proved the first term is not 0. By hypothesis, the second term is at most one dimensional. Hence the third term is 0. But it obviously contains the

−→ coker

projection map N u. Then this projection map is zero and coker

u = 0.

−→ N

−→ 0 we get: In the same way, from 0 −→ ker

u −→ M 0 −→ HomE[D] (N, A,t,r ⊗E M) = HomH,t,r [D] (Q∗ , A,t,r ) −→ HomE[D] (M, A,t,r ⊗E M) −→ HomH,t,r [D] (ker

u, A,t,r ⊗E M) 1

, A,t,r ⊗E M) = Ext1 −→ Ext (N (Q∗ , A,t,r ) = 0 H,t,r [D]

H,t,r [D]

in which , by hypothesis, the second term is one dimensional. Hence the first term is at most one dimensional. But we have seen that the first term is not zero, hence the

this inclusion is third one must be zero. As it contains the inclusion map ker

u −→ M, zero and ker

u = 0. Hence we have proved

u to be a horizontal isomorphism between

and N.

M u−1 belongs to the Thus, any u = 0 in HomE[D] (M, A,t,r ⊗E N) is invertible and

one dimensional differential module HomE[D] (N, A,t,r ⊗E M), so the proposition is proved.

432

Gilles Christol

4 Frobenius structures In this section, we will assume ρ = 1, K discretely valued and the residue field k of K perfect. Let σ be a continuous automorphism of K that lifts the Frobenius automorphism of k.

4.1 Weak Frobenius structure   4.1.1. For f = an x n in R, let ϕ(f ) = σ (an )x pn (instead of x pn we could use ψ(x)n for any ψ in H such that |x p − ψ| < 1). For M an R-differential module, let ϕ ∗ (M) be the inverse image of M by ϕ. Namely
if M is isomorphic  to (R, G) (by choosing a basis) then ϕ ∗ (M) is isomorphic to R, px p−1 Gσ (x p ) . It is rather easy to check that ϕ ∗ is a functor from MCS(R) to itself. The following is much deeper Theorem 4.1. The functor ϕ ∗ is an auto-equivalence of the category MCS(R). This is basically the result of [4]. Unfortunately when that paper was written, the basic role of R was not brought out and properties of the functor ϕ ∗ were only stated over closed annuli. However one can extend the result to open annuli and then to R. 4.1.2. As a consequence of Theorem 4.1, solvability of an R-differential module M is characterized by the existence of a (unique) infinite sequence of solvable R-differential modules M(h) such that ϕ ∗h (M(h) ) = M. Following Dwork we will call such a sequence the weak Frobenius structure of M. Actually in [10] the original definition of Dwork goes in the opposite way (namely M(h) = ϕ ∗h (M)). Since the existence of a successor is obvious while, on the contrary, the existence of an antecedent (i.e., the functor is essentially surjective) is not, Dwork added global conditions. 4.1.3. R-differential modules with slope 0 (i.e. such that M = M ≤0 ) are characterized by a further property of their weak Frobenius structure: for any r > 0 and for h large enough, the antecedent M(h) comes from an A(r,1) -differential module. More precisely, if M = R ⊗A(η,1) Mη then M = M ≤0 if and only if, for each h there exists an A(ηph ,1) -differential module Mη,(h) such that M(h) = R ⊗A ph Mη,(h) . (η

,1)

Curiously, this result is of no help when defining M ≤0 . However it is basic in the definition of exponents of M: that is clear in [6] but also in [12] even if less obvious.

4.2 Strong Frobenius structure 4.2.1. Following Dwork, we will say that a solvable R-differential module M has a strong Frobenius structure if its weak Frobenius structure is periodic: for some h ≥ 1, one has M(h) = M. In other words, for some h ≥ 1, ϕ ∗h (M) = M. This

Thirty years later

433

isomorphism gives an action of ϕ ∗h on M which is compatible with the derivation, h namely D · ϕ ∗h (m) = ph x p −1 ϕ ∗h (D · m). It is quite difficult to decide whether or not a given M has a strong Frobenius structure. Fortunately, differential modules coming from geometry have, by construction, a strong Frobenius structure. 4.2.2. In [9] Dwork made the conjecture “If none of the irrational exponents of L lie in Zp then L has a strong Frobenius structure.” Merely to state this conjecture precisely requires a knowledge of p-adic differential equations that was not available when Dwork first formulated it. In particular, p-adic exponents must be precisely defined which is not at all obvious for differential modules of rank bigger than one. 4.2.3. Very early Dwork noticed that K(x)-differential modules endowed with a strong Frobenius structure are solvable in A,t,ρ . Indeed this was even one deep reason for introducing the generic disk. By a slight generalization, R-differential modules endowed with a strong Frobenius structure are proven to be solvable. That the functor ϕ ∗ multiplies exponents by p and that exponents are defined modulo Z is easy to verify both when using a “naïve” definition or the true definition for exponents. It follows directly that exponents of differential modules endowed with a strong Frobenius structure are rational. 4.2.4. Let M be an R-differential module of rank µ > 1 endowed with a strong Frobenius structure. Beyond solvability and rationality of its exponents, it has two other properties.  Firstly, its µ-th exterior power det(M) = µ M is also endowed with a strong Frobenius structure. In particular the exponent of this one dimensional differential module is rational. For instance, if α is an irrational p-adic number, a solvable differential module M = (Rµ , G) and its twist x α M = (Rµ , αx I + G), which is also solvable, cannot both be endowed with a strong Frobenius structure. Secondly HomR (M, M) is endowed with a strong Frobenius structure and hence has rational exponents. More generally, for any R-differential submodules P and Q of M, HomR (P , Q) is a subquotient of HomR (M, M) and thus has rational exponents. Hence if Q = x α P then α is rational. The point is that such an α appears as an exponent of HomR (M, M) but not as a difference of exponents of M when P , and hence Q, have a non zero slope. 4.2.5. With the above obstacles in mind, we propose an up to date version of Dwork’s conjecture. Conjecture 4.2. Let M be a solvable R-differential module. Then M has a strong Frobenius structure if (and only if ): a) exponents of M are rational,

434

Gilles Christol

b) exponents of HomR (M, M) are rational, c) for any finite extension K of K and any irreducible RK -differential submodule N of RK ⊗R M the exponent of det(N) is rational. This conjecture is proved in [2] for one dimensional differential modules (in that case conditions b) and c) are nugatory). In the general case, Theorems 2.13 and 3.1 reduce the above conjecture to the following one. Conjecture 4.3. Let M be a solvable entirely irreducible R-differential module such that the exponent of det(M) is zero. Then M is endowed with a strong Frobenius structure. The case of one dimensional differential modules being known, it remains to consider entirely irreducible modules with a strictly positive slope. In a forthcoming paper we hope to prove the conjecture for entirely irreducible modules of rank p and slope k/p with k prime to p. 4.2.6. Any solvable R-differential module can be split into one dimensional modules ([13] there are also proofs by André and Kedlaya but they seem less suitable for our purpose): Theorem 4.4 (Strong p-adic Turrittin theorem). Let M be a solvable R-differential module endowed with a strong Frobenius structure. There exists a finite étale extension R of R such that the differential module R ⊗R M can be obtained by successive extensions of one dimensional R -differential modules. Indeed the splitting can even be into trivial modules: Theorem 4.5 (Crew’s conjecture). Let M be a solvable R-differential module endowed with a strong Frobenius structure. There exists a finite étale extension R of R such that the differential module R ⊗R M can be obtained by successive extensions of differential modules isomorphic to R . But this is of no direct help for proving Conjecture 4.2. Actually, finite étale extensions do not respect slope: for instance (R, πx −2 ) with π p−1 = −p has slope 1 but becomes trivial, hence of slope 0, under the étale extension y −1 − y −p = x −1 . By the way, Robba’s exponentials enable one to generalize this situation to any solvable one dimensional R-differential module and so to deduce Crew’s conjecture from the strong Turrittin p-adic theorem. Taking the inverse image by an étale extension can then increase the part of slope 0 and thus can introduce new exponents. It is quite difficult to manage these new exponents. The only known way to do so is to suppose that M is endowed with a strong Frobenius structure. But this is obviously ineffective when trying to prove Conjecture 4.2!

Thirty years later

435

Remark 4.6. Theorem 4.4 is strictly contained in Theorem 4.5 and just a little weaker. However there are two reasons to cite it. Firstly, it is the strict analog of the classical Turrittin theorem and furthermore it does insist upon differences between p-adic and complex theories. Secondly, examples show that the extension needed for Theorem 4.4 is considerably smaller than that for Theorem 4.5. It is then hoped that properties of the differential module can be maintained, in some sense, when passing to the reduced form of 4.4 although they are quite entirely forgotten in passage to the “quasi unipotent” form of 4.5.

References [1]

V. G. Berkovich, p-analytic spaces. Doc. Math. extra volume ICM 1998, II, 141–151.

[2]

B. Chiarellotto, G. Christol, Overconvergent isocrystals and F-isocrystals. Compositio. Math. 100 (1996), 77–99.

[3]

G. Christol, Un théorème de transfert pour les disques singuliers réguliers. Astérisque 119–120 (1984), 151–168.

[4]

G. Christol, B. Dwork, Modules différentiels sur des couronnes. Ann. Inst. Fourier 44 (1994), 689–720.

[5]

G. Christol, Z. Mebkhout, Sur le théorème de l’indice des équations différentielles p-adiques I. Ann. Inst. Fourier 44 (1994), 663–701.

[6]

G. Christol, Z. Mebkhout, Sur le théorème de l’indice des équations différentielles p-adiques II. Ann. of Math. 146 (1997), 345–410.

[7]

G. Christol, Z. Mebkhout, Sur le théorème de l’indice des équations différentielles p-adiques III. Ann. of Math. 151 (2000), 385–457.

[8]

G. Christol, Z. Mebkhout, Sur le théorème de l’indice des équations différentielles p-adiques IV. Invent. Math. 143 (2001), 629–672.

[9]

B. Dwork, On p-adic differential equations I. Bull. Soc. Math. France, Mémoire 39-40 (1974), 27–37.

[10]

B. Dwork, On p-adic differential equations II. Ann. of Math. 98 (1973), 366–376.

[11]

B. Dwork, On p-adic differential equations III. Invent. Math. 20 (1973), 295–316.

[12]

B. Dwork, On exponents of p-adic differential modules. J. Reine Angew. Math. 484 (1997), 85–126.

[13]

Z. Mebkhout, Analogue p-adique du théorème de Turrittin et le théorème de la monodromie p-adique. Invent. Math. 148 (2002), 319–351.

[14]

S. E. Remmal, Equations différentielles p-adiques aux points singuliers irréguliers et principe de transfert. Thèse de doctorat d’Etat, Fès, Maroc (1991).

[15]

P. Robba, On the Index of p-adic Differential Operators I. Ann. of Math. 101 (1975), 280–316.

436 [16]

Gilles Christol P. Robba, Indice d’un opérateur différentiel p-adique IV. Cas des systèmes. Mesure de l’irrégularité dans un disque. Ann. Inst. Fourier 35 (1985), 13–55.

Gilles Christol, Théorie des nombres, Université Paris 6, 175 rue du Chevaleret, 75013 Paris, France E-mail: [email protected]

Approximation of eigenforms of infinite slope by eigenforms of finite slope Robert F. Coleman and William A. Stein

Contents 1

Introduction

437

2 Approximating Teichmüller twists of finite slope eigenforms 439 2.1 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 3 An infinite slope eigenform that is not approximable 442 3.1 An extension of a theorem of Hatada . . . . . . . . . . . . . . . . . . . . . . . 442 3.2 Another eigenform that conjecturally cannot be approximated . . . . . . . . . . 443 4

Computations about approximating infinite slope eigenforms 445 4.1 A question about families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 4.2 An approximation bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 4.3 Some data about approximations . . . . . . . . . . . . . . . . . . . . . . . . . 447

1 Introduction Fix a prime p. Consider a classical newform 
 an q n ∈ Sk 1 (Npt ), Qp F = n≥1

where k and N are positive integers and p
N is a prime (by a newform we mean a Hecke eigenform that lies in the new subspace and is normalized so that a1 = 1). The slope of F is ordp (ap ), where ordp (p) = 1. By [Shi94, Prop. 3.64], the twist  χ(n)an q n Fχ = of F by any Dirichlet character χ of conductor dividing p is an eigenform on 1 (Npmax {t+1,2} ). This twist has infinite slope.

438

Robert F. Coleman and William A. Stein

In Section 2, we prove that if F has finite slope then it is possible to approximate F χ arbitrarily closely by (classical) finite slope eigenforms. Assuming refinements of standard conjectures, the best estimate we obtain for the smallest weight of an approximating eigenforms is exponential in the approximating modulus pA . Section 4 contains computations that suggest that the best estimates should have weight that is linear in p A . One motivation for the question of approximation of infinite slope eigenforms by finite slope eigenforms is the desire to understand the versal deformation space of a residual modular representation [Maz89] (the deformation space of an irreducible representation is universal [Maz89] as is the deformation space of a residual pseudorepresentation [CM98]). In [GM98] (see also [Maz97], and [Böc01] for a generalization), it was shown that the Zariski closure of the locus of finite slope modular deformations of an absolutely irreducible “totally unobstructed” residual modular representation is Zariski dense in the associated representation space but very little is known about the topological closure of this locus. For example, it is not known if it contains any nonempty open sets. Our result implies that it contains tamely ramified twists of modular deformations. We also show in Section 3.1 that a result of Hatada implies that in at least one (albeit not irreducible) case it does not contain all modular deformations. Our investigation began with our answer in Section 2 to a question of Jochnowitz. The idea of studying the p-adic variation of modular forms began with Serre [Ser73] and was since developed by Katz [Kat75] and Hida [Hid86] (see also [Gou88] for a sketch of the theory). It follows, in particular, from their work, that one can approximate all forms on X0 (p n ) with forms on the j -line X0 (1), but not necessarily with eigenforms. We prove the above result about twists in Section 2, then state some questions about approximation by finite slope forms in Section 2.1. We explain how to reinterpret Hatada’s result in Section 3.1, then present the results of our computations in Section 4. Based on the results and computations discussed in this article, Mazur has suggested that it may be the case that an infinite slope eigenform can be approximated by finite slope eigenforms only if the corresponding representation is what he calls tamely semistable (i.e., semistable, in the sense of [CF00], after a tame extension). Acknowledgments. The authors thank Naomi Jochnowitz for provoking this line of thought and for interesting conversations, Barry Mazur for helpful comments and questions, Frank Calegari for conversations, Loïc Merel for his comments on an early draft of this paper, and the referee for a brilliant report.

Approximation of eigenforms of infinite slope by eigenforms of finite slope

439

2 Approximating Teichmüller twists of finite slope eigenforms This section is the theoretical heart of the paper. We prove that the infinite slope eigenforms obtained as twists of finite slope eigenforms by powers of the Teichmüller character can always be approximated by finite slope eigenforms. We first show that certain overconvergent eigenforms of sufficiently close weight are congruent and have the same slope. Then we use the θ operator on overconvergent forms to deduce the main result (Theorem 2.1) below. Let p be a prime. All eigenforms in this section will be cusp forms with coefficients in Qp normalized so that a1 = 1. Suppose F = n≥1 an q n is an eigenform and χ : (Z/MZ)∗ → C∗p is a Dirichlet character with modulus M, which we extend to Z/MZ by setting χ (n) = 0 if (n, M) = 1. Then the twist of F by χ is the eigenform  Fχ = χ (n)an q n . n≥1

Let ω : (Z/pZ)∗ → Z∗p be the Teichmüller character (so ω(n) ≡ n (mod p)). The following theorem concerns finite slope approximations of twists of F by powers of ω. For example, it concerns the twist  0 an (F )q n Fω = (n,p)=1

of F by the trivial character mod p, which we call the “p-deprivation” of F and which has infinite slope. Theorem 2.1. Suppose F is a classical eigenform on X1 (Npt ), t ≥ 1, over Qp of weight k, character ψ, and finite slope at p. Let A ∈ Z>0 and r, s ∈ Z≥0 with r, s < p − 1. Then there exists a classical finite slope eigenform G on X1 (Npt ) with r G(q) ≡ F ω (q) (mod p A ) such that G has weight congruent to k + 2r − s modulo p − 1 and character ψ · ωs . r

(The slope of G will be at least A, since the pth Fourier coefficient of F ω is 0.) Let q = 4 if p = 2 and p otherwise. Let τ : Z∗p → C∗p be the character  of finite order such that a ≡ τ (a) (mod q). We only need to assume that F = n≥1 an q n is an overconvergent eigenform of tame level N of finite slope with arithmetic weightcharacter κ : a → χ(a)ak , where χ is a character of finite order whose conductor divides Np t , k is a possibly negative integer, and a = a/τ (a). (For example, if F is a classical eigenform of weight k and character ψ, then χ = ψωk .) Recall that the collection of continuous characters on Z∗p is a metric space, with d(ρ, ψ) = max{|ρ(a) − ψ(a)| : a ∈ Z∗p }, where | | is the absolute value on Cp normalized so that |p| = 1/p. We need,

440

Robert F. Coleman and William A. Stein

Proposition 2.2. Suppose L ∈ Z≥0 and H is an overconvergent eigenform of tame level N , finite slope and weight-character κ. Then if γ is a weight-character sufficiently close to κ there exists an overconvergent eigenform R of weight-character γ with the same slope as H such that H (q) ≡ R(q)

(mod p L ).

Proof. We will use the notation of the “R-families” section (in §B5) of [Col97b]. In particular, B is an affinoid disk in weight space containing κ and X is an affinoid finite over B such that A(X) is generated by the images of the “Hecke operators” T (n). Moreover, if x ∈ X and ηx : A(X) → Cp is the corresponding homomorphism, then  ηx (T (n))q n Fx (q) = n≥1

is the q-expansion of an overconvergent finite slope eigenform and finally there is a point y ∈ X such that Fy (q) = H (q). Note that X is a subdomain of the eigencurve of tame level N (although the eigencurves of level N > 1 are not yet defined in the literature). The ring A0 (X) is finite over A0 (B) by Corollary 6.4.1/5 of [BGR84]. Let f1 , . . . , fn be generators. Let f0 be a uniformizing parameter on B so that A(B) = Cp f0 , where Cp f0  is the ring of power series in f0 whose coefficients tend to 0 with their degree. Let ZL (y) be the following Weiersträss subdomain of X: {x ∈ X : |fi (x) − fi (y)| ≤ p −L , 0 ≤ i ≤ n}. Since the functions x → ηx (T (n)) lie in A0 (X), it follows that if x ∈ ZL (y), then Fx (q) ≡ H (q)

(mod pL ).

Finally, since ZL (y) is a subdomain of X and X is finite over B, the map from ZL (y) to B is quasi-finite. It follows from Proposition A5.5 of [Col97b] that its image in B is a subdomain. Since κ is the image of y, its image contains a disk around y.   Proof of Theorem 2.1. Let α be the slope of F . It follows from Proposition 2.2 that if m ∈ Z is sufficiently small p-adically there exists an overconvergent eigenform K of tame level N, weight-character χ ·  k−m and slope α such that K(q) ≡ F (q) (mod p A ). Suppose m ≥ k. Then, by Proposition 4.3 of [Col96] (see also [Col97a]) if F1 = θ m−k+1 K, then F1 is an overconvergent eigenform of weight-character κ1 := ω2(m−k+1) · χ ·  k1 , where k1 = m − k + 2, and F1 has finite slope α1 = α + m − k + 1. Applying this same process to F1 , for  ∈ Z sufficiently small p-adically such that  ≥ k1 , we obtain an overconvergent finite slope eigenform F2 of weight-character κ2 , where κ2 = ω2 · χ ·  k2 and where k2 =  − k1 + 2 = k +  − m, such that if

Approximation of eigenforms of infinite slope by eigenforms of finite slope

F2 (q) =



n≥1 bn q

n,

441

then bn ≡ n−k1 +1 nm−k+1 an ≡ n an

(mod p A ).

The latter is congruent to ωr (n)an (mod p A ) if  ≡ r (mod ϕ(p A )) and  + v(ap ) ≥ A. It follows from [Col96, §8], [Col97a], and [Col97b] that if c is an integer sufficiently small p-adically, such that c + k2 > v(bp ) + 1 (note that v(bp ) is the slope of F2 so is finite) there exists a classical eigenform G on X1 (Np t ) of weight k2 +c = k+−m+c, r slope v(bp ) and character ωm+r−c · ψ such that G(q) ≡ F2 (q) ≡ F ω (q) (mod p A ). We can choose c so that m + r − c ≡ s (mod p − 1) and then k2 + c ≡ k + 2r − s mod (p − 1).   The following corollary addresses a question of Jochnowitz, which motivated this entire investigation: Corollary 2.3. Suppose R is a classical eigenform of weight k on X1 (N ), let A ∈ Z>0 , and let r ∈ Z≥0 with r < p − 1. Then there exists a classical eigenform S on X1 (N ) r of weight congruent to k + 2r modulo p − 1 such that S(q) ≡ R ω (q) (mod p A ). Proof. Suppose the F in Theorem 2.1 is one of the old eigenforms associated to R on X1 (Np) and s = 0. Let G be a classical eigenform of weight c + k2 as mentioned in the proof of the theorem, but suppose c + k2 > 2v(bp ) + 1. Then G is old of weight congruent to k mod (p − 1) and G is congruent to an eigenform S of the same weight   on X1 (N ) modulo pv(bp ) . Since bp ≡ 0 (mod p A ), we obtain the corollary. Remark 2.4. Assuming a natural refinement of the Gouvêa-Mazur conjectures, the best estimate we obtain for the weight of H in the above proof is exponential in pA . Computational evidence suggests that the best estimates should have weights that are linear in p A (see Section 4). Remark 2.5. Jochnowitz and Mazur have independently observed that the above argument can be used to prove the following result: Suppose F is an overconvergent eigenform of arithmetic weight-character κ, which is a limit of overconvergent eigenforms of finite slope. If ι : Z∗p → Z∗p is the identity character, then the twist F ι/κ (q) of F by ι/κ, which is the q-expansion of a convergent eigenform of weight-character ι2 /κ, is the limit of overconvergent eigenforms of finite slope. Remark 2.6. One can also approach the p-deprivation (the twist by the 0th power of Teichmüller) of a finite slope eigenform F by using the evil twins of eigenforms approaching F .

442

Robert F. Coleman and William A. Stein

2.1 Questions Some natural questions arise: 1. Is every p-adic convergent eigenform which is the limit of finite slope overconvergent eigenforms an overconvergent eigenform? (We can show the twist of an overconvergent eigenform by a Dirichlet character is overconvergent.) 2. Which infinite slope eigenforms are limits of finite slope eigenforms? 3. If F (q) is the q-expansion of an overconvergent eigenform of weight-character κ, is F ι/κ (q) the q-expansion of an overconvergent eigenform of weight-character ∼ → Z∗p )? Another closely related ι2 /κ (recall that ι is the identity character Z∗p − question is as follows: Suppose ρ is the representation of the absolute Galois group of Q attached to an overconvergent eigenform and let χ denote the cyclotomic character. Then is the representation ρ ⊗ χ · det(ρ)−1 attached to an overconvergent eigenform?

3 An infinite slope eigenform that is not approximable In Section 3.1, we prove an extension to higher level of a theorem of Hatada about the possibilities for systems of Hecke eigenvalues modulo 8. We use this result to deduce that the normalized weight 2 cusp form on X0 (32) is not 2-adically approximable by normalized eigenforms of tame level 1 and finite slope. In Section 3.2 we give an example of an infinite slope eigenform of level 27 that computer computations suggest cannot be approximated by finite slope forms. For related investigations, see [CE03].

3.1 An extension of a theorem of Hatada  Theorem 3.1. If F = an q n is a normalized cuspidal newform over C2 of finite n slope on X0 (2 ), then a2 ≡ 0 (mod 8) and ap ≡ p + 1 (mod 8) for all odd primes p. Proof. Suppose F has weight k and finite slope α. The assumption that F has finite slope implies n ≤ 1. If n = 0 the assertion of Theorem 3.1 was proved by Hatada in [Hat79], so we may assume that n = 1 and α = (k − 2)/2 (in general, the slope of a newform on 0 (p) of weight k is (k − 2)/2). Note that α ≥ 3 since there are no newforms on X0 (2) of weight < 8. It follows from Theorems A of [Col97b] (see §B2 of [Col97b] for the extension to p = 2) and Theorem B5.7 of [Col97b] that if j is an integer sufficiently close 2-adically to k, then there exists a classical normalized cuspidal eigenform G on X0 (2) of weight j and slope α such that G(q) ≡ F (q)

(mod 8).

Approximation of eigenforms of infinite slope by eigenforms of finite slope

443

If in addition we assume that j > 2(α + 1), then G must be old (since the slope of a newform of weight j is (j − 2)/2 = α). Thus there is a cuspidal eigenform  H = bn q n of level 1 such that G is a linear combination of H (q) and H (q 2 ). More precisely, G(q) = H (q) − ρH (q 2 ) where ρ is a root of P (X) = X2 − b2 X + 2j −1 . By Hatada’s theorem ord2 (b2 ) ≥ 3, and j ≥ 12, so the slopes of the Newton polygon of P (X) at 2 are both at least 3. Thus G(q) ≡ H (q) (mod 8), which proves the theorem because H has level 1.   Corollary 3.2. Let G be the normalized weight 2 cusp form on X0 (32). Then G is not 2-adically approximable by normalized eigenforms of tame level 1 and finite slope. F Proof. If F32 were approximable there would have to be a normalized  eigenform n where, on X0 (2) such that F32 (q) ≡ F (q) (mod 8). However, F32 (q) = ∞ a q n=1 n  2x if p = x 2 + y 2 , written so x + y ≡ x 2 (mod 4) ap = 0 otherwise. As a3 = 0 ≡ 4 (mod 8), we see from Theorem 3.1 that F does not exist.

 

Remark 3.3. If p ≡ 1 (mod 4) then the coefficient of ap in F32 agrees modulo 8 with p + 1. If p is 3 mod 4 it does not because for F32 the coefficient vanishes. What is happening is that there is a reducible mod 8 pseudo-representation (namely the trivial one-dimensional representation plus the cyclotomic character) such that any finite slope level 2n form gives this pseudo-representation mod 8. Conversely the mod 8 representation associated to F32 is the direct sum of the quadratic character associated to Q(i) and the cyclotomic character. Hence the congruence works when p = 1 mod 4 but not otherwise.

3.2 Another eigenform that conjecturally cannot be approximated In this section we consider an infinite slope eigenform that is not a Teichmüller twist of a finite slope eigenform. We conjecture that this eigenform cannot be approximated arbitrarily closely by finite slope eigenforms. Conjecture 3.4. There are exactly five residue classes in (Z/9Z)[[q]] of normalized eigenforms in Sk (0 (N)) where k ≥ 1 and N = 1, 3, 9. They are given in the following table, where the indicated weight is the smallest weight where that system of eigenvalues occurs (the level is 1 in each case):

444

Robert F. Coleman and William A. Stein

Weight 12 16 20 24 32

[ a2 , a3 , . . . , a43 mod 9 ] [ 3, 0, 6, 5, 3, 8, [ 0, 0, 0, 2, 0, 2, [ 6, 0, 3, 8, 6, 5, [ 6, 0, 3, 5, 6, 8, [ 3, 0, 6, 8, 3, 5,

0, 0, 0, 0, 0,

2, 2, 2, 2, 2,

6, 0, 3, 3, 6,

3, 0, 6, 6, 3,

8, 2, 5, 8, 5,

2, 2, 2, 2, 2,

6, 0, 3, 3, 6,

5 2 8 5 8

] ] ] ] ]

The system of eigenvalues mod 9 associated to the weight 2 form F on X0 (27) is [ 0, 0, 0, 8, 0, 5, 0, 2, 0, 0, 5, 2, 0, 8 ], so we conjecture that there is no eigenform f on 0 (N ) with N | 9 such that f ≡ F (mod 9). As evidence, we verified that each of the mod 9 reductions of each newform of level 1 and weight k ≤ 74 has one of the five systems of Hecke eigenvalues listed in the table. We also verified that all newforms of levels 3 and 9 and weight k ≤ 40 have corresponding system of eigenvalues mod 9 in the above table. We checked using the method described in Section 4 that there is no newform of level 1 with weight k ≤ 300 that approximates the weight 2 form on X0 (27) modulo 9. We now make some remarks about pseudo-representations when p = 3. Let χ : Z/27Z → Z/9Z be the mod 9 cyclotomic character, so χ has order 6 and if gcd(n, 3) = 1 then χ(n) = n ∈ Z/9Z. The pseudo-representation corresponding to a form of weight k giving the system of eigenvalues in the table in Conjecture 3.4 are Weight 12 16 20 24 32 S2 (0 (27))

Pseudo-representation χ2 ⊕ χ3 1 ⊕ χ3 χ3 ⊕ χ4 1 ⊕ χ5 1⊕χ χ2 ⊕ χ5

Note that the square of any pseudo-representation of level 1 in the above table has 1 as an eigenvalue, but the square of the pseudo-representation attached to S2 (0 (27)) does not have 1 as an eigenvalue. Also, F ≡ f16 ⊗ χ 2

(mod 9),

where f16 is of weight 16. The order of χ 2 is 3, so χ 2 is not a power of the Teichmüller character (which has order 2) and Theorem 2.1 does not apply. Further computations suggest that the pseudo-representations attached to forms of level 1 with coefficients in Z9 are

Approximation of eigenforms of infinite slope by eigenforms of finite slope

Weight k ≡ 0 (mod 6) k ≡ 2 (mod 6) k ≡ 4 (mod 6)

445

Pseudo-representations 1 ⊕ χ 5, χ 2 ⊕ χ 3 1 ⊕ χ, χ 3 ⊕ χ 4 1 ⊕ χ3

The pseudo-representations attached to forms of level 27 with coefficients in Z9 seem to be Weight k ≡ 0 (mod 6) k ≡ 2 (mod 6) k ≡ 4 (mod 6)

Pseudo-representations χ ⊕ χ4 χ2 ⊕ χ5 χ ⊕ χ 2, χ 4 ⊕ χ 5

Also note that if χ i ⊕ χ j is one of the pseudo-representations of level 27 in the table, then the sum of the orders of χ i and χ j is 9, whereas at level 1 the sum of the orders is at most 7.

4 Computations about approximating infinite slope eigenforms In this section, we investigate computationally how well certain infinite slope form can be approximated by finite slope eigenforms.

4.1 A question about families The following question is an analogue of [GM92, §8] but for eigenforms of infinite slope. Fix a prime p and an integer N with (N, p) = 1. Question 4.1. Suppose f ∈ Sk0 (0 (Npr )) is an eigenform having infinite slope (note that f need not be a newform). Is there a “family” of eigenforms {fk }, with fk ∈ Sk (0 (Np)), where the weights k run through an arithmetic progression k ∈ K = {k0 + mp ν (p − 1) for m = 1, 2, . . .} for some integer ν, such that fk ≡ f

(mod pn ),

where n = ordp (k − k0 ) + 1? (When p = 2 set n = ord2 (k − k0 ) + 2.) Our question differs from the one in [GM92, §8] because there the form being approximated has finite slope, whereas our form f does not. We know, as discussed in the previous section, that our question sometimes has a negative answer since it might not be possible to approximate f at all.

446

Robert F. Coleman and William A. Stein

4.2 An approximation bound Let f =



an q n ∈ K[[q]]

n≥1

be a q-expansion with coefficients that generate a number field K. Fix a prime p and an even integer k ≥ 2. In order to gather some data about Question 4.1, we now define a reasonably easy to compute upper bound on how well f can be approximated by an eigenform in Sk (0 (p)). Suppose  ≥ 1, let F be the characteristic polynomial of T acting on the space Sk (0 (p)) of classical cusp forms of weight k and tame level 1, and let H be the characteristic polynomial of a ∈ K. Let G be the resultant of F (Y ) and H (X + Y ) with respect to the variable Y , normalized so that G is monic. Thus the roots of G are the differences α − β where α runs through the roots of F and β runs through the Gal(Q/Q)-conjugates of a . We can easily compute the p-valuations of the roots of G without finding the roots, because the p-valuations of the roots are the slopes of the Newton polygon of G. Let m ∈ Q ∪ {∞} be the maximum of the slopes of the Newton polygon of G. Let ck (f, r) = min{m :  ≤ r is prime}. We note that computing ck (f, r) requires knowing only the characteristic polynomials of Hecke operators T on Sk (0 (p)) and of a for primes  ≤ r. Proposition 4.2. If there is a normalized eigenform g ∈ Sk (0 (p)) such that f ≡ g (mod p A ), then A ≤ ck (f, r) for any r. Proof. To see this observe that ck (f, r) is the minimum of the ordp (an (f ) − an (g)) where 1 ≤ n ≤ r and g runs through all normalized eigenforms in Sk (0 (p)), and we   run through all possible embeddings of f and g into Zp [[q]]. The motivation for our definition of ck (f, r) is that it is straightforward to compute from characteristic polynomials of Hecke operators, even when the coefficients of f lie in a complicated number field. The number ck (f, r) could overestimate the true extent to which f is approximated by an eigenform in Sk (0 (p)) in at least two ways: 1. There is an r > r such that ck (f, r ) < ck (f, r). 2. No single eigenform g is congruent to f , but each coefficient of f is congruent to some coefficient of some eigenform g.

Approximation of eigenforms of infinite slope by eigenforms of finite slope

447

4.3 Some data about approximations Let p be a prime and f ∈ Sk0 (0 (pr )) be a newform of infinite slope. Suppose that the answer to Question 4.1 for f is yes. If k is a weight (in the arithmetic progression) then there should be an eigenform fk ∈ Sk (0 (p)) such that fk ≡ f (mod p n+1 ) where n = ordp (k − k0 ). Thus we should have ordp (k − k0 ) + 1 ≤ ck (f, r) for all r > 1 and all k in an arithmetic progression K = {k0 + mpν (p − 1) for m = 0, 1, 2, . . .}. (When p = 2 we should have ord2 (k − k0 ) + 2 ≤ ck (f, r).) The following or the results of some computations of ck (f, r). p = 2: 1. For k0 = 6, 10, 12, 14, 16, 20 let f ∈ Sk0 (0 (4)) be the unique newform. Then for all k with k0 < k ≤ 100 we have ck (f, 47) = ord2 (k − k0 ) + 2. 2. For k0 = 18, 22 let f ∈ Sk0 (0 (4)) be the unique, up to Galois conjugacy, newform. Then for all k with k0 < k ≤ 100 we have ck (f, 7) = ord2 (k−k0 )+2. 3. Let f ∈ S4 (0 (8)) be the unique newform. For most 4 < k ≤ 100 we have ck (f, 47) = ord2 (k − k0 ) + 2. However, in this range if ord2 (k − k0 ) ≥ 4 then ck (f, 47) = 5. Since ord2 (68 − 4) + 2 = 8, this is a problem; perhaps this form is not approximated. Very similar behavior occurs for the newforms in S6 (0 (8)), S8 (0 (8)), and S4 (0 (16)). 4. For the two newforms f ∈ S6 (0 (16)), we have ck (f, 47) ≤ 3 for all k < 100, so these f probably can not be approximated by finite slope forms. 5. Let f be the 2-deprivation of the unique normalized eigenform in Sk0 (0 (1)) for k0 = 12, 16, 18, 20, 22, 26. Then ck (f, 47) = ord2 (k − k0 ) + 2 for 12 < k ≤ 100. Same statement for k0 = 24, 28 for the 2-deprivation of one of the Galois conjugates and ck (f, 47) replaced by ck (f, 7). p = 3: 1. Suppose f is a newform in Sk0 (0 (9)) for k0 ≤ 12. Then for k0 < k ≤ 100 we have ck (f, 47) = ord3 (k − k0 ) + 1, except possibly for the nonrational form of weight 8, where we have only checked that ck (f, 7) ≥ ord3 (k − k0 ) + 1. 2. Let f be the twist of a newform in Sk0 (0 (1)) by ω3 for k0 ≤ 32. Then ck (f, 7) ≥ ord3 (k − k0 ) + 1 for k0 < k ≤ 100, with equality usually. 3. Let f be the newform in S2 (0 (45)) of tame level 5. Then c2+(3−1)3n (f, 7) = n + 1 for n = 0, 1, 2, 3 (here we are testing congruences with forms in Sk (0 (15))). p = 5: 1. Let f = q + q 2 + · · · ∈ S4 (0 (25)) be a newform. Then c4+4 (f, 7) = 1, c4+4·5 (f, 7) = 2, and c4+4·52 (f, 7) = 3. Same result for the newform f = q + 4q 2 + · · · ∈ S4 (0 (25)).

448

Robert F. Coleman and William A. Stein

2. Let f = q −q 2 +· · · ∈ S2 (0 (2 ·25)). Then c2+4 (f, 7) = 1 and c2+4·5 (f, 7) = 2, where we are testing congruences with forms in Sk (0 (10)). 3. Let f be one of the newforms in S2 (0 (53 )) defined over a quadratic extension of Q. Then c2+4 (f, 7) = c2+4·5 (f, 7) = c2+4·52 (f, 2) = 1/2. Thus it seems unlikely that f can be approximated by forms of finite slope. p = 7: 1. Let f ∈ S2 (0 (49)) be the newform. Then c2+6 (f, 7) = 1 and c2+6·7 (f, 7) = 2. Same statement for the form f = q − q 2 ∈ S4 (0 (49)) at weights 4 + 6 and 4 + 6 · 7. The data and results of this paper suggests the following: Guess 4.3. Let p be a prime and N an integer coprime to p. Then the eigenforms on X0 (Npt ) that can be approximated by finite-slope eigenforms are exactly the eigenforms on X0 (Np2 ). Suppose f is an infinite slope eigenform that can be approximated by finite slope eigenforms and f has weight k0 . Then for any k > k0 with k ≡ k0 (mod p − 1), there is an eigenform fk on X0 (Np) of weight k such that f ≡ fk (mod p n ) where n = ordp (k − k0 ) + 1 (or +2 if p = 2). (In general one might have to restrict to n sufficiently large.)

References [Böc01]

G. Böckle, On the density of modular points in universal deformation spaces. Amer. J. Math. 123 (5) (2001), 985–1007.

[BCP97]

W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (3–4) (1997), 235–265.

[BGR84]

S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis: A systematic approach to rigid analytic geometry. Springer-Verlag, Berlin 1984.

[CE03]

F. Calegari and M. Emerton, The Hecke Algebra Tk has Large Index. Preprint, 2003.

[CM98]

R. Coleman and B. Mazur, The Eigencurve. In Galois representations in arithmetic algebraic geometry (Durham, 1996), Cambridge University Press, Cambridge 1998, 1–113.

[Col96]

R. F. Coleman, Classical and overconvergent modular forms. Invent. Math. 124 (1–3) (1996), 215–241.

[Col97a]

R. F. Coleman, Classical and overconvergent modular forms of higher level. J. Théor. Nombres Bordeaux 9 (2) (1997), 395–403.

[Col97b]

R. F. Coleman, p-adic Banach spaces and families of modular forms. Invent. Math. 127 (3) (1997), 417–479.

Approximation of eigenforms of infinite slope by eigenforms of finite slope

449

[CF00]

P. Colmez and J.-M. Fontaine, Construction des représentations p-adiques semistables. Invent. Math. 140 (1) (2000), 1–43.

[GM92]

F. Gouvêa and B. Mazur, Families of modular eigenforms. Math. Comput. 58 (198) (1992), 793–805.

[GM98]

F. Q. Gouvêa and B. Mazur, On the density of modular representations. In Computational perspectives on number theory (Chicago, IL, 1995), Amer. Math. Soc., Providence, RI, 1998, 127–142.

[Gou88]

F. Q. Gouvêa, Arithmetic of p-adic modular forms. Springer-Verlag, Berlin 1988.

[Hat79]

K. Hatada, Eigenvalues of Hecke operators on SL(2, Z). Math. Ann. 239 (1) (1979), 75–96.

[Hat01]

K. Hatada, On classical and l-adic modular forms of levels Nl m and N . J. Number Theory 87 (1) (2001), 1–14.

[Hid86]

H. Hida, Iwasawa modules attached to congruences of cusp forms. Ann. Sci. École Norm. Sup. (4) 19 (2) (1986), 231–273.

[Kat73]

N. M. Katz, p-adic properties of modular schemes and modular forms. In Modular functions of one variable III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972). Lecture Notes in Math. 350, Springer-Verlag, Berlin 1973, 69–190.

[Kat75]

N. M. Katz, Higher congruences between modular forms. Ann. of Math. (2) 101 (1975), 332–367.

[Maz89]

B. Mazur, Deforming Galois representations. In Galois groups over Q (Berkeley, CA, 1987), Springer-Verlag, New York 1989, 385–437.

[Maz97]

B. Mazur, An “infinite fern” in the universal deformation space of Galois representations. Journées Arithmétiques (Barcelona, 1995), Collect. Math. 48 (1–2) (1997), 155–193.

[Ser73]

J-P. Serre, Formes modulaires et fonctions zêta p-adiques. In Proceedings of the International Summer School (University of Antwerp, RUCA, July 17–August 3, 1972), Lecture Notes in Math. 350, Springer-Verlag, Berlin 1973, 191–268.

[Shi94]

G. Shimura, Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan, Kanô Memorial Lectures 11 (1), Princeton University Press, Princeton, NJ, 1994.

[Stu87]

J. Sturm, On the congruence of modular forms. In Number theory (New York, 1984–1985), Springer-Verlag, Berlin 1987, 275–280.

Robert F. Coleman, Department of Mathematics, 970 Evans Hall #3840, University of California, Berkeley, CA 94720-3840, U.S.A. E-mail: [email protected] William A. Stein, Department of Mathematics, 325 Science Center, Harvard University, One Oxford Street, Cambridge, MA 02138, U.S.A. E-mail: [email protected]

Crystalline cohomology of singular varieties Richard Crew∗

Abstract. Let V be a complete discrete valuation ring of mixed characteristic p, absolute ramification index e, and residue field k. If m is an integer such that p m ≥ e/(p − 1), · Berthelot has defined a cohomology theory Hcris,m (_ /V) on the category of k-schemes, which for m = 0 is ordinary crystalline cohomology. If e ≤ p − 1 and X/k is proper and smooth, · Hcris,m (X/V) ⊗ Q is isomorphic to crystalline cohomology tensored with Q, but for singular X these spaces can have infinite dimension. We show that if X/k is proper and m sufficiently · large, the quotient of Hcris,m (X/V)⊗Q by the subspace of elements annihilated by some power i (X). of Frobenius is isomorphic to the rigid cohomology Hrig

2000 Mathematics Subject Classification: 14F30

1 Introduction · (X/K) of The subject of this article is the relation between the rigid cohomology Hrig a complete but possibly singular variety X over a field k of characteristic p > 0, and · (X/V), for sufficiently the corresponding crystalline cohomology of level m, Hcris,m large m. This latter theory was discovered by Berthelot in the course of developing the · (X/V) theory of arithmetic D-modules. For any separated X/k of finite type, Hcris,m is a graded module over a complete discrete valuation ring V with residue field k and fraction field K, defined whenever pm ≥ e/(p − 1), e being the absolute ramification index of V. When e ≤ p − 1, we can take m = 0 and the result is ordinary crystalline cohomology. The level m theory is functorial in the pair (X, V), and for m ≥ m there is a natural map · · Hcris,m  (X/V) → Hcris,m (X/V).

(1.1)

· and showed that In a letter to Illusie [3], Berthelot sketched the construction of Hcris,m when X is proper, the rigid cohomology of X is a derived category inverse limit of level m crystalline cohomology of X tensored with Q: ∼

R lim Rcris,m (X/V)Q −−→ Rrig (X/K). ← m

∗ Partially supported by the NSA

(1.2)

452

Richard Crew

Here the right hand side is the derived category object computing the rigid cohomology of X, and the subscript Q indicates (here and afterwards) a tensor product with Q. If X/k is smooth and proper, the maps 1.1 become isomorphisms when tensored with Q, and it follows from 1.2 that rigid cohomology is isomorphic to crystalline cohomology of any level tensored with Q. This cannot happen if X is singular, since · (X/K) are of finite dimension and the H · the Hrig cris,m (X/V)Q , in general, are not, as Berthelot showed long ago when m = 0. Nonetheless there is a fairly close relation between crystalline and rigid cohomology, even in the singular case. Choose a lifting σ : V → V of the absolute Frobenius of k; we use the same symbol to denote its · (X/V) is functorial in extension to K. As level m crystalline cohomology Hcris,m the pair (X, V), the absolute Frobenius of X induces a σ -linear endomorphism F · · (X/V)Q . Denote by F ∞ Hcris,m (X/V)Q the F -torsion of the K-vector space Hcris,m · subspace of Hcris,m (X/V)Q , i.e. the set of elements annihilated by some power of F ; · (X/V)Q . it is a K-subspace of Hcris,m Theorem 1.1. Suppose that X/k is proper and p m ≥ e/(p − 1). If m ≥ 2, the · (X/V)Q is annihilated by a fixed power of Frobenius, F -torsion subspace F ∞ Hcris,m · · and the quotient Hcris,m (X/V)Q /F ∞ Hcris,m (X/V)Q has finite dimension over K. If k is perfect, the condition m ≥ 2 can be dropped. For m >> 0, there is a natural isomorphism ∼

· · · (X/K) −−→ Hcris,m (X/V)Q /F ∞ Hcris,m (X/V)Q Hrig

(1.3)

· The proof will show that the inverse system {Hcris,m (X/V)Q }m is Mittag-Leffler, so that 1.2 yields an isomorphism ∼

· · (X/K) −−→ lim Hcris,m (X/V)Q . Hrig ←

(1.4)

m

For most of this article we will assume in addition that X is embeddable, i.e. that there a closed immersion X → Y over V, where Y is a formal scheme proper over V and formally smooth in a neighborhood of X (this will be the case, for example, if X/k is projective). Both the definition of rigid cohomology and the arguments for theorem 1.1 in the general case require results on cohomological descent, which we briefly sketch at the end of this article. Suppose now that k is algebraically closed, V is a vector space over K (not necessarily of finite dimension), and F is a σ -linear endomorphism of V (not necessarily an isomorphism). The slope λ part V λ of V can be defined in the same way as for finite-dimensional V : if λ = r/s is a nonzero rational number expressed in lowest terms, then V λ is the K-span of the kernel of F s − p r . Corollary 1.2. Suppose, in addition to the hypotheses of 1.1, that k is algebraically closed. Then for any λ ∈ Q and all sufficiently large m, the isomorphism 1.3 induces

Crystalline cohomology of singular varieties

453

an isomorphism ∼

· · (X/K)λ −−→ Hcris,m (X/V)λQ . Hrig

(1.5)

· Proof. It is enough to check that F s − pr is an isomorphism on F ∞ Hcris,m (X/V)Q , · s r for then it follows that the kernel of F − p on Hcris,m (X/V)Q maps bijectively to · · (X/V)Q /F ∞ Hcris,m (X/V)Q . But by theorem 1.1, the kernel of F s − pr on Hcris,m · ns nr F − p acts as an isomorphism on Hcris,m (X/V)Q for n >> 0, and as F ns − p nr · is (as an endomorphism of Hcris,m (X/V)Q ) divisible by F s − p r on both the left and s r   on the right, F − p act as an isomorphism as well.

The proof of 1.1 gives no effective control on the value of m for which 1.3 or 1.5 is an isomorphism. For λ = 0 however, 1.5 is an isomorphism for all m · (X/V)0Q is canonically isomorphic to the p-adic étale cohomology because Hcris,m · Het (X, Q) ⊗ K, as was shown by Bloch and Illusie when m = 0 and X/k is proper and smooth, and by Etesse and Le Stum [11] in general. Following [11], one can introduce, for X/k separated and of finite type, a sort of crystalline cohomology of level m with compact supports, by choosing an embedding X → X into a proper scheme over k; if we set Z = X − X and denote by i : Z → X the canonical embedding, then we set Rc,m (X ⊂ X/V) = R((X/V)(m) , OX/V → i∗cris i cris∗ OX/V ) · (X ⊂ X/V) = H · (R and Hc,m c,m (X ⊂ X/V)). This is not necessarily independent of the choice of embedding. The excision exact sequence for rigid cohomology with compact support and theorem 1.1 immediately imply the following:

Corollary 1.3. If X/k is separated and of finite type and m is sufficiently large, there is a natural isomorphism ∼

· · (X ⊂ X/V)Q /F ∞ Hc,m (X ⊂ X/V)Q . Hc· (X/K) −−→ Hc,m

The next section summarizes the construction of the higher crystalline sites and some of their basic properties, such as their relation to rigid cohomology, not all of the details of which have made it into print. Our main sources are an unpublished letter of Berthelot [3], and the articles [6], [11]. The proof of 1.1 is described in section 3. Acknowledgements. I would like to thank the organizers of the “Dwork Trimester in Italy,” especially P. Berthelot and B. Chiarellotto, who organized the week devoted to “Rigid Cohomology and Isocrystals.” The results described here were not known at the time of the conference, and the essential idea arose in a conversation with P. Berthelot immediately after I had given my lecture there. I am grateful to him, and also to B. Le Stum, for a number of helpful discussions on these matters, particularly the treatment of the non-embeddable case. Finally I would like to thank the mathematics departments at the Université de Rennes I and the Università di Padova for their hospitality.

454

Richard Crew

2 Rigid and crystalline cohomology Crystalline cohomology of level m ≥ 0 uses an extension of the notion of divided powers in its construction, so we first briefly review the latter, referring the reader to [8] and [5] for full details. Let m be a natural number, A a Z(p) -algebra, and I ⊂ A an ideal. A partial divided power structure of level m on I is an ideal with divided powers (J, γ ), where γ is compatible with the canonical divided power structure on (p) ⊂ Zp , such that J ⊂ I,

I (p

m)

+ pI ⊂ J

(p m )

denotes the ideal generated by the pm powers of the elements of I . The where I triple (I, J, γ ) is called an m-PD-ideal. If m = 0, this is the usual notion of ideal with divided powers used in crystalline cohomology. For example, if V is a complete discrete valuation ring of mixed characteristic p, then the maximal ideal of V has a partial divided power of level m if the absolute ramification index e of V satisfies p m ≥ e/(p − 1). Observe that an m-PD-ideal (I, J, γ ) is automatically an m -PDideal for any m > m. If A is a Z(p) -algebra and I is any ideal in A, there is an A-algebra P (m) (I ) and an m-PD-ideal (I , I˜, [−] ) in P (m) (I ) that is universal for homomorphisms A → A sending I to an m-PD-ideal; it is called the divided power envelope of level m of (A, I ). Its formation is compatible with flat base change, and it sheafifies in the obvious way. Suppose now that S is a p-adic formal scheme, or a scheme on which p is nilpotent, and fix an m-PD-structure (J,
, γ ) on S. If X is an S-scheme, the crystalline site of level m Cris(X/S)(m) is defined in the same way as the usual cristalline site (e.g. as in [1] III §1), but with divided powers of level m replacing the usual notion; thus an object (U, T , δ) of Cris(X/S)(m) consists of a Zariski-open U ⊂ X, an S-scheme T on which p is nilpotent, a closed immersion U → T , and finally an m-PD-structure δ on the ideal of U → T , compatible with (J,
, γ ) (c.f. [5] 1.3.2 for this latter notion). An object (U, T , δ) is called an “m-PD-thickening of U .” Morphisms are defined in the usual way, by an obvious commutative diagram. A covering family is a collection of morphisms (Ui , Ti , δi ) → (U, T , δ) such that (Ti → T ) is a cover in the Zariski topology. The associated topos will be denoted (X/S)(m) . As an example of a sheaf on Cris(X/S)(m) , we have the crystalline structure sheaf, defined as in the usual case by OX/S (U, T , δ) = (T , OT ). It is a crystal of OX/S -modules, where this latter notion is defined in the same way as for m = 0 (e.g. in [1] III §3.2) We recall some other facts and constructions regarding these topoi: • If m ≥ m, then an m-PD-thickening is automatically an m -PD-thickening, and  thus Cris(X/S)(m) is naturally a subcategory of Cris(X/S)(m ) . The inclusion  functor extends to a morphism of topoi (X/S)(m) → (X/S)(m ) , which satisfies an obvious transitivity condition when m ≥ m ≥ m.

455

Crystalline cohomology of singular varieties (m)

(m)

• There is a morphism of topoi uX/S : (X/S)cris → XZar , the “projection onto the (m)

Zariski topos”, such that R((X/S)(m) , ) R(X, )  RuX/S . The associated direct image functor associates to a sheaf E on (X/S)(m) the sheafification of the presheaf U  → ((U/S)(m) , E|U ). For the case m = 0, see [1].

• The theory of cohomological descent for a Zariski-open covering that is described in [1] V §3.4 for m = 0 extends basically without change to the case of arbitrary m. Berthelot shows in [3] that if p is locally nilpotent on S, the direct image (m) RuX/S∗ OX/V can be calculated up to isogeny by de Rham cohomology, in the sense that if X → Y is an embedding into an S-scheme smooth in a neighborhood of X, and P is the level m divided power neighborhood of X in Y, then there is a natural map (m)

RuX/S∗ OX/V → OP ⊗ ·Y/V

(2.1)

whose kernel and cokernel are annihilated by p (dim(X/S)+1)m (this is one point where the theory for level m > 0 differs from the case m = 0, for which the de Rham complex computes this direct image exactly). If S is a p-adic formal scheme, we take Y to be a formal scheme, formally smooth in a neighborhood of X, and replace P by the inductive limit Pˆ , in the category of formal schemes, of the level m divided power envelopes of X in the Y ⊗ V/p n V. We will call it the “completed divided power envelope of X in Y” (abusively, since it is not defined as the completion of a V-scheme). From now on we fix a complete discrete valuation ring V of mixed characteristic p, with residue field k, fraction field K, and absolute ramification index e, and m will always be a positive integer satisfying pm ≥ e/(p − 1), so that the maximal ideal (π ) of V has a unique m-PD-structure compatible with the canonical one on Z(p) . The preceding theory then applies with S = Spf(V), and the isogeny 2.1 yields an isomorphism (m)

RuX/V∗ OX/V ⊗ Q = OPˆ ⊗ ·Y/V ⊗ Q

(2.2)

where Pˆ is the formal scheme described above. Following [3], we can now explain, in the embeddable case, the relation between the cohomology of the sites (X/V)(m) and the rigid cohomology of X (c.f. also [6] 1.9 for the case m = 0). The general case requires cohomological descent and will be described later. Suppose, then, that X → Y is an embedding into a formal V-scheme formally −m smooth in a neighborhood of X, and let [X]m denote the closed tube of radius |p|p about X. If Y = Spf(A) (which we can always assume, working locally), and I = (f1 , . . . , fr ) is the ideal of X ⊂ Y, then [X]m = Max(Bm ⊗ K) with pm

pm

Bm = A{T1 , . . . , Tr }/(pT1 − f1 , . . . , pTr − fr ).

456

Richard Crew

From this it is clear that the ideal I Bm has an m-PD-structure, given by pBm ⊂ I Bm endowed with its canonical divided powers. By the universal property of divided power envelopes, there is a map of the level m divided power envelope of (A, I ) to p m−1

)[p] defines a map from Bm . On the other hand, the assignment Ti  → (p − 1)!(fi Bm to the level m − 1 divided power envelope of (A, I ), since p!x [p] = x p . What we get in the end are OY ⊗ Q-algebra maps (m) (m−1) PˆQ → sp∗ O[X]m → PˆQ

(2.3)

where sp : Yan → Y ⊗ k is the specialization map [2]. The composite of the maps in 2.3 is the canonical map Pˆ (m) → Pˆ (m−1) , and the same holds for the composite sp∗ O[X]m → sp∗ O[X]m−1 . Tensoring 2.3 with the de Rham complex and passing to the inverse limit yield an isomorphism ∼

R lim(OPˆ (m) ⊗ ·Y/V ⊗ Q) −−→ R lim sp∗ ·[X]m ← ← m

(2.4)

m

Yan

where ]X[⊂ is the (open) tube of X ⊂ Y (for the sense in which such derived inverse limits are to be understood, see e.g. the discussion in [9] 7.17–26). If we observe that for any locally free sheaf E on [X]m , the map on sections over an affinoid induced by the inclusion [X]m → [X]m+1 has dense image, then we conclude from the topological Mittag-Leffler condition of [EGA 0III 13.2.4] that the derived inverse limit on the left hand side of 2.5 coincides with the ordinary inverse limit. We thus obtain an isomorphism ∼

R lim sp∗ ·[X]m −−→ sp∗ ·]X[ ←

(2.5)

m

which, combined with 2.4, results in a quasi-isomorphism ∼

R lim(OPˆ (m) ⊗ ·Y ⊗ Q) −−→ sp∗ ·]X[ . ←

(2.6)

m

The complex of global sections over ]X[ of the target of 2.6 computes the rigid cohomology of X, so the result of applying the global sections functor to 2.6 can be written ∼

R lim Rcris,m (X, OX/V )Q −−→ Rrig (X). ←

(2.7)

m

In these last two quasi-isomorphisms it is essential that the tensor product with Q goes inside the inverse limit. One can show, at least if X/k is smooth, that the moving the tensor product outside the limit results in the slope zero part of the rigid cohomology.

Crystalline cohomology of singular varieties

457

3 The inverse limit and the action of Frobenius The three basic steps in the proof of theorem 1.1 are the three lemmas 3.1-3.3 below. The first (3.1), on the behavior of the inverse limit in 2.7, is an application of the finiteness theorems of Grosse-Klönne for the de Rham cohomology of dagger spaces. · (X/V). The remaining two concern the behaviour of the Frobenius map on Hcris,m One (3.2) uses a technical result that is at the base of Berthelot’s “Frobenius descent” theorem for arithmetic D-modules; the other (3.3) uses the nondegeneracy of the (linearized) Frobenius, which was proven long ago by Monsky in the context of Washnitzer-Monsky cohomology [13], and has been shown by Tsuzuki [14] to hold for rigid cohomology in general. Lemma 3.1. If X/k is proper, then the natural map · · Hcris,m+2 (X/V)Q → Hcris,m (X/V)Q

has finite K-rank. In the embeddable case the proof is as follows. Let X → Y be a closed immersion, with Y proper and formally smooth over V in a neighborhood of X. Let π be a uniformizer of V. Recall that if X ⊂ Y is defined by the sheaf of ideals I = (fi ) and  λ ∈ |K × | satisfies |π | < λ ≤ 1, the open tube ]X[λ ⊂ Yan is the locus of |fi | < λ; is independent of the choice of generators of I ([4] 1.1.8). We now choose λ such that |p|p

−(m+1)

< λ < |p|p

−(m+2)

(3.1)

so that there are maps of OY ⊗ Q-algebras (m+2) (m) PˆQ → sp∗ O[X]m+2 → sp∗ O]X[λ → sp∗ O[X]m+1 → PˆQ

(3.2)

analogous to 2.3. Tensoring with the de Rham complex of Y and taking global coho· · mology, we find that the projection Hcris,m+2 (X/V) → Hcris,m (X/V) factors · · · Hcris,m+2 (X/V)Q → HdR (]X[λ ) → Hcris,m (X/V)Q

where the middle term is rigid-analytic de Rham cohomology. Thus it is enough to · (]X[ ) has finite dimension, but this follows from the finiteness theorem show that HdR λ of Grosse-Klönne ([12] Theorem A). This concludes the proof of 3.1 in the embeddable case; the proof in the general case will be sketched later. · It follows from 3.1 that the inverse system {Hcris,m (X/V)Q }m is Mittag-Leffler, and this together with 2.7 yields the isomorphism ∼

· · lim Hcris,m (X/V)Q −−→ Hrig (X/K) ← m

promised in §1.

(3.3)

458

Richard Crew

Fix s > 0 and denote by : X → X the relative ps -power Frobenius, and by g : X  → X the canonical projection; then FX = g  and FX =  g are the is the absolute p s -power Frobenius morphisms of X and X respectively. Finally, let · · (X/V)Q → Hcris,m (X/V)Q be the canonical projection. p : Hcris,m+s ∼

· · Lemma 3.2. There is an isomorphism δ : Hcris,m (X /V)Q −−→ Hcris,m+s (X/V)Q ∗ ∗ compatible with absolute Frobenius, such that the induced maps FX , FX factor

FX∗ = δ  g ∗  p,

FX∗  = g ∗  p  δ.

(3.4)

Proof. It is enough to construct an isomorphism δ satisfying ∗ = p  δ and FX∗  δ = δ  FX∗  , for then FX∗  δ = δ  FX∗  = δ  g ∗  ∗ = δ  g ∗  p  δ and the equalities in 3.4 follow immediately. The map δ is basically the one constructed in [7] 2.2.8(i). By cohomological descent we can work locally, so we can assume there is a closed immersion X → Y of X into a formal V-scheme formally smooth around X, and a lifting of the absolute p-power Frobenius of Y ⊗ k to Y. Let φ : Y → Y be the corresponding lifting of the relative Frobenius. Finally, we denote as before the (m) completed divided power envelopes of level m of X ⊂ Y and X ⊂ Y by PˆX , resp. (m) PˆX . By the Poincaré lemma 2.2, it suffices to see that there is an isomorphism (m) ∼ (m+s) φ ∗ PˆX −−→ PˆX

through which the action of Frobenius factors. In fact if we set X˜ = X ×Y Y, the (m+s) (m) construction of the divided power envelopes shows that PˆX Pˆ ˜ , while on the X other hand, the natural map (m)

(m) X

φ ∗ PˆX → Pˆ ˜

is an isomorphism since Y → Y is flat, and the formation of divided power envelopes of any level commutes with flat base change. The compatibility with Frobenius is evident from the construction.   · For the last step in the proof of theorem 1.1 we set Mm = Hcris,m (X/V)Q and · M = lim Mm Hrig (X) (the last isomorphism is 3.3). It follows from lemma 3.2 ←m that the kernel of the projection Mm+s → Mm is contained in the kernel of the absolute p s -power Frobenius FX∗ : Mm+s → Mm+s , and from this it follows that the induced map

Mm+s /F ∞ Mm+s → Mm /F ∞ Mm injective. On the other hand, lemma 3.1 shows that the projection Mm+2 → Mm has finite rank, and therefore FX2∗ : Mm+2 → Mm+2 has finite rank too. Since Mm+2 /Ker(FX2∗ ) has finite dimension, its Frobenius torsion is killed by a fixed power

Crystalline cohomology of singular varieties

459

of FX , and the same must be true for Mm+2 . This proves the first assertion of 1.1. Furthermore if k is perfect, g : X → X is an isomorphism compatible with absolute Frobenius. We can then use the isomorphism δ with s = 2 to reduce the cases m = 0, 1 to the case m ≥ 2. Lemma 3.3. lim F ∞ Mm = 0. ←

(3.5)

m

Proof. Set M  = limm→∞ F ∞ Mm , so that M  ⊂ M. By [14] 8.1.1, the linearization fs : σ s∗ M → M of FXs∗ : M → M is bijective. The subspace M  is evidently FX∗ -stable, and as M  has finite dimension, fs : σ s∗ M  → M  must be a bijection as well. By what has been proven so far we have FXs∗ (F ∞ Mm ) = 0 for some s > 0.  Denote by πm : M → Mm the canonical projection, and choose an element
x ∈ M .   Since fs is bijective on M , there are ai ∈ K, yi ∈ M such that y = i ai ⊗ yi satisfies fs (y) = x. Since fs (πm (yi )) = 0 for all i, we have πm (x) = 0 as well, and consequently x = 0. Since x ∈ M  was arbitrary, we have M  = 0.   Since every term in the exact sequence 0 → F ∞ Mm → Mm → Mm /F ∞ Mm → 0

(3.6)

satisfies the Mittag-Leffler condition, 3.3 and lemma 3.3 show that · (X) lim Mm /F ∞ Mm . Hrig ← m

Since the transition maps in {Mm /F ∞ Mm }m are injective and the individual terms have · (X) M / ∞ M for all finite dimension for m ≥ 2, we have an isomorphism Hrig m F m sufficiently large m, which is the last assertion in 1.1. It remains to explain how we prove 3.1 in the case when X is not necessarily embeddable. The main task is to find a replacement for the tube of radius λ used in the proof of 3.1; the idea is to construct a variant of rigid cohomology which uses such tubes of radius λ systematically. This can be done by following, step by step, Berthelot’s construction [4] in the embeddable case, and those of Chiarellotto and Tsuzuki [10] to construct a theory of cohomological descent for “rigid cohomology with radius λ.” We will just sketch the argument, and leave to the reader the rather extensive task of checking that everything really works. Once the construction is complete, it remains to construct maps analogous to those in 2.3 and 3.2. By cohomological descent, this is a purely local problem. The essential points in the construction of rigid cohomology with radius λ are the following:

460

Richard Crew

• In the embeddable case, the construction is as follows. Suppose that X → X is an open immersion of separated k-schemes of finite type, and X → Y is an closed immersion into a formal V-scheme, proper and formally smooth in a neighborhood of X. As before, fix λ such that |π| < λ ≤ 1. An admissible open V ⊂]X[λ is a λ-strict neighborhood of ]X[λ if ]X[λ ⊂ V and {V , ]X − X[λ } is an admissible cover of ]X[λ . If E is a sheaf on ]X[λ , we define the sheaf j † E by setting j † E(W ) = lim E(W ∩ V ) → V

for any affinoid subset W ⊂]X[λ ; the direct limit is over λ-strict neighborhoods of ]X[λ in ]X[λ . If X is complete, we define the radius λ rigid cohomology of X by · Hrig,λ (X/K) = H · (]X[λ , j † ·]X[/K ).

The fibration theorems of [4] (c.f. also [2]) are valid for open tubes of radius λ such that |π| < λ ≤ 1 (c.f. [4] 1.3.1 and the remark after 1.3.2; note that the arguments of [4] use only the assumption |π | < λ ≤ 1). It follows that the right hand side of the above definition is independent of the choices of X ⊂ X ⊂ Y. If λ = 1, this is simply Berthelot’s original definition. Note that Grosse-Klönne’s argument for the finite-dimensionality of rigid cohomology (in the embeddable · (X/K) has finite case, c.f. [12] 3.6 and 3.8) can also be used to prove that Hrig,λ dimension as well. • For the general definition we follow Chiarellotto and Tsuzuki [10], [14]. Let X → X be as before, choose a Zariski cover U → X if X, and set U = X ×X U (here U is disjoint union of open subsets of X). We can assume that this has been done so that there is a closed immersion U → U such that U → V is formally smooth in a neighborhood of U (this is always possible: take U to be a union of affine open subsets, for example). We then define simplicial (formal) schemes U· = cosk X 0 U,

U · = cosk X 0 U,

U· = cosk V 0U

and observe that there is are closed immersions U· → U · → U· of simplicial formal schemes. One can show that U· is, in the terminology of [14], a universally de Rham descendable hypercovering of (X, X) (or rather: the analogous concept for radius λ). One can then define the radius λ rigid cohomology of X by · Hrig,λ (X/K) = H · (]U · [λ , j † ·]U

· [/K

).

(3.7)

Following the method of [10] and [14], one has to check that radius λ rigid cohomology, so defined, is independent of the choices of X, U , and U, and that it coincides with the previous definition in the case that X is embeddable.

Crystalline cohomology of singular varieties

461

· (X/K) has finite dimension (c.f. [14] 8.1.1). • The next step is to show that Hrig,λ This can be done by means of a Mayer–Vietoris spectral sequence for radius λ rigid cohomology relative to an open covering of X (c.f. [14] 7.1.2). One can then prove the finite dimensionality of by reducing  to the embeddable case. If {Ui } is an open covering of X and X· = cosk 0X ( i Ui ), then a modification of the simplicial construction used above leads to a spectral sequence pq

p

p+q

E1 = Hrig,λ (Xq /K) ⇒ Hrig,λ (X/K). We can choose the Ui to be embeddable (e.g. affine), in which case the E1 terms have finite dimension; the same is then true of the abutment. To complete the proof of 3.1 in the general case, we have to show that the canonical projection factors 

· · · (X/V)Q → Hrig,λ (X/K) → Hcris,m (X/V)Q Hcris,m+2

(3.8)

for any λ ∈ |K × | satisfying 3.1. Since X is proper, we can take X = X, U = U in the above construction. But now the maps in 3.8 can be constructed by cohomological descent, so the construction can be made locally, i.e. on the components of U . But now the situation relative to U → U is the same as in §2, so the same construction can be used. This concludes the proof of 3.1, and with it the proof of 1.1. · (X/K) has finite dimension over K, and argument similar to Remark. Since Hrig,λ the one given for crystalline cohomology shows that there is an isomorphism ∼

· · (X/K) −−→ Hrig (X/K) lim Hrig,λ ←

λ→1 · (X/K) Hrig

· · Hrig,λ (X/K)/F ∞ Hrig,λ (X/K) for λ sufficiently close to one. and that · · (X/K) for such λ? Equivalently, Is there, in fact, an isomorphism Hrig,λ (X/K) Hrig · is Frobenius bijective on Hrig,λ (X/K)?

References [1]

P. Berthelot, Cohomologie cristalline des schémas de caractéristique p > 0. Lecture Notes in Math 407, Springer-Verlag, Berlin 1974.

[2]

P. Berthelot, Géometrie rigide et cohomologie des variétés algebriques de caracteristiques p, Mem. Soc. Math. France 23 (1986), 7–32.

[3]

P. Berthelot, letter to Illusie, May 1990.

[4]

P. Berthelot, Cohomologie rigide et cohomologie rigide à supports propre, Première partie. Preprint IRMAR Jan. 1996.

462

Richard Crew

[5]

P. Berthelot, D-modules arithmétiques I. Opérateurs différentiels de niveau fini. Ann. Sci. École Norm. Sup. 29 (1996), 185–272.

[6]

P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide. Invent. Math. 128 (1997) 329–377.

[7]

P. Berthelot, D-modules arithmétiques II. Descente par Frobenius. Mem. Soc. Math. France 81 (2000).

[8]

P. Berthelot, Introduction à la théorie arithmetique des D-modules. In Cohomologie padiques et applications arithmétiques II, Astérisque 279 (2002) 1–80.

[9]

P. Berthelot and A. Ogus, Notes on Crystalline Cohomology, Princeton Univiversity Press, 1978.

[10] B. Chiarellotto and N. Tsuzuki, Cohomological descent of rigid cohomology for etale coverings. Rend. Sem. Mat. Univ. Padova, to appear. [11] J.-Y. Etesse and B. Le Stum, Fonctions L associées aux F -isocristaux surconvergents II: Zéros et pôles unités. Invent. Math. 127 (1997), 1–31. [12] E. Grosse-Klönne, Finiteness of de Rham cohomology in rigid analysis. Duke Math J. 113 (1) (2002) 57–91. [13] P. Monsky and G. Washnitzer, Formal cohomology: I. Ann. of Math. 88 (1968) 181–217. [14] N. Tsuzuki, Cohomological descent in rigid cohomology. In Geometric Aspects of Dwork Theory (A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz and F. Loeser, eds.), Walter de Gruyter, Berlin 2004, 931–981. Richard Crew, Dept. of Mathematics, 358 Little Hall, The University of Florida, Gainesville, FL 32611, U.S.A. E-mail: [email protected]

Stacks of twisted modules and integral transforms Andrea D’Agnolo∗ and Pietro Polesello∗∗

Abstract. Stacks were introduced by Grothendieck and Giraud and are, roughly speaking, sheaves of categories. Kashiwara developed the theory of twisted modules, which are objects of stacks locally equivalent to stacks of modules over sheaves of rings. In this paper we recall these notions, and we develop the formalism of operations for stacks of twisted modules. As an application, we state a twisted version of an adjunction formula which is of use in the theory of integral transforms for sheaves and D-modules. 2000 Mathematics Subject Classification: 14A20, 32C38, 35A22

Contents 1

2

3

Stacks of twisted modules 1.1 Prestacks . . . . . . . . . 1.2 Stacks . . . . . . . . . . . 1.3 Constructions of stacks . . 1.4 Operations on stacks . . . 1.5 Linear stacks . . . . . . . 1.6 Operations on linear stacks 1.7 Stacks of twisted modules

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

465 466 469 471 472 473 474 474

Operations 2.1 Morita theory I. Functors admitting an adjoint 2.2 Internal product of stacks of twisted modules 2.3 Morita theory II. Relative case . . . . . . . . 2.4 Pull-back of stacks of twisted modules . . . . 2.5 Twisted sheaf-theoretical operations . . . . . 2.6 Derived twisted operations . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

475 476 477 480 480 483 484

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

Descent 485 3.1 Morita theory III. Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . 485

∗A.D’A. had the occasion to visit the Research Institute for Mathematical Sciences of Kyoto University during the preparation of this paper. Their warm hospitality is gratefully acknowledged. ∗∗ P.P. was partially supported by INdAM during the preparation of this paper.

464 3.2 3.3 3.4 3.5 3.6 4

Andrea D’Agnolo and Pietro Polesello Twisting data on an open covering . . . . Twisting data . . . . . . . . . . . . . . . Classification of stacks of twisted modules Operations in terms of twisting data . . . Complex powers of line bundles . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

486 487 492 493 495

Examples and applications 4.1 Twisted modules over commutative local rings 4.2 Twisting by inner forms . . . . . . . . . . . . 4.3 Azumaya algebras . . . . . . . . . . . . . . . 4.4 Twisted differential operators . . . . . . . . . 4.5 Twisted D-module operations . . . . . . . . 4.6 Twisted adjunction formula . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

495 496 497 498 500 503 504

Introduction Stacks are, roughly speaking, sheaves of categories. They were introduced by Grothendieck and Giraud [14] in algebraic geometry where some special stacks, called gerbes, are now commonly used in moduli problems to describe objects with automorphisms (see for example [2, 29]). Recently, gerbes have infiltrated differential geometry and mathematical physics (see for example [7, 31, 15, 6]). We are interested here in the related notion of twisted modules, which are objects of stacks locally equivalent to stacks of modules over sheaves of rings. The simplest example is that of stacks of twisted R-modules on a locally ringed space (X, R). These can be considered as higher cohomological analogues to line bundles. More precisely, line bundles on X are sheaves of R-modules locally isomorphic to R, and their isomorphism classes describe the cohomology group H 1 (X; R× ). Stacks of twisted R-modules are R-linear stacks on X locally equivalent to the stack Mod(R) of R-modules and, as we shall recall, their equivalence classes describe the cohomology group H 2 (X; R× ). As line bundles correspond to principal R× -bundles, so stacks of twisted R-modules correspond to gerbes with band R× . However, this correspondence no longer holds for the more general type of stacks of twisted modules that we consider here. Twisted modules appear in works by Kashiwara on representation theory [18] and on quantization [20]. In the first case, they were used to describe solutions on flag manifolds to quasi-equivariant modules over rings of twisted differential operators (see also [26]). In the second case, twisted modules turned out to be the natural framework for a global study of microdifferential systems on a holomorphic contact manifold (see also [28, 30]). Rings of microdifferential operators can be locally defined on a contact manifold, but do not necessarily exist globally. Kashiwara proved that there exists

Stacks of twisted modules and integral transforms

465

a globally defined C-linear stack which is locally equivalent to the stack of modules over a ring of microdifferential operators. Twisted modules induced by Azumaya algebras are used in [8, 11] in relation with the Fourier–Mukai transform. Motivated by Kashiwara’s work on quantization, we consider here twisted modules over sheaves of rings which are not necessarily commutative nor globally defined. More precisely, let X be a topological space, or more generally a site, and R a sheaf of commutative rings on X. Then M is a stack  of R-twisted modules on X if it is R-linear and there exist an open covering X = i∈I Ui , sheaves of R|Ui -algebras Ai , and R|Ui -equivalences M|Ui ≈ Mod(Ai ), where Mod(Ai ) denotes the stack of left Ai -modules on Ui . We review the notions of stack and stack of twisted modules in Section 1, restricting to the case of topological spaces for simplicity of exposition. Morita theory is the basic tool to deal with stacks of R-twisted modules, and we use it to develop the formalism of operations, namely duality (·)−1 , internal product R , and inverse image f  by a continuous map f : Y − → X. If A and A are sheaves of R-algebras on X, these operations satisfy Mod(A)−1 ≈ Mod(Aop ), Mod(A) R Mod(A ) ≈ Mod(A ⊗R A ), and f  Mod(A) ≈ Mod(f −1 A). With this formalism at hand, we then describe Grothendieck’s six operations for derived categories of twisted modules over locally compact Hausdorff topological spaces. This is the content of Section 2. Morita theory is used again in Section 3 to describe effective descent data attached to semisimplicial complexes. In particular, we get a Cech-like classification of stacks of R-twisted modules, with invertible bimodules as cocycles, which is parallel to the bitorsor description of gerbes in [6]. In Section 4, assuming that R is a commutative local ring, we recall the above mentioned classification of stacks of twisted R-modules in terms of H 2 (X; R× ). We then consider the case of twisted modules induced by ordinary modules over an inner form of a given R-algebra. This allows us to present in a unified manner the examples provided by modules over Azumaya algebras and over rings of twisted differential operators. Finally, we state a twisted version of an adjunction formula for sheaves and D-modules, which is of use in the theory of integral transforms with regular kernel, as the Radon–Penrose transform. This paper is in part a survey and in part original. The survey covers material from Kashiwara’s papers [18, 20], from his joint works [26, 27], and from the last chapter of his forthcoming book with Pierre Schapira [25]. The main original contribution is in establishing the formalism of operations for stacks of twisted modules. It is a pleasure to thank Masaki Kashiwara for several useful discussions and insights. We also wish to thank him and Pierre Schapira for allowing us to use results from a preliminary version of their book [25].

466

Andrea D’Agnolo and Pietro Polesello

1 Stacks of twisted modules The theory of stacks is due to Grothendieck and Giraud [14]. We review it here restricting for simplicity to the case of stacks on topological spaces (thus avoiding the notions of site and of fibered category). Finally, we recall the notion of stack of twisted modules, considering the case of modules over rings which are not necessarily commutative nor globally defined. Our main references were [18, 20, 24, 25].

1.1 Prestacks We assume that the reader is familiar with the basic notions of category theory, as those of category, functor between categories, transformation between functors (also called morphism of functors), and equivalence of categories. If C is a category, we denote by Ob(C) the set1 of its objects, and by Hom C (c, d) the set of morphisms between the objects c and d. The identity of Hom C (c, c) will be denoted by idc . Denote by Cop the opposite category, which has the same objects as C and reversed morphisms Hom Cop (c, d) = Hom C (d, c). If D is another category, denote by Hom (C, D) the category of functors from C to D, with transformations as morphisms. Let X be a topological space, and denote by X the category of its open subsets with inclusion morphisms. Recall that the category of presheaves on X with values in a category C is the category Hom (Xop , C) of contravariant functors from X to C. In particular, presheaves of sets are obtained by taking C = Set, the category of sets2 and maps of sets. Considering C = Cat, the category of categories3 and functors, one v has a notion of presheaf of categories. This a functor F : Xop − → Cat, and if W − → u V − → U are inclusions of open sets, the restriction functors F (u) : F (U ) − → F (V ) and F (v) : F (V ) − → F (W ) are thus required to satisfy the equality F (v)  F (u) = F (u  v). Such a requirement is often too strong in practice, and the notion of prestack is obtained by weakening this equality to an isomorphism of functors, i.e. to an invertible transformation. In other words, prestacks are the 2-categorical4 version of presheaves of categories. However, we prefer not to use the language of 2-categories, giving instead the unfolded definition of prestack. 1 Following Bourbaki’s appendix in [SGA4], one way to avoid the paradoxical situation of dealing with the set of all sets is to consider universes, which are “big” sets of sets stable by most of the set-theoretical operations. We assume here to be given a universe U and, unless otherwise stated, all categories C will be assumed to be U-categories, i.e. categories such that Ob(C) ⊂ U and Hom C (c, d) ∈ U for every pair of objects. 2 More precisely, Set denotes the U-category of sets belonging to the fixed universe U. 3 More precisely, let V be another fixed universe with U ∈ V. Then Cat denotes the V-category whose objects are U-categories. From now on we will leave to the reader who feels that need the task of making the universes explicit. 4 Roughly speaking, a 2-category (refer to [33, §9] for details) C is a “category enriched in Cat”, i.e. a category whose morphism sets are the object sets of categories Hom C (c, d), such that composition is a

Stacks of twisted modules and integral transforms

467

Definition 1.1. A prestack P on X consists of the data (a) for every open subset U ⊂ X, a category P(U ), u

→ U of open subsets, a functor P(u) : P(U ) − → P(V ), (b) for every inclusion V − called restriction functor, and v

u

→V − → U of open subsets, invertible transformations (c) for every inclusion W − P(v, u) : P(v)  P(u) ⇒ P(u  v) of functors from P(U ) to P(W ), subject to the conditions (i) if U is an open subset, then P(idU ) = idP(U ) and P(idU , idU ) = ididP(U ) ; w

v

u

→ W − → V − → U are inclusions of open subsets, then the following (ii) if Y − diagram of transformations of functors from P(U ) to P(Y ) commutes P(w)  P(v)  P(u)

P(w,v)idP(u)

idP(w) P(v,u)

 P(w)  P(u  v)

+3 P(v  w)  P(u) P(vw,u)

P(w,uv)

 +3 P(u  v  w).

In particular, P(u, idU ) = P(idV , u) = idP(u) . u

For F ∈ P(U ) and V − → U an open inclusion, one usually writes F |V instead of P(u)(F ). One denotes by P|U the natural restriction of P to U given by V → P(V ) for V ⊂ U . Definition 1.2. Let P and Q be prestacks on X. A functor of prestacks ϕ : P − →Q consists of the data (a) for any open subset U ⊂ X, a functor ϕ(U ) : P(U ) − → Q(U ), u

→ U , an invertible transformation ϕ(u) : ϕ(V )  (b) for any open inclusion V − P(u) ⇒ Q(u)  ϕ(U ) of functors from P(U ) to Q(V ), subject to the condition functor. Morphisms in the category Hom C (c, d) are called 2-cells. The basic example is the 2-category Cat which has categories as objects, functors as morphisms, and transformations as 2-cells. There is a natural notion of pseudo-functor between 2-categories, preserving associativity for the composition functor only up to an invertible 2-cell. Then a prestack (see [SGA1, exposé VI]) is a pseudo-functor Xop − → Cat, where Xop is the 2-category obtained by trivially enriching Xop with identity 2-cells. Functors of prestacks and their transformations are transformations and modifications of pseudo-functors, respectively. Note that Corollary 9.2 of [33] asserts that any prestack is equivalent, in the 2-category of pseudo-functors, to a presheaf of categories. However, this equivalence is not of practical use for our purposes.

468

Andrea D’Agnolo and Pietro Polesello v

u

(i) if W − →V − → U are inclusions of open subsets, then the following diagram of transformations of functors from P(U ) to Q(W ) commutes ϕ(W )  P(v)  P(u)

ϕ(v)idP(u)

+3 Q(v)  ϕ(V )  P(u)

idQ(v) ϕ(u)

+3 Q(v)  Q(u)  ϕ(U )

idϕ(W ) P(v,u)

Q(v,u)idϕ(U )

 ϕ(W )  P(u  v)

 +3 Q(u  v)  ϕ(U ).

ϕ(uv)

In particular, ϕ(idU ) = idϕ(U ) . Definition 1.3. Let ϕ, ψ : P − → Q be functors of prestacks. A transformation α : ϕ ⇒ ψ of functors of prestacks consists of the data (a) for any open subset U ⊂ X, a transformation α(U ) : ϕ(U ) ⇒ ψ(U ) of functors from P(U ) to Q(U ), such that u

(i) if V − → U is an inclusion of open subsets, then the following diagram of transformations of functors from P(U ) to Q(V ) commutes ϕ(V )  P(u)

α(V )idP(u)

+3 ψ(V )  P(u) ψ(u)

ϕ(u)

  idQ(u) α(U ) +3 Q(u)  ψ(U ). Q(u)  ϕ(U ) An example of prestack is the prestack PShX of presheaves of sets on X. It associates to an open subset U ⊂ X the category Hom (Uop , Set) of presheaves of → sets on U , and to an open inclusion V ⊂ U the restriction functor PShX (U ) − PShX (V ), F  → F |V . For open inclusions W ⊂ V ⊂ U one has F |V |W = F |W , so that PShX is in fact a presheaf of categories. If P and Q are prestacks, one gets another prestack Hom (P, Q) by associating to an open subset U ⊂ X the category Hom (P|U , Q|U ) of functors of prestacks from P|U to Q|U , with transformations of functors of prestacks as morphisms, and with the natural restriction functors. Note that Hom (P, Q) is actually a presheaf of categories. (Pre)stacks which are not (pre)sheaves of categories will appear in Section 2.

1.2 Stacks The analogy between presheaves and prestacks goes on for sheaves and stacks. Let X be a topological space. Given a family of subsets {Ui }i∈I of X, let us use the notations Uij = Ui ∩ Uj ,

Uij k = Ui ∩ Uj ∩ Uk ,

etc.

469

Stacks of twisted modules and integral transforms

Recall that a presheaf of sets F on X is called a sheaf if for any open subset U ⊂ X, and any open covering {Ui }i∈I of U , the natural sequence given by the restriction maps



/ // i,j ∈I F (Uij ) F (U ) i∈I F (Ui ) is exact, i.e. if for any family of sections si ∈ F (Ui ) satisfying si |Uij = sj |Uij there is a unique section s ∈ F (U ) such that s|Ui = si . Similarly to the definition of sheaf, a prestack S on X is called a stack if for any open subset U ⊂ X, and any open covering {Ui }i∈I of U , the natural sequence given by the restriction functors



//

/ // i,j ∈I S(Uij ) S(U ) / i,j,k∈I S(Uij k ) i∈I S(Ui ) is exact in the sense of [SGA1, exposé XIII], i.e. if the category S(U ) is equivalent to the category whose objects are families of objects Fi of S(Ui ) and of isomorphisms ∼ → Fi |Uij which are compatible in the triple intersections, in a natural θij : Fj |Uij − sense. More explicitly, recall that a descent datum for S on U is a triplet F = ({Ui }i∈I , {Fi }i∈I , {θij }i,j ∈I ),

(1.1) ∼

→ Fi |Uij are where {Ui }i∈I is an open covering of U , Fi ∈ S(Ui ), and θij : Fj |Uij − isomorphisms such that the following diagram of isomorphisms commutes Fj |Uij k o O

S

θij |Uij k

Fj |Uij |Uij k

/ Fi |Uij |Uij k

S

S

/ Fi |Uij k O S

Fj |Uj k |Uij k fMMM MMM MM θj k |Uij k MMM Fk |Uj k |Uij k

S

/ Fk |Uij k o

S

Fi |Uik |Uij k q8 q q q q qqq qqq θik |Uij k Fk |Uik |Uij k .

The descent datum F is called effective if there exist F ∈ S(U ) and isomorphisms ∼ → Fi for each i, such that the following diagram of isomorphisms commutes θ i : F | Ui − F |Uj |Uij

S

/ F |Uij o

θj |Uij

 Fj |Uij

S

F |Ui |Uij θi |Uij

θij

 / Fi |Uij .

To S a prestack on X is attached a bifunctor of prestacks → PShX , Hom S : Sop × S −

470

Andrea D’Agnolo and Pietro Polesello

associating to F , G ∈ S(U ) the presheaf of sets on U ⊂ X given by Hom S|U (F , G) : V  → Hom S(V ) (F |V , G|V ). Definition 1.4. (i) A prestack S on X is called separated if for any open subset U , and any F , G ∈ S(U ), the presheaf Hom S|U (F , G) is a sheaf. (ii) A stack is a separated prestack such that any descent datum is effective. (iii) Functors and transformations of stacks are functors and transformations of the underlying prestacks, respectively. For example, the prestack ShX of sheaves of sets, associating to U ⊂ X the category of sheaves of sets on U , is actually a stack. As another example, if S and T are stacks, then the prestack Hom (S, T) is a stack. (Note that both ShX and Hom (S, T) are in fact sheaves of categories.) One says that a functor of stacks ϕ : S − → T is an equivalence if there exists a functor ψ : T − → S, called a quasi-inverse to ϕ, and invertible transformations ϕ  ψ ⇒ idT and ψ  ϕ ⇒ idS . One says that ϕ admits a right adjoint if there exists a functor of stacks ψ : T − → S, called a right adjoint to ϕ, and an invertible transformation Hom T (ϕ(·), ·) ⇒ Hom S (·, ψ(·)). Similarly for left adjoint. Finally, → ShX is representable if there exists F ∈ S(X), if T = ShX , one says that ϕ : S − called a representative of ϕ, and an invertible transformation ϕ ⇒ Hom S (F , ·). Lemma 1.5. For a functor of stacks to be an equivalence (resp. to admit a right or left adjoint, resp. to be representable) is a local property. Proof. Right or left adjoints and representatives are unique up to unique isomorphisms, and hence glue together globally. As for equivalences, assume that ϕ is locally an equivalence. Then we have to show that for each open subset U ⊂ X the functors ϕ(U ) are fully faithful and essentially surjective. Being fully faithful is a local property already for separated  prestacks. Assume that ϕ(Ui ) are essentially surjective for a covering U = i Ui . Let G ∈ Ob(T(U )), and choose Fi ∈ Ob(S(Ui )) with ∼ → G|Ui . Since ϕ is fully faithful, the restriction morphisms isomorphisms ϕ(Ui )(Fi ) − of G|Ui give descent data for Fi . Finally, since S is a stack, one gets F ∈ Ob(S(U )) ∼ → G.   with ϕ(U )(F ) −

1.3 Constructions of stacks The forgetful functor, associating to a sheaf of sets its underlying presheaf, has a left adjoint, associating a sheaf P + to a presheaf P . There is a similar construction

471

Stacks of twisted modules and integral transforms

associating a stack P+ to a prestack P. This is done in two steps as follows. Consider first the separated prestack Pa , with the same objects as P and morphisms Hom Pa (U ) (F , G) = Γ (U ; Hom+P| (F , G)). U

Then let P+ (U ) be the category whose objects are descent data for Pa on U , and → ({Vj }, {Gj }, {jj }) consist of morphisms whose morphisms ({Ui }, {Fi }, {θii }) − ϕj i : Fi |Ui ∩Vj − → Gj |Ui ∩Vj such that j j  ϕj i = ϕj i  θi i on Uii ∩ Vjj . Since sheaves of sets form a stack, descent data for sheaves are effective. Similarly, it is possible to patch stacks together. More precisely, a descent datum for stacks on X is a quadruplet S = ({Ui }i∈I , {Si }i∈I , {ϕij }i,j ∈I , {αij k }i,j,k∈I ),

(1.2) ≈

→ Si |Uij where {Ui }i∈I is an open covering of X, Si are stacks on Ui , ϕij : Sj |Uij − are equivalences of stacks, and αij k : ϕij  ϕj k ⇒ ϕik are invertible transformations of functors from Sk |Uij k to Si |Uij k , such that for any i, j, k, l ∈ I , the following diagram of transformations of functors from Sl |Uij kl to Si |Uij kl commutes ϕij  ϕj k  ϕkl

αij k idϕkl

+3 ϕik  ϕkl

idϕij αj kl

 ϕij  ϕj l

αij l



(1.3)

αikl

+3 ϕil .

Proposition 1.6. Descent data for stacks are effective, meaning that given a descent datum for stacks S as in (1.2), there exist a stack S on X, equivalences of stacks ≈ → Si , and invertible transformations of functors αij : ϕij  ϕj |Uij ⇒ ϕi |Uij ϕi : S|Ui − such that αij |Uij k  αj k |Uij k = αik |Uij k  αij k . The stack S is unique up to equivalence. Sketch of proof. For U ⊂ X open, denote by S(U ) the category whose objects are triplets F = ({Vi }i∈I , {Fi }i∈I , {ξij }i,j ∈I ), → Fi |Vij are isomorwhere Vi = U ∩ Ui , Fi ∈ Ob(Si (Vi )), and ξij : ϕij (Fj |Vij ) − phisms such that for i, j, k ∈ I the following diagram commutes αij k

/ ϕik (Fk |Vij k )

ξij |Vij k



ϕij (ϕj k (Fk |Vij k )) 

ϕij (ξj k |Vij k )

ϕij (Fj |Vij k )

ξik |Vij k

/ Fi |Vij k .

472

Andrea D’Agnolo and Pietro Polesello

For G = ({Vi }, {Gi }, {ηij }), a morphisms γ : F − → G in S(U ) consists of morphisms → Gi in Si (Vi ) such that the following diagram commutes γi : Fi − ξij

ϕij (Fj |Vij )

ϕij (γj |Vij )

 ϕij (Gj |Vij )

ηij

/ Fi |Vij 

γi |Vij

/ Gi |Vij .

Then one checks that the prestack S : U − → S(U ) is a stack satisfying the requirements in the statement.  

1.4 Operations on stacks Let us recall the stack-theoretical analogue of internal and external operations for sheaves. Given two stacks S and S on X, denote by S × S the prestack S × S (U ) = S(U ) × S (U ). This is actually a stack. We already noticed that the prestack Hom (S, S ) is a stack. If S

is another stack, there is a natural equivalence ≈

→ Hom (S, Hom (S , S

)). Hom (S × S , S

) − Let f : Y − → X be a continuous map of topological spaces. If T is a stack on Y , denote by f∗ T the prestack f∗ T(U ) = T(f −1 U ), which is actually a stack. If S is a stack on X, denote by f −1 S = (f ∼ S)+ the stack associated with the prestack ∼ f ∼ S defined as follows. For V ⊂ Y , f S(V ) is the category whose objects are the disjoint union U : f −1 U ⊃V Ob(S(U )), and whose morphisms are given by

Hom f ∼ S(V ) (F U , F U ) =

lim − →

U

: U

⊂U ∩U , f −1 U

⊃V

Hom S(U

) (F U |U

, F U |U

),

for F U ∈ Ob(S(U )) and F U ∈ Ob(S(U )). There is a natural equivalence ≈

→ Hom (S, f∗ T). f∗ Hom (f −1 S, T) −

1.5 Linear stacks As a matter of conventions, in this paper rings are unitary, and ring homomorphisms preserve the unit. If R is a commutative ring, we call R-algebra a not necessarily commutative ring A endowed with a ring homomorphism R − → A whose image is in the center of A. Let R be a commutative ring. An R-linear category, that we will call R-category for short, is a category C whose morphism sets are endowed with a structure of R-module such that composition is R-bilinear. An R-functor is a functor which is R-linear at the

Stacks of twisted modules and integral transforms

473

level of morphisms. Transformations of R-functors are simply transformations of the underlying functors. Note that if D is another R-category, the category Hom R (C, D) of R-functors and transformations is again an R-category, the R-module structure on the sets of transformations being defined object-wise. For each c ∈ Ob(C) the set of endomorphisms End C (c) has a natural structure of R-algebra, with product given by composition. In particular, note that R-algebras are identified with R-categories with a single object. Let us denote for short by End (idC ) the R-algebra End End (C) (idC ). It is a commutative5 R-algebra, called the center of R C. Note that C is an R-category if and only if C is a Z-category (also called preadditive category) endowed with a ring homomorphism R − → End (idC ). Definition 1.7. (a) An R-linear stack, that we will call R-stack for short, is a stack S such that S(U ) is an R-category for every open subset U , and whose restrictions are R-functors. An R-functor of R-stacks is a functor which is linear at the level of morphisms. No additional requirements are imposed on transformations of R-functors. (b) Let R be a sheaf of commutative rings on X. An R-linear stack, that we will call R-stack for short, is a Z-stack S whose center E nd (idS ) is a sheaf of commutative R-algebras6 . There is a natural notion of R-functor7 , and transformations of R-functors are just transformations of the underlying functors. One says that an R-functor ϕ : S − → T is an equivalence (resp. admits a right or a left adjoint) if it is so forgetting the R-linear structure. Note that a quasi-inverse to ϕ (resp. its right or left adjoint) is necessarily an R-functor itself. One says that ϕ : S − → Mod(R) is representable if there is an invertible transformation ϕ ⇒ Hom S (F , ·) for some F ∈ S(X). 5 Let α, β : id ⇒ id be transformations, and c ∈ Ob(C). By definition of transformation, applying α C C to the morphism β(c) we get a commutative diagram

c β(c)



c

α(c) α(c)

/c 

β(c)

/ c,

→ End C (c), so that αβ = βα. Note that for each c ∈ Ob(C) there is a natural morphism End (idC ) − α  → α(c). 6 By definition, this means that there is a morphism of sheaves of rings µ : R − → E nd (idS ). Note that the data of µ is equivalent to the requirement that for every open subset U ⊂ X, and any F , G ∈ S(U ) the sheaf Hom S|U (F , G) has a structure of R|U -module compatible with restrictions, and such that composition is R-bilinear. 7 Given µ : R − → E nd (idS ) and µ : R − → E nd (idS ), an R-functor ϕ : S − → S is a functor of Z-stacks such that ϕ(µ(r)(F )) = µ (r)(ϕ(F )), as endomorphisms of ϕ(F ), for any U ⊂ X, r ∈ R(U ), and F ∈ Ob(S(U )).

474

Andrea D’Agnolo and Pietro Polesello

1.6 Operations on linear stacks Let S and S be R-stacks. The stack Hom R (S, S ) of R-functors and transformations is an R-stack. The product S ⊗R S is the stack associated with the prestack ∼

∼

S ⊗R S defined as follows. At the level of objects, Ob(S ⊗R S ) = Ob(S)×Ob(S ). At the level of morphisms, Hom

∼

S⊗ S (U ) R

((F1 , F1 ), (F2 , F2 )) = Hom S(U ) (F1 , F2 ) ⊗R Hom S (U ) (F1 , F2 ).

If S

is another R-stack, there is a natural R-equivalence ≈

→ Hom R (S, Hom R (S , S

)). Hom R (S ⊗R S , S

) − Let f : Y − → X be a continuous map of topological spaces, S an R-stack on X, and T an f −1 R-stack on Y . Then f∗ T is an R-stack, f −1 S is an f −1 R-stack, and there is a natural R-equivalence ≈

→ Hom R (S, f∗ T). f∗ Hom f −1 R (f −1 S, T) −

(1.4)

1.7 Stacks of twisted modules Let X be a topological space, R a sheaf of commutative rings on X, and A a sheaf of not necessarily commutative R-algebras. Let Mod(A) be the category of A-modules and A-linear morphisms. Unless otherwise stated, by A-module we mean here left A-module. The prestack Mod(A) of A-modules on X is defined by U  → Mod(A|U ), with natural restriction functors. It is clearly an R-stack. Definition 1.8. (a) A stack of R-twisted modules is an R-stack which is locally R-equivalent to stacks of modules over R-algebras. More precisely, an R-stack M is a stack of R-twisted modules if there exist an open covering {Ui }i∈I of X, R|Ui -algebras → Mod(Ai ). Ai on Ui , and R|Ui -equivalences ϕi : M|Ui − (b) A stack of R-twisted A-modules is an R-stack which is locally R-equivalent to Mod(A). (c) A stack of twisted R-modules is a stack of R-twisted R-modules. If M is a stack of R-twisted modules (resp. a stack of R-twisted A-modules, resp. a stack of twisted R-modules), objects of M(X) are called R-twisted modules (resp. R-twisted A-modules, resp. twisted R-modules). Recall that a stack M is called additive if the categories M(U ) and the restriction functors are additive. A stack M is called abelian if the categories M(U ) are abelian, and the restriction functors are exact. Since stacks of modules over R-algebras are abelian, stacks of R-twisted modules are also abelian.

Stacks of twisted modules and integral transforms

475

Remark 1.9. The stacks constructed in [20, 28, 30] provide examples of stacks of twisted modules which are of an intermediate nature between (a) and (b) of Definition 1.8. With notations as in (a), denote by ψi a quasi-inverse to ϕi . These are stacks of R-twisted modules for which the equivalences ϕj  ψi |Uij are induced by ∼

→ Aj |Uij . This is related to non-abelian isomorphisms of R|Uij -algebras Ai |Uij − cohomology as in [14], and we will discuss these matters in [9]. Recall that A-modules are sheaves of R-modules F endowed with a morphism of sheaves of R-algebras m : A − → E nd R (F ). Definition 1.10. If A is an R-algebra and S an R-stack, one considers the Rstack ModR (A; S) whose objects on an open subset U ⊂ X are pairs of an object F ∈ S(U ) and a morphism of R|U -algebras m : A|U − → E nd S|U (F ), and whose morphisms are those morphisms in S(U ) commuting with m. We denote by ModR (A; S) the category ModR (A; S)(X). Let A and B be R-algebras. Recall that an A ⊗R B-module is the same as a B-module M endowed with an R-algebra morphism A − → E nd B (M). Hence, there is an R-equivalence ModR (A; Mod(B)) ≈ Mod(A ⊗R B).

(1.5)

In particular, if M is a stack of R-twisted modules (resp. of twisted R-modules), then ModR (A; M) is a stack of R-twisted modules (resp. of R-twisted A-modules).

2 Operations Using Morita theory, we develop the formalism of operations for stacks of twisted modules. We then obtain Grothendieck’s six operations for derived categories of twisted modules over locally compact Hausdorff topological spaces.

2.1 Morita theory I. Functors admitting an adjoint Morita theory describes, in terms of bimodules, functors between categories of modules which admit an adjoint (references are made to [1, 12]). We are interested in the local analogue of this result, dealing with stacks of modules over sheaves rings. Our reference was [25], where only the case of equivalences is discussed. We thus adapt here their arguments in order to deal with functors admitting an adjoint. Let R be a sheaf of commutative rings on a topological space X, and let A be a sheaf of not necessarily commutative R-algebras. Denote by Aop the opposite algebra to A, given by Aop = {a op : a ∈ A} with product a op bop = (ba)op . Note that left (resp. right) Aop -modules are but right (resp. left) A-modules.

476

Andrea D’Agnolo and Pietro Polesello

For S and S two R-stacks, denote by HomrR (S, S ) the full R-substack of Hom R (S, S ) of functors that admit a right adjoint. This is equivalent to the opposite of the stack of R-functors from S to S that admit a left adjoint. Proposition 2.1. Let A and B be R-algebras. The functor → HomrR (Mod(B), Mod(A))  : Mod(A ⊗R B op ) − given by L → L ⊗B (·) is an R-equivalence. It follows that R-functors Mod(A) − → Mod(B) which admit a left adjoint are of the form Hom A (L, ·), for an A ⊗R B op -module L. Proof. (We follow here arguments similar to those in the proof of Morita theorem given in [25].) One checks that  is fully faithful. Let us show that it is essentially surjective. Let ϕ : Mod(B) − → Mod(A) be an R-functor admitting a right adjoint. The A-module L = ϕ(B) inherits a compatible B op -module structure by that of B itself, and we set ϕ (·) = L ⊗B (·). A transformation α : ϕ ⇒ ϕ is defined as follows. For U ⊂ X and N ∈ Mod(B|U ), the morphism → ϕ(N ) α(N ) : ϕ(B)|U ⊗B|U N − → N denotes the map b  → bn. We have is given by l ⊗ n  → ϕ( n)(l), where n : B|U − to prove that α(N ) is an isomorphism. The B|U -module N admits a presentation   B − → B − → N − → 0, where one sets (·)U = u! u−1 for u : U − → X the U U j i j i 

B . Since ϕ and ϕ open inclusion. We may then assume that N = i i U    admit a 

right adjoint, one has ϕ( i BUi )  i ϕ(B)Ui , and ϕ ( i BUi )  i ϕ (B)Ui by ∼ → Lemma 2.2. Hence we are reduced to prove the isomorphism ϕ(B|U ) ⊗B|U BU − ϕ(B|U ), which is obvious.   Lemma 2.2. Let A and B be R-algebras, and let ϕ : Mod(B) − → Mod(A) be an R-functor admitting a right adjoint. Then for any family  of open subsets {Ui }i∈I of U ⊂ X, and any N ∈ Mod(B|U ), one has ϕ( i NUi )  i ϕ(N )Ui . Proof. The proof is straightforward. We leave it to the reader to check that a functor admitting a right adjoint commutes with inductive limits, and in particular with direct → U . Let sums. Let us check that ϕ(NV )  ϕ(N )V for an open inclusion v : V − ψ be a right adjoint to ϕ. Note that the proper direct image v! is left adjoint to the

Stacks of twisted modules and integral transforms

477

restriction functor v −1 (·) = (·)|V . For every M ∈ Mod(A|U ) one has Hom A|U (ϕ(NV ), M) = Hom A|U (ϕ(v! (N |V )), M)  Hom B|V (N |V , ψ(M)|V )  Hom B|V (N |V , ψ(M|V ))  Hom A|U (v! (ϕ(N |V )), M)  Hom A|U (v! (ϕ(N )|V ), M) = Hom A|U (ϕ(N )V , M), where the second and fourth isomorphisms follow from the fact that ψ and ϕ, respectively, are functors of stacks.  

2.2 Internal product of stacks of twisted modules We are now ready to define duality and internal product for stacks of twisted modules. Let R be a sheaf of commutative rings on a topological space X. Recall that for S and S two R-stacks, we denote by HomrR (S, S ) the stack of R-functors that admit a right adjoint. Definition 2.3. Let S and S be R-stacks on X. Set S−1 = HomrR (S, Mod(R)),

S R S = HomrR (S−1 , S ).

Remark 2.4. Note that S−1 does not depend on the base ring R, up to equivalence8 . As a consequence of Proposition 2.1 and equivalence (1.5), we have Proposition 2.5. If A and A are R-algebras, there are R-equivalences Mod(A)−1 ≈ Mod(Aop ), Mod(A) R Mod(A ) ≈ Mod(A ⊗R A ). ≈ ModR (A; Mod(A )).

(2.1) (2.2)

In particular, if M and M are stacks of R-twisted modules on X, then M−1 and M R M are stacks of R-twisted modules on X. Let us list some properties of these operations. 8 Using arguments as in [9], denote by R+ the full substack of line bundles in Mod(R). Then Mod(R) ≈ Hom Z (R+ , Mod(ZX )), and S−1 ≈ HomrR (S, Hom Z (R+ , Mod(ZX ))) ≈ HomrR (R+ ⊗Z S, Mod(ZX )) ≈ HomrZ (S, Mod(ZX )).

478

Andrea D’Agnolo and Pietro Polesello

Lemma 2.6. Let A be an R-algebra, and S and S be R-stacks. Then there are natural R-functors Mod(A) R S − → ModR (A; S),

(2.3)

−1 −1

S− → (S HomrR (S, S )

− →

) = S R Mod(R), r

−1 Hom R (S , S−1 ) = S R

(2.4) −1

S

.

(2.5)

Proof. In the identification Mod(A) R S ≈ HomrR (Mod(Aop ), S), the functor φ

→ (2.3) is given by φ  → (F , m), where F = φ(Aop ), and m : A  E nd Aop (Aop ) − op E nd S (φ(A )). The functor (2.4) is given by F  → (φ  → φ(F )), using the identification (S−1 )−1 = HomrR (HomrR (S, Mod(R)), Mod(R)). Finally, the functor (2.5) is given by ϕ → (ψ  → ψ  ϕ), using the identification  HomrR (S −1 , S−1 ) = HomrR (HomrR (S , Mod(R)), HomrR (S, Mod(R))).  We need the following lemma from [25]. Lemma 2.7. For M a stack of R-twisted modules, there is a natural R-functor → M. ⊗R : Mod(R) × M − Proof. For M ∈ Mod(R) and F ∈ M(X), the functor Hom R (M, Hom M (F , ·)) : M − → Mod(R) is locally (and hence globally) representable, and we denote by M ⊗R F a representative.   Proposition 2.8. Let M, M , and M

be stacks of R-twisted modules. Then there is a natural R-equivalence ≈

HomrR (M R M , M

) − → HomrR (M, HomrR (M , M

)). Proof. The above R-functor is given by ϕ  → (F  → (F  → ϕ(ψF ,F ))), where ψF ,F ∈ M R M = HomrR (HomrR (M, Mod(R)), M ) is defined by ψF ,F (η) = η(F ) ⊗R F . Here we used the R-functor ⊗R described in Lemma 2.7. We are then left to prove that this functor is a local equivalence. We may then assume that M ≈ Mod(A), M ≈ Mod(A ), and M

≈ Mod(A

) for some R-algebras A, A ,   and A

. In this case both terms are equivalent to Mod(Aop ⊗R A op ⊗R A

).

479

Stacks of twisted modules and integral transforms

Proposition 2.9. Let A be an R-algebra, and let M, M , and M

be stacks of Rtwisted modules. Then there are natural R-equivalences Mod(A) R M ≈ ModR (A; M), −1 −1

M ≈ (M

)

M R M ≈ M R M,

−1

(M R M )

−1

= M R Mod(R),

(2.7) (2.8)

−1

≈M R M ,

(M R M ) R M ≈ M R (M R M ).

(2.6)



(2.9) (2.10)

Proof. Equivalences (2.6) and (2.7) follow by noticing that the functors (2.3) and (2.4) are local equivalences for S = M. The equivalence (2.8) follows by noticing that the functor (2.5) is locally an equivalence for S = M−1 and S = M . The equivalence (2.9) follows from the chain of equivalences HomrR (M R M , Mod(R)) ≈ HomrR (M, HomrR (M , Mod(R))) ≈ HomrR ((M−1 )−1 , M −1 ).

The equivalence (2.10) follows from the chain of equivalences HomrR ((M R M )−1 , M

) ≈ HomrR (M−1 R M −1 , M

)

≈ HomrR (M−1 , HomrR (M −1 , M

)).

 

Let us describe a couple of other functors. There is a natural R-functor Mod(R) − → M R M−1 ,

(2.11)

given by F  → F ⊗R (·), in the identification M R M−1 ≈ HomrR (M, M). Locally, M ≈ Mod(A) for some R-algebra A, and the above functor coincides with Mod(R) − → Mod(A ⊗R Aop ), F  → F ⊗R A. This has a right adjoint Mod(A ⊗R op A ) − → Mod(R), M  → Z(M), where Z(M) = Hom A⊗ Aop (A, M) = {m ∈ R M : am = ma, ∀a ∈ A}. Hence there is a right adjoint to (2.11) M R M−1 − → Mod(R).

(2.12)

Note also that the forgetful functor →M ModR (A; M) − has (locally, and hence globally) a right adjoint → ModR (A; M). A ⊗R (·) : M −

2.3 Morita theory II. Relative case In order to describe the pull-back functor for stacks of twisted modules, we need the following relative versions of the results in Section 2.1.

480

Andrea D’Agnolo and Pietro Polesello

Let f : Y − → X be a continuous map of topological spaces, R a sheaf of commutative rings on X, S an R-stack, and T an f −1 R-stack. Denote by Hom f −1∗ R (f −1 S, T) r-f

the full f −1 R-substack of Hom f −1 R (f −1 S, T) of functors ψ whose image by (1.4) belongs to HomrR (S, f∗ T). Proposition 2.10. Let f : Y − → X be a continuous map of topological spaces, B an R-algebra on X, and C an f −1 R-algebra on Y . The functor  : Mod(C ⊗f −1 R f −1 B op ) − → Hom f −1∗ R (f −1 Mod(B), Mod(C)) r-f

given by F  → F ⊗f −1 B (·) is an f −1 R-equivalence. Proof. The proof of this proposition is similar to that of Proposition 2.1, and we only show the essential surjectivity of . Let ψ : f −1 Mod(B) − → Mod(C) be an f −1 R-functor such that f∗ ψ admits a right adjoint. Set F = ψ(f −1 B) and ψ (·) = F ⊗f −1 B (·). For V ⊂ Y and N ∈ f −1 Mod(B)(V ), we have to check that the morphism → ψ(N ), β(N ) : ψ(f −1 B|V ) ⊗f −1 B|V N − defined as the morphism α in Proposition 2.1, is an isomorphism.  −1 By the definition of Mk − → N , where Mk pull-back for stacks, N locally admits a presentation kf  are objects of Mod(B) and k means that the sum is finite. Thus any y ∈ Y has an open neighborhood W ⊂ V such that there is a presentation

 f −1 B U j k |W − → f −1 BUik |W − → N |W − → 0. j

k

i

k

Since f∗ ψ admits a right adjoint, one has
 f −1 BUik |W ) = ψ(f −1 B|W )f −1 (Uik )∩W . ψ( k

i

k

ψ ,

i

ψ

A similar formula holds for since also f∗ admits a right adjoint. Hence we are ∼ → ψ(f −1 B|W ), reduced to prove the isomorphism ψ(f −1 B|W ) ⊗f −1 B|W f −1 B|W − which is obvious.  

2.4 Pull-back of stacks of twisted modules We can now define the pull back of stacks of twisted modules. Let f : Y − → X be a continuous map of topological spaces, R a sheaf of commutative rings on X, S an R-stack, and T an f −1 R-stack. Recall that we denote by r-f Hom f −1∗ R (f −1 S, T) the f −1 R-stack of functors ψ whose image by (1.4) admits a right adjoint.

Stacks of twisted modules and integral transforms

481

Definition 2.11. With the above notations, set f  S = Hom f −1∗ R (f −1 (S−1 ), Mod(f −1 R)). r-f

Remark 2.12. As in Remark 2.4, note that f  S does not depend on the base ring R, up to equivalence. As a consequence of Proposition 2.10, we have Proposition 2.13. Let f : Y − → X be a continuous map of topological spaces, and A an R-algebra on X. Then, there is an f −1 R-equivalence f  Mod(A) ≈ Mod(f −1 A).

(2.13)

In particular, if M is a stack of R-twisted modules, then f  M is a stack of f −1 Rtwisted modules. Let us list some properties of this operation. Proposition 2.14. If S is an R-stack, there is a natural R-functor → f∗ f  S. f −1 : S − Proof. The usual sheaf-theoretical pull-back operation gives an R-functor → f∗ f  Mod(R) ≈ f∗ Mod(f −1 R). f −1 : Mod(R) − The functor in the statement is then obtained as the composition (2.4)

S −−→ S R Mod(R) idS  f −1

−−−−−−−→ S R f∗ Mod(f −1 R) R

≈ HomrR (S−1 , f∗ Mod(f −1 R)) ≈ f∗ Hom f −1∗ R (f −1 (S−1 ), Mod(f −1 R)) r-f

≈ f∗ f  S.

 

Proposition 2.15. Let M be a stack of R-twisted modules, and N a stack of f −1 Rtwisted modules. Then there is a natural R-equivalence ≈

→ HomrR (M, f∗ N). f∗ Homrf −1 R (f  M, N) − Proof. The functor f −1 : M − → f∗ f  M of Proposition 2.14 is locally the usual sheaftheoretical pull-back, which has a right adjoint in the sheaf-theoretical push-forward. Moreover, it induces by (1.4) an f −1 R-functor → f  M. f −1 M −

482

Andrea D’Agnolo and Pietro Polesello

Hence we get a functor → Hom f −1∗ R (f −1 M, N). Homrf −1 R (f  M, N) − r-f

This is a local (and hence global) equivalence. We thus have the chain of equivalences f∗ Homrf −1 R (f  M, N) ≈ f∗ Hom f −1∗ R (f −1 M, N) r-f

≈ HomrR (M, f∗ N).

 

Proposition 2.16. Let M and M be stacks of R-twisted modules. Then there are natural f −1 R-equivalences f  (M−1 ) ≈ (f  M)−1 , 



(2.14) 

f (M R M ) ≈ f M f −1 R f M .

(2.15)

Proof. The equivalence (2.14) follows from the chain of equivalences Hom f −1∗ R (f −1 ((M−1 )−1 ), Mod(f −1 R)) ≈ Hom f −1∗ R (f −1 M, Mod(f −1 R)) r-f

r-f

≈ Homrf −1 R (f  M, Mod(f −1 R)). To prove (2.15), note that, by functoriality of f  , to any R-stacks S and S is associated an R-functor → f∗ Homrf −1 R (f  S, f  S ). f  : HomrR (S, S ) − For S = M−1 and S = M this is locally the sheaf-theoretical pull-back functor → f∗ Mod(f −1 (A ⊗R A )), f −1 : Mod(A ⊗R A ) − which has a right adjoint. Hence f  has a right adjoint, i.e.
 f  ∈ HomrR HomrR (M−1 , M ), f∗ Homrf −1 R (f  (M−1 ), f  M ) . By Proposition 2.15 we get a functor → Homrf −1 R (f  (M−1 ), f  M ). f  HomrR (M−1 , M ) − This is locally, and hence globally, an equivalence.

 

2.5 Twisted sheaf-theoretical operations Let us now show how the usual operations of sheaf theory extend to the twisted case. For the classical non-twisted case, that we do not recall here, we refer e.g. to [22].

Stacks of twisted modules and integral transforms

483

Proposition 2.17. Let f : Y − → X be a continuous map of topological spaces, and M and M be stacks of R-twisted modules. Then there exist R-functors → M R M , ⊗R : M × M −

Hom R : (M−1 )op × M − → M R M , f −1 : M − → f∗ f  M, f ∗ : f∗ f  M − → M. If moreover X and Y are locally compact Hausdorff topological spaces, there exists an R-functor → M. f! : f∗ f  M − If U ⊂ X is an open subset where M|U ≈ Mod(A) and M |U ≈ Mod(A ) for some R|U -algebras A and A , then the restrictions to U of the above functors coincide with the usual sheaf operations. Proof. The functor → M R M = HomrR (HomrR (M, Mod(R)), M ) ⊗R : M × M − is defined by (F , F ) → (φ  → φ(F ) ⊗R F ), using Lemma 2.7. For F an object of M−1 there is a natural functor → M M R M −

(2.16)

given by φ  → φ(F ) in the identification M R M = HomrR (M−1 , M ). Locally → Mod(A ), M  → F ⊗A M, for F ∈ this corresponds to the functor Mod(A⊗R A ) − Mod(Aop ). If N is an A -module, there is a functorial isomorphism Hom A (F ⊗A M, N )  Hom A⊗ A (M, Hom R (F , N )). Hence (2.16) admits a right adjoint, R

that we denote by Hom R (F , ·). This construction is functorial in F , and hence we get the bifunctor Hom R (·, ·). The functor f −1 was constructed in Proposition 2.14. The functor f∗ is obtained by noticing that if M is a stack of R-twisted modules, then f −1 is locally the usual sheaf-theoretical pull-back, which admits a right adjoint. Assume that f : Y − → X is a continuous map of locally compact Hausdorff topological spaces. Recall that for an f −1 A-module G on Y one denotes by f! G the subsheaf of f∗ G of sections s ∈ f∗ G(U ) such that f |supp(s) is proper. Such a condition is local on X, and hence for a stack of R-twisted modules M there is an R-functor → M locally given by the usual proper direct image functor for sheaves f ! : f∗ f  M − just recalled.  

484

Andrea D’Agnolo and Pietro Polesello

2.6 Derived twisted operations Let us now deal with the twisted version of Grothendieck’s formalism of six operations for sheaves over locally compact Hausdorff topological spaces. We do not recall here such formalism for the classical non-twisted case, referring instead e.g. to [22]. Let M be a stack of R-twisted modules, and denote by D(M) the derived category of the abelian category M(X). Let Db (M) (resp. D+ (M), resp. D− (M)) be the full triangulated subcategory of D(M) whose objects have bounded (resp. bounded below, resp. bounded above) amplitude. Lemma 2.18. The category M(X) has enough injective objects. Proof. The classical proof, found e.g. in [22, Proposition 2.4.3], adapts as follows. Consider the natural map p : Xˆ − → X, where Xˆ is the set X endowed with the discrete topology. For F ∈ M(X), the adjunction morphism F − → p∗ p−1 F is injective, and → I , where I the functor p∗ is left exact. It is thus enough to find an injection p−1 F − ˆ Since Xˆ is discrete, p M is equivalent to a stack is an injective object in p  M(X). of (non twisted) modules.9   Let f : Y − → X be a continuous map of topological spaces. Deriving the functors f −1 , f∗ , and H om R , one gets functors → D±,b (f  M), f −1 : D±,b (M) − Rf ∗ : D+ (f  M) − → D+ (M), RHom R : D− (M−1 )op × D+ (M ) − → D+ (M R M ). Assuming that the weak global dimension of R is finite, one gets that M(X) has enough flat objects. Deriving ⊗R one gets a functor L : D±,b (M) × D±,b (M ) − → D±,b (M R M ). ⊗R

Assuming that f is a map between locally compact Hausdorff topological spaces, one can derive the functor f! , and get → D+ (M). Rf ! : D+ (f  M) − Assume that f! has finite cohomological dimension. The usual construction of Poincaré–Verdier duality (cf. e.g. [22, §3.1]) extends to the twisted case as follows10 . Let L ∈ Mod(ZX ), and consider the functor → M. f! (· ⊗Z L) : f∗ f  M − 9Another proof is obtained by applying Grothendieck’s criterion, stating that a category has enough

injective objects if it admits small filtrant inductive limits, which are exact, and if it admits a generator. Let ≈ → M|Ui be R-equivalences for some {Ui }i∈I be an open covering of X, let ϕi : Mod(Ai ) −  R-algebras Ai , and let Gi be generators of Mod(Ai ). Then a generator of M(X) is given by G = i ji! ϕi (Gi ), where ji : Ui − → X are the open inclusions. 10 The existence of f ! also follows from Brown representability theorem

Stacks of twisted modules and integral transforms

485

Denote by I(M) the full substack of M of injective objects. Assuming that L is flat and f -soft, there exists a functor → I(f∗ f  M) fL! : I(M) − characterized by the isomorphism, functorial in I and G, H om M (f! (G ⊗Z L), I )  f∗ H om f  M (G, fL! I ). In fact, the above isomorphism shows that the existence of fL! is a local problem, and locally this is the classical construction. As in the classical case, one finally gets a functor → D+ (f  M) f ! : D+ (M) − by letting f ! F be the simple complex associated to the double complex fL! • I • , where I • ∈ K + (I(M)(X)) is quasi-isomorphic to F , and L• is a (non twisted) bounded, flat, f -soft resolution of ZY . One proves that the usual formulas relating the six operations above, like adjunction, base-change, or projection formulas, hold.

3 Descent Effective descent data for stacks of twisted modules, called twisting data, are considered in [18, 26, 25], and we recall here this notion using the language of semisimplicial complexes. We then describe in terms of twisting data equivalences, operations, and the example of twisted modules associated with a line bundle.

3.1 Morita theory III. Equivalences In Section 2.1 we recalled how functors between stacks of modules admitting an adjoint are described in term of bimodules. We discuss here the particular case of equivalences. (References are again made to [1, 12, 25].) Two R-algebras A and B are called Morita equivalent if Mod(A) and Mod(B) are R-equivalent. Let us recall how such equivalences are described in terms of A ⊗R B op -modules. Proposition 3.1. Let L be an A ⊗R B op -module. Then the following conditions are equivalent: (i) There exists a B ⊗R Aop -module L , such that L ⊗B L  A as A ⊗R Aop modules and L ⊗A L  B as B ⊗R B op -modules.

486

Andrea D’Agnolo and Pietro Polesello

(ii) For L∗A = Hom A (L, A), the canonical morphism L ⊗B L∗A − → A is an isomorphism of A ⊗R Aop -modules, and L∗A ⊗A L  B as B ⊗R B op modules. (iii) L is a faithfully flat A-module locally of finite presentation, and there is an ∼ → E nd A (L). R-algebra isomorphism B op − (iv) L is a faithfully flat B op -module locally of finite presentation, and there is an ∼ → E nd B op (L). R-algebra isomorphism A − → Mod(A) is an R-equivalence. (v) L ⊗B (·) : Mod(B) − → Mod(B) is an R-equivalence. (vi) Hom A (L, ·) : Mod(A) − Definition 3.2. An A ⊗R B op -module L is called invertible if the equivalent conditions in Proposition 3.1 are satisfied. The ring A itself is the invertible A ⊗R Aop -module corresponding to the identity functor of Mod(A). Note that invertible A ⊗R Aop -modules are not necessarily locally isomorphic to A as A-modules, even if A is a commutative ring. Theorem 3.3 (Morita). If ϕ : Mod(B) − → Mod(A) is an equivalence of R-stacks, then L = ϕ(B) is an invertible A ⊗R B op -module, and ϕ  L ⊗B (·). Moreover, a quasi-inverse to ϕ is given by Hom A (L, ·)  L∗A ⊗A (·). Proof. Let ψ be a quasi-inverse to ϕ. Since ψ is right adjoint to ϕ, by Proposition 2.1 L = ϕ(B) is an A ⊗R B op -module such that ϕ  L ⊗B (·). Interchanging the role of ϕ and ψ there also exists a B ⊗R Aop -module L such that ψ  L ⊗A (·). Since ϕ  ψ and ψ  ϕ are isomorphic to the identity functors, L is invertible and L  L∗A . Finally, since L is a flat A-module locally of finite presentation, one has   L∗A ⊗A (·)  Hom A (L, ·).

3.2 Twisting data on an open covering By definition, if M is a stack of R-twisted modules there exist an open covering {Ui }i∈I of X, R|Ui -algebras Ai on Ui , and R|Ui -equivalences ϕi : M|Ui − → Mod(Ai ). Let ψi be a quasi-inverse of ϕi , and let αi : ψi  ϕi ⇒ idM be an invertible transformation. By (the R-linear analogue of) Proposition 1.6, the following descent datum for stacks is enough to reconstruct M ({Ui }i∈I , {Mod(Ai )}i∈I , {ϕij }i,j ∈I , {αij k }i,j,k∈I ).

(3.1)

Here ϕij = ϕi |Uij  ψj |Uij , and αij k : ϕij  ϕj k ⇒ ϕik is induced by αj , so that they satisfy condition (1.3). Functors as ϕij are described by Morita’s Theorem 3.3, so that the descent datum (3.1) is replaced by t = ({Ui }i∈I , {Ai }i∈I , {Lij }i,j ∈I , {aij k }i,j,k∈I ),

(3.2)

Stacks of twisted modules and integral transforms

487

where {Ui }i∈I is an open covering of X, Ai is an R|Ui -algebra on Ui , Lij are invertible op → Lik |Uij k are isomorphisms Ai ⊗R Aj |Uij -modules, and aij k : Lij ⊗Aj Lj k |Uij k − op

of Ai ⊗R Ak |Uij k -modules satisfying the analogue of condition (1.3). As in the proof of Proposition 1.6, up to equivalence a twisted module F ∈ M(X) is thus described by a pair ({Fi }i∈I , {mij }i,j ∈I ),

where Fi ∈ Mod(Ai ), and mij : Lij ⊗Aj Fj |Uij − → Fi |Uij is an isomorphism of Ai |Uij -modules on Uij such that the following diagram on Uij k commutes Lij ⊗Aj Lj k ⊗Ak Fk

aij k ⊗idFk

/ Lik ⊗A Fk k

idLij ⊗mj k

 Lij ⊗Aj Fj

mik

 / Fi .

mij

This is actually the definition of twisted modules given in [18]. It is also an example of twisting data, of which we now give a more general definition.

3.3 Twisting data We shall use here the language of semisimplicial complexes. On the one hand, this allows one to consider more general situations than open coverings, on the other hand, it provides a very efficient bookkeeping of indices. Recall that semisimplicial complexes are diagrams of continuous maps of topological spaces11 q0[3] ,...,q3[3]

X[3]

//

//

X[2]

q0[2] ,q1[2] ,q2[2]

/

//

X[1]

q0[1] ,q1[1]

//

X[0]

q0[0] =q [0] =q

/ X[−1] = X, (3.3)

satisfying the commutativity relations qj[r]  qi[r+1] = qi[r]  qj[r+1] +1 , for 0 ≤ i ≤ j ≤ r. In the coskeleton construction, one considers the topological space X [r]+1 = {(x0 , . . . , xr+1 ) ∈ (X[r] )r+2 : qj[r] (xi ) = qi[r] (xj +1 ) for 0 ≤ i ≤ j ≤ r}, 11 In dealing with stacks, we will only need the terms X [r] with r ≤ 3.

488

Andrea D’Agnolo and Pietro Polesello

[r+1] and let q [r+1] : X [r+1] − → X [r]+1 be the map x  → (q0[r+1] (x), . . . , qr+1 (x)). Hence there are commutative diagrams for 0 ≤ i ≤ r + 1 qi[3]

q [2]

q [1]

q i i / X[2] / X [1] / X[0] /X 7 7 7 o o rrrr p o o p r o o r p r pp rrrrr ooo ooo q [3] q [2] q [1] q [0] r ppp ooo ooo rrr r p o o r r p o o  o  o  p  rrrrr X X [2]+1 X[1]+1 X[0]+1

X [3]

where the diagonal arrows are the projection to the ith factor. Example 3.4. (a) Let us say that a semisimplicial complex is coskeletal if X [r+1]  X [r]+1 for r ≥ 0. In other words, X[r] = Y ×X · · · ×X Y is the (r + 1)-fold fibered product of a continuous map q : Y − → X, and qi[r] the projection omitting the ith factor. (a1) A particular case of coskeletal semisimplicial complex is the one attached to an  open covering {Ui }i∈I of X. In this case, Y = i∈I Ui is the disjoint union of the Ui ’s,and q is the natural map (which is a local homeomorphism). Note that X [r] = i0 ,...,ir ∈I Ui0 ···ir . (a2) Another particular case of coskeletal semisimplicial complex is obtained when q: Y − → X is a principal G-bundle, for G a topological group. Denoting by m: G×Y − → Y the group action, this semisimplicial complex is identified with qi[3]

G×G×G×Y

//

// G × G × Y

qi[2]

//

/ G×Y

qi[1]

// Y

q

/ X,

where qr[r] = idGr−1 ×m, q0[r] is the projection omitting the 0th factor, and qi[r] (g0 , . . . , gr−1 , y) = (g0 , . . . , gi−1 gi , . . . , gr−1 , y) for 0 < i < r. (b) Other examples of semisimplicial complexes are the ones attached to hyper[r]+1 . These open covering of X coverings, where X [r+1] is  induced by an   α [0] [1] are of the form X = = i Ui for X = i∈I Ui , X i,j,α Uij , for   γ β ξ α α [2] = Uij = α∈Aij Uij , X i,j,k,α,β,γ ,ξ Uij kαβγ for Uij ∩ Ukj ∩ Uki =  ξ αβγ Uij kαβγ , and so on. ξ ∈ ij k

Let s > r, 0 ≤ i0 < · · · < ir ≤ s, and 0 ≤ ir+1 < · · · < is ≤ s, be such that {i0 , . . . , is } = {0, . . . , s}. If F is a sheaf on X[r] , we denote by Fi0 ···ir =  · · ·  qi[s] )−1 F its sheaf-theoretical pull-back to X[s] , and we use the same (qi[r+1] s r+1 notations for morphisms of sheaves.12 12 In the coskeletal case, F

i0 ···ir is the pull-back of F by the projection to the (i0 , . . . , ir )th factors.

489

Stacks of twisted modules and integral transforms

Definition 3.5. (i) An R-twisting datum on X is a quadruplet13 q

t = (X [•] − → X, A, L, a),

(3.4)

q

where X[•] − → X is a semisimplicial complex, A is a q −1 R-algebra on X [0] , L op → L02 is an is an invertible A0 ⊗R A1 -module on X [1] , and a : L01 ⊗A1 L12 − op [2] isomorphism of A0 ⊗R A2 -modules on X such that the following diagram on X [3] commutes14 L01 ⊗A1 L12 ⊗A2 L23 

a012 ⊗idL23

/ L02 ⊗A L23 2

idL01 ⊗a123

(3.5)

a023



a013

L01 ⊗A1 L13

/ L03 .

(ii) A coskeletal R-twisting datum on X is an R-twisting datum whose associated semisimplicial complex is coskeletal. One can now mimic the construction in the sketch of proof of Proposition 1.6. Denote by Mod(t) the category whose objects are pairs (F , m), where F is an A→ F0 is an isomorphism of A0 -modules on X [1] module on X [0] , and m : L ⊗A1 F1 − 13 This notion was discussed in [18] for semisimplicial complexes attached to open coverings, and in [26] for coskeletal semisimplicial complexes. 14 Let us denote by L[r] the sheaf L on X [r] . Then one should pay attention to the fact that in X [3] ij ij [3]

[3]

[2]

one has isomorphisms like L01  (q3 )−1 L01 , but not equalities. Thus, much as in Definition 1.4 (iv), one should write (3.5) more precisely as [3]

[2]

[3] ∼

[2]

(q3 )−1 (L01 ⊗A1 L12 ) ⊗A2 L23 a⊗id [3] L

[3]

23



[2]

[3]

[3]

[3] ∼

L01 ⊗A1 L12 ⊗A2 L23

[3]

[3]

id [3] ⊗a L

[3]

[3]

(q3 )−1 L02 ⊗A2 L23 ∼



01

[3]

[2]

∼

[2]

[3] [3] L01 ⊗A1 L13 ∼ [2]

[3]

(q1 )−1 (L01 ⊗A1 L12 ) a

[2]

L01 ⊗A1 (q0 )−1 L02

[3] [3] L02 ⊗A2 L23 ∼ [3]

[2]

L01 ⊗A1 (q0 )−1 (L01 ⊗A1 L12 )



[3] [2] (q1 )−1 L02

[2]

[2]

(q2 )−1 (L01 ⊗A1 L12 ) a ∼

[3] L03

∼



[3] [2] (q2 )−1 L02 .

Such a level of precision is both quite cumbersome and easy to attain, so we prefer a sloppier but clearer presentation.

490

Andrea D’Agnolo and Pietro Polesello

such that the following diagram on X [2] commutes L01 ⊗A1 L12 ⊗A2 F2

a⊗idF2

/ L02 ⊗A F2 2

idL01 ⊗m12

 L01 ⊗A1 F1

(3.6)

m02

 / F0 ,

m01

and whose morphisms α : (F , m) − → (F , m ) consists of morphisms of A-modules

α: F − → F , such that the following diagram on X[1] commutes L ⊗A1 F1 

m

/ F0

m

 /F . 0

idL ⊗α1

L ⊗A1 F1

α0

Definition 3.6. Let t be an R-twisting datum on X. We denote by Mod(t) the prestack on X defined by U  → Mod(t|U ), which is in fact an R-stack. Here, t|U denotes the R|U -twisting datum on U naturally induced by t. Note that if B is an R-algebra on X, then Mod(B) ≈ Mod(1B ) for id

1B = (X − → X, B, B, ·) ∼

→ B. the trivial R-twisting datum, with · being the canonical isomorphism B ⊗B B − We spend the rest of this section to show that Mod(t) is actually a stack of Rtwisted modules, using arguments adapted from those in [26]. In order to get this result → X [r−1]+1 admit local sections for it is natural to assume that the maps q [r] : X[r] − r = 0, 1, 2, 3. However, for the sake of simplicity, we will consider here the stronger assumption the maps q [r] , for r = 0, 1, 2, 3, admit sections locally on X.

(3.7)

Note that for coskeletal semisimplicial complexes this reduces to the assumption → X admits local sections, q : X[0] −

(3.8)

which holds true for semisimplicial complexes attached to open coverings or to principal G-bundles, as in Example 3.4 (a1) and (a2). In general, (3.7) does not hold for semisimplicial complexes attached to hypercoverings, as in Example 3.4 (b). We refer to e.g. [5] for a discussion of this case. q

→ X, A , L , a ) be another R-twisting datum on X. Let t = (X [•] −

491

Stacks of twisted modules and integral transforms

Definition 3.7. (i) A refinement of R-twisting data ρ : t − → t consists of commutative diagrams X [3] 

ρ [3]

X[3]

qk [3]

qk[3]

/ X [2] 

ρ [2]

/ X[2]

qj [2]

qj[2]

/ X [1] 

ρ [1]

/ X[1]

qi [1]

qi[1]

/ X [0] 

q

/X

q

/X

ρ [0]

/ X[0] ∼

of an isomorphism of q −1 R-algebras (ρ [0] )−1 A − → A , and of an isomorphism ∼

op → L compatible with a and a . of A 0 ⊗R A1 -modules (ρ [1] )−1 L − → t one associates the functor ρ −1 : Mod(t) − → Mod(t ), given by (ii) To ρ : t − [0] −1 [1] −1 (F , m)  → ((ρ ) F , (ρ ) m). q

→ X, A, L, a) be such that the maps q [r] admit global Lemma 3.8. Let t = (X [•] − [r] sections s . Then → t, (i) there is a refinement of R-twisting data s˜ : 1(s [0] )−1 A − (ii) the functor s˜ −1 : Mod(t) − → Mod((s [0] )−1 A) is an equivalence. → X[r] by induction as15 Proof. Define the maps s˜ [r] : X − s˜ [0] (x) = s [0] (x),

s˜ [r+1] (x) = s [r+1] (˜s [r] (x), . . . , s˜ [r] (x)).

Since qi[2]  s˜ [2] = s˜ [1] , one has isomorphisms (˜s [2] )−1 Lj k  (˜s [1] )−1 L. Then, a gives an isomorphism (˜s [1] )−1 L ⊗(s [0] )−1 A (˜s [1] )−1 L  (˜s [1] )−1 L.

(3.9)

Since L is invertible, there is an A1 ⊗R A0 -module L such that L ⊗A1 L  A0 . Applying the functor (·) ⊗(s [0] )−1 A (˜s [1] )−1 L to (3.9), we get an isomorphism of (s [0] )−1 A ⊗R (s [0] )−1 Aop -modules (˜s [1] )−1 L  (s [0] )−1 A. This proves (i). → X [r+1] by induction as16 To prove (ii), let us define the maps σ [r] : X [r] − 
σ [−1] = s [0] , σ [r] (x) = s [r+1] σ [r−1] (q0[r] (x)), . . . , σ [r−1] (qr[r] (x)), x . op

→ Mod(t), given by Using the maps σ [r] one gets a functor σ −1 : Mod((s [0] )−1 A) − G  → ((σ [0] )−1 L ⊗q −1 (s [0] )−1 A q −1 G, (σ [1] )−1 a). This is well-defined, since (3.6) is obtained by applying (σ [2] )−1 to (3.5). One checks that σ −1 is a quasi-inverse   to s˜ −1 . 15 For coskeletal semisimplicial complexes, one has s˜ [r] = δ [r]  s [0] , where δ [r] : X [0] − → X[r] is the diagonal embedding. 16 For coskeletal semisimplicial complexes, one has σ [r] (x) = (x, s [0] (q(x))) ∈ X [r] × X [0] = X [r+1] , X [j ] → X is the composite of the qi ’s maps. where q : X[r] −

492

Andrea D’Agnolo and Pietro Polesello

Proposition 3.9. (i) Let t be an R-twisting datum on X satisfying (3.7). Then Mod(t) is a stack of R-twisted modules. (ii) Any stack of R-twisted modules on X is R-equivalent to Mod(t) for some coskeletal R-twisting datum t satisfying (3.8). Proof. By definition, the maps q [r] ’s admit local sections on X. Hence part (i) follows   from Lemma 3.8. As for (ii), it is enough to take t as in (3.2). Proposition 3.10. Let ρ : t − → t be a refinement of coskeletal R-twisting data on X → Mod(t ) is an R-equivalence. satisfying (3.8). Then the functor ρ −1 : Mod(t) − q

q

Proof. Let t = (X [•] − → X, A, L, a) and t = (X [•] − → X, A , L , a ). Proving that −1 →X ρ is an equivalence is a local problem, and we may thus assume that q : X [0] − admits a global section. Then s = ρ [0]  s is a global section of q : X[0] − → X. With the notations of Lemma 3.8, one has s˜ = ρ  s˜ . Hence there is a diagram of functors commuting up to an invertible transformation ρ −1

/ Mod(t ) Mod(t)P PPP n PPP nnn PPP nnn n n PP' s˜ −1 wnnn s˜ −1

[0] −1 Mod((s ) A ),  

whose diagonal arrows are equivalences.

3.4 Classification of stacks of twisted modules q

One may consider coskeletal R-twisting data t = (X [•] − → X, A, L, a) as a kind of Cech cocycles attached to the covering q, with (3.5) playing the role of the cocycle condition. There is also a straightforward analogue to the notion of coboundary, given q → X, B, M, b) be another coskeletal by Morita theorem as follows. Let u = (X [•] − R-twisting datum attached to the same covering q as t. Let us say that t and u differ by a coboundary if there exist a pair (E , e) where E is an invertible A ⊗R B op -module ∼ op → E0 ⊗B0 M is an isomorphism of A0 ⊗R B1 -modules on X [0] , and e : L ⊗A1 E1 − on X [1] such that the following diagram on X[2] commutes L01 ⊗A1 L12 ⊗A2 E2

idL01 ⊗e12

/ L01 ⊗A E1 ⊗B M12 1 1

a⊗idE2

 L02 ⊗A2 E2

e02

e01 ⊗idM12

/ E0 ⊗B M01 ⊗B M12 0 1 

idE0 ⊗b

/ E0 ⊗B M02 . 0

Stacks of twisted modules and integral transforms

493

→ Mod(t) given by (G, n)  → (E ⊗B In this case, there is an R-equivalence Mod(u) − G, (idE0 ⊗n)  (e ⊗ idG1 )). Note that R-equivalence classes of stacks of R-twisted modules are in one-toone correspondence with this “cohomology”. (The analogue correspondence appears in [32] for the case of bundle gerbes, and in [6] for general gerbes.) In fact, one q r → X, A, L, a) and u = (Y [•] − → X, B, M, b) are arbitrary checks that if t = (X[•] − coskeletal R-twisting data, then Mod(t) and Mod(u) are R-equivalent if and only if t and u differ by a coboundary on a common refinement. This means that there exist → t and u − → u such that t and u are refinements of coskeletal R-twisting data t − attached to the same covering, and differ by a coboundary.

3.5 Operations in terms of twisting data Operations for stacks of twisted C-modules were described in [26] using twisting data. We give here a similar description for general twisted modules. q

Let t = (X[•] − → X, A, L, a) be an R-twisting datum on the topological space X. Its opposite is the R-twisting datum q

top = (X [•] − → X, Aop , L−1 , a −1 ),

(3.10)

where L−1 = Hom A0 (L, A0 ), and a −1 is the inverse of the following chain of isomorphisms L−1 02 = Hom A0 (L02 , A0 ) a

− → Hom A0 (L01 ⊗A1 L12 , A0 )  Hom A1 (L12 , Hom A0 (L01 , A0 ))  Hom A1 (L12 , A1 ) ⊗A1 Hom A0 (L01 , A0 ) −1 op = L−1 01 ⊗A L12 , 1

where in the last isomorphism holds because L12 is a flat A1 -module locally of finite presentation. q

→ X, A , L , a ) be another R-twisting datum on X. Consider the Let t = (X [•] − p → X, and denote by π [•] : X [•] ×X X [•] − → X [•] semisimplicial complex X[•] ×X X [•] −

[•] [•]

[•]

[•] [r] and π : X ×X X − → X the natural maps. If F is a sheaf on X and F is a

[r] sheaf on X , write for short F ⊗R F = (π [r] )−1 F ⊗R (π [r] )−1 F on X[r] ×X X [r] . The product of t and t is the R-twisting datum on X p t ⊗R t = (X [•] ×X X [•] − → X, A ⊗R A , L ⊗R L , a ⊗R a ).

(3.11)

Let f : Y − → X be a continuous map of topological spaces. Consider the semisimr → Y , and denote by f [•] : Y ×X X [•] − → X [•] the natural plicial complex Y ×X X[•] −

494

Andrea D’Agnolo and Pietro Polesello

maps. The pull-back of t by f is the f −1 R-twisting datum on Y r

→ Y, (f [0] )−1 A, (f [1] )−1 L, (f [2] )−1 a). f −1 t = (Y ×X X [•] −

(3.12)

One checks that, for t and t satisfying (3.8), there are two R-equivalences and one

f −1 R-equivalence

Mod(t)−1 ≈ Mod(top ), Mod(t) R Mod(t ) ≈ Mod(t ⊗R t ), f  Mod(t) ≈ Mod(f −1 t).

Recall that a topological manifold X is a paracompact Hausdorff topological space locally homeomorphic to Rn . In particular, X is locally compact. In the context of twisting data, the sheaf theoretical operations of Proposition 2.17 are easily described under the assumption that f : Y − → X is a morphism of topological manifolds, and q

q

→ X and X [•] − → X are semisimplicial complexes of topological manifolds X [•] − with submersive maps. (Note that this last requirement is automatically fulfilled for twisting data as in (3.2).) For example, let us describe the direct image functor f∗ . With the same notations as in (3.12), consider the Cartesian squares ri[1]

Y ×X X[1] 

f [1]

qi[1]

X [1]

/ Y × X[0] X 

r

f [0]

/ X[0]

/Y f

q

 / X.

If (G, n) is an object of Mod(f −1 t), then f∗ (G, n) = (f∗[0] G, f∗ n), where f∗ n is the composite L ⊗A1 (q1[1] )−1 f∗[0] G  L ⊗A1 f∗[1] (r1[1] )−1 G  f∗[1] ((f [1] )−1 L ⊗(f [1] )−1 A1 (r1[1] )−1 G) ∼

− → f∗[1] (r0[1] )−1 G n

 (q0[1] )−1 f∗[0] G. Here, the first and last isomorphisms hold because the maps qi[1] ’s are submersive (and hence so are the ri[1] ’s), while the second isomorphism is due the fact that L is a flat A1 -module locally of finite presentation, and hence locally a direct summand of a free A1 -module of finite rank.

Stacks of twisted modules and integral transforms

495

3.6 Complex powers of line bundles Let us discuss the example of twisting data attached to line bundles. Let X be a complex analytic manifold, and denote by OX its structural sheaf. Let π: F − → X be a line bundle, let q : Y = F \ X − → X be the associated principal C× bundle obtained by removing the zero-section, and denote by F the sheaf of sections of π. As in Example 3.4 (a2), consider the semisimplicial complex where X [r] is the (r + 1)-fold fibered product of Y . For λ ∈ C, one has a local system on X [1] Lλ = p−1 Ct λ , where p : X [1] − → C× is the map (x, y)  → x/y, and Ct λ ⊂ OC× is the local system × on C generated by t λ . This defines a CX -twisting datum q

tλ = (X [•] − → X, CY , Lλ , a),

where a is given by (c (x/y)λ , d (y/z)λ ) → cd (x/z)λ . Denote by OY (λ) the subsheaf of OY of λ-homogeneous functions, i.e. solutions of eu − λ, where eu is the infinitesimal generator of the action of C× on the fibers of q. It is a q −1 OX -module locally constant along the fibers of q, and there is a → (q0[1] )−1 OY (λ) on X[1] given by natural isomorphism m : Lλ ⊗ (q1[1] )−1 OY (λ) − (c (x/y)λ , ϕ(y))  → c ϕ(x). This gives an object F −λ = (OY (λ), m) ∈ Mod(OX ; tλ ). The choice of sign is due to the fact that there is an isomorphism ∼

F − → q∗ OY (−1), given by ϕ → (x  → ϕ(q(x))/x), with inverse ψ  → (x  → ψ(x) x).

4 Examples and applications Giraud [14] uses gerbes to define the second cohomology of a sheaf of not necessarily commutative groups G,17 and if G is abelian this provides a geometric description of the usual cohomology group H 2 (X; G). We consider here the case of a sheaf of commutative local rings R, and recall how R-equivalence classes of stacks of twisted R-modules are in one-to-one correspondence with H 2 (X; R× ). We also discuss the examples of stacks of twisted modules associated with inner forms of an R-algebra, considering in particular the case of Azumaya algebras and TDO-rings. 17 We will discuss in [9] the linear analogue, where G is replaced by a not necessarily commutative R-algebra

496

Andrea D’Agnolo and Pietro Polesello

As an application, we state a twisted version of an adjunction formula for sheaves and D-modules in the context of Radon-type integral transforms.

4.1 Twisted modules over commutative local rings Let R be a sheaf of commutative rings. With the terminology of Definition 3.2, an R-module is called invertible if it is invertible as R ⊗R Rop -module. Denote by Pic(R) the set of isomorphism classes of invertible R-modules, endowed with the abelian group law given by tensor product over R. This is called the Picard group of R. Proposition 4.1. Let M be a stack of twisted R-modules. The group of isomorphism classes of R-equivalences of M to itself is isomorphic to Pic(R). Proof. To an invertible R-module L, one associates the R-functor ϕ = L ⊗R (·). To an R-equivalence ϕ of M to itself, one associates the invertible R-module L =   Hom (idM , ϕ). Let M be a stack of twisted R-modules, and denote by [M] its R-equivalence class. The multiplication [M][M ] = [M R M ] is well defined, with identity [Mod(R)] and inverse [M]−1 = [M−1 ]. Let us denote by Tw(R) the set of Requivalence classes of stacks of twisted R-modules endowed with this abelian group structure. Definition 4.2. Let us say that R is Picard good if invertible R-modules are locally isomorphic to R itself. Recall that a sheaf of commutative rings R is called local if for any U ⊂ X and any r ∈ R(U ) there exists an open covering {Vi }i∈I of U such that for any i ∈ I either R/Rr = 0 or R/R(1 − r) = 0 on Vi . Sheaves of commutative local rings are examples of Picard good rings. In the rest of this section we assume that R is Picard good. Denote by R× the multiplicative group of invertible elements in R. Proposition 4.3. (i) There is a group isomorphism Pic(R)  H 1 (X; R× ). (ii) There is a group isomorphism Tw(R)  H 2 (X; R× ). Part (i) easily follows from the definition of Picard good. Part (ii) of the above proposition is proved as the analogue result for gerbes discussed e.g. in [5, §2.7]. Recall that H 2 (X; R× ) is calculated using hypercoverings, and coincides with Cech cohomology if X is Hausdorff paracompact.

Stacks of twisted modules and integral transforms

497

Definition 4.4. Let M be a stack of twisted R-modules. We say that F ∈ M(X) is a locally free twisted R-module of finite rank if there exists a covering {Ui }i∈I of → Mod(R|Ui ), such that ϕi (F |Ui ) is a locally X, and R|Ui -equivalences ϕi : M|Ui − free R|Ui -module of finite rank. More generally, for an R-algebra A we will speak of locally free R-twisted A-modules of finite rank in ModR (A; M). Note that if F is a locally free twisted R-module of finite rank, then for any R|U equivalence ϕ : M|U − → Mod(R|U ), ϕ(F ) is a locally free R-module of finite rank. Note also that the rank of F is a well defined locally constant function. Proposition 4.5. Let M be a stack of twisted R-modules. (i) M is R-equivalent to Mod(R) if and only if M(X) has a locally free twisted R-module of rank 1. (ii) More generally, M is R-equivalent to another stack of twisted R-modules N if and only if M−1 R N(X) has a locally free twisted R-module of rank 1. (iii) If M(X) has a locally free twisted R-module of rank n, then n-fold product Mn = M R · · · R M is R-equivalent to Mod(R). Proof. To a locally free twisted R-module L of rank 1 in M(X) one associates the R-equivalence L ⊗R (·) : Mod(R) − → M. To an R-equivalence ϕ : Mod(R) − → M, one associates the locally free twisted R-module of rank one ϕ(R). This proves (i). (ii) follows from (i). As for (iii), let F ∈ M(X) be a locally free twisted R-module   of rank n. Then det F is a locally free twisted R-module of rank 1 in Mn .

4.2 Twisting by inner forms Let R be a Picard good sheaf of commutative rings, and let A be an R-algebra. Denote by Aut R-alg (A) the sheaf of groups of automorphisms of A as an R-algebra, and by
nn (A) its normal subgroup of inner automorphisms, i.e. the image of the adjunction → Aut R-alg (A), a  → (b  → aba −1 ). morphism ad : A× − Definition 4.6. An R-algebra B is called an inner form of A if there exist an open ∼ → B|Ui of R-algebras such that covering {Ui }i∈I of X and isomorphisms θi : A|Ui − the automorphisms θj−1  θi of A|Uij are inner. Isomorphism classes of inner forms of A are classified by H 1 (X;
nn (A)). Assume that A is a central R-algebra, i.e. that its center Z(A) is equal to R. (If A is not central, the following discussion still holds by replacing R with Z(A).) Then the exact sequence ad

1− → R× − → A× − →
nn (A) − →1

(4.1)

498

Andrea D’Agnolo and Pietro Polesello

induces the exact sequence of pointed sets γ

δ

→ H 1 (X;
nn (A)) − → H 2 (X; R× ). H 1 (X; A× ) −

(4.2) (L∗ )].

If L is a locally free A-module of rank one, then γ ([L]) = [E nd Aop If B is an inner form of A, then δ([B]) = [MB ], where MB is the stack of twisted R-modules described in the following proposition. Proposition 4.7. Let A be a central R-algebra, and B an inner form of A. Then there exists an R-equivalence ϕ : Mod(B) − → ModR (A; MB ), where MB is a stack of twisted R-modules. Moreover, ϕ  LB ⊗B (·) where LB = ϕ(B) is a locally free R-twisted A-module of rank one in ModR (A ⊗R B op ; MB ), and there is an ∗A isomorphism of R-algebras B  E nd Aop (L∗A B ), where LB = Hom A (LB , A) ∈ −1 ModR (B ⊗R Aop ; MB ). Proof. Since B is an inner form of A, there exist an open covering {Ui }i∈I of X, ∼ → B|Ui of R-algebras such that θj−1  θi are inner. Let and isomorphisms θi : A|Ui − ϕi : Mod(B|Ui ) − → Mod(A|Ui ) be the induced R|Ui -equivalences, denote by ψi a quasi-inverse to ϕi , set ϕij = ϕi |Uij  ψj |Uij , and let αij k : ϕij  ϕj k ⇒ ϕik be the associated invertible transformations. One checks that ϕij  idMod(A|Uij ) , so that αij k ∈ End (idMod(A|Uij k ) )×  Γ (Uij k ; R× ). By Proposition 4.3 (ii), this is thus an R-twisting datum defining a stack of twisted R-modules MB . The equivalences ϕi glue together, giving an equivalence ϕ : Mod(B) − → ModR (A; MB ). The rest of the statement is a twisted version of Morita theorem. LB = ϕ(B) is a locally free R-twisted A-module of rank one in ModR (A; MB ) which inherits a compatible B op -module structure by that of B itself, and is such that B    E nd Aop (L∗A B ).

4.3 Azumaya algebras We consider here modules over Azumaya algebras as natural examples of twisted Rmodules. Refer to [14, 13] for more details. See also [8, 11], where a twisted version of the Fourier–Mukai transform is discussed, and [16], for applications to mathematical physics. In this section we assume that R is a sheaf of commutative local rings on X. Definition 4.8. An Azumaya R-algebra18 is an R-algebra locally isomorphic to the endomorphism algebra of a locally free R-module of finite rank. If the rank of such 18 The definition that we give here is good for the analytic topology, or for the étale topology. With this

definition, if A is an Azumaya R-algebra, then the morphism of R-algebras A ⊗R Aop − → E nd R (A)

Stacks of twisted modules and integral transforms

499

modules is constant and equal to n, then one says that the Azumaya R-algebra has rank n2 . If F is a locally free R-module of finite rank, then R and E nd R (F ) are Morita equivalent. This is a basic example of Morita equivalence, and is proved by noticing that F itself is an invertible R ⊗R E nd R (F )op -module (in fact, one has natural isomorphisms F ∗ ⊗R F  E nd R (F ) and F ⊗E nd (F ) F ∗  R, where F ∗ = R

Hom R (F , R)). It follows that if A is an Azumaya R-algebra then Mod(A) provides an example of stack of twisted R-modules. Moreover, the Skolem–Noether theorem (see e.g. [14, Lemme V.4.1]) asserts

Proposition 4.9. Any R-algebra automorphism of an Azumaya R-algebra is inner. In particular, Azumaya R-algebras of rank n2 are inner forms of the central R-algebra Mn (R) = E nd R (Rn ). Set GLn (R) = Mn (R)× , and P GLn (R) = GLn (R)/R× . Then the set of R-algebra isomorphism classes of Azumaya R-algebras of rank n2 is isomorphic to H 1 (X; P GLn (R)). Proposition 4.10. Let A be an Azumaya R-algebra of rank n2 . Then Mod(A) ≈ MA is a stack of twisted R-modules, and there exists a locally free twisted R-module FA op of rank n in M−1 A (X) ≈ Mod(A ) such that A  E nd R (FA ) as R-algebras. Proof. By Proposition 4.7 there exists a stack of twisted R-modules MA , and an R-equivalence ϕ : Mod(A) − → ModR (Mn (R); MA ). Rn

The functor ⊗Mn (R) (·) gives an R-equivalence Mod(Mn (R)) − → Mod(R). By (2.6), this induces an R-equivalence ψ : ModR (Mn (R); MA ) − → MA . Since A is locally isomorphic to Mn (R), ψ(ϕ(A)) is locally isomorphic to Rn . Set   FA = Hom R (ψ(ϕ(A)), R). With these notations, (4.1) and (4.2) read → GLn (R) − → P GLn (R) − → 1, 1− → R× − and γn

δn

H 1 (X; GLn (R)) − → H 1 (X; P GLn (R)) − → H 2 (X; R× ),

(4.3)

given by a ⊗ b  → (c  → acb) is an isomorphism. For algebraic manifolds with the Zariski topology, it is this property which is sometimes used to define Azumaya R-algebras when R is the sheaf of rings of regular functions.

500

Andrea D’Agnolo and Pietro Polesello

respectively. If F is a locally free R-module of rank n, then γn ([F ]) = [E nd R (F )]. If A is an Azumaya R-algebra of rank n2 , then δn ([A]) = [Mod(A)]. One says that two Azumaya R-algebras A and A are equivalent if there exist two locally free (non twisted) R-modules of finite rank F and F such that A ⊗R E nd R (F )  A ⊗R E nd R (F ). Lemma 4.11. Two Azumaya R-algebras are equivalent if and only if they are Morita R-equivalent. Proof. Since E nd R (F ) and R are Morita equivalent, so are A ⊗R E nd R (F ) and A by (2.6). On the other hand, if A and A are Morita equivalent, then there is an R-equivalence ϕ : Mod(R) − → Mod(A) R Mod(A )−1 ≈ Mod(A ⊗R A op ).

op Hence A ⊗R A  E nd R (F ) for F = ϕ(R). Tensoring with A we finally get an   isomorphism A ⊗R E nd R (A )  A ⊗R E nd R (F ). Denote by [A] the equivalence class of A. The multiplication [A][A ] = [A ⊗R is well defined, with identity [R] and inverse [A]−1 = [Aop ]. Denote by Br(R) the set of equivalence classes of Azumaya R-algebras endowed with this abelian group law, which is called Brauer group of R. By the Skolem–Noether theorem one has a group isomorphism A ]

Br(R)  lim H 1 (X; P GLn (R)). − → n The limit of the maps δn in (4.3) gives a group homomorphism δ : Br(R) − → Tw(R)

(4.4)

which is described by [A]  → [Mod(A)]. Proposition 4.12. The homomorphism δ is injective, and its image is contained in the torsion part of Tw(R). Proof. Injectivity follows from Lemma 4.11. As for the description of the image, let A be an Azumaya R-algebra of rank n2 , and let FA be the locally free twisted R-module of rank n in Mod(Aop ) of Proposition 4.10. By Proposition 4.5 (iii) one   has n · [Mod(A)] = −[Mod(Aop )n ] = 0.

4.4 Twisted differential operators Rings of twisted differential operators (TDO-rings for short) were introduced in a representation theoretical context in [3, 4]. Modules over TDO-rings provide another example of twisted modules, and we recall here these facts following the presentation in [18] (see also [27]). Since we deal with complex analytic manifolds, as opposed to algebraic varieties, many arguments are simpler than in loc. cit.

Stacks of twisted modules and integral transforms

501

Let X be a complex analytic manifold, and denote by OX its structural sheaf of holomorphic functions. Recall that an OX -ring is a C-algebra A endowed with a → A. Morphisms of OX -rings are morphisms of morphism of C-algebras β : OX − C-algebras compatible with β. Denote by DX the sheaf of differential operators on X. Recall that DX is a simple OX -ring with center CX . Definition 4.13. A TDO-ring on X, short for ring of twisted differential operators, is an OX -ring locally isomorphic to DX as OX -ring. A TDO-ring A has a natural increasing exhaustive filtration defined by induction by F−1 A = 0, Fm+1 A = {P ∈ A : [P , a] ∈ Fm A ∀a ∈ OX }, where [P , Q] = P Q − QP is the commutator. Note that Fm+1 A = F1 AFm A for m ≥ 0, and that the associated graded algebra GA is naturally isomorphic to SOX (X ), the symmetric algebra of vector fields over OX . Proposition 4.14. There are group isomorphisms Aut OX -ring (A)  dOX 
nn (A). In particular, TDO-rings are inner forms of the central C-algebra DX . × Proof. One has A× = OX . Hence the short exact sequence d log

× → OX −−→ dOX − →0 1− → C× X −

gives a group isomorphism
nn (A)  dOX . This proves the second isomorphism. To prove the first, note that any OX -ring automorphism ϕ of A preserves the filtration. Let ω ∈ dOX , P ∈ F1 A, and denote by σ1 (P ) ∈ X its symbol of order one. Then P  → P + σ1 (P ), ω extends uniquely to an OX -ring automorphism of A. On the other hand, to an OX -ring automorphism ϕ of A one associates the closed form   θ → ϕ(θ˜ ) − θ˜ , where θ˜ ∈ F1 (A) is such that σ1 (θ˜ ) = θ . Let F be a locally free OX -module of rank one, and set F ∗ = H om OX (F , OX ). Then the basic example of TDO-ring is given by D F = F ⊗ DX ⊗ F ∗ , O

O

where (s ⊗ P ⊗ · (t ⊗ Q ⊗ = s ⊗ P t, s ∗ Q ⊗ t ∗ . Equivalently, DF is the sheaf of differential endomorphisms of F , i.e. C-endomorphisms ϕ such that for any s ∈ F there exists P ∈ DX with ϕ(as) = P (a)s for any a ∈ OX . More generally, for λ ∈ C one has the TDO-ring s∗)

t ∗)

DF λ = F λ ⊗ DX ⊗ F −λ , O

O

where F λ was described in Section 3.6. By definition, sections of DF λ are locally of the form s λ ⊗ P ⊗ s −λ , where s is a nowhere vanishing local section of F , with the

502

Andrea D’Agnolo and Pietro Polesello

gluing condition s λ ⊗ P ⊗ s −λ = t λ ⊗ Q ⊗ t −λ if and only if Q = (s/t)λ P (s/t)−λ . This is independent from the choice of a branch for the ramified function (s/t)λ . Proposition 4.15. Let A be a TDO-ring. Then there exists a stack of twisted CX modules MA such that Mod(A) is C-equivalent to ModC (DX ; MA ). Moreover, in −1 ModC (OX ; MA ) there exists a locally free CX -twisted OX -module of rank one OA , such that A  DOA as OX -rings. Proof. By Proposition 4.7 there exists a stack of twisted CX -modules MA , and a C-equivalence ϕ : Mod(A) − → ModC (DX ; MA ). × , any locally free CX -twisted DX -module of Since DX× = OX phic to DX ⊗OX F for a locally free CX -twisted OX -module

particular, ϕ(A)  DX ⊗OX FA for some FA OA = Hom OX (FA , OX ).

rank one is isomorof rank one F . In ∈ ModC (OX ; MA ), and we set  

Conjecture 4.16. Any stack of CX -twisted DX -modules is C-equivalent to a stack of the form ModC (DX ; M) for some stack of twisted CX -modules M. Denote by X the sheaf of differential forms of top degree, and recall that there op is a natural isomorphism of OX -rings DX  DX . To the TDO-rings A and A one associates the TDO-rings A  A = E nd A⊗A (A ⊗ A ), C

O

A−1 = ∗X ⊗ Aop ⊗ X , O

O

(4.5)

where in the right-hand-side of the first equation A and A are regarded as OX modules by left multiplication and A ⊗ A is regarded as an A ⊗ A -module by right O

C

multiplication. Note that if F and F are locally free C-twisted OX -modules of rank one, then −1

DF  DF ∗ , DF  DF  DF ⊗F , O

∗

where F = Hom OX (F , OX ). Let us denote by [A] the OX -ring isomorphism class of A. The multiplication [A][A ] = [A  A ] is well defined, with identity [DX ] and inverse [A]−1 = [A−1 ]. Let us denote by TDO(OX ) the set of OX -ring isomorphism classes of TDO-rings, endowed with this abelian group law. As a corollary of Proposition 4.14, we get Proposition 4.17. There is a group isomorphism TDO(OX )  H 1 (X; dOX ). For inner forms of DX , the short exact sequence (4.1) reads d log

× → OX −−→ dOX − → 0. 1− → C× X −

Stacks of twisted modules and integral transforms

503

It induces the long exact cohomology sequence × × H 1 (X; C× → H 1 (X; OX )− → H 1 (X; dOX ) − → H 2 (X; C× → H 2 (X; OX ), X) − X) −

which may be written as the exact sequence of groups γ

δ

Pic(CX ) − → Pic(OX ) − → TDO(OX ) − → Tw(CX ) − → Tw(OX ), where γ ([F ]) = [DF ] and δ([A]) = [MA ]. Note that (4.5) implies the relations [Aop ] = γ ([X ]) − [A], and [MA ] = −[MAop ]. Note also that the complex span of the image of γ is described by λ · γ ([F ]) = [DF λ ], for λ ∈ C. Example 4.18. Let X = P be a complex finite dimensional projective space. Then the above long exact sequence reads 0 − →Z− →C− → C/Z − → 0. Denote by OP (−1) the tautological line bundle, and for λ ∈ C set OP (λ) = (OP (−1))−λ . Then any TDOring on P is of the form DOP (λ) for some λ, and [MDO (λ) ] = [MDO (µ) ] if and only P P → Mod(DOP (µ) ) is given if λ − µ ∈ Z. In this case, an equivalence Mod(DOP (λ) ) − by OP (µ−λ) ⊗OX (·).

4.5 Twisted D-module operations We recall here the twisted analogue of D-module operations, following [18, 27]. (We do not recall here the classical formalism of operations for D-modules, referring instead to [21, 19].) Besides the internal operations for TDO-rings recalled in (4.5), there is an external operation defined as follows. Let f : Y − → X be a morphism of complex analytic manifolds. To a TDO-ring A on X one associates the TDO-ring on Y f  A = E nd f −1 A (OY ⊗f −1 OX f −1 A), where OY ⊗f −1 OX f −1 A is regarded as a right f −1 A-module. One has f  (A−1 )  (f  A)−1 , f  (A  A )  f  A  f  A . Moreover, if F is a locally free C-twisted OX -module of rank one, then f  DF  Df ∗ F , where f ∗ F = OY ⊗f −1 OX f −1 F . Let f : Y − → X be a morphism of complex analytic manifolds, and A a TDO-ring on X. Consider the transfer modules AY →X = OY ⊗f −1 OX f −1 A,

an f  A ⊗CY (f −1 A)op -module,

AX←Y = f −1 A ⊗f −1 OX f ,

an f −1 A ⊗CY (f  A)op -module,

504

Andrea D’Agnolo and Pietro Polesello

where f = Y ⊗ f ∗ ∗X , ∗X denoting the dual H om OX (X , OX ) of X . Note O

that if F is a locally free C-twisted OX -module of rank one, then (DF )Y →X  f ∗ F ⊗OY DY →X ⊗f −1 OX f −1 F ∗ , (DF )X←Y  f −1 F ⊗f −1 OX DX←Y ⊗OY f ∗ F ∗ , where DY →X and DX←Y are the classical transfer bimodules. Let M and M be two stacks of twisted CX -modules, A and A two TDOrings on X. Denote by Db (A; M) the bounded derived category of ModC (A; M) = ModC (A; M)(X). The usual operations for D-modules extend to the twisted case, yielding the functors D

⊗ : Db (A; M) × Db (A ; M ) − → Db (A  A ; M C M ),

Df ∗ : Db (A; M) − → Db (f  A; f  M),

Df ∗ : Db (f  A; f  M) − → Db (A; M), D

L

defined by M ⊗ M = M ⊗ M , Df ∗ M = AY →X ⊗fL−1 A f −1 M, and Df ∗ N = O

Rf ∗ (AX←Y ⊗fL A N ). The usual formulas, like base-change or projection formula, hold. Moreover, all local notions like those of coherent module, of characteristic variety, or of regular holonomic module, still make sense. We will also consider the functor → Db (M C M ). RHom A : Db (A; M−1 )op × Db (A; M ) −

4.6 Twisted adjunction formula An adjunction formula for sheaves and D-modules in the context of Radon-type integral transforms was established in [10]. We briefly explain here how such formula generalizes to the twisted case. Note that a twisted adjunction formula for Poisson-type integral transforms was established in [26], where the group action and the topology of functional spaces are also taken into account. Let X and Y be complex analytic manifolds, M a stack of twisted CX -modules, N a stack of twisted CY -modules, A a TDO-ring on X, and B a TDO-ring on Y . We will use the notations MA and OA from Proposition 4.15. Consider the natural projections π2

π1

X ←− X × Y −→ Y, and set ∗ N = π1 M C π2 N, M




 B = π A  π B. A
1 2

505

Stacks of twisted modules and integral transforms

∗ N−1 ) one associates the functor To K ∈ Db (M
→ Db (M), K  (·) : Db (N) −

G  → Rπ1! (K ⊗ π2−1 G).

 B; M−1
∗ N) one associates the functor To K ∈ Db (A−1
D

→ Db (B; N), (·)  K : Db (A; M) −

D

M  → Dπ2∗ (Dπ1∗ M ⊗ K).

To F ∈ Db (MA C M) one associates the objects of Db (A; M) defined by  F ⊗ OA for  = ω,    w  for  = ∞, F ⊗ OA C  (F ) =

 T Hom (F , OA ) for  = −∞,    RHom (F , OA ) for  = −ω, w

where F = RHom (F, CX ), and T H om and ⊗ are the functors of formal and temperate cohomology of [17, 23]. (One checks that the construction in [23] of the functors of formal and temperate cohomology, starting from exact functors defined on the underlying real analytic manifolds, extends to the twisted case.) Hence, for  = ±∞ we have to assume that F is R-constructible.  B; M−1
∗ N), and G ∈ Db (MB C N). Let M ∈ Db (A; M), K ∈ Db (A−1
Consider the solution complex K = RH om A−1
 B (K, OA−1
 B ) of K, which is ∗ (MB C N)−1 ). an object of Db ((MA C M)
Theorem 4.19. With the above notations, assume that M is coherent, and K is regular holonomic, so that K is C-constructible. If  = ±ω, assume that π2 is proper on supp(K), and that char(K) ∩ (T ∗ X × TY∗ Y ) is contained in the zero-section of T ∗ (X × Y ). If  = ±∞, assume instead that G is R-constructible. Then, there is an isomorphism in Db (C) D

RHom A (M, C ± (K  G))[dX ]  RHom B (M  K, C ± (G)), where [dX ] denotes the shift by the complex dimension of X. We do not give here the proof, which follows the same lines as the one for the non-twisted case given in [10, 23].

References [1]

H. Bass, Algebraic K-theory. Benjamin, New York, Amsterdam 1968.

[2]

K. Behrend, Localization and Gromov-Witten invariants. In: Quantum cohomology (CIME, Cetraro, 1997), Lecture Notes in Math. 1776, Springer-Verlag, Berlin 2002, 3–38.

506

Andrea D’Agnolo and Pietro Polesello

[3]

A. Beilinson and J. Bernstein, Localisation de g-modules. C. R. Acad. Sci. Paris Sér. I Math. 292 (1) (1981), 15–18.

[4]

—, A proof of Jantzen conjectures. In: I. M. Gelfand Seminar, Adv. Soviet Math. 16 (Part 1), Amer. Math. Soc., Providence, RI, 1993, 1–50.

[5]

L. Breen, On the classification of 2-gerbes and 2-stacks. Astérisque 225 (1994).

[6]

L. Breen and W. Messing, Differential geometry of gerbes. E-print (2001), arXiv:math.AG/0106083.

[7]

J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization. Progr. Math. 107, Birkhäuser, Boston, MA, 1993.

[8]

A. H. Caldararu, Nonfine moduli spaces of sheaves on K3 surfaces. Int. Math. Res. Not. 20 (2002), 1027–1056.

[9]

A. D’Agnolo and P. Polesello, Stacks of algebroids. In preparation.

[10]

A. D’Agnolo and P. Schapira, Leray’s quantization of projective duality. Duke Math. J. 84 (2) (1996), 453–496.

[11]

R. Donagi and T. Pantev, Torus fibrations, gerbes, and duality. E-print (2003), arXiv:math.AG/0306213.

[12]

C. Faith, Algebra. I. Rings, modules, and categories. Grundlehren Math. Wissen. 190, Springer-Verlag, Berlin, New York 1981.

[13]

O. Gabber, Letter to Bogomolov (1988).

[14]

J. Giraud, Cohomologie non abélienne, Grundlehren Math. Wiss. 179, SpringerVerlag, Berlin, New York 1971.

[15]

N. J. Hitchin, Lectures on Special Lagrangian Submanifolds. Lectures given at the ICTP School on differential geometry, e-print (1999), arXiv:math.DG/ 9907034.

[16]

A. Kapustin, D-branes in a topologically nontrivial B-fields. Adv. Theor. Math. Phys. 4 (1) (2000), 127–154.

[17]

M. Kashiwara, The Riemann-Hilbert problem for holonomic systems. Publ. Res. Inst. Math. Sci. 20 (2) (1984), 319–365.

[18]

—, Representation theory and D-modules on flag varieties. In Orbites unipotentes et représentations, III, Astérisque 173-174 (1989), 55–109.

[19]

—, Algebraic study of systems of partial differential equations. Mém. Soc. Math. France (N.S.) 63 (1995), xiv+72, Kashiwara’s Master’s Thesis, Tokyo University 1970, translated from the Japanese by A. D’Agnolo and J.-P. Schneiders.

[20]

—, Quantization of contact manifolds. Publ. Res. Inst. Math. Sci. 32 (1) (1996), 1–7.

[21]

—, D-modules and microlocal calculus. Transl. Math. Monogr. 217, Amer. Math. Soc., 2003, translated from the 2000 Japanese original by M. Saito.

[22]

M. Kashiwara and P. Schapira, Sheaves on manifolds. Grundlehren Math. Wiss. 292, Springer-Verlag, Berlin 1990.

Stacks of twisted modules and integral transforms

507

[23]

—, Moderate and formal cohomology associated with constructible sheaves. Mém. Soc. Math. France 64 (1996).

[24]

—, Ind-sheaves. Astérisque 271 (2001).

[25]

—, Categories and sheaves. Preliminary version of a book in preparation (8/2002).

[26]

M. Kashiwara and W. Schmid, Equivariant derived category and representation of semisimple Lie groups. Preliminary version (1994); see the announcement: Quasiequivariant D-modules, equivariant derived category, and representations of reductive Lie groups, in Lie theory and geometry, Birkhäuser, Boston, MA, 1994, 457–488.

[27]

M. Kashiwara and T. Tanisaki, Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II. Nonintegral case. Duke Math. J. 84 (3) (1996), 771–813.

[28]

M. Kontsevich, Deformation quantization of algebraic varieties. In EuroConférence Moshé Flato, Part III (Dijon, 2000), Lett. Math. Phys. 56 (39) (2001), 271–294.

[29]

G. Laumon and L. Moret-Bailly, Champs algébriques. Ergeb. Math. Grenzgeb. 39, Springer-Verlag, Berlin 2000.

[30]

P. Polesello and P. Schapira, Stacks of quantization-deformation modules on complex symplectic manifolds. E-print (2003), to appear in Internat. Math. Res. Notices. arXiv:math.AG/0305171.

[31]

M. Murray, Bundle gerbes. J. London Math. Soc. 54 (1996), 403–416.

[32]

D. Stevenson, The geometry of bundle gerbes. Ph.D. Thesis (Adelaide Univ. 2000), e-print (2000), arXiv:math.DG/0004117.

[33]

R. Street, Categorical structures. In: Handbook of algebra, Volume 1, North-Holland, Amsterdam 1996, 529–577.

[SGA1] A. Grothendieck, Revêtements étales et groupe fondamental. With contributions by M. Raynaud. Lecture Notes in Math. 224, Springer-Verlag, Berlin 1971. [SGA4] M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. Lecture Notes in Math. 269, Springer-Verlag, Berlin 1972. Andrea D’Agnolo, Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via G. Belzoni, 7, 35131 Padova, Italy E-mail: [email protected] Pietro Polesello, Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via G. Belzoni, 7, 35131 Padova, Italy; and: Analyse Algébrique, Institut de Mathématiques, 175, rue du Chevaleret, 75013 Paris, France E-mail: [email protected]

On some rational generating series occuring in arithmetic geometry Jan Denef and François Loeser

To Bernie’s memory, with gratitude

Introduction The main purpose of the present paper is to illustrate the following motto: “rational generating series occuring in arithmetic geometry are motivic in nature”. More  in Z. We shall say precisely, consider a series F = n∈N an T n with coefficients  F is motivic in nature if there exists a series Fmot = n∈N An T n , with coefficients An in some Grothendieck ring of varieties, or some Grothendieck ring of motives, such that an is the number of rational points of An in some fixed finite field, for all n ≥ 0. Furthermore, we require Fmot to be canonically attached to F . Of course, such a definition is somewhat incomplete, since one can always take for An the disjoint union of an points. In the present paper, which is an update of a talk by the second author at the Conference “Geometric Aspects of Dwork’s theory” that took place in Bressanone in July 2001, we consider the issue of being motivic in nature for the following three types of generating series: Hasse–Weil series, Igusa series and Serre series. In Section 4, we consider the easiest case, that of Igusa type series, for which being motivic in nature follows quite easily from Kontsevich’s theory of motivic integration as developed in [7], [8]. The Serre case is more subtle. After a false start in Section 5, we explain in Section 6 how to deal with it by using the work in [9] on arithmetic motivic integration. Finally, in Section 7, we consider the case of Hasse–Weil series, which still remains very much open. Here there is a conjecture, which is due to M. Kapranov [21] and can be traced back to insights of Grothendieck cf. p. 184 of [5]. Since a “counterexample” to the conjecture recently appeared [22], we spend some time to explain the dramatic effects of inverting the class of the affine line in the Grothendieck group of varieties. This gives us the opportunity of reviewing some interesting recent work of Poonen [27], Bittner [3] and Larsen and Lunts [22], [23] and allows us to propose a precised form of Kapranov’s conjecture that escapes Larsen and Lunts’ counterexample.

510

Jan Denef and François Loeser

It was one of Bernie’s insights that most, if not all, functions occuring in Number Theory should be of geometric origin. So we hope the present contribution will not be too inadequate as an homage to his memory.

1 Conventions and preliminaries In this paper, by a variety over a ring R, we mean a reduced and separated scheme of finite type over Spec R. Let A be a commutative ring. The ring of rational formal series with coefficients in A is the smallest subring of A[[T ]] containing A[T ] and stable under taking inverses (when they exist in A[[T ]]).

2 Some classical generating series 2.1 The Hasse–Weil series Let X be a variety over Fq . We set Nn := |X(Fq n )|, for n ≥ 1. Theorem 1 (Dwork [11]). The Hasse–Weil series   N n n T Z(T ) := exp n n≥1

is rational.

2.2 The Igusa series Let K be a finite extension of Qp with ring of integers OK and uniformizing parameter π. Let X be a variety over OK . We set N˜ n := |X(OK /π n+1 )|, for n ≥ 0. Theorem 2 (Igusa [20]). The series Q(T ) :=



N˜ n T n

n≥0

is rational. Strictly speaking this result is due to Igusa [20] in the hypersurface case and to Meuser in general [25]. However, as mentioned in the review MR 83g:12015 of [25], a trick by Serre allows to deduce the general case from the hypersurface case.

On some rational generating series occuring in arithmetic geometry

511

2.3 The Serre–Oesterlé series Let K be a finite extension of Qp with ring integers OK and uniformizing parameter π. We keep the notations of 2.2. For n ≥ 0 we denote by N n the cardinality of the image of X(OK ) in X(OK /π n+1 ). In other words, N n is the number of points in X(OK /π n+1 ) (approximate solutions modulo π n+1 ) that may be lifted to points in X(OK ) (actual solutions in OK ). Clearly, N n is finite. Furthermore, when X is smooth, then N˜ n = N n for every n. Theorem 3 (Denef [6]). The series P (T ) :=



N nT n

n≥0

is rational. Remark. The problem of proving the analogue of Theorems 2 and 3 when K is a finite extension of Fq [[t]] still remains very much an open issue, but the level of difficulty seems quite different for Q(T ) or P (T ). While rationality of Q(T ) for function fields would follow using Igusa’s proof once Hironaka’s strong form of resolution of singularities is known in characteristic p, proving rationality of P (T ) for function fields would require completely new ideas, since no general quantifier elimination Theorem is known, or even conjectured, in positive characteristic.

3 Additive invariants of algebraic varieties 3.1 Additive invariants Let R be a ring. We denote by Var R the category of algebraic varieties over R. An additive invariant λ : Var R −→ S, with S a ring, assigns to any X in Var R an element λ(X) of S such that λ(X) = λ(X ) for X X , λ(X) = λ(X ) + λ(X \ X ), for X closed in X, and λ(X × X ) = λ(X)λ(X ) for every X and X .

512

Jan Denef and François Loeser

Let us remark that additive invariants λ naturally extend to take their values on constructible subsets of algebraic varieties.

3.2 Examples 3.2.1 Euler characteristic. Here  R = k is a field. When k is a subfield of C, the Euler characteristic Eu(X) := i (−1)i rkHci (X(C), C) give rise to an additive invariant Eu : Var k → Z. For general k, replacing Betti cohomology with compact support by -adic cohomology with compact support, = char k, one gets an additive invariant Eu : Var k → Z, which does not depend on . 3.2.2 Hodge polynomial. Let us assume R = k is a field of characteristic zero. Then it follows from Deligne’s Mixed Hodge Theory that there is a unique additive invariant H : Var k → Z[u, v], which assigns to a smooth projective variety X over k its usual Hodge polynomial (−1)p+q hp,q (X)up v q , H (u, v) := p,q p

with hp,q (X) = dim H q (X, X ) the (p, q)-Hodge number of X. 3.2.3 Virtual motives. More generally, when R = k is a field of characteristic zero, there exists by Gillet and Soulé [14], Guillen and Navarro-Aznar [17], a unique additive invariant χc : Var k → K0 (CHMotk ), which assigns to a smooth projective variety X over k the class of its Chow motive, where K0 (CHMotk ) denotes the Grothendieck ring of the category of Chow motives over k (with rational coefficients). 3.2.4 Counting points. Counting points also yields additive invariants. Assume k = Fq , then Nn : X → |X(Fq n )| gives rise to an additive invariant Nn : Var k → Z. Similarly, if R is (essentially) of finite type over Z, for every maximal ideal P of R with finite residue field k(P), we have an additive invariant NP : Var R → Z, which assigns to X the cardinality of (X ⊗ k(P))(k(P)).

3.3 Grothendieck rings There exists a universal additive invariant [_] : Var R → K0 (Var R ) in the sense that composition with [_] gives a bijection between ring morphisms K0 (Var R ) → S and additive invariants Var R → S. The construction of K0 (Var R ) is quite easy: take the free abelian group on isomorphism classes [X] of objects of Var R and mod out by the relation [X] = [X ] + [X \ X ] for X closed in X. The product is now defined by [X][X ] = [X ×R X ]. Here the product X ×R X is considered with its underlying reduced structure.

On some rational generating series occuring in arithmetic geometry

513

We shall denote by L the class of the affine line A1R in K0 (Var R ). An important role will be played by the ring MR := K0 (Var R )[L−1 ] obtained by localization with respect to the multiplicative set generated by L. This construction is analogous to the construction of the category of Chow motives from the category of effective Chow motives by localization with respect to the Lefschetz motive. (Remark that the morphism χc of 3.2.3 sends L to the class of the Lefschetz motive.) One should stress that very little is known about the structure of the rings K0 (Var R ) and MR even when R is a field. Let us just quote a result by Poonen [27] saying that when k is a field of characteristic zero the ring K0 (Var k ) is not a domain (we shall explain this result with more details in §7.3). For instance, even for a field k, it is not known whether the localization morphism (Var k ) → Mk is injective or not (although the whole point of §7.5 relies on the guess it should not). Remark. In fact, the ring K0 (Var k ) as well as the canonical morphism χc : K0 (Var k ) → K0 (CHMotk ), were already considered by Grothendieck in a letter to Serre dated August 16, 1964, cf. p. 174 of [5].

4 Geometrization of Q(T ) 4.1 Arcs Let k be a field. For every variety X over k, we denote by L(X) the corresponding space of arcs. It is a k-scheme, which satisfies L(X)(K) = X(K[[t]]) for every field K containg k. More precisely L(X) is defined as the inverse limit L(X) := lim Ln (X), where Ln (X) represents the functor from k-algebras to sets ←− sending a k-algebra R to X(R[[t]]/t n+1 R[[t]]). We shall always consider L(X) as endowed with its reduced structure. We shall denote by πn the canonical morphism L(X) → Ln (X).

4.2 Motivic Igusa series We consider the generating series Qgeom (T ) :=



[Ln (X)] T n

n≥0

in Mk [[T ]]. Theorem 4 (Denef–Loeser). Assume char k = 0.

514

Jan Denef and François Loeser

1) The series Qgeom (T ) in Mk [[T ]] is rational of the form R(T ) , (1 − La T b )



with R(T ) in Mk [T ], a in Z and b in N \ {0}. 2) If X is defined over some number field K, then, for almost all finite places P, NP (Qgeom (T )) = QX⊗OKP (T ). Here we should explain what we mean by NP (Qgeom (T )). For X a variety over K, NP (X) makes sense for almost all finite places P, by taking some model over OKP . Now we apply this termwise to the series Qgeom (T ). This is possible since the series is rational by 4 1). The Theorem is proved for hypersurfaces in [7] , and the general case is similar and may also be deduced from general results in [8] and [9]. Oversimplified sketch of proof of rationality. Let us first recall Igusa’s proof of Theorem 2 when X is an hypersurface defined by f = 0 in Am OK . The basic idea is to express the series Q(T ) as the integral

|f |s |dx|, I (s) := m OK

up to trivial factors, with T = q −s , q the cardinality of the residue field. Then one may use Hironaka’s resolution of singularities to reduce the computation of I (s) to the case where f = 0 is locally given by monomials for which direct calculation is easy. Our proof of the rationality of Qgeom (T ) follows similar lines. One express first our series as an integral, but here p-adic integration is replaced by motivic integration. If Y is a variety over k, motivic integration assigns to certain subsets A of the arc space L(Y ) a motivic measure µ(A) in Mk (or sometimes, but this will not be considered here, a measure in a certain completion of Mk ). Then, to be able to use Hironaka’s resolution of singularities to reduce to the locally monomial case as in Igusa’s proof, we have to use the fundamental change of variable formula established in §3 of [8].  

5 Geometrization of P (T ): I In view of the previous section, it is natural to consider now the image πn (L(X)) of L(X) in Ln (X). Thanks to Greenberg’s Theorem on solutions of polynomial systems in Henselian rings, we know that πn (L(X)) is a constructible subset of Ln (X), hence we may consider its class [πn (L(X))] in Mk . We consider the generating series [πn (L(X))] T n Pgeom (T ) := n≥0

in Mk [[T ]].

On some rational generating series occuring in arithmetic geometry

515

Theorem 5 (Denef–Loeser [8]). Assume char k = 0. The series Pgeom (T ) in Mk [[T ]] is rational of the form R(T ) , (1 − La T b )



with R(T ) in Mk [T ], a in Z and b in N \ {0}. Oversimplified sketch of proof. Let us first recall the strategy of the proof [6] of Theorem 3 in the p-adic case. One reduces to the case where X is a closed subvariety of Am OK . Then one express the series P (T ) as the integral

J (s) := d(x, X)s |dx|, m OK

up to trivial factors, with T = q −s , similarly as in Igusa’s case, where d(x, X) is the function “distance to X”. Here an essential new feature appears, the function d(x, X) being in general not a polynomial function, but only a definable or semi-algebraic function. Then one is able to use Macintyre’s quantifier elimination Theorem [24], a p-adic analogue of Tarski–Seidenberg’s theorem, to prove rationality. In the present setting our proof follows a similar pattern, replacing p-adic integration by motivic integration and the theory of p-adic semi-algebraic sets by a theory of k[[t]]-semi-algebraic sets built off from a quantifier elimination Theorem due to Pas [26].   When X is defined over a number field K, a quite natural guess would be, by analogy with what we have seen so far, that, for almost all finite places P, NP (Pgeom (T )) = PX⊗OKP (T ). But such a statement cannot hold true. This is due to the fact that, in the very definition of P (T ), one is concerned in not considering extensions of the residue field, while in the definition of Pgeom (T ) extensions of the residue field k are allowed. To remedy this, one needs to be more careful about rationality issues concerning the residue field, and for that purpose it is convenient to introduce definable subassignments as we do in the next section.

6 Geometrization of P (T ): II 6.1 Subassignments Fix a ring R. We denote by FieldR the category of R-algebras that are fields. For an R-scheme X, we denote by hX the functor which to a field K in FieldR assigns the set hX (K) := X(K). By a subassignment h ⊂ hX of hX we mean the datum, for every field K in FieldR , of a subset h(K) of hX (K). We stress that, contrarly to subfunctors, no compatibility is required between the various sets h(K).

516

Jan Denef and François Loeser

All set theoretic constructions generalize in an obvious way to the case of subassignments. For instance if h and h are subassignments of hX , then we denote by h ∩ h the subassignment K → h(K) ∩ h (K), etc. Also, if π : X → Y is a morphism of R-schemes and h is a subassignment of hX , we define the subassignment π(h) of hY by π(h)(K) := π(h(K)) ⊂ hY (K).

6.2 Definable subassignments Let R be a ring. By a ring formula ϕ over R, we mean a first order formula in the language of rings with coefficients in R and free variables x1 , . . . xn . In other words ϕ is built out from boolean combinations (“and”, “or”, “not”) of polynomial equations over R and existential and universal quantifiers. For example (∃x)(x 2 + x + y = 0 and 4y = 1) is a ring formula over Z with free variable y. To a ring formula ϕ over R with free variables x1 , . . . xn one assigns the subassignment hϕ of hAnR defined by   hϕ (K) := (a1 , . . . , an ) ∈ K n | ϕ(a1 , . . . , an ) holds in K ⊂ K n = hAnR (K). (6.1) Such a subassignment of hAnR is called a definable subassignment. More generally, using affine coverings, cf. [9], one defines definable subassignments of hX for X a variety over R. It is quite easy to show that if π : X → Y is an R-morphism of finite presentation, π(h) is a definable subassignment of hY if h is a definable subassignment of hX . In our situation, we are concerned with the subassignment π(hL(X) ) ⊂ hLn (X) . Remark that πn : L(X) → Ln (X) is not of finite type. Nevertheless, we have the following: Proposition 1 ([9]). π(hL(X) ) is a definable subassignment of hLn (X) .

6.3 Formulas and motives Let k be a field of characteristic zero. It follows from 3.2.3 that we have a canonical morphism χc : K0 (Var k ) → K0 (CHMotk ). We shall denote by K0mot (Var k ) the image of K0 (Var k ) in K0 (CHMotk ) under this morphism. Remark that the image of L in K0mot (Var k ) is not a zero divisor since it is invertible in K0 (CHMotk ). Let us explain now how to assign in a canonical way to a ring formula ϕ over k an element χc ([ϕ]) of K0mot (Var k ) ⊗ Q. Let ϕ be a formula over a number field K. For almost all finite places P with residue field k(P), one may extend the definition in (6.1) to give a meaning to hϕ (k(P)). If

On some rational generating series occuring in arithmetic geometry

517

ϕ and ϕ are formulas over K, we set ϕ ≡ ϕ if hϕ (k(P)) = hϕ (k(P)) for almost all finite places P. It follows from a fundamental result of J. Ax [2] that ϕ ≡ ϕ if and only if hϕ (L) = hϕ (L ) for every pseudo-finite field L containing K. Let us recall that a pseudo-finite field is an infinite perfect field that has exactly one field extension of any given finite degree, and over which every geometrically irreducible variety has a rational point. Historically, the above result of Ax was one of the main motivation for introducing that notion. One way of constructing pseudo-finite fields is by taking infinite ultraproducts of finite fields. Let us now introduce the Grothendieck ring of formulas over R, K0 (FieldR ), and K0 (PFFR ) the Grothendieck ring of the theory of pseudo-finite fields over R. The ring K0 (FieldR ) (resp. K0 (PFFR )) is the group generated by symbols [ϕ], where ϕ is any ring formula over R, subject to the relations [ϕ1 or ϕ2 ] = [ϕ1 ] + [ϕ2 ] − [ϕ1 and ϕ2 ], whenever ϕ1 and ϕ2 have the same free variables, and the relations [ϕ1 ] = [ϕ2 ], whenever there exists a ring formula ψ over k that, when interpreted in any field (resp. any pseudo-finite field) K in FieldR , yields the graph of a bijection between the tuples of elements of K satisfying ϕ1 and those satisfying ϕ2 . The ring multiplication is induced by the conjunction of formulas in disjoint sets of variables. There is a canonical morphism K0 (FieldR ) −→ K0 (PFFR ). We can now state the following: Theorem 6 (Denef–Loeser [9],[10]). Let k be a field of characteristic zero. There exists a unique ring morphism χc : K0 (PFFk ) −→ K0mot (Var k ) ⊗ Q satisfying the following two properties: (i) For any formula ϕ which is a conjunction of polynomial equations over k, the element χc ([ϕ]) equals the class in K0mot (Var k ) ⊗ Q of the variety defined by ϕ. (ii) Let X be a normal affine irreducible variety over k, Y an unramified Galois cover 1 of X, and C a cyclic subgroup of the Galois group G of Y over X. For such data we denote by ϕY,X,C a ring formula, whose interpretation in any field K containing k, is the set of K-rational points on X that lift to a geometric point on Y with decomposition group C (i.e. the set of points on X that lift to a K-rational point of Y/C, but not to any K-rational point of Y/C with C a proper subgroup of C). Then χc ([ϕY,X,C ]) =

|C| χc ([ϕY,Y /C,C ]), |NG (C)|

1 Meaning that Y is an integral étale scheme over X with Y /G ∼ X, where G is the group of all = endomorphisms of Y over X.

518

Jan Denef and François Loeser

where NG (C) is the normalizer of C in G. Moreover, when k is a number field, for almost all finite places P, NP (χc ([ϕ])) is equal to the cardinality of hϕ (k(P)). The above theorem is a variant of results in §3.4 of [9]. A sketch of proof is given in [10]. Some ingredients in the proof. Uniqueness uses quantifier elimination for pseudofinite fields (in terms of Galois stratifications, cf. the work of Fried and Sacerdote [13], [12, §26]), from which it follows that K0 (PFFk ) is generated as a group by classes of formulas of the form ϕY,X,C . Thus by (ii) we only have to determine χc ([ϕY,Y /C,C ]), with C a cyclic group. But this follows directly from the following recursion formula: |C| [Y/C] = |A|χc ([ϕY,Y /A,A ]). (6.2) A subgroup of C

This recursion formula is a direct consequence of (i), (ii), and the fact that the formulas ϕY,Y /C,A yield a partition of Y/C. The proof of existence is based on work of del Baño Rollin and Navarro Aznar [4] who associate to any representation over Q of a finite group G acting freely on an affine variety Y over k, an element in the Grothendieck group of Chow motives over k. By linearity, we can hence associate to any Q-central function α on G (i.e. a Qlinear combination of characters of representations of G over Q), an element χc (Y, α) of that Grothendieck group tensored with Q. Using Emil Artin’s Theorem, that any Q-central function α on G is a Q-linear combination of characters induced by trivial representations of cyclic subgroups, one shows that χc (Y, α) ∈ K0mot (Var k ) ⊗ Q. For X := Y /G and C any cyclic subgroup of G, we define χc ([ϕY,X,C ]) := χc (Y, θ ), where θ sends g ∈ G to 1 if the subgroup generated by g is conjugate to C, and else to 0. With some more work we prove that the above definition of χc ([ϕY,X,C ]) extends   by additivity to a well-defined map χc : K0 (PFFk ) −→ K0mot (Var k ) ⊗ Q. Clearly χc (ϕ) depends only on hϕ and the construction easily extends by additivity to definable subassignments of hX , for any variety X over k. So, to any such definable subassignment h, we may associate χc (h) in K0mot (Var k ) ⊗ Q. Proposition 2 (Denef–Loeser). Let k be a field of characteristic zero. For any definable subassignment h, Eu(χc (h)) belongs to Z. Proof. It is enough to show that Eu(χc (ϕY,X,C )) belongs to Z for every Y , X and C. Consider first the case C is the trivial subgroup e of G. We have χc (ϕY,X,e ) =

1 1 χc (ϕY,Y,e ) = [Y ]. |G| |G|

It follows that Eu(χc (ϕY,X,e )) =

1 Eu(Y ) = Eu(X) ∈ Z. |G|

On some rational generating series occuring in arithmetic geometry

519

When C is a non trivial cyclic subgroup of G, by induction on |C|, it follows from the   recursion formula (6.2) that Eu(χc (ϕY,X,C )) = 0. Let us give an easy example. Let n be a integer ≥ 1 and assume k contains all n-roots of unity. Consider the formula ϕn : (∃y)(x = y n and x = 0) ; then uv−1 χc (ϕn ) = L−1 n . In particular Eu(χc (ϕn )) = 0 and H (χc (ϕn )) = n . This example contradicts the example on page 430 line -2 of [9] (page 3 line 4 in the preprint) which is unfortunately incorrect. Remark. It is the place to correct the following errors in the published version of [10]. On line 18 of the third page, after the word “motives” one has to insert “, and by killing all L-torsion”. Once this correction is made, it is easily checked that K0mot (Var k ) becomes the same in the present paper and in [10]. On line 6 of the eighth page, one has to delete the last sentence.

6.4 The series Par We now consider the series Par (T ) :=



χc (πn (hL(X) )) T n

n≥0

in K0mot (Var k ) ⊗ Q. Theorem 7 (Denef–Loeser [9]). Assume char k = 0. 1) The series Par (T ) in K0mot (Var k ) ⊗ Q is rational of the form R(T ) , (1 − La T b )



with R(T ) in (K0mot (Var k ) ⊗ Q)[T ], a in Z and b in N \ {0}. 2) If X is defined over some number field K, then, for almost all finite places P, NP (Par (T )) = PX⊗OKP (T ). In the proof of Theorem 7, one uses in an essential way arithmetic motivic integration a variant of motivic integration developed in [9]. The specialization statement 2) in Theorem 7 is a special case of the following results, which states that “natural p-adic integrals are motivic”. Theorem 8 (Denef–Loeser [9]). Let K be a number field. Let ϕ be a first order formula in the language of valued rings with coefficients in K and free variables x1 , . . . , xn . Let f be a polynomial in K[x1 , . . . , xn ]. For P a finite place of K, denote

520

Jan Denef and François Loeser

by KP the completion of K at P. Then there exists a canonical motivic integral which specializes to

|f |sP |dx|P hϕ (KP )

for almost all finite places P. The formulation here is somewhat unprecise and we refer to [9] for details. Let us just say that formulas in the language of valued rings include expressions like orda ≤ ordb or ordc ≡ b mod e. Theorem 8 applies in particular to integrals occuring in p-adic harmonic analysis, like orbital integrals. This has led recently Tom Hales [18] to propose that many of the basic objects in representation theory should be motivic in nature and to develop a beautiful conjectural program aiming to the determination of the virtual Chow motives that should control the behavior of orbital integrals and leading to a motivic fundamental lemma (see [15] and [19] for recent progress on these questions).

7 Geometrization of Z(T ) 7.1 The Kapranov series Let k be a field and let X be a variety over k. For n ≥ 0, we denote bt X (n) the n-fold symmetric product of X, i.e. the quotient of the cartesian product Xn by the symmetric group of n elements. Note that X (0) is isomorphic to Spec k. Following Kapranov [21], we define the motivic zeta function of X as the power series ∞ [X(n) ] T n Zmot (T ) := n=0

in K0 (Var k )[[T ]]. Also, when α : K0 (Var k ) → A is a morphism of rings, we denote by Zmot,α (T ) (n) n the power series ∞ n=0 α([X ]) T in A[[T ]]. We shall write L for α(L). Proposition 3. If k = Fq , and we write N (S) = N1 (S) = |S(k)| for S a variety over k, then Zmot,N (T ) is equal to the Hasse–Weil zeta function considered in 2.1. Proof. Rational points of X(n) over k correspond to degree n effective zero cycles of X, hence the result follows from the usual inversion formula between the number of effective zero cycles of given degree on X and the number of rational points of X over finite extensions of k.   In his paper [21], Kapranov proves the following result:

On some rational generating series occuring in arithmetic geometry

521

Theorem 9 (Kapranov [21]). Let X be a smooth projective irreducible curve of genus g. Let α : K0 (Var k ) → A be a morphism of rings with A a field, such that L is non zero in A. Assume also there exists a degree 1 line bundle on X. Then: 1) The series Zmot,α (T ) is rational. It is the quotient of a polynomial of degree 2g by (1 − T )(1 − LT ). 2) The function Zmot,α (T ) satisfies the functional equation Zmot,α (L−1 T −1 ) = L1−g T 2−2g Zmot,α (T ). The proof follows the lines of F. K. Schmidt’s classical proof [28] of rationality and functional equation for the Hasse–Weil zeta function of a smooth projective curve. Remark. In fact, in the curve case, the rationality of Zmot holds already in K0 (Var k )[[T ]], cf. [23]. In the same paper, Kapranov states “it is natural to expect that the motivic zeta functions are rational and satisfy similar functional equations?”. Remark. A generating series similar to Zgeom and the question of its rationality were already considered by Grothendieck in a letter to Serre dated September 24, 1964, cf. p. 184 of [5].

7.2 Stable birational invariants We now give a new presentation by generators and relations of K0 (Var k ) due to F. Bittner [3]. We denote by K0bl (Var k ) the quotient of the free abelian group on isomorphism classes of irreducible smooth projective varieties over k by the relations [BlY X] − [E] = [X] − [Y ], for Y and X irreducible smooth projective over k, Y closed in X, BlY X the blowup of X with center Y and E the exceptional divisor in BlY X. As for K0 (Var k ), cartesian product induces a product on K0bl (Var k ) which endowes it with a ring structure. There is a canonical ring morphism K0bl (Var k ) → K0 (Var k ), which sends [X] to [X]. Theorem 10 (Bittner [3]). Assume k is of characteristic zero. The canonical ring morphism K0bl (Var k ) → K0 (Var k ) is an isomorphism. The proof is based on Hironaka resolution of singularities and the weak factorization Theorem of Abramovich, Karu, Matsuki and Włodarczyk [1]. One deduces easily the following result, first proved by Larsen and Lunts [22].

522

Jan Denef and François Loeser

Theorem 11 (Larsen and Lunts [22]). Let us assume k is algebraically closed of characteristic zero. Let A be the monoid of isomorphism classes of smooth projective irreducible varieties over k and let : A → G be a morphism of commutative monoids such that 1) If X and Y are birationally equivalent smooth projective irreducible varieties over k, then ([X]) = ([Y ]). 2) ([P nk ]) = 1. Then there exists a unique morphism a rings  : K0 (Var k ) :−→ Z[G] such that ([X]) = ([X]) when X is smooth projective irreducible. We assume from now on that k is algebraically closed of characteristic zero. We denote by SB the monoid of equivalence classes of smooth projective irreducible varieties over k under stably birational equivalence2 . It follows from Corollary 11 that there exists a universal stable birational invariant SB : K0 (Var k ) :−→ Z[SB]. Proposition 4 (Larsen and Lunts [22]). The kernel of the morphism SB : K0 (Var k ) :→ Z[SB] is the principal ideal generated by L = [A1k ]. Sketch of proof. It is clear that L lies in the kernel of SB . Conversely, take α =  [X ] − i 1≤i≤r 1≤j ≤s [Y SB , with Xi and Yj smooth, projective j ] in the kernel of and irreducible. Since 1≤i≤r [Xi ] = 1≤j ≤s [Yj ] in Z[SB], r = s and, after renumbering the Xi ’s, we may assume Xi is stably birational to Yi for every i. Hence it is enough to show that if X and Y are smooth, projective and irreducible stably birationally equivalent, then [X] − [Y ] belongs to LK0 (Var k ). Since [X] − [P rk × X] belongs to LK0 (Var k ), we can even assume X and Y are birationally equivalent and then the result follows easily from the weak factorization Theorem.  

7.3 Back to Poonen’s result As promised, we shall now give some explanations concerning the proof of Poonen’s Theorem 12. Key-Lemma 7.3.1 (Poonen [27]). Let k be a field of characteristic zero. There exists abelian varieties A and B over k such that A×A is isomorphic to B ×B but Ak ∼ = Bk . 2 X and Y are called stably birational if X × P r is birational to Y × P s for some r, s ≥ 0. k k

On some rational generating series occuring in arithmetic geometry

523

The proof relies on the following lemma: Lemma 7.3.2 (Poonen [27]). Let k be a field of characteristic zero. There exists an abelian variety A over k such that Endk (A) = Endk (A) ∼ = O, with O the ring of integers of a number field of class number 2. √When k = C, one may take A an elliptic curve with complex multiplication by Z[ −5]. The general case is much more involved and necessitates the use of modular forms and Eichler–Shimura Theory as well as some table checking, see [27]. Let us now explain how Poonen deduces from the Key-Lemma the following: Theorem 12. The ring K0 (Var k ) is not a domain, for k a field of characteristic zero. Proof. Take A and B as in the Key-Lemma. We have ([A] + [B])([A] − [B]) = 0 in K0 (Var k ). To check that [A] + [B] and [A] − [B] are nonzero in K0 (Var k ), it is enough to check that they have a nonzero image under the composition K0 (Var k ) → K0 (Var k ) → Z[SBk ] → Z[AVk ], where AVk is the monoid of isomorphism classes of abelian varities over k and the last morphism is induced by the Albanese functor assigning to a smooth irreducible variety its Albanese variety (which is indeed a stable birational invariant). To conclude we just have to remark that the Albanese variety of an abelian variety is equal to itself.   Remark. Poonen’s proof does not tell us anything about zero divisors in Mk . Indeed, it relies on the use of stable birational invariants, and after inverting L no (non trivial) such invariant is left.

7.4 Non rationality results Larsen and Lunts proved [22] the following non rationality Theorem: Theorem 13 (Larsen–Lunts). Let X be a smooth projective complex irreducible surface with geometric genus pg (X) ≥ 2. Then there exists a morphism α : K0 (Var k ) → F , with F a field, such that the zeta function Zmot,α attached to X is not rational. Some ideas from the proof. Larsen and Lunts consider, for X a smooth  projective complex irreducible variety of dimension d, the polynomial h (X) := 1≤i≤d hi,0 t i . Remark pg (X) is the leading coefficient of h (X). It is well known h is a stable birational invariant, hence by Corollary 11 it gives rise to a ring morphism h : K0 (Var k ) −→ Z[C], with C the multiplicative monoid of polynomials in Z[t] with positive leading coefficient. Larsen and Lunts show that Z[C] is a domain and take for α the composition

524

Jan Denef and François Loeser

of h with the localization morphism from Z[C] to its fraction field F . Note that h (and also α) factors through K0 (Var k )/LK0 (Var k ), since h is a stable birational invariant. A key ingredient in the proof is the fact surfaces
that, for smooth projective (n) ) is by defini(X . Here p with geometrical genus r ≥ 2, pg (X(n) ) = r+n−1 g r−1 tion the geometric genus of any smooth projective variety birational to X (n) . The Hilbert scheme X [n] parametrizing closed zero-dimensional subschemes of length n of X is such
a variety  and it follows from results of Göttsche and Soergel [16] that pg (X[n] ) = r+n−1 r−1 . Some more work is needed in order to deduce non rationality   for the series Zmot,α . Since the first version of the present paper was written, Larsen and Lunts wrote a very interesting sequel [23] of [22]. One of the main result in [23] is the following improvement of Theorem 13: Theorem 14. Let X be a complex surface. Then Zmot (T ) is rational in K0 (Var k )[[T ]] if and only if X has Kodaira dimension −∞. For proving this result,  Larsen and Lunts consider instead of just h (X), the whole family of invariants 1≤i≤d dim H 0 (X n iX )t i , where n denotes the n-th Adams operation.

7.5 Rationality conjectures In view of Theorems 13 and 14, one cannot hope for the series Zmot to be in general rational in K0 (Var k )[[T ]]. Nevertheless, since all invariants used in [22] and [23] to prove non rationality factor through K0 (Var k )/LK0 (Var k ), they do not survive in Mk , so we can still believe the following to be true: Rationality conjecture (strong form) 7.5.1. Let X be a variety over a field k. Then the series Zmot attached to X is rational in Mk [[T ]]. Rationality conjecture (weak form) 7.5.2. Let X be a variety over a field k. Then, for every morphism α : Mk → F , with F a field, the series Zmot,α attached to X is rational in F [[T ]]. Remarks. 1) A posteriori it is not so surprising that we have to invert L in order for rationality to conjecturally hold. Indeed, the guess that the motivic series should be rational comes from analogy with Dwork’s Theorem 1. But counting rational points is certainly not a birational invariant! 2) When X is smooth and proper, one can conjecture strong and weak forms of functional equations.

On some rational generating series occuring in arithmetic geometry

525

References [1]

D. Abramovich, K. Karu, K. Matsuki, J. Włodarczyk, Torification and factorization of birational maps. J. Amer. Math. Soc. 15 (2002), 531–572.

[2]

J. Ax, The elementary theory of finite fields. Ann. of Math. 88 (1968), 239–271.

[3]

F. Bittner, The universal Euler characteristic for varieties of characteristic zero. math.AG/0111062, to appear in Compositio Math.

[4]

S. del Baño Rollin, V. Navarro Aznar, On the motive of a quotient variety. Collect. Math. 49 (1998), 203–226.

[5]

P. Colmez, J.-P. Serre (editors), Correspondance Grothendieck-Serre. Documents Mathématiques 2, Société Mathématique de France, Paris 2001.

[6]

J. Denef, The rationality of the Poincaré series associated to the p-adic points on a variety. Invent. Math. 77 (1984), 1–23.

[7]

J. Denef, F. Loeser, Motivic Igusa zeta functions. J. Algebraic Geom. 7 (1998), 505–537.

[8]

J. Denef, F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135 (1999), 201–232.

[9]

J. Denef, F. Loeser, Definable sets, motives and p-adic integrals. J. Amer. Math. Soc. 14 (2001), 429–469.

[10]

J. Denef, F. Loeser, Motivic Integration and the Grothendieck Group of Pseudo-Finite Fields. In Proceedings of the International Congress of Mathematicians, Beijing 2002, Volume 2, Higher Education Press, Beijing 2002, 13–23.

[11]

B. Dwork, On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82 (1960), 631–648.

[12]

M. Fried, M. Jarden, Field arithmetic. Ergeb. Math. Grenzgeb. (3) 11, Springer-Verlag, Berlin 1986.

[13]

M. Fried, G. Sacerdote, Solving diophantine problems over all residue class fields of a number field and all finite fields. Ann. Math. 100 (1976), 203–233.

[14]

H. Gillet, C. Soulé, Descent, motives and K-theory. J. Reine Angew. Math. 478 (1996), 127–176.

[15]

J. Gordon, T. Hales, Virtual Transfer Factors. math.RT/0209001.

[16]

L. Göttsche, W. Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces. Math. Ann. 296 (1993), 235–245.

[17]

F. Guillén,V. NavarroAznar, Un critère d’extension d’un foncteur défini sur les schémas lisses. Inst. Hautes Études Sci. Publ. Math. 95 (2002), 1–91.

[18]

T. Hales, Can p-adic integrals be computed? math.RT/0205207.

[19]

T. Hales, Orbital Integrals are Motivic. math.RT/0212236.

[20]

J. Igusa, Complex powers and asymptotic expansions. II. Asymptotic expansions. J. Reine Angew. Math. 278/279 (1975), 307–321.

526

Jan Denef and François Loeser

[21]

M. Kapranov, The elliptic curve in the S-duality theory and Eisenstein series for KacMoody groups. math.AG/0001005.

[22]

M. Larsen, V. Lunts, Motivic measures and stable birational geometry. math.AG/ 0110255.

[23]

M. Larsen, V. Lunts, Rationality criteria for motivic zeta-functions. math.AG/0212158.

[24]

A. Macintyre, On definable subsets of p-adic fields. J. Symbolic Logic 41 (1976), 605–610.

[25]

D. Meuser, On the rationality of certain generating functions. Math. Ann. 256 (1981), 303–310.

[26]

J. Pas, Uniform p-adic cell decomposition and local zeta functions. J. Reine Angew. Math. 399 (1989), 137–172.

[27]

B. Poonen, The Grothendieck ring of varieties is not a domain. math.AG/0204306.

[28]

F. K. Schmidt, Analytische Zahlentheorie in Körpern der Charakteristik p. Math. Z. 33 (1931), 1–32.

Jan Denef, University of Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium E-mail: [email protected] Home page: http://www.wis.kuleuven.ac.be/wis/algebra/denef.html François Loeser, École Normale Supérieure, Département de mathématiques et applications, UMR 8553 du CNRS, 45 rue d’Ulm, 75230 Paris Cedex 05, France E-mail: [email protected] Home page: http://www.dma.ens.fr/˜loeser

Compactifications arithmétiques des variétés de Hilbert et formes modulaires de Hilbert pour 1 (c, n) Mladen Dimitrov

Soit F un corps de nombres totalement réel de degré dF , d’anneau des entiers o, de différente d et de discriminant F = NF/Q (d). On abrégera N = NF/Q . On se donne un groupe algébrique D/ Q , intermédiaire entre Gm et ResFQ Gm , ∗ connexe : Gm → D → ResFQ Gm . On définit le groupe algébrique GD / Q (resp. G/ Q ) comme le produit fibré de D (resp. Gm ) et de ResFQ GL2 au-dessus de ResFQ Gm . On a le diagramme cartésien suivant :    / G∗  / GD  / ResF GL2 ResFQ SL2  Q  1

 / Gm  

  / D

ν / ResF Gm , Q

où la flèche ν : ResFQ GL2 → ResFQ Gm est donnée par la norme réduite. Le sous-groupe de Borel standard de GD , son radical unipotent et son tore maximal standard sont notés B, U et T , respectivement. On pose T1 = T ∩ ker(ν). Pour toute Q-algèbre R et pour tout groupe algébrique H sur Q, on note HR le groupe de ses R-points. Soit n un idéal de o premier à F et ne divisant ni 2, ni 3 et soit c un idéal fractionnaire de F , que l’on peut supposer premier à n.Alors le groupe de congruences  = 1D (c, n), défini dans la partie 3, est sans torsion et l’espace de modules de variétés abéliennes 1 ]-schéma, lisse de Hilbert–Blumenthal correspondant M = M1D (c, n) est un Z[ N(n) 1 au-dessus de Z[  ], où  = N(dn) (voir la partie 4 pour une définition précise de l’espace de modules M). Cet article décrit les compactifications arithmétiques de M et donne quelques unes de leurs propriétés. Les principales références sont les articles [11] de M. Rapoport et [2] de C.-L. Chai, où les compactifications toroïdales et minimale sont construites pour le sous-groupe de congruence principal de niveau N(n), lorsque D = Gm . Par ailleurs, Rapoport explique comment on peut obtenir une compactification partielle de M aux pointes non-ramifiées. La contribution principale de ce travail est qu’il fournit les cartes locales servant à compactifier les pointes ramifiées. Une application immédiate est le “principe du q-développement” en ces pointes ramifiées.

528

Mladen Dimitrov

Les résultats de cet article sont utilisés dans un article commun avec J. Tilouine [6], où figurent aussi différentes applications aux formes modulaires de Hilbert. En vue de ces applications, il est important de disposer de compactifications toroïdales lisses de M, puisque l’on sait prolonger les fibrés automorphes à celles-ci. Le groupe auxiliaire D nous permet de traiter simultanément le cas du groupe modulaire de Hilbert et celui de sa version étendue, qui sont d’égale importance et correspondent à D = Gm et D = ResFQ Gm , respectivement (voir [1]). Je remercie tous ceux qui m’ont consacré du temps pour discuter, et en particulier Y. Henrio, qui a eu la gentillesse de m’expliquer le théorème de descente formelle de Rapoport, ainsi que A. Abbès, D. Barsky, G. Chenevier, H. Hida, A. Mokrane, M. Raynaud et E. Urban. Je voudrais exprimer toute ma gratitude à J. Tilouine parce qu’il m’a initié à ce sujet de recherche passionnant et constamment encouragé au cours de la préparation de ce travail. Enfin, je remercie les rapporteurs pour leurs remarques intéressantes. Nous rappelons d’abord brièvement la construction générale de variétés semi-abéliennes, donnée par D. Mumford dans le cas totalement dégénéré [10]. Nous introduisons ensuite la notion de (R, n)-pointe, version algébrique de la -pointe. Cela nous permet de construire, en suivant [11], les cartes locales, qui seront utilisées pour les compactifications toroïdales arithmétiques.

1 La construction de Mumford Soit R un anneau excellent, intégralement √ clos, noethérien, complet pour la topologie I -adique, pour un idéal radiciel I = I . Soit K le corps des fractions de R. Soit S = Spec(R), η son point générique et S0 = Spec(R/I ) le sous-schéma fermé défini par I . Définition 1.1. Un S-schéma en groupes commutatif, lisse et de type fini G est dit semi-abélien, si ses fibres géométriques sont des extensions d’une variété abélienne par un tore.  = Grm × S de rang r sur S. Soit b un sous-groupe Considérons le tore déployé G η . L’objet de cette section est d’esquisser la construction d’un discret polarisable de G  par b. La stratégie est la suivante : schéma semi-abélien G/S, comme “quotient” de G  → P  telle que l’action de b s’étende à P  (i) Construire une “compactification” G  × S0 (pour la topologie de Zariski). et que b agisse librement et discontinument sur P S

(ii) Suivre les flèches du diagramme :  G  G



ouvert

/P o

complétion

 P

quotient formel par b ouvert

/P o

algébrisation

 P .

Compactifications arithmétiques des variétés de Hilbert

529

, que (iii) Enfin, montrer que G est semi-abélien sur S, indépendant du choix de P r  Gη est abélien, et que G0 = G0 = Gm × S0 .  Pour α ∈ a, Périodes et polarisation. Soit a = Zr le groupe des caractères de G. α 0  notons X ∈ H (G, OG  ) le caractère associé. Alors de manière canonique :  = Spec(R[Xα ; α ∈ a]). G η isomorphe à Zr . Définition 1.2. Un ensemble de périodes est un sous-groupe b ⊂ G Définition 1.3. Une polarisation pour b est un homomorphisme φ : b → a tel que :  (i) Xφ(β) (β  ) = Xφ(β ) (β), pour tout β, β  ∈ b, (ii) Xφ(β) (β) ∈ I , pour tout β ∈ b \{0}. Lemme 1.4. Pour tout α ∈ a, il existe un entier n ≥ 1 avec Xnφ(β)+α (β) ∈ R pour tout β ∈ b. η muni Modèles relativement complets. Étant donné un ensemble de périodes b ⊂ G d’une polarisation φ, Mumford donne la  par rapport à (b, φ), est la donné Définition 1.5. Un modèle relativement complet de G, des éléments suivants : , localement de type fini sur R, (a) Un schéma intègre P  → P , (b) Une immersion ouverte i : G ,  sur P (c) Un faisceau inversible L  sur P  et L, → P  et Sg∗ : L  notée Sg : P  → L,  pour tout (d) Une action du tore G  point fonctoriel g de G,  et L, → P  et T ∗ : L  notée Tβ : P  → L,  pour tout (e) Une action de b sur P β β ∈ b, satisfaisant aux conditions suivantes :     (i) Il existe un ouvert G-invariant U ⊂ P de type fini sur S et tel que P = β∈b Tβ (U ).  et qui est (ii) Pour toute valuation v sur le corps des fonctions rationnelles sur G positive sur R, on a :  ⇐⇒ pour tout α ∈ a, il existe β ∈ b avec v(Xα (β)Xα ) ≥ 0. v a du centre sur P  et b sur P  prolongent leurs actions par translation sur G η . (iii) Les actions de G  et b sur L  vérifient la condition de compatibilité suivante : (iv) Les actions de G  Sg∗ Tβ∗ = Xφ(β) (g)Tβ∗ Sg∗ , pour tout β ∈ b et tout point fonctoriel g de G. , au sens que les compléments des lieux des zéros des sections  est ample sur P (v) L 0 ⊗n  .  globales H (P , L ), n ≥ 1, forment une base de la topologie de Zariski de P

530

Mladen Dimitrov

Définition 1.6. Une étoile  de a est un sous-ensemble fini de a tel que 0 ∈ ,  = − et  contient une base de a. Soit l’anneau gradué : R =

∞

k=0 K[X

α; α

∈ a] · θ k .

 ∗  Tβ (c) = c, pour c ∈ K, On définit une action du groupe b sur R par : Tβ∗ (Xα ) = Xα (β)Xα , pour α ∈ a,   ∗ Tβ (θ ) = Xφ(β) (β)X2φ(β) θ. Définition 1.7. Soit  une étoile de a ; on note Rφ, le sous anneau de R engendré sur R par les éléments Tβ∗ (Xα θ) pour β ∈ b et α ∈ , i.e. : Rφ, = R[Xφ(β)+α (β)X2φ(β)+α θ ]β∈b,α∈ . D’après le lemme 1.4 on peut supposer, quitte à remplacer φ par nφ, que Rφ, ⊂ R[Xα θ ]α∈a .  On montre alors que Proj(Rφ, ) est un modèle relativement complet pour G. Comme Rφ, est un anneau gradué engendré par ses éléments de degré 1, Proj(Rφ, ) est muni d’un faisceau très ample inversible canonique, qui est le O(1). On obtient ainsi le :  un tore déployé sur S, b ⊂ G η un groupe Théorème 1.8 (Mumford [10]). Soit G de périodes et φ : b → a une polarisation. Alors, pour toute étoile  de a, quitte à  = Proj(Rφ, ), muni de son faisceau canonique remplacer φ par nφ (n ∈ Z, n 0), P  sur S, par rapport à (b, 2φ). O(1), est un modèle relativement complet pour G η = P η . On remarque que G La construction du quotient procède en deux temps : Mumford forme d’abord le  le long du bord, par b. Ce quotient est un schéma quotient P du complété formel de P formel projectif et de type fini, donc s’algébrise en un schéma projectif de type fini noté P .    ⊂ P . Soit B  = P −  Considérons l’ouvert β∈b Tβ (G) β∈b Tβ (G) le sousschéma réduit, et B le quotient par b de son complété formel. C’est la complétion formelle d’un sous-schéma réduit B ⊂ P . Posons G = P \ B. Par construction les  sont canoniquement isomorphes. complétions I -adiques de G et G Théorème 1.9 (Mumford [10]). Le schéma G/S est semi-abélien, Gη est une variété abélienne et G0 est un tore déployé de rang r. Le schéma G/S ne dépend que du tore  et du groupe de périodes b, et il est indépendant de la fonction de polarisation φ et G . La construction de G/S est fonctorielle en G/S  et du modèle relativement complet P en b.

Compactifications arithmétiques des variétés de Hilbert

531

2 Construction de VAHB dégénérantes On applique la construction de Mumford pour construire des variétés abéliennes de Hilbert–Blumenthal dégénérantes. Afin d’éviter des répétitions avec la partie 2 de [6], nous n’allons donner la définition d’une variété abélienne de Hilbert–Blumenthal que dans le cas où le discriminant F du corps F est inversible. Définition 2.1. Une variété abélienne de Hilbert–Blumenthal (abrégé VAHB) sur un Z[ 1F ]-schéma S est la donnée d’un schéma abélien f : A → S de dimension relative dF et d’une injection ι : o → End(A/S) tels que le faisceau ω = f∗ 1A/S soit localement libre de rang 1 sur o ⊗ OS , pour la topologie de Zariski. Pour tout idéal fractionnaire f de F on pose f∗ = f−1 d−1 . On a un accouplement parfait Tr F/Q : f × f∗ → Z. Soit X un idéal fractionnaire de F , muni de sa positivité X+ = X ∩ (F ⊗ R)+ . L’anneau de base S σ . Soit R = Z[q ξ ; ξ ∈ X]. Soit S = Spec(R) = Gm ⊗ X ∗ le tore de groupe de caractères X. ∗ et soit S → S , l’immersion torique Soit  un éventail complet lisse de X+  associée. On rappelle qu’elle est obtenue en recollant, pour σ ∈ , les immersions toriques affines S → Sσ = Spec(Rσ ), où Rσ = Z[q ξ ; ξ ∈ X∩ σˇ ]. Soit Sσ∧ le complété ∧ le complété de S le long de S ∞ := S \S. de Sσ le long de Sσ∞ := Sσ \S et S    Pour écrire les choses plus explicitement, donnons nous une base ξ1∗ ,. . .,ξr∗ de σ que l’on complète en une base ξ1∗ ,. . .,ξd∗ de X∗ . Soit ξ1 ,. . .,ξd la base duale de X et posons ± Zd± ] et Sσ∞ est le diviseur à croisements Zi = q ξi . Alors Rσ = Z[Z1 , . . . , Zr , Zr+1 normaux de Sσ défini par l’équation Z1 . . . Zr = 0. On a Sσ∧ = Spf(Rσ∧ ), où Rσ∧ est le complété de Rσ en l’idéal principal radiciel (Z1 . . . Zr ). d   Pour décrire ce complété, on décompose tout n = (n 1 , . . . , nd ) ∈ Z en (n , n ) ∈ nd n1 r d−r Z × Z . Disons qu’une série de Laurent formelle n∈Zd cn Z1 . . . Zd à coefficients cn ∈ Z est (Z1 . . . Zr )-entière si (i) pour tout n , cn ,n = 0, si n  ∈ Nr , / [H, ∞[r × Zd−r . (ii) pour tout H ≥ 1 on a cn ,n = 0, pour presque tout (n , n ) ∈  Le complété Rσ∧ s’identifie alors à l’ensemble des séries n∈Zd cn Z1n1 . . . Zdnd qui sont (Z1 . . . Zr )-entières. C’est un anneau normal. On voit ainsi que Rσ∧ est aussi le complété de Rσ par rapport à la topologie suivante : q ξi → 0 ⇐⇒ Tr F/Q (ξi ξ ∗ ) → +∞,

∀ξ ∗ ∈ σ.

(1)

L’anneau de base sur lequel nous effectuons la construction de Mumford ici est Rσ∧ . Soit S σ = Spec(Rσ∧ ) ; posons S 0σ = S × S σ = Spec(Rσ∧ ⊗Rσ R). C’est l’ouvert Sσ

de S σ obtenu en rendant inversible q ξ pour tout élément ξ de X ∩ σˇ 0 (où σˇ 0 désigne l’intérieur du cône dual σˇ de σ ). Soit S σ 0 := S σ \ S 0σ muni de la structure réduite. Si σ  ⊂ σ , on a une flèche S σ  → S σ .

532

Mladen Dimitrov


Soit a ( = P ∗ dans les notations de Rapoport [11]) un idéal du corps de Le tore G.  := (Gm ⊗ a∗ ) × S σ le S σ -tore de groupe des nombres totalement réel F et soit G 
 caractères a. Explicitement : G = Spec Rσ∧ [Xα ; α ∈ a] . L’ensemble des périodes b. Soit b ( = N dans les notations de Rapoport [11]) un idéal fractionnaire de F , tel que ab−1 = c et ab ⊂ X.  par le morphisme Pour chaque β ∈ b on définit un S 0σ -point de G, Rσ∧ [Xα ; α ∈ a] → Rσ∧ ⊗Rσ R,

Xα  → q αβ .

Ceci définit un homomorphisme o-équivariant de S 0σ -schémas en groupes q : b →  (où b désigne le schéma en groupes constant). Gm ⊗ a∗ = G, La polarisation φ. Se donner une polarisation o-linéaire φ : b → a (voir la définition 1.3) revient à se donner un élément [φ] ∈ c+ = c ∩ (F ⊗ R)+ . La construction de Mumford donne un schéma semi-abélien Gσ sur S σ . Propriétés du schéma semi-abélien Gσ . − La restriction de Gσ à S 0σ est une VAHB, notée G0σ .

− Tout élément [φ] ∈ c donne une flèche naturelle Gm ⊗ a∗ → Gm ⊗ b∗ , d’où, par fonctorialité de la construction, une flèche symétrique φ de la variété abélienne G0σ = (Gm ⊗ a∗ )/q(b) vers sa duale (G0σ )t = (Gm ⊗ b∗ )/q(a). Si [φ] ∈ c+ , alors φ est une polarisation. − Par le lemme du serpent, appliqué à la multiplication par n dans Gm ⊗a∗ , on trouve la n-torsion de G0σ (qui est le sous-schéma en groupes réduit, intersection des noyaux des multiplications par les éléments de n) au milieu de la suite exacte 1 → (a/na)(1) → G0σ [n] → n−1 b/b → 0.

(2)

− La restriction de Gσ à S σ 0 est égale au tore (Gm ⊗a∗ ) × S σ 0 . − La construction est fonctorielle en les σ ∈  et compatible avec l’action de o× , i.e. pour tout σ  ⊂ σ et pour tout u ∈ o× on a des diagrammes cartésiens : Gσ   Sσ 

/ Gσ

Gσ

 /S σ

 Sσ

∼

/ Gu2 σ

∼

/S

 u2 σ

.

3 R-pointes et (R, n)-pointes × Pour tout idéal f ⊂ o on note o× f le sous-groupe de o formé des unités congrues à 1 × modulo f. On note o+ le groupe des unités totalement positives de o.

Compactifications arithmétiques des variétés de Hilbert

533

Pour tout o-réseau L de F 2 notons G+ (L) le stabilisateur de L dans GD Q + (pour D −1 l’action à gauche donnée par γ · l = lγ , pour tout γ ∈ GQ et l ∈ L). On a

 o c∗  ν(γ ) ∈ o× ∩ D G+ (o ⊕ c∗ ) = γ ∈ Q . + cd o Posons  =

1D (c, n)

 =

a c

b d



  ∈ G (o ⊕ c ) c ∈ cdn, d ≡ 1(mod n) . +

∗

Cette partie étudie la combinatoire des pointes d’une variété modulaire de Hilbert– Blumenthal en niveau 1D (c, n) et servira à la construction de cartes locales pour les compactifications toroïdales. Cette étude a été déjà effectuée par Rapoport en niveau  D (c, n) et en niveau 1D (c, n) pour une pointe non-ramifiée, lorsque n est un entier naturel et D = Gm (voir [11]). Par ailleurs, lorsque F = Q, l’étude est faite par Deligne et Rapoport [5], en niveau (n), et par Katz et Mazur [8] en général. Soit c un idéal fractionnaire de F , muni de sa de positivité c+ = c ∩ (F ⊗ R)+ . Les objets combinatoires considérés dans cette partie sont inspirés par les structures de niveau des VAHB : une VAHB c-polarisée complexe admet une uniformisation de la forme F ⊗ C /L, où L est un o-réseau de F 2 tel que ∧2o L = c∗ . Or, un tel réseau s’écrit L = b ⊕ a∗ , avec a et b deux idéaux fractionnaires de F tels que a∗ b = c∗ . La µn structure de niveau sur une telle VAHB est donnée alors par un homomorphisme injectif de o-modules β : n−1 d−1 /d−1 → n−1 L/L. Par ailleurs tout o-module projectif de rang 2 est isomorphe à un o-réseau de F 2 . La définition suivante est une variante de celle donnée par Rapoport dans le cas D = Gm . Définition 3.1. Une R-pointe C (resp. une classe d’isomorphisme de R-pointes) est une classe d’équivalence de sextuplets (a, b, L, i, j, λ), où (i) a et b sont deux idéaux fractionnaires de F tels que a∗ b = c∗ , (ii) L est un o-réseau de F 2 tel que l’on a une suite exacte o-modules j

i

0 → a∗ → L → b → 0, (iii) λ : ∧2o L → c∗ est un isomorphisme o-linéaire (polarisation),

pour la relation d’équivalence suivante : (a, b, L, i, j, λ) et (a , b , L , i  , j  , λ ) sont équivalents, si a = a , b = b (resp. a = ξ a et b = ξ b avec ξ ∈ F ) et s’il existe un diagramme commutatif de o-modules : 0

/ a∗

0

 / a ∗

i i

/L

j

/b

/0

 / L

j

 / b

/0,

∼ ∧2 L où les flèches verticales sont des isomorphismes et tel que l’isomorphisme ∧2o L = o   ∗ (déduit de L ∼ = L ) induise, via λ et λ , un automorphisme de c , donné par un élément × de o× D+ := o+ ∩ DQ .

534

Mladen Dimitrov

L’application qui à une R-pointe C = (a, b, L, i, j, λ) associe l’idéal b est une bijection entre l’ensemble des R-pointes et l’ensemble
F des idéaux fractionnaires de F . En effet, par (i) la donnée de b détermine a = bc, et deux suites exactes courtes (ii), correspondant au même idéal b, sont équivalentes, car toutes les deux sont scindées. La notion d’isomorphisme de R-pointes correspond alors à celle d’homothétie des idéaux. On obtient par passage au quotient un isomorphisme entre les classes d’isomorphisme de R-pointes et le groupe ClF des classes d’idéaux de F . Une R-pointe est déterminée par son o-réseau L de F 2 (en effet, la donnée d’un tel réseau détermine les idéaux a∗ := L ∩ ({0} × F ) et b = ca−1 , et donc la R-pointe × C, à équivalence près). Le groupe GoQ := {γ ∈ GD Q | ν(γ ) ∈ o+ } agit transitivement sur ces réseaux. Le stabilisateur du réseau o ⊕ c∗ dans GoQ est égal à G+ (o ⊕ c∗ ). De plus, deux réseaux L et L donnent la même R-pointe C, si et seulement s’ils sont dans la même TZ UQ -orbite. Le diagramme commutatif suivant, traduit la correspondance entre les R-pointes et les pointes classiques dans P1 (F ) pour le sous-groupe de congruence G+ (o ⊕ c∗ )
F  ClF

∼

∼/

/ R−pointes

∼

 R−pointes/isom.

/ TZ UQ \ Go /G+ (o ⊕ c∗ ) Q

∼

 / BQ \ GD /G+ (o ⊕ c∗ ) Q



∼

/ G+ (o ⊕ c∗ ) \ F 2 − {0}/o×

∼

 / G+ (o ⊕ c∗ ) \ P1 (F ),

a b ∈ GoQ la double classe BQ γ −1 G+ (o ⊕ c∗ ) s’envoie d’une c d part sur la pointe classique G+ (o ⊕ c∗ )γ ∞ et d’autre part sur l’idéal b = ao + cc∗ (voir [6] Lemme 1.7).

où pour tout γ =

Définition 3.2. (i) Une (R, n)-pointe C (resp. une classe d’isomorphisme de (R, n)pointes) est la donnée d’une classe d’équivalence de paires formées d’un sextuplet (a, b, L, i, j, λ) (comme dans la définition 3.1) et d’un morphisme injectif de o-modules β : n−1 d−1 /d−1 → n−1 L/L, pour la relation d’équivalence suivante : C est équivalent à C  , s’il existe un isomorphisme de o-modules L ∼ = L induisant une égalité (resp. un isomorphisme) des R-pointes sous-jacentes et dont la réduction modulo n rend le diagramme suivant commutatif : n−1 L/L iR RRR RRR V 6 β

∼

/ n−1 L /L 5 k kk (  kkkk 

n−1 d−1 /d−1 .

β

On associe à C l’idéal fractionnaire b ⊃ b tel que b /b = j (im(β)). (ii) Une (R, n)-pointe est dite non-ramifiée lorsque la flèche β : n−1 d−1 /d−1 → −1 n L/L se factorise par la flèche naturelle n−1 a∗ /a∗ → n−1 L/L (ou si de manière équivalente b = b).

535

Compactifications arithmétiques des variétés de Hilbert

(iii) Soit une (R, n)-pointe C et soit n l’exposant du groupe b /b. Une (R, n)-pointe est dite appartenir à la même (R, n)-composante que C (resp. à une (R, n)-composante isomorphe), s’il existe a ∈ (Z /n)× et un isomorphisme de o-modules L ∼ = L induisant une égalité (resp. un isomorphisme) des R-pointes sous-jacentes et dont la réduction ψ modulo n fait commuter le diagramme suivant C

n−1 L/L

∼ ϕ

/ n−1 L/L

∼ ψ

/ n−1 L /L 5 iRRR k RRR  kkkkk k ( R V 6 β β n−1 d−1 /d−1 ,

où la flèche ϕ est un automorphisme o-linéaire de n−1 L/L, induisant l’identité sur n−1 a∗ /a∗ et la multiplication par a sur n−1 b/b. Soit y0 tel que o = n + y0 c. On munit la R-pointe L0 = o ⊕ c∗ de la structure de ·y0 niveau β0 : n−1 d−1 /d−1 −→ n−1 c∗ /c∗ → n−1 L0 /L0 . Le groupe GoQ agit transitivement sur ces réseaux munis de structures de niveau et le stabilisateur de (L0 , β0 ) est . De plus, deux réseaux L et L donnent la même R-pointe C, si et seulement s’ils sont dans la même TZ UQ -orbite. D’où le diagramme suivant : (R, n)−pointes  (R, n)−pointes/isom.

∼

/ TZ UQ \ GoQ /  ∼

 / BQ \ GD /  . Q

γ −1 , γ



a b = ∈ GoQ . c d

Proposition 3.3. Soit une (R, n)-pointe C, donnée par TZ UQ Alors, (i) L’idéal b, correspondant à la R-pointe sous-jacente à C est donné par ao + cc∗ et sa classe ne dépend que de la classe d’isomorphisme de la pointe C.

Quitte à changer γ , en le multipliant par un élément de

UQ , ce qui ne change pas ∗ b (bc) . Sous cette hypothèse : sa classe double, on suppose que γ ∈ GoQ ∩ bcd b−1 (y0 c,y0 d)

(ii) La structure de niveau de C est donnée par β : n−1 d−1 /d−1 −−−−−−→n−1 L/L, où L = b ⊕ a∗ , avec a = bc. (iii) L’idéal b de la définition 3.2(i) est contenu dans n−1 b et sa classe ne dépend que de la classe d’isomorphisme de la pointe C. De plus b = ao + c(cn)∗ . La pointe C est non-ramifiée, si et seulement si, c ∈ nbcd. (iv) Le groupe d’automorphismes de la (R, n)-pointe C est égal à γ −1 γ ∩ BQ . La suite exacte 1 → U → B → T → 1, donne une suite exacte : 0 → X∗ → γ −1 γ ∩ BQ → o× C → 1, × ×  −1  −1 }. En où X = cbb et o× C = {(u, ) ∈ o × oD+ | u − 1 ∈ nb b , u − 1 ∈ bb × ×  −1 ) ∩ (1 + nb b−1 )}. particulier, on a o× C,1 := oC ∩ T1 = {u ∈ o | u ∈ (1 + bb

536

Mladen Dimitrov

(v) L’ensemble des (R, n)-pointes est fibré au-dessus de
F . La fibre de l’idéal b est isomorphe à (G+ (b ⊕ a∗ ) ∩ TZ UQ ) \ G+ (b ⊕ a∗ )/γ −1 γ , où a = bc, L = b ⊕ a∗ . Elle s’identifie avec l’ensemble :

   u ξ ∗ × × ∗ 2 ∗  (n−1 L/L)prim ,  ∈ o , ξ ∈ (cb ) u ∈ o , D+ 0 u−1 où (n−1 L/L)prim désigne l’ensemble des vecteurs primitifs du o/n-module n−1 L/L,  × et son cardinal est égal à #(o/bb −1 )× #(o/nb b−1 )× /[(o× × o× D+ ) : oC ]. n−1 b⊃b ⊃b

(vi) L’ensemble des (R, n)-composantes est fibré au-dessus de
F . La fibre de l’idéal b s’identifie avec l’ensemble :

  au ξ ∗  × −1 × × ∗ 2 ∗  u ∈ o ,  ∈ oD+ , a ∈ (Z /n) , ξ ∈ (cb ) (n L/L)prim 0 u−1  × qui est de cardinal #(o/bb −1 )× #(o/nb b−1 )× /#(Z /n)× [(o× × o× D+ ) : oC ], n−1 b⊃b ⊃b

où n est égal à l’exposant du groupe b /b. De plus   −1 = (u, ) ∈ o× × o× o× D+ | u − 1 ∈ nb b , C

u ∈ (Z /n)× + bb

−1 

,

 −1  −1 o× = u ∈ o× | u ∈ (1 + nb b ) ∩ ((Z /n)× + bb ) . C,1 Démonstration. (i) La R-pointe sous-jacente à C correspond à la classe double a  TZ UQ γ −1 G+ (o ⊕ c∗ ) et donc à la G+ (o ⊕ c∗ )-pointe γ ∞ = . Par le diagramme c qui précède la définition 3.2 la R-pointe C correspond à l’idéal b = ao + cc∗ . (ii), (iii) La structure de niveau β de L est obtenue en faisant agir γ −1 sur la structure de niveau β0 de L0 . Or, par le choix que nous avons fait de γ , on a L0 γ = (cy0 ,dy0 )

b ⊕ a∗ = L et donc β : n−1 d−1 /d−1 −−−−−−→b /b ⊕ n−1 a∗ /a∗ → n−1 L/L. La pointe est donc non-ramifiée si, et seulement, si cy0 n−1 d−1 ⊂ b, i.e. c ∈ nbcd. Enfin b = b + cy0 d−1 n−1 = ao + cc∗ + cc∗ n−1 = ao + c(cn)∗ . L’indépendance des classes de b et b découle du lemme 1.7 de [6] . (iv) Pour le calcul du groupe d’automorphismes γ −1 γ

∩ BQ de la (R, n)-pointe ∗ u ξu, C, on remarque qu’il est formé de matrices , avec u ∈ o× ,  ∈ o× D+ , 0 u−1 ξ ∗ ∈ (cb2 )∗ (c’est la forme générale d’un automorphisme de la R-pointe sous-jacente) qui respectent en plus la structure de niveau β. Ceci équivaut au système

(u − 1)c ∈ nbcd (3) ∗ c ∈ ncda∗ = nb−1 . (u − 1)d −  −1 ξu, En posant u =  = 1 on retrouve que X∗ est formé des ξ ∗ ∈ c−1 nb−1 ∩ (cb2 )∗ = (cb)∗ ((c(cn∗ )−1 ∩ b−1 ) = (cbb )∗ , i.e. X = cbb .

Compactifications arithmétiques des variétés de Hilbert

537

Pour le calcul de o× C on remarque que la première condition de (3) équivaut à u − 1 ∈ c−1 nbcd ∩ o = b(c(cn)∗−1 ∩ b−1 ) = bb −1 . La deuxième condition équivaut à u−1 ∈ d −1 (nb−1 +c(cb2 )∗ ) = (db)−1 nb b−1 . Par ailleurs u−1 ∈ o ⊂ c−1 nb cd = (c(bc)∗ )−1 nb b−1 . Comme (db)−1 ∩ (c(bc)∗ )−1 = (db + c(bc)∗ )−1 = o, par le choix de γ , on en déduit que la deuxième condition de (3) équivaut à u − 1 ∈ nb b−1 . −1 ∗ −1 Notons que pour tout u ∈ o× + d(bb −1 ∩ nb b−1 )) ⊂ (cbb )∗ + C , ξu, ∈ c (nb db((cb2 )∗ ∩ (ncb 2 )∗ ) ⊂ (cbb )∗ + (cb2 )∗ ∩ (ncb 2 )∗ , et ce dernier est un idéal inclus (parfois strictement !) dans (cb2 )∗ (voir l’exemple à la fin de l’article). (v), (vi) Comme γ transforme o ⊕ c∗ en b ⊕ a∗ et γ −1 G+ (o ⊕ c∗ )γ = G+ (b ⊕ a∗ ), la fibre de l’idéal b est isomorphe à (G+ (b ⊕ a∗ ) ∩ TZ UQ ) \ G+ (b ⊕ a∗ )/γ −1 γ , L’ensemble G+ (b⊕a∗ )/γ −1 γ s’identifie avec celui des vecteurs primitifs du o/n-module n−1 L/L. Le calcul du cardinal de la fibre se fait en analysant la condition sous laquelle deux vecteurs primitifs correspondent à la même (R, n)-pointe. La démonstration du (vi) est tout a fait analogue. Comme par définition no ⊂ bb −1 ⊂ o, l’ensemble (Z /n)× + bb −1 est bien : o× une réunion de classes de o, modulo l’idéal entier bb −1 Notons que [o× C ] diC × vise #(Z /n) et le quotient représente le nombre de (R, n)-pointes dans la (R, n)
composante C. Exemple 3.4. On pose c = o (polarisation principale) et G = G∗ (o× D+ = {1}). (i) Si F = Q, n = p Z, avec p un nombre premier, on a p − 1 (R, n)-pointes, au-dessus de la R-pointe ∞ (b = Z), dont × = o× − (p−1)/2 non-ramifiées, avec b = Z et oC C = {1}. Chacune de ces pointes est seule dans sa (R, n)-composante. = {±1} ⊃ o× − (p−1)/2 ramifiées, avec b = p−1 Z et o× C = {1}, contenues dans C une seule (R, n)-composante. (ii) Si n = p2 , avec p un idéal premier de o de degré résiduel 1 (N(p) = p, avec p un nombre premier), on a 3 types de (R, n)-pointes, au-dessus de la R-pointe ∞ (b = o) : × × × − si b = o, on a n = 1, o× = o× C = op2 , et donc on a p(p−1)/[o : op2 ] pointes C non-ramifiées, chacune seule dans sa (R, n)-composante. × × 2 × − si b = p−1 , on a n = p, o× = o× C = op , et donc on a (p − 1) /[o : op ] C pointes peu ramifiées, partagées par groupes de (p −1), en (p −1)/[o× : o× p ] (R, n)composantes. × × × = o× , o× − si b = p−2 , on a n = p2 , o× C = op2 , et donc on a p(p−1)/[o : op2 ] C pointes très ramifiées, contenus dans une seule (R, n)-composante. (iii) Si n = p, avec p un idéal premier de o de degré résiduel 2 (N(p) = p2 , avec p un nombre premier), on a 2 types de (R, n)-pointes, au-dessus de la R-pointe ∞ (b = o) : × × 2 × = o× − si b = o, on a n = 1, o× C = op , et donc on a (p −1)/[o : op ] pointes C non-ramifiées, chacune seule dans sa (R, n)-composante.

538

Mladen Dimitrov

× × × p − si b = p−1 , on a n = p, o× C = op oC = {u ∈ o | u − u ∈ p}, et donc on a × × (p2 −1)/[o× : o× p ] pointes peu ramifiées, partagées par groupes de (p−1)/[oC : op ], × en (p + 1)/[o× : oC ] (R, n)-composantes.

4 Construction des cartes locales Soit c un idéal de F , muni de sa positivité naturelle c+ = c ∩ (F ⊗ R)+ . Posons

u 0 .  = F N(n) = N(dn). Nous identifions T1 × D et T par (u, ) → 0 u−1 1 ]On on considère le foncteur contravariant M 1 (resp. M) de la catégorie des Z[ N(n) schémas vers celle des ensembles, qui à un schéma S associe l’ensemble des classes d’isomorphisme de quadruplets (A, ι, λ, α)/S (resp. (A, ι, λ, α)/S), où (A, ι) est une VAHB (voir [6] Déf.2.2), λ est une c-polarisation sur A (resp. λ est une classe de c-polarisations ; voir [6] Déf.2.3), et α : (o/n)(1) → A[n] est une µn -structure de niveau (voir [6] Déf.2.5). 1 ]-schéma quasi-projectif, normal, Le foncteur M 1 est représentable par un Z[ N(n) 1 géométriquement connexe M de dimension dF , qui est lisse au-dessus de Z[ 1 ] et muni d’un quadruplet universel (A, ι, λ, α) (voir [6] Thm.4.1). 1 ] quasi-projectif, Le foncteur M admet un schéma de modules grossier M sur Z[ N(n) 1 normal, géométriquement connexe et lisse au-dessus de Z[  ] (voir [6] Cor.4.2). × ×2 Le schéma M est le quotient de M 1 par le groupe fini o× D+ /(oD+ ∩ on ) qui agit proprement et librement par [] : (A, ι, λ, α)/S  → (A, ι, λ, α)/S. Le but de cette partie est de munir les VAHB construites dans la partie 2 de différentes µn -structures de niveau, et ainsi fournir les cartes locales servant à compactifier la variété modulaire de Hilbert M. A chaque (R, n)-composante C, on peut associer par la Déf.3.2 et la Prop.3.3 des idéaux b, b et X = cbb , un entier n égal à l’exposant du groupe b /b, des groupes × × × × × × × d’unités o× C , oC , oC,1 , oC,1 et des sous-groupes HC = oC /oC , HC,1 = oC,1 /oC,1 du groupe (Z /n Z)× (ces objets sont a priori associés à une (R, n)-pointe, mais sont constants au sein d’une (R, n)-composante). Soit une (R, n)-composante C et considérons le tore S = SC = Gm ⊗X ∗ . Soit C ∗ . Soit σ ∈  C . La construction de la partie précédente,  un éventail complet de X+ appliquée à (X, a, b), nous donne alors un schéma semi-abélien Gσ /S σ , muni d’une action de o et dont la restriction à G0σ /S 0σ est une VAHB c-polarisée. En appliquant une deuxième fois la construction de la partie précédente, cette fois à (X, a, b ), on obtient un schéma semi-abélien Gσ /S σ , muni d’une action de o et dont 0   −1 -polarisée. Par fonctorialité on a une la restriction G0 σ /S σ est une VAHB c = ab flèche Gσ → Gσ , dont la restriction G0σ → G0 σ est une isogénie. On en déduit la suite exacte : q

0 → b /b → G0σ [n] → G0 σ [n] → 1.

(4)

539

Compactifications arithmétiques des variétés de Hilbert

Considérons d’abord le cas où C est non-ramifiée. On a alors b = b et donc X = ab. La variété abélienne G0σ associée à une (R, n)-composante non-ramifiée est naturellement munie d’une µn -structure de niveau (o/n)(1) ∼ = (a/na)(1) → G0σ [n], −1 −1 où la première flèche vient de l’isomorphisme β : n d /d−1 ∼ = n−1 a∗ /a∗ et la deuxième du (2). Passons maintenant au cas où C est ramifiée. Afin de munir G0σ d’une µn -structure de niveau, on doit : − choisir un relèvement de b /b dans im(β) (appelé uniformisation de C), 1 − se placer dans ce cas au-dessus de Spec(Z[ N(n) , ζC ]), où ζC désigne une racine de l’unité d’ordre égal à l’exposant n du groupe abélien b /b. 1 Au-dessus de Spec(Z[ N(n) , ζC ]) on a un isomorphisme canonique b∗ /b ∗ ∼ =  0 (b /b)(1), d’où une µn -structure de niveau sur Gσ : (o/n)(1)



 / (a/na)(1) × (b∗ /b ∗ )(1) ∼ = (a/na)(1) × b /b

où la première inclusion vient de la flèche β :

n−1 d−1 /d−1

→

(2)(4)

/ G0 [n],

n−1 a∗ /a∗

σ

× b /b.

Proposition 4.1. (i) Pour toute (R, n)-composante uniformisée C et pour tout cône σ ∈  C la construction ci-dessus donne un carré cartésien : 1 G0σ × Spec(Z[ N(n) , ζC ])

/A

1 S 0σ × Spec(Z[ N(n) , ζC ])

 / M1

/M.

(ii) Changer l’uniformisation de la pointe C revient à se donner un élément x ∈ (ab)∗ /(ab )∗ = Hom(b /b, n−1 a∗ /a∗ ) et correspond donc à l’automorphisme de S 0σ × n Tr

(ξ x)

1 , ζC ]) qui envoie q ξ sur ζC F/Q q ξ (ξ ∈ ab ). Spec(Z[ N(n) (iii) Soient C1 , C2 deux (R, n)-composantes uniformisées et soient deux cônes ∗ , i = 1, 2. Supposons qu’il existe σi ⊂ Xi,R ∼ C2 (d’où ξ ∈ F × tel que a∗ = ξ a∗ , − un isomorphisme de (R, n)-composantes C1 = 2 1 ∗ ∗ −1 2 ∗ b2 = ξ b1 et X2 = ξ X1 ) induisant sur c (via les polarisations de L et L ), la multiplication par une unité  ∈ o× D+ ,

× h 2 2 − des éléments (u, ) ∈ o× C1 = oC2 et h ∈ HC , tels que σ2 = u ξ σ1 et ζC2 = ζC1 . 1 1 Alors, on a un isomorphisme S 0σ1 ×Spec(Z[ N(n) , ζC1 ]) ∼ , ζC2 ]) = S 0σ2 ×Spec(Z[ N(n) 1 0 qui complète les deux flèches S σi × Spec(Z[ N(n) , ζCi ]) → M (i = 1, 2) du (i) en un triangle commutatif.

Le (i) et (ii) découlent de ce qui précède. Le (iii) utilise la fonctorialité de la construction de G0σ en σ et sa compatibilité avec l’action de o× C (voir fin de la partie 2 et la Prop.3.3(iv)).
Avant de décrire la construction des compactifications toroïdales arithmétiques, on doit la préparer. C’est l’objet des deux parties suivantes.

540

Mladen Dimitrov

5 Un théorème de descente formelle de Rapoport La construction d’une compactification toroïdale peut être vue comme l’ajout d’un bord à M. On a un schéma formel de type fini candidat pour ce bord, à savoir l’analogue algébrique de : 
∧ C× ⊗(cbb )∗  C /o× Xan = C. pointes C/∼

Le but de cette partie est de donner un critère abstrait, trouvé par Rapoport [11], pour résoudre le problème de “Descente Formelle”, en l’occurrence, le problème d’existence et unicité du schéma recollement Y d’un ouvert Y 0 et d’un schéma formel X : Y 0 → Y ← X. Il repose en partie sur un critère d’immersion ouverte de Rapoport dont on rappellera l’énoncé. Le problème de Descente Formelle sera en fait d’abord posé dans la catégorie des espaces algébriques. On verra dans la partie 7 que les conditions d’application du critère sont satisfaites dans notre cas. Dans cette partie V désignera un anneau de valuation discrète complet, de corps des fractions K et de corps résiduel k. S désigne un V -schéma. Soit Aff /S la catégorie des S-schémas affines, munie de la topologie étale. Un faisceau d’ensembles sur Aff /S s’appelle un S-espace. Définition 5.1. Une relation d’équivalence étale sur un S-schéma U1 est donnée par une immersion fermée quasi-compacte U2 → U1 ×S U1 de S-schémas dont les deux projections sont étales et qui définit une relation d’équivalence : pour tout Y ∈ Aff /S, U2 (Y ) → U1 (Y ) ×S(Y ) U1 (Y ) est une relation d’équivalence. Un S-espace algébrique est un S-espace qui est quotient d’un schéma U1 , appelé un atlas étale, par une relation d’équivalence étale. L’ensemble Alg /S des S-espaces algébriques muni des flèches de S-espaces forme une catégorie. On définit de même pour un schéma formel S ∧ la catégorie des S ∧ espaces algébriques formels, notée Form /S ∧ . Définition 5.2. Soit f : X → X un morphisme dans Form /S ∧ . On dit que f est un éclatement admissible de X si f est un éclatement X → X dans Form /S ∧ , par rapport à un idéal qui contient une puissance de l’idéal de définition de X. La catégorie des espaces rigides Rig /S est la catégorie localisée de Form /S, par rapport aux éclatements admissibles. Définition 5.3. Un épaississement de (K, V ) est un couple (R, R (0) ) tel que : − R est un anneau local artinien de corps résiduel K. On note RV l’image réciproque de V dans R. − R (0) ⊂ RV ⊂ R est un sous-anneau noethérien tel que le morphisme R (0) → V soit surjectif et la localisation de R (0) au point générique de V soit égale à R (c’est à dire R est le localisé de R (0) en J = ker(R (0) → V )).

Compactifications arithmétiques des variétés de Hilbert

541

Soit  π un élément de R (0) qui se projette sur une uniformisante  de(i)V . Pour tout J (0) ⊂ R (1) ⊂ · · · ⊂ R et i ≥ 1 on pose R (i) = R (0) [  ]. Alors R = RV . V i iR π (0) On a Sprig (K) = Spf(V )rig et Sprig (R) := Spf(R )rig (= Spf(R (i) )rig ), car Spf(R (i) ) est obtenu par éclatement (admissible) de Spf(R (0) ), par rapport à l’idéal ( π i ) + J (car J est nilpotent). Exemple 5.4. Soit l’anneau local artinien R = K[t]/(t 2 ). Le sous-anneau RV = V +K·t n’est pas noethérien. Considérons le sous-anneau noethérien R (0) = V [t]/(t 2 ). t et donc Alors (R, R (0) ) est un épaississement de (K, V ). On a R (i) = V + V ·  πi  (i) = RV . iR A toute flèche frig : Xrig → Yrig on peut associer un modèle formel f : X → Y, défini à éclatement admissible près. Définition 5.5. frig est une immersion ouverte, s’il existe un modèle formel f qui est une immersion ouverte. M. Rapoport a démontré le critère d’immersion ouverte suivant, qui est utilisé pour démontrer le résultat de recollement abstrait que l’on a en vue. Théorème 5.6 (Théorème 3.15 de [11]). frig est une immersion ouverte, si et seulement si, les deux conditions suivantes sont satisfaites : frig ∗

(i)rig Pour tout corps K, discrètement valué, l’application Hom(Sprig (K), Xrig ) −→ Hom(Sprig (K), Yrig ) est injective. (ii)rig Pour tout épaississement (R, R (0) ) de (K, V ) on peut compléter de façon unique le diagramme commutatif suivant : / Xrig q8 q   qq / Yrig . Sprig (R)

Sprig (K)

Remarque 5.7. L’anneau V étant principal, il n’admet pas d’éclatements admissibles. La condition (i)rig peut s’écrire donc Hom(Spf(V ), X) → Hom(Spf(V ), Y), alors que le diagramme dans la condition (ii)rig devient (pour i assez grand) : Spf(V )  Spf(R (0) ) o

r (i) Spf(R )

r

r

/ r8 X  /Y.

Soit S un schéma affine, de type fini sur le spectre d’un corps ou d’un anneau de Dedekind excellent (pour les applications aux compactifications toroïdales, il suffit de prendre S de type fini sur Z).

542

Mladen Dimitrov

Soit A un anneau noethérien complet pour la topologie I -adique, définie par un idéal I ⊂ A. Soit U = Spf(A) le schéma formel affine correspondant. Posons U = Spec(A), U 0 = Spec(A/I ) = l’âme de U et U 0 = U \ U 0 . Lemme 5.8 (EGA III.5). Soit Y un espace algébrique de type fini sur S et Y0 ⊂ Y un sous-espace fermé. On suppose que U = Spec(A) est un S-schéma et on se donne un S-morphisme formel adique f : U → Y |Y0 . Alors, il existe un unique morphisme f : U → Y dont le complété formel est f. Définition 5.9. Un morphisme g 0 : Spec(K) → U 0 sera dit permis, s’il vient (via le lemme 5.8) d’un morphisme formel de type fini g : Spf(V ) → U. Plus généralement (si U est un S-schéma), un morphisme f 0 : U 0 → Y 0 dans un espace algébrique de type fini sur S sera dit permis, s’il existe une immersion ouverte de Y 0 dans un S-espace algébrique propre Y , telle que : pour tout morphisme permis Spec(K) → U 0 , l’unique extension à Spec(V ) du morphisme composé Spec(K) → Y , envoie le point spécial dans Y \ Y 0 . Un morphisme f 0 , provenant par restriction d’un morphisme f : U → Y , est permis, s’il existe un morphisme formel f : U → Y = Y |Y0 qui fait commuter le diagramme suivant : U

f

/Y

 U

f

 /Y

 U0

f0

 / Y0 .

En d’autres termes, un morphisme est permis s’il “envoie le bord sur le bord”. Définition 5.10. Soit X un S-espace algébrique formel, séparé et de type fini. Un découpage de X est la donnée : − d’un atlas affine U2 = Spf(A2 ) ⇒ U1 = Spf(A1 ) → X, et − d’un espace algébrique Y 0 de type fini sur S, tel que les deux composés suivants f0

soient égaux : U 02 ⇒ U 01 → Y 0 , où U 1 = Spec(A1 ) et U 2 = Spec(A2 ) et les flèches U 2 ⇒ U 1 viennent, via le lemme 5.8, des flèches U2 ⇒ U1 . Le découpage est dit effectif, s’il existe un S-espace algébrique de type fini Y , une ∼ immersion ouverte j : Y 0 → Y et un isomorphisme ϕ : X → Y, où Y est le complété formel de Y le long de Y \ Y 0 , tels que le morphisme f : U 1 → Y , venant (via le

Compactifications arithmétiques des variétés de Hilbert

543

∼

lemme 5.8) du morphisme f : U1 → X → Y, induise f 0 : U 01 → Y 0 sur U 01 ⊂ U 1 . // U

U2

f

/X∼ =Y

1

 UO 2 //

 UO 1

? U 02

 // ? 0 U1

 /Y O

f

f0

 / Y? 0

Théorème 5.11 (Théorème 3.5 de [11])). Soit un découpage. On suppose : − U 01 est schématiquement dense dans U 1 (i.e. OU 1 → OU 0 ). 1

− Y 0 est compactifiable (i.e. il existe une S-immersion ouverte Y 0 → Y ∗ avec Y ∗ propre sur S). − Le morphisme f 0 : U 01 → Y 0 est permis. − Pour tout anneau de valuation discrète complet V , de corps des fractions K : (i )rig la suite U 02 (K)permis ⇒ U 01 (K)permis → Y 0 (K)permis est exacte, et

(ii )rig pour tout épaississement (R, R (0) ) de (K, V ) on peut compléter de façon unique le diagramme commutatif suivant : Spec(K)  r Spec(R)

permis

r

r

/ 0 r8 U 1

permis

 / Y0 .

Alors le découpage est effectif.

6 La construction de Raynaud Pour pouvoir vérifier les conditions (i )rig et (ii )rig ci-dessus dans la situation où l’ouvert Y 0 est l’espace de modules M 1 et le schéma formel X est celui donné par les cartes locales de la proposition 4.1, on a besoin de la construction suivante (donnée par Raynaud dans [12] et reprise par Rapoport dans le cas d’une VAHB [11]). Il est à noter que l’on a besoin de cette construction non seulement sur un corps mais aussi sur un épaississement artinien, auquel cas l’argument donné par Raynaud reste valable. Soit V un anneau de valuation discrète complet de corps des fractions K, et soit (R, R (0) ) un épaississement de (K, V ). Définition 6.1. Une variété abélienne A sur R (resp. sur K) est dite à réduction semistable (déployée) s’il existe un schéma en groupes lisse sur R (i) , pour un certain i ≥ 0, (resp. sur V ), prolongeant A et dont la fibre spéciale est une extension d’une variété abélienne par un tore (déployé).

544

Mladen Dimitrov

Pour des raisons de dimension, si une VAHB sur R (ou sur K) est à mauvaise réduction semi-stable déployée, alors la fibre spéciale est un tore déployé. Dans ce cas la description rigide-analytique de Raynaud devient : Théorème 6.2 (Raynaud). Soit A une VAHB sur R (ou sur K) à mauvaise réduction semi-stable déployée. Alors : Arig = (Gm ⊗a∗ )rig /brig , où a et b sont des idéaux de F . De plus : − on a une suite exacte 0 → (a/na)(1) → A[n] → n−1 b/b → 0, − la forme bilinéaire  ,  : a × b → Val(K) ∼ = Z (α, β)  → val(Xα (β)) vérifie aα, β = α, aβ pour tout a ∈ o, et donc définit un unique élément ξ ∗ ∈ (ab)∗ , à × Q× + près et à l’action de o près, − le cône positif des polarisations sur A, P (A) ⊂ Symo (A, At ) = ab−1 est obtenu comme produit de l’unique positivité sur ab pour laquelle ξ ∗ > 0 et de la positivité naturelle sur b−2 .

7 Compactifications toroïdales arithmétiques Construction des compactifications toroïdales. Définition 7.1. Un éventail -admissible  = ( C )C est la donnée pour chaque ∗ , stable par o× et contenant un (R, n)-composante C d’un éventail complet  C de X+ C nombre fini d’éléments modulo cette action, de sorte que les données soient compatibles aux isomorphismes de (R, n)-composantes C ∼ = C. Voici l’analogue du résultat principal de l’article [11] dans le cas de groupe de niveau  (on rappelle que  est sans torsion). Théorème 7.2. Soit  = { C }C un éventail -admissible. 1 (i) Il existe une immersion ouverte j : M 1 → M1 de Spec(Z[ N(n) ])-schémas et un isomorphisme de schémas formels 
∧  ∼ 1 HC,1 ) −→ S C /o× M1∧ , ϕ: C,1 × Spec(Z[ N(n) , ζC ] (R,n)−composantes/∼

(où M1∧ est le complété formel de M1 le long de sa partie à l’infini), tels que pour toute (R, n)-composante C et pour tout σ ∈  C on a la propriété suivante : l’image ré1 , ζC ]) → ciproque de la VAHB universelle sur M 1 par le morphisme S σ × Spec(Z[ N(n) 1 1 ∧ M (déduit par le lemme 5.8 du morphisme formel Sσ × Spec(Z[ N(n) , ζC ]) → M1∧ construit à l’aide de ϕ), soit la VAHB c-polarisée avec µn -structure de niveau 1 1 G0σ × Spec(Z[ N(n) , ζC ]) sur S 0σ × Spec(Z[ N(n) , ζC ]) construite dans la proposition 4.1(i). Le couple (j, ϕ) est unique, à unique isomorphisme près.

Compactifications arithmétiques des variétés de Hilbert

545

1 ])-schémas et (ii) Il existe une immersion ouverte j : M → M de Spec(Z[ N(n) un isomorphisme de schémas formels 
∧  ∼ 1 ∧ HC S C /o× ϕ: C × Spec(Z[ N(n) , ζC ] ) −→ M . (R,n)−composantes/∼

Démonstration. (i) Il y a un nombre fini de (R, n)-composantes C modulo isomorphisme. Soit {σiC } un ensemble fini de représentants des cônes de l’éventail  C , modulo l’action de o× C,1 .  ∧ 1 S C × Spec(Z[ N(n) , ζC ]). Il est Considérons le schéma formel affine U1 := C/∼ i

σi

de type fini sur Z et muni d’un morphisme toroïdales” et quotient  
∧étale×(“immersions 1 S C /oC,1 × Spec(Z[ N(n) , ζC ]HC,1 ). étale par le groupe HC,1 ) dans X := C/∼  1 S σ C × Spec(Z[ N(n) , ζC ]). Posons U 1 = C/∼ i

i

D’après la proposition 4.1(i) on a un morphisme f 0 : U 01 → M 1 , qui est permis, car toute variété abélienne obtenue comme image réciproque, par un morphisme permis 1 1 , ζC ]), de la variété abélienne G0σ ×Spec(Z[ N(n) , ζC ]) Spec(K) → S 0 C ×Spec(Z[ N(n) σi

est à mauvaise réduction d’après la partie 2. Posons U2 := U1 ×X U1 = Spf(A2 ) et U 2 = Spec(A2 ). Les deux flèches composées U 02 ⇒ U 01 → M 1 sont égales par compatibilité de la construction de Mumford avec les inclusions σ  ⊂ σ , avec l’action de o× C,1 et avec l’action de HC,1 (appliquer la proposition 4.1(iii) dans le cas D = Gm ). Vérifions la condition (i )rig du théorème 5.11 : Soient g10 , g20 : Spec(K) → U 01 deux morphismes permis avec f 0  g10 = f 0  g20 . 1 Chaque morphisme gj0 se factorise par un certain S 0σj × Spec(Z[ N(n) , ζCj ]), où σj C

désigne un des σi j et détermine ainsi : − une (R, n)-composante Cj (à laquelle sont attachés des objets aj , bj , bj , Xj , βj ), (j )

− une racine de l’unité ζC ∈ K, d’ordre l’exposant nj du groupe bj /bj , − un cône σj de  Cj et un morphisme ψj : Rσ∧j → V , d’où un élément ξj∗ ∈ σj ∩Xj∗ , déterminé par la propriété suivante : pour tout ξ ∈ σˇ j ∩Xj on a val(ψj (q ξ )) = Tr F/Q (ξ ξj∗ ). Le morphisme permis f 0 gj0 fournit une VAHB A sur K munie d’une c-polarisation et µn -structure de niveau, à mauvaise réduction semi-stable déployée. L’uniformisation de Raynaud–Tate de la VAHB A, décrite dans la partie 6, donne alors : − deux idéaux a et b, tels que Arig = (Gm ⊗a∗ )rig /brig et c = Symo (A, At ) = ab−1 (ceci nous donne une R-pointe C, bien définie modulo isomorphisme). Comme la construction de Mumford et celle de Raynaud sont inverses l’une à l’autre (i.e. le 1-motif associé par Raynaud à la VAHB sur K construite par Mumford est le 1-motif du départ), les R-pointes sous-jacentes de C1 et C2 sont isomorphes à C.

546

Mladen Dimitrov

− une suite exacte : 0 → (a/na)(1) → A[n] → n−1 b/b → 0. Ainsi, la µn structure de niveau sur A détermine-t-elle bien une une (R, n)-composante C audessus de la R-pointe C et une racine de l’unité ζC . De nouveau par compatibilité de la construction de Mumford et celle de Raynaud on déduit que ζC1 et ζC2 sont conjuguées sous HC,1 et que les (R, n)-composantes C1 et C2 sont isomorphes, et donc égales, car elles vivent dans un ensemble de représentants modulo isomorphisme. − un élément ξ ∗ ∈ (ab)∗+ bien défini modulo o× C,1 . Un dernière fois par compatibilité des constructions de Mumford et de Raynaud, on trouve que ξ1∗ ∈ σ1 et ξ2∗ ∈ σ2 ∗ ∗ sont dans la même o× C,1 -orbite. Par conséquent ξ1 = ξ2 et, par exemple σ1 ⊂ σ2 . On en déduit qu’il existe un morphisme permis h0 : Spec(K) → U 02 tel que 0 g1 = p10  h0 et g20 = p20  h0 , ce qui termine la vérification du (i )rig . Vérifions la condition (ii )rig du théorème 5.11 : Les morphismes permis Spec(K) → U 01 et Spec(R) → M 1 nous donnent deux VAHB A/K et A /R à mauvaise réduction, avec A ∼ = A ×R K. Comme dans la 0  vérification de (i )rig , la flèche Spec(K) → U 1 détermine les des donnés combinatoires C = (a, b, X, β), ζC , ξ ∗ ∈ X∗ . Par la théorie de Raynaud–Tate A et A admettent des uniformisation rigides analytiques Arig = (Gm ⊗a∗ )rig /brig (compatibilité entre la construction de Mumford et celle de Raynaud) et Arig = (Gm ⊗a ∗ )rig /brig . Comme Arig = Arig ×R K, on en déduit que l’on peut prendre a = a , b = b , ζC = ζC  et ξ ∗ = ξ  ∗ , d’où le (ii )rig . Nous sommes maintenant en mesure d’appliquer le théorème 5.11 qui nous donne le couple cherché (j, ϕ), dont on admet l’unicité. × (ii) Comme  C est stable par o× C (et non-seulement par oC,1 ), le groupe fini × ×2 1 o× D+ /(oD+ ∩ on ) du revêtement galoisien étale M → M agit proprement et librement sur M1 et la construction du (i) passe au quotient. La flèche M1 → M est encore étale.
Remarque 7.3. Soit  = ( b )b∈
F , où pour tout idéal b,  b est un éventail o× -admissible de (cb2 )∗+ . On aurait pu tenter de définir M comme la normalisation dans M de la compactification de Rapoport M(c) de l’espace de modules M(c). Le problème est que le schéma M ainsi défini n’est jamais lisse. En effet, pour compactifier chaque (R, n)-pointe C qui est au-dessus de la R-pointe correspondant à b on utilise le même éventail  b . Or, si bb −1  = no (n ∈ Z),  b ne peut pas être un éventail lisse pour (cb2 )∗+ et (cbb )∗+ simultanément. Propriétés des compactifications toroïdales. Dans la suite, pour alléger les notations, nous écrirons M à la place de M , en gardant en tête la dépendance du système d’éventails . 1 Corollaire 7.4. Localement pour la topologie étale sur Spec(Z[ N(n) ]), j : M → M C est isomorphe à SC → Sσ pour un certain couple C, σ ∈  .

547

Compactifications arithmétiques des variétés de Hilbert

En particulier, pour tout cône σ ∈  C \{0}, et tout corps algébriquement clos k de caractéristique p ne divisant pas N(n), l’ensemble des k-points de la strate M(σ ) de M s’identifie à celui des k-points de la strate fermée S(σ ) de l’immersion torique affine S → Sσ . Ceci résulte du fait que o× C opère librement sur l’ensemble des strates non-ouvertes ∧ /o× (et donc M ∧ ) est de SC → S C , et donc localement pour la topologie étale S C C isomorphe à Sσ∧ , pour un certain σ ∈  C . Corollaire 7.5. Quitte à raffiner l’éventail , on obtient un schéma M qui est lisse au-dessus de Spec(Z[ 1 ]). Proposition 7.6. Il existe un unique schéma en groupes semi-abélien f : G → M 1 qui prolonge la VAHB universelle f : A → M 1 . Ce schéma en groupes est muni d’une action de o et c’est un tore au-dessus de M 1 \ M 1 . Démonstration. L’unicité est montrée dans un cadre beaucoup plus général dans le chapitre I du livre de Chai et Faltings [7]. Pour l’existence on considère le diagramme suivant : _ _ _ _ _ _ _ _ _7/ G o  oo7 A o o oo o o  1 / Gσ [ 1 , ζC ] , ζC ] G0σ [ N(n)  N(n)   / 1o M1 pp8 M pp8 p p p p p p   / S σ [ 1 , ζC ] o S 0σ [ 1 , ζC ] N(n)

N(n)

M1 oo7 o o o

1 Sσ∧ [ N(n) , ζC ] .

∧




1 ])-schéma M est propre. Théorème 7.7. Le Spec(Z[ N(n)

Démonstration. L’idée, comme dans [11], est d’appliquer le critère valuatif de propreté tel qu’il est énoncé dans [3] (voir Théorème 4.19 et le commentaire qui suit). Il suffit de vérifier la propreté de M 1 . Soit V un anneau de valuation discrète de corps de fractions K. Comme M 1 est ouvert et dense dans M 1 , il suffit de vérifier que tout morphisme g 0 : Spec(K) → M 1 , s’étend en un morphisme g : Spec(V ) → M 1 . Supposons que g 0 ne s’étend pas déjà en un morphisme g : Spec(V ) → M 1 . La VAHB A/K donnée par f 0 est donc à mauvaise réduction (voir Deligne–Pappas [4]). Quitte à remplacer K par une extension finie et V par sa normalisation, on peut supposer que A/K est à mauvaise réduction semi-stable. Nous sommes alors en mesure d’appliquer à A/K la théorie de géométrie rigide de Raynaud, qui nous fournit (voir la partie 6) : − deux idéaux a et b, tels que Arig = (Gm ⊗a∗ )rig /brig et c = Symo (A, At ) = ab−1 (ceci définit une R-pointe).

548

Mladen Dimitrov

− une suite exacte 0 → (a/na)(1) → A[n] → n−1 b/b → 0. La µn -structure de niveau (o/n)(1) → A[n] définit alors une (R, n)-composante C au-dessus de la R-pointe définie précédemment (à laquelle on peut associer un idéal b ⊃ b) et une racine de l’unité ζC d’ordre l’exposant du groupe b /b. − un élément ξ ∗ ∈ (ab)∗+ (bien défini modulo l’action de o× C ), venant de la forme bilinéaire o-équivariante  ,  : a × b → Val(K) ∼ Z (α, β)  → val(Xα (β)). = × ∗ Un translaté de ξ par le groupe oC,1 appartient à un certain cône σiC ∈  C , parmi les cônes choisis dans la démonstration du théorème. Le morphisme g 0 se factorise 1 , ζC ]) → M 1 . Le morphisme composé alors par la carte locale S 0 C × Spec(Z[ N(n) σi

1 g : Spec(V ) → S σ C × Spec(Z[ N(n) , ζC ]) → M 1 étend le morphisme g 0 . i




8 Formes de Hilbert et compactification minimale arithmétiques Nous savons qu’une forme modulaire de Hilbert classique (sur C) est uniquement déterminée par son q-développement en une pointe C, que la condition d’holomorphie à l’infinie est automatiquement satisfaite si dF > 1 (Principe de Koecher) et qu’il n’y a pas de séries d’Eisenstein en poids non-parallèle. Le but de cette partie est de décrire, en suivant [11], les propriétés du q-développement d’une forme de Hilbert arithmétique. C’est le point de départ de la construction de la compactification minimale arithmétique de M (voir [2]). Formes modulaires de Hilbert arithmétiques. Pour la définition de l’espace des formes modulaires de Hilbert, nous suivons le paragraphe 6.8 de [11], rédigé par P. Deligne. Considérons le schéma en groupes T1 = ResoZ Gm sur Z, dont la fibré générique est le tore ResFQ Gm de groupes de caractères Z[JF ], où JF désigne l’ensemble des plongements de F dans R. On suppose dans cette partie que dF > 1. Par définition de la VAHB universelle, le faisceau ωA/M 1 = f∗ 1A/M 1 est un o-fibré inversible sur M 1 × Spec(Z[ 1 ]).

Soit κ ∈ Z[JF ] = X(T1 ) un poids et soit F  un corps de nombres, contenant les valeurs du caractère κ : F × → C× . On peut prendre, par exemple, F  = Q et poids parallèle, ou bien F  = F gal et poids quelconque.

Soit o l’anneau des entiers de F  . Le morphisme de groupes algébriques   κ : ResFQ Gm → ResFQ Gm , se prolonge en un morphisme ResoZ Gm → ResoZ Gm , qui équivaut (par la formule d’adjonction) à un morphisme de groupes algébriques sur o , ResoZ Gm × Spec(o ) → Gm × Spec(o ), noté encore κ. A l’aide de κ, on peut découper dans ω un fibré inversible sur M 1 × Spec(o [ 1 ]), κ noté ωκ . Soit o l’anneau des entiers de F  = F  ( 1/2 ,  ∈ o× D+ ). Alors ω descend en

549

Compactifications arithmétiques des variétés de Hilbert

un fibré inversible sur M × Spec(o [ 1 ]), noté encore ωκ (voir la partie 4 de [6] pour une présentation plus détaillée). Pour tout Z[ 1 ]-schéma Y , on pose Y  = Y ×Spec(Z[ 1 ]) Spec(o [ 1 ]). 

Définition 8.1. Soit R une o [ 1 ]-algèbre. Une forme modulaire de Hilbert arithmétique de poids κ, de niveau  et à coefficients dans R, est une section globale de ωκ sur M ×Spec(Z[ 1 ]) Spec(R). On note Gκ (c, n; R)geom := H0 (M ×Spec(Z[ 1 ]) Spec(R), ωκ )   l’espace de ces formes modulaire de Hilbert.  Remarque 8.2. 1) Le faisceau ωt (t = τ ∈JF τ ) n’est autre que le faisceau ∧dF ω = det(ω) sur M, et ωkt - sa puissance k-ième. Les formes modulaires de Hilbert de poids parallèle k ≥ 1, s’écrivent donc Gkt (c, n)geom = H0 (M, (∧dF ω)⊗k ). 2) Si F  ⊃ F gal , l’action de o permet de décomposer ω ∼ = o ⊗ OM  en somme τ  directe de fibrés inversibles ω sur M , indexés par les différents plongements τ de F  dans F gal . Si κ = kτ τ , on a ωκ = τ (ωτ )⊗kτ . Soit f : G → M 1 le schéma semi-abélien au-dessus d’une compactification toroïdale M 1 de M 1 , comme dans la partie précédente. Le faisceau ωG/M 1 := e∗ 1 1 , G/M

où e : M 1 → G désigne la section unité, prolonge le faisceau ωA/M 1 . En outre ωG/M 1 coïncide avec le faisceau (f ∗ 1

G/M 1

)G des G-invariants de f ∗ 1

cartes formelles, on voit comme dans [11], qu’au-dessus de

. En passant aux

G/M 1 1 Z[  ], le faisceau ωG/M 1 

est un o-fibré inversible. Le fibré ωG/M 1 descend en un o-fibré inversible sur M , noté ω (voir la partie 7 de [6]). Pour tout κ ∈ Z[JF ], on peut ainsi prolonger le fibré inversible  ωκ en un fibré inversible sur M , noté encore ωκ . D’après la partie 2 pour toute (R, n)-composante uniformisée C, tout cône σ ∈  C et pour toute o [ 1 , ζC ]-algèbre R on a ω|Sσ∧ ×Spec(R) (a ⊗ OSσ∧ ×Spec(R) ), d’où ωκ |S ∧C ×Spec(R) (a ⊗ OS ∧C ⊗ R)−κ = (a ⊗ o [ 1 ])−κ ⊗ (o ⊗ OS ∧C ⊗ R)−κ (5) 





1 o [  ]

(κ)

∧ × Spec(R))/o× , ωκ ) = a(κ) ⊗ (κ) = Par conséquent H0 ((S C C o [ 1 ] RC (R), où a 

(a ⊗ o [ 1 ])−κ est un o [ 1 ]-module inversible et    ∗ )  n Tr F/Q (ξ uξu, (κ) RC (R) := aξ q ξ  aξ ∈ R, au2 ξ =  κ/2 uκ ζC aξ , ∀(u, ) ∈ o× C . ξ ∈X+ ∪{0}

∗ est un élément de b b−1 d−1 , bien défini modulo d−1 , et donc Notons que ξ uξu, ∗ ) ∈ Z /n Z (on rappelle que n Z = Z ∩bb −1 et n = ord(ζ )). n Tr F/Q (ξ uξu, C
∧ o× (κ) 0 −κ C On a RC (R) = H S C × Spec(R), (o ⊗ OS ∧C ⊗ R) . 

550

Mladen Dimitrov (κ)

Le diagramme suivant montre comment l’anneau RC (R) se situe par rapport aux différents anneaux déjà considérés dans la partie 2 :  / Rσ ⊗ R = R[q ξ ]ξ ∈X∩σˇ R[q ξ ]ξ ∈X  _ + ∪{0} _ (κ)

RC (R)



 / R[[q ξ ]]ξ ∈X

+ ∪{0}

 / R∧ ⊗ R . σ



Principe de Koecher. Théorème 8.3 (Principe de Koecher [11] 4.9). Soit M une compactification toroïdale de M. Alors

 H0 M × Spec(R), ωκ = H0 M × Spec(R), ωκ Démonstration. Le problème est local et il suffit de le vérifier après complétion, le long d’une (R, n)-composante C. D’après la trivialisation (5) du fibré inversible ωκ , il s’agit de voir que les sections ∧ × Spec(R), qui sont globales méromorphes du faisceau (o ⊗ OS ∧C ⊗ R)−κ sur S C (κ)



appartiennent à RC (R). o× C -invariantes, × ∧ ξ 0 ∧ Soit f = ⊗ R)−κ ) une ξ ∈X aξ q ∈ Hmer ((S C × Spec(R))/oC , (o ⊗ OS C telle section. Supposons que aξ0  = 0 avec ξ0 non-totalement positif. Il existe donc ∗ ∗ ξ0∗ ∈ XR + avec Tr F/Q (ξ0 ξ0 ) strictement négatif. Comme dF > 1, on peut choisir des × unités u ∈ oC,1 de manière à rendre la quantité Tr F/Q (u2 ξ0 ξ0∗ ) arbitrairement proche de −∞. Soit σ un cône polyédral de  C contenant ξ0∗ . Par définition de Sσ∧ , on voit (κ)
que f n’est pas méromorphe sur Sσ∧ . Contradiction. Donc f ∈ RC (R). q-développement. Le paragraphe précédent montre que l’on peut associer à une (R, n)-composante uniformisée C et à une forme modulaire de Hilbert f de poids κ, niveau , et à coefficients dans une o [ 1 , ζC ]-algèbre R, un élément : (κ)

fC ∈ a(κ) ⊗o [ 1 ] RC (R). 

Définition 8.4. L’élément fC est appelé le q-développement de la forme f en la (R, n)composante uniformisée C. On note evC,κ l’application f  → fC . Le principe du q-développement s’énonce alors : Proposition 8.5. Soient κ un poids, C une (R, n)-composante uniformisée et R une o [ 1 , ζC ]-algèbre (contenant les valeurs de κ). Alors (i) l’application evC,κ est injective,

(κ)

(ii) pour toute R-algèbre R  et f ∈ Gκ (c, n; R  ), si evC,κ (f ) ∈ a(κ) ⊗o [ 1 ] RC (R),  alors f ∈ Gκ (c, n; R),

Compactifications arithmétiques des variétés de Hilbert

551

(iii) s’il existe f ∈ Gκ (c, n; R) tel que le terme constant de evC,κ (f ) ne soit pas nul, alors  κ/2 uκ − 1 est un diviseur de zéro dans R, pour tout (u, ) ∈ o× C. Le cas de l’anneau nul R = 0 redonne une formulation classique du principe. Pour démonstration du (i) et du (ii) voir la partie 7 de [6]. Le (iii) est clair.
Compactification minimale. La compactification minimale est la contrepartie arithmétique de la compactification de Satake sur C. Contrairement au cas complexe, dans le cas arithmétique, la construction de la compactification minimale utilise les compactifications toroïdales. Voici l’analogue en niveau  de l’énoncé donné par C.-L. Chai dans [2]. k0 t Théorème 8.6. (i) Il existe k0 ∈ N∗ tel que le faisceau ωA/M 1 , soit engendré par ses

sections globales sur M 1 . (ii) Le morphisme canonique π : M 1 → M 1∗ := ProjZ[


 1 N(n) ]

k≥0

 H0 (M 1 , ωkt ) , A/M 1

1 est surjectif. Le Z[ N(n) ]-schéma M 1∗ est indépendant du choix de  (on rappelle que

M 1 = M1 ). (iii) L’anneau gradué



k≥0 H

0 (M 1 , ωkt ) A/M 1

1 est de type fini sur Z[ N(n) ] et M 1∗

× 1 ×2 est un Z[ N(n) ]-schéma projectif, normal, de type fini. Le groupe o× D+ /(oD+ ∩ on ) 1 ]-schéma du revêtement fini étale M 1 → M agit sur M 1∗ et le quotient est un Z[ N(n) ∗ projectif, normal, de type fini M , muni d’un morphisme surjectif π : M → M ∗ .

(iv) π |M induit un isomorphisme sur un ouvert dense de M ∗ , noté encore M. M ∗ \M 1 ] et en fait isomorphe à : est fini et étale sur Z[ N(n)  1 Spec(Z[ N(n) , ζC ]HC ). (R,n)−composantes/∼

Les composantes connexes de M ∗ \M sont appelées les pointes de M. Cependant celles-ci ne sont des points fermés que pour les (R, n)-composantes non-ramifiées. (v) L’image réciproque π −1 (C) de chaque pointe C de M, est une composante connexe de M\M. La complétion formelle de M le long de l’image réciproque d’une com∧ /o× ) × Spec(Z[ 1 , ζ ]HC ). posante π −1 (C), est canoniquement isomorphe à (S C C N(n) C En particulier, la complétion formelle de M le long de l’image réciproque π −1 (C) d’une (R, n)-composante non-ramifiée C, est canoniquement isomorphe à × ∧ × 1 (S C /(on × oD+ )) × Spec(Z[ N(n) ]).

(vi) Pour tout κ ∈ Z[JF ] le faisceau ωκ s’étend en un faisceau inversible sur M ∗ si et seulement si κ est parallèle.

552

Mladen Dimitrov

Démonstration. Nous suivons la méthode de C.-L. Chai [2]. D’après [9] Chap.IX Thm.2.1 (voir aussi [7] Chap.V Prop.2.1), il existe k0 ≥ 1 tel que le faisceau inversible k0 t 1 ωA/M 1 soit engendré par ses sections globales sur M . Ceci nous fournit un morphisme M 1 → ProjZ[




1 N(n) ]


 k0 t Sym• H0 M 1 , ωA/M . 1



 kk0 t  0 M 1 , ωkk0 t . Le Soit B • la normalisation de Sym• H0 M 1 , ωA/M 1 dans ⊕ H A/M 1 k≥0

morphisme associé π : M 1 → ProjZ[

1 N(n) ]

(B • ) est birationnel, surjectif et satisfait

k0 t π ∗ O(1) = ωA/M 1 . Le Théorème de Connexité de Zariski implique alors que les fibres de π sont connexes. D’après [7] Chap.V Prop.2.2, la partie abélienne est constante dans chaque fibre géométrique de π, et par conséquent les fibres géométriques de π sont − soit des points géométriques de M 1 , − soit des composantes géométriques connexes de M 1 \ M 1 . k0 kt k0 t Comme pour tout k ≥ 1, π ∗ O(k) = ωA/M 1 et ω A/M 1 est engendré par ses sections
kk0 t  0 • globales sur M 1 , on obtient H0 M 1 , ωA/M 1 = H (Proj(B ), O(k)). Par conséquent
kk0 t  1 B • = ⊕ H0 M 1 , ωA/M 1 et c’est une Z[ N(n) ]-algèbre de type fini. Or, l’algèbre k≥0

kk0 t  ⊕ H0 M 1 , ωkt est entière sur ⊕ H0 M 1 , ωA/M 1 , engendrée par les éléments A/M 1 k≥0

k≥0

de degré plus petit que k0 . Il en résulte que ⊕ H0 (M 1 , ωkt ) est de type fini sur A/M 1 k≥0 
1 Z[ N(n) ], et que M 1∗ := Proj( ⊕ H0 M 1 , ωkt ) = Proj(B • ). Par le principe de A/M 1 k≥0

Koecher, le schéma M 1∗ est indépendant du choix particulier de la compactification × ×2 1∗ ∗ toroïdale M 1 de M 1 . Le groupe o× D+ /(oD+ ∩on ) agit sur M et on définit M comme 1∗ ∗ le quotient. Notons qu’en général M → M n’est pas étale, car les pointes peuvent avoir des stabilisateurs non-triviaux. On a donc (i),(ii),(iii) et la première partie de (iv). Le calcul de la complétion formelle de M, le long de l’image réciproque d’une composante connexe de M ∗ \ M découle du Théorème des Fonctions Formelles de Grothendieck. Enfin, examinons à quelle condition ωκ 1 descend en un fibré inversible sur M 1∗ . Comme (π∗ ωκ

G/M

G/M

)| 1 = ωκA/M 1 et codim(M 1∗ \ M 1 ) ≥ 2, le faisceau π∗ ωκ 1 M

est

G/M 1 ∧ /o× × cohérent. Il est inversible si et seulement si ωκ 1 peut être trivialisé sur S C C,1 G/M ∧ ×Spec(R) est canoniquement trivial Spec(R). D’après (5) le pull-back de ωκ 1 à S C G/M × et oC,1 agit sur ce pull-back à travers κ, d’où (vi).


Exemples de q-développement en une pointe ramifiée. Nous nous proposons de dé(κ) crire explicitement dans le cas particulier de l’exemple 3.4(ii)(iii) les anneaux RC (R) contenant les q-développements des formes modulaires de Hilbert de poids κ et ni-

Compactifications arithmétiques des variétés de Hilbert

553

veau . Rappelons que o désigne les entiers d’un corps de nombres contenant les valeurs du caractère κ. On suppose que ClF = {1}. Plaçons nous dans le cas (ii). Le bord M 1∗ \ M 1 s’écrit alors 

          o×/o×  p2 1 1 1 Spec Z N(n) Spec Z N(n) , ζp Spec Z N(n) , ζp2

(R,n)−comp. non-ramifiés/∼

(R,n)−comp. peu ramifiés/∼

(R,n)−comp. très ramifiés/∼

− Si la pointe C est non-ramifiée, pour toute o [ 1 ]-algèbre R, on a   (κ) RC (R) = aξ q ξ | aξ ∈ R, au2 ξ = uκ aξ , ∀u ∈ o× . 2 p ξ ∈o+

− Si la pointe C est peu ramifiée, pour toute o [ 1 , ζp ]-algèbre R, on a (κ)

RC (R) =

 

p Tr F/Q (ξ uξu∗ )

aξ q ξ | aξ ∈ R, au2 ξ = uκ ζp

 aξ , ∀u ∈ o× p .

ξ ∈p−1 +

− Si la pointe C est très ramifiée, pour toute o [ 1 , ζp2 ]-algèbre R, on a (κ)

RC (R) =

  ξ ∈p−2 +

p 2 Tr F/Q (ξ uξu∗ )

aξ q ξ | aξ ∈ R, au2 ξ = uκ ζp2

 aξ , ∀u ∈ o× . p2

En fait, d’après la démonstration de la Prop.3.3(iv), on a ξu∗ ∈ p2 d−1 et donc Tr F/Q (ξ uξu∗ ) ∈ Z (alors qu’à priori il appartient juste à p−2 Z !). On en déduit que    (κ) RC (R) = aξ q ξ | aξ ∈ R, au2 ξ = uκ aξ , ∀u ∈ o× , 2 p ξ ∈p−2 +

ce qui est compatible avec le fait que ζp2 n’appartient pas au corps de définition de la pointe C, qui est Q(ζp2 )

o×/o×2 p

(notons que −1 ∈ o×/o× ). p2

Plaçons nous dans le cas (iii). Le bord M 1∗ \ M 1 s’écrit alors 

   1 Spec Z N(n)

(R,n)−comp. non-ramifiés/∼



  o× /o×p  1 Spec Z N(n) , ζp C .

(R,n)−comp. ramifiés/∼

− Si la pointe C est non-ramifiée, pour toute o [ 1 ]-algèbre R, on a   (κ) RC (R) = aξ q ξ | aξ ∈ R, au2 ξ = uκ aξ , ∀u ∈ o× p . ξ ∈o+

554

Mladen Dimitrov

− Si la pointe C est ramifiée, pour toute o [ 1 , ζp ]-algèbre R, on a    p Tr (ξ uξ ∗ ) (κ) RC (R) = aξ q ξ | aξ ∈ R, au2 ξ = uκ ζp F/Q u aξ , ∀u ∈ o× p . ξ ∈p−1 +

En fait, d’après la démonstration de la Prop.3.3(iv), on a ξu∗ ∈ pd−1 et donc Tr F/Q (ξ uξu∗ ) ∈ Z (alors qu’à priori il appartient juste à p−1 Z !). On en déduit que    (κ) aξ q ξ | aξ ∈ R, au2 ξ = uκ aξ , ∀u ∈ o× RC (R) = p , ξ ∈p−1 +

ce qui est compatible avec le fait que ζp n’appartient pas au corps de définition de la pointe C, qui est Q(ζp )

o× /o× p C

(notons que −1 ∈ o× /o× ). C p

Références [1] J.-L. Brylinski and J.-P. Labesse, Cohomologie d’intersection et fonctions L de certaines variétés de Shimura. Ann. Sci. École Norm. Sup. 17 (1984), 361–412. [2] C.-L. Chai, Arithmetic minimal compactification of the Hilbert-Blumenthal moduli space. Ann. of Math. 131 (1990), 541–554. [3] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. [4] P. Deligne and G. Pappas, Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant. Compositio Math. 90 (1994), 59–79. [5] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques. In Modular functions of one variable II, Lecture Notes in Math. 349, Springer-Verlag, Berlin 1972, 143–316. [6] M. Dimitrov and J. Tilouine, Variétés et formes modulaires de Hilbert arithmétiques pour 1 (c, n). In Geometric Aspects of Dwork Theory (A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz and F. Loeser, eds.), Walter de Gruyter, Berlin 2004, 555–614. [7] G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties. Ergeb. Math. Grenzgeb. (3) 22, Springer-Verlag, Berlin 1990. [8] N. Katz and B. Mazur, Arithmetic moduli of elliptic curves. Ann. of Math. Stud. 108, Princeton University Press, Princeton, NJ, 1985. [9] L. Moret-Bailly, Pinceaux de variétés abéliennes. Astérisque 129 (1985). [10] D. Mumford, An analytic construction of degenerating abelian varieties over complete rings. Compositio Math. 24 (1972), 239–272. [11] M. Rapoport, Compactification de l’espace de modules de Hilbert-Blumenthal. Compositio Math. 36 (1978), 255–335. [12] M. Raynaud, Variétés abéliennes et géométrie rigide. In Actes du Congrès Internat. Math (Nice, 1970), Tome 1, Gauthier-Villars, Paris 1971, 473–477. Mladen Dimitrov, LAGA, Institut Galilée, Université Paris 13, 99, avenue J.-B. Clément, 93430 Villetaneuse, France E-mail: [email protected]