L-Functions and Galois Representations (London Mathematical Society Lecture Note Series)

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L-functions and Galois Representations

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N. J. Hitchin, Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at www.cambridge.org/mathematics 159 160 161 163 164 166 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 194 195 196 197 198 199 200 201 202 203 204 205 207 208 209 210 211 212 214 215 216 217 218 220 221

Groups St Andrews 1989 volume 1, C.M. CAMPBELL & E.F. ROBERTSON (eds) Groups St Andrews 1989 volume 2, C.M. CAMPBELL & E.F. ROBERTSON (eds) ¨ Lectures on block theory, B. KULSHAMMER Topics in varieties of group representations, S.M. VOVSI Quasi-symmetric designs, M.S. SHRIKANDE & S.S. SANE Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) Boolean function complexity, M.S. PATERSON (ed) Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK Squares, A.R. RAJWADE Algebraic varieties, G.R. KEMPF Discrete groups and geometry, W.J. HARVEY & C. MACLACHLAN (eds) Lectures on mechanics, J.E. MARSDEN Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE & A. PITTS (eds) Lower K- and L-theory, A. RANICKI Complex projective geometry, G. ELLINGSRUD et al Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT Geometric group theory I, G.A. NIBLO & M.A. ROLLER (eds) Geometric group theory II, G.A. NIBLO & M.A. ROLLER (eds) Shintani zeta functions, A. YUKIE Arithmetical functions, W. SCHWARZ & J. SPILKER Representations of solvable groups, O. MANZ & T.R. WOLF Complexity: knots, colourings and counting, D.J.A. WELSH Surveys in combinatorics, 1993, K. WALKER (ed) Local analysis for the odd order theorem, H. BENDER & G. GLAUBERMAN Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY Polynomial invariants of finite groups, D.J. BENSON Finite geometry and combinatorics, F. DE CLERCK et al Symplectic geometry, D. SALAMON (ed) Independent random variables and rearrangement invariant spaces, M. BRAVERMAN Arithmetic of blowup algebras, W. VASCONCELOS ¨ Microlocal analysis for differential operators, A. GRIGIS & J. SJOSTRAND Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI et al The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN Invariant potential theory in the unit ball of C”, M. STOLL The Grothendieck theory of dessins d’enfant, L. SCHNEPS (ed) Singularities, J.-P. BRASSELET (ed) The technique of pseudodifferential operators, H.O. CORDES Hochschild cohomology of von Neumann algebras, A. SINCLAIR & R. SMITH Combinatorial and geometric group theory, A.J. DUNCAN, N.D. GILBERT & J. HOWIE (eds) Ergodic theory and its connections with harmonic analysis, K. PETERSEN & I. SALAMA (eds) Groups of Lie type and their geometries, W.M. KANTOR & L. DI MARTINO (eds) Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD & W.M. OXBURY (eds) ´ & N. YUI Arithmetic of diagonal hypersurfaces over finite fields, F.Q. GOUVEA Hilbert C*-modules, E.C. LANCE Groups 93 Galway / St Andrews I, C.M. CAMPBELL et al (eds) Groups 93 Galway / St Andrews II, C.M. CAMPBELL et al (eds) Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders, V. KOWALENKO et al Number theory 1992–93, S. DAVID (ed) Stochastic partial differential equations, A. ETHERIDGE (ed) Quadratic forms with applications to algebraic geometry and topology, A. PFISTER Surveys in combinatorics, 1995, P. ROWLINSON (ed) Algebraic set theory, A. JOYAL & I. MOERDIJK Harmonic approximation., S.J. GARDINER

222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287

Advances in linear logic, J.-Y. GIRARD, Y. LAFONT & L. REGNIER (eds) Analytic semigroups and semilinear initial boundary value problems, KAZUAKI TAIRA Computability, enumerability, unsolvability, S.B. COOPER, T.A. SLAMAN & S.S. WAINER (eds) A mathematical introduction to string theory, S. ALBEVERIO, et al Novikov conjectures, index theorems and rigidity I, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Novikov conjectures, index theorems and rigidity II, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Ergodic theory of Zd actions, M. POLLICOTT & K. SCHMIDT (eds) Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & E.V. FLYNN Semigroup theory and its applications, K.H. HOFMANN & M.W. MISLOVE (eds) The descriptive set theory of Polish group actions, H. BECKER & A.S. KECHRIS Finite fields and applications, S. COHEN & H. NIEDERREITER (eds) Introduction to subfactors, V. JONES & V.S. SUNDER Number theory 1993–94, S. DAVID (ed) The James forest, H. FETTER & B. G. DE BUEN Sieve methods, exponential sums, and their applications in number theory, G.R.H. GREAVES et al Representation theory and algebraic geometry, A. MARTSINKOVSKY & G. TODOROV (eds) Stable groups, F.O. WAGNER Surveys in combinatorics, 1997, R.A. BAILEY (ed) Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds) Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds) Model theory of groups and automorphism groups, D. EVANS (ed) Geometry, combinatorial designs and related structures, J.W.P. HIRSCHFELD et al p-Automorphisms of finite p-groups, E.I. KHUKHRO Analytic number theory, Y. MOTOHASHI (ed) Tame topology and o-minimal structures, L. VAN DEN DRIES The atlas of finite groups: ten years on, R. CURTIS & R. WILSON (eds) Characters and blocks of finite groups, G. NAVARRO Gr¨obner bases and applications, B. BUCHBERGER & F. WINKLER (eds) ¨ Geometry and cohomology in group theory, P. KROPHOLLER, G. NIBLO & R. STOHR (eds) The q-Schur algebra, S. DONKIN Galois representations in arithmetic algebraic geometry, A.J. SCHOLL & R.L. TAYLOR (eds) Symmetries and integrability of difference equations, P.A. CLARKSON & F.W. NIJHOFF (eds) ¨ Aspects of Galois theory, H. VOLKLEIN et al An introduction to noncommutative differential geometry and its physical applications 2ed, J. MADORE Sets and proofs, S.B. COOPER & J. TRUSS (eds) Models and computability, S.B. COOPER & J. TRUSS (eds) Groups St Andrews 1997 in Bath, I, C.M. CAMPBELL et al Groups St Andrews 1997 in Bath, II, C.M. CAMPBELL et al Analysis and logic, C.W. HENSON, J. IOVINO, A.S. KECHRIS & E. ODELL Singularity theory, B. BRUCE & D. MOND (eds) New trends in algebraic geometry, K. HULEK, F. CATANESE, C. PETERS & M. REID (eds) Elliptic curves in cryptography, I. BLAKE, G. SEROUSSI & N. SMART Surveys in combinatorics, 1999, J.D. LAMB & D.A. PREECE (eds) ¨ Spectral asymptotics in the semi-classical limit, M. DIMASSI & J. SJOSTRAND Ergodic theory and topological dynamics, M.B. BEKKA & M. MAYER Analysis on Lie groups, N.T. VAROPOULOS & S. MUSTAPHA Singular perturbations of differential operators, S. ALBEVERIO & P. KURASOV Character theory for the odd order theorem, T. PETERFALVI Spectral theory and geometry, E.B. DAVIES & Y. SAFAROV (eds) The Mandlebrot set, theme and variations, TAN LEI (ed) Descriptive set theory and dynamical systems, M. FOREMAN et al Singularities of plane curves, E. CASAS-ALVERO Computational and geometric aspects of modern algebra, M.D. ATKINSON et al Global attractors in abstract parabolic problems, J.W. CHOLEWA & T. DLOTKO Topics in symbolic dynamics and applications, F. BLANCHARD, A. MAASS & A. NOGUEIRA (eds) Characters and automorphism groups of compact Riemann surfaces, T. BREUER Explicit birational geometry of 3-folds, A. CORTI & M. REID (eds) Auslander-Buchweitz approximations of equivariant modules, M. HASHIMOTO Nonlinear elasticity, Y. FU & R.W. OGDEN (eds) ¨ (eds) Foundations of computational mathematics, R. DEVORE, A. ISERLES & E. SULI Rational points on curves over finite, fields, H. NIEDERREITER & C. XING Clifford algebras and spinors 2ed, P. LOUNESTO Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE et al

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Surveys in combinatorics, 2001, J. HIRSCHFELD (ed) Aspects of Sobolev-type inequalities, L. SALOFF-COSTE Quantum groups and Lie theory, A. PRESSLEY (ed) Tits buildings and the model theory of groups, K. TENT (ed) A quantum groups primer, S. MAJID Second order partial differential equations in Hilbert spaces, G. DA PRATO & J. ZABCZYK Introduction to the theory of operator spaces, G. PISIER Geometry and Integrability, L. MASON & YAVUZ NUTKU (eds) Lectures on invariant theory, I. DOLGACHEV The homotopy category of simply connected 4-manifolds, H.-J. BAUES Higher operads, higher categories, T. LEINSTER Kleinian Groups and Hyperbolic 3-Manifolds Y. KOMORI, V. MARKOVIC & C. SERIES (eds) Introduction to M¨obius Differential Geometry, U. HERTRICH-JEROMIN Stable Modules and the D(2)-Problem, F.E.A. JOHNSON Discrete and Continuous Nonlinear Schr¨odinger Systems, M. J. ABLORWITZ, B. PRINARI & A. D. TRUBATCH Number Theory and Algebraic Geometry, M. REID & A. SKOROBOGATOV (eds) Groups St Andrews 2001 in Oxford Vol. 1, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) Groups St Andrews 2001 in Oxford Vol. 2, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) Peyresq lectures on geometric mechanics and symmetry, J. MONTALDI & T. RATIU (eds) Surveys in Combinatorics 2003, C. D. WENSLEY (ed.) Topology, geometry and quantum field theory, U. L. TILLMANN (ed) Corings and Comodules, T. BRZEZINSKI & R. WISBAUER Topics in Dynamics and Ergodic Theory, S. BEZUGLYI & S. KOLYADA (eds) ¨ Groups: topological, combinatorial and arithmetic aspects, T. W. MULLER (ed) Foundations of Computational Mathematics, Minneapolis 2002, FELIPE CUCKER et al (eds) ¨ Transcendental aspects of algebraic cycles, S. MULLER-STACH & C. PETERS (eds) Spectral generalizations of line graphs, D. CVETKOVIC, P. ROWLINSON & S. SIMIC Structured ring spectra, A. BAKER & B. RICHTER (eds) Linear Logic in Computer Science, T. EHRHARD et al (eds) Advances in elliptic curve cryptography, I. F. BLAKE, G. SEROUSSI & N. SMART Perturbation of the boundary in boundary-value problems of Partial Differential Equations, D. HENRY Double Affine Hecke Algebras, I. CHEREDNIK ´ (eds) L-Functions and Galois Representations, D. BURNS, K. BUZZARD & J. NEKOVAR Surveys in Modern Mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N. C. SNAITH (eds) Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) Singularities and Computer Algebra, C. LOSSEN & G. PFISTER (eds) Lectures on the Ricci Flow, P. TOPPING Modular Representations of Finite Groups of Lie Type, J. E. HUMPHREYS Fundamentals of Hyperbolic Manifolds, R. D. CANARY, A. MARDEN & D. B. A. EPSTEIN (eds) Spaces of Kleinian Groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) Noncommutative Localization in Algebra and Topology, A. RANICKI (ed) Foundations of Computational Mathematics, Santander 2005, L. PARDO, A. PINKUS, E. SULI & M. TODD (eds) ¨ Handbook of Tilting Theory, L. ANGELERI HUGEL, D. HAPPEL & H. KRAUSE (eds) Synthetic Differential Geometry 2ed, A. KOCK The Navier-Stokes Equations, P. G. DRAZIN & N. RILEY Lectures on the Combinatorics of Free Probability, A. NICA & R. SPEICHER Integral Closure of Ideals, Rings, and Modules, I. SWANSON & C. HUNEKE Methods in Banach Space Theory, J. M. F. CASTILLO & W. B. JOHNSON (eds) Surveys in Geometry and Number Theory, N. YOUNG (ed) Groups St Andrews 2005 Vol. 1, C.M. CAMPBELL, M. R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) Groups St Andrews 2005 Vol. 2, C.M. CAMPBELL, M. R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) Ranks of Elliptic Curves and Random Matrix Theory, J. B. CONREY, D. W. FARMER, F. MEZZADRI & N. C. SNAITH (eds) Elliptic Cohomology, H. R. MILLER & D. C. RAVENEL (eds) Algebraic Cycles and Motives Vol. 1, J. NAGEL & C. PETERS (eds) Algebraic Cycles and Motives Vol. 2, J. NAGEL & C. PETERS (eds) Algebraic and Analytic Geometry, A. NEEMAN Surveys in Combinatorics, 2007, A. HILTON & J. TALBOT (eds) Surveys in Contemporary Mathematics, N. YOUNG & Y. CHOI (eds)

L-functions and Galois Representations Edited by D AV I D B U R N S , KEVIN BUZZARD AND ´ R ˇ JA N N E KOV A

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org c Cambridge University Press 2007 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2007 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data ISBN 978-0-521-69415-5 paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface List of participants

page ix xi

Stark–Heegner points and special values of L-series Massimo Bertolini, Henri Darmon and Samit Dasgupta

1

Presentations of universal deformation rings Gebhard B¨ockle

24

Eigenvarieties Kevin Buzzard

59

Nontriviality of Rankin-Selberg L-functions and CM points Christophe Cornut and Vinayak Vatsal

121

A correspondence between representations of local Galois groups and Lie-type groups Fred Diamond 187 Non-vanishing modulo p of Hecke L–values and application Haruzo Hida

207

Serre’s modularity conjecture: a survey of the level one case Chandrashekhar Khare

270

Two p-adic L-functions and rational points on elliptic curves with supersingular reduction Masato Kurihara and Robert Pollack 300 From the Birch and Swinnerton-Dyer Conjecture to non-commutative Iwasawa theory via the Equivariant Tamagawa Number Conjecture - a survey Otmar Venjakob 333 vii

viii

Contents

The Andr´e-Oort conjecture - a survey Andrei Yafaev

381

Locally analytic representation theory of p-adic reductive groups: a summary of some recent developments Matthew Emerton

407

Modularity for some geometric Galois representations - with an appendix by Ofer Gabber Mark Kisin

438

The Euler system method for CM points on Shimura curves Jan Nekov´arˇ

471

Repr´esentations irr´eductibles de GL(2, F ) modulo p Marie-France Vign´eras

548

Preface

The London Mathematical Society symposium on L-functions and Galois representations took place at the University of Durham from the 19th to the 30th of July, 2004; this book is a collection of research articles in the areas covered by the conference, in many cases written by the speakers or audience members. There were series of lectures in each of the following subject areas: • • • • • •

Local Langlands programme Local p-adic Galois representations Modularity of Galois representations Automorphic forms and Selmer groups p-adic modular forms and eigenvarieties The Andr´e-Oort conjecture

In practice it is becoming harder to distinguish some of these areas from others, because of major recent progress, much of which is documented in this volume. As well as these courses, there were 19 individual lectures. The organisers would like to thank the lecturers, and especially those whom we persuaded to contribute to this volume. The symposium received generous financial support from both the EPSRC and the London Mathematical Society. These symposia now command a certain reputation in the number theory community and the organisers found it easy to attract many leading researchers to Durham; this would not have been possible without the financial support given to us, and we would like to heartily thank both organisations. The conference could not possibly have taken place if it had not been for the efforts of John Bolton, James Blowey and Rachel Duke of the Department of Mathematics at the University of Durham, and for the hospitality of Grey College. We are grateful to both these institutions for their help in making the operation run so smoothly. ix

x

Preface

The feedback from the participants to the organisers seemed to indicate that many participants found the symposium mathematically stimulating; and the organisers can only hope that this volume serves a similar purpose. David Burns Kevin Buzzard Jan Nekov´aˇr

List of participants

Viktor Abrashkin (Durham) Amod Agashe (Missouri) Adebisi Agboola (UCSB) Konstantin Ardakov (Cambridge) Julian Arndts (Cambridge) Joel Bella¨ıche (IPDE - Roma I) Denis Benois (Bordeaux I) Laurent Berger (IHES) Massimo Bertolini (Milano) Amnon Besser (Be´er Sheba) Bryan Birch (Oxford) Thanasis Bouganis (Cambridge) Christophe Breuil (IHES) Manuel Breuning (King’s College London) David Burns (King’s College London) Colin Bushnell (King’s College London) Kevin Buzzard (Imperial College) Nigel Byott (Exeter) Frank Calegari (Harvard) Laurent Clozel (Orsay, Paris-Sud) Pierre Colmez (Jussieu, Paris 6) James Cooper (Oxford) Christophe Cornut (Jussieu, Paris 6) Anton Deitmar (Exeter) Rob de Jeu (Durham) Daniel Delbourgo (Nottingham) Ehud de Shalit (Hebrew) Fred Diamond (Brandeis) Vladimir Dokchitser (Cambridge) Neil Dummigan (Sheffield) Matthew Emerton (Northwestern) Ivan Fesenko (Nottingham) Tom Fisher (Cambridge) Jean-Marc Fontaine (Orsay, Paris-Sud) Kazuhiro Fujiwara (Nagoya) Toby Gee (Imperial College) Sasha Goncharov (MPI (Bonn), Brown) Ralph Greenberg (Washington) Hannu Harkonen (Cambridge) Guy Henniart (Orsay, Paris-Sud) Haruzo Hida (UCLA) Richard Hill (University College London) Ben Howard (Harvard) Susan Howson (Oxford) Annette Huber-Klawitter (Leipzig) Frazer Jarvis (Sheffield) Adam Joyce (Imperial College)

Chandrashekhar Khare (Utah) Mark Kisin (Chicago) Bruno Klingler (Chicago) Bernhard Koeck (Southampton) Stephen Kudla (Maryland) Masato Kurihara (Tokyo Metropolitan) Mathias Lederer (Bielefeld) Stephen Lichtenbaum (Brown) Ron Livn´e (Hebrew) Jayanta Manoharmayum (Sheffield) Ariane M´ezard (Orsay, Paris-Sud) Jan Nekov´aˇr (Jussieu, Paris 6) Wieslawa Nizioł (Utah) Rachel Ollivier (Jussieu, Paris 6) Louisa Orton (Paris 13) Pierre Parent (Bordeaux I) Vytautas Paskunas (Bielefeld) Mark Pavey (Exeter) Karl Rubin (Stanford) Mohamed Saidi (MPI (Bonn)) Takeshi Saito (Tokyo) Kanetomo Sato (Nottingham) Michael Schein (Harvard) Peter Schneider (Munster) Tony Scholl (Cambridge) Alexei Skorobogatov (Imperial College) Paul Smith (Nottingham) Victor Snaith (Southampton) David Solomon (King’s College London) Noam Solomon (Be´er Sheba) Michael Spiess (Bielefeld) Nelson Stephens (Royal Holloway) Shaun Stevens (East Anglia) Peter Swinnerton-Dyer (Cambridge) Martin Taylor (UMIST) Richard Taylor (Harvard) Jacques Tilouine (Paris 13) Douglas Ulmer (Arizona) Eric Urban (Columbia) Otmar Venjakob (Heidelberg) Marie-France Vign´eras (Jussieu, Paris 7) Stephen Wilson (Durham) Christian Wuthrich (Cambridge) Andrei Yafaev (University College London) Atsushi Yamagami (Kyoto) Sarah Zerbes (Cambridge) Shou-Wu Zhang (Columbia)

xi

Stark–Heegner points and special values of L-series Massimo Bertolini Dipartimento di Matematica Universita’ degli Studi di Milano Via Saldini 50 20133 Milano, Italy [email protected]

Henri Darmon McGill University Mathematics Department, 805 Sherbrooke Street West Montreal, QC H3A-2K6 CANADA [email protected]

Samit Dasgupta Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, MA 02138, U.S.A., [email protected]

Introduction Let E be an elliptic curve over Q attached to a newform f of weight two on Γ0 (N ). Let K be a real quadratic field, and let p||N be a prime of multiplicative reduction for E which is inert in K, so that the p-adic completion Kp of K is the quadratic unramified extension of Qp . Subject to the condition that all the primes dividing M := N/p are split in K, the article [Dar] proposes an analytic construction of “Stark–Heegner points” in E(Kp ), and conjectures that these points are defined over specific class fields of K. More precisely, let R :=

a b c d

∈ M2 (Z[1/p]) such that M divides c

be an Eichler Z[1/p]-order of level M in M2 (Q), and let Γ := R1× denote the group of elements in R of determinant 1. This group acts by M¨obius transformations on the Kp -points of the p-adic upper half-plane Hp := P1 (Kp ) − P1 (Qp ), 1

2

Massimo Bertolini, Henri Darmon and Samit Dasgupta

and preserves the non-empty subset Hp ∩ K. In [Dar], modular symbols attached to f are used to define a map Φ : Γ\(Hp ∩ K) −→ E(Kp ),

(0.1)

whose image is conjectured to consist of points defined over ring class fields of K. Underlying this conjecture is a more precise one, analogous to the classical Shimura reciprocity law, which we now recall. Given τ ∈ Hp ∩ K, the collection Oτ of matrices g ∈ R satisfying τ τ g = λg for some λg ∈ K, (0.2) 1 1 is isomorphic to a Z[1/p]-order in K, via the map g → λg . This order is also equipped with the attendant ring homomorphism η : Oτ −→ Z/M Z sending g to its upper left-hand entry (taken modulo M ). The map η is sometimes referred to as the orientation at M attached to τ . Conversely, given any Z[1/p]-order O of discriminant prime to M equipped with an orientation η, the set HpO of τ ∈ Hp with associated oriented order equal to O is preserved under the action of Γ, and the set of orbits Γ\HpO is equipped with a natural simply transitive action of the group G = Pic+ (O), where Pic+ (O) denotes the narrow Picard group of oriented projective O-modules of rank one. Denote this action by (σ, τ ) → τ σ , for σ ∈ G and τ ∈ Γ\HpO . Class field theory identifies G with the Galois group of the narrow ring class field of K attached to O, denoted HK . It is conjectured in [Dar] that the points Φ(τ ) belong to E(HK ) for all τ ∈ HpO , and that Φ(τ )σ = Φ(τ σ ),

for all σ ∈ Gal(HK /K) = Pic+ (O).

(0.3)

In particular it is expected that the point PK := Φ(τ1 ) + · · · + Φ(τh ) should belong to E(K), where τ1 , . . ., τh denote representatives for the distinct orbits in Γ\HpO . The article [BD3] shows that the image of PK in E(Kp ) ⊗ Q is of the form t · PK , where (i) t belongs to Q× ; (ii) PK ∈ E(K) is of infinite order precisely when L (E/K, 1) = 0; provided the following ostensibly extraneous assumptions are satisfied (i) P¯K = ap PK , where P¯K is the Galois conjugate of PK over Kp , and ap is the pth Fourier coefficient of f . (ii) The elliptic curve E has at least two primes of multiplicative reduction.

Stark–Heegner points and special values

3

The main result of [BD3] falls short of being definitive because of these two assumptions, and also because it only treats the image of PK modulo the torsion subgroup of E(Kp ). The main goal of this article is to examine certain “finer” invariants associated to PK and to relate these to special values of L-series, guided by the analogy between the point PK and classical Heegner points attached to imaginary quadratic fields. In setting the stage for the main formula, let E/Q be an elliptic curve of conductor M ; it is essential to assume that all the primes dividing M are split in K. This hypothesis is very similar to the one imposed in [GZ] when K is imaginary quadratic, where it implies that L(E/K, 1) vanishes systematically because the sign in its functional equation is −1. In the case where K is real quadratic the “Gross-Zagier hypothesis” implies that the sign in the functional equation for L(E/K, s) is 1 so that L(E/K, s) vanishes to even order and is expected to be frequently non-zero at s = 1. Consistent with this expectation is the fact that the Stark–Heegner construction is now unavailable, in the absence of a prime p||M which is inert in K. The main idea is to bring such a prime into the picture by “raising the level at p” to produce a newform g of level N = M p which is congruent to f . The congruence is modulo an appropriate ideal λ of the ring Og generated by the Fourier coefficients of g. Let Ag denote the abelian variety quotient of J0 (N ) attached to g by the Eichler-Shimura construction. The main objective, which can now be stated more precisely, is to relate the local behaviour at p of the Stark–Heegner points in Ag (Kp ) to the algebraic part of the special value of L(E/K, 1), taken modulo λ. The first key ingredient in establishing such a relationship is an extension of the map Φ of (0.1) to arbitrary eigenforms of weight 2 on Γ0 (M p) such as g, and not just eigenforms with rational Fourier coefficients attached to elliptic curves, in a precise enough form so that phenomena related to congruences between modular forms can be analyzed. Let T be the full algebra of Hecke operators acting on the space of forms of weight two on Γ0 (M p). The theory presented in Section 1, based on the work of the third author [Das], produces a torus T over Kp equipped with a natural T-action, whose character group (tensored with C) is isomorphic as a T ⊗ C-module to the space of weight 2 modular forms on Γ0 (M p) which are new at p. It also builds a Hecke-stable lattice L ⊂ T (Kp ), and a map Φ generalising (0.1) Φ : Γ\(Hp ∩ K) −→ T (Kp )/L.

(0.4)

It is conjectured in Section 1 that the quotient T /L is isomorphic to the rigid analytic space associated to an abelian variety J defined over Q. A strong

4

Massimo Bertolini, Henri Darmon and Samit Dasgupta

partial result in this direction is proven in [Das], where it is shown that T /L is isogenous over Kp to the rigid analytic space associated to the p-new quotient J0 (N )p-new of the jacobian J0 (N ). In Section 1, it is further conjectured that the points Φ(τ ) ∈ J(Kp ) satisfy the same algebraicity properties as were stated for the map Φ of (0.1). Letting Φp denote the group of connected components in the N´eron model of J over the maximal unramified extension of Qp , one has a natural Heckeequivariant projection ∂p : J(Cp ) −→ Φp .

(0.5)

The group Φp is described explicitly in Section 1, yielding a concrete description of the Hecke action on Φp and a description of the primes dividing the cardinality of Φp in terms of “primes of fusion” between forms on Γ0 (M ) and forms on Γ0 (M p) which are new at p. This description also makes it possible to attach to E and K an explicit element ¯ p, L(E/K, 1)(p) ∈ Φ ¯ p is a suitable f -isotypic quotient of Φp . Thanks to a theorem of Popa where Φ [Po], this element is closely related to the special value L(E/K, 1), and, in particular, one has the equivalence L(E/K, 1) = 0

⇐⇒

L(E/K, 1)(p) = 0 for all p.

Section 2 contains an exposition of Popa’s formula. Section 3 is devoted to a discussion of L(E/K, 1)(p) ; furthermore, by combining the results of Sections 1 and 2, it proves the main theorem of this article, an avatar of the Gross-Zagier formula which relates Stark–Heegner points to special values of L-series. Main Theorem. For all primes p which are inert in K, ∂p (PK ) = L(E/K, 1)(p) . Potential arithmetic applications of this theorem (conditional on the validity of the deep conjectures of Section 1) are briefly discussed in Section 4. Aknowledgements. It is a pleasure to thank the anonymous referee, for some comments which led us to improve our exposition.

Stark–Heegner points and special values

5

1 Stark–Heegner points on J0 (M p)p-new Heegner points on an elliptic curve E defined over Q can be defined analytically by certain complex line integrals involving the modular form f :=

∞

an (E)e2πinz

n=1

corresponding to E, and the Weierstrass parametrization of E. To be precise, let τ be any point of the complex upper half plane H := {z ∈ C|z > 0}. The complex number τ Jτ := 2πif (z)dz ∈ C ∞

gives rise to an element of C/ΛE ∼ = E(C), where ΛE is the N´eron lattice of E, and hence to a complex point Pτ ∈ E(C). If τ also lies in an imaginary quadratic subfield K of C, then Pτ is a Heegner point on E. The theory of complex multiplication shows that this analytically defined point is actually defined over an abelian extension of K, and it furthermore prescribes the action of the Galois group of K on this point. The Stark–Heegner points of [Dar], defined on elliptic curves over Q with multiplicative reduction at p, are obtained by replacing complex integration on H with a double integral on the product of a p-adic and a complex upper half plane Hp × H. We now very briefly describe this construction. Let E be an elliptic curve over Q of conductor N = M p, with p M . The differential ω := 2πif (z)dz and its anti-holomorphic counterpart ω ¯ = −2πif (¯ z )d¯ z give rise to two elements in the DeRham cohomology of X0 (N )(C): ω ± := ω ± ω ¯. To each of these differential forms is attached a modular symbol y ± −1 m± {x → y} := (Ω ) ω ± , for x, y ∈ P1 (Q). E E x

± Here Ω± E is an appropriate complex period chosen so that mE takes values in Z and in no proper subgroup of Z. The group Γ defined in the Introduction acts on P1 (Qp ) by M¨obius transformations. For each pair of cusps x, y ∈ P1 (Q) and choice of sign ±, a Z-valued additive measure μ± {x → y} on P1 (Qp ) can be defined by −1 μ± {x → y}(γZp ) = m± x → γ −1 y}, E {γ

(1.1)

where γ is an element of Γ. Since the stabilizer of Zp in Γ is Γ0 (N ), equation (1.1) is independent of the choice of γ by the Γ0 (N )-invariance of m± E . The

6

Massimo Bertolini, Henri Darmon and Samit Dasgupta

motivation for this definition, and a proof that it extends to an additive measure on P1 (Qp ), comes from “spreading out” the modular symbol m± E along the Bruhat-Tits tree of PGL2 (Qp ) (see [Dar], [Das], and Section 1.2 below). For any τ1 , τ2 ∈ Hp and x, y ∈ P1 (Qp ), a multiplicative double integral on Hp ×H is then defined by (multiplicatively) integrating the function (t − τ1 )/(t − τ2 ) over P1 (Qp ) with respect to the measure μ± {x → y}: τ2 y t − τ2 × ω± := × dμ± {x → y}(t) t − τ1 τ1 x P1 (Qp ) ± tU − τ2 μ {x→y}(U) = lim ∈ C× (1.2) p. ||U ||→0 tU − τ 1 U ∈U

Here the limit is taken over uniformly finer disjoint covers U of P1 (Qp ) by open compact subsets U , and tU is an arbitrarily chosen point of U . Choosing special values for the limits of integration, in a manner motivated by the classical Heegner construction described above, one produces special elements in C× p . These elements are transferred to E using Tate’s p-adic uniformization ∼ C× p /qE = E(Cp ) to define Stark–Heegner points. In order to lift the Stark–Heegner points on E to the Jacobian J0 (N )p-new , one can replace the modular symbols attached to E with the universal modular symbol for Γ0 (N ). In this section, we review this construction of Stark–Heegner points on J0 (N )p-new , as described in fuller detail in [Das].

1.1 The universal modular symbol for Γ0 (N ) The first step is to generalize the measures μ± {x → y} on P1 (Qp ). As we will see, the new measure naturally takes values in the p-new quotient of the homology group H1 (X0 (N ), Z). Once this measure is defined, the construction of Stark–Heegner points on J0 (N )p-new can proceed as the construction of Stark– Heegner points on E given in [Dar]. The Stark–Heegner points on J0 (N )p-new will map to those on E under the modular parametrization J0 (N )p-new → E. We begin by recalling the universal modular symbol for Γ0 (N ). Let M := Div0 P1 (Q) be the group of degree zero divisors on the set of cusps of the complex upper half plane, defined by the exact sequence 0 → M → Div P1 (Q) → Z → 0.

(1.3)

The group Γ acts on M via its action on P1 (Q) by M¨obius transformations. For any abelian group G, a G-valued modular symbol is a homomorphism m : M −→ G; we write m{x → y} for m([x] − [y]). Let M(G) denote the

Stark–Heegner points and special values

7

left Γ-module of G-valued modular symbols, where the action of Γ is defined by the rule (γm){x → y} = m{γ −1 x → γ −1 y}. Note that the natural projection onto the group of coinvariants M −→ MΓ0 (N) = H0 (Γ0 (N ), M) is a Γ0 (N )-invariant modular symbol. Furthermore, this modular symbol is universal, in the sense that any other Γ0 (N )-invariant modular symbol factors through this one. One can interpret H0 (Γ0 (N ), M) geometrically as follows. Given a divisor [x]− [y] ∈ M, consider any path from x to y in the completed upper half plane H ∪ P1 (Q). Identifying the quotient Γ0 (N )\(H ∪ P1 (Q)) with X0 (N )(C), this path gives a well-defined element of H1 (X0 (N ), cusps, Z), the singular homology of the Riemann surface X0 (N )(C) relative to the cusps. Manin [Man] proves that this map induces an isomorphism between the maximal torsion-free quotient H0 (Γ0 (N ), M)T and H1 (X0 (N ), cusps, Z). Furthermore, the torsion of H0 (Γ0 (N ), M) is finite and supported at 2 and 3. The projection M → MΓ0 (N) → H1 (X0 (N ), cusps, Z) is called the universal modular symbol for Γ0 (N ). The points of X0 (N ) over C correspond to isomorphism classes of pairs (E, CN ) of (generalized) elliptic curves E/C equipped with a cyclic subgroup CN ⊂ E of order N. To such a pair we can associate two points of X0 (M ), namely the points corresponding to the pairs (E, CM ) and (E/Cp , CN /Cp ), where Cp and CM are the subgroups of CN of size p and M , respectively. This defines two morphisms of curves f1 : X0 (N ) → X0 (M ) and f2 : X0 (N ) → X0 (M ),

(1.4)

each of which is defined over Q. The map f2 is the composition of f1 with the Atkin-Lehner involution Wp on X0 (N ). Write f∗ = f1 ∗ ⊕ f2 ∗ and f ∗ = f1∗ ⊕ f2∗ (resp. f∗ and f ∗ ) for the induced maps on singular homology (resp. relative singular homology): f∗ :

H1 (X0 (N ), Z) → H1 (X0 (M ), Z)2

f∗ :

H1 (X0 (N ), cusps, Z) → H1 (X0 (M ), cusps, Z)2

f∗ :

H1 (X0 (M ), Z)2 → H1 (X0 (N ), Z)

f∗ :

H1 (X0 (M ), cusps, Z)2 → H1 (X0 (N ), cusps, Z).

8

Massimo Bertolini, Henri Darmon and Samit Dasgupta

The abelian variety J0 (N )p-new is defined to be the quotient of J0 (N ) by the images of the Picard maps on Jacobians associated to f1 and f2 . Define H and H to be the maximal torsion-free quotients of the cokernels of f ∗ and f ∗ , respectively: H := (Cokerf ∗ )T and H := (Cokerf ∗ )T . If we write g for the dimension of J0 (N )p-new , the free abelian groups H and H have ranks 2g + 1 and 2g, respectively, and the natural map H → H is an injection ([Das, Prop. 3.2]). The groups H and H have Hecke actions generated by T for N , U for

|N , and Wp . We omit the proof of the following proposition. Proposition 1.1 The group (H/H)T ∼ = Z is Eisenstein; that is, T acts as +1 for N , U acts as for |M , and Wp acts as −1. Proposition 1.1 implies that it is possible to choose a Hecke equivariant map ψ : H → H such that the composition ψ

H −→ H −→H

(1.5)

has finite cokernel. For example, we may take ψ to be the Hecke operator (p2 − 1)(Tr − (r + 1)) for any prime r N . We fix a choice of ψ for the remainder of the paper. 1.2 A p-adic uniformization of J0 (N )p-new For any free abelian group G, let Meas(P1 (Qp ), G) denote the Γ-module of G-valued measures on P1 (Qp ) with total measure zero, where Γ acts by (γμ)(U ) := μ(γ −1 U ). In order to construct a Γ-invariant Meas(P1 (Qp ), H)-valued modular symbol, we recall the Bruhat-Tits tree T of PGL2 (Qp ). The set of vertices V(T ) of T is identified with the set of homothety classes of Zp -lattices in Q2p . Two vertices v and v are said to be adjacent if they can be represented by lattices L and L such that L contains L with index p. Let E(T ) denote the set of oriented edges of T , that is, the set of ordered pairs of adjacent vertices of T . Given e = (v1 , v2 ) in E(T ), call v1 = s(e) the source of e, and v2 = t(e) the target of e. Define the standard vertex v o to be the class of Z2p , and the standard oriented edge eo = (v o , v) to be the edge whose source is v o and whose stabilizer in Γ is equal to Γ0 (N ). Note that E(T ) is equal to the disjoint union of the Γ-orbits of eo and e¯o , where e¯o = (v, v o ) is the opposite edge of eo . A half line of T is a sequence (en ) of oriented edges such that t(en ) = s(en+1 ). Two half lines are said to be equivalent if they have in common all but a finite

Stark–Heegner points and special values

9

number of edges. It is known that the boundary P1 (Qp ) of the p-adic upper half plane bijects onto the set of equivalence classes of half lines. For an oriented edge e, write Ue for the subset of P1 (Qp ) whose elements correspond to classes of half lines passing through e. The sets Ue are determined by the rules: (1) Ue¯o = Zp , (2) Ue¯ = P1 (Qp ) − Ue , and (3) Uγe = γUe for all γ ∈ Γ. The Ue give a covering of P1 (Qp ) by compact open sets. Finally, recall the existence of a Γ-equivariant reduction map r : (Kp − Qp ) −→ V(T ), defined on the Kp -points of Hp . (As before, Kp is an unramified extension of Qp .) See [GvdP] for more details. Define a function κ{x → y} : E(T ) −→ H as follows. When e belongs to the Γ-orbit of eo and γ ∈ Γ is chosen so that γe = eo , let κ{x → y}(e) be ψ applied to the image of γ −1 ([x] − [y]) in H under the universal modular symbol for Γ0 (N ). Let κ{x → y}(e) be the negative of this value when the relation γe = e¯o holds. The function κ{x → y} is a harmonic cocycle on T , that is, it obeys the rules (i) κ{x → y}(¯ e) = −κ{x → y}(e) for all e ∈ E(T ), and (ii) s(e)=v κ{x → y}(e) = 0 for all v ∈ V(T ), where the sum is taken over the p + 1 oriented edges e whose source s(e) is v. Furthermore, we have the Γ-invariance property κ{γx → γy}(γe) = κ{x → y}(e) for all γ ∈ Γ. The natural bijection between Meas(P1 (Qp ), H) and the group of harmonic cocycles on T valued in H shows that the definition μ{x → y}(Ue ) := κ{x → y}(e) yields a Γ-invariant Meas(P1 (Qp ), H)-valued modular symbol μ ([Das, Prop. 3.1]). When m = [x] − [y] ∈ M, we write μm for μ{x → y}. We can now define, for τ1 , τ2 ∈ Hp and m ∈ M, a multiplicative double integral attached to the universal modular symbol for Γ0 (N ): τ2 t − τ2 × ω := × dμm (t) t − τ1 τ1 m P1 (Qp ) tU − τ2 = lim ⊗ μm (U) ∈ C× p ⊗Z H, tU − τ1 ||U ||→0 U ∈U

10

Massimo Bertolini, Henri Darmon and Samit Dasgupta

with notations as in (1.2). One shows that this integral is Γ-invariant: γτ2 τ2 × ω=× ω for γ ∈ Γ. γτ1

γm

τ1

m

Letting T denote the torus T = Gm ⊗Z H, we thus obtain a homomorphism ((Div0 Hp ) ⊗ M)Γ

→ T τ2 ([τ1 ] − [τ2 ]) ⊗ m → × ω. τ1

(1.6)

m

Consider the short exact sequence of Γ-modules defining Div0 Hp : 0 → Div0 Hp → Div Hp → Z → 0. After tensoring with M, the long exact sequence in homology gives a boundary map δ1 : H1 (Γ, M) → ((Div0 Hp ) ⊗ M)Γ .

(1.7)

The long exact sequence in homology associated to the sequence (1.3) defining M gives a boundary map δ2 : H2 (Γ, Z) → H1 (Γ, M).

(1.8)

Define L to be the image of H2 (Γ, Z) under the composed homomorphisms in (1.6), (1.7), and (1.8): H2 (Γ, Z) → T (Qp ). Note that the Hecke algebra T of H acts on T . Theorem 1.2 ([Das], Thm. 3.3) Let Kp denote the quadratic unramified extension of Qp . The group L is a discrete, Hecke stable subgroup of T (Qp ) of rank 2g. The quotient T /L admits a Hecke-equivariant isogeny over Kp to the rigid analytic space associated to the product of two copies of J0 (N )p-new . Remark 1.3 If one lets the nontrivial element of Gal(Kp /Qp ) act on T /L by the Hecke operator Up , the isogeny of Theorem 1.2 is defined over Qp . Remark 1.4 As described in [Das, §5.1], Theorem 1.2 is a generalization of a conjecture of Mazur, Tate, and Teitelbaum [MTT, Conjecture II.13.1] which was proven by Greenberg and Stevens [GS]. Theorem 1.2 implies that T /L is isomorphic to the rigid analytic space associated to an abelian variety J defined over a number field (which can be embedded in Qp ). We now state a conjectural refinement of Theorem 1.2.

Stark–Heegner points and special values

11

Conjecture 1.5 The quotient T /L is isomorphic over Kp to the rigid analytic space associated to an abelian variety J defined over Q. Presumably, the abelian variety J will have a natural Hecke action, and the isomorphism of Conjecture 1.5 will be Hecke equivariant; furthermore we expect that if one lets the nontrivial element of Gal(Kp /Qp ) act on T /L by the Hecke operator Up , the isomorphism will be defined over Qp . The abelian variety J breaks up (after perhaps an isogeny of 2-power degree) into a product J + × J − , where the signs represent the eigenvalues of complex conjugation on H, and Theorem 1.2 (or rather its proof) implies that each of J ± admits an isogeny denoted ν± to J0 (N )p-new . Throughout this article, we will need to avoid a certain set of bad primes. Let S denote a finite set of primes containing those dividing 6ϕ(M )(p2 − 1) or the size of the cokernel of the composite map (1.5). We say that two abelian varieties (or two analytic spaces) are S-isogenous if there is an isogeny between them whose degree is divisible only by primes in S. We expect that ν± may be chosen to be S-isogenies defined over Q, but as we will not need this result in the current article, we refrain from stating it as a formal conjecture.

1.3 Stark–Heegner points on J and J0 (N )p-new Fix τ ∈ Hp and x ∈ P1 (Q). The significance of the subgroup L is that it is the smallest subgroup of T such that the cohomology class in H 2 (Γ, T /L) given by the 2-cocycle γ1−1 τ dτ,x (γ1 , γ2 ) := × τ

γ2 x

ω

(mod L)

x

vanishes (the cohomology class of this cocycle, and hence the smallest trivializing subgroup L, is independent of τ and x). Thus there exists a map βτ,x : Γ → T /L such that γ1−1 τ βτ,x (γ1 γ2 ) − βτ,x (γ1 ) − βτ,x (γ2 ) = × τ

γ2 x

ω

(mod L).

(1.9)

x

The 1-cochain βτ,x is defined uniquely up to an element of Hom(Γ, T /L). The following proposition, which follows from the work of Ihara and whose proof is reproduced in [Das, Prop. 3.7], allows us to deal with this ambiguity. Proposition 1.6 The abelianization of Γ is finite, and any prime dividing its size divides 6ϕ(M )(p2 − 1).

12

Massimo Bertolini, Henri Darmon and Samit Dasgupta

We may now define Stark–Heegner points on J and J0 (N )p-new . Let K be a real quadratic field such that p is inert in K, and choose a real embedding σ of K. For each τ ∈ Hp ∩ K, consider its associated order Oτ as defined in (0.2). Let γτ be the generator of the group of units in Oτ× of norm 1 whose associated λγ (see (0.2)) is greater than 1 under σ. Finally, choose any x ∈ P1 (Q), and let t denote the exponent of the abelianization of Γ. We then define the Stark–Heegner point associated to τ by Φ(τ ) := t · βτ,x (γτ ) ∈ T (Kp )/L. The multiplication by t ensures that this definition is independent of the choice of βτ,x , and one also checks that Φ(τ ) is independent of x. Furthermore, the point Φ(τ ) depends only on the Γ-orbit of τ , so we obtain a map Φ : Γ\(Hp ∩ K) → T (Kp )/L = J(Kp ).

(1.10)

Following [Das], we conjecture that the images of Φ satisfy explicit algebraicity properties analogous to those mentioned in the Introduction. Fix a Z[1/p]-order O in K, and let HK be the narrow ring class field of K attached to O, whose Galois group is canonically identified by class field theory with Pic+ (O). If h is the size of this Galois group, there are precisely h distinct Γ-orbits of points in Hp ∩ K whose associated order is O. Let τ1 , . . . , τh be representatives for these orbits. Conjecture 1.7 The points Φ(τi ) are global points defined over HK : Φ(τi ) ∈ J(HK ).

(1.11)

They are permuted simply transitively by Gal(HK /K), so the point PK := Φ(τ1 ) + · · · + Φ(τh )

(1.12)

lies in J(K). While a proof of this conjecture (particularly, of equation (1.11)) seems far from the methods we have currently developed, one may still hope to glean some information from the p-adic invariants of Stark–Heegner points, and it seems of independent interest to relate such invariants to special values of Rankin L-series. Let PK = J(Kp ) be as in (1.12). The goal of Section 3 is to relate PK to a certain algebraic part of L(E/K, 1), the latter being defined in terms of a formula of Popa that is explained in Section 2. This approach lends itself to generalisations to linear combinations of the points Φ(τi ) associated to the complex characters of Gal(HK /K) (see Section 4 for more details). We now conclude this section by remarking that Stark–Heegner points on J0 (N )p-new are defined by composing the map Φ of (1.10) with the maps ν±

Stark–Heegner points and special values

13

resulting from Theorem 1.2. In [Das] it is conjectured that Stark–Heegner points on J0 (N )p-new are defined over HK ; Conjecture 1.7 may thus be viewed as a refinement.

2 Popa’s formula Let D denote the discriminant of K, and fix an orientation η : OK −→ Z/M Z of the ring of integers O := OK of K. With notations as in the Introduction, there are exactly h = #G different R0 (M )× conjugacy classes of oriented optimal embeddings of O into the order R0 (M ) of matrices in M2 (Z) which are upper triangular modulo M . Let Ψ1 , . . . , Ψh denote representatives for these classes of embeddings. After fixing a fundamental unit K of K of norm one, normalised so that K > 1 with respect to the fixed real embedding of K, set γj := Ψj (K ) ∈ Γ0 (M ).

(2.1)

Let f be the normalised eigenform attached to E. Then we have Proposition 2.1 (Popa) The equality ⎛ h 1/2 2 ⎝ L(E/K, 1) · (D /4π ) = j=1

⎞2 γj z0

f (z)dz ⎠

z0

holds, for any choice of z0 in the extended complex upper half plane. Proof See Theorem 6.3.1 of [Po]. Remark 2.2 The result of Popa, which is stated here for simplicity in the case of the trivial character, deals more generally with twists of the L-series of E/K by complex characters of Pic+ (O) (and even with twists by complex characters attached to more general orders of K). In order to formulate the result in this more general form, one needs to define an action of Pic(OK ) on the set of conjugacy classes of oriented optimal embeddings of O into the order R0 (M ). See [Po] for more details. The eigenform f determines an algebra homomorphism ϕf : T −→ Z satisfying ϕf (Tn ) = an (f ), for (n, N ) = 1,

ϕf (U ) = a (f ), for |N.

14

Massimo Bertolini, Henri Darmon and Samit Dasgupta

Write If for the kernel of ϕf . For a T-module A, let Af := A/If A be the largest quotient of A on which T acts via ϕf . Note that H1 (X0 (M ), Z)f is a Zmodule of rank 2. Given a finite set of primes S, let ZS denote the localization of Z in which the primes of S are inverted. By possibly enlarging S, we may assume that H1 (X0 (M ), ZS )f is torsion-free, and hence a free ZS -module of rank 2. For any such S, denote by [γj ] ∈ H1 (X0 (M ), ZS ) the homology class corresponding to γj , and set [γK ] :=

h

[γj ].

j=1

Define the algebraic part of L(E/K, 1) by the formula

L(E/K, 1) = L(E/K, 1)S := the natural image of [γK ] in H1 (X0 (M ), ZS )f . Proposition 2.1 directly implies the following Corollary 2.3 L(E/K, 1) = 0 if and only if L(E/K, 1) = 0.

3 The main formula The goal of this section is to compute the image of the Stark-Heegner point PK in the group of connected components at p of the abelian variety J introduced in Section 1, and relate it to L(E/K, 1).

3.1 The p-adic valuation The image of the multiplicative double integral under the p-adic valuation map has a simple combinatorial description. Proposition 3.1 ([BDG], Lemma 2.5 or [Das], Lemma 4.2) For τ1 , τ2 ∈ Kp − Qp , and x, y ∈ P1 (Q), the equality τ2 y ordp × ω= κ{x → y}(e) τ1

x

e:v1 →v2

holds in H, where vi ∈ V(T ), i = 1, 2 is the image of τi by the reduction map, and the sum is taken over the edges in the path joining v1 to v2 .

Stark–Heegner points and special values

15

This proposition implies: Proposition 3.2 ([Das], Props. 4.1, 4.9) The image of L under ∂p = ordp ⊗ Id : T (Qp ) = Q× p ⊗H →Z⊗H =H is equal to the image of ker f¯∗ by the composition of ψ with the natural ¯ projection H1 (X0 (N ), cusps, Z) −→ H.

3.2 Connected components and primes of fusion Let Φp denote the quotient Cokerf ∗ / ker f∗ . Let S be a finite set of primes chosen as at the end of section 1.2, that is, S contains the primes dividing 6ϕ(M )(p2 − 1) or the size of the cokernel of the composite map (1.5). The group Φp is finite, and the primes dividing the cardinality of Φp ⊗ ZS are “congruence primes.” This will be discussed further below. Let Φp,S denote the ZS -module Φp ⊗ ZS . By Proposition 3.2, combined with the results of [Das], pp. 438-441, for any unramified extension Kp of Qp , the p-adic valuation gives a well-defined homomorphism ∂p,S : T (Kp )/L → Φp,S . By the theory of p-adic uniformisation of abelian varieties, the group of connected components of the N´eron model of J over the maximal unramified extension of Qp , tensored with ZS , is equal to Φp,S . ˜ resp. T denote the Hecke algebra acting faithfully on Let T, H1 (X0 (N ), Z), resp. H1 (X0 (M ), Z). ˜q for This algebra is generated by the Hecke operators T˜q for q N and U q | N , resp. Tq for q M and Uq for q | M . Identify H1 (X0 (M ), Z)2 with a submodule of H1 (X0 (N ), Z) via f ∗ . Note that H1 (X0 (M ), Z)2 is sta˜ For (n, p) = 1, the action of the operator T˜n ∈ T ˜ ble for the action of T. 2 on H1 (X0 (M ), Z) is equal to the diagonal action of Tn ∈ T; moreover, the

16

Massimo Bertolini, Henri Darmon and Samit Dasgupta

Tp p that Up and Tp (with Tp acting diagonally) satisfy the relation ˜ is equal to that of the operator Up := ˜p ∈ T action of U

−1 0

. Note

Up2 − Tp Up + p = 0. ˜ acting on H1 (X0 (M ), Z)2 is called the p-old The maximal quotient of T ˜ ˜ p-old . quotient of T, and is denoted T ˜ Proposition 3.3 There is a T-equivariant isomorphism Φp,S H1 (X0 (M ), ZS )2 /Im(f∗ ◦ f ∗ ).

(3.1)

Proof The module Φp,S is isomorphic to the quotient of H1 (X0 (N ), ZS ) by the ZS -submodule generated by the image of f ∗ and the kernel of f∗ . It follows from a result of Ribet that the size of the cokernel of f∗ divides ϕ(M ) (see [Rib1, Thm 4.3]). Thus, having tensored with ZS , we find that Φp,S is isomorphic to the cokernel of the endomorphism f∗ ◦ f ∗ of H1 (X0 (M ), ZS )2 .

Corollary 3.4 There is an isomorphism Φp,S ∼ = H1 (X0 (M ), ZS )/(Tp2 − (p + 1)2 ),

(3.2)

˜ resp. Tn ∈ which is compatible for the action of the Hecke operators T˜n ∈ T, T, for (n, p) = 1, on the left-, resp. right-hand side. Proof The endomorphism f∗ ◦ f ∗ is given explicitly by the matrix p+1 Tp f∗ ◦ f ∗ = . Tp p+1 Since p + 1 is invertible in ZS , the cokernel of this map is isomorphic to H1 (X0 (M ), ZS )/(Tp2 − (p + 1)2 ).

Remark 3.5 In this remark, assume as in Section 2 that f is the normalised eigenform attached to E, and write an (f ) ∈ Z for the n-th Fourier coefficient of f . Let S = Sf be a finite set of primes containing those dividing 6ϕ(M )(p2 − 1)(ar (f ) − (r + 1)), for a prime r not dividing N . An argument similar to the proof of Corollary 3.4 shows that there is an isomorphism (Φp,S )f = Φp,S /If Φp,S ∼ = H1 (X0 (M ), ZS )f /(ap (f )2 − (p + 1)2 ). (3.3)

Stark–Heegner points and special values

17

Let λ be a maximal ideal of T belonging to the support of the module H1 (X0 (M ), ZS )/(Tp2 − (p + 1)2 ), and let be the characteristic of the finite field T/λT. The algebra homomorphism π : T −→ T/λT is identified with the reduction in characteristic of a modular form f on Γ0 (M ). (The normalised eigenform attached to an elliptic curve of conductor M has Fourier coefficients in Z, and therefore arises in this way.) ˜ be a maximal ideal of T ˜ arises ˜ compatible with λ, in the sense that λ Let λ p-old ¯ ˜ ¯ from a maximal ideal λ of T , and both λ and λ are contained in a maximal ˜ is guaranteed by the ideal of the Hecke ring T[Up ]. (Note that the existence of λ ˜ going-up theorem of Cohen-Seidenberg.) The isomorphism (3.1) shows that λ is p-new (besides being p-old), since it appears in the support of the component ˜ acts via its maximal p-new quotient. Therefore, λ ˜= group Φp,S , on which T ˜ λg corresponds to the reduction in characteristic of a p-new modular form g on Γ0 (N ). ˜ is an ideal of fusion between the p-old and In the terminology of Mazur, λ the p-new subspaces of modular forms on Γ0 (N ). The forms f and g are called congruent modular forms. For more details on these concepts, see [Rib2].

3.3 Specialisation of Stark-Heegner points This section computes the image ∂p,S (Φ(τ )) of a Stark-Heegner point Φ(τ ) in the group of connected components Φp,S . Assume that S contains the primes dividing 6ϕ(M )(p2 −1)(ar (f )−(r+1)), for a prime r not dividing N , and is such that H1 (X0 (M ), ZS )f is a free ZS -module of rank 2. We begin by imitating the definition of the map κ, with the group Γ0 (M ) ˜ = R× , where R is the order appearing in the replacing Γ0 (N ). Let Γ Introduction. Define a function κ ¯ {x → y} : V(T ) −→ H1 (X0 (M ), cusps, ZS ) ˜ is chosen by setting κ ¯ {x → y}(v) = image of ([γx] − [γy]), where γ ∈ Γ ˜ is Γ0 (M ), and the natural so that γv = v o . Since the stabilizer of v o in Γ homomorphism from M to H1 (X0 (M ), cusps, ZS ) factors through Γ0 (M ), it follows that the map κ ¯ is well defined. ˜g ⊂ T ˜ introduced in Section Recall the compatible ideals λf ⊂ T and λ 3.2. Assume in the sequel that f is the eigenform with rational coefficients ˜ maps to 1 in the ˜p ∈ T attached to E, and that the p-th Hecke operator U ˜ ˜ ˜ g ; the condition ˜ ˜ ˜ λ quotient ring T/λg . (Since λg is p-new, Up maps to ±1 in T/

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Massimo Bertolini, Henri Darmon and Samit Dasgupta

we are imposing is equivalent to requiring that λf belongs to the support of the module H1 (X0 (M ), ZS )/(Tp − (p + 1)).) The identification H1 (X0 (M ), cusps, ZS )/λf H1 (X0 (M ), cusps, ZS ) ˜ g H = Φp,S /λ ˜ g Φp,S , = H/λ which follows from Remark 3.5, implies that the reduction modulo λf of κ ¯ can ˜ g Φp,S -valued function. also be viewed as a Φp,S /λ Lemma 3.6 The relation κ ¯ {x → y}(v ) − κ ¯ {x → y}(v) = κ{x → y}(e)

˜ g Φp,S ) (mod λ

holds for all the oriented edges e = (v, v ). Proof By our choice of S, the reduction modulo λf of κ ¯ yields a map ηf : V(T ) −→ H1 (X0 (M ), ZS )/λf H1 (X0 (M ), ZS ) F2 , where F is the finite field with elements. The map ηf : E(T ) −→ F2 given by the rule ηf (e) = ηf (v ) − ηf (v), for e = (v, v ), defines the p˜p η = η , where stabilised eigenform associated to ηf . It satisfies the relation U f f ˜p acts by sending an oriented edge e to the formal sum of the the operator U ˜ g is a prime of fusion, it follows that oriented edges originating from e. Since λ ˜g . η coincides with the reduction of κ modulo λ f

Let v ∈ V(T ) denote the reduction of τ . Define a 1-cochain ˜ g Φp,S β¯τ,x : Γ → Φp,S /λ by the rule β¯τ,x (γ) = κ ¯ {x → γx}(v). A direct calculation using the equation κ ¯ {x → y}(γv) = κ ¯ {γ −1 x → γ −1 y}(v) along with Lemma 3.6 and Proposition 3.1 shows that −1 γ1 τ ¯ ¯ ¯ βτ,x (γ1 γ2 ) − βτ,x (γ1 ) − βτ,x (γ2 ) = ordp × τ

γ2 x

x

ω

(3.4)

Stark–Heegner points and special values

19

˜ g Φp,S . From equation (1.9) defining βτ,x and the fact that βτ,x is in Φp,S /λ unique up to translation by a homomorphism from Γ, it follows that t · ordp (βτ,x (γ)) = t · β¯τ,x (γ) ˜ g Φp,S . In particular, we have in Φp,S /λ Proposition 3.7 The equality ∂p,S (Φ(τ )) = t · κ ¯ {x → γτ x}(v) ˜ g Φp,S . holds in Φp,S /λ With notations as in the Introduction, let τ = τ1 , . . . , τh be representatives for the distinct Γ-orbits of points in Hp ∩ K corresponding to a real quadratic order O. Define the Stark-Heegner point PK := Φ(τ1 ) + . . . + Φ(τh ). Write vj , j = 1, . . . , h for the image of τj by the reduction map, and γj ∈ O1× for the element appearing in the definition of Φ(τj ). Normalise the τj so that the associated γj are defined as in equation (2.1). This implies that the vertices vj all coincide with the standard vertex v o . One finds Corollary 3.8 The equality ∂p,S (PK ) =

h

t·κ ¯ {x → γi x}(v o )

i=1

˜ g Φp,S . holds in Φp,S /λ Recall the homology element L(E/K, 1) ∈ H1 (X0 (M ), ZS ) defined in Section 2. In light of Proposition 3.3, let L(E/K, 1)(p) denote the natural ˜ g Φp,S . (Note that t is a unit in ZS by image of t · L(E/K, 1) in Φp,S /λ Proposition 1.6, so that L(E/K, 1)(p) is non-zero if and only if the image ˜ g Φp,S is non-zero.) Then, combining Corollary 3.8 of L(E/K, 1) in Φp,S /λ with Proposition 2.1 yields the main theorem of the Introduction: Theorem 3.9 For all primes p which are inert in K, ∂p,S (PK ) = L(E/K, 1)(p) .

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Remark 3.10 The element L(E/K, 1)(p) depends on the choice of primes p and (the residue characteristic of λf ), and of the set S. Given a rational prime p which is inert in K, it is certainly possible that the module H1 (X0 (M ), ZS )f / (ap (f ) − (p + 1)) be zero, and that no modular form ˜ g Φp,S is nong, congruent to f , be available for which the quotient Φp,S /λ zero. For such a choice of p, the statement of Theorem 3.9 amounts to a trivial equality. However, the Chebotarev density theorem can be used to produce infinitely many p (for a fixed ) for which the equality of Theorem 3.9 is non-trivial. Assume that L(E/K, 1) is non-zero, or equivalently by Corollary 2.3, that L(E/K, 1) = L(E/K, 1)S is non-zero, where S is such that H1 (X0 (M ), ZS )f is a free ZS -module of rank 2. By a theorem of Serre, for all but finitely many primes , the element L(E/K, 1) is non-zero modulo

and the Galois representation ρE, attached to E[ ]—the -torsion of E—is surjective. A standard application of the Chebotarev density theorem (see for example [BD2]) shows that there exist infinitely many primes p which are inert in K and such that the following conditions are satisfied: (i) divides the integer ap (f ) − (p + 1), (ii) does not divide 6ϕ(M )(p2 − 1)(ar (f ) − (r + 1)), for a prime r M p. Enlarge the set S above by including all the primes dividing the quantity 6ϕ(M )(p2 −1)(ar (f )−(r+1)). For such an S, the results of Section 3.2—see in particular Remark 3.5—show that there exists a congruent form g for which ˜ g Φp,S . (Note that in this case, L(E/K, 1)(p) is a non-zero element of Φp,S /λ ˜ g Φp , since all the primes in S are the latter quotient is identified with Φp /λ units modulo . Thus, the formula of Theorem 3.9 can be written by omitting a reference to S.)

4 Arithmetic applications The Shimura reciprocity law implies that the points Φ(τ ), as τ varies over Hp ∩ K, satisfy the same norm-compatibility properties as classical Heegner points attached to an imaginary quadratic K, and it is expected that they should yield an “Euler system” in the sense of Kolyvagin. (Cf. [BD1], Prop. 6.18). Theorem 3.9 gives a relationship between Stark–Heegner points and special values of related Rankin L-series, and one might ask whether this result could have applications to the arithmetic of elliptic curves analogous to those of the Gross-Zagier theorem. For example, assume Conjecture 1.7 that PK belongs to J(K). Theorem 3.9 then shows that, when L(E/K, 1) = 0, the points PK are of infinite order for infinitely many p (in fact, for precisely those p for which L(E/K, 1)(p) = 0).

Stark–Heegner points and special values

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The PK can then be used to construct a large and well-behaved supply of cohomology classes in H 1 (K, Ep ). Following the methods of Kolyvagin, such classes could be used to prove the following theorem: Theorem 4.1 Assume Conjecture 1.7. If L(E/K, 1) = 0, the Mordell-Weil group and Shafarevich-Tate group of E over K are finite. We omit the details of the proof, but point out that such a proof would follow that same strategy as in [BD2], but with Stark-Heegner points replacing the classical Heegner points that are used in [BD2]. (See also [Lo] where a similar strategy is used to prove the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank 0 defined over totally real fields which do not necessarily arise as quotients of modular or Shimura curves.) Theorem 4.1 has the drawback of being conditional on Conjecture 1.7—a limitation that appears all the more flagrant when one notes that the conclusion of this theorem already follows, unconditionally, from earlier results of Kolyvagin (or of Kato) applied in turn to E/Q and to the twist of E by the even Dirichlet character associated to K. However, greater generality could be achieved by introducing a ring class character χ : Gal(HK /K) −→ C× and considering twisted special values of L(E/K, χ, 1) along with related eigencomponents of the Mordell-Weil group E(HK ) and of the ShafarevichTate group X := X(E/HK ): E(HK )χ := {P ∈ E(HK )⊗C such that σP = χ(σ)P, ∀ σ ∈ Gal(HK /K)} Xχ := {x ∈ X ⊗ Z[χ] such that σx = χ(σ)x, ∀ σ ∈ Gal(HK /K)}. In light of Proposition 2.1 and Remark 2.2, Theorem 3.9 generalises directly to a relation between the special value L(E/K, χ, 1) and the images of the χparts of Stark-Heegner points in connected components. Furthermore, when χ is not a quadratic character an unconditional proof of the following theorem would appear to lie beyond the scope of the known Euler systems discovered by Kolyvagin and Kato, and would yield a genuinely new arithmetic application of the conjectural Euler system made from Stark–Heegner points: Theorem 4.2 Assume conjecture 1.7 of Section 1. If L(E/K, χ, 1) = 0, then the Mordell-Weil group E(HK )χ is trivial and the Shafarevich-Tate group Xχ is finite.

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Part of the inspiration for Theorem 3.9 is the strong analogy between it and the “first explicit reciprocity law” of chapter 4 of [BD2] in the setting where K is imaginary. (See in particular the displayed formula in lemma 8.1 of [BD2].) In [BD2] this first explicit reciprocity law was used in conjunction with a “second explicit reciprocity law” to prove one divisibility in the anticyclotomic Main Conjecture of Iwasawa Theory (for K imaginary quadratic). Such a main conjecture has no counterpart when K is real quadratic (since K has no “anticyclotomic Zp -extension”) but versions of this statement over ring class fields of K (of finite degree) remain non-trivial and meaningful. Note in this connection that it may be interesting to formulate a convincing substitute for the “second explicit reciprocity law” of [BD2] describing the local behaviour of the point PK at primes different from p.

Bibliography [BD1] M. Bertolini, H.Darmon, The p-adic L-functions of modular elliptic curves. in Mathematics unlimited—2001 and beyond, 109–170, Springer, Berlin, 2001. [BD2] M. Bertolini, H. Darmon, The main conjecture of Iwasawa theory for elliptic curves over anticyclotomic Zp -extensions. Ann. of Math. (2) 162 (2005), no. 1, 1–64. [BD3] M. Bertolini, H. Darmon, The rationality of Stark-Heegner points over genus fields of real quadratic fields. Ann. of Math., to appear. [BDG] M. Bertolini, H. Darmon, P. Green, Periods and points attached to quadratic algebras. MSRI Publ. 49, Cambridge Univ. Press, 323-367, 2004. [Dar] H. Darmon, Integration on Hp ×H and arithmetic applications. Ann. of Math. (2) 154 (2001), no. 3, 589–639. ´ [Das] S. Dasgupta, Stark–Heegner points on modular Jacobians. Ann. Scient. Ec. Norm. Sup., 4e s´er., 38 (2005), 427-469. [DG] H. Darmon, P. Green, Elliptic curves and class fields of real quadratic fields: algorithms and verifications. Experimental Mathematics, 11:1, 2002, 37-55. [GvdP] L. Gerritzen, M. van der Put, Shottky Groups and Mumford Curves. Lecture Notes in Mathematics, 817. Springer, Berlin, 1980. [GS] R. Greenberg, G. Stevens, p-adic L-functions and p-adic periods of modular forms. Invent. Math. 111 (1993), no. 2, 407–447. [GZ] B. H. Gross, D. B. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84 (1986), no. 2, 225–320. [Lo] M. Longo, On the Birch and Swinnerton-Dyer for modular elliptic curves over totally real fields, Ann. Inst. Fourier (Grenoble) 56 (2006) no. 3, 689–733. [Man] J.I. Manin, Parabolic Points and Zeta Functions of Modular Curves. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), no. 1, 19–66. [MTT] B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84 (1986), no. 1, 1–48. [Po] A. Popa, Central values of Rankin L-series over real quadratic fields. Compos. Math. 142 (2006) no. 4, 811–866.

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[Rib1] K. Ribet, Congruence Relations Between Modular Forms. Proceedings of the International Congress of Mathematicians, Warsaw, August 16-24, 1983. ¯ [Rib2] K. Ribet, On modular representation of Gal(Q/Q) arising from modular forms, Invent. Math. 100 (1990) 431-476. [Se] J.P. Serre, Trees. Springer, Berlin, 1980. [W] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. of Math. 141 (1995) 443-551.

Presentations of universal deformation rings Gebhard B¨ockle Fachbereich Mathematik Universit¨at Duisburg-Essen Campus Essen 45117 Essen Germany [email protected]

Abstract Let F be a finite field of characteristic > 0, F a number field, GF the absolute Galois group of F and let ρ¯ : GF → GLN (F) be an absolutely irreducible continuous representation. Suppose S is a finite set of places containing all places above and above ∞ and all those at which ρ¯ ramifies. Let O be a complete discrete valuation ring of characteristic zero with residue field F. In such a situation one may consider all deformations of ρ¯ to O-algebras which are unramified outside S and satisfy certain local deformation conditions at the places in S. This was first studied by Mazur, [16], and under rather general hypotheses, the existence of a universal deformation ring was proven. In [2] I studied, among other things, the number of generators needed for an ideal I in a presentation of such a universal deformation ring as a quotient of a power series ring over O by I. The present manuscript is an update of this part of [2]. The proofs have been simplified, the results slightly generalized. We also treat = 2, more general groups than GLN , and cases where not all relations are local. The results in [2] and hence also in the present manuscript are one of the (many) tools used in the recent attacks on Serre’s conjecture by C. Khare and others. 1

Introduction Let us consider the following simple lemma from commutative algebra: Lemma 0.1 Suppose that a ring R has a presentation of the form R = W (F)[[T1 , . . . , Tn ]]/(f1 , . . . , fm ). If R/( ) is finite, and if n ≥ m, then n = m and R is a complete intersection that is finite flat over W (F). 1 1991 Mathematics Subject Classification. Primary 11F34, 11F70, Secondary 14B12.

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Presentations of universal deformation rings

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If the ring R in the above lemma is a universal deformation ring for certain deformation types of a given residual representation, then the conclusion of the lemma would provide one with a lift to characteristic zero of this deformation type. This observation was first made by A.J. de Jong, [9], (3.14), in 1996. Using obstruction theory and Galois cohomology, in [2] we investigated the existence of presentations of universal deformation rings of the type required in the lemma. In many cases such a presentation was found. However the finiteness of the ring R/( ) seemed to be out of reach. This was changed enormously by the ground breaking work [22] of R. Taylor where a potential version of Serre’s conjecture was proved. The results of Taylor do allow one in many cases to prove the finiteness of R/( ). I first learned about this from C. Khare soon after [22] was available. This gives a powerful tool to construct -adic Galois representations that are (potentially) semistable or ordinary at and have prescribed ramification properties at primes away from . Besides the deep modularity results for such representations provided by Wiles, Taylor, Skinner et al., the results in [2] were one of the ingredients of the recent proof of Serre’s conjecture for conductor N = 1 and arbitrary weight by Khare in [10] (cf. also [11]). This is based on previous joint work between Khare and Wintenberger [12], and a result by Dieulefait [7]. Dieulefait also has some partial results on Serre’s conjecture [8].

The present manuscript is an update of those parts of [2] which study the number of generators needed for an ideal I in a presentations of a given universal deformation ring as a quotient of a power series ring over O by I. The proofs have been simplified and the results generalized. We also treat = 2, more general groups than GLN and cases where not all relations are local. A key improvement is that the use of auxiliary primes has been avoided entirely. We hope that this will be useful for the interested reader. We now give a summary of the individual sections. In Section 1, we start by briefly recalling Mazur’s fundamental results on universal deformations with the main emphasis on presentations of universal deformation rings. In Section 2 we give a first link between the ideals of presentations of local and of global deformation rings in the setting of Mazur adapted to global number fields. The discrepancy is measured by W2S of the adjoint representation of the given residual representation. It is natural to put further local restrictions on the initial deformation problem studied by Mazur. To obtain again a representable functor the local conditions need to be relatively representable. In Section 3 we present a possibly useful variant of this notion.

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The core of the present article is Section 4, cf. Corollary 4.3. Here we study presentations of (uni)versal deformation rings for deformations in the sense of Mazur that moreover satisfy a number of local conditions that follow the axiomatics in Section 3. The obstruction module W2S is replaced by a naturally occurring dual Selmer group. The main novelty of the present paper is that unlike in [2] we do not require that this dual Selmer group vanishes. Instead we incorporate it into the presentation of the corresponding ring. The final three sections investigate consequences of our results. In Section 5 we make some general comments and study the case of GL2 in detail. In particular, we present the numerology for local ordinary deformation rings over arbitrary local fields (of any characteristic). In Section 6 we compare our results to those in [15, 23] of Mauger and Tilouine. The last section, Section 7, is dedicated to deriving a presentation of a global universal deformation rings as the quotient of a power series ring over the completed tensor product over all local versal deformation rings. The main result here is due to M. Kisin [13]. We show how to derive it using the results of Section 4. Notation: For the rest of this article, we fix the following notation: F is a finite field of characteristic . The ring of Witt vectors of F is denoted W (F). For a local ring R its maximal ideal is denoted by mR . By O we denote a complete discrete valuation ring of characteristic zero with residue field F, so that in particular O is finite over W (F). The category of complete noetherian local O-algebras R with a fixed isomorphism R/mR ∼ = F will be CO . Here and in the following F[ε]/(ε2 ) is an O-algebra via O → F → F[ε]/(ε2 ). For a ring R of CO its mod mO tangent space is defined as tR := HomO (R, F[ε]/(ε2 )) ∼ = HomF (mR /(m2R + mO R), F). For J an ideal of a ring R in CO , we define gen(J) := dimF J/(mR J). By Nakayama’s lemma, gen(J) is the minimal number of generators of J as an ideal in R. By F we denote a number field and by S a finite set of places of F . We always assume that S contains all places of F above and ∞. The maximal outside S unramified extension of F inside a fixed algebraic closure F alg of F is denoted FS . It is a Galois extension of F whose corresponding Galois group is GF,S := Gal(FS /F ). For each place ν of F let Fν be the completion of F at ν, let Gν be the absolute Galois group of Fν , and Iν ⊂ Gν the inertia subgroup. Choosing for each such ν a field homomorphism FS → Fνalg , we obtain induced group homomorphisms Gν → GF,S .

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Acknowledgments: This article owes many ideas and much inspiration to the work of Mazur, Wiles, Taylor, de Jong, and many others. Many thanks go to C. Khare for constantly reminding me to write un ‘update’ of the article [2] and for many comments. Many thanks also to Mark Kisin for having made available [13] and for some interesting related discussions.

1 A simple deformation problem In this section we recall various basic notions and concepts from [16]. In terms of generality, we follow [23], and so we fix a smooth linear algebraic group G over O. By ZG we denote the center of G, by T we denote a smooth affine algebraic group over O that is a quotient of G via some surjective homomorphism d : G → T of algebraic groups over O. The kernel of d is denoted G0 . The Lie algebras over O corresponding to G and G0 will be g and g0 , respectively. Example 1.1 (i) d := det : G := GLN → T := GL1 . Then g = MN (F) and g0 ⊂ g is the subset of trace zero matrices. (ii) G is the Borel subgroup of GLN formed by the set of upper triangular N matrices, T := GLN 1 , and d : G → GL1 is the the quotient homomorphism of G modulo its unipotent radical. The corresponding Lie algebras are the obvious ones. Throughout this section let Π be a profinite group such that the pro- completion of every open subgroup is topologically finitely generated. (This is the finiteness condition Φ of [16], Def. 1.1.) Let us fix a continuous (residual) representation ρ¯ : Π → G(F). The adjoint representation of Π on g(F) is denoted by adρ¯, its subrepresentation on g0 (F) ⊂ g(F) by ad0ρ¯. For M an F[Π]-module, we define its dimension as hi (Π, M ) := dimF H i (Π, M ). Following Mazur we first consider the following simple deformation problem: A lifting of ρ¯ to R ∈ CO is a continuous representation ρ : Π → G(R), such that ρ (mod mR ) = ρ. ¯ A deformation of ρ¯ to R is a strict equivalence class [ρ] of liftings ρ of ρ¯ to R, where two liftings ρ1 and ρ2 from Π to G(R) are strictly equivalent, if there exists an element in the kernel of G(R) → G(F) which conjugates one into the other.

28

Gebhard B¨ockle We consider the functor Def O,Π : CO → Sets : R → {[ρ] | [ρ] is a deformation of ρ¯ to R}.

Theorem 1.2 ([16]) Suppose Π and ρ¯ are as above. Then (i) The functor Def O,Π has a versal hull, which we denote by ρρ,O ¯ : Π → G(Rρ,O ). ¯ (ii) If furthermore the centralizer of Im(¯ ρ) in G(F) is contained in ZG (F), then Def O,Π is representable by the above pair (Rρ,O ¯ , ρρ,O ¯ ). 1 ∼ (iii) tRρ,O H (Π, ad ). = ρ¯ ¯ ∼ (iv) Rρ,O has a presentation Rρ,O = O[[T1 , . . . , Th ]]/J for some ideal ¯ ¯ J ⊂ O[[T1 , . . . , Th ]], where h = h1 (Π, adρ¯) and gen(J) ≤ h2 (Π, adρ¯). Proof The proof is essentially contained in [16] §1.2, §1.6, where a criterion of Schlessinger is verified. For (ii) Mazur originally assumed that ρ¯ was absolutely irreducible. It was later observed by Ramakrishna, [20], that this could be weakened to the condition given. A proof for GLN instead of a general group G in the precise form above can be found in [2], Thm. 2.4. The adaption to general G is obvious, and so we omit details. Since this will be of importance later, we remark that the proofs in [16] or [2] show that there is a canonical surjective homomorphism H 2 (Π, adρ¯)∗ −→ → J/mO[[T1 ,...,Th ]] J.

(1.1)

of vector spaces over F 1 Remark 1.3 If Rρ,O ¯ /( ) is known to have Krull dimension h (Π, adρ¯) − 2 h (Π, adρ¯), then Theorem 1.2, which is obtained entirely by the use of obstruction theory, implies that Rρ,O is flat over O of relative dimension ¯ 1 2 h (Π, adρ¯) − h (Π, adρ¯), and a complete intersection. So in this situation Theorem 1.2 has some strong ring-theoretic consequences for Rρ,O ¯ . In the generality of the present section, it can not be expected that the Krull 1 2 dimension of Rρ,O ¯ /( ) is always equal to h (Π, adρ¯)−h (Π, adρ¯)+1. Recent work [1] by Bleher and Chinburg shows that this fails for finite groups Π and also for Galois groups Π = GF,S in case S does not contain all the primes above . If G = GLn and ρ¯ is absolutely irreducible, and if Π = GF,S and S contains all primes above , or if Π = GFν (cf. Remark 5.2), all evidence suggests that Rρ,O is a complete intersection, flat over O and of relative dimension ¯

Presentations of universal deformation rings

29

h1 (Π, adρ¯) − h2 (Π, adρ¯). But the amount of evidence is small. On the one hand, there is Leopoldt’s conjecture, cf. [16], 1.10. On the other, if one is bootstrapping the content of [4] in light of the recent modularity results of Taylor et al., e.g. [22], there is some evidence for odd two-dimensional representations ρ¯ of GF,S where F is a totally real field. One often considers the following subfunctor of Def Π,O : Let η : Π → T (O) be a fixed lift of the residual representation d ◦ ρ¯ : Π → T (F). Then one defines the subfunctor Def ηΠ,O of Def Π,O as the functor which to R ∈ CO assigns the set of all deformations [ρ] of ρ¯ to R for which the composite d ◦ ρ : Π → T (R) and the composite τ ◦ η of η with the canonical homomorphism τ : T (O) → T (R) only differ by conjugation inside T (R). One obtains the following analog of Theorem 1.2: Theorem 1.4 ([16]) Suppose Π and ρ¯ are as above. Then: (i) The functor Def ηΠ,O has a versal hull, which we denote by ρηρ,O ¯ : Π → G(Rηρ,O ). ¯ (ii) If furthermore the centralizer of Im(¯ ρ) in G(F) is contained in ZG (F), η then Def ηO,Π is representable by the above pair (Rρ,O ¯ ). ¯ , ρρ,O 0 1 η 1 1 ∼ η (iii) tRρ,O = H (Π, adρ¯) := Im(H (Π, adρ¯) → H (Π, adρ¯)). ¯ η ∼ (iv) Rηρ,O has a presentation Rρ,O = O[[T1 , . . . , Thη ]]/J η for some ¯ ¯ η ideal J ⊂ O[[T1 , . . . , Thη ]], where hη = dimF H 1 (Π, ad0ρ¯)η and gen(J η ) ≤ dimF H 2 (Π, ad0ρ¯). Remark 1.5 From the long exact cohomology sequence for 0 −→ g0 −→ g −→ g/g0 −→ 0 0 it follows that H 1 (Π, adρ¯)η ∼ = H 1 (Π, adρ¯) is an isomorphism if and only if

H 0 (Π, adρ¯) −→ → H 0 (Π, adρ¯/ad0ρ¯) is surjective. To measure the discrepancy between hη and h1 (Π, ad0ρ¯), we define δ(Π, adρ¯) = 0 and δ(Π, adρ¯)η := dimF Coker(H 0 (Π, adρ¯) −→ H 0 (Π, adρ¯/ad0ρ¯)) = h1 (Π, ad0ρ¯) − hη .

(1.2)

As an example consider the case d = det : G = GLN → T = GL1 . If

does not divide N , then adρ¯ = ad0ρ¯ ⊕ F, where here F denotes the trivial representation of Π, and so δ(Π, adρ¯)η = 0. However for |N and absolutely irreducible ρ¯, one finds that δ(Π, adρ¯)η = 1.

30

Gebhard B¨ockle

If η = 1, one can in fact consider two deformation functors: (i) the functor that arises from considering deformations into G0 instead of G, and (ii) the functor Def ηO,Π considered above. If δ(Π, adρ¯)η = 0, the two agree. Otherwise, the functor for G0 is less rigid, and in fact its mod mO tangent space has a larger dimension (the difference being given by δ(Π, adρ¯)η ). Note also that the bound for gen(J η ) in part (iv) is solely described in terms of ad0ρ¯.

2 A first local to global principle For the remainder of this article we fix a residual representation ρ¯ : GF,S → G(F). Whenever it makes sense, we fix a lift η : GF,S → T (O) of d ◦ ρ¯. (If T (F) is of order prime to , such a lift always exists.) As in the previous section, the adjoint representation of GF,S on g(F) is denoted by adρ¯, its subrepresentation on g0 (F) ⊂ g(F) by ad0ρ¯. To ρ¯ we can attach the following canonical deformation functors: First, we define Def S,O := Def GF,S ,O ,

and

Def ηS,O := Def ηGF,S ,O .

The first functor parameterizes all deformations of ρ¯ which are unramified outside S, the second (sub)functor moreover fixes the chosen determinant η. Let ν be any place of F . The restriction of ρ to Gν defines a residual representation Gν → G(F), the restriction of η to Gν a lift Gν → T (O) of d ◦ ρ¯ restricted to Gν . Thus we may define local deformation functors by Def ν,O := Def Gν ,O ,

and

Def ην,O := Def ηGν ,O .

Notational convention: In the sequel we often write ?(η) in formulas. This expresses two assertions at once: First, the formula is true if the round brackets are missing throughout. Second, the formula is also true if (η) is entirely omitted throughout the formula. Corresponding to the above cases the usage (0) of adρ¯ has to be interpreted as follows: If brackets around (η) are omitted, (0) then they are to be omitted around (0) in adρ¯ , too; if (η) is omitted, then (0) (0)

in adρ¯ is to be omitted, as well. By global, respectively local class field theory, the groups GF,S and Gν satisfy the conditions imposed on the abstract profinite group Π in Section 1. Therefore Theorems 1.2 and 1.4 are applicable to ρ¯ and its restriction to the

Presentations of universal deformation rings

31 (η)

groups Gν . The resulting (uni)versal global deformations are denoted by ρS,O : (η)

(η)

(η)

GF,S → RS,O , and the local ones by ρν,O : Gν → Rν,O . We also set h := h1 (GF,S , adρ¯), hη := h1 (GF,S , adρ¯)η , hν := h1 (Gν , adρ¯), hην := h1 (Gν , adρ¯)η . With the above notation, Theorem 1.2 shows that there exist presentations (η)

0 −→ Jν(η) −→ O[[Tν,1, . . . , Tν,h(η) ]] −→ Rν,O −→ 0, ν

(η)

0 −→ J (η) −→ O[[T1 , . . . , Th(η) ]] −→ RS,O −→ 0.

(2.1)

(2.2)

The restriction Gν → GF,S applied to deformations, induces a natural transformation of functors Def S,O → Def ν,O . ν∈S

This yields a ring homomorphism Rν(η) −→ RS(η) , ⊗

ν∈S

we denote the completed tensor product over the ring O. Using where by ⊗, the smoothness of O[[T1 , . . . , Th(η) ]], and the above presentations, we obtain a commutative diagram with inserted dashed arrows, where α is a product of (η) local maps αν : O[[Tν,1, . . . , Tν,h(η) ]] → O[[T1 , . . . , Th(η) ]] and where Jν ν

(η)

denotes the ideal generated by the Jν in / O[[Tν,1 , . . . , Tν,h(η) | ν ∈ S]] Jν(η) : ν ∈ S ν α= αν / O[[T1 , . . . , Th(η) ]] J (η)

//

ˆ ν∈S R(η) ⊗ ν

//

(2.3)

(η)

RS,O .

For any F[GF,S ] module M , define W2S (M ) = Ker(H 2 (GF,S , M ) −→ ⊕ν∈S H 2 (Gν , M )). Our first result on a local to global relation is the following simple consequence of Theorems 1.2 and 1.4: (η)

Theorem 2.1 The ideal J (η) is generated by the images of the ideals Jν , (0) ν ∈ S, together with at most dimF W2S (adρ¯ ) further elements. In particular, if the corresponding W2S (. . .) vanishes, then all relations in (η) J are local.

32

Gebhard B¨ockle

Proof By (1.1), there is a surjection H 2 (GF,S , adρ¯ )∗ −→ → J (η) /mO[[T1 ,...,Th(η) ]] J (η) , (0)

and similarly for the local terms. Comparing local and global terms yields the commutative diagram

/

H 2 (Gν , adρ¯ )∗ (0)

ν∈S

(η) (η) ν∈S Jν /mO[[T1 ,...,T (η) ]] Jν hν

/

∗ / / W2S (ad(0) ρ ¯ )

H 2 (GF,S , adρ¯ )∗ (0)

J (η) /mO[[T1 ,...,T

]] h(η)

J (η)

where the lower horizontal homomorphism is induced from the ring homomorphism α of the previous diagram, and where the vertical homomorphisms are surjective. By Nakayama’s Lemma, any subset of J (η) whose image generates (η) J /mO[[T1 ,...,Th(η) ]] J (η) forms a generating system for J (η) . Therefore the assertion of the theorem follows immediately from the above diagram. Remark 2.2 An obvious consequence of Theorem 2.1 is the inequality (0) gen(J (η) ) ≤ dimF W2S (adρ¯ ) + gen(Jν(η) ). ν∈S

In general, this inequality is not best possible, since one has the exact sequence (0)

(0)

0 −→ W2S (adρ¯ ) −→ H 2 (GF,S , adρ¯ ) −→ ⊕ν∈S H 2 (Gν , adρ¯ ) −→ H 0 (GF,S , (adρ¯ )∨ )∗ −→ 0. (0)

(0)

3 Local conditions For the applications to modularity questions, the functors considered in the previous section are too general. At places ν above the prime modular Galois representations are potentially semistable; at places ν away from , one often wants to prescribe a certain behavior of the local Galois representations in (η) (η) question. This leads one to consider subfunctors Def of the functors Def ν,O

ν,O

that describe a certain type of local deformation. An important requirement on these subfunctors is that the resulting global deformation problems should have a versal hull. There are various approaches to achieve this. We find it most convenient to work with the notion of relative representability, which is basically described in [17], § 19.

Presentations of universal deformation rings

33

Let us recall from [2], § 2, the relevant notion of relative representability: Following Schlessinger a homomorphism π : A → C of Artin rings in CO is called a small extension if π is surjective and if the kernel of π is isomorphic to the A-module F. In [17], p. 277, in the definition of small, the requirement of surjectivity is left out. Therefore the statement of Schlessinger’s Theorem as given there is weaker than that given in [21]. The statement in [17], p. 277, is also true if small morphisms are assumed to be surjective. A covariant functor F : CO → Sets is called continuous, if for any directed inverse system (Ai )i∈I of Artin rings in CO with limit A := lim Ai in CO , one ←− has F (A) = lim F (Ai ). ←− Definition 3.1 Given two covariant continuous functors F, G : CO → Sets such that G is a subfunctor of F , we say that G is relatively representable if (i) G(k) = ∅, and (ii) for all small surjections f1 : A1 → A0 and maps f2 : A2 → A0 of artinian rings CO , the following is a pullback diagram: G(A1 ×A0 A2 )

/ G(A1 ) ×G(A0 ) G(A2 )

F (A1 ×A0 A2 )

/ F (A1 ) ×F (A0 ) F (A2 ).

Remark 3.2 The definition of relative representability given in [17] seems at the outset more restrictive. However, by a reduction procedure similar to that of Schlessinger in [21], our definition might be equivalent to the the one given in [17]. The property from [17] is the one that is satisfied for essentially all subfunc(η) (η) tors Def ⊂ Def that have been considered in deformation problems ν,O

ν,O

for Galois representations. Hence in all of these cases, the local deformation problems are relatively representable in the above sense. Proposition 3.3 Suppose F, Fi , Gi : CO → Sets, i ∈ I, I a finite set, are covariant continuous functors. Suppose for each i ∈ I that Gi is a relatively representable subfunctor of Fi . Then the following holds: (i) If Fi has a hull, i.e., Fi satisfies conditions (H1 ), (H2 ) and (H3 ) of Schlessinger, [21], Thm. 2.11, or [17], § 18, then so does Gi . If Fi is representable, then so is Gi .

34

Gebhard B¨ockle (ii) The product i∈I Gi is a continuous subfunctor of i Fi which is relatively representable. (iii) Suppose the Fi have a versal hull. Let α : F → i Fi be a natural transformation, and let G be defined as the pullback of G _ _ _/ Gi / Fi . F Then, if F has a versal hull, then so does G, and if F is representable, then so is G.

The proof exploits the representability criterion of Schlessinger. It is a simple exercise in diagram chasing, and left to the reader. After the above detour on general representability criteria, let us come back to the deformation functors we introduced in the previous section. The functors (η) (η) Def S,O and Def ν,O are continuous. To work with finer local conditions, for each place ν in S we fix relatively representable subfunctors (η)

(η) Def ν,O ⊂ Def ν,O . (η)

We also define Def S,O as the pullback of functors in the diagram (η) _ _ _/ (η) Def ν∈S Def ν,O S,O (η) (η) / Def S,O Def ν,O .

By Proposition 3.3, we obtain: (η)

(η) Proposition 3.4 The functors Def ν,O : Gν → ν,O have a versal hull ρ (η)

). The functor Def ). G(R S,O : Gν → G(R S,O is representable, say by, ρ ν,O S,O (η) (η) (η) (η) The induced ring homomorphisms R → R and R → R are (η)

(η)

ν,O

ν,O

(η)

S,O

S,O

surjective.

4 A refined local to global principle (η)

We keep the hypotheses of the previous sections that the subfunctors Def ν,O (η)

⊂ Def ν,O are relatively representable. In this section, we want to derive an

Presentations of universal deformation rings

35

analog of Theorem 2.1, i.e., some kind of local to global principle for the (η) . The necessary substitute for W2 (adρ¯) refined deformation problem Def S,O S is a certain dual Selmer group. In our exposition of generalized Selmer groups, we follow Wiles, cf. also [18], (8.6.19) and (8.6.20). (η) (η) is an epimorphism, Let us consider a place ν of S. Since Rν,O → R ν,O there is an inclusion of mod tangent spaces tR (η) → tR(η) . Via the isomorν,O

ν,O

(η) phism H 1 (Gν , adρ¯)(η) ∼ = tR(η) this yields a subspace Lν ⊂ H 1 (Gν , adρ¯)(η) ν,O

(η)

(η) . Its dimension will be denoted canonically attached to Def hν . From the ν,O (η)

interpretation of Lν as a mod mO tangent space, we deduce the existence of a presentation (η) −→ 0. 0 −→ Jν(η) −→ O[[Tν,1, . . . , Tν,h(η) ]] −→ R ν,O ν

(4.1)

(η)

The collection (Lν )ν∈S is often abbreviated by L(η) . Let us also denote by L0ν ⊂ H 1 (Gν , ad0ρ¯) the inverse image of Lην ⊂ H 1 (Gν , adρ¯)η under the surjection H 1 (Gν , ad0ρ¯) −→ → H 1 (Gν , adρ¯)η . Convention on notation: For the refined deformation problems, the universal ring and the ideals in a presentation, and the dimensions of the mod mO tangent spaces are given a tilde. For the corresponding subspaces of H 1 (. . .)(η) we stick to the commonly used notation L?ν . We denote by χ ¯cyc the mod cyclotomic character. For any finite F[GF,S ]module M , we define M (i) := M ⊗ χ ¯icyc and denote by M ∨ the Cartier dual ∨ of M as an F[GF,S ]-module, i.e., M = HomF (M, F)(1). Example 4.1 Any simple Lie algebra is self-dual via the Killing form. This (0) (0) often proves adρ¯ ∼ = (adρ¯ )∨ . For instance consider d = det : G = GLN → 0 GL1 . If | N , then g0 is simple, and so (ad0ρ¯)∨ ∼ = adρ¯. This self-duality can be realized quite explicitly by the perfect trace pairing (A, B) → Tr(AB) on MN (F) (which also shows that adρ¯ is self-dual for G = GLN ). If | N this pairing restricts to a non-degenerate pairing on g0 (F). For |N , the pairing 0 is degenerate on the traceless matrices MN (F), but induces a non-degenerate 0 pairing on MN (F) modulo the subrepresentation of scalar matrices. The obvious pairing M ×M ∨ → F(1) yields the perfect Tate duality pairing H 2−i (Gν , M ) × H i (Gν , M ∨ ) → H 2 (Gν , F(1)) ∼ = F, ∨ 1 i ∈ {0, 1, 2}. Applied to M = adρ¯, one defines L⊥ ν ⊂ H (Gν , adρ¯ ) as the 1 annihilator of Lν ⊂ H (Gν , adρ¯) under this pairing for M = adρ¯, and one

36

Gebhard B¨ockle

0 η,⊥ sets L⊥ := (L⊥ as the annihilator of L0ν ν )ν∈S . For M = adρ¯, one defines Lν η,⊥ η,⊥ under this pairing, and one sets L := (Lν )ν∈S . It is now standard to define the Selmer group HL1 (GF,S , adρ¯) as the pullback of the diagram HL1 (GF,S , adρ¯) _ _ _ _ _/ ν∈S Lν _ _ res / 1 1 H (Gν , adρ¯), H (GF,S , adρ¯) ν∈S

where the lower horizontal map is the restriction on cohomology. The analo⊥ gous diagram with ad∨ ρ¯ in place of adρ¯ and Lν in place of Lν defines the dual ∨ 1 Selmer group HL⊥ (GF,S , adρ¯ ). By analogy, we define HL1 (GF,S , adρ¯)η as the pullback of the diagram HL1 (GF,S , adρ¯)η _ _ _ _ _/ ν∈S Lην _ _ res / 1 η 1 H (G H (GF,S , adρ¯)η ν , adρ¯) . ν∈S The space HL1 (GF,S , adρ¯)(η) is readily identified with the tangent space of (η) (η) . For its dimension we write R h . Thus we have presentations: S,O 0 −→ J(η) −→ O[[T1 , . . . , Th(η) ]] −→ R S,O −→ 0. (η)

(4.2)

Note that Im(H 0 (GF,S , adρ¯/ad0ρ¯) → H 1 (GF,S , ad0ρ¯)) injects under the canonical restriction homomorphism into each of the H 1 (Gν , ad0ρ¯). From this and our definition of the L0ν , one deduces that there is a short exact sequence 0 → Im(H 0 (GF,S , adρ¯/ad0ρ¯) → H 1 (GF,S , ad0ρ¯)) → HL1 0 (GF,S , ad0ρ¯) → HL1 (GF,S , adρ¯)η → 0.

(4.3)

For the proof of Theorem 4.2 below, we recall the following consequence of Poitou-Tate global duality, [18], (8.6.20): For M ∈ {adρ¯, ad0ρ¯} and L the usual L or L0 , respectively, there is a five term exact sequence 0 −→ HL1 (GF,S , M ) −→ H 1 (GF,S , M ) −→ H 1 (Gν , M )/Lν ν∈S

−→

HL1 ⊥ (GF,S , M ∨ )∗

−→ W2S (M ) −→ 0.

By our definition of L0ν , we have H 1 (Gν , ad0ρ¯)/L0ν ∼ = H 1 (Gν , adρ¯)η /Lην . From the exact sequence (4.3) and the above 5-term sequence we thus obtain

Presentations of universal deformation rings

37

the 5-term exact sequence 0 → HL1 (GF,S , adρ¯)(η) → H 1 (GF,S , adρ¯)(η) →

H 1 (Gν , adρ¯)(η) /L(η) ν

ν∈S

→

(0) HL1 (η),⊥ (GF,S , (adρ¯ )∨ )∗

(0)

→ W2S (adρ¯ ) → 0.

(4.4)

As in Section 2, one can compare local and global presentations of deformation rings also for the more restricted deformation problems. Jν

(η)

: ν ∈ S

/ O[[Tν,1, . . . , T (η) | ν ∈ S]] ν,h

//⊗ (η) ˆ ν∈S R ν,O

/ O[[T1 , . . . , T (η) ]] h

//R (η) . S,O

ν

J(η)

Theorem 4.2 As an ideal, J(η) is generated by the images of the ideals Jν , (0) ν ∈ S together with at most dimF HL1 (η),⊥ (GF,S , adρ¯ ) other elements. In particular (0) gen(J(η) ) ≤ gen(Jν(η) ) + dimF HL1 (η),⊥ (GF,S , adρ¯ ). (η)

ν∈S

Proof Let us first consider the local situation. The following diagram compares (η) (η) , the local presentations (2.1) and (4.1) for the functors Def and Def ν,O

ν,O

respectively: / O[[Tν,1, . . . , T (η) ]] ν,h

/ / R(η) ν,O

(η) Jν

/ O[[Tν,1, . . . , T (η) ]] ν,h

//

(η)

Jν

_

(η) Jν

ν

ν

(η) R ν,O

πν

/ O[[Tν,1, . . . , T (η) ]] ν,h ν

//R (η) . ν,O

The ideal Jν is the kernel of the composite O[[Tν,1, . . . , Tν,h(η) ]] → Rν,O → ν (η) . The epimorphism πν is chosen so that the lower right square comR (η)

(η)

ν,O

mutes. We may rearrange the variables in such a way that πν is concretely (η) given by mapping Tν,i to Tν,i , for i ≤ hν , and by mapping Tν,i to zero (η) for i > hν . Let moreover fν,1, . . . , fν,rν denote a minimal set of elements

38

Gebhard B¨ockle

(η) in O[[Tν,1, . . . , Tν,h(η) ]] whose images in O[[Tν,1, . . . , Tν,h(η) ]] generate Jν . ν ν (η) Then a set of generators of Jν is formed by the elements

fν,1 , . . . , fν,rν , Tν,h(η) +1 , . . . , Tν,h(η) . ν

ν

Now we turn to the global situation. By Theorem 2.1 the relation ideal in the (η) presentation (2.2) of RS,O is generated by local relations together with at most r := dimF W2S (M ) further elements f1 , . . . , fr . Let the αν and α = ν αν be (η) we have the following homomorphisms as in diagram (2.3). For the ring R S,O (η)

two presentations. First, since Def S,O ⊂ Def S,O is defined by imposing local (η)

(η)

conditions, we may take the presentation of RS,O and consider its quotient by further local relations. Second, we have the presentation (4.2). We obtain the following commutative diagram with exact rows {αν (Jν(η) ); ν ∈ S} ∪ {f1 , . . . , fr }

J(η)

/

/

O[[T1 , . . . , Th(η) ]]

//

(η) R S,O

//

(η) . R S,O

π

O[[T1 , . . . , T (η) ]] h

(η) Since h = dim tR (η) , the homomorphism π is surjective. By properly S,O

choosing the coordinate functions Ti , we may thus assume that π is given as (η) (η) Ti → Ti for i = 1 . . . , h and Ti → 0 for i > h . To further understand π, we interprete the F-dual of sequence (4.4) as an assertion on the variables of our local and global presentations. Defining Δ via 0 −→ W2S (adρ¯ )∗ −→ HL1 (η),⊥ (GF,S , (adρ¯ )∨ ) −→ Δ −→ 0, (0)

(0)

we have 0 −→ Δ −→

∗ 1 (η) ∗ (H 1 (Gν , M )(η) /L(η) ) ν ) −→ (H (GF,S , M )

ν∈S

−→ (HL1 (GF,S , M )(η) )∗ −→ 0. For R ∈ CO we have t∗R = mR /(mO + m2R ). This gives an interpretation for the H 1 (. . .)∗ -terms: • The (images of the) elements T1 , . . . , Th(η) form an F-basis of (HL1 (GF,S , M )(η) )∗ . • The (images of the) elements T1 , . . . , Th(η) form an F-basis of (H 1 (GF,S , M )(η) )∗ .

Presentations of universal deformation rings

39

• The (images of the) elements Tν,l(η) +1 , . . . , Tν,h(η) form an F-basis of ν

ν

(H 1 (Gν , M )(η) /Lν )∗ . Thus in the set V := ν∈S αν ({Tν,h(η) +1 , . . . , Tν,h(η) }) we may choose ν ν (η) h(η) − h elements which form a basis of the F-span of {Th(η) +1 , . . . , Th(η) }. Using the freedom we have in choosing the variables Th(η) +1 , . . . , Th(η) , we may assume that these are precisely the chosen ones from V . Hence under π, these chosen variables all map to zero. We may therefore conclude the following: The ideal J(η) is spanned by the (η) images of the relations fν,j , ν ∈ S, j = 1, . . . , lν , i.e., the local relations in (η) , together with the images of the elements fj , a minimal presentation of R ν,O j = 1, . . . , r, and together with the (η) (η) d := (h(η) − h(η) ) ν − hν ) − (h (η)

ν∈S

further elements in V which may or may not map to zero under π. Since d = (0) dimF Δ, and d + r = dimF HL1 (η),⊥ (GF,S , adρ¯ ), the assertion of the theorem is shown. Corollary 4.3 For the presentation 0 −→ J(η) −→ O[[T1 , . . . , Th(η) ]] −→ R S,O −→ 0 (η)

one has h(η) − gen(J(η) ) ≥ h0 (GF,S , adρ¯ ) − h0 (GF,S , (adρ¯ )∨ ) − δ(GF,S , adρ¯)(η) (0) (η) 0 (η) + h(η) + δ(G , ad ) − h (G , ad ) − gen( J ) . ν ρ ¯ ν ρ¯ ν ν (0)

(0)

(4.5)

ν∈S

Proof Following Wiles, cf. [18] (8.6.20), and using (4.3) we have (0) h(η) + δ(GF,S , adρ¯)(η) − dimF HL1 (η),⊥ (GF,S , (adρ¯ )∨ ) (0) (0) (0) 0 = h0 (GF,S , adρ¯ ) − h0 (GF,S , (adρ¯ )∨ ) + ( h(0) ν − h (Gν , adρ¯ )). ν∈S

By our definition of the bound for gen(J desired estimate.

(0) hν (η)

(0) hν

(η) hν

we have = + δ(Gν , adρ¯)(η) . Subtracting (η) ) from Theorem 4.2 from the quantity h yields the

Remark 4.4 Because of Remark 2.2, we expect the above estimate to be opti(0) mal in the case that h0 (GF,S , (adρ¯ )∨ ) = 0. If F contains -th roots of unity,

40

Gebhard B¨ockle

the same remark shows that for adρ¯ the above estimate will not be optimal. (For G = GL1 , i.e, for class field theory, the reader may easily verify this.) If S,O adρ¯ = ad0ρ¯⊕F this problem can be remedied since then the universal ring R η with the deformation ring for oneis the completed tensor product of R S,O dimensional representations. By class field theory (and Leopoldt’s conjecture) the latter is well-understood.

5 General remarks and the case G = GL2 The aim of this section is to analyze the terms occurring in estimate (4.5) given in Corollary 4.3 for the number of variables minus the number of relations in a (η) . After some initial general remarks we shall soon focus presentation of R S,O on the case G = GL2 . The main result is Theorem 5.8. (η)

For many naturally defined subfunctors Def ν,O ⊂ Def ν,O (for ν ∈ S) (for instance for the examples presented below) one has the following: (η)

(0) (0) (η) (i) If ν | , then hν − h0 (Gν , adρ¯ ) − gen(Jν ) ≥ 0. (ii) If one imposes a suitable semistability condition on deformations at places ν| , and a suitable parity condition at places above ∞, then (0) 0 (η) h(0) ν − h (Gν , adρ¯ ) − gen(Jν ) ≥ 0. ν| or ν|∞

The estimate in (i) is typically easy to achieve, and without any requirements on the restriction of ρ¯ to Gν . This is presently not so for (ii) at places ν| : If ρ¯ satisfies some ordinariness condition at ν, then the ring parameterizing deformations satisfying a similar ordinariness conditions is relatively well understood. If on the other hand ρ¯ is flat at ν, then suitable deformation rings are only well understood and well-behaved if the order of ramification of ρ¯ at ν is relatively small. If (i), resp. (ii) are satisfied, then in all known cases the corresponding local deformation ring is a complete intersection, finite flat over O thus of Krull (η) (η) dimension hν − gen(Jν ). One may ask what consequences known properties of the global universal ring will have on the local universal rings. To discuss this, suppose (i) and (ii) (η) (η) is equal to are satisfied and that the dimension of R h −gen(J(η) ). Since S,O typically the term (ii) arises from precise estimates for all ν| , ∞, having equal(η) ity in (4.5) yields explicit expressions for all the terms gen(Jν ). Hence each of them will be equal to some “expected” expression. If furthermore the term

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41

(η) (η) is a complete gen(Jν ) is at most 1, then one can even conclude that R S,O intersection.

We now turn to some examples, first for the local situation: Example 5.1 ν | , ∞: (i) At such places one has h1 (Gν , M ) − h0 (Gν , M ) − h2 (Gν , M ) = 0 for the local Euler-Poincar´e characteristic for any finite F[Gν ]-module (η) (0) (0) (η) M . Thus for Def ν,O one obtains hν −h0 (Gν , adρ¯ )−gen(Jν ) ≥ 0. (ii) For the local deformation problems defined by Ramakrishna in [19], η is smooth over O of relative dimension Prop. 1, p. 122, the ring R ν,O 0 0 1 h (Gν , adρ¯) = h (Gν , ad0ρ¯) − h2 (Gν , ad0ρ¯) over O; cf. the remark in [19], p. 124. Here h0ν − h0 (Gν , ad0ρ¯) − gen(Jνη ) = 0. (iii) The local deformation problem defined in [10], Prop. 3.4, is smooth of relative dimension 1 over O and again one has h0ν − h0 (Gν , ad0ρ¯) − gen(Jνη ) = 0. (iv) The local deformation problem in [6], p. 141, in the definition of R at places ν ∈ P , i.e., at prime number p with p ≡ −1 (mod h) again defines a local deformation problem with versal representing ring smooth of relative dimension 1 over O. As in the previous cases one has h0ν − h0 (Gν , ad0ρ¯) − gen(Jνη ) = 0. Remark 5.2 Let ρ¯ : GFν → GL2 (F) be arbitrary. Building on previous work of Boston, Mazur and Taylor-Wiles it is shown for ν | in [3] that the Krull η dimension of Rν,O is for any choice of ρ¯ equal to h1 (Gν , ad0ρ¯) − h2 (Gν , ad0ρ¯). In [3] the same is shown for ν| for all possible ρ¯. Thus by Remark 1.3 in these η cases the rings Rν,O are known to be complete intersections of the expected dimensions. So the estimate in part (i) above is optimal for d = det : G = GL2 → T = GL1 . For parts (ii), (iii) and (iv) the same can be shown (e.g. by explicit calculation). Example 5.3 ν|∞, > 2: If ν is real, then Gν is generated by a complex conjugation cν (of order 2). For > 2 and R ∈ CO , the group ring R[Z/(2)] has idempotents for the two R-projective irreducible representations of cν . Hence for any deformation [ρ] of ρ¯ to R, the lift of ρ¯Gν is unique up to isomorphism. (η) (η) (0) Therefore hν = 0 = gen(Jν ), and h0 (Gν , adρ¯ ) depends on the action (0) of cν on adρ¯ , more precisely on the conjugacy class of ρ¯(cν ). For = 2 the problem is more subtle, cf. Example 5.4.

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Gebhard B¨ockle

(η) (η) If ν is complex, then Gν acts trivially, and again hν = 0 = gen(Jν ) (this (0) (0) also holds for = 2). Clearly one has h0 (Gν , adρ¯ ) = dimF adρ¯ .

For cases with ν| we refer to Examples 5.5 and 6.1. For a case with ν|∞ and

= 2, we refer to Example 5.4. For the remainder of this section, we assume that d = det : G = GL2 → T = GL1 . One calls a residual representation ρ¯ odd, if for any real place ν of F one has det ρ¯(cν ) = −1. Note that for = 2, the condition det ρ¯(cν ) = −1 is vacuous. Example 5.4 Suppose ν is real and ρ¯ is odd. If = 2, then h0 (Gν , ad0ρ¯) = 1, and so from the remarks in Example 5.3 it is clear that h0ν − h0 (Gν , ad0ρ¯) − gen(Jνη ) = −1. If = 2, the main interest lies in deformations which are odd, i.e., for which the image of cν is non-trivial. The following two cases are the important ones for G = GL2 , and say with η fixed: Case I: ρ¯(cν ) is conjugate to then given by

0 1 1 0

. The versal hull for this problem is

ρην,O : Z/(2) = cν −→ GL2 (O) : cν →

0 1 1 0

.

We have hην = 0, h0 (Gν , ad0ρ¯) = 2 and gen(Jνη ) = 0, so that δ(Gν , adρ¯)η + hην − h0 (Gν , ad0ρ¯) − gen(Jνη ) = 1 + 0 − 2 − 0 = −1. Case II: ρ¯(cν ) is conjugate to 10 01 . Then the versal hull of a good deformation problem at ν (so that the deformations are odd whenever this is reasonable) is given by ρην,O : Z/(2) = cν −→ GL2 (O[[a, b, c]]/(a2 + 2a + bc)) b cν → 1+a c −1−a . We have hην = 3, h0 (Gν , ad0ρ¯) = 3, gen(Jνη ) = 1 and (!) δ(Gν , adρ¯)η = 0, so that δ(Gν , adρ¯)η + hην − h0 (Gν , ad0ρ¯) − gen(Jνη ) = 0 + 3 − 3 − 1 = −1.

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Example 5.5 We now turn to the case ν| . Case I: Fν = Q , h0 (Gν , ad0ρ¯) = 0, and ρ¯GK is flat at ν for some finite extension K of Q of ramification degree at most − 1 so that the corresponding group scheme and its Cartier dual are both connected. Then by [5] and [20], η η ∼ one has R ν,O = O[[T ]] for a suitable flat deformation functor Def ν,O . Because

= 2 by Remark 1.5 we have δ(Gν , adρ¯)η = 0. Hence δ(Gν , adρ¯)η + hην − h0 (Gν , ad0ρ¯) − gen(Jνη ) = 1 = [Fν : Q ]. Case II: ρ¯ is ordinary at ν. We recall the computation of the obstruction theoη η retic invariants for Def ν,O ⊂ Def ν,O (much of these computations is contained in work of Mazur and Wiles): Suppose we are given a residual representation ¯ χ ¯ b ρ¯ : Gν → GL2 (F) : σ → 0 χ¯−1 η¯ν , (5.1) where χ ¯ is unramified and η¯ν denotes the mod mO reduction of ην = η|Gν . We make the following (standard) hypotheses: The image Im(¯ ρ) is not con−1 tained in the set of scalar matrices, and, if χ ¯ = χ ¯ η¯ν , then (after possibly twisting by a character) we assume that χ ¯=χ ¯−1 η¯ν is the trivial character. In particular this means that if ¯b = 0, then χ ¯2 = η¯ν . By an ordinary lift of fixed determinant we mean a lift of the form χ b ρ : Gν → GL2 (R) : σ → 0 χ−1 η¯ν , where χ is unramified. Since Im(¯ ρ) is not contained in the set of scalar matrices, passing to strict equivalence classes of such lifts defines a relatively η η representable subfunctor Def ν,O ⊂ Def ν,O in the sense of Definition 3.1. η , and To compute the mod mO tangent space of the corresponding ring R ν,O

a bound on the number of relations in a minimal presentation we distinguish several subcases: (i) = 2 (ii) = 2 and χ ¯2 = η¯ν , (iii) = 2 and χ ¯2 = η¯ν (and so by our assumptions on ρ¯, we have χ ¯ = η¯ν = 1.). Subcase (i), = 2: Let us denote by ρ a lift to F[ε]/(ε2 ), and use ρ¯ also to denote the trivial lift. Then σ → ρ(σ)¯ ρ(σ)−1 −1 = χ(σ)χ¯0 (σ)

¯(σ)b(σ)) η ¯ν−1 (σ)(b(σ)χ(σ)−χ ¯ χ−1 (σ)χ(σ) ¯

=:

1+εc1 (σ) εc2 (σ) 0 1−εc1 (σ)

(5.2)

defines a 1-cocycle into the upper triangular matrices in ad0ρ¯. Because we assume = 2, one may in fact verify that the matrix entries c1 and c2 are also 1cocycles for a suitable module. In fact they yield classes [c1 ] ∈ H 1 (Gν /Iν , F) ˆ one has H 1 (Gν /Iν , F) ∼ and [c2 ] ∈ H 1 (Gν , χ ¯2 η¯ν−1 ). Since Gν /Iν ∼ = Z, = F.

44

Gebhard B¨ockle

If ρ and ρ are lifts to F[ε]/(ε2 ) of the required form, such that ρ and ρ are conjugate by 1 + εa for some a ∈ ad0ρ¯ which is upper triangular, then 1-cocycles for ρ and ρ give rise to the same cohomology classes. Conversely, if to a given pair of classes, one chooses different 1-cocycles, the resulting lifts ρ , ρ differ by conjugation by a 1 + εa for some a ∈ ad0ρ¯ which is upper triangular. Regarding strict equivalence one has the following easy if tedious result: For arbitrary a ∈ ad0ρ¯ the conjugate of any lift ρ to F[ε]/(ε2 ) under 1 + εa is again a lift of the required form if and only if one of the following happens: (i) a ∈ ad0ρ¯ is arbitrary if χ ¯2 = η¯ν and ¯b(Iν ) = {0} 0 (ii) a ∈ adρ¯ is upper triangular otherwise. Case (i) means that the image of ρ¯ is an -group and that ρ¯ is unramified. We define δν,unr to be 1 in case (i) and 0 in case (ii). Then we have h0ν = dimF tR η

ν,O

= 1 + h1 (Gν , χ ¯2 η¯ν−1 ) − δν,unr .

Similarly, one can compute the obstruction to further lift a representation χ b ρ : Gν → GL2 (R) : σ → 0 χ−1 ην to a representation ρ given by

χ b 0 (χ )−1 ην

for a small surjection R → R.

Letting χ be an unramified character which lifts χ (and always exists since ˆ is of cohomological dimension one) and b a set-theoretic Gν /Iν ∼ = Z continuous lift, as is standard, one shows that (s, t) → ρ (st)ρ (t)−1 ρ (s)−1 =: 10 cs,t 1 defines a 2-cocycle of Gν with values in χ ¯2 η¯ν−1 , and so we obtain a class in η 2 2 −1 H (Gν , χ ¯ η¯ν ). This gives the bound gen(Jν,O ) ≤ h2 (Gν , χ ¯2 η¯ν−1 ). As a last ingredient, we compute h0 (Gν , ad0ρ¯). This leads to the identity ¯ ! β bχ ¯−1 −2αχ ¯¯ b¯ ην−1 +β χ ¯2 η ¯ν−1 −γ ¯ b2 η ¯ν−1 α β ρ¯ αγ −α ρ¯−1 = α+γ = , −2 −1 ¯ γ −α γχ ¯ η ¯ν −α−γ bχ ¯ and so gives the conditions γ¯b = 0, γ(1 − η¯ν χ ¯−2 ) = 0, β(1 − η¯ν χ ¯−2 ) = 2αχ ¯−1¯b = 0. Since under our hypotheses we cannot have ¯b = 0 and η¯ν = χ ¯2 simultaneously, we obtain γ = 0. From the last condition we see that the vanishing of β depends on η¯ν = χ ¯2 or not. So we find h0 (Gν , ad0ρ¯) = 1 + h0 (Gν , χ ¯2 η¯ν−1 ).

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45

Using the formula for the local Euler-Poincar´e characteristic at a place ν|

2 one obtains i=0 hi (Gν , χ ¯2 η¯ν−1 ) = −[Fν : Q ]. Hence h0ν − h0 (Gν , ad0ρ¯) − gen(Jνη ) ≥ h0 (Gν , χ ¯2 η¯ν−1 ) − h0 (Gν , ad0ρ¯) + [Fν : Q ] + 1 − δν,unr [Fν : Q ] − δν,unr

=

η . for the corresponding ring R ν,O From now on, we assume = 2. In this case δ(Gν , adρ¯)η = 0 by Remark 1.5, and so h0ν = hην . Now for = 2, the 1-cocycle defined in (5.2) cannot be decomposed in two independent 1-cocycles, and so one proceeds differently: Let (n ⊂)b ⊂ ad0ρ¯ denote the subrepresentations on (strictly) upper triangular matrices of ad0ρ¯. Following Wiles, we see that the cocycle defines a cohomology class in 1 Hstr := Ker(H 1 (Gν , b) −→ H 1 (Iν , b/n)).

One observes that H 1 (Gν , b) −→ H 1 (Iν , b/n) factors via H 1 (Gν , b/n), and that the action of Gν on b/n ∼ = F is trivial. Using the left exact inflation1 restriction sequence one finds that Hstr is the pullback of the diagram H 1 (Gν /Iν , b/n)

H 1 (Gν , b)

/ H 1 (Gν , b/n).

Case (ii), = 2 and χ ¯2 = η¯ν . We claim that H 1 (Gν , b) → H 1 (Gν , b/n) is surjective: Using the long exact sequence of cohomology it suffices to show that H 2 (Gν , n) −→ H 2 (Gν , b) −→ H 2 (Gν , b/n) −→ 0 (0)

is also exact on the left. Using Tate local duality, one has h2 (Gν , adρ¯ ) = h0 (Gν , (adρ¯ )∨ ). The formulas 1 if χ ¯2 = η¯ν h2 (Gν , n) = 0 else (0)

1 if χ ¯cyc is trivial 0 else in b by column vectors α β ,

h2 (Gν , b/n) =

β follow readily. Representing matrices α0 −α the representation of σ ∈ Gν on b is given by 1 0 α → −2χ(σ) ¯ β ¯ b(σ) η ¯−1 (σ)χ ¯2 (σ) ν

α β

.

46

Gebhard B¨ockle

The invariants of the ∨-dual of b are the solutions (α, β) in F2 to the equations 1 2χ ¯−1 (σ)¯ b(σ)¯ ην (σ) α α −1 = χ ¯ (σ) −2 cyc β β , 0 η ¯ν (σ)χ ¯ (σ) where σ ranges over all elements of Gν . For fixed σ, the dimension of the solution space is 2 minus the rank of the matrix χ ¯cyc (σ)−1 2χ ¯−1 (σ)χ ¯cyc (σ)¯ b(σ)¯ ην (σ) . 0 η ¯ν (σ)χ ¯−2 (σ)χ ¯cyc (σ)−1 If χ ¯cyc (σ) is non-trivial and different from η¯ν−1 (σ)χ ¯2 (σ), it follows that 0 ∨ 0 H (Gν , (adρ¯) ) = 0. For varying σ, the maximal rank has to be non-zero, since otherwise we would have 1 = χ ¯cyc = η¯ν−1 χ ¯2 , contradicting our hypotheses. One concludes that if one of the identities 1 = χ ¯cyc or 1 = η¯ν−1 χ ¯2 holds, 0 ∨ 0 then the rank of H (Gν , (adρ¯) ) is 1, and otherwise, it is 2. It follows that h2 (Gν , b) − h2 (Gν , n) − h2 (Gν , b/n) = 0, and so the claim is shown. By the claim the horizontal homomorphism in the above pullback diagram is surjective. By the inflation restriction sequence the vertical homomorphism H 1 (Gν /Iν , b/n) → H 1 (Gν , b/n) is injective. As H 1 (Gν /Iν , b/n) ∼ = ˆ F) ∼ H 1 (Z, = F, we therefore deduce that 1 dimF Hstr = h1 (Gν , b) − h1 (Gν , b/n) + 1.

Using the local Euler-Poincar´e formula and the above results, we find that 1 dimF Hstr

=

[Fν : Q ] + 1 + h2 (Gν , b) − h2 (Gν , F) + h0 (Gν , b) − h0 (Gν , F)

=

[Fν : Q ] + h2 (Gν , n) + h0 (Gν , b).

One easily shows that h0 (Gν , b) = h0 (Gν , ad0ρ¯). To compute hην there is as in case (i) the question about strict equivalence. 1 The definition of Hstr only takes conjugation by upper triangular elements into account. However, from χ2 = η¯ν , one may easily deduce (as in case (i)) that conjugating by a matrix of the form 1 + εa, a ∈ ad0ρ¯, preserves the upper 1 diagonal form of a lift to F[ε]/(ε2 ) only if a lies in b. Hence in fact Hstr does η describe the mod mO tangent space of the versal hull of Def ν,O ⊂ Def ην,O . Thus hη = dimF H 1 . ν

str

The computation of possible obstructions proceeds as in case (i). The analη ysis given there does not depend on = 2. Again one finds gen(Jν,O ) ≤ 2 h (Gν , n). Combining the above results, we find that η hην − gen(Jν,O ) − h0 (Gν , ad0ρ¯) ≥ [Fν : Q ].

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47

Case (iii), = 2 and χ ¯2 = η¯ν . In this case the image of ρ¯ is an elementary of Gν . abelian -group. Therefore all the lifts factor via the pro- quotient G ν This group is known to a sufficient degree, as to yield a precise estimate for η hην − gen(Jν,O ) − h0 (Gν , ad0ρ¯) by direct computation. This could be deduced from [3]. But for completeness we chose to give a simple direct argument. There are two cases. Suppose first that Fν does not contain a primitive -th ν is a free pro- group on root of unity. Then by [14], Thm. 10.5, the group G (topological) generators s, t1 , . . . , tn , n = [Fν : Q ], such the normal closure ν . By our hypothesis on ρ¯, for any lift of the ti form the inertia subgroup of G to some ring R in CO one has 1+α β 1 τ1 1 τn s → , t → , . . . , t → −1 1 n 0 ην (t1 ) 0 ην (tn ) . 0 (1+α) ην (s) Obviously there are no obstructions to lifting. The element α lies in mR . The elements τi lie in mR precisely if ρ(t ¯ i ) is trivial. Not having taken strict equivalence into account we have therefore n + 2 independent variables. The analysis of the effect of strict equivalence proceeds as in case (i) and leads to the same cases (a) and (b) as described there. Hence one finds hην = n + 2 − (1 + δν,unr ) = [Fν : Q ] + 1 − δν,unr . Moreover h0 (ad0ρ¯) = 1, and since there are no obstructions to lifting, we find η hην − gen(Jν,O ) − h0 (Gν , ad0ρ¯) = [Fν : Q ] − δν,unr .

Let us now assume that Fν does contain a primitive -th root of unity, and let q be the largest -power so that Fν contains a primitive q-th root of unity. Let F be the free pro- group on (topological) generators s, t0 , . . . , tn , n = [Fν : Q ]. For closed subgroups H, K of F , let [H, K] denote the closed subgroup of F generated by commutators [h, k] = h−1 k −1 hk, h ∈ H, k ∈ K, and denote by 2 N the closed normal subgroup [F, F ]∩F q [F q , F ][F q [F, F ], F ]. Then by [14], ν is the quotient of F by the closed normal subgroup Thm. 10.9, the group G generated by the element r := tq00 tq11 [t0 , s][t1 , t2 ] . . . [tn−1 , tn ]r for some -powers q0 , q1 which are divisible by q, and for some element r ∈ N . The isomorphism may be chosen, so that the closed normal subgroup ν . By our hypothesis on generated by the ti maps to the inertia subgroup of G ρ¯, for any lift to some ring R in CO one has 1+α β 1 τ1 1 τn s → 0 ην (t0 ) , . . . , tn → 0 ην (tn ) . 0 (1+α)−1 ην (s) , t0 →

48

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If the variables α, β and τi are chosen arbitrarily, the image of r in GL2 (R) is

of the form 10 x1 for some x ∈ mR . The reason is as follows: The image of such a ρ is upper triangular. Passing to the quotient modulo the unipotent upper triangular normal subgroup gives a representation into R∗ × R∗ , which , by our hypothesis factors via Gν /Iν . Now r lies in the inertia subgroup of G ν and so its image in R∗ × R∗ is zero. In fact the expression x is computable in terms of α, β and the τi up to ˜ some error coming from r . It is useful to write each variable ξ as ξ0 + ξ, where ξ0 is the Teichm¨uller lift of the reduction mod mR of ξ, and ξ˜ ∈ mR ˜ τ˜0 , . . . , τ˜n ]]. Working modulo m3 , one can evaluate x and for R = O[[˜ α, β, R show that it does not vanish. It may happen that x lies in mR (m2R , mO ). η one has gen(R η ) ≤ 1, Therefore in a minimal presentation (4.1) of R ν,O ν,O η ) is n + 3 − 1 = n + 2 minus the number of and moreover hη − gen(R ν

ν,O

variables that disappear when passing from lifts to deformations, i.e., to strict equivalence classes of lifts. The analysis of the effect of strict equivalence is as in the case where no primitive -th root of unity lies in Fν , and so we need to subtract (1 + δν,unr ) from n + 2. This yields η ) = n + 1 − δ ,unr . hην − gen(R ν ν,O Since h0 (Gν , ad0ρ¯) = 1 in case (iii), we found, independently of a primitive

-th root of unity being in Fν or not, that η hην − gen(Jν,O ) − h0 (Gν , ad0ρ¯) = n − δν,unr = [Fν : Q ] − δν,unr .

Let us summarize our results: η

Proposition 5.6 Suppose ρ¯ is ordinary at ν, i.e., of the form (5.1). Let Def ν,O ⊂ η Def ν,O denote the subfunctor of ordinary deformations with fixed determinant. Define δν,unr to be 1 if at the same time ρ¯ is unramified and Im(¯ ρ) is an -group, and to be zero otherwise. Then η hην − gen(Jν,O ) − h0 (Gν , ad0ρ¯) ≥ [Fν : Q ] − δν,unr .

Example 5.7 Lastly, we need to discuss the global terms in the estimate (4.5). Let G denote the quotient of Im(¯ ρ) modulo its intersection with the center of GL2 (F), and assume that G is non-trivial. We first give the results for = 2. There, independently of F , by explicit computation one finds: 2 if G is abelian, h0 (GF,S , adρ¯) = h0 (GF,S , (adρ¯)∨ ) = 1 otherwise.

Presentations of universal deformation rings h

0

(GF,S , ad0ρ¯)

=

2 1

49

if G is a 2-group (and hence abelian), otherwise.

η

δ(GF,S , adρ¯) = ⎧ 2 ⎪ ⎪ ⎨ 1 h0 (GF,S , (ad0ρ¯)∨ ) = ⎪ ⎪ ⎩ 0

0 1

if G is of order prime to 2, otherwise.

if G is a 2-group if G is dihedral or of Borel type and not a 2-group otherwise.

For = 2, the result depends on F , because χ ¯cyc will in general be alg non-trivial. If ρ¯ when considered over F is reducible, let χ ¯2 denote the character of GF,S on a one-dimensional quotient and χ ¯1 on the corresponding 1-dimensional subspace, and define χ ¯=χ ¯1 χ ¯−1 2 . One finds: 1 if G is abelian, h0 (GF,S , adρ¯) − 1 = h0 (GF,S , ad0ρ¯) = 0 otherwise. h0 (GF,S , (adρ¯)∨ ) = h0 (GF,S , (ad0ρ¯)∨ ) + h0 (GF,S , F(1)), and ⎧ ∼ ¯cyc = χ ¯ ⎪ ⎪ 2 if G = Z/(2) and χ ⎪ ⎪ ∼ ⎪ 1 if G Z/(2) and χ ¯ = 1 = cyc ⎪ ⎪ ⎪ ⎪ Z/(2) is abelian and χ ¯cyc ∈ {χ, ¯ χ ¯−1 , 1} = ⎨ 1 if G ∼ 0 ∨ 0 1 if G is non-abelian of Borel type and χ ¯cyc = χ ¯ h (GF,S , (adρ¯) ) = ⎪ ⎪ ⎪ 1 if G is dihedral, [F (ζ ) : F ] = 2 and ρ¯|GF (ζ ) is ⎪ ⎪ ⎪ ⎪ reducible ⎪ ⎪ ⎩ 0 otherwise. (5.3) Combining the above results, we obtain the following general theorem in the case G = GL2 : Theorem 5.8 Suppose F is totally real and ρ¯ is odd. Suppose further that (i) At ν | , ∞ the local deformation problem satisfies hη − h0 (Gν , ad0 ) − ν

ρ¯

gen(Jνη ) ≥ 0. (ii) At ν|∞ we choose either of the versal hulls in Example 5.4 depending on whether ρ(c ¯ ν ) is trivial or not. (iii) At ν| , either (i) Fν = Q , and ρ¯ satisfies the requirements in 5.5 case η I, and Def ¯ is ordinary ν,O is the functor of “flat deformations”, or (ii) ρ η and δ ,unr = 0, and Def is the functor of ordinary deformations ν

ν,O

with fixed determinant. (iv) h0 (GF,S , (ad0ρ¯)∨ ) = 0. (cf. Example 5.7 for explicit conditions.)

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η has a presentation O[[T1 , . . . , Tn ]]/(f1 , . . . , fn ) for suitable fi ∈ Then R S,O O[[T1 , . . . , Tn ]]. Proof Example 5.7 and (iv) imply that h0 (GF,S , ad0ρ¯) − δ(GF,S , ad0ρ¯)η = 0. Hence, in view of (i) it suffices to show that the joint contribution in (4.5) from the places above and ∞ under the stated hypotheses is zero. But using Proposition 5.6 and Example 5.4 yields h0ν − h0 (Gν , ad0ρ¯) − gen(Jνη ) = [Fν : Q ] − 1 ν| or ν|∞

ν|

ν|∞

= [F : Q] − [F : Q] = 0.

6 Comparison to the results of Tilouine and Mauger In this section we will apply the estimate from Corollary 4.3 to obtain another approach to the results by Tilouine and Mauger in [23, 15] on presentations for universal deformations. While their main interest was in representations into (0) symplectic groups, their results are rather general. If h0 (GF,S , (adρ¯ )∨ ) = 0, we completely recover their results with fewer hypothesis. If not, a comparison is less clear. It seems however, that in most cases where their results (0) are applicable, the term h0 (GF,S , (adρ¯ )∨ ) will be zero. Our main result is Theorem 6.6.

Example 6.1 Let d : G → T be arbitrary and let Sord ⊂ S be a set of places of F which contains all places above and none above ∞. For each ν ∈ Sord , we fix a smooth closed O-subgroup scheme Pν ⊂ G. For each place ν in Sord , we consider the subfunctor Def ν-n.o. ν,O ⊂ Def ν,O of deformations [ρν : Gν → G(R)] such that there exists some gν ∈ G(R), whose reduction mod mR is the identity, such that gν ρν gν−1 (Gν ) ⊂ Pν (R). For this subfunctor to make sense, one obviously requires that ρ¯(Gν ) ⊂ Pν (F). Following [23], a deformation [ρ] is called P-nearly ordinary (at Sord ) (where P stands for the family (Pν )ν∈Sord ) if for each ν ∈ Sord the restriction S -n.o. 0 [ρ|G ] satisfies the above condition. By Def ⊂ Def S,O we denote the ν

S,O

global deformation functor of deformations which are P-nearly ordinary at

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51

Sord ⊂ S, and are described by some other relatively representable functors η η Def ν,O ⊂ Def ν,O at places ν ∈ S Sord . If one furthermore fixes a lift η : GF,S −→ T (O) of d ◦ ρ¯ : GF,S → G(F) → T (F), the corresponding subfunctor is S -n.o.,η

0 Def S,O

S0 -n.o.

:= Def S,O

∩ Def ηS,O .

For each ν ∈ Sord , let pν ⊂ g denote the Lie-subalgebra of g which corresponds to Pν ⊂ G. It carries a natural Pν -action, so that g/pν (F) is a finite Pν -module. Again following [23], we define the condition (Reg) : For all ν ∈ Sord one has h0 (Gν , g/pν (F)) = 0. One has the following simple result whose proof we omit: Lemma 6.2 If the condition (Reg) holds, for all ν ∈ Sord , the subfunctor Def ν-n.o. ⊂ Def ν,O is relatively representable. Hence in this case ν,O S -n.o.,(η)

0 Def S,O

has a versal hull S -n.o.,(η)

0 ρS,O

S -n.o.,(η)

0 : GF,S → G(RS,O

).

(η)

Locally at ν ∈ Sord denote by Def Pν ,O the functor of deformations for representations of Gν into Pν (possibly with the additional condition that the deformations are compatible with the chosen η.) Let (η)

(η)

ρPν ,O : GF,S → Pν (RPν ,O ) denote a corresponding versal hull and define p0ν to be the Lie-Algebra of the d kernel of the composite Pν → G −→ T . By Theorems 1.2 and 1.4 we find: (η)

Proposition 6.3 The mod mO tangent space of RPν ,O is isomorphic to 1 H 1 (Gν , pν )(η) := Im(H 1 (Gν , p(0) ν ) → H (Gν , pν )). (η)

Let hPν := dimF H 1 (Gν , pν )(η) . Then there exists a presentation (η)

(η)

0 −→ JPν −→ O[[T1 , . . . , Th(η) ]] −→ RPν ,O −→ 0 Pν

(η)

(η)

(0)

for some ideal JPν ⊂ O[[T1 , . . . , Th(η) ]] with gen(JPν ) ≤ dimF H 2 (Gν , pν ). Pν

ν-n.o.,(η)

(η)

The two functors Def ν,O and Def Pν ,O essentially describe the same deformation problem, except that a priori they work with a different notion of strict equivalence.

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Lemma 6.4 The obvious surjection (η)

ν-n.o.,(η)

Def Pν ,O (F[ε]/(ε2 )) −→ → Def ν,O

(F[ε]/(ε2 ))

is a bijection provided that (Reg) holds. Proof Clearly, by definition, every lift ρ of ρ¯ to F[ε]/(ε2 ) whose class ν-n.o.,(η) lies in Def ν,O (F[ε]/(ε2 )) can be conjugated to take its image inside (η)

Pν (F[ε]/(ε2 )). Moreover the notion of strict equivalence for Def Pν ,O is an ν-n.o.,(η) Def ν,O ,

a priori weaker one than for so that the orbits under the second notion of strict equivalence may be larger. This shows that the map in the lemma is well-defined and surjective. Let us now show injectivity, i.e., that the orbits under both notions of strict equivalence agree. Let ρ = (1 + εa)¯ ρ be a lift of ρ¯ to F[ε]/(ε2 ) with image inside Pν (F[ε]/(ε2 )), so that a : Gν → pν is a 1-cocycle. Let g = 1 + εb be arbitrary with b ∈ g. We need to show that the set of those b for which gρg−1 lies in Pν (F[ε]/(ε2 )) (for all a as above) is exactly the set pν : One computes explicitly gρg −1 = (1 + ε(a + gbg−1 − b))¯ ρ. So independently of a, the element gbg −1 − g must lie in pν for all g ∈ ρ¯(Gν ). Equivalently, the image of b under the surjection g −→ → g/pν must lie in H 0 (Gν , g/pν ). By (Reg) the latter set is zero, and so b lies indeed in pν = Ker(g −→ → g/pν ). Recall that 0 ≤ δ(Gν , pν )η = h1 (Gν , p0ν ) − dim H 1 (Gν , p)η . The formula for the local Euler-Poincar´e characteristic yields: (η)

ν-n.o.,(η)

Proposition 6.5 For ν ∈ S0 and the functor Def ν,O = Def ν,O

one has

(0) (η) h(η) − h0 (Gν , adρ¯ ) − gen(Jν(η) ) ν + δ(Gν , pν ) 0, if ν | , = [Fν : Q ] dimF pν , if ν | .

In [23, 15] a lift of d ◦ ρ¯ is never chosen. Hence the term δ(Gν , pν )(η) is not present in their formulas. Combining the above with Corollary 4.3 shows: Theorem 6.6 Fix P = (Pν )ν∈S0 as above, and assume that: S -n.o.,(η)

0 (i) ρ¯ ∈ Def S,O

(F).

Presentations of universal deformation rings

53

(0) (0) (ii) At ν ∈ S (Sord ∪ {ν : ν|∞}) we have hν − h0 (Gν , adρ¯ ) − (η) gen(Jν ) ≥ 0. (iii) The condition (Reg) is satisfied.

Then for the presentation 0 0 −→ J(η) −→ O[[T1 , . . . , Th(η) ]] −→ R S,O

S -n.o.,(η)

−→ 0

one has (0) h(η) − gen(J (η) ) ≥ h0 (GF,S , adρ¯ )

− h0 (GF,S , (adρ¯ )∨ ) − δ(GF,S , adρ¯)(η) + (0)

[Fν : Q ] dimF p(0) ν

ν|

(0) (η) + h(η) − h0 (Gν , adρ¯ ) − gen(Jν(η) ) . ν + δ(Gν , adρ¯) ν|∞

If = 2, or if no constraints are imposed for the deformation at the infinite places, then their contribution in the above formula simplifies to give (0) − ν|∞ h0 (Gν , adρ¯ ). Remark 6.7 Since in [23] or [15] no homomorphism η is fixed, and there are no conditions at ∞, the above is (philosophically) the same formula as that in (0) [23], Prop. 7.3 or [15], Prop. 3.9, except for the term −h0 (GF,S , (adρ¯ )∨ ). As noted in Remark 4.4, we expect that usually this term is not present in the formula – but that technically we are not able to remove it. By ‘philosophically’ we mean that their formula was used primarily to bound the Krull dimension of some deformation ring. Our formula can obviously serve the same purpose. Our hypotheses and those in [23, 15] are however different. If (0) h0 (GF,S , (adρ¯ )∨ ) = 0 our result holds under much weaker hypotheses, namely without the hypothesis (Reg ) in [15], Prop. 3.9. The latter seems to be rather hard to verify in practice. (0) If h0 (GF,S , (adρ¯ )∨ ) is non-zero the comparison is less clear. The nonvanishing either means that we are in the case adρ¯ and F contains a primitive

-th root of unity, or that ad0ρ¯ surjects onto a one-dimensional quotient representation on which GF,S acts by the inverse of the mod -cyclotomic character. In the former case we’d expect that the pν typically also contain a trivial sub(0) (0) representation, and then the terms h0 (Gν , (pν )∨ ) = h2 (Gν , pν ) would be non-zero, so that the hypothesis (Reg ) in [15], Prop. 3.9, would not be satisfied. In the latter case it is not clear to us whether this one-dimensional quotient (0) will typically also occur as a quotient of one of the pν . In any case, if ρ¯ is

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‘highly irreducible’ which is the generic case, the second case is unlikely to occur.

7 Relative presentations In this last section we deduce some results on presentations of global deformation rings as quotients of power series rings over the completed tensor product of the corresponding local versal deformation rings from the results in Section 4. This is inspired by M. Kisin’s theory of framed deformations. The results below are due to Kisin [13], who has given a different more direct approach. This section makes no reference to Sections 5 and 6. We let the notation be as in Section 4. Lemma 7.1 There exists a natural number s and a presentation 0 −→ J −→

ν∈S

(η) [[U1 , . . . , Us ]] −→ R (η) −→ 0 R ν,O S,O

being bounded from above by with gen(J) s + δ(GF,S , adρ¯)(η) − h0 (GF,S , adρ¯ ) + h0 (GF,S , (adρ¯ )∨ ) (0) + h0 (Gν , adρ¯ ) − δ(Gν , adρ¯)(η) . (0)

(0)

ν∈S

Proof The proof of Theorem 4.2 yields the following commutative diagram Jloc

π(Jloc ) ∪ {g1 , . . . , gr }

/

O[[Tν,j : ν ∈ S, j = 1, . . . , h(η) ν ]]

/

Rloc

//

(η) R S,O

π

O[[T1 , . . . , T (η) ]] h

(η) where we set Jloc := {Jν : ν ∈ S}, Rloc :=

r =

//

!

(η) ν∈S Rν,O

,

(0) dimF HL1 (η),⊥ (GF,S , (adρ¯ )∨ )

and the gj are suitable elements of (η) can be O[[T1 , . . . , Th(η) ]]. The failure of the surjectivity of Rloc → R S,O measured by considering the induced homomorphism on mod mO tangent spaces. Let s denote the dimension of the cokernel of tRloc → tR (η) . (It S,O

(0)

is not difficult to show that s = dim W1 (adρ¯ ), but we do not need this.) (η) for Then there is a surjective homomorphism Rloc [[U1 , . . . , Us ]] → R S,O

Presentations of universal deformation rings

55

(η) variables Ui . Abbreviating Sloc := O[[Tν,j : ν ∈ S, j = 1, . . . , hν ]], there is a commutative diagram / Sloc [[U1 , . . . , Us ]] / Rloc [[U1 , . . . , Us ]] Jloc

{˜ π (Jloc ) ∪ {g1 , . . . , gr }

/

π ˜

O[[T1 , . . . , T (η) ]] h

//

(η) R S,O

with surjective middle and right vertical homomorphisms. Since Sloc is a power series ring over O, the kernel of π ˜ is generated by (η) u := s + h(η) ν −h ν∈S

elements H1 , . . . , Hu . Because π ˜ is smooth, we may choose elements G1 , . . . , Gr in the ring Sloc [[U1 , . . . , Us ]] whose images in O[[T1 , . . . , Th(η) ]] agree (η) is the quotient of Rloc [[U1 , . . . , Us ]] by the ideal with g1 , . . . , gr . Thus R S,O

J generated by the images of the elements G1 , . . . , Gr , H1 , . . . , Hu . Using the first formula in the proof of Corollary 4.3, we have ∨ (η) + dimF H 1 (η),⊥ (GF,S , (ad(0) −s = gen(J) h(η) ρ¯ ) ) ν −h L ν∈S

= δ(GF,S , adρ¯)(η) −h0 (GF,S , adρ¯ )+h0 (GF,S , (adρ¯ )∨ ) (0) + h0 (Gν , adρ¯ )−δ(Gν , adρ¯)(η) . (0)

(0)

ν∈S

If R is flat over O, its relative Krull dimension over O is denoted by dimKrull/O R. Corollary 7.2 Suppose that (η) /( ) is finite. (i) R S,O

(η) , ν ∈ S, are flat over O. (ii) The rings R ν,O (η) (η) ≥ h0 (Gν , ad(0) (iii) dimKrull/O R for ν | , ∞. ρ¯ ) − δ(Gν , adρ¯) ν,O (iv) One has (0) (η) ≥ dimKrull/O R (h0 (Gν , adρ¯ )−δ(Gν , adρ¯)(η) ). ν,O ν| or ν|∞

ν| or ν|∞

(v) δ(GF,S , adρ¯)(η) − h0 (GF,S , adρ¯ ) + h0 (GF,S , (adρ¯ )∨ ) = 0. (0)

(0)

(η) is finite, and R (η) modulo its -torsion is non-zero Then the -torsion of R S,O S,O over O. Hence this quotient is non-zero and finite flat over O.

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(η) is noetherian the -torsion submodule of R (η) is finitely Proof Since R S,O S,O generated. Therefore there exists some m ≥ 0 such that the -torsion submod(η) /( m ). By condition (i) (and noetherianess of R (η) ) the ule injects into R S,O S,O latter is finite. To complete the proof of the corollary, it suffices to show that (η) is at least one. the Krull dimension of R S,O We first compute the relative Krull dimension of the middle term in the presentation of Lemma 7.1: (ii) (η) (η) dimKrull/O R = dimKrull/O R ν,O ν,O ν∈S

ν∈S

(iii),(iv) ≥

(0)

(h0 (Gν , adρ¯ ) − δ(Gν , adρ¯)(η) ).

ν∈S

Using (v), the relative Krull dimension of therefore at least

!

(η) ν∈S Rν,O [[U1 , . . . , Us ]]

over O is

s + δ(GF,S , adρ¯)(η) − h0 (GF,S , adρ¯ ) + h0 (GF,S , (adρ¯ )∨ ) (0) + h0 (Gν , adρ¯ ) − δ(Gν , adρ¯)(η) . (0)

(0)

ν∈S

in the presentation of Lemma 7.1. Now the This is also the bound on gen(J) quotient of a local ring by a number of relations decreases the Krull dimension of the ring by at most this number (unless the quotient is zero). Since the Krull dimension is one more than the relative Krull dimension over O, it follows that the Krull dimension of (η) ∼ (η) [[U1 , . . . , Us ]]/J R R = S,O ν,O ν∈S

is at least one, as was to be shown. Remark 7.3 The following is a ring theoretic example which shows that without any further hypotheses, one cannot rule out the possibility of -torsion in (η) : the ring R S,O (η) ∼ Suppose F = Q, S = { , ∞}, R = Z [[S, T ]]/(( + S)T, T 2 ), that (iv) ,O

(η) is the quotient of R (η) and (v) of the corollary are satisfied, and that R S,O ,O (η) ∼ by the ideal (S). Then R = Z [[T ]]/( T, T 2 ) has -torsion, although the ,O

remaining assertions (i) and (ii) of the corollary hold ((iii) holds trivially). However, if in addtion to (i)–(v) one imposes the further condition that ! (η) ν∈S Rν,O is Cohen-Macaulay, then from standard results in commutative (η) is flat over O. This was pointed out algebra one may indeed deduce that R S,O

by M. Kisin.

Presentations of universal deformation rings

57

We now apply the previous corollary to the situation of Theorem 5.8, where however we relax the condition at the places above : Theorem 7.4 Suppose d = det : G = GL2 → T = GL1 , F is totally real and ρ¯ is odd. Suppose further that η /( ) is finite. (i) R S,O η , ν ∈ S, are flat over O. (ii) The rings R ν,O

(iii) At ν | , ∞ the local deformation problem satisfies h0ν − h0 (Gν , ad0ρ¯) − η gen(Jν ) ≥ 0. (iv) At ν|∞ we choose either of the versal hulls in Example 5.4 depending on whether ρ(c ¯ ν ) is trivial or not. 0 0 η (v) At ν| , one has dimKrull/O R ν,O = [Fν : Q ] + h (Gν , adρ¯)− δ(Gν , adρ¯)η . (vi) h0 (GF,S , (ad0ρ¯)∨ ) = 0. (cf. Example 5.7 for explicit conditions.) η has finite -torsion, and its quotient modulo -torsion is non-zero Then R S,O and finite flat over O. Proof It suffices to verify the hypothesis of Corollary 7.2. Conditions (i), (ii), (iii) and (vi) imply conditions (i), (v), (ii) and (iv) of Corollary 7.2, respec(η) = tively. At the infinite places ν, condition (iv) implies dimKrull/O R ν,O 0 h0 (Gν , adρ¯) − δ(Gν , adρ¯)η − 1. Because of the identity ν| [Fν : Q ] = [F : Q] = ν|∞ 1, the latter observation combined with condition (v) implies condition (iii) of Corollary 7.2.

Bibliography [1] F. Bleher, T. Chinburg, Universal deformation rings need not be complete intersection rings, C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 229–232. [2] G. B¨ockle, A local-to-global principle for deformations of Galois representations, J. Reine Angew. Math. 509 (1999), 199–236. [3] G. B¨ockle, Demuˇskin groups with group actions and applications to deformations of Galois representations, Compositio Math. 121 (2000), no. 2, 109–154. [4] G. B¨ockle, On the density of modular points in universal deformation spaces, Amer. J. Math. 123 (2001), no. 5, 985–1007. [5] B. Conrad, Ramified deformation problems, Duke Math. J. 97 (1999), no. 3, 439– 513. [6] F. Diamond, The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997), no. 2, 379–391. [7] L. Dieulefait, Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture, J. Reine Angew. Math. 577 (2004), 147–151.

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[8] L. Dieulefait, The level 1 weight 2 case of Serre’s conjecture, preprint available at http://arxiv.org/abs/math/0412099. [9] A. J. de Jong, A conjecture on arithmetic fundamental groups, Israel J. Math. 121 (2001), 61–84. [10] C. Khare, Serre’s Modularity Conjecture: The Level One Case, Duke Math. J. 134 (2006) no. 3, 557–589. [11] C. Khare, Serre’s modularity conjecture: a survey of the level one case, this volume, pp. 270-299. [12] C. Khare, J.-P. Wintenberger, On Serre’s reciprocity conjecture for 2-dimensional mod p representations of the Galois group of Q, preprint 2004, http://arXiv.org/abs/math/0412076 [13] M. Kisin, Modularity of Potentially Barsotti-Tate Galois Representations, to appear in ‘Current Developments in Mathematics, 2005’, International Press, Somerville, U.S.A. [14] H. Koch, Galois Theory of p-Extensions, Springer Monographs in Mathematics, Springer Verlag, Berlin 2002. [15] D. Mauger, Alg`ebra de Hecke quasi-ordinaire universelle d’un group r´eductive, thesis, 2000. [16] B. Mazur, Deforming Galois Representations, in Galois groups over Q, ed. Ihara et al., Springer-Verlag 1987. [17] B. Mazur, An introduction to the deformation theory of Galois representations, in “Modular forms and Fermat’s last theorem” (Boston, MA, 1995), pp. 243–311, Springer, New York, 1997. [18] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields, Grundlehren der Math. Wiss. 323, Springer, Berlin, 2000. [19] R. Ramakrishna, Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur, Ann. of Math. (2) 156 (2002), no. 1, 115–154. [20] R. Ramakrishna, On a variation of Mazur’s deformation functor, Compositio Math. 87 (1993), 269–286. [21] M. Schlessinger, Functors of Artin rings, Trans. A. M. S. 130 (1968), 208–222. [22] R. Taylor, Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu 1 (2002), no. 1, 125–143. [23] J. Tilouine, Deformations of Galois representations and Hecke algebras, Publ. Mehta Research Institute, Narosa Publishing House, New Delhi, 1996.

Eigenvarieties Kevin Buzzard Department of Mathematics Imperial College London U.K. [email protected]

Abstract We axiomatise and generalise the “Hecke algebra” construction of the Coleman-Mazur Eigencurve. In particular we extend the construction to general primes and levels. Furthermore we show how to use these ideas to construct “eigenvarieties” parametrising automorphic forms on totally definite quaternion algebras over totally real fields.

1 Introduction In a series of papers in the 1980s, Hida showed that classical ordinary eigenforms form p-adic families as the weight of the form varies. In the non-ordinary finite slope case, the same turns out to be true, as was established by Coleman in 1995. Extending this work, Coleman and Mazur construct a geometric object, the eigencurve, parametrising such modular forms (at least for forms of level 1 and in the case p > 2). On the other hand, Hida has gone on to extend his work in the ordinary case to automorphic forms on a wide class of reductive groups. One might optimistically expect the existence of nonordinary families, and even an “eigenvariety”, in some of these more general cases. Anticipating this, we present in Part I of this paper (sections 2–5) an axiomatisation and generalisation of the Coleman-Mazur construction. In his original work on families of modular forms, Coleman in [10] developed Riesz theory for orthonormalizable Banach modules over a large class of base rings, and, in the case where the base ring was 1-dimensional, constructed the local pieces of a parameter space for normalised eigenforms. There are two places where we have extended Coleman’s work. Firstly, we set up Coleman’s Fredholm theory and Riesz theory (in sections 2 and 3 respectively) in a slightly more general 59

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Kevin Buzzard

situation, so that they can be applied to spaces such as direct summands of orthonormalizable Banach modules; the motivation for this is that at times in the theory we meet Banach modules which are invariants of orthonormalizable Banach modules under the action of a finite group; such modules are not necessarily orthonormalizable, but we want to use Fredholm theory anyway. And secondly we show in sections 4–5 that given a projective Banach module and a collection of commuting operators, one of which is compact, one can glue the local pieces constructed by Coleman to form an eigenvariety, in the case where the base ring is an arbitrary reduced affinoid. At one stage we are forced to use Raynaud’s theory of formal models; in particular this generalisation is not an elementary extension of Coleman’s ideas. The resulting machine can be viewed as a construction of a geometric object from a family of Banach spaces equipped with certain commuting linear maps. Once one has this machine, one can attempt to feed in Banach spaces of “overconvergent automorphic forms” into the machine, and get “eigenvarieties” out. We extend the results of [9] in Part II of this paper (sections 6 and 7), constructing an eigencurve using families of overconvergent modular forms, and hence removing some of the assumptions on p and N in the main theorems of [9]. Note that here we do not need the results of section 4, as weight space is 1-dimensional and Coleman’s constructions are enough. There are still technical geometric problems to be resolved before one can give a definition of an overconvergent automorphic form on a general reductive group, but one could certainly hope for an elementary definition if the group in question is compact mod centre at infinity, as the geometry then becomes essentially non-existent. As a concrete example of this, we propose in Part III (sections 8–13) a definition of an overconvergent automorphic form in the case when the reductive group is a compact form of GL2 over a totally real field, and apply our theory to this situation to construct higher-dimensional eigenvarieties. Chenevier has constructed Banach spaces of overconvergent automorphic forms for compact forms of GLn over Q and one can feed his spaces into the machine also to get eigenvarieties for these unitary groups. This work began in 2001 during a visit to Paris-Nord, and the author would like to thank Jacques Tilouine for the invitation and Ahmed Abbes for several useful conversations. In fact the author believes that he was the first to coin the phrase “eigenvarieties”, in 2001. Part I of this paper was written at that time, as well as some of Part III. The paper then remained in this state for three years, and the author most sincerely thanks Gaetan Chenevier for encouraging him to finish it off. In fact Theorem 4.6 of this paper is assumed both by Chenevier

Eigenvarieties

61

in [8], and Yamagami in [17], who independently announced results very similar to those in Part III of this paper, the main difference being that Yamagami works with the U operator at only one prime above p and fixes weights at the other places, hence his eigenvarieties can have smaller dimension than ours, but they see more forms (they are only assumed to have finite slope at one place above p). My apologies to both Chenevier and Yamagami for the delay in writing up this construction; I would also like to thank both of them for several helpful comments. A lot has happened in this subject since 2001. Matthew Emerton has recently developed a general theory of eigenvarieties which in many cases produces cohomological eigenvarieties associated to a large class of reductive algebraic groups. As well as Coleman and Mazur, many other people (including Emerton, Ash and Stevens, Skinner and Urban, Mazur and Calegari, Kassaei, Kisin and Lai, Chenevier, and Yamagami), have made contributions to the area, all developing constructions of eigenvarieties in other situations. We finish this introduction with an explanation of the relationship between Emerton’s work and ours. Emerton’s approach to eigenvarieties is more automorphic and more conceptual than ours. His machine currently needs a certain spectral sequence to degenerate, but this degeneration occurs in the case of the Coleman-Mazur eigencurve and hence Emerton has independently given a construction of this eigencurve for arbitrary N and p as in Part II of this paper. However, Emerton’s construction is less “concrete” and in particular the results in [6] and [7] rely on the construction of the 2-adic eigencurve presented in this paper. On the other hand Emerton’s ideas give essentially the same construction of the eigenvariety associated to a totally definite quaternion algebra over a totally real field, in the sense that one can check that his more conceptual approach, when translated down, actually becomes equivalent to ours. We would like to thank Peter Schneider for pointing out an error in an earlier version of Lemma 5.6, Elmar Grosse-Kloenne for pointing out a simplification in the definition of our admissible cover, and the referee for several helpful remarks, in particular for pointing out that flatness was necessary in Lemma 5.5. PART I: The eigenvariety machine.

2 Compact operators on K-Banach modules In this section we collect together the results we need from the theory of commutative Banach algebras. A comprehensive source for the terminology we use is [1]. Throughout this section, K will be a field complete with respect

62

Kevin Buzzard

to a non-trivial non-archimedean valuation |.|K , and A will be a commutative Noetherian K-Banach algebra. That is, A is a commutative Noetherian K-algebra equipped with a function |.| : A → R≥0 , and satisfying • |1| ≤ 1, and |a| = 0 iff a = 0, • |a + b| ≤ max{|a|, |b|}, • |ab| ≤ |a||b|, • |λa| = |λ|K |a| for λ ∈ K, and such that A is complete with respect to the metric induced by |.|. For elementary properties of such algebras we refer the reader to [1], §3.7 and thereafter. Such algebras are Banach algebras in the sense of Coleman [10]. Later on we shall assume (mostly for simplicity) that A is a reduced K-affinoid algebra with its supremum norm, but this stronger assumption does not make the arguments of this section or the next any easier. From the axioms one sees that either |1| = 0 and hence A = 0, or |1| = 1, in which case the map K → A is injective and the norm on A extends the norm on K. Fix once and for all ρ ∈ K × with |ρ|K < 1. Such ρ exists as we are assuming the valuation on K is non-trivial. We use ρ to “normalise” vectors in several proofs. If A0 denotes the subring {a ∈ A : |a| ≤ 1} then one easily checks that the ideals of A0 generated by ρn , n = 1, 2, . . ., form a basis of open neighbourhoods of zero in A. Note that A0 may not be Noetherian (for example if A = K = Cp ). Let A be a commutative Noetherian K-Banach algebra. A Banach Amodule is an A-module M equipped with |.| : M → R≥0 satisfying • |m| = 0 iff m = 0, • |m + n| ≤ max{|m|, |n|}, • |am| ≤ |a||m| for a ∈ A and m ∈ M , and such that M is complete with respect to the metric induced by |.|. Note that A itself is naturally a Banach A-module, as is any closed ideal of A. In fact all ideals of A are closed, by Proposition 3.7.2/2 of [1]. If M and N are Banach A-modules, then we define a norm on M ⊕ N by |m ⊕ n| = Max{|m|, |n|}. This way M ⊕ N becomes a Banach A-module. In particular we can give Ar the structure of a Banach A-module in a natural way. By a finite Banach A-module we mean a Banach A-module which is finitelygenerated as an abstract A-module. We use the following facts several times in what follows:

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Proposition 2.1 (a) (Open Mapping Theorem) A continuous surjective K-linear map between Banach K-modules is open. (b) The category of finite Banach A-modules, with continuous A-linear maps as morphisms, is equivalent to the category of finite A-modules. In particular, any A-module homomorphism between finite Banach A-modules is automatically continuous, and if M is any finite A-module then there is a unique (up to equivalence) complete norm on M making it into a Banach A-module. Proof (a) is Th´eor`eme 1 in Chapter I, §3.3 of [3] (but note that “homomorphisme” here has the meaning assigned to it in §2.7 of Chapter III of [2], and in particular is translated as “strict morphism” rather than “homomorphism”). (b) is proved in Propositions 3.7.3/2 and 3.7.3/3 of [1]. Note that by (b), a finite A-module M has a canonical topology, induced by any norm that makes M into a Banach A-module. We call this topology the Banach topology on M . As an application of these results, we prove the following useful lemma: Lemma 2.2 If M is a Banach A-module, and P is a finite Banach A-module, then any abstract A-module homomorphism φ : P → M is continuous. Proof Let π : Ar → P be a surjection of A-modules, and give Ar its usual Banach A-module norm. Then π is open by the Open Mapping Theorem, and φπ is bounded and hence continuous. So φ is also continuous. If I is a set, and for every i ∈ I we have ai , an element of A, then by the statement limi→∞ ai = 0, we simply mean that for all > 0 there are only finitely many i ∈ I with |ai | > . This is no condition if I is finite, and is the usual condition if I = Z≥0 . For general I, if limi→∞ ai = 0 then only countably many of the ai can be non-zero. We also mention here a useful convention: occasionally we will take a max or a supremum over a set (typically a set of norms) which can be empty in degenerate cases (e.g., if a certain module or ring is zero). In these cases we will define the max or the supremum to be zero. In other words, throughout the paper we are implicitly taking suprema in the set of non-negative reals rather than the set of all reals. Let A be a non-zero commutative Noetherian K-Banach algebra, let M be a Banach A-module, and consider a subset {ei : i ∈ I} of M such that |ei | = 1 for all i ∈ I. Then for any sequence (ai )i∈I of elements of A with limi→∞ ai = 0, the sum i ai ei converges. We say that a Banach A-module

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M is orthonormalizable, or ONable for short, if there exists such a subset {ei : i ∈ I} of M with the following two properties: • Every element m of M can be written uniquely as i∈I ai ei with limi→∞ ai = 0, and • If m = i ai ei then |m| = maxi∈I |ai |. Such a set of elements {ei } is called an orthonormal basis, or an ON basis, for M . Note that the second condition implies that |ei | = 1 for all i ∈ I. Again assume A = 0. If I is a set, we define cA (I) to be A-module of functions f : I → A such that limi→∞ f (i) = 0. Addition and the A-action are defined pointwise. We define |f | to be Max{|ai | : i ∈ I}. With respect to this norm, cA (I) becomes a Banach A-module. If i ∈ I and we define ei to be the function sending j ∈ I to 0 if i = j, and to 1 if i = j, and if furthermore A = 0, then it is easily checked that the ei are an ON basis for cA (I), and we call {ei : i ∈ I} the canonical ON basis for cA (I). If M is any ONable Banach A-module, then to give an ON basis {ei : i ∈ I} for M is to give an isometric (that is, metric-preserving) isomorphism M ∼ = cA (I). Note also that cA (I) has the following universal property: if M is any Banach A-module then there is a natural bijection between HomA (cA (I), M ) and the set of bounded maps I → M , given by sending φ : cA (I) → M to the map i → φ(ei ). If A = 0 then the only A-module is M = 0, and we regard this module as being ONable of arbitrary rank. We have chosen to ignore this case in the definitions above because if we had included it then we should have to define an ONable Banach module as being a collection of ei as above but with |ei | = |1| and so on; however this just clutters notation. There is no other problem with the zero ring in this situation. We will occasionally assume A = 0 in proofs, and leave the interested reader to fill in the trivial details in the case A = 0. We recall some basic results on “matrices” associated to endomorphisms of Banach A-modules. The proofs are elementary exercises in analysis. Let M and N be Banach A-modules, and let φ : M → N be an A-module homomorphism. Then a standard argument (see Corollary 2.1.8/3 of [1]) using the fact that one can use ρ to renormalise elements of M , shows that φ is continuous iff it is bounded, and in this case we define |φ| = sup0=m∈M |φ(m)| |m| (this set of reals is bounded above if φ is continuous). Now assume that M is ONable, with ON basis {ei : i ∈ I}. One easily checks that if φ is continuous and φ(ei ) = ni , then the ni are a bounded collection of elements of N which uniquely determine φ. Furthermore, if ni are an arbitrary bounded collection of elements of N there is a unique continuous map φ : M → N such that φ(ei ) = ni for all i, and |φ| = supi∈I |ni |.

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Now assume that N is also ONable, with basis {fj : j ∈ J}. If φ : M → N is a continuous A-module homomorphism, we can define its associated matrix coefficients (ai,j )i∈I,j∈J by1 φ(ei ) = ai,j fj . j∈J

One checks easily from the arguments above that the collection (ai,j ) has the following two properties: • For all i, limj→∞ ai,j = 0. • There exists a constant C ∈ R such that |ai,j | ≤ C for all i, j. In fact C can be taken to be |φ|, and furthermore we have |φ| = supi,j |ai,j |. Conversely, given a collection (ai,j )i∈I,j∈J of elements of A, satisfying the two conditions above, there is a unique continuous φ : M → N with norm supi,j |ai,j | whose associated matrix is (ai,j ). As a useful consequence of this, we see that if φ and ψ : M → N are continuous, with associated matrices (ai,j ) and (bi,j ), then |φ − ψ| ≤ iff |ai,j − bi,j | ≤ for all i and j. Let A be a commutative Noetherian K-Banach algebra and let M , N be Banach A-modules. The A-module Hom(M, N ) of continuous A-linear homomorphisms from M to N is then also a Banach A-module: completeness follows because if φn is a Cauchy sequence in Hom(M, N ) then for all m ∈ M , φn (m) is a Cauchy sequence in N , and one can define φ(m) as its limit; then φ is the limit of the φn . A continuous A-module homomorphism M → N is said to be of finite rank if its image is contained in a finitely-generated A-submodule of N . The closure in Hom(M, N ) of the finite rank homomorphisms is the set of compact homomorphisms (many authors use the term “completely continuous”). Let M and N be ONable Banach modules, and let φ : M → N be a continuous homomorphism, with associated matrix (ai,j ). We wish to give a simple condition which is expressible only in terms of the ai,j , and which is equivalent to compactness. Such a result is announced in Lemma A1.6 of [10] for a more general class of rings A, but the proof seems to be incomplete. This is not a problem with the theory however, as the proof can be completed in all cases of interest without too much trouble. We complete the proof here in the case of commutative Noetherian K-Banach algebras. We start with some preliminary results. As ever, A is a Noetherian K-Banach algebra. If M is an ONable Banach A-module, with ON basis {ei : i ∈ I}, and if S ⊆ I is a finite subset, then we define AS to be the submodule ⊕i∈S Aei , and we define the projection 1 Here we follow Serre’s conventions in [15], rather than writing aj,i for ai,j .

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πS : M → AS to be the map sending i∈I ai ei to i∈S ai ei . Note that this projection is norm-decreasing onto a closed subspace of M . Lemma 2.3 Let M be an ONable Banach A-module, with basis {ei : i ∈ I}, and let P a finite submodule of M . (a) There is a finite set S ⊆ I such that πS : M → AS is injective on P . (b) P is a closed subset of M , and hence is complete. (c) For all > 0, there is a finite set T ⊆ I such that for all p ∈ P we have |πT (p) − p| ≤ |p|. Proof Say P is generated by m1 , . . . , mr and for 1 ≤ α ≤ r we have mα = i aα,i ei . (a) For i ∈ I let vi be the element (aα,i )1≤α≤r of Ar . The A-submodule of Ar generated by the vi is finitely-generated, as A is Noetherian, and hence there is a finite set S ⊆ I such that this module is generated by {vi : i ∈ S}. It is now an easy exercise to check that this S works, because if πS ( α bα mα ) is zero, then α bα aα,i is zero for all i ∈ S and hence for all i ∈ I. (b) P is a finite A-module, and hence there is, up to equivalence, a unique complete A-module norm on P . Let Q denote P equipped with this norm. The algebraic isomorphism Q → P induces a map Q → M which is continuous by Lemma 2.2, and hence the algebraic isomorphism Q → P is continuous. On the other hand, if S is chosen as in part (a), then the injection P → AS induces a continuous injection from P onto a submodule of AS which is closed by Proposition 3.7.3/1 of [1], and this submodule is algebraically isomorphic to Q, and hence isomorphic to Q as a Banach A-module. We hence have continuous maps Q → P → Q, which are algebraic isomorphisms, and hence the norms on P and Q are equivalent. So the maps are also homeomorphisms, and P is complete with respect to the metric induced from M , and is hence a closed submodule of M . (c) By (b), P is complete and hence a K-Banach space. The map Ar → P sending (aα ) to α aα mα is thus a continuous surjection between K-Banach spaces, and hence by the open mapping theorem there exists δ > 0 such that r if p ∈ P with |p| ≤ δ then p = α=1 aα mα with |aα | ≤ 1 for all α. Choose T such that for all mα we have |πT (mα ) − mα | ≤ δ|ρ|. This T works: if p ∈ P is arbitrary, then either p = 0 and hence the condition we are checking is automatic, or p = 0. In this case, there exists some n ∈ Z such that |ρ|δ < |ρ|n |p| ≤ δ, and then ρn p = α aα mα with |aα | ≤ 1 for all α.

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" " " " " " |πT (p) − p| = " ρ−n aα (πT (mα ) − mα )" " α " ≤ |ρ|−n δ|ρ| ≤ |p|

and we are done. Proposition 2.4 Let M, N be ONable Banach A-modules, with ON bases {ei : i ∈ I} and {fj : j ∈ J}. Let φ : M → N be a continuous Amodule homomorphism, with basis (ai,j ). Then φ is compact if and only if limj→∞ supi∈I |ai,j | = 0. Proof If the matrix of φ satisfies limj→∞ supi∈I |ai,j | = 0, then φ is easily seen to be compact: for any > 0 there is a finite subset S ⊆ J such that |φ − πS φ| ≤ . The other implication is somewhat more delicate. It suffices to prove the result when φ has finite rank. If φ = 0 then the result is trivial, so assume 0 = φ and φ(M ) ⊆ P , where P ⊆ N is finite. By part (c) of Lemma 2.3, for any > 0 we may choose T such that |πT (p) − p| ≤ |p|/|φ|, and hence |πT φ − φ| ≤ . Hence |ai,j | ≤ if j ∈ T , and we are home because was arbitrary. Remark If we allow A to be non-Noetherian then we do not know whether the preceding proposition remains true. From this result, it easily follows that a compact operator φ : M → M , where M is an ONable Banach A-module, has a characteristic power series det(1 − Xφ) = n≥0 cn X n ∈ A[[X]], defined in terms of the matrix coefficients of φ using the usual formulae, which we recall from §5 of [15] for convenience: firstly choose an ON basis {ei : i ∈ I} for M , and say φ has matrix (ai,j ) with respect to this basis. If S is any finite subset of I, then define cS = sum ranges over σ:S→S sgn(σ) i∈S ai,σ(i) , where the n all permutations of S, and for n ≥ 0 define cn = (−1) S cS , where the sum is over all finite subsets of I of size n. One easily checks that this sum converges, using Proposition 2.4. Furthermore, again using Proposition 2.4 and following Proposition 7 of [15], one sees that the resulting power series det(1 − Xφ) = n cn X n converges for all X ∈ A. However, from our definition it is not clear to what extent the power series depends on the choice of

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ON basis for M . We now investigate to what extent this is the case. We begin with some observations in the finite-dimensional case. If P is any finite free ONable Banach A-module, with ON basis (ei ), and φ : P → P is any A-module homomorphism, then det(1 − Xφ), defined as above with respect to the ei , is the usual algebraically-defined det(1 − Xφ), because the definition above coincides with the usual classical definition, which is independent of choice of basis. Next we recall the well-known fact that if P and Q are both free A-modules of finite rank, and u : P → Q and v : Q → P are A-module homomorphisms, then det(1 − Xuv) = det(1 − Xvu). Now let M be an ONable Banach A-module, with ON basis {ei : i ∈ I}. We use this fixed basis for computing characteristic power series in the lemma below. Lemma 2.5 (a) If φn : M → M , n = 1, 2, . . . are a sequence of compact operators that tend to a compact operator φ, then limn det(1 − Xφn ) = det(1 − Xφ), uniformly in the coefficients. (b) If φ : M → M is compact, and furthermore if the image of φ is contained in P := ⊕i∈S Aei , for S a finite subset of I, then det(1 − Xφ) = det(1 − Xφ|P ), the right hand determinant being the usual algebraically-defined one. (c) (strengthening of (b)) If φ : M → M is compact, and if the image of φ is contained within an arbitrary submodule Q of M which is free of finite rank, then det(1 − Xφ) = det(1 − Xφ|Q ), where again the right hand side is the usual algebraically-defined determinant. Proof (a) This follows mutatis mutandis from [15], Proposition 8. (b) If (ai,j ) is the matrix of φ then ai,j = 0 for j ∈ S and the result follows immediately from the definition of the characteristic power series. (c) Choose > 0. By Lemma 2.3(c), there is a finite set T ⊆ I such that πT : Q → P := AT has the property that |πT − i| ≤ , where i : Q → M is the inclusion. Define φT = πT φ : M → P ⊆ M . By (b) we see that det(1 − XφT ) equals the algebraically-defined polynomial det(1 − XφT |P ). Furthermore, by consideration of the maps φ : P → Q and πT : Q → P , we see that this polynomial also equals the algebraically-defined det(1 − XφT ), where φT = φπT : Q → Q. One can compute this latter determinant with respect to an arbitrary algebraic A-basis of Q. By Lemma 2.3(b), Q with its subspace topology is complete, and hence the topology on Q is the Banach topology. Now as tends to zero, φT : Q → Q tends to φ : Q → Q and φT : M → M tends to φ : M → M , and the result follows by part (a).

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Corollary 2.6 Let M be an A-module, and let |.|1 and |.|2 be norms on M both making M into an ONable Banach A-module, and both inducing the same topology on M . Then an A-linear map φ : M → M is compact with respect to |.|1 iff it is compact with respect to |.|2 , and furthermore if {ei : i ∈ I} and {fj : j ∈ J} are ON bases for (M, |.|1 ) and (M, |.|2 ) respectively, then the definitions of det(1 − Xφ) with respect to these bases coincide. Proof All one has to do is to check that φ can be written as the limit as maps φn which have image contained in free modules of finite rank, and then the result follows from parts (a) and (c) of the Lemma. To do this, one can simply use Lemma 2.3 to construct φ as a limit of πTn φ, for Tn running through appropriate finite subsets of I. Note that φn will then tend to φ with respect to both norms (recall that two norms on M are equivalent iff they induce the same topology, because the valuation on K is non-trivial) and the result follows. The corollary enables us to conclude that the notion of a characteristic power series only depends on the topology on M , when A is a commutative Noetherian K-Banach algebra. In particular it does not depend on the choice of an orthonormal basis for M . Coleman proves in corollary A2.6.1 of [10] that the definition of the characteristic power series only depends on the topology on M when A is semi-simple; on the other hand neither of these conditions on A implies the other. Next we show that the analogue of Corollaire 2 to Proposition 7 of [15] is true in this setting. Coleman announces such an analogue in Proposition A2.3 of [10] but again we have not been able to complete the proof in the generality in which Coleman is working. We write down a complete proof when A is a commutative Noetherian K-Banach algebra and remark that it is actually slightly delicate. We remark also that in the case where A is a reduced affinoid, which will be true in the applications, one can give an easier proof by using Corollary 2.10 to reduce to the case treated by Serre. Lemma 2.7 If M and N are ONable Banach A-modules, if u : M → N is compact and v : N → M is continuous, then uv and vu are compact, and det(1 − Xuv) = det(1 − Xvu). Proof If there exist finite free sub-A-modules F of M and G of N such that u(M ) is contained in G and v(G) is contained in F , then u : F → G and v : G → F , and by Lemma 2.5(c) it suffices to check that the algebraicallydefined characteristic polynomials of uv : G → G and vu : F → F are the same, which is a standard result. We reduce the general case to this case by

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several applications of Lemma 2.3, the catch being that it is not clear (to the author at least) whether any finite submodule of an ONable Banach module is contained within a finite free submodule. We return to the general case. Write u as a limit of finite rank operators un . Then un v and vun are finite rank, so uv and vu are both the limit of finite rank operators and are hence compact. By Lemma 2.5(a), it suffices to prove that det(1 − Xun v) = det(1 − Xvun ) for all n, and hence we may assume that u is finite rank. Let Q ⊆ N denote a finite A-module containing the image of u. Choose ON bases {ei : i ∈ I} for M and {fj : j ∈ J} for N . Now for any positive integer n we may, by Lemma 2.3(c), choose a finite subset Tn ⊆ J such that |πTn q − q| ≤ n1 |q| for all q ∈ Q. It follows easily that |πTn u − u| ≤ |u|/n and hence limn→∞ πTn u = u. Hence vπTn u → vu and πTn uv → uv and again by Lemma 2.5(a) we may replace u by πTn u and in particular we may assume that the image of u is contained in a finite free A-submodule of N . Let G denote this submodule. Now P = v(G) is a finite submodule of M , and for any positive integer n we may, as above, choose a finite subset Sn ⊆ I such that |πSn p − p| ≤ n1 |p| for all p ∈ P . It is unfortunately not the case that πSn v → v as n → ∞, as v is not in general compact. However we do have πSn vu → vu and hence the characteristic power series of πSn vu tends (uniformly in the coefficients) to the characteristic power series of vu. Also, the image of uv : N → N and uπSn v are both contained within G and hence the characteristic power series of uv (resp. uπSn v) is equal to the algebraically-defined characteristic power series of uv : G → G (resp. uπSn v : G → G). Once one has restricted to G, one does have uπSn v → uv, and hence the characteristic power series of uπSn v tends to the characteristic power series of uv. We may hence replace v by πSn v and in particular may assume that the image of v is contained within a finite free A-submodule F of M . We have now reduced to the algebraic case dealt with at the beginning of the proof. Corollary 2.6 also enables us to slightly extend the domain of definition of a characteristic power series: if M is a Banach A-module, then we say that M is potentially ONable if there exists a norm on M equivalent to the given norm, for which M becomes an ONable Banach A-module. Equivalently, M is potentially ONable if there is a bounded collection {ei : i ∈ I} of elements of M with the following two properties: firstly, every element m of M can be uniquely written as i ai ei with limi→∞ ai = 0, and secondly there exist positive constants c1 and c2 such that for all m = i ai ei in M , we have c1 supi |ai | ≤ |m| ≤ c2 supi |ai |. We call the collection {ei : i ∈ I} a potentially ON basis for M . Being potentially ONable is probably a more natural

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notion than being ONable, because it is useful to be able to work with norms only up to equivalence, whereas ONability of a module really depends on the precise norm on the module. Note that to say a module is ONable is equivalent to saying that it is isometric to some cA (I), and to say that it is potentially ONable is just to say that it is isomorphic to some cA (I) (in the category of Banach modules, with continuous maps as morphisms). If M is potentially ONable then one still has the notion of the characteristic power series of a compact operator on M , defined by choosing an equivalent ONable norm and using this norm to define the characteristic power series. By Corollary 2.6, this is independent of all choices. We note that certainly there can exist Banach A-modules which are potentially ONable but not ONable, √ for example if A = K = Qp and M = Qp ( p) with its usual norm, then |M | = |A| and so M is not ONable, but is potentially ONable. A useful result is Lemma 2.8 If h : A → B is a continuous morphism of Noetherian K-Banach A B is algebras, and M is a potentially ONable Banach A-module, then M ⊗ a potentially ONable Banach B-module, and furthermore if {ei : i ∈ I} is a potentially ON basis for M , then {ei ⊗ 1 : i ∈ I} is a potentially ON basis for A B. M⊗ Proof Set N = cB (I), and let {fi : i ∈ I} be its canonical ON basis. Then there is a natural A-bilinear bounded map M × B → N sending ( i ai ei , b) A B → N . On the other to i bh(ai )fi , which induces a continuous map M ⊗ hand, if n ∈ N , one can write n as a limit of elements of the form i∈S bi fi , where S is a finite subset of I. The element i∈S ei ⊗ bi of M ⊗A B has norm bounded above by a constant multiple of maxi∈S |bi | and hence as S increases, the resulting sequence i∈S ei ⊗ bi is Cauchy and so its image in A B tends to a limit. This construction gives a well-defined continuous M⊗ A B which is easily checked to be an A-module homomorphism N → M ⊗ A B → N , and now everything follows. inverse to the natural map M ⊗ Note that because we are only working in the “potential” world, we do not need to assume the map A → B is contractive, although in the applications we have in mind it usually will be. Corollary 2.9 If h : A → B is a continuous morphism of commutative Noetherian K-Banach algebras, M and N are potentially ONable Banach A-modules with potentially ON bases (ei ) and (fj ), and φ : M → N is com AB → N ⊗ A B is also compact and pact, with matrix (ai,j ), then φ ⊗ 1 : M ⊗

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if (bi,j ) is the matrix of φ ⊗ 1 with respect to the bases (ei ⊗ 1) and (fj ⊗ 1) then bi,j = h(ai,j ). Proof Compactness of φ ⊗ 1 follows from Proposition 2.4 and the rest is easy. Corollary 2.10 With notation as above, if det(1 − Xφ) = det(1 − X(φ ⊗ 1)) = n h(cn )X n .

n cn X

n

then

Proof Immediate. In practice we need to extend the notion of the characteristic power series of a compact operator still further, to the natural analogue of projective modules in this setting. Let us say that a Banach A-module P satisfies property (P r) if there is a Banach A-module Q such that P ⊕ Q, equipped with its usual norm, is potentially ONable. I am grateful to the referee for pointing out the following universal property for such modules: P has property (P r) if and only if for every surjection f : M → N of Banach A-modules and for every continuous map α : P → N , α lifts to a map β : P → M such that f β = α. The proof is an elementary application of the Open Mapping Theorem; the key point is that if P = cA (I) for some set I, then to give α : P → N is to give a bounded map I → N , and such a map lifts to a bounded map I → M by the Open Mapping Theorem. Note however that it would be perhaps slightly disingenuous to call such modules “projective”, as there are epimorphisms in the category of Banach A-modules whose underlying module map is not surjective. One can easily check that if P is a finite Banach A-module which is projective as an A-module, then P has property (P r). The converse is also true: Lemma 2.11 If P is a finite Banach A-module with property (P r) then P is projective as an A-module. Proof Choose a surjection An → P for some n and then use the universal property above. Note that potentially ONable Banach A-modules have property (P r), but in general the converse is false—for example if there are finite A-modules which are projective but not free then such modules, equipped with any complete Banach A-module norm, will satisfy (P r) but will not be potentially ONable. Say P satisfies property (P r) and φ : P → P is a compact morphism. Define det(1 − Xφ) thus: firstly choose Q such that P ⊕ Q is potentially

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ONable, and define det(1 − Xφ) = det(1 − X(φ ⊕ 0)); note that φ ⊕ 0 : P ⊕ Q → P ⊕ Q is easily seen to be compact. This definition may a priori depend on the choice of Q, but if R is another Banach A-module such that P ⊕ R is also potentially ONable, then so is P ⊕ Q ⊕ P ⊕ R, and the maps φ ⊕ 0 ⊕ 0 ⊕ 0 and 0 ⊕ 0 ⊕ φ ⊕ 0 are conjugate via an isometric A-module isomorphism, and hence have the same characteristic power series. Now the fact that det(1−Xφ) is well-defined independent of choice of Q follows easily from the fact that if M and N are ONable A-modules, and φ : M → M is compact, then the characteristic power series of φ and φ⊕0 : M ⊕N → M ⊕N coincide. Many results that we have already proved for potentially ONable Banach A-modules are also true for modules with property (P r), and the proofs are typically easy, because one can reduce to the potentially ONable case without too much difficulty. Indeed the trick used in the example above is typically the only idea one needs. One sometimes has to also use the following standard ingredients: Firstly, if R is any commutative ring, P is a finite projective Rmodule, and φ : P → P is an R-module homomorphism, then there is an algebraically-defined det(1 − Xφ), defined either by localising and reducing to the free case, or by choosing a finite projective R-module Q such that P ⊕Q is free, and defining det(1 − Xφ) to be det(1 − X(φ ⊕ 0)). And secondly, if M and N both have property (P r) and φ : M → M and ψ : N → N are compact, then det(1 − X(φ ⊕ ψ)) = det(1 − Xφ) det(1 − Xψ). Finally, we leave it as an exercise for the reader to check the following generalisations of Lemma 2.7 and Lemma 2.8–Corollary 2.10. Lemma 2.12 If M and N are Banach A-modules with property (P r), if u : M → N is compact and v : N → M is continuous, then uv and vu are compact, and det(1 − Xuv) = det(1 − Xvu). Lemma 2.13 If M is a Banach A-module with property (P r), φ : M → M is compact, and h : A → B is a continuous morphism of commutative Noethe A B has property (P r) as a B-module, rian K-Banach algebras, then M ⊗ φ ⊗ 1 is compact, and det(1 − X(φ ⊗ 1)) is the image of det(1 − Xφ) under h.

3 Resultants and Riesz theory We wish now to mildly extend the results in sections A3 and A4 of [10] to the case where A is a Noetherian K-Banach algebra and M is a Banach Amodule satisfying property (P r). Fortunately much of what Coleman proves already applies to our situation, or can easily be modified to do so. We make

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what are hopefully some helpful comments in case the reader wants to check the details. This section is not self-contained, and anyone wishing to check the details should read it in conjunction with §A3 and §A4 of [10]. Section A3 of [10] applies to commutative Noetherian K-Banach algebras already (apart from the comments relating to semi-simple algebras, because in general a commutative Noetherian K-Banach algebra may contain nilpotents). We give some hints for following the proofs in this section of [10]. We define the ring A{{T }} to be the subring of A[[T ]] consisting of power series n |cn |Rn → 0 n≥0 cn T with the property that for all R ∈ R>0 , we have as n → ∞. One could put a norm on A{{T }}, for example | n cn T n | = Maxn |cn |, but A{{T }} is not in general complete with respect to this norm. One very useful result about this ring is that if H(T ) ∈ A{{T }} and D(T ) is a monic polynomial of degree d ≥ 0 then H(T ) = Q(T )D(T ) + R(T ) with Q(T ) ∈ A{{T }} and R(T ) a polynomial of degree less than d. Furthermore, Q(T ) and R(T ) are uniquely determined. A word on the proof: uniqueness uses the kind of trick in Lemma A3.1 of [10]. For existence one reduces to the case where all the coefficients of D have norm at most 1 and proves the result for polynomials first, and then takes a limit. If Q ∈ A[T ] is a monic polynomial, and P ∈ A{{T }}, then Coleman defines the resultant Res(Q, P ) on the top of p434 of [10]. Many of the formulae that Coleman needs are classical when P is a polynomial, and can be extended to the power series case using the following trick: straight from the definition it follows that if u ∈ A× then Res(Q, P ) = Res(u−n Q(uT ), P (uT )). This normalisation can be used to renormalise either Q or P into A0 T , where A0 := {a ∈ A : |a| ≤ 1}. If Sn denotes the symmetric group acting naturally on A0 T1 , . . . , Tn then the subring left invariant by the action is A0 e1 , . . . , en , where the ei are the elementary symmetric functions of the Ti . Hence if P, Q ∈ A0 T then Res(Q, P ) ∈ A0 . If Q ∈ A0 [T ] is monic then one can check that the definition of a resultant makes sense for P ∈ AT , and furthermore that Res(Q, −) is locally uniformly continuous in the second variable (in the sense that for all M ∈ R, Res(Q, −) is a uniformly continuous function from {P ∈ AT : |P | ≤ M } to A). Coleman defines a function D sending a pair B, P ∈ A[X] to an element D(B, P ) ∈ A[T ]. In fact if P has degree n and we define P ∗ (X) = X n P (X −1 ), then D(B, P ) = Res(P ∗ (X), 1 − T B(X)) where the resultant is computed in AT {{X}}. One can check that D(B(uX), P (u−1 X)) = D(B, P ) if u ∈ A× , and that D is locally uniformly continuous in the B variable. It is also locally uniformly continuous in the P variable, because Res(X, C(X)) = 1 if C(0) = 1. This is enough to check that Coleman’s definition of D(B, P ) makes sense when B and P are in A{{T }}. In fact

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we shall only need it when B is a polynomial. Another useful formula is that D(uB, P )(T ) = D(B, P )(uT ) for u ∈ A× . In §A4 of [10] Coleman assumes his hypothesis (M), which tends not to be true for affinoids over K if K is not algebraically closed. Coleman also assumes that he is working with an ONable Banach A-module. We work in our more general situation. Hence let A denote a commutative Noetherian K-Banach algebra, let M be a Banach A-module satisfying property (P r) and let φ : M → M be a compact morphism, with characteristic power series P (X) = det(1 − Xφ). We define the Fredholm resolvant of φ to be P (X)/(1 − Xφ) ∈ A[φ][[X]]. Exactly as in Proposition 10 of [15], m one can prove that if F (X) = then for all R ∈ R>0 the n≥0 vm X m sequence |vm |R tends to zero, where vm is thought of as being an element of Hom(M, M ). Lemma A4.1 of [10] goes through unchanged, and we recall it here (Notation: if Q(X) is a polynomial of degree n then Q∗ (X) denotes X n Q(X −1 )): Lemma 3.1 With A, M , φ and P as above, if Q(X) ∈ A[X] is monic then Q and P generate the unit ideal in A{{X}} if and only if Q∗ (φ) is an invertible operator on M . Before we continue, let us make some remarks on zeroes of power series. If f = a T n is in A[[T ]] and s ∈ Z≥0 then we define Δs f = #n+s$n≥0 n n T ∈ A[[T ]]. If f, g ∈ A[[T ]] then it is possible to check n≥0 s an+s s i s−i that Δs (f g) = (g). One also easily checks that if A is a i=0 Δ (f )Δ Noetherian K-Banach algebra then Δs sends A{{T }} to itself. We say that a ∈ A is a zero of order h of H ∈ A{{T }} if (Δs H)(a) = 0 for s < h and (Δh H)(a) is a unit. If h ≥ 1 and H = 1 + a1 T + . . . then this implies that −1 = a(a1 + a2 a + . . .) and hence that a is a unit. One now checks by induction on h that H(T ) = (1 − a−1 T )h G(T ), where G ∈ A{{T }}, and then that G(a) is a unit. Again let M be a Banach A-module with property (P r) and let φ : M → M be a compact morphism, with characteristic power series P (T ). Say a ∈ A is a zero of P (T ) of order h. Proposition 3.2 There is a unique decomposition M = N ⊕ F into closed φ-stable submodules such that 1 − aφ is invertible on F and (1 − aφ)h = 0 on N . The submodules N and F are defined as the kernel and the image of a projector which is in the closure in Hom(M, M ) of A[φ]. Moreover, N is projective of rank h, and assuming h > 0 then a is a unit and the characteristic power series of φ on N is (1 − a−1 T )h .

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Proof We start by following Proposition 12 of [15], much of which goes through unchanged in our setting. We find that there are maps p and q in Hom(M, M ), both in the closure of A[φ], such that p2 = p, q 2 = q and p + q = 1, and if we consider the decomposition M = N ⊕ F corresponding to these projections, N = ker(p), then (1 − aφ)h = 0 on N , and (1 − aφ) is invertible on F . The decomposition is visibly unique, as if ψ = (1 − aφ)h then N = ker(ψ) and F = Im(ψ). We now diverge from Proposition 12 of [15]. It is clear that N satisfies (P r), but furthermore we have (1 − aφ)h = 0 on N which implies that the identity is compact on N . An elementary argument (change the metric on N to an equivalent one if necessary and reduce to a computation of matrices) shows that if β ∈ Hom(N, N ) has sufficiently small norm, then |β n | → 0 and hence 1 − β is invertible. Because 1 is compact, we can choose α : N → N of finite rank such that 1 − α is sufficiently small, and hence α is invertible and so N is finitely-generated. By Lemma 2.11, N is projective. If h = 0 then N = 0 and F = M , as can be seen from Lemma 3.1, and we are home. So assume for the rest of the proof that h > 0. Then P (a) = 0 and this implies that a must be a unit. If PN and PF denote the characteristic power series of φ on N and F respectively, then P = PN PF and by Lemma 3.1 we see that (T − a)h and PF generate the unit ideal in A{{T }}. Hence (1 − a−1 T )h divides PN in A{{T }}. Moreover, PN is a polynomial because N is finitely-generated, and hence (1 − a−1 T )h divides PN in A[T ]. Moreover, (1 − a−1 T )h has constant term 1 and is hence not a zero-divisor in A{{T }}, hence if PN (T ) = D(T )(1 − a−1 T )h and P (T ) = (1 − a−1 T )h G(T ) then D(T ) divides G(T ) in A{{T }} and so D(a) is a unit. We know that D(T ) is a polynomial. Furthermore, because (1 − aφ)h = 0 on N we see that φ has an inverse on N and hence that the determinant of φ is in A× . Hence the leading term of D is a unit. Reducing the situation modulo a maximal ideal of A we see that the reduction of PN must be a power of the reduction of (1 − a−1 T ) and this is enough to conclude that D = 1. Hence the characteristic power series of φ on N is (1 − a−1 T )h . Finally, the fact that φ is invertible on N means that the rank of N at any maximal ideal must equal the degree of PN modulo this ideal, and hence the rank is h everywhere. Keep the notation: M has (P r) and φ : M → M is compact, with characteristic power series P (T ). Theorem 3.3 Suppose P (T ) = Q(T )S(T ), where S = 1+. . . ∈ A{{T }} and Q = 1+. . . is a polynomial of degree n whose leading coefficient is a unit, and which is relatively prime to S. Then there is a unique direct sum decomposition

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M = N ⊕ F of M into closed φ-invariant submodules such that Q∗ (φ) is zero on N and invertible on F . The projectors M → N and M → F are elements of Hom(M, M ) which are in the closure of A[φ]. Furthermore, N is projective of rank n and the characteristic power series of φ on N is Q(T ). Proof We follow Theorem A4.3 of [10]. The operator v = 1 − Q∗ (φ)/Q∗ (0) has a characteristic power series which has a zero at T = 1 of order n. Applying the previous proposition to v, we see M = N ⊕ F , where N and F are defined as the kernel and the image of a projector in the closure of A[v] and hence in the closure of A[φ]. Hence both N and F are φ-stable. Unfortunately, by the end of the proof of Theorem A4.3 of [10] one can only deduce that Q∗ (φ)n is zero on N and invertible on F , so we are not quite home yet. However, by Proposition 3.2, N is projective of rank n. Moreover, the characteristic power series of φ on F is coprime to Q, by Lemma 3.1. Hence if G(T ) is the characteristic power series of φ on N , we see that Q divides G. But G and Q have degree n and the same constant term, and furthermore the leading coefficient of Q is a unit. This is enough to prove that G = Q.

4 An admissible covering The key aim in this section is to generalise some of the results of section A5 of [10] (especially Proposition A5.8) to the case where the base is an arbitrary reduced affinoid. In fact almost all of Coleman’s results go through unchanged, but there are some differences, which we summarise here. Firstly it is not true in general that the image of an affinoid under a quasi-finite map is still affinoid. However if one works with finite unions of affinoids then one can deal with the problems that this causes. Secondly Coleman uses the notion of a strict neighbourhood of a subspace of the unit disc. We slightly modify this notion to one which suits our purpose. Lastly we need some kind of criterion for when a quasi-finite map of rigid spaces of constant degree is finite. We use a theorem of Conrad whose proof invokes Raynaud’s theory of admissible formal models of rigid spaces. We set up some notation. Let K be a field with a complete non-trivial nonArchimedean valuation. Let R denote a reduced K-affinoid algebra, and let B = Max(R) be the associated affinoid variety. We equip R with its supremum semi-norm, which is a norm in this case. Let R{{T }} denote the ring of power series n≥0 an T n with an ∈ R such that for all real r > 0 we have |an |rn → 0 as n → ∞. Then R{{T }} is just the ring of functions on B ×K A1,an , where here A1,an denotes the analytification of affine 1-space over K.

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Let P (T ) = n≥0 rn T n ∈ R{{T }} be a function with r0 = 1. Our main object of study is the rigid space cut out by P (T ), that is, the space Z ⊆ B ×K A1,an defined by the zero locus of P (T ). In practice, P (T ) will be the characteristic power series of a compact endomorphism of a Banach Rmodule. Certainly Z is a rigid analytic variety, equipped with projection maps f : Z → B and % g : Z → A1,an . We frequently make use of the following cover of Z: If r ∈ |K × | (that is, some power of r is the norm of a non-zero element of K) then let B[0, r] denote the closed affinoid disc over K of radius r, considered as an admissible open subspace of A1,an . Let Zr denote the zero locus of P (T ) on the space B ×K B[0, r]. Then Zr is an affinoid, and the Zr admissibly cover Z. Let fr : Zr → B denote the canonical projection. Note that any affinoid subdomain of Z will be admissibly covered by its intersections with the Zr , which are affinoids, and hence will be contained within some Zr . Now let C denote the set of affinoid subdomains Y of Z with the following property: there is an affinoid subdomain X of B (depending on Y ) with the property that Y ⊆ ZX := f −1 (X), the induced map f : Y → X is finite and surjective, and Y is disconnected from its complement2 in ZX , that is, there is a function e ∈ O(ZX ) such that e2 = e and Y is the locus of ZX defined by e = 1. Our goal is (c.f. Proposition A5.8 of [10]) Theorem C is an admissible cover of Z. The reason we want this result is that in later applications Z will be a “spectral variety”, and the Y ∈ C are exactly the affinoid subdomains of Z over which one can construct a Hecke algebra, and hence an eigenvariety, without any technical difficulties. We prove the theorem after establishing some preliminary results. Lemma 4.1 fr : Zr → B is quasi-finite and flat. Proof By increasing r if necessary, we may assume r ∈ |K × | and hence, by rescaling, that r = 1. The situation we are now in is as follows: R is an affinoid and P (T ) = n≥0 rn T n ∈ RT with r0 = 1, and we must show that R → RT /(P (T )) is quasi-finite and flat. Quasi-finiteness is immediate from the Weierstrass preparation theorem. For flatness observe that RT is 2 Elmar Grosse-Kloenne has pointed out that this condition in fact follows from the others; one can use 9.6.3/3 and 9.5.3/5 of [1] to check that Y → ZX is both an open and a closed immersion.

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flat3 over R and that if P is any maximal ideal of RT then P := P ∩ R is a maximal ideal of R. Hence RT /(PRT ) = (R/PR)T is an integral domain and the image of P (T ) in (R/PR)T is non-zero, as its constant term is non-zero. Hence the image of P (T ) is not a zero-divisor and flatness now follows from Theorem 22.6 of [14]. Corollary 4.2 If Y ⊆ Z is an affinoid then f : Y → B is quasi-finite and flat. Proof Y is affinoid and hence Y ⊆ Zr for some r ∈ follows from the previous lemma.

% |K × |, so the result

Corollary 4.3 If Y ⊆ Z is an affinoid, and X ⊆ B is an admissible open such that Y ⊆ f −1 (X), and if there is an integer d ≥ 0 such that all fibres of the induced map f : Y → X have degree d, then f : Y → X is finite and flat. Proof f : Y → X is flat by Corollary 4.2. It is also quasi-compact and separated, so finiteness follows from Theorem A.1.2 of [11]. Note that this latter result uses the full force of Raynaud’s theory of formal models. % Lemma 4.4 If r ∈ |K × | and fr : Zr → B, then for i ≥ 0 define Ui := {x ∈ B : deg(fr−1 (x)) ≥ i}. Then each Ui is a finite union of affinoid subdomains of B, and Ui is empty for i sufficiently large. Proof The sequence |rn |r n tends to zero as n → ∞, and hence for any x ∈ B, the set {|rn (x)|r n : n ≥ 0} has a maximum, denoted Mx , which is attained. Note that |r0 (x)| = 1 and hence Mx ≥ 1, and in particular if N is an integer such that |rn |rn < 1 for all n ≥ N then Mx = Max{|rn (x)|r n : 0 ≤ n < N } and Mx > |rn (x)|rn for all n ≥ N . For i ≥ 0 let Si denote the affinoid subdomain of B defined by {x ∈ B : |ri (x)|r i = Mx }. Then Si is empty for i ≥ N . A calculation on the Newton polygon shows that Ui = ∪j≥i Sj and the result follows. Definition If S and T are admissible open subsets of the affinoid B, such that both S and T are finite unions of affinoid subdomains of B, then we say that T 3 One can prove flatness by using the Open Mapping Theorem and mimicking the proof of the result stated in Exercise 7.4 of [14], noting that the solution to the exercise is on p289 of loc. cit.

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is a strict neighbourhood of S (in B) if S ⊆ T and there is an admissible open subset U of B with the following properties: • U is a finite union of affinoid subdomains of B, • U ∩ S is empty, • U ∪ T = B. The intersection of two affinoid subdomains of B is an affinoid subdomain of B. Hence if U and V are admissible open subsets of B which are both the union of finitely many affinoid subdomains, then so is U ∩ V . As a consequence, we see that if Tα is a strict neighbourhood of Sα for 1 ≤ α ≤ n then ∪α Tα is a strict neighbourhood of ∪α Sα . We now prove the key technical lemma that we need. % Lemma 4.5 Suppose r ∈ |K × | and V ⊆ B is an affinoid subdomain with the property that fr : fr−1 (V ) → V is finite of constant degree d > 0. Then there is an affinoid of V % subdomain X of B which is a strict neighbourhood −1 × in B, and s ∈ |K | with s > r such that the affinoid Y = fs (X) contains fr−1 (V ), lies in C, and is finite flat of degree d over X. Proof (c.f. Lemma A5.9 of [10]). If x ∈ V then let Px (T ) = n≥0 rn (x)T n denote the specialisation of P (T ) to K(x){{T }}. The statement that the degree of fr−1 (x) is d translates by the theory of the Newton polygon to the statement that for all x ∈ V we have |rd (x)| = 0 (so rd is a unit in Oan (V )) and furthermore that for all integers n ≥ 0 we have − log(|rn (x)|) ≥ (n − d) log(r) − log(|rd (x)|), with strict inequality when n > d. Here log is the usual logarithm, with the usual convention that − log(0) = +∞. Because P (T ) is entire, there exists an integer N > d such that for n ≥ N we have − log |rn (x)| > n log(r + 1) for all x ∈ B and hence − log(|rn (x)|) ≥ (n − d) log(r + 1) − log(|rd (x)|) for all n ≥ N . For d < n < N we have − log(|rn (x)|) > (n − d) log(r) − log(|rd (x)|) and hence |rn (x)/rd (x)| < r d−n for all x % ∈ V . Because functions on affinoids attain their bounds, there is some t ∈ |K × | with r < t < r + 1 and |rn (x)/rd (x)| < td−n for d < n < N , and hence for all x ∈ V we have − log(|rn (x)|) ≥ (n − d) log(t) − log(|rd (x)|) %for alln ≥ 0, with equality iff n = d. Now choose γ1 , γ2 , δ1 , δ2 ∈ log |K × | such that δ2 > − log |rd (x)| > δ1 for all x ∈ V and log(r) < γ1 < γ2 < log(t). Let X be the affinoid subdomain of B defined by the N equations δ1 ≤ − log |rd (x)| ≤ δ2 ,

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− log |rn (x)| + log |rd (x)| ≥ (n − d)γ1 for 0 < n < d, and − log |rn (x)| + log |rd (x)| ≥ (n − d)t for d < n < N. These equations define X as a Laurent subdomain of B, and if x ∈ V then not only are these equations satisfied, but strict inequality holds in every case. Hence V ⊂ X and moreover if we consider the N affinoids, each defined by one of the N equations δ1 ≥ − log |rd (x)|, − log |rd (x)| ≥ δ2 , − log |rn (x)| + log |rd (x)| ≤ (n − d)γ1 , 0 < n < d, − log |rn (x)| + log |rd (x)| ≤ (n − d)t, d < n < N, and let W be the union of these N affinoids, then X ∩W is empty and X ∪V = B. Hence X is a strict neighbourhood of V in B, in the sense we defined above. Let s = exp(γ2 ) and set Y = fs−1 (X). Then Y is an affinoid in Z and by the previous corollary and the way we have arranged the Newton polygons, f : Y → X is finite of degree d (note that by our choice of t we have − log |rn (x)| + log |rd (x)| ≥ (n − d)t for all n ≥ N, with strict inequality for x ∈ V ). Furthermore if x ∈ X then no slope of the Newton polygon of Px (T ) can equal s, and hence the projection from f −1 (X) to A1,an contains no elements of norm s. Hence Y is disconnected from its complement in f −1 (X), and in particular is an affinoid subdomain of Z, so Y ∈ C. We are now ready to prove the theorem. Theorem 4.6 C is an admissible cover of Z. Proof Again % we follow Coleman. We know that Z is admissibly covered by the Zr , r ∈ |K × |, and hence it suffices to prove that for every Zr , there is a finite collection of affinoids in C whose union contains Zr . Recall that for i ≥ 0, Ui is the subset of points in B such that deg(fr−1 (a)) ≥ i, and that Ui is a finite union of affinoids. Furthermore, clearly Ui+1 ⊆ Ui . If U1 is empty there is nothing to prove, so let us assume that it is not. Let d denote the largest i such that Ui is non-empty. For 1 ≤ i ≤ d let H(i) denote the following statement:

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H(i): “There is a finite set Y1 , Y2 . . . Yn(i) of affinoid subdomains of Z, and a finite set X1 , X2 . . . Xn(i) of affinoid subdomains of B, such that for 1 ≤ α ≤ n(i) we have f : Yα → Xα is finite flat and surjective, Yα ∈ C, n(i) n(i) fr−1 (Ui ) ⊆ ∪α=1 Yα , and ∪α=1 Xα is a strict neighbourhood of Ui in B.” If we can establish H(1) then we are home because fr−1 (U1 ) = Zr . We firstly establish H(d), and then show that H(i) implies H(i − 1) for i ≥ 2, and this will be enough. For H(d) we cover Ud by finitely many affinoid subdomains V1 , V2 ,. . . Vn(1) of B (in fact it is not difficult to show that Ud is itself an affinoid, but we shall not need this). By Corollary 4.3 we know that fr−1 (Vα ) → Vα is finite and flat. Now applying Lemma 4.5 to Vα we get a strict affinoid neighbourhood Xα of Vα , and if Yα = fs−1 (Xα ) (s as in the Lemma) then H(d) follows immediately. Now let us assume H(i), i ≥ 2. Then choose a finite union of affinoid subn(i) domains W ⊂ B such that W ∩ Ui is empty and W ∪ α=1 Xα = B. Then W ∩ Ui−1 is a finite union Vn(i)+1 , Vn(i)+2 . . . Vn(i)+m of affinoid subdomains. Set n(i − 1) = n(i) + m. Note that fr : fr−1 (Vα ) → Vα is finite of degree i − 1 for n(i) < α ≤ n(i − 1), hence one is in a position to apply Lemma 4.5 to get Yα → Xα finite flat of degree i − 1, Yα ∈ C, fr−1 (Vα ) ⊆ Yα , and Xα a strict neighbourhood of Vα , for n(i) < α ≤ n(i−1). n(i−1) We now show that α=1 Xα is a strict neighbourhood of Ui−1 . We know n(i−1) n(i−1) that α=n(i)+1 Xα is a strict neighbourhood of α=n(i)+1 Vα , so choose a n(i−1) finite union of affinoid subdomains W such that W ∩ ( α=n(i)+1 Vα ) is n(i−1) empty and W ∪( α=n(i)+1 Xα ) = B. Now set W = W ∩W . Then W is a n(i−1) finite union of affinoids, W ∩Ui−1 = W ∩W ∩Ui−1 = W ∩( α=n(i)+1 Vα ) n(i−1) is empty, and W ∪ ( α=1 Xα ) = B, and we are done. 5 Spectral varieties and eigenvarieties Let R be a reduced affinoid K-algebra equipped with its supremum norm, let M be a Banach R-module satisfying (P r), and let T be a commutative R-algebra equipped with an R-algebra homomorphism to EndR (M ), the continuous R-module endomorphisms of M . In practice T will be a polynomial R-algebra generated by (typically infinitely many) Hecke operators. We frequently identify t ∈ T with the endomorphism of M associated to it. Fix once and for all an element φ ∈ T, and assume that the induced endomorphism φ : M → M is compact. Let F (T ) = 1 + n≥1 cn T n be the characteristic power series of φ. We define the spectral variety Zφ associated to φ to be the closed subspace of the rigid space Max(R) × A1 cut out by F . The spectral variety is a geometric object parametrising, in some sense, the reciprocals of

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the non-zero eigenvalues of φ. Its formulation is compatible with base change, by Lemma 2.13. Our main goal in this section is to write down a finite cover of this spectral variety, the eigenvariety associated to the data (R, M, T, φ). Points on the eigenvariety will correspond to systems of eigenvalues for all the operators in T, such that the eigenvalue for φ is non-zero. The construction is just an axiomatisation of Chapter 7 of [9] and is really not deep (in fact by far the deepest part of the entire construction is the fact that the cover C of Section 4 is admissible, as this appealed to the theory of formal models at one point). Unfortunately the construction does involve a lot of bookkeeping. We begin with a finite-dimensional example, where φ is invertible and hence where we may avoid the technicalities of §4. Let R be a reduced affinoid K-algebra, and let M denote a finitely-generated projective R-module of rank d. Let T be an arbitrary R-algebra equipped with an R-algebra homomorphism to EndR (M ), and let φ be an element of T. Assume furthermore that φ : M → M has an inverse, that is, there is an R-linear φ−1 : M → M such that φ◦φ−1 = φ−1 ◦φ is the identity on M . Define P (T ) = det(1−T φ) = 1+ . . . ∈ R[T ]; then the leading term of P , that is, the coefficient of T d , is a unit. Let Zφ denote the zero locus of P (T ) regarded as a function on Max(R)×A1 . Then R[T ]/(P (T )) is a finite R-algebra and hence an affinoid algebra, and Zφ is the affinoid rigid space associated to this affinoid algebra. Let T(Zφ ) denote the image of T in EndR (M ); then T(Zφ ) is a finite R-algebra and hence an affinoid algebra. By the Cayley-Hamilton theorem we have φ−1 ∈ T(Zφ ), and furthermore there is a natural map R[T ]/P (T ) → T(Zφ ) sending T to φ−1 . Set Dφ = Max(T(Zφ )). Then the maps R → R[T ]/P (T ) → T(Zφ ) of affinoids give maps Dφ → Zφ → Max(R). We call Zφ the spectral variety and Dφ the eigenvariety associated to this data. As a concrete example, consider 2 the case where M as # 1 XR$ = KX, Y , M #0 = $R , T = R[φ, t], where φ acts on Y the matrix 0 1 , and t acts as 0 0 . Then det(1 − T φ) = (1 − T )2 so Zφ is non-reduced, and T(Zφ ) is the ring R ⊕ I, with I = (X, Y ) and 2 = 0. Note that in this case the maps Dφ → Zφ and Dφ → Max(R) are not flat, and Dφ is not reduced either. It would not be unreasonable to say that what follows in this section is just a natural generalisation of this set-up, the main complication being that M is not necessarily finitely-generated, the purpose of the admissible cover C being to remedy this. See also Chenevier’s thesis, where he develops essentially the same theory in essentially the same way (assuming Theorem 4.6 of this paper). The example above shows that in this generality one cannot expect Dφ and Zφ to have too many “good” geometric properties; however one can hope that the examples of spectral and eigenvarieties arising “in nature” are better behaved.

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Let us now go back to our more general situation, where M is a Banach Rmodule satisfying property (P r) and T is a commutative R-algebra equipped with an R-algebra map T → EndR (M ), such that the endomorphism of M induced by φ ∈ T is compact. Let Zφ be the closed subspace of Max(R) × A1 defined by the zero locus of the characteristic power series of φ. Let C be the admissible cover of Zφ constructed in section 4. Let Y be an element of this admissible cover, with image X ⊆ Max(R). By definition, X is an affinoid subdomain of Max(R), so set A = O(X). Then A is reduced by R A and, for t ∈ T, let tA denote Corollary 7.3.2/10 of [1]. Define MA := M ⊗ the A-linear continuous endomorphism of MA induced by t : M → M . Note that φA : MA → MA is still compact, by Lemma 2.13. Let FA (T ) be the characteristic power series of φA on MA . Again by Lemma 2.13, FA is just the image of F in A{{T }}. Let us assume first that X is connected. Then we wish to associate to Y a factor of FA (T ) so that we are in a position to apply Theorem 3.3. We do this as follows. We know that O(Y ) is a finite flat A-module, and hence it is projective of some rank d. The element T of O(Y ) is a root of its characteristic polynomial Q , which is monic of degree d, and hence gives us a map A[T ]/(Q (T )) → O(Y ). In fact, Y is a closed subspace of X × B[0, r] for some r, and hence if S is some appropriate K-multiple of T then the natural map AS → O(Y ) is surjective. By Proposition 3.7.4/1 of [1] and its proof, any residue norm on O(Y ) will be equivalent to any of the Banach norms that O(Y ) inherits from being a finite complete A-module. One can deduce from this that the map A[T ]/(Q (T )) → O(Y ) is surjective. Hence A[T ]/(Q (T )) → O(Y ) is an isomorphism, because both sides are locally free A-modules of rank d. This means that the image of FA (T ) in A{{T }}/(Q (T )) is zero, and hence that Q divides FA in A{{T }}. Comparing constant terms, we see that Q = a0 + a1 T + . . . with a0 a unit, and hence we can define Q = a−1 0 Q and we are in a position to invoke Theorem 3.3 to give a decomposition MA = N ⊕ F where N is projective of rank d over A. Note that in general N will not be free. Because the projector MA → N is in the closure of A[φ], it commutes with all the endomorphisms of MA induced by elements of T, and hence N is t-invariant for all t ∈ T. Define T(Y ) to be the A-sub-algebra of EndA (N ) generated by all the elements of T. Now EndA (N ) is a finite A-module, and hence T(Y ) is a finite A-algebra and hence an affinoid. Let D(Y ) denote the associated affinoid variety. We know that Q∗ (φ) is zero on N , and hence T(Y ) is naturally a finite A[S]/(Q∗ (S))algebra, via the map sending S to φ. Because the constant term of Q∗ is a unit, there is a canonical isomorphism A[S]/(Q∗ (S)) = O(Y ) sending S to T −1 .

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Hence T(Y ) is a finite O(Y )-algebra, and thus there is a natural finite map D(Y ) → Y . For general Y ∈ C, the image of Y in Max(R) may not be connected, but Y can be written as a finite disjoint union Y = ∪Yi corresponding to the connected components of the image of Y in Max(R). We define D(Y ) as the disjoint union of the D(Yi ). This construction gives us, for each Y ∈ C, a finite cover D(Y ) of Y . We wish to glue together the D(Y ), as Y ranges through all elements of C, and the resulting curve D, which will be a finite cover of Zφ , will be the eigenvariety associated to the data (R, M, T, φ). We firstly establish a few lemmas. Lemma 5.1 If Y ∈ C with image X ⊆ Max(R), and X is an affinoid subdomain of X then Y , the pre-image of X under the map Y → X, is in C, and is an affinoid subdomain of Y . Furthermore, D(Y ) is canonically isomorphic to the pre-image of Y under the map D(Y ) → Y . Proof Y is the pre-image of X under the map Y → X and is hence an affinoid subdomain of Y by Proposition 7.2.2/4 of [1]. The map Y → X is finite and surjective, and if e is the idempotent in O(ZX ) showing that Y is disconnected from its complement (that is, e|Y = 1 and e|ZX \Y = 0), then the restriction of e to O(ZX ) will do the same for Y . Hence Y ∈ C. It is now O(X) O(X ) and hence D(Y ) is elementary to check that T(Y ) = T(Y )⊗ the pre-image of Y under the map D(Y ) → Y . Lemma 5.2 If Y1 , Y2 ∈ C then Y := Y1 ∩ Y2 ∈ C. Furthermore for 1 ≤ i ≤ 2, Y is an affinoid subdomain of Yi , and D(Y ) is canonically isomorphic to the pre-image of Y under the map D(Yi ) → Yi . Proof Let Xi denote the image of Yi in Max(R). Then the Xi are affinoid subdomains of Max(R), and hence so is their intersection. Let X denote a component of X1 ∩ X2 . It suffices to prove the assertions of the lemma with Y replaced by Y ∩ ZX , so let us re-define Y to be Y ∩ ZX . Let Yi denote the pre-image of X under the map Yi → Xi . Then Yi is an affinoid subdomain of Yi containing Y and by Lemma 5.1 we have Yi ∈ C with D(Yi ) the pre-image of Yi under the map D(Yi ) → Yi . Now Y = Y1 ∩ Y2 is finite and flat over Y1 and hence finite and flat over X. Let ei ∈ O(ZX ) be the idempotent associated to Yi , and set e = e1 e2 . Then Y is the subset of ZX defined by e = 1, and hence Y is locally free of finite rank over X. One easily checks that Y is a union of components of Yi for 1 ≤ i ≤ 2, in fact. If Y is empty then the rest of the lemma is clear. If not then the map

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Y → X is surjective, and Y ∈ C. Finally, for 1 ≤ i ≤ 2, the idempotents e and ei (1 − e3−i ) sum to 1 on Yi showing that D(Y ) is actually a union of connected components of D(Yi ), pulling back the inclusion Y ⊆ Yi . Now by Proposition 9.3.2/1 of [1] we can glue the D(Y ) for Y ∈ C to get a rigid space Dφ (the cocycle conditions are satisfied because they are satisfied for the cover C of Zφ ), and by Proposition 9.3.3/1 of [1] we can glue the maps D(Y ) → Y to get a map Dφ → Zφ . We say that the rigid space Dφ is the eigenvariety associated to the data (R, M, T, φ). We have already seen in the finite-dimensional case that the map Dφ → Zφ might not be flat, and that Zφ and Dφ may be non-reduced. We summarise the obvious positive results about Zφ and Dφ that come out of their construction: Lemma 5.3 Dφ and Zφ are separated, and the map Dφ → Zφ is finite. Proof Zφ is separated by, for example, Proposition 9.6/7 of [1] (applied to the admissible covering {Zr } of Zφ defined in the previous section). The construction of Dφ over Zφ shows that Dφ → Zφ is finite; hence Dφ → Zφ is separated, which implies that Dφ is separated. We need to establish further functorial properties of this construction. As before, let R be a reduced K-affinoid algebra equipped with its supremum norm, let M be a Banach R-module satisfying (P r), and let T be a commutative R-algebra equipped with a distinguished element φ and an R-algebra homomorphism T → HomR (M, M ), such that the image of φ is a compact endomorphism. We now consider what happens when we change R. More specifically, let R denote another reduced K-affinoid algebra equipped with a map R → R , and let M , T , φ denote the obvious base extensions. The constructions above give us maps Dφ → Zφ → Max(R) and Dφ → Zφ → Max(R ). The map R → R gives us a map Max(R ) → Max(R). Lemma 5.4 Zφ → Max(R ) is canonically isomorphic to the pullback of Zφ → Max(R) to Max(R ). Proof This is an immediate consequence of Lemma 2.13. In particular there is a natural map Zφ → Zφ . Lemma 5.5 If R → R is flat then Dφ → Zφ is canonically isomorphic to the pullback of Dφ → Zφ under the map Zφ → Zφ .

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Proof Let C and C denote usual the admissible covers of Zφ and Zφ . If Y ∈ C and Y is the pullback of Y to Zφ then one checks without too much difficulty that Y ∈ C . It is not immediately clear whether every element of C arises in this way (although this is the case if Max(R ) ⊆ Max(R) is an affinoid subdomain, which will be the only case we are interested in in practice). However, this does not matter because the elements of C which arise in this way still form an admissible covering of Zφ , as they cover the separated space Zφ , and all the elements of C are affinoids so one can use Proposition 9.1.4/2 of [1]. Hence we may construct Dφ by gluing the D(Y ) for all Y which arise in this way. One checks that D(Y ) is the pullback of D(Y ) under the map Y → Y (this is where we use flatness, the point being that without flatness one cannot deduce that the natural map from D(Y ) to the pullback of D(Y ) is an isomorphism) and that everything is compatible with gluing, and after this somewhat tedious procedure one deduces that the maps D(Y ) → D(Y ) identify Dφ with the pullback of Dφ as indicated. We will only be applying the above lemma in the case where Max(R ) is an affinoid subdomain of Max(R) and in particular R → R is flat in this case. I am grateful to the referee for pointing out that flatness is necessary for this lemma to be true. Indeed, one checks that in the example at the beginning of this section (with R = KX, Y ), if R = K and the map R → R sends X and Y to zero, then the pullback of the eigenvariety is not isomorphic to the eigenvariety associated to the pullback. On the other hand, one can use the q-expansion principle to check that the construction of the cuspidal Coleman-Mazur eigencurve (see part II of this paper) will commute with any base change of reduced affinoids. Had we set up the theory for non-reduced bases one could no doubt even check that the construction commutes with arbitrary base change. The same arguments do not work for the eigenvarieties associated to totally definite quaternion algebras over totally real fields (see part III of this paper) and for Chenevier’s unitary group eigenvarieties [8]. One does not have a q-expansion principle in these cases, and whether construction of these eigenvarieties commutes with all base changes (even those coming from the inclusion of a point into weight space) seems to be an open question, related to multiplicity one issues for overconvergent automorphic eigenforms in these settings. See Lemma 5.9 for a partial result. Another way of setting up the foundations of the theory of eigenvarieties might be to construct the eigenvariety as a limit of spectral varieties such as those in sublemma 6.2.3 of [9]; such a construction might well commute with all base changes, but would not see subtleties such as non-semisimplicity of eigenspaces at a fixed weight.

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We now analyse how eigenvarieties change under a specific type of change of module. As always, let R be a reduced affinoid, let M and M denote Banach modules satisfying property (P r), let T be a commutative R-algebra equipped with maps T → EndR (M ) and T → EndR (M ), such that our chosen element φ ∈ T acts compactly on both M and M . In practice we are interested only in modules M and M which are related in a specific way, which we now axiomatise. We say that a continuous R-module and T-module homomorphism α : M → M is a “primitive link” if there is a compact R-linear and T-linear map c : M → M such that φ : M → M is α ◦ c and φ : M → M is c ◦ α. Note that these assumptions force the characteristic power series of φ on M and M to coincide, by Lemma 2.12. Note also that the identity map M → M is a primitive link (take c = φ). We say that a continuous R-module and T-module homomorphism α : M → M is a “link” if one can find a sequence M = M0 , M1 , M2 , . . . , Mn = M of Banach Rmodules satisfying property (P r) with T-actions, and continuous R-module and T-module maps αi : Mi → Mi+1 such that each αi is a primitive link, and α is the compositum of the αi . We apologise for this terrible notation but the underlying notion is what occurs in applications; our motivation is the study of r-overconvergent modular forms as r changes. More precisely, with notation as in Part II of this manuscript, if 0 < r ≤ r < p/(p + 1) and α is the inclusion from r -overconvergent forms to r-overconvergent forms, then α will be a primitive link if r ≤ pr, but if r > pr then α may only be a link. Perhaps all of this can be avoided if one sets up the theory with a slightly more general class of topological modules. Lemma 5.6 Let R, M , M , T, φ be as above, and assume that we are given a link α : M → M in the sense above. Let Dφ denote the eigenvariety associated to (R, M, T, φ) and let Dφ denote the eigenvariety associated to (R, M , T, φ). Then Dφ and Dφ are isomorphic. Proof This is clear if α is an isomorphism, so we may assume that α is a primitive link, and thus that there is a compact c : M → M such that αc and cα are equal to the endomorphisms of M and M induced by φ. We use a dash to indicate the analogue of one of our standard constructions, applied to M (for example Zφ , C and so on). By Lemma 2.13, Zφ and Zφ are equal, as are C and C (as their construction does not depend on the underlying Banach module). Choose Y ∈ C with connected image X ⊆ Max(R). It will suffice to prove that α induces an isomorphism D (Y ) = D(Y ) that commutes with all the glueing data on both sides, and this will follow if we can show that, after base extension to A = O(X), α induces an isomorphism between the finite

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flat sub-R-modules N and N of M and M corresponding to Y , and hence that α (which recall is T-linear) induces an isomorphism T(Y ) = T (Y ). Recall that there is a polynomial Q = 1 + . . . associated to Y as in the definition of D(Y ), such that the leading term of Q is a unit, and such that N and N are the kernels of Q∗ (φ) on M and M respectively. From this one can conclude that α maps N to N , that c maps N to N , and that there is an element ψ of R[φ] ⊆ T, such that ψ is an inverse to φ on both N and N . We now see that α : N → N must be an R-module isomorphism, because it is elementary to check that ψ ◦ c is a two-sided inverse (recall that ψ is a polynomial in φ and hence commutes with c). Now everything else follows without too much trouble. I thank Peter Schneider for pointing out a problem with the proof of the above lemma in the initial version of this manuscript. We now have enough for our eigenvariety machine. The data we are given is the following: we have a reduced rigid space W, a commutative R-algebra T, and an element φ ∈ T. For any admissible affinoid open X ⊆ W, with O(X) = RX (equipped with its supremum norm), we have a Banach RX module MX satisfying (P r), and an R-module homomorphism T → HomRX (MX , MX ), denoted t → tX , such that φX is compact. Finally, if Y ⊆ X ⊆ W are two admissible affinoid opens, then we have a continuous O(Y )-module RX RY which is a “link” in the above sense, homomorphism α : MY → MX ⊗ and such that if X1 ⊆ X2 ⊆ X3 ⊆ W are all affinoid subdomains then O(Xi ) O(Xj ). α13 = α23 α12 where αij denotes the map MXi → MXi ⊗ Construction 5.7 (eigenvariety machine) To the above data we may canonically associate the eigenvariety Dφ , a rigid space equipped with a map to W, with the property that for any affinoid open X ⊆ W, the pullback of Dφ to X is canonically isomorphic to the eigenvariety associated to the data (RX , MX , T, φX ). There is very little left to check in this construction. If Y ⊆ X are affinoid subdomains of W then by Lemma 5.6 the eigenvarieties associated to RX RY , T, φX ) are isomorphic. By Lemma (RY , MY , T, φY ) and (RY , MX ⊗ RX RY , T, φX ) is isomorphic to 5.5 the eigenvariety associated to (RY , MX ⊗ the pullback to Y of the eigenvariety that is associated to (RX , MX , T, φX ). The assumption on compatibility of the α ensures that the cocycle condition is satisfied, and hence the Dφi glue together to give an eigenvariety Dφ over W whose restriction to Xi is Dφi . As we have seen earlier, one cannot expect Dφ to be reduced or flat over W in this generality. However, here are some positive results.

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Lemma 5.8 Assume W is equidimensional of dimension n. Then Dφ is also equidimensional of dimension n. The finite map Dφ → Zφ has the property that each irreducible component of Dφ maps surjectively to an irreducible component of Zφ . Moreover, the image in W of each irreducible component of Dφ is Zariski-dense in a component of W. Proof This is Proposition 6.4.2 of [8]. We now explain how the points of Dφ are in bijection with systems of eigenvalues of T. For L a complete extension of K, say that a map λ : T → L is an L-valued system of eigenvalues if there is an affinoid X = Max(RX ) ⊆ W, RX L such a point in X(L) (giving a map RX → L) and 0 = m ∈ MX ⊗ that tm = λ(t)m for all t ∈ T. Say that an L-valued system of eigenvalues is φ-finite if λ(φ) = 0. Lemma 5.9 There is a natural bijection between φ-finite systems of eigenvalues and L-points of Dφ . Proof Because Dφ is separated, no pathologies occur when base extending to L and hence we may assume L = K. Recall that Dφ is covered by the D(Y ) for Y ∈ C; choose Y ∈ C and let X ⊆ W be its image in W. Choose a K-point P of X. This K-point corresponds to a map RX → K and it suffices to construct a bijection between the K-points of D(Y ) lying above P and the φ-finite systems of eigenvalues coming from eigenvectors in N ⊗RX K, where N ⊆ MX is the subspace corresponding to Y . The result then follows from the following purely algebraic lemma. Lemma 5.10 Let R be a commutative Noetherian ring and let N be a projective module of finite rank over R. Let T be a commutative subring of EndR (N ). Let m denote a maximal ideal of R, and let S denote the image of the natural map T /mT → EndR/m (N/mN ). Then the natural map T /mT → S induces a bijection between the prime ideals of T /mT and the prime ideals of S. Proof It suffices to show that the kernel of the map T /mT → S is nilpotent. After localising at m we may assume that N is free; choose a basis for N . Let t be an element of T whose image in EndR/m (N/mN ) is zero. Then all the matrix coefficients of t with respect to this basis are in m. Thinking of t as a matrix with coefficients in R, we see that t is a root of its characteristic

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polynomial, which is monic and all of whose coefficients other than its leading term are in mR. Hence t is nilpotent in T /mT and we are home. Part II: The Coleman-Mazur eigencurve.

6 Overconvergent modular forms. One can say much more about the eigenvariety Dφ in the specific case for which all this machinery was originally invented, namely the Coleman-Mazur eigencurve. Again we shall not give a complete treatment of this topic, but will refer to [9] for many of the basic definitions and results we need. The paper [9] gives constructions of two objects, called C and D, both in the case of level 1 and p > 2. The results in sections 2–3 of this paper are enough for us to be able to extend the construction of D to the case of an arbitrary level and an arbitrary prime p, and we shall give details of the construction here. Note that we do not need the results in sections 4–5 of this paper here, because the eigenvarieties constructed are over a 1-dimensional base, and the rigid analytic results that Coleman develops in section A5 of [10] are sufficient. Fix a prime p, set K = Qp , and let W be weight space, that is the rigid space whose Cp -points are naturally the continuous group homomorphisms × Z× p → Cp (see section 2 of [5] for more details on representability of such functors). Then W is the disjoint union of finitely many open discs, and there is a natural affinoid covering of W which on each component is a cover of the open disc by countably many closed discs. Coleman and Mazur restrict to the case p > 2, and for an affinoid Y in weight space define MY to be the space of r-overconvergent modular forms of level 1, for some appropriate real number r. As Y gets bigger one has to consider forms which overconverge less and less; this is why we must include cases where (in the notation of section 5) the “links” α are not the identity. Finally the map φ is chosen to be the Hecke operator Up , which is compact. See [9] for rigorous definitions of the above objects, and verification that they satisfy the necessary criteria for the machine to work. In [9] it is proved that (for N = 1 and p > 2) the resulting eigencurve Dφ is reduced, and flat over Zφ (see [9], Proposition 7.4.5 and the remarks before Theorem 7.1.1 respectively). Our Lemma 5.9 is just the statement that points on the eigencurve are overconvergent systems of finite slope eigenvalues, and the existence of q-expansions assures us that, at least in the cuspidal case, points on the eigencurve correspond bijectively with normalised overconvergent eigenforms. In fact much more is proved in [9], where two rigid spaces are constructed for each odd prime p: a curve C, constructed via deformation theory and the

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theory of pseudorepresentations, and a curve D constructed via glueing Hecke algebras as above. In Theorem 7.5.1 of [9] it is proven that D is isomorphic to the space C red . Since the paper [9] appeared, various authors have assumed that the constructions in it would generalise to the cases N > 1 and p = 2. In fact, it seems to us that the following are the main reasons that N = 1 and p > 2 are assumed in [9]. Firstly, the theory of pseudorepresentations does not work quite so well in the case p = 2. Secondly, there are some issues to be resolved when writing down the local conditions at primes dividing N on the deformation theory side (we remark that Kisin tells us that both of these issues can be resolved without too much trouble). And thirdly one sometimes has to deal with eigenspaces for the action of (Z/4Z)× on a 2-adic Banach module when p = 2 on the Hecke algebra side, causing problems when looking for orthonormal bases. The first two issues will not concern us in this paper, as we do not talk about the construction of C, and the results in sections 2 and 3 of this paper are enough to deal with the third issue. In fact Chenevier has pointed out to us that one can also avoid the troubles caused by the third issue when constructing eigencurves for p = 2 by appealing to the corollary of lemma 1 in [15]. We do not construct a generalisation of C here, but we do show how to construct an eigencurve D for a general prime p and level N prime to p. We first establish some generalities and notation. All our rigid spaces will be over K = Qp in this section. Recall that W is the rigid space over Qp representing maps Z× p → Gm . Define q = p if p > 2, and q = 4 if p = 2. Define D = (Z/qZ)× , regarded as a quotient of Z× p in the natural way. Define to be the set of group homomorphisms D → C× . Set γ = 1 + q ∈ Z× . The D p p natural surjection Z× p → D has kernel 1 + qZp , which is topologically isomorphic to Zp , and is topologically generated by γ. The map Z× p → D induces an isomorphism between D and the roots of unity in Z× , and hence the surjection p ∼ splits and we have an isomorphism Z× D × Z ; we shall thus identify Z× p p = p with D × Zp . If χ ∈ D then the composite of χ with the natural projection Z× p → D is an element of W, and one easily checks that distinct elements are in different components of W. Hence this construction establishes a of D Let Wχ denote the bijection between the components of W and the group D. Let 1 denote the trivcomponent of W corresponding to the character χ ∈ D. ial character of D (sending everything to 1) and let B denote the component then multiplication by χ gives an isomorphism W1 of W. Note that if χ ∈ D W 1 → Wχ . For n ≥ 1 let Xn denote the affinoid subdomain of W corresponding to × pn−1 group homomorphisms ψ : Z× − 1| ≤ |q|. p → Cp such that |ψ(1 + q) Xn ∩ Wχ is an affinoid disc, that It is easily checked that for any χ ∈ D,

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X1 ⊆ X2 ⊆ . . ., and that the Xi give an admissible cover of W. The inclusion Xi ⊂ W induces a bijection of the connected components of Xi with the then write Xi,χ for the closed disc connected components of W; if χ ∈ D Xi ∩ Wχ . We remark here that if k ∈ Z and χ : (Z/qpn−1 Z)× → C× p then pn−1 k χ(1 + q) = 1 and hence the map ψ ∈ W defined by ψ(x) = x χ(x) is in Xn (Cp ). Define Ri = O(Xi ). Then Ri is an affinoid for each i, and Ri = ⊕χ Ri,χ , and Ri,χ = O(Xi,χ ). where χ runs through D In preparation for the application of our eigenvariety machine, we have to choose a family of radii of overconvergence. Fortunately Coleman and Mazur have done enough for us here, even if p = 2. We give a brief description of the modular curves and affinoids that we shall use. Let Af denote the finite adeles. For a compact open subgroup Γ ⊂ GL2 (Af ) that contains the principal congruence subgroup ΓN for some N prime to p, we define the compact modular curve X(Γ) over Qp in the usual way. Let us firstly assume that Γ is sufficiently small to ensure that the associated moduli problem on generalised elliptic curves has no non-trivial automorphisms (we will remove this assumption below). Now recall from section 3 of [4], for example, that for an elliptic curve E over a finite extension of Qp , there is a measure v(E) of its supersingularity, and that v(E) < p2−m /(p + 1) implies that E possesses a canonical subgroup of order pm . So for r ∈ Q with 0 ≤ r < p/(p + 1) we define X(Γ)≥p−r to be the affinoid subdomain of the rigid space over Qp associated to X(Γ) whose non-cuspidal points parametrise elliptic curves E with a level Γ structure and such that v(E) ≤ r. For example if r = 0 then X(Γ)≥p−r is the ordinary locus of X(Γ). For m ≥ 1 there is a fine moduli space X(Γ, Γ1 (pm )) (resp. X(Γ, Γ0 (pm ))) over Qp whose non-cuspidal points parametrise elliptic curves equipped with a level Γ structure and a point (resp. cyclic subgroup) of order pm over Qp schemes. There are natural forgetful functors X(Γ, Γ1 (pm )) → X(Γ, Γ0 (pm )) → X(Γ). If 0 ≤ r < p2−m /(p + 1) and E is an elliptic curve over a finite extension of Qp with v(E) ≤ r then, as mentioned above, E has a canonical subgroup of order pm . For r in this range we define X(Γ, Γ0 (pm ))≥p−r to be the components of the pre-image of X(Γ)≥p−r in X(Γ, Γ0 (pm )) whose noncuspidal points parametrise elliptic curves with the property that their given cyclic subgroup of order pm equals their canonical subgroup, and we define the rigid space X(Γ, Γ1 (pm ))≥p−r to be the pre-image of X(Γ, Γ0 (pm ))≥p−r in X(Γ, Γ1 (pm )).

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All these spaces are affinoids; this follows from the fact that X(Γ)≥p−r is an affinoid, being the complement of a non-zero finite number of open discs in a complete curve. There is a natural action of the finite group (Z/pm Z)× on X(Γ, Γ1 (pm )) and on X(Γ, Γ1 (pm ))≥p−r via the (weight 0) Diamond operators. Finally if Γ is a compact open subgroup of GL2 (Af ) containing ΓN for some N prime to p, but which is not “sufficiently small”, then choose some prime l 2N p and define Γ := Γ ∩ Γl ; then Γ is a normal subgroup of Γ and Γ is sufficiently small. Hence one may apply all the constructions above to Γ and then define X(Γ)≥p−r , X(Γ, Γ1 (pm ))≥p−r and so on by taking Γ/Γ -invariants. The resulting objects are only coarse moduli spaces but this will not trouble us. A standard argument shows that this construction is Γ1 (pm ))≥p−r and independent of l. We define X1 (pm )≥p−r := X(GL2 (Z), m m X0 (p )≥p−r := X(GL2 (Z), Γ0 (p ))≥p−r . Similarly if N ≥ 1 is prime to p then we define X0 (N pm )≥p−r := X(Γ0 (N ), Γ0 (pm ))≥p−r , where Γ0 (N ) is which are upper triangular mod N . Note that as usual the matrices in GL2 (Z) X1 (q)≥1 is the curve that Coleman and Mazur refer to as Z1 (q), and that the quotient of X1 (pm )≥p−r by the action of (Z/pm Z)× is X0 (pm )≥p−r . Note also that X0 (pm )≥p−r is “independent of m”, in the sense that the natural (forgetful) map X0 (pm )≥p−r → X0 (p)≥p−r is an isomorphism (as both rigid spaces represent the same functor). We now come to the definition of the radii of overconvergence ri . Note that these numbers depend only on p and not on any level structure. We let Ep denote the function on X1 (q)≥1 ×B defined in Proposition 2.2.7 of [9] (briefly, Ep is the function which, when restricted to a classical even weight k ≥ 4 in B, corresponds to the function Ek (q)/Ek (q p ), where Ek (q) is the p-deprived ordinary old Eisenstein series of weight k and level p). In Proposition 2.2.7 of [9] it is proved that Ep is overconvergent over B. The specialisations to a classical weight of Ep are fixed by the weight 0 Diamond operators, and hence Ep descends to an overconvergent function on X0 (q)≥1 × B = X0 (p)≥1 × B. Furthermore, the assertions about the q-expansion coefficients of Ep made in Proposition 2.2.7 of [9], and the q-expansion principle, are enough to ensure that Ep has no zeroes on X0 (p)≥1 × W1 . Hence the inverse of Ep is a function on X0 (p)≥1 × B and it is elementary to check that it is also overconvergent over B. In particular, for any i ≥ 1 there exists a rational 0 < ri < 1/(p + 1) such that the restrictions of both Ep and E−1 p to X0 (p)≥1 × Xi,1 extend to functions on X0 (p)≥p−ri × Xi,1 . We choose a sequence of rationals r1 ≥ r2 ≥ r3 ≥ . . . ≥ 0 such that each ri has the aforementioned property. We

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may furthermore assume (for technical convenience) that ri < p2−i /q(p + 1) (although this may be implied by the other assumptions). Remark It is not important for us to establish concrete values for the ri , so we do not. However for other applications where one wants to have explicit knowledge about how far one can extend overconvergent modular forms, it may in future be important to understand exactly how the ri behave. For example, machine computations for p ∈ {3, 5, 7, 13} and i = 1 show that it is not the case that r1 can be taken to be an arbitrary rational number less than 1/(p + 1), because there are classical Eisenstein eigenforms of level p which have zeroes in ∪r<1/(p+1) X1 (p)≥p−r . Can one take r1 to be any positive rational less than (p − 1)/p(p + 1)? There are other interesting questions here, which we shall not attempt to answer here.4 We are now ready to begin our definitions of the Banach modules of overconvergent forms. These modules depend on an auxiliary level structure, which we now choose. Fix a compact open subgroup Γ ⊆ GL2 (Af ) that contains the principal congruence subgroup ΓN for some N prime to p. For each n ≥ 1 we wish now to define a Banach module Mn = MΓ,n over Rn := O(Xn ). Recall that Rn = ⊕χ∈D Rn,χ . If χ ∈ D then we define the Banach module Bn,χ to be Qp O(X(Γ, Γ1 (q))≥p−rn ) Bn,χ := Rn,χ ⊗ where rn is the radius of convergence defined previously. Note that Bn,χ is naturally a potentially ONable Banach Rn,χ -module, because the ring O(X(Γ, Γ1 (q))) can be viewed as a Banach space over a discretely-valued field and is hence potentially ONable by Proposition 1 of [15] and the remarks before it. We give the module Bn,χ an Rn,χ -linear action of the group (Z/qZ)× by letting it act trivially on Rn,χ and via the (weight 0) Diamond operators on O(X(Γ, Γ1 (q))). We define Mn,χ to be the direct summand of Bn,χ where (Z/qZ)× acts via χ. Note that by definition Mn,χ satisfies property (P r) (one can check that it is even potentially ONable, but we shall not need this because of our extension of Coleman’s theory). We remark in passing that if then Mn,χ = 0 if χ(−1) = 1. Finally we define Mn = MΓ,n to Γ = GL2 (Z) be the module over Rn = ⊕χ∈D Rn,χ whose Rn,χ -part is Mn,χ . We give Mn an Rn -linear action of (Z/qZ)× by letting it act via χ on Mn,χ . We will shortly see that the fibre of Mn at a point κ ∈ Xn can be naturally identified with the 4 See however [7] for explicit calculations in the case p = 2, and note that these calculations may well generalise to odd primes p.

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rn -overconvergent forms of weight κ (although there are caveats regarding compatibility of this map with Diamond operators; see below). The modules Mn we have described have the following functorial property: if Γ1 and Γ2 both satisfy the conditions imposed on Γ above (that is, they are of level prime to p), if γ ∈ GL2 (Af ) has determinant which is a unit at p, and if γΓ1 γ −1 ⊆ Γ2 , then there is a natural induced finite flat map X(Γ1 ) → X(Γ2 ) and the assumption on the determinant of γ means that if 0 ≤ r ≤ r1 then X(Γ1 , Γ1 (q))≥p−r is the pre-image of X(Γ2 , Γ1 (q))≥p−r . Hence there is a natural inclusion MΓ2 ,n → MΓ1 ,n . If furthermore γΓ1 γ −1 = Γ2 then this inclusion has an inverse and is hence an isomorphism. One can check using these ideas that if Γ1 is a normal subgroup of Γ2 then MΓ2 ,n is the Γ2 /Γ1 invariants of MΓ1 ,n . Note also that for n ≥ 1 there is a natural map Mn → Rn+1 Rn , induced by restriction, which we shall later see is a link, in Mn+1 ⊗ the sense of Part I. We now make explicit the relation between these spaces and Katz’ spaces of overconvergent modular forms. Firstly we recall the definitions (in the form due to Coleman). If Γ ⊂ GL2 (Af ) is a sufficiently small compact open subgroup then there is a sheaf commonly denoted ω on X(Γ) which, on noncuspidal points, is the pushforward of the differentials on the universal elliptic curve. A weight k modular form of level Γ, defined over Qp , is a global section of ω ⊗k . If L is a field extension of Qp then a weight k modular form of level Γ defined over L is a global section of ω ⊗k on the base change of X(Γ) to L. If Γ contains ΓN for some positive integer N prime to p and 0 ≤ r < p/(p + 1) then a p−r -overconvergent modular form of level Γ and weight k defined over Qp is a section of (the analytic sheaf associated to) ω ⊗k on the rigid space X(Γ)≥p−r , and similarly for L a complete extension of Qp . If Γ is not sufficiently small then one can still make sense of these definitions by replacing Γ by a sufficiently small normal subgroup Γ and then taking Γ/Γ -invariants, as Γ/Γ will act on everything. Finally, if 0 ≤ r < p2−n /(p + 1) then one can define p−r -overconvergent modular forms of weight k and level Γ ∩ Γ1 (pn ) as sections of ω ⊗k on X(Γ, Γ1 (pn ))≥p−r (again using the standard tricks if Γ is not sufficiently small). There is a natural action of (Z/pn Z)× on these spaces via the (weight k) Diamond operators. Fix n ≥ 1 and let L be the field Qp (ζpn−1 ) generated by a primitive pn−1 th root of unity ζpn−1 . Fix k ∈ Z, and a character ε : (Z/qpn−1 Z)× → L× . × k Define ψ : Z× p → L by ψ(x) = x ε(x). Then ψ is an L-valued point of such that ψ ∈ Wχ . weight space and in fact ψ ∈ Xn (L). Choose χ ∈ D Define κ = ψ/χ ∈ W1 and let Eκ = 1 + · · · be the associated Eisenstein

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series (see p39 of [9] and note that κ ∈ W1 = B so there are no problems with zeros of p-adic L-functions). Proposition 6.1 The q-expansion Eκ is the q-expansion of a p−rn - overconvergent weight k modular form of level Γ1 (qpn−1 ) defined over L, which is non-vanishing on X1 (qpn−1 )≥p−rn . Furthermore Eκ is in the ε/χ-eigenspace for the Diamond operators. Proof The fact that the q-expansion Eκ is the q-expansion of a section of ω ⊗k on X1 (qpn−1 )≥p−r for some r > 0 is Corollary 2.2.6 of [9]. Now the fact that Eκ is an eigenvector for Up , which increases overconvergence, implies that Eκ extends to a section of ω ⊗k on of X1 (qpn−1 )≥p−r for any rational r with 0 ≤ r < p3−n /q(p + 1). For r in this range, there is a map Frob : X1 (qpn−1 )≥p−r/p → X1 (qpn−1 )≥p−r which is finite and flat of degree p and which induces, by pullback, the map F (q) → F (q p ) on modular forms (on non-cuspidal points the map sends (E, P ) to (E/C, Q) where C is the canonical subgroup of order p of E and Q is the image in E/C of any generator Q of the canonical subgroup of order qpn of E such that pQ = P ). By the q-expansion principle, Eκ has no zeroes on the ordinary locus X1 (qpn−1 )≥1 . Let S denote the set of zeroes of Eκ on the non-ordinary locus of X1 (qpn−1 )≥p−rn . It suffices to show that S is empty. We know that S is finite, because X1 (qpn−1 )≥p−rn is a connected affinoid curve and Eκ = 0 (as its q-expansion is non-zero). We also know that Eκ (q)/Eκ (q p ) has no zeroes on X1 (qpn−1 )≥p−rn , by definition of rn , and that rn < p2−n /q(p + 1). Hence any zero of Eκ is also a zero of Eκ (q p ) = Frob∗ Eκ . But if S is non-empty then let P denote a point of S which is “nearest to the ordinary locus” (that is, such that v (P ) is minimal, where v is the composite of the natural projection X1 (qpn−1 ) → X0 (p) and the function denoted v in section 4 of [4]). Then P is also a zero of Eκ (q p ) and hence if P = Frob(Q) for some point Q ∈ X1 (qpn−1 )≥p−rn /p then Q is also a zero of Eκ (q), and furthermore Q is closer to the ordinary locus than P (in fact v (Q) = 1p v (P ) < v (P ) by Theorem 3.3(ii) of [4]), a contradiction. The assertion about the Diamond operators is classical if k ≥ 2 (see for example Proposition 7.1.1 of [13]). For general k it can be deduced as follows: by Theorem B4.1 of [10] applied with i = 0, the function on X1 (q)≥1 × B × B denoted Eα (q)Eβ (q)/Eαβ (q) in Theorem 2.2.2 of [9], when restricted to X1 (q)≥1 × B ∗ × B∗ (in the notation of the proof of Theorem 2.2.2 of [9]) is invariant under the natural action of (Z/qZ)× (acting trivially on B × and via Diamond operators on X1 (q)). But this action is continuous, so (Z/qZ)× acts trivially on Eα (q)Eβ (q)/Eαβ (q). Hence, the function Eα (q)Eβ (q)/Eαβ (q)

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descends to a function on X0 (q)≥1 × B × B = X0 (qpn−1 )≥1 × B × B. The result for general k now follows from the result for k ≥ 2. Corollary 6.2 Let k, ε, ψ, χ, κ be as above, and let ψ : Rn → L also denote the homomorphism corresponding to the L-point of Xn induced by ψ. Then multiplication by Eκ induces an isomorphism between MΓ,n ⊗Rn L (the tensor product formed via the homomorphism ψ : Rn → L) and the space of p−rn overconvergent modular forms of level Γ, weight k and character ε defined over L. Proof Say ψ ∈ Wχ . Then unravelling the definitions gives that MΓ,n ⊗Rn L = Mn,χ ⊗Rn,χ L is equal to the χ-eigenspace of O(X(Γ, Γ1 (q))≥p−rn ) ⊗Qp L, and hence the χ-eigenspace of O(X(Γ, Γ1 (qpn−1 ))≥p−rn ) ⊗Qp L, where χ here is regarded as a character of (Z/qpn−1 Z)× (note that the forgetful functor X(Γ, Γ1 (qpn−1 ))≥p−rn → X(Γ, Γ1 (q))≥p−rn is the map induced by quotienting X(Γ, Γ1 (qpn−1 ))≥p−rn out by the group (1 + qZp /1 + qpn−1 Zp )). The result now follows from the fact that Eκ has weight k, character ε/χ and is non-vanishing on X(Γ, Γ1 (qpn−1 ))≥p−rn . Motivated by this Corollary, we define the q-expansion of an element of Mn to be the following element of Rn [[q]], as follows: it suffices to attach a q-expansion in Rn,χ [[q]] to an element of Mn,χ , and hence it suffices to attach a q-expansion in Rn,χ [[q]] to an element of Bn,χ . Now an element of O(X(Γ, Γ1 (q))≥p−rn ) has a q-expansion in Qp [[q]] in the usual way, and hence an element of Bn,χ has a q-expansion in Rn,χ [[q]]. This is not the qexpansion that we are interested in however—this q-expansion corresponds to a family of weight 0 overconvergent forms. we twist this q-expansion by multiplying it by E, the q-expansion of the restricted Eisenstein family defined in Section 2.2 of [9]. Note that the restricted Eisenstein family is a family over B and so we have to explain how to regard it as a family over Xn,χ ; we do this by pulling back via the composite of the natural inclusion Xn,χ → Wχ and the natural isomorphism Wχ → W1 . The resulting power series is defined to be the q-expansion of m, and with this normalisation, the isomorphism of the previous corollary preserves q-expansions. As we shall see in the next section, we will be defining Hecke operators on MΓ,n (at least for certain choices of Γ) so that they agree with the standard Hecke operators on overconvergent modular forms, via the isomorphism of the previous Corollary. We finish this section by remarking that the isomorphism of the previous corollary is however not compatible with Diamond operators

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in general, for two reasons—firstly, overconvergent forms of classical weightcharacter ψ naturally have an action of the group (Z/qpn−1 Z)× , but the full space Mn only has an action of (Z/qZ)× , and secondly even the actions of (Z/qZ)× do not in general coincide, as one is defined in weight 0 and the other in weight k so the two actions differ (at least for p > 2) by the kth power of the Teichm¨uller character.

7 Hecke operators and classical eigencurves We now restrict to the case where Γ is either the congruence subgroup Γ0 (N ) consisting of matrices which are upper triangular (the subgroup of GL2 (Z) mod # ∗ ∗ $N ) or Γ1 (N ) (the subgroup of Γ0 (N ) consisting of matrices of the form 0 1 mod N ) of GL2 (Af ), for some N prime to p. The associated moduli problems are then just the usual problems of representing cyclic subgroups or points of order N . We now define Hecke operators Tm for m prime to p, and a compact operator Up , on the spaces MΓ,n for n ≥ 1. Almost all of the work has been done for us, in section B5 of [10] (where Hecke operators are defined on overconvergent forms over B∗ ) and in section 3.4 of [9] (where this work is extended to B). Note that the arguments in these references do not assume p > 2 or N = 1. We do not reproduce the arguments here, we just mention that the key point is that because the argument is not a geometric one, the construction of Tm is done at the level of q-expansions and the resulting definitions initially go from forms of level N to forms of level N m. However one can prove that the resulting maps do in fact send forms of level N to forms of level N by noting that the result is true for forms of classical weight, where the Hecke operator can be defined via a correspondence, and deducing the general case by considering a trace map and noting that a family of forms that vanishes at infinitely many places must be zero. The one lacuna in the arguments in the references above is that in both cases the operators are defined as endomorphisms of overconvergent forms, rather than r-overconvergent forms for some fixed r. What we need to do is to prove that the Hecke operators defined by Coleman and Mazur send roverconvergent forms to r-overconvergent forms, for some appropriate choice of r. Let E denote the restricted Eisenstein family (that is, the usual family of Eisenstein series over B), and for = p a prime number, let E (q) denote the ratio E(q)/E(q ) as in Proposition 2.2.7 of [9], thought of as an overconvergent function on X0 ( p)≥1 × B (note that Coleman and Mazur only assert that this function lives on X1 (q, )≥1 × B but it is easily checked to be invariant under the Diamond operators at p).

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Lemma 7.1 The restriction of E (q) to X0 ( p)≥1 × Xn,1 extends to a nonvanishing function on X0 ( p)≥p−rn × Xn,1 . Proof For simplicity in this proof, we say that a function on X0 (p)≥1 × Xn,1 is r-overconvergent if it extends to a function on X0 (p)≥p−r × Xn,1 , and similarly we say that such a function is r-overconvergent and non-vanishing if it extends to a non-vanishing function on X0 (p)≥p−r × Xn,1 . We first prove that E (q) is rn -overconvergent. Observe that Proposition 2.2.7 of [9] tells us that E (q) is r-overconvergent for some r > 0. Let us assume r < rn and explain how to analytically continue E (q) a little further. By definition of rn , we know that Ep (q) is rn -overconvergent and non-vanishing. Furthermore the non-trivial degeneracy map X0 ( p) → X0 (p) which on q-expansions sends F (q) to F (q ) induces a morphism of rigid spaces X0 ( p)≥p−rn → X0 (p)≥p−rn and hence Ep (q ) is also an rn overconvergent non-vanishing function. The ratio Ep (q )/Ep (q) is therefore also an rn -overconvergent non-vanishing function. But Ep (q )/Ep (q) = E (q p )/E (q) and E (q) is r-overconvergent, and thus E (q p ) is also r-overconvergent. Let r = min{pr, rn }. We claim that E (q) is r -overconvergent and this clearly is enough because repeated applications of this idea will analytically continue E (q) until it is rn -overconvergent, which is what we want. But it is a standard fact that the U operator increases overconvergence by a factor of p, and hence if E (q p ) is r-overconvergent, then E (q) = U (E (q p )) is r -overconvergent. Finally we show that E (q) is non-vanishing on X0 ( p)≥p−rn × Xn,1 and this follows from an argument similar to the non-vanishing statement proved in Proposition 6.1—if E (q) had a zero then choose a zero (x, κ) and then specialise to weight κ; we may assume that x is a zero closest to the ordinary locus in weight κ, and then E (q p ) has a zero closer to the ordinary locus in weight κ and hence E (q)/E (q p ) would have a pole in weight κ, contradicting the fact that E (q)/E (q p ) = Ep (q)/Ep (q ) is rn -overconvergent. We now have essentially everything we need to apply our eigenvariety machine. The preceding lemma and the arguments in section 3.4 of [9] can be used to define Hecke operators Tm (m prime to p) and Up on rn -overconvergent forms over Xn . If X is any admissible affinoid open subdomain of W then there exists some n ≥ 1 such that X ⊆ Xn ; we choose the smallest such n and define the Banach module MX to be the pullback to O(X) of the O(Xn )-module Mn . We let T denote the abstract polynomial algebra over R generated by the Hecke operators Tm for m prime to p, the

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operator φ = Up , and the Diamond operators at N if Γ = Γ1 (N ). These Hecke operators are well-known to commute, as can be seen by checking on classical points. We need to check that the natural restriction maps α between spaces of overconvergent forms of different radii are all links, but this follows easily from the technique used in the standard proof that the characteristic power series of Up on r-overconvergent forms is independent of the choices of r > 0. The key point is that the endomorphism Up of r-overconvergent forms can be checked to factor as a continuous map from r-overconvergent forms to s-overconvergent forms, for some s > r, followed by the (compact) restriction map from s-overconvergent forms to r-overconvergent forms. In fact one can take s to be anything less than both pr and p/(p + 1). One deduces that for any 0 < r < r < p/(p + 1), the natural map from r -overconvergent forms to r-overconvergent forms is a “link”. Our conclusion is that the construction of the “D” eigencurve in [9] can be generalised to all p and N ≥ 1 prime to p. PART III: Eigenvarieties for Hilbert modular forms.

8 Thickenings of K-points and weight spaces Let K be a non-archimedean local field (that is, a field either isomorphic to a finite extension of Qp or to the field of fractions of k[[T ]] with k finite). Let O denote the integers of K, and let V denote the closed affinoid unit disc over K. As an example of a construction which will be used many times in the sequel, we firstly show that is not difficult to construct a sequence U1 ⊃ U2 ⊃ . . . of affinoid subdomains of V , defined over K, with the property that &# $ Ut (L) = V (K) = O (∗) t≥1

for all complete extensions L of K. Note that (∗) implies that Ut (K) = V (K) for all t (set L = K), but also that no non-empty K-affinoid subdomain of V can be contained in all of the Ut . The Ut should be thought of as a system of affinoid neighbourhoods of V (K) = O in V . The construction of the Ut is simple: Let π ∈ K be a uniformiser and define Ut = α∈Xt B(α, |π|t ), where Xt is a set of representatives in O for O/(π)t , and B(α, |π|t ) is the closed affinoid disc with centre α and radius |π|t . Note that Xt is finite because K is a local field, and hence Ut is an affinoid subdomain of V : it is a finite union of affinoid subdiscs of V of radius |π|t . It is easy to check moreover that the Ut satisfy (∗) (use the fact that O is compact to get the harder inclusion). We in fact need a “twisted” n-dimensional version of this construction, which is more technical to state but which requires essentially no new ideas.

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Before we explain this generalisation, we make an observation about the possible radii of discs defined over non-archimedean fields. Let K be an arbitrary field complete with respect to a non-trivial non-archimedean norm. If L is a finite extension of K then there is a unique way to extend the norm on K to a norm on L, and hence there is a unique" way" to extend the norm ' on K to a×norm ( " ×" on an algebraic closure K of K. Let "K " denote the set |x| : x ∈ K . It is easily that this set is just {|y|1/d : y ∈ K × , d ∈ Z≥1 }. Note that for " checked " " ×" r ∈ "K " the closed disc B(0, r) with centre zero and radius r is a rigid space defined over K: if r ≤ 1 then r = |y|1/d for some y ∈ K × and d ∈ Z≥1 , and one can construct B(0, r) as the space associated to the affinoid algebra KT, S/(T d − yS); the general case can be reduced to this by scaling. By using products of these discs, one sees that one can construct polydiscs with “fractional” radii over K. Let M/M0 be a fixed finite extension of non-archimedean local fields, and assume that the restriction of the norm on M is the norm on M0 . Let K denote any complete extension of M0 , again with the norm on K assumed to extend the norm on M0 . Assume moreover that K has the property that the image of any M0 -algebra homomorphism M → K (an algebraic closure of K) lands in K. The Ut above will correspond to the case M0 = M = K of the construction below. Later on M0 will be Qp but we do not need to assume that we are in mixed characteristic yet. Let I denote the set of M0 -algebra homomorphisms M → K. We will use |.| to denote the norms on both M and K, and there is of course no ambiguity here because any M0 -algebra map i : M → K will be norm-preserving. Let O now denote the integers of M (in particular O is no longer the integers of K), and let π be a uniformiser of M . Let V be the unit polydisc over K of dimension |I|, the number of elements of I, and think of the coordinates of V as being × indexed by elements of I. Let NK denote the set {|x| : x ∈ K , |x| ≤ 1} and × × let NK denote NK \{1} = {|x| : x ∈ K , |x| < 1}. If α = (α1 , α2 , . . . , ) ∈ K I and r ∈ NK then we define B(α, r) to be the K-polydisc whose L-points, for L any complete extension of K, are B(α, r)(L) = {(x1 , x2 , . . .) ∈ LI : |xi − αi | ≤ r for all i}. For example we have V = B(0, 1). Note that B(α, r) is defined over K by the comments above. There is a natural map M → K I which on the ith component sends m to i(m), and we frequently write mi for i(m). Furthermore, we implicitly identify m ∈ M with its natural image (mi )i∈I in K I . In particular, if α ∈ O then α can be thought of as a K-point of V .

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Definition If r ∈ NK then define Br := α∈O B(α, r), and if furthermore × r ∈ NK then define B× r := α∈O × B(α, r). We see that Br and B× r are finite unions of polydiscs, because B(α, r) = B(β, r) if |α − β| ≤ r, and O is compact. In particular Br and B× r are Kaffinoid subdomains of V . The space Br should be thought of as a thickening × of O in V ; similarly B× r should be thought of as a thickening of O . One can check that for all complete extensions L of K, we have & Br (L) = O r∈NK

and

&

× B× r (L) = O .

× r∈NK

The proof follows without too much difficulty from the fact that O and O× are compact subsets of the metric space K I . One can also check that for any r ∈ NK , the space Br is an affinoid subgroup of (A1 )I , the product of I copies × of the additive group, and that if r ∈ NK then B× r is an affinoid subgroup of I Gm , the product of I copies of the multiplicative group. An example of the idea continually used in the argument is that if (yi ) ∈ B× r (L) for L some complete extension of K, then (yi ) is close to some element α of O× , and hence (yi−1 ) is close to α−1 . See the lemma below for other examples of this type of argument. We record elementary properties of Br and B× r that we shall use later. # a b some $ Let γ = c d be an element of M2 (O) with |c| < 1, |d| = 1 and det(γ) = 0. Define m ∈ NK by | det(γ)| = m. Choose r ∈ NK , and t ∈ Z>0 such that |c| ≤ |π t |. Lemma 8.1 (a) There is a map of rigid spaces Br → B× r|π t | which on points sends (zi ) to (ci zi + di ) (where here as usual ci denotes the image of c ∈ M in K via the map i and so on). (b) There is a map of rigid spaces Br → Brm which on points sends (zi ) to ai zi +bi ci zi +di . Proof (a) Clearly there is a map V → V sending (zi ) to (ci zi + di ); we must check that the image of Br is contained within B× r|π t | . Because all the rigid spaces in question are finite unions of affinoid polydiscs, it suffices to check this on L-points, for L any complete extension of K. So let (zi ) be an L-point of Br . Then there exists α ∈ O such that |zi − αi | ≤ r ≤ 1 for all i ∈ I. In particular |zi | ≤ 1 so |ci zi + di | = 1, and so ci zi + di

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is invertible. Note now that β := cα + d ∈ O× and for i ∈ I we have |(ci zi + di ) − βi | = |ci (zi − αi )| ≤ |c|r ≤ |π t |r, which is what we wanted. (b) A similar argument works, the key point being that if (zi ) ∈ Br (L) and α ∈ O is chosen such that |zi − αi | ≤ r for all i, then defining β = aα+b cα+d ∈ O, " " " ai zi +bi " we see that " ci zi +di − βi " = | det(γ)i (zi − αi )| ≤ | det(γ)|r = mr (as |ci zi + di | = 1 = |cα + d| as in (a)) and this is enough.

Now let M0 , M , K be as before, and let p > 0 denote the residue characteristic of K. Let Γ be a profinite abelian group containing an open subgroup topologically isomorphic to Zdp for some d. If U is a rigid space over K, let O(U ) denote the ring of rigid functions on U . Note that O is still the integer ring of M but this should not cause confusion. We say that a group homomorphism Γ → O(U ) (resp. Γ → O(U )× ) is continuous if, for all affinoid subdomains X of U , the induced map Γ → O(X) (resp. Γ → O(X)× ) is continuous. We recall some results on representability of certain group functors. By a K-group we mean a group object in the category of rigid spaces over K. Lemma 8.2 (a) The functor from K-rigid spaces to groups, sending a space X to the group O(X) under addition (resp. the group O(X)× under multiplication) is represented by the K-group A1 (resp. Gm ), the analytification of the affine line (resp. the affine line with zero removed). (b) If Γ is as above, then there is a separated K-group XΓ representing the functor which sends a K-rigid space U to the group of continuous group homomorphisms Γ → O(U )× . Moreover XΓ is isomorphic to the product of an open unit polydisc and a finite rigid space over K. (c) If L is a complete extension of K then the base change of XΓ to L represents the functor on L-rigid spaces sending U to the group of continuous group homomorphisms Γ → O(U )× . Proof (a) If X is a rigid space and f ∈ O(X) then for all affinoid subdomains U ⊆ X, f induces a unique map U → A1 because there exists 0 = λ ∈ K such that λf is power-bounded on U , and then there is a unique map KT → O(U ) sending T to λf by Proposition 1.4.3/1 of [1]; everything is compatible and glues, and the result for A1 follows easily. Moreover, if f is in O(X)× then for every affinoid subdomain U of X, it is possible to find an affinoid annulus in Gm containing f (U ), as both f (U ) and (1/f )(U ) lie in an affinoid subdisc of A1 . Hence f ∈ O(X)× gives a map X → Gm , and the result in the multiplicative case now follows without too much difficulty.

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(b) The existence of XΓ is Lemma 2(i) of [5]. We recall the idea of the proof: the structure theorem for topologically finitely-generated profinite abelian groups shows that Γ is topologically isomorphic to a product of groups which are either finite and cyclic, or copies of Zp . Now by functorial properties of products in the rigid category, it suffices to show representability in the cases Γ = Zp and Γ a finite cyclic group. The case Γ = Zp is treated in Lemma 1 of [5] and the remarks after it, which show that the functor is represented by the open unit disc centre 1, and the case of Γ cyclic of order n is represented by the analytification of μn over K. That χΓ has the stated structure is now clear. (c) By functoriality it is enough to verify these base change properties in the two cases Γ = Zp and Γ finite cyclic; but in both of these cases the result is clear. We now assume that M0 = Qp , and hence that M is a finite extension of Qp . We assume (merely for notational ease) that the norms on M0 , M and K are all normalised such that |p| = p−1 . We remind the reader that if t is an element of an affinoid K-algebra and |t| < 1 then the power series for log(1 + t) converges, and if |t| < p−1/(p−1) then the power series for exp(t) converges; furthermore log and exp give isomorphisms of rigid spaces from the open disc with centre 1 and radius p−1/(p−1) to the open disc with centre 0 and radius p−1/(p−1) . We would like to use logs to analyse B× r and hence are × −1/(p−1) particularly interested in the spaces Br for r < p ; we call such r “sufficiently small”. For these r, we see that the component of B× r containing 1 is isomorphic, via the logarithm on each coordinate, to the component of Br containing 0. Recall that O is the integers of M , and hence Γ = O× satisfies the conditions just before Lemma 8.2. If n ∈ ZI then there is a group homomorphism O× → K × = Gm (K), which sends α to i αni i . It is easily checked (via exp and log) that if r is sufficiently small then this map is the K-points of a map of K-rigid spaces B× r → Gm . For more arithmetically complicated continuous maps from O× to invertible functions on affinoids, we might have to make r smaller still before such an analytic extension exists, but the proposition below shows that we can always do this. An important special case of this proposition is the case of an arbitrary continuous homomorphism O× → K × , but the proof is essentially no more difficult if K × is replaced by the invertible functions on an arbitrary affinoid, so we work in this generality. Proposition 8.3 If X is a K-affinoid space and n : O× → O(X)× is a × continuous group homomorphism, then there is at least one r ∈ NK and, for

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any such r, a unique map of K-rigid spaces βr : B× r ×X → Gm , such that for all α ∈ O× , n(α) is the element of O(X)× corresponding (via Lemma 8.2(a)) to the map X → Gm obtained by evaluating βr at the K-valued point α of B× r . Definition We call βr a thickening of n. Proof of Proposition. It suffices to prove that there exists at least one sufficiently small r such that if B is the component of B× r containing the K-point 1, then there is a unique β : B × X → Gm with a property analogous to that of βr above for all α ∈ Δ := {α ∈ O × : |α − 1| ≤ r} (or even in some subgroup of finite index). Via the logarithm map one sees that Δ is isomorphic to O. It hence suffices to prove that for any continuous group homomorphism χ : O → O(X)× there exists N ∈ Z≥0 such that the induced homomorphism pN O → O(X)× is induced by a unique map of K-rigid spaces B(0, p−N )×X → Gm . Here B(0, p−N ) denotes the polydisc in V with radius p−N , and pN O is embedded as a subset of the K-points of B(0, p−N ) in the usual way. Now observe that if d = [M : Qp ] then as a topological group, O is isomorphic to Zdp . Hence, if one fixes a K-Banach algebra norm on O(X) and a Zp -basis e1 , e2 , . . . , ed of O, one sees using Lemma 1 of [5] that there exists a positive integer N such that |χ(pN ej ) − 1| < p−1/(p−1) for all j. Observe now that log(χ) is a continuous group homomorphism pN O → O(X), and O(X) is a K-vector space; furthermore, the image of pN O will land in a finitedimensional K-subspace of O(X). It is a standard fact (linear independence of distinct field embeddings) that the continuous group homomorphisms O → K form a finite-dimensional K-vector space with basis the set I, now regarded as the ring homomorphisms O → K, and hence there exists f1 , f2 , . . . , fd ∈ O(X)× such that for all α ∈ pN O we have χ(α) = exp ( i αi fi ), where α i(α) ∈ K. By increasing N if necessary, we may assume that " iNdenotes " "p fi " < p−1/(p−1) for all i, and we claim that this N will work. To construct β : B(0, p−N ) × X → Gm it suffices, by# Lemma 8.2(a), to$ construct a unit in the affinoid O(X)T1 , T2 , . . . , Td = O B(0, p−N ) × X which speN N cialises χ(α) via # to N $ the map sending Ti to αi /p , for all α ∈ p O. The unit exp i p Ti fi is easily seen to do the trick. For uniqueness it suffices (again via exp and log) to prove that a map of rigid spaces f : B1 → A1 which sends every element of O to 0 must be identically 0, that is, that O is Zariski-dense in B1 . It suffices to show that f vanishes on a small polydisc centre 0, and one can check this on points. Again choose a Zp -basis (e1 , e2 , . . . , ed ) of O as a Zp -module. It suffices to prove that for all complete extensions L of K, f is zero on all L-points of B1 of

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the form z1 e1 + z2 e2 + . . . + zd ed with zi ∈ OL , as this contains all the L-points of a small polydisc in B1 by a determinant calculation. Note that if all the zβ are in Zp then certainly f (z1 e1 + z2 e2 + . . .) = 0. Now fix zβ ∈ Zp for β ≥ 2 and consider the function on the affinoid unit disc over L sending z1 to f (z1 e1 + z2 e2 + . . .). This is a function on a closed 1-ball that vanishes at infinitely many points, and hence it is identically zero. Now fix z1 ∈ OL and zβ ∈ Zp for β ≥ 3, and let z2 vary, and so on, to deduce that f is identically 0. For applications, we want to consider products of the Br and B× r constructed above. Let F denote a number field, with integers OF , and let p be a prime. Let Op denote OF ⊗ Zp , the product of the integer rings in the completions of F at all the primes above p, and let K0 be the closure in Qp of the compositum of the images of all the field homomorphisms F → Qp . Then K0 is a finite Galois extension of Qp . Then K0 contains the image of any field homomorphism Fv → Qp , where v is any place of F above p, so we are in a position to apply the previous constructions with M0 = Qp , M = Fv , and K any complete extension of K0 . Let J denote the set of places of F above p, and let I denote the set of field homomorphisms F → K. Note that each i ∈ I extends naturally to a map i : Fp → K where Fp = F ⊗ Qp = ⊕j∈J Fj . For j ∈ J, let Ij denote the subset of I consisting of i : Fp → K which factor through the completion J F → Fj . Then I is the disjoint union of the Ij . For r ∈ (NK ) and j ∈ J write rj for the component of r at j. Let Br (resp. B× r if rj < 1 for all j) denote the rigid space over K which is the product over j ∈ J of the rigid × spaces Brj (resp. B× rj ) defined above. Then Br (resp. Br ) is a thickening of × Op (resp. Op ) in the unit g-polydisc over K, where now g = [F : Q]. Indeed, it is easily checked that for all complete extensions L of K we have Br (L) = {z ∈ LI : there is α ∈ Op with |zi − αi | ≤ ri } and, when ri < 1 for all i, I × B× r (L) = {z ∈ L : there is α ∈ Op with |zi − αi | ≤ ri }

just as before, where, for α ∈ Op , αi denotes i(α) ∈ K. Now assume that F is totally real. Let G denote a subgroup of OF× of finite index, and let ΓG be the quotient of Op× × Op× by the closure of the#image of$ G via the map γ → (γ, γ 2 ). Then ΓG is topologically isomorphic to Op× /G × Op× , so its dimension is related to the defect of Leopoldt’s conjecture for the pair (F, p) (in particular, the dimension is at least g + 1 and conjecturally equal to g + 1). Let XΓG be the rigid space associated to ΓG in Lemma 8.2(b), and let W to be the direct limit lim XΓG as G varies over the set of subgroups of

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finite index of OF× , partially ordered by inclusion. The fact that W exists is an easy consequence of Lemma 2(iii) of [5], which shows that the transition morphisms are closed and open immersions: if G1 ⊆ G2 ⊆ OF× are subgroups of OF× of finite index and Γi = ΓGi , then there is a surjection Γ1 → Γ2 with finite kernel, and the corresponding map XΓ2 → XΓ1 is a closed immersion which geometrically identifies XΓ2 with a union of components of XΓ1 . In particular we see that the XΓG , as G varies through the subgroups of OF× of finite index, form an admissible cover of W. A K-point of W corresponds to a continuous group homomorphism Op× × Op× → K × whose kernel contains a subgroup of OF× of finite index (we always regard OF× as being embedded in Op× × Op× via the map γ → (γ, γ 2 )). More generally, we define a weight to be a continuous group homomorphism κ : Op× × Op× → O(X)× , for X any affinoid, such that the kernel of κ contains a subgroup of OF× of finite index. If U is an affinoid K-space and U → W is a map of rigid spaces, then (because the XΓG cover W admissibly) there is a subgroup G of finite index of OF× such that the image of U is contained within XΓG . In particular, by the universal property of XΓG there is an induced continuous group homomorphism ΓG → O(U )× , which induces a continuous group homomorphism κ : Op× × Op× → O(U)× . By composing this map with the map Op× → Op× × Op× sending γ to (γ, 1), we get a continuous group homomorphism n : Op× → O(U)× , which can be written as a product over j ∈ J of continuous group homomorphisms nj : OF×j → O(U)× . Hence by Propo# × $J sition 8.3 there exists r ∈ NK and a map B× r × U → Gm giving rise to n. We call such a map a thickening of n. Because we have only set up our Fredholm theory on Banach modules, we will have to somehow single out one such thickening, which we do (rather arbitrarily) in the definition below. First # × $J we single out a discrete subset Nd× of NK as follows: let πj denote a uni# × $J × formiser of Fj and define Nd ⊂ NK to be the product over j ∈ J of the sets {|πjt | : t ∈ Z>0 }. We equip Nd× with the obvious partial ordering. Definition Let X be an affinoid and let κ = (n, v) : Op× × Op× → O(X)× be a weight. We define r(κ) to be the largest element of Nd× such that the t construction above works. Explicitly, we choose r(κ)j = |πjj | with tj ∈ Z>0 , and the tj are chosen as small as possible such that the maps nj : OF×j → O(X)× are induced by maps B× r(κ)j × X → Gm and hence the map n : × × Op → O(X) is induced by a map B× r(κ) × X → Gm . This construction applies in particular when X is an affinoid subdomain of W (the inclusion X → W induces a map κ as above). Note however that

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as the image of X in W gets larger, the r(κ)j will get smaller—there is in general no universal r and map B× r × W → Gm . Note also that the construction applies if X is a point, and in this case κ corresponds to a point of W.

9 Classical automorphic forms. Our exposition of the classical theory follows [12] for the most part. We recall the notation of the latter part of the previous section, and add a little more. Recall that F is a totally real field of degree g over Q, and OF is the integers ∼ of F . We fix an isomorphism C − → Qp ; then we can think of I as the set of all infinite places of F , or as all the field embeddings F → Qp . Note that any such map i extends to a map i : Fp := F ⊗ Qp → Qp . Recall that K is a complete extension of K0 , the compositum of the images i(F ) of F as i runs through I, and we may also think of I as the set of all field homomorphisms F → K. We let J denote the set of primes of OF dividing p. If j ∈ J then let Fj denote the completion of F at j and let Oj denote the integers in Fj . We set Op := OF ⊗ Zp ; then Fp = ⊕j∈J Fj and Op = ⊕j∈J Oj . Choose once and for all uniformisers πj of the local fields Fj for all j, and let π ∈ Fp denote the element whose jth component is πj . We will also use π to denote the ideal of OF which is the product of the prime ideals above p. Note that some constructions (for example the Hecke operators Uπj defined later) will depend to a certain extent on this choice, but others (for example the eigenvarieties we construct) will not. Any i ∈ I gives a map Fp → K and this map factors through the projection Fp → Fj for some j := j(i) ∈ J; hence we get a natural surjection I → J. If S is any set then this surjection induces a natural injection S J → S I , where as usual S I denotes the set of maps I → S. We continue to use the following very useful notation: if (aj ) ∈ S J and i ∈ I then by ai we mean aj for j = j(i). Now let D be a quaternion algebra over F ramified at all infinite places. Let us assume that D is split at all places above p.5 Let OD denote a fixed maximal order of D, and fix an isomorphism OD ⊗OF OFv = M2 (OFv ) for all finite places v of F where D splits (here Fv is the completion of F at v and OFv is the integers in this completion). In particular we fix an isomorphism OD ⊗OF Op = M2 (Op ), and this induces an isomorphism Dp := D ⊗F Fp = M2 (Fp ). 5 One can almost certainly develop some of the theory as long as at least one place above p is split, although one might have to fix the weights at the ramified places.

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We recall the classical definitions of automorphic forms for D. If n ∈ ZI≥0 then we define Ln to be the K-vector space with basis the monomials mi I i∈I Zi , where m ∈ Z≥0 , 0 ≤ mi ≤ ni , and where the Zi are independent indeterminates. If t ∈ ZJ≥1 then define Mt to be the elements (γj ) # aj bj $ of M2 (Op ) = then j∈J M2 (Oj ) with the property that if γj = cj dj t

det(γj ) = 0, πjj divides cj , and πj does not divide dj . Then Mt is a monoid under multiplication. By M1 we mean the monoid Mt for t = (1, 1, . . . , 1). If v ∈ ZI and n ∈ ZI≥0 then define the right M1 -module Ln,v to be the Kvector space with the action of M1 defined by letting (γj ) = # Ln equipped mi mi aj bj $ ni vi ai Zi +bi send i Zi to i (ci Zi + di ) (ai di − bi ci ) cj dj ci Zi +di j∈J

and extending K-linearly (note that here we are using the notation ai for the image of aj(i) in K via the map i, as explained above). Note that in fact the same definition gives an action of GL2 (Fp ) on Ln,v , but we never use this action. The natural maps OF → Op → M2 (Op ) (via the diagonal embedding) induce an embedding from OF× into M1 . An easy check shows that the totally positive units in OF× act trivially on Ln,v if n + 2v ∈ Z. Define AF,f to be the finite adeles of F and Df := D ⊗F AF,f . If x ∈ Df then let xp ∈ Dp = M2 (Fp ) denote the projection onto the factor of Df at p. If t ∈ ZJ≥1 then we say that a compact open subgroup U ⊂ Df× has wild level ≥ π t if the projection U → Dp× is contained within Mt . If t = (1, 1, . . . , 1) then we drop it from the notation and talk about compact open subgroups of wild level ≥ π. )μ Say Df× = λ=1 D× τλ U . Then the groups Γλ := τλ−1 D × τλ ∩ U are × finitely-generated and moreover τλ Γλ τλ−1 ⊂ D× is commensurable with OD and hence with OF× . Hence Γλ is also commensurable with OF× . If n is an ideal of OF which is coprime to Disc(D) then we define U0 (n) (resp. U1 (n)) # in$ × which are congruent to ∗0 ∗∗ the usual way as being matrices in (O ⊗ Z) D # $ (resp. ∗0 1∗ ) mod n. Note that for many such choices of U we see that the Γλ are all contained within OF× (see, for example, Lemma 7.1 of [12] and the i observation that, in Hida’s notation, the groups Γ (U ) are finite because D is totally definite). However we do not need to assume this because of our generalisation of Coleman’s Fredholm theory. Say t ∈ ZJ≥1 , U is a compact open of wild level ≥ π t , and A is any right Mt -module, with action written (a, m) → a.m. If f : Df× → A and u ∈ U then define f |u : Df× → A by (f |u)(g) := f (gu−1 ).up . Now set ' ( L(U, A) := f : D× \Df× → A : f |u = f for all u ∈ U .

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Note that f ∈ L(U, A) is determined by f (τλ ) for 1 ≤ λ ≤ μ, and one checks easily that the map f → (f (τλ ))1≤λ≤μ induces an isomorphism L(U, A) →

μ

AΓλ .

λ=1

In particular, the functor L(U, −) is left exact. We remark that in the circumstances that will interest us later on, A will be an ONable Banach module over an affinoid in characteristic zero, the Γλ will all act via finite groups, and the invariants will hence be a Banach module with property (P r). Indeed, this phenomenon was the main reason for extending Coleman’s theory from ONable modules to modules with property (P r). If η ∈ Df× and ηp ∈ Mt then one can define an endomorphism [U ηU ] of ) L(U, A) as follows: decompose U ηU = i U xi (a finite union) and define f |[U ηU ] := f |xi . i

This operator is called the Hecke operator associated to η. Now let n ∈ ZI≥0 and v ∈ ZI be such that n + 2v ∈ Z. Set k = n + 2 and w = v + n + 1. Then k − 2w ∈ Z and k ≥ 2 (that is, ki ≥ 2 for all i), and conversely given k and w with these properties one can of course recover n and v. We finish by recalling the definition of classical automorphic forms for D in this context. Let U ⊂ Df× be a compact open subgroup of wild level ≥ π. D Definition The space of classical automorphic forms Sk,w (U ) of weight (k, w) and level U for D is the space L(U, Ln,v ).

This space is a finite-dimensional K-vector space. It is not, in the strict sense, a classical space of forms, because we have twisted the weight action from infinity to p. On the other hand if one chooses a field homomorphism D K → C then Sk,w (U) ⊗K C is isomorphic to a classical space of Hilbert modular forms, as described in, for example, section 2 of [12]. We remark also that because the full group GL2 (Fp ) acts naturally on Ln,v , our assumption that U has wild level ≥ π is unnecessary at this point. However, the forms that we shall p-adically interpolate will always have wild level ≥ π, because of the standard phenomenon that to p-adically analytically interpolate forms on GL2 one has to drop an Euler factor.

10 Overconvergent automorphic forms. Let X be an affinoid over K, and let κ = (n, v) : Op× × Op× → O(X)× be a weight. In this section we will define O(X)-modules of r-overconvergent

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automorphic forms of weight κ. In this generality, κ really is a family of weights; one important case to keep in mind is when X is a point, so n and v are continuous group homomorphisms Op× → K × , the resulting spaces will then be Banach spaces over K and will be automorphic forms of a fixed weight. One important special case of this latter situation is when n and v are i of the form α → i αm where the mi are integers; the resulting spaces of i automorphic forms will have a “classical weight” and there will be a natural finite-dimensional subspace corresponding to a space of classical automorphic forms as defined in the previous section. The group homomorphism κ : Op× × Op× → O(X)× induces a map f : X → W by Lemma 8.2(b) and (c). We extend the map v : Op× → O(X)× to a group homomorphism v : Fp× → O(X)× by defining v(πj ) = 1 for all j ∈ J. Note that this extension depends on our choice of πj (it is analogous to Hida’s choices of {xv } in [12]) but subsequent definitions will not depend seriously on this choice (in particular the eigenvariety we construct will not depend on this choice). Note also that the supremum semi-norm of every element in the image of n or v is 1. J Now for r ∈ (NK ) define Aκ,r to be the K-Banach algebra O(Br × X). Note that Aκ,r does not yet depend on κ but we will define a monoid action below which does. Let us assume for simplicity that X is reduced (this is not really necessary, but will be true in practice and also gives us a canonical choice of norm on O(X), namely the supremum norm). Endow Aκ,r with the supremum norm. As usual write κ = (n, v) and n = j∈J nj with nj : OF×j → O(X)× . Definition We say that t = (tj )j∈J ∈ ZJ>0 is good for the pair (κ, r) if for each j ∈ J there is thickening of nj to a map B× "" tj "" × X → Gm . rj "πj "

Equivalently, t ∈ ZJ>0 is good if r|π t | <= r(κ) in (NK )J with the obvious partial order. Given any (κ, r) as above, there will exist good t ∈ ZJ>0 by Proposition 8.3 (indeed, there will exist a unique minimal good t). The point of the definition is that if t is good for (κ, r) then we can define a# right $ action of Mt on Aκ,r (we denote the action by a dot) by letting γ = ac db ∈ Mt act as follows: if h ∈ Aκ,r and (z, x) ∈ Br (L) × X(L) for L any complete extension of K then (h.γ)(z, x) := n(cz + d, x) (v (det(γ)) (x)) h ((az + b)/(cz + d), x) . This is really a definition “on points” but it is easily checked that h.γ ∈ Aκ,r , using Lemma 8.1, and that the definition does give an action. It is elementary to check that for fixed γ ∈ Mt , the map Aκ,r → Aκ,r defined by h → h.γ is a

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continuous O(X)-module homomorphism (but it is not in general a ring homomorphism if κ is non-trivial). The fact that n and v take values in elements of O(X)× with supremum norm 1 easily implies that γ : Aκ,r → Aκ,r is normdecreasing. One also checks using Lemma 8.1(b) that if | det(γj )| = mj then γ induces a continuous norm-decreasing O(X)-module homomorphism from Aκ,rm to Aκ,r . We now have enough to define our Banach modules of overconvergent modular forms. This definition is ultimately inspired by [16], a preprint which sadly may well never see the light of day but which contained the crucial idea of beefing up a polynomial ring to a restricted power series ring in order to move from the classical to the overconvergent setting. Definition Let X be a reduced affinoid over K and let X → W be a morphism J of rigid spaces, inducing κ : Op× × Op× → O(X)× . If r ∈ (NK ) , if t is good for (κ, r), and if U is a compact open subgroup of Df× of wild level ≥ π t , then define the space of r-overconvergent automorphic forms of weight κ and level U to be the O(X)-module SD κ (U; r) := L(U, Aκ,r ).

We remark that, just as in the case of “classical” overconvergent modular forms, the hypotheses of the definition imply that if κ is a weight near the boundary of weight space (that is, such that r(κ) is small), then r|π t | must be small and hence for each j either rj is small or there must be some large power of πj in the level. If f ∈ SD κ (U ; r) then f is determined by f (τλ ) for λ = 1, . . . , μ. Moreover, if u ∈ U then so is u−1 , and hence both up and u−1 p are in Mt . In particular, both up and its inverse are norm-decreasing, and hence up is norm-preserving. We deduce that for d ∈ D× , τ ∈ Df× and u ∈ U we have |f (dτ u)| = |f (τ ).up | = |f (τ )| and hence |f (g)| ≤ maxλ f (τλ ) for all g ∈ Df× . In particular we can define a norm on SD κ (U ; r) by |f | = maxg∈Df× |f (g)|, and the isomorphism SD κ (U ; r) →

μ

(Aκ,r )

Γλ

λ=1

defined by f → (f (tλ ))λ is norm-preserving. Next observe that the group Γλ contains, with finite index, a subgroup of OF× of finite index, and hence Γλ acts on Aκ,r via a finite quotient. Hence SD κ (U ; r) is a direct summand of an ONable Banach O(X)-module and our Fredholm theory applies.

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Fix n ∈ and v ∈ ZI such that n+2v ∈ Z. Set k = n+2 and w = v+n+1 as usual. Define κ : Op× × Op× → K × by κ(α, β) = i αni i βivi . Note that κ is trivial on the totally positive units in OF× (embedded via γ → (γ, γ 2 ) as usual) and hence κ is a K-point of W. With notation as above, we are taking L = K and X a point. The map n : Op× → K × defined by α → i αni i extends to a # × $J map of rigid spaces B× , so r(κ)j = |πj | for all r → Gm for any r ∈ NK J j ∈ J. If r ∈ (NK ) then there is a natural injection Ln,v → Aκ,r = O(Br ) induced from the natural inclusion Br ⊂ (A1 )I and one checks easily that this is an M1 -equivariant inclusion. If U ⊂ Df× is a compact open subgroup of level ≥ π then we get an inclusion ZI≥0

D Sk,w (U ) = L(U, Ln,v ) ⊆ L(U, Aκ,r ) = SD κ (U ; r)

between the finite-dimensional space of classical forms and the typically infinite-dimensional space of overconvergent ones. This relationship between classical and overconvergent modular forms is however not quite the one that we want in general. When we construct our eigenvarieties in this setting, we will want the level structure at p to be U0 (π), and hence we need to explain how to interpret finite slope classical forms of conductor π 2 and above, or forms with non-trivial character at p, as forms of level π and some appropriate weight. Briefly, the trick is that we firstly load the character at p into the weight of the overconvergent form, thus reducing us to level U0 (π n ), and then decrease r and decrease the level to U0 (π). Note that a variant of this trick is used to construct the classical eigencurve—although there the level structure is reduced only to Γ1 (p) (or Γ1 (4) if p = 2) because the p-adic zeta function may have zeroes on points of weight space which are not contained in the identity component. Let us explain these steps in more detail. Let U0 be a compact open subgroup of Df× of the form U × GL2 (Op ), choose t ∈ ZJ≥1 , and let U1 denote the group U0 ∩ U1 (π t ). Then U1 is a normal subgroup of U0 ∩ U0 (π t ); #let Δ$ denote the quotient group. The map Op× → GL2 (Op ) ⊂ U0 sending d to 10 d0 identifies Δ with the quotient (Op /π t )× . If L is a complete extension of K, if n ∈ ZI≥0 and v ∈ ZI are chosen such that n + 2v ∈ Z, and if k = n + 2 D and w = v + n + 1 as usual, then for f ∈ Sk,w (U1 ) = L(U1 , Ln,v ) and t −1 D D u ∈ U0 ∩ U0 (π ) we have f |u ∈ Sk,w (uU1 u−1 ) = Sk,w (U1 ) and hence t there is a left action of U0 ∩ U0 (π ) on the finite-dimensional L-vector space D Sk,w (U1 ) defined by letting u act by f → f |u−1 . This action is easily seen to factor through Δ, and is just the Diamond operators at primes above p in D this setting. If L contains enough roots of unity then Sk,w (U1 ) is a direct sum

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of eigenspaces for this action. Choose a character ε : Δ → L× and let ε also denote the induced character of Op× . Now define κ : Op× × Op× → L× by κ(α, β) = ε(α) i αni i βivi . The fact that n + 2v ∈ Z means that κ vanishes on a subgroup of OF× of finite index, and hence κ is a weight. One checks that J |π t | ≤ r(κ) and hence that if r ∈ (NK ) then t is good for (r, κ), so the spaces L(U0 , Aκ,r ) and L(U1 , Aκ,r ) are well-defined. Moreover, the natural map Ln,v → Aκ,r for the action of the submonoid of Mt consisting # is equivariant $ t of matrices ac db with πjj |(dj − 1). Hence if L(U1 , Ln,v )(ε) denotes the εeigenspace of L(U1 , Ln,v ) under the action of Δ, then we get an induced map L(U1 , Ln,v )(ε) → L(U1 , Aκ,r ) and unravelling the definitions one checks easily that the image of L(U1 , Ln,v )(ε) is in fact contained in L(U0 , Aκ,r ) (the point being that the map Ln,v → Aκ,r is not U0 (π t )-equivariant, and the two actions differ by ε). This construction embeds classical forms with nontrivial character at primes above p into overconvergent forms with U0 (π t ) level structure at p, and should be thought of as the replacement in this setting of the construction of moving from a classical form of level pn and character ε to an overconvergent function of level pn and trivial character, by dividing by an appropriate Eisenstein series with character . Note the phenomenon, also D present in the classical case, that forms in distinct eigenspaces of Sk,w (U1 ) for the Diamond operators above p actually become overconvergent eigenforms of distinct weights in this setting. We now explain the relationship between forms of level U0 (π) and forms of level U0 (π r ) for any r ≥ 1. Let X → W be a map from a reduced affinoid to weight space, and let κ = (n, v) : Op× × Op× → O(X) be the induced weight. Let U be a compact open subgroup of Df× of the form U × GL2 (Op ). Say r ∈ J

(NK ) , s ∈ ZJ≥0 and t ∈ ZJ≥1 are chosen such that there is a thickening of n s to B× r|π s+t | ×X → Gm . Then we have defined spaces of r|π |-overconvergent weight κ automorphic forms of level U ∩ U0 (π t ) and also r-overconvergent weight κ automorphic forms of level U ∩ U0 (π t+s ). We now show that these spaces are canonically isomorphic. Proposition 11.1 There is a canonical isomorphism L(U ∩ U0 (π t ), Aκ,r|πs | ) ∼ = L(U ∩ U0 (π t+s ), Aκ,r ). Remark We will see later that this isomorphism preserves the action of various Hecke operators when U = U1 (n) or U0 (n). Proof The proof is an analogue of [5], Lemma 4, part 4, in this setting. We explain the construction of maps in both directions; it is then easy to check

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s that these maps are well-defined and inverse to one another. As usual # slet$π sj π 0 denote the element of Op whose component at j ∈ J is πj . Then 0 1 is an element of GL2 (Op ) and hence we can think of it as an element of Df× . If f ∈ L(U ∩ U (π t ), Aκ,r|πs | ) then define h : Df× → Aκ,r by h(g) = # # −s $$ # s 0$ # s $ f g π 0 01 . π0 01 ; note that if φ ∈ O(Br|πs | × X) then φ. π0 01 can be thought of as an element of O(Br ×X) as if z ∈ Br (L) then π s z ∈ Br|πs | (L). One checks that h ∈ L(U ∩ U0 (π t+s ), Aκ,r ). Slightly harder work is the map the other way. First note that Br|πs | is the disjoint union of π s Br + α as α ∈ Op runs through a set of coset represen tatives S for Op /π s ; hence Aκ,r = α∈S O((π s Br + α) × X). Now if h ∈ L(U ∩U0 (π t+s ), Aκ,r ) then define f : Df× → Aκ,r|πs | as follows: g ∈ Df× # # sfor$$ s s π α we define f (g) on (π Br +α)×X by f (g)(π z +α, x) = h g 0 1 (z, x). One checks easily that this is well-defined (that is, independent of choice of coset representatives S). A little trickier is that f ∈ L(U ∩ U0 (π t ), Aκ,r|πs | ), the hard part being to check that f |u = f for u ∈ U ∩ U0 (π t ). We give a sketch of the idea, # $which is just algebra. Firstly one checks easily that f |u = f if u = 10 γ1 with γ ∈ Op . Now say u ∈ U ∩ U0 (π t ), and choose α ∈ Op . We must check that (f (g))(π s z + α, x) = ((f |u)(g))(π s z + α, x) for all x ∈ X and z ∈ Br (again we present the argument on points but of course this suffices). unravel the right hand # 1The $trick # 1 αis$knowing how to −α t side. Because u := u ∈ U ∩ U (π ), there exists β ∈ Op 0 0 1 0 1 #1 β$ #a b$ such that u = v with v = satisfying π s |b. In particular p c d # −s $ # 0s 1 $ π 0 v π 0 = v ∈ U ∩ U (π t+s ). Hence 0 0 1 0 1 # # $$ ((f |u)(g))(π s z + α, x) = f |v 10 −α (g)(π s z + α, x) 1 # # 1 α $$ s = (f |v) g 0 1 (π z, x)

and by expanding out the definition of f |v and then using definition of f in # # the $$ s terms of h, one checks readily that this equals (h|v ) g π0 α1 (z, x). We are now home, as h|v = h. Finally one checks easily that the above associations f → h and h → f are inverse to one another. No doubt one can now mimic the constructions of section 7 of [5] to deduce the existence of various canonical maps between spaces of overconvergent forms, and relate the kernels of these maps to spaces of classical forms; these maps, analogous to Coleman’s θ k−1 operator, will not be considered here for reasons of space, as they are not necessary for the construction of eigenvarieties. The reader interested in these things might like, as an exercise, to verify that overconvergent forms of small slope are classical in this setting, following section 7 of [5].

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12 Hecke operators. Let X be a reduced affinoid over K and let κ = (n, v) : X → W be a J morphism of rigid spaces. If r ∈ (NK ) , if ρ = |π t | ∈ Nd× is such that n has a thickening to Brρ , and if U is a compact open subgroup of Df× of wild level ≥ π t , then we have defined the r-overconvergent automorphic forms of weight κ and level U . If v is a finite place of F where D splits then we define ηv ∈ Df× to# be the $ element which is the identity at all places away from v, and the matrix π0v 10 at v, where πv ∈ Fv is a uniformiser. If v is prime to p then will not matter which uniformiser we choose, but if v|p then for simplicity we use the uniformiser which we have already chosen earlier (this is really just for notational convenience though—a different choice would only change the operators we define by units). Let us assume in this section that U is a compact open of the form U0 (n) ∩ U1 (r) for some integral ideals n and r of OF , both prime to Disc(D), with π|n and π coprime to r. In this case, the resulting Hecke operator Tv = [U ηv U ], acting on L(U, Aκ,r ), is easily checked to be independent of the choice of πv , as long as v is prime to p. If furthermore v is prime to nr then we may regard πv as an element of the centre of Df× and we define Sv to be the resulting Hecke operator [U πv U ]. A standard argument shows that the endomorphisms Tv and Sv all commute with one another. Furthermore, we have Lemma 12.1 The isomorphism of Proposition 11.1 is Hecke equivariant. Proof For the Hecke operators away from p this is essentially immediate. At primes above p things are slightly more delicate, because for $tj ≥ 1 the t # t natural left coset decomposition of the double coset U0 (πjj ) π0j 10 U0 (πjj ) is ) tj # πj 0 $ t α∈Oj /πj U0 (πj ) απ j 1 which depends on tj . However, one checks easily j

that if (in the notation of Proposition 11.1) f ∈ L(U ∩ U0 (π t ), Ar|πs | ) and h is the element of L(U ∩ U0 (π t+s ), Ar ) associated to f in the proof, then Tv h is indeed associated to Tv f , for all v|p, the calculation boiling down to the fact that s πj 0 πj 0 πs 0 π 0 = . t t +s 0 1 0 1 απjj 1 απjj j 1

e If p factors in F as j pj j then let Uj denote the Hecke operator Tpj , let Uπ denote j∈J Uj , and let ηj denote the matrix ηpj .

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D Lemma 12.2 The map Uπ : SD κ (U ; r) → Sκ (U ; r) is the composite of the natD D ural inclusion Sκ (U ; r) → Sκ (U ; r|π|) and a continuous norm-decreasing D D D map SD κ (U; r|π|) → Sκ (U ; r). The inclusion Sκ (U ; r) → Sκ (U ; r|π|) is norm-decreasing and compact, and hence Uπ , considered as an endomorphism of SD κ (U ; r), is also norm-decreasing and compact.

Proof One checks easily that Uπ is the Hecke operator [U ηU ] associated to the matrix η := j ηj . If one decomposes U ηU into a finite disjoint union ) δ U xδ of cosets, then det((xδ )p )/ det(ηp ) is a unit at all places of F above p, and hence by Lemma 8.1(b) the endomorphism of Aκ,r induced by (xδ )p can be factored as the inclusion Aκ,r ⊂ Aκ,r|π| followed by a norm-decreasing map Aκ,r|π| → Aκ,r . The inclusion Aκ,r ⊂ Aκ,r|π| is induced by the inner inclusion of affinoids Br|π| → Br and is hence compact and norm-decreasing; the result now follows easily.

13 The characteristic power series of Uπ . We now have enough data to define the ingredients for our eigenvariety machine in this case. Let n be an integral ideal of OF prime to p and to Disc(D) (this latter hypothesis is not really necessary, but we enforce it for simplicity’s sake), and set U0 = U0 (n) or U1 (n). Define U = U0 ∩ U0 (π); then U has wild level ≥ π. If X ⊂ W is an affinoid subdomain, then set RX = O(X), let κ : Op× × Op× → O(X)× denote the corresponding weights, and let r = r(κ). Let T be the set of Hecke operators Tv (for v running through all the finite places of F where D splits) and Sv (for v running through all the finite places of F prime to np where D splits) defined above, and let φ denote the operator Uπ . Define MX = SD κ (U ; r) = L(U, Aκ,r ). If Y ⊆ X is an affinoid subdomain and κ is the weight corresponding to Y then r(κ) ≤ r(κ ) and hence there is an inclusion Br(κ) ⊆ Br(κ ) . There RX RY = L(U, Aκ ,r(κ) ), and the inclusion is a canonical isomorphism MX ⊗ Br(κ) → Br(κ ) induces an injection Aκ ,r(κ ) → Aκ ,r(κ) and hence an injec RX RY . It is easy to check that this injection commutes tion α : MY → MX ⊗ with the action of all the Hecke operators Tv and Sv . We now check that α is a link; the argument is a slight variant on the usual one because we have allowed non-parallel radii of convergence in our definitions and hence have to make essential use of r-overconvergent forms with r ∈ Nd× . Lemma 13.1 If U = U0 ∩ U0 (π) as above, if Y ⊆ W is a reduced affinoid with corresponding weight κ : Op× × Op× → O(Y )× , and if r, r ∈ (NK )J

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with r, r ≤ r(κ) and rj ≤ rj for all j, then the natural map SD κ (U ; r ) → D Sκ (U ; r) is a link.

Proof It suffices to prove that α is a primitive link when rj |πj | < rj ≤ rj for all j. But this is not too hard: let c be the compositum of the (compact) restricD tion map SD κ (U ; r) → Sκ (U ; r |π|) and the continuous norm-decreasing map D D β : Sκ (U ; r |π|) → Sκ (U ; r ) in the statement of Lemma 12.2; then it is not hard to check that αc and cα are both Uπ as endomorphisms of their respective spaces. We may now apply our eigenvariety machine, and deduce the existence of an eigenvariety parametrising systems of Hecke eigenvalues on overconvergent automorphic forms, and in particular p-adically interpolating classical automorphic forms for D. The eigenvariety itself is a rigid space, the geometry of which we know very little about—indeed if we do not know Leopoldt’s conjecture then we do not even know its dimension. If one were to check (and it is no doubt not difficult, following the ideas of section 7 of [5]) that overconvergent forms of small slope were classical, then the existence of the eigenvariety implies results of Gouvˆea-Mazur type for classical Hilbert modular forms over F , if [F : Q] is even (although there are probably more elementary ways of attacking analogues of the Gouvˆea-Mazur conjectures in this setting—see for example the recent thesis of Aftab Pande).

Bibliography [1] S. Bosch, U. G¨untzer, and R. Remmert. Non-Archimedean analysis, volume 261 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1984. ´ ements de math´ematique. Part I. Les structures fondamentales de [2] N. Bourbaki. El´ l’analyse. Livre III. Topologie g´en´erale. Chapitres III et IV. Actual. Sci. Ind., no. 916. Hermann & Cie., Paris, 1942. [3] N. Bourbaki. Espaces vectoriels topologiques. Chapitres 1 a` 5. Masson, Paris, ´ ements de math´ematique. [Elements of mathematics]. new edition, 1981. El´ [4] K. Buzzard. Analytic continuation of overconvergent eigenforms. J. Amer. Math. Soc., 16(1):29–55 (electronic), 2003. [5] K. Buzzard. On p-adic families of automorphic forms. Progress in Mathematics, 224:23–44, 2004. [6] K. Buzzard and F. Calegari. The 2-adic eigencurve is proper. Documenta Math. Extra Volume: John H. Coates’ Sixtieth Birthday (2006) 211–232. [7] K. Buzzard and L. J. P. Kilford. The 2-adic eigencurve at the boundary of weight space. Compos. Math., 141(3):605–619, 2005. [8] G. Chenevier. Familles p-adiques de formes automorphes pour GL(n). Journal f¨ur die reine und angewandte Mathematik, 570:143–217, 2004.

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[9] R. Coleman and B. Mazur. The eigencurve. In Galois representations in arithmetic algebraic geometry (Durham, 1996), volume 254 of London Math. Soc. Lecture Note Ser., pages 1–113. Cambridge Univ. Press, Cambridge, 1998. [10] R. Coleman. p-adic Banach spaces and families of modular forms. Invent. Math., 127(3):417–479, 1997. [11] B. Conrad. Modular curves and rigid-analytic spaces. Pure Appl. Math. Q., 2(1):29–110, 2006. [12] H. Hida. On p-adic Hecke algebras for GL2 over totally real fields. Ann. of Math. (2), 128(2):295–384, 1988. [13] H. Hida. Elementary theory of L-functions and Eisenstein series, volume 26 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1993. [14] H. Matsumura. Commutative ring theory, volume 8 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1989. Translated from the Japanese by M. Reid. [15] J-P. Serre. Endomorphismes compl`etement continus des espaces de Banach p´ adiques. Inst. Hautes Etudes Sci. Publ. Math., (12):69–85, 1962. [16] G. Stevens. Overconvergent modular symbols. Preprint. [17] A. Yamagami. On p-adic families of Hilbert cusp forms of finite slope. Preprint.

Nontriviality of Rankin-Selberg L-functions and CM points Christophe Cornut Institut de Math´ematiques de Jussieu 4 place Jussieu F-75005 Paris France [email protected]

Vinayak Vatsal University of British Columbia Department of Mathematics Vancouver, BC V6T 1Z2 Canada [email protected]

Contents 1

2

3

4

Introduction 1.1 Rankin-Selberg L-functions 1.2 Gross-Zagier Formulae 1.3 The indefinite case 1.4 The definite case 1.5 Applications 1.6 Sketch of proof 1.7 Notations The Galois group of K[P ∞ ]/K 2.1 The structure of G(∞) 2.2 A filtration of G0 2.3 A formula Shimura Curves 3.1 Shimura curves 3.2 Connected components 3.3 Related group schemes 3.4 Hecke operators 3.5 The Hodge class and the Hodge embedding 3.6 Differentials and automorphic forms 3.7 The P -new quotient 3.8 CM Points The Indefinite Case 4.1 Statement of the main results 4.2 An easy variant 4.3 Proof of Theorem 4.2 4.4 Changing the level 4.5 Geometric Galois action 4.6 Chaotic Galois action

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122 122 126 129 132 134 134 137 138 138 139 140 142 142 143 144 145 146 149 151 152 153 153 155 157 158 160 163

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6

The definite case 5.1 Automorphic forms and representations 5.2 The exceptional case 5.3 CM points and Galois actions 5.4 Main results Appendix: Distribution Relations 6.1 Orders 6.2 The δ = 0 case 6.3 The δ = 1 case 6.4 The δ ≥ 2 case 6.5 Predecessors and degeneracy maps

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1 Introduction 1.1 Rankin-Selberg L-functions Let π be an irreducible cuspidal automorphic representation of GL2 over a totally real number field F . Let K be a totally imaginary quadratic extension of × F . Given a quasi-character χ of A× K /K , we denote by L(π, χ, s) the RankinSelberg L-function associated to π and π(χ), where π(χ) is the automorphic representation of GL2 attached to χ – see [19] and [18] for the definitions. This L-function, which is first defined as a product of Euler factors over all places of F , is known to have a meromorphic extension to C with functional equation L(π, χ, s) = (π, χ, s)L(˜ π , χ−1 , 1 − s) where π ˜ is the contragredient of π and (π, χ, s) is a certain -factor. × Let ω : A× → C× be the central quasi-character of π. The condition F /F χ·ω =1

× on A× F ⊂ AK

(1.1)

implies that L(π, χ, s) is entire and equal to L(˜ π , χ−1 , s). The functional equation thus becomes L(π, χ, s) = (π, χ, s)L(π, χ, 1 − s) and the parity of the order of vanishing of L(π, χ, s) at s = 1/2 is determined by the value of def

(π, χ) = (π, χ, 1/2) ∈ {±1}. We say that the pair (π, χ) is even or odd, depending upon whether (π, χ) is +1 or −1. It is expected that the order of vanishing of L(π, χ, s) at s = 1/2 should ’usually’ be minimal, meaning that either L(π, χ, 1/2) or L (π, χ, 1/2) should be nonzero, depending upon whether (π, χ) is even or odd.

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Calculation of sign For the computation of (π, χ), one first writes it as the product over all places v of F of the local signs (πv , χv ) which are attached to the local components of π and χ, normalized as in [11, Section 9]. Let η be the quadratic Hecke character of F attached to K/F , and denote by ηv and ωv the local components of η and ω. Then (π, χ) = (−1)#S(χ)

(1.2)

S(χ) = {v; (πv , χv ) = ηv · ωv (−1)}.

(1.3)

where def

Indeed, S(χ) is finite because (πv , χv ) = 1 = ηv · ωv (−1) for all but finitely many v’s, and (1.2) then follows from the product formula: η · ω(−1) = 1 = v ηv · ωv (−1). The various formulae for the local -factors that are spread throughout [19] and [18] allow us to decide whether a given place v of F belongs to S(χ), provided that the local components πv and π(χv ) of π and π(χ) are not simultaneously supercuspidal. At the remaining places, one knows that χ ramifies and we may use a combination of [19, Proposition 3.8] and [18, Theorem 20.6] to conclude that, when χ is sufficiently ramified at v, v does not belong to S(χ). For our purposes, we just record the following facts. For any finite place v of F in S(χ), Kv is a field and πv is either special or supercuspidal. Conversely, if v is inert in K, χ is unramified at v and πv is either special or supercuspidal, then v belongs to S(χ) if and only if the v-adic valuation of the conductor of πv is odd. Finally an archimedean (real) place v of F belongs to S(χ) if χv = 1 and πv is the holomorphic discrete series of weight kv ≥ 2. Ring class characters In this paper, we regard the automorphic representation π and the field K as being fixed and let χ vary through the collection of ring class characters of P -power conductor, where P is a fixed maximal ideal in the ring of integers OF ⊂ F . Here, we say that χ is a ring class character if there exists some OF -ideal C such that χ factors through the finite group × × × A× K /K K∞ OC Pic(OC ) def

where K∞ = K ⊗ R and OC = OF + COK is the OF -order of conductor C in K. The conductor c(χ) of χ is the largest such C. Note that this definition

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differs from the classical one — the latter yields an ideal c (χ) of OK such that c (χ) | c(χ)OK . Equivalently, a ring class character is a finite order character whose restriction to A× F is everywhere unramified. In view of (1.1), it thus make sense to × require that ω is a finite order, everywhere unramified character of A× F /F . n Then, there are ring class characters of conductor P satisfying (1.1) for any sufficiently large n. Concerning our fixed representation π, we also require that π is cuspidal of parallel weight (2, · · · , 2) and level N , and the prime-to-P part N of N is relatively prime to the discriminant D of K/F .

In this situation, we can give a fairly complete description of S(χ). Lemma 1.1 For a ring class character χ of conductor P n , S(χ) = S or S ∪ {P }, where S is the union of all archimedean places of F , together with those finite places of F which do not divide P , are inert in K, and divide N to an odd power. Moreover, S(χ) = S if either P does not divide N , or P splits in K, or n is sufficiently large. Remark 1.2 Note that πv is indeed special or supercuspidal for any finite place v of F which divides N to an odd power, as the conductor of a principal series representation with unramified central character is necessarily a square. It follows that the sign of the functional equation essentially does not depend upon χ, in the sense that for all but finitely many ring class characters of P power conductor, (π, χ) = (−1)|S| = (−1)[F :Q] η(N ). If P N or splits in K, this formula even holds for all χ’s. We say that the triple (π, K, P ) is definite or indefinite depending upon whether this generic sign (−1)[F :Q] η(N ) equals +1 or −1. Exceptional cases In the definite case, it might be that the L-function L(π, χ, s) actually factors as the product of two odd L-functions, and therefore vanishes to order at least 2. This leads us to what Mazur calls the exceptional case. Definition 1.3 We say that (π, K) is exceptional if π π ⊗ η. This occurs precisely when π π(α) for some quasi-character α of × A× K /K in which case L(π, χ, s) = L(αχ, s)·L(αχ , s) where χ is the outer twist of χ by Gal(K/F ). Moreover, both factors have a functional equations

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with sign ±1 and it can and does happen that both signs are −1, in which case L(π, χ, s) has at least a double zero. It is then more natural to study the individual factors than the product; this is the point of view taken in [23]. In this paper, we will always assume that (π, K) is not exceptional when (π, K, P ) is definite.

Mazur’s conjectures With this convention, we now do expect, in the spirit of Mazur’s conjectures in [20], that the order of vanishing of L(π, χ, 1/2) should generically be 0 in the definite case and 1 in the indefinite case. Let us say that χ is generic if it follows this pattern. We will show that there are many generic χ’s of conductor P n for all sufficiently large n. More precisely, let K[P n ]/K be the abelian extension of K associated by × × class field theory to the subgroup K × K∞ OP n of A× K , so that def × × × n G(n) = Gal(K[P n ]/K) A× K /K K∞ OP n Pic(OP ).

Put K[P ∞ ] = ∪K[P n ], G(∞) = Gal(K[P ∞ ]/K) = lim G(n) and let G0 ←− be the torsion subgroup of G(∞). It is shown in section 2 below that G0 is a finite group and G(∞)/G0 is a free Zp -module of rank [FP : Qp ], where p is the residue characteristic of P . Moreover, the reciprocity map of K maps × A× F ⊂ AK onto a subgroup G2 Pic(OF ) of G0 (the missing group G1 will make an appearance latter). Using this reciprocity map to identify ring class characters of P -power conductor with finite order characters of G(∞), and ω with a character of G2 , we see that the condition (1.1) on χ is equivalent to the requirement that χ · ω = 1 on G2 . Conversely, a character χ0 of G0 induces a character on A× F , and it make × sense therefore to require that χ0 · ω = 1 on AF . Given such a character, we denote by P (n, χ0 ) the set of characters of G(n) which induce χ0 on G0 and do not factor through G(n − 1) – these are just the ring class characters of conductor P n which, beyond (1.1), satisfy the stronger requirement that χ = χ0 on G0 . Theorem 1.4 Let the data of (π, K, P ) be given and definite. Let χ0 be any character of G0 with χ0 · ω = 1 on A× F . Then for all n sufficiently large, there exists a character χ ∈ P (n, χ0 ) for which L(π, χ, 1/2) = 0. For the indefinite case, we obtain a slightly more restrictive result.

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Theorem 1.5 Let the data of (π, K, P ) be given and indefinite. Suppose also that ω = 1, and that N , D and P are pairwise coprime. Let χ0 be any character of G0 with χ0 = 1 on A× F . Then for all n sufficiently large, there exists a character χ ∈ P (n, χ0 ) for which L (π, χ, 1/2) = 0. We prove these theorems using Gross-Zagier formulae to reduce the nonvanishing of L-functions and their derivatives to the nontriviality of certain CM points. The extra assumptions in the indefinite case are due to the fact that these formulae are not yet known in full generality, although great progress has been made by Zhang [32, 31, 33] in extending the original work of Gross and Zagier. We prove the relevant statements about CM points without these restrictions.

1.2 Gross-Zagier Formulae Roughly speaking, the general framework of a Gross-Zagier formula yields a discrete set of CM points on which the Galois group of the maximal abelian extension of K acts continuously, together with a function ψ on this set with values in a complex vector space such that the following property holds: a character χ as above is generic if and only if def a(x, χ) = χ(σ)ψ(σ · x)dσ = 0 (1.4) Galab K

where x is any CM point whose conductor equals that of χ – we will see that CM points have conductors. Note that the above integral is just a finite sum. In the indefinite case, the relevant set of CM points consists of those special points with complex multiplication by K in a certain Shimura curve M defined over F , and ψ takes its values in (the complexification of) the Mordell-Weil groups of a suitable quotient A of J = Pic0M/F . In the definite case, a finite set M plays the role of the Shimura curve. The CM points project onto this M and the function ψ is the composite of this projection with a suitable complex valued function on M . Quaternion algebras In both cases, these objects are associated to a quaternion algebra B over F whose isomorphism class is uniquely determined by π, K and P . To describe this isomorphism class, we just need to specify the set Ram(B) of places of F where B ramifies. In the definite case, the set S of Lemma 1.1 has even order and we take Ram(B) = S, so that B is totally definite. In the indefinite case, S is odd but it still contains all the archimedean (real) places of F . We fix arbitrarily a real place τ of F and take Ram(B) = S − {τ }.

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We remark here that in both cases, B splits at P . Moreover, the JacquetLanglands correspondence implies that there is a unique cuspidal automorphic representation π on B associated to π = JL(π ), and π occurs with multiplicity one in the space of automorphic cuspforms on B – this is the space denoted by A0 (G) in [30]. Finally, since Kv is a field for all v’s in S, we may embed K into B as a maximal commutative F -subalgebra. We fix such an embedding. def Let G = ResF/Q (B × ) be the algebraic group over Q whose set of points on a commutative Q-algebra A is given by G(A) = (B ⊗ A)× . Thus, G def

is a reductive group with center Z = ResF/Q (F × ) and the reduced norm nr : B → F induces a morphism nr : G → Z which also identifies Z with the cocenter G/[G, G] of G. Our chosen embedding K → B allows us to view def T = ResF/Q (K × ) as a maximal subtorus of G which is defined over Q. CM points For any compact open subgroup H of G(Af ), we define a set of CM points by def

CMH = T (Q)\G(Af )/H. There is an action of T (Af ) on CMH , given by left multiplication in G(Af ). This action factors through the reciprocity map recK : T (Af ) Galab K and thus defines a Galois action on CMH . For x = [g] in CMH (with g in G(Af )), the stabilizer of x in T (Af ) equals U (x) = T (Q) · (T (Af ) ∩ gHg −1 ) def

and we say that x is defined over the abelian extension of K which is fixed by × for some OF -order R ⊂ B, T (Af ) ∩ gHg −1 = recK (U (x)). When H = R × for some OF -order O(x) ⊂ K, and we define the conductor of x to be O(x) that of O(x). In particular, a CM point of conductor P n is defined over K[P n ]. We shall also need a somewhat more technical notion, namely that of a good CM point.

× as above, and that the P Definition 1.6 Assume therefore that H = R component of R is an Eichler order of level P δ in BP M2 (FP ). Then R is uniquely expressed as the (unordered) intersection of two OF -orders R1 and R2 in B, which are both maximal at P but agree with R outside P . We say that a CM point x = [g] ∈ CMH is good if either δ = 0 or KP ∩ gP R1 gP−1 = KP ∩ gP R2 gP−1 , and we say that x is bad otherwise.

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It is relatively easy to check that if CMH contains any CM point of P -power conductor, then it contains good CM points of conductor P n for all sufficiently large n. Automorphic forms Let S denote the space of automorphic forms on B in the definite case, and the space of automorphic cuspforms on B in the indefinite case. As a first step towards the construction of the function ψ of (1.4), we shall now specify a certain line C · Φ in the realization S(π ) of π in S. We will first define an admissible G(Af )-submodule S2 of S, using the local behavior of π at infinity. The line we seek then consists of those vectors in S2 (π ) = S2 ∩S(π ) which are fixed by a suitable compact open subgroup H of G(Af ). We refer to [10] for a more comprehensive discussion of these issues. Recall from [19] that S and π are representations of G(Af ) × H∞ where H∞ is a certain sort of group algebra associated to G(R). As a representation of H∞ , π is the direct sum of copies of the irreducible representation π∞ = ⊗v|∞ πv of H∞ . Let V∞ be the representation space of π∞ . We claim that V∞ is one dimensional in the definite case, while V∞ has a “weight decomposition” V∞ = ⊕k∈2Z−{0} V∞,k

(1.5)

into one dimensional subspaces in the indefinite case. Indeed, the compatibility of the global and local Jacquet-Langlands correspondence, together with our assumptions on π = JL(π ), implies that for a real place v of F , πv is the trivial one dimensional representation of Bv× if v ramifies in B, while for v = τ in the indefinite case, πv πv is the holomorphic discrete series of weight 2 which is denoted by σ2 in [6, section 11.3], and the representation space of σ2 is known to have a weight decomposition similar to (1.5). Remark 1.7 In the indefinite case, the above decomposition is relative to the choice of an isomorphism between Bτ and M2 (R). Given such an isomorphism, the subspace V∞,k consists of those vectors in V∞ on which SO2 (R) acts by the character cos(θ) sin(θ) 2kiθ . − sin(θ) cos(θ) → e Definition 1.8 We denote by S2 the admissible G(Af )-submodule of S which is the image of the G(Af )-equivariant morphism HomH∞ (V∞ , S) → S :

ϕ → ϕ(v∞ )

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where v∞ is any nonzero element of V∞ in the definite case, and any nonzero element of V∞,2 (a lowest weight vector) in the indefinite case. By construction, the G(Af )-submodule S2 (π ) = S2 ∩ S(π ) of S2 is isomorphic to HomH∞ (V∞ , S(π )). It is therefore irreducible. Level subgroups Turning now to the construction of H, let δ be the exponent of P in N , so that N = P δ N . Let R0 ⊂ B be an Eichler order of level P δ such that the conductor of the OF -order O = OK ∩ R0 is a power of P . The existence of R0 is given by [29, II.3], and we may even require that O = OK if P does not divide N or splits in K. On the other hand, recall that the reduced discriminant of B/F is the squarefree product of those primes of F which are inert in K and divide N to an odd power. We may thus find an ideal M in OK such that NormK/F (M) · DiscB/F = N . We then take def × H = R

where

def

R = O + M ∩ O · R0 .

(1.6)

Note that R is an OF -order of reduced discriminant N in B. Since RP = R0,P is an Eichler order (of level P δ ), we have the notion of good and bad CM points on CMH . Since x = [1] is a CM point of P -power conductor (with O(x) = O), there are good CM points of conductor P n for all sufficiently large n. We claim that S2 (π )H is 1-dimensional. Indeed, for every finite place v of × F , (πv )Rv is 1-dimensional: this follows from [3, Theorem 1] when v does not divide N (including v = P ) and from [10, Proposition 6.4], or a mild generalization of [32, Theorem 3.2.2] in the remaining cases.

1.3 The indefinite case Suppose first that (π, K, P ) is indefinite, so that B⊗R = Bv M2 (R) × H[F :Q]−1 v|∞

where H is Hamilton’s quaternion algebra and the M2 (R) factor corresponds to v = τ . We fix such an isomorphism, thus obtaining an action of G(R) GL2 (R) × (H× )[F :Q]−1 def

on X = C − R by combining the first projection with the usual action of GL2 (R) on X.

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For any compact open subgroup H of G(Af ), we then have a Shimura curve ShH (G, X) whose complex points are given by ShH (G, X)(C) = G(Q)\ (G(Af )/H × X) . The reflex field of this curve is the subfield τ (F ) of C, and its pull-back to F is a smooth curve MH over F whose isomorphism class does not depend upon our choice of τ . When S = ∅, F = Q, G = GL2 and the MH ’s are the classical (affine) modular curves over Q. These curves can be compactified by ∗ adding finitely many cusps, and we denote by MH the resulting proper curves. ∗ In all other cases, MH is already proper over F and we put MH = MH . We denote by JH the connected component of the relative Picard scheme of ∗ MH /F . Let x be the unique fixed point of T (R) in the upper half plane X + ⊂ X. The map g → (g, x) then defines a bijection between CMH and the set of special points with complex multiplication by K in MH . It follows from Shimura’s theory that these points are defined over the maximal abelian extension K ab of K, and that the above bijection is equivariant with respect to the Galois actions on both sides. On the other hand, there is a natural G(Af )-equivariant isomorphism between the subspace S2 of S and the inductive limit (over H) of the spaces ∗ of holomorphic differentials on MH . This is well-known in the classical case where S = ∅ – see for instance [6], section 11 and 12. For the general case, we sketch a proof in section 3.6 of this paper. In particular, specializing now to the level structure H defined by (1.6), we obtain a line S2 (π )H = C · Φ in the space S2H of holomorphic differentials on ∗ MH , a space isomorphic to the cotangent space of JH /C at 0. By construction, this line is an eigenspace for the action of the universal Hecke algebra TH , with coefficients in Z, which is associated to our H. Since the action of TH on the cotangent space factors through EndF JH , the annihilator of C · Φ in TH cuts out a quotient A of JH : def

A = JH /AnnTH (C · Φ) · JH . The Zeta function of A is essentially the product of the L-function of π together with certain conjugates – see [32, Theorem B] for a special case. The function ψ of (1.4) is now the composite of ∗ • the natural inclusions CMH → MH → MH , ∗ • a certain morphism ιH ∈ Mor(MH , JH ) ⊗ Q, and • the quotient map JH A.

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∗ In the classical case where S = ∅, ιH is a genuine morphism MH → JH ∗ which is defined using the cusp at ∞ on MH . In the general case, one has to use the so-called Hodge class. For a discussion of the Hodge class, we refer to [33, section 6], or [11, section 23]. A variant of this construction, adapted to our purposes, is given in section 3.5 below.

Statement of results Now, let χ be a ring class character of conductor P n such that χ·ω = 1 on A× F. Suppose also that (π, χ) = −1: this holds true for any n ≥ 0 if P N or P splits in K, but only for n 0 in the general case. Then L(π, χ, 1/2) = 0 and the Birch and Swinnerton-Dyer conjecture predicts that the χ−1 -component of A(K[P n ]) ⊗ C should be non-trivial. If moreover L (π, χ, 1/2) = 0, the Gross-Zagier philosophy tells us more, namely that this non-triviality should be accounted for by the CM points of conductor P n : if x is such a point, there should exists a formula relating L (π, χ, 1/2) to the canonical height of def

a(x, χ) =

1 χ(σ)ψ(σx) ∈ A(K[P n ]) ⊗ C, |G(n)| σ∈G(n)

thereby showing that L (π, χ, 1/2) is nonzero precisely when a(x, χ) is a nonzero element in the χ−1 -component of A(K[P n ]) ⊗ C. Unfortunately, such a formula has not yet been proven in this degree of generality. For our purposes, the most general case of which we are aware is Theorem 6.1 of Zhang’s paper [33], which gives a precise formula of this type under the hypotheses that the central character of π is trivial and that N , D and P are pairwise prime to each other. Remark 1.9 We point out that Zhang works with the Shimura curves attached to G/Z instead of G, and uses a(x, χ−1 ) instead of a(x, χ). The first distinction is not a real issue, and the second is irrelevant, as long as we are restricting our attention to the anticyclotomic situation where χ = ω = 1 −1 on A× is then equal to the outer twist of χ by Gal(K/F ), F . Indeed, χ so that L(π, χ, s) = L(π, χ−1 , s) and any lift of the non-trivial element of Gal(K/F ) to Gal(K[P n ]/F ) interchanges the eigenspaces for χ and χ−1 in A(K[P n ]) ⊗ C. One has to be more careful when χ is non-trivial on A× F . To be consistent with the BSD conjecture, a Gross-Zagier formula should relate L (π, χ, 1/2) to a point in the χ−1 -component of A(K[P n ]) ⊗ C.

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In any case, Zhang’s Gross-Zagier formula implies that Theorem 1.5 is now a consequence of the following result, which itself is a special case of Theorem 4.1 in the text. Theorem 1.10 Let χ0 be any character of G0 such that χ0 · ω = 1 on A× F. n Then, for any good CM point x of conductor P with n sufficiently large, there exists a character χ ∈ P (n, χ0 ) such that a(x, χ) = 0.

1.4 The definite case Suppose now that the triple (π, K, P ) is definite, so that π∞ is the trivial 1dimensional representation of G(R) = Bv× (H× )[F :Q] . v|∞

Then S(π ) is contained in S2 , and the latter is simply the subspace of S on which G(R) acts trivially; this is the space of all smooth functions φ : G(Q)\G(A)/G(R) = G(Q)\G(Af ) −→ C, with G(Af ) acting by right translation. Note that the G(Af )-module underlying S(π ) = S2 (π ) is admissible, infinite dimensional and irreducible; it contains no nonzero function which factors through the reduced norm, because any such function spans a finite dimensional G(Af )-invariant subspace. For any compact open subgroup H of G(Af ), we may identify S2H with the set of complex valued functions on the finite set def

MH = G(Q)\G(Af )/H, and any such function may be evaluated on CMH = T (Q)\G(Af )/H. Specializing now to the H which is defined by (1.6), let ψ be the function induced on CMH by some nonzero element Φ in the 1-dimensional space S2 (π )H = S(π )H = C · Φ: Φ

ψ : CMH → MH −→ C. For a ring class character χ of conductor P n such that χ · ω = 1 on A× F, the Gross-Zagier philosophy predicts that there should exist a formula relating 2 L(π, χ, 1/2) to |a(x, χ)| , for some CM point x ∈ CMH of conductor P n , with 1 def a(x, χ) = χ(σ)ψ(σ · x) ∈ C. |G(n)| σ∈G(n)

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Such a formula has indeed been proven by Zhang [33, Theorem 7.1], under the assumption that ω = 1, and that N , D and P are pairwise coprime. On the other hand, there is a more general theorem of Waldspurger which, although it does not give a precise formula for the central value of L(π, χ, s), still gives a criterion for its non-vanishing. Statement of results Thus, let χ be any character of T (Q)\T (A) such that χ · ω = 1 on A× F . Such a character yields a linear form χ on S(π ), defined by def

χ (φ) = χ(t)φ(t)dt Z(A)T (Q)\T (A)

where dt is any choice of Haar measure on T (A). By a fundamental theorem of Waldspurger [30, Th´eor`eme 2], this linear form is nonzero on S(π ) if and only if L(π, χ, 1/2) = 0 and certain local conditions are satisfied. The results of Tunnell and Saito which are summarized in [11, Section 10] show that these local conditions are satisfied if and only if the set S(χ) of (1.3) is equal to the set S of places where B ramifies. For a ring class character χ of P -power conductor, Lemma 1.1 shows that S(χ) = S if and only if (π, χ) is even. We thus obtain the following simple criterion. Theorem 1.11 [Waldspurger] For a ring class character χ of P -power conductor such that χ · ω = 1 on A× F, L(π, χ, 1/2) = 0

⇔

∃ φ ∈ S(π ) : χ (φ) = 0.

Remark 1.12 Waldspurger’s theorem does not give a precise formula for the value of L(π, χ, 1/2), and it does not specify a canonical choice of φ (a test vector in the language of [12]) on which to evaluate the linear functional χ . The problem of finding such a test vector φ and a Gross-Zagier formula relating χ (φ) to L(π, χ, 1/2) is described in great generality in [11], and explicit formulae are proven in [9] (for F = Q) and for a general F in [31, 33], under various assumptions. A leisurely survey of this circle of ideas may be found in [28]. Recall that ψ is the function which is induced on CMH by some nonzero Φ in S(π )H . For φ = g · Φ ∈ S(π ), with g ∈ G(Af ) corresponding to a CM point x = [g] ∈ CMH whose conductor P n equals that of χ, we find that, up to a nonzero constant,

χ (φ) ∼ a(x, χ).

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Theorem 1.4 therefore is a consequence of the following result, which itself is a special case of Theorem 5.10 in the text. Theorem 1.13 Let χ0 be any character of G0 such that χ0 · ω = 1 on A× F. Then, for any good CM point x of conductor P n with n sufficiently large, there exists a character χ ∈ P (n, χ0 ) such that a(x, χ) = 0. 1.5 Applications As we have already explained, the present work was firstly motivated by a desire to prove non-vanishing of L-functions and their derivatives. However, the results we prove on general CM points have independent applications to Iwasawa theory, even when they are not yet known to be related to L-functions. For instance, Theorem 1.10 implies directly that certain Euler systems of Bertolini-Darmon (when F = Q) and Howard (for general F ) are actually non-trivial. The non-triviality of these Euler systems is used by B. Howard in [17] to establish half of the relevant Main Conjecture, for the anti-cyclotomic Iwasawa theory of abelian varieties of GL(2)-type. It is also used by J. Nekov´aˇr in [21] to prove new cases of parity in the Bloch-Kato conjecture, for Galois representations attached to Hilbert modular newforms over F with trivial central character and parallel weight (2k, · · · , 2k), k ≥ 1. 1.6 Sketch of proof We want to briefly sketch the proof of our nontriviality theorems for CM points. The basic ideas are drawn from our previous papers [4, 26, 27] with a few simplifications and generalizations. × Thus, let χ0 be a character of G0 such that χ0 · ω = 1 on A× F , let H = R be the compact open subgroup of G(Af ) defined by (1.6), and let x be a CM point of conductor P n in CMH . We have defined a function ψ on CMH with values in a complex vector space, and we want to show that 1 def a(x, χ) = χ(σ)ψ(σ · x) |G(n)| σ∈G(n)

is nonzero for at least some χ ∈ P (n, χ0 ), provided that n is sufficiently large. The analysis of such sums proceeds in a series of reductions.

From G(n) to G0 To prove that a(x, χ) = 0 for some χ ∈ P (n, χ0 ), it suffices to show that the sum of these values is nonzero. A formal computation in the group algebra of

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G(n) shows that this sum, def

b(x, χ0 ) =

a(x, χ)

χ∈P (n,χ0 )

is given by b(x, χ0 ) =

1 χ0 (σ)ψ∗ (σ · x ˜) q |G0 | σ∈G0

where ψ∗ is the extension of ψ to Z[CMH ] and x ˜ = q · x − TrZ(n) (x) = x − σ · x. σ∈Z(n)

Here, Z(n) = Gal(K[P n ]/K[P n−1 ]) and q = |Z(n)| = |OF /P |.

Distribution relations To deal with x ˜, we have to use distribution relations and Hecke correspondences, much as in the case of F = Q treated in our previous works. However, there are numerous technicalities to overcome, owing to the fact that we are now working over a more general field, with automorphic forms that may have a nontrivial central character, and with a prime P that may divide the level. Although the necessary arguments are ultimately quite simple, the details are somewhat tedious, and we request forgiveness for what might seem to be a rather opaque digression. To avoid obscuring the main lines of the argument, we have banished the discussion of distribution relations to the appendices. Basically, these distribution relations will produce for us a level structure H + ⊂ H, a function ψ + on CMH + and a CM point x+ ∈ CMH + of conductor P n such that ∀σ ∈ Galab K : ×

ψ + (σ · x+ ) = ψ∗ (σ · x ˜).

In fact, H + = R+ for some OF -order R+ ⊂ B which agrees with R outside P , and is an Eichler order of level P max(δ,2) at P . Here we remind the reader that δ is defined in Definition 1.6. Note that this part of the proof is responsible for the goodness assumptions in our theorems. Indeed, the bad CM points simply do not seem to satisfy any distribution relations, and the above construction may therefore only be applied to a good CM point x. We also mention that this computation would not work with a more general function ψ: one needs ψ to be new at P , in some suitable sense.

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We now have b(x, χ0 ) =

1 χ0 (σ)ψ + (σ · x+ ). q |G0 | σ∈G0

Using the fact that χ0 · ω = 1 on G2 , we prove that 1 b(x, χ0 ) = χ0 (σ)ψ + (σ · x+ ). q |G0 /G2 | σ∈G0 /G2

Indeed, the map σ → χ0 (σ)ψ + (σ · x+ ) factors through G0 /G2 .

From G0 /G2 to G0 /G1 We can reduce the above sum to something even simpler. Indeed, it turns out that there is a subgroup G1 ⊂ G0 , containing G2 , such that the Galois action of the elements in G1 can be realized by geometric means. In fact, G1 is the maximal such subgroup, and G1 /G2 is generated by the classes of the frobeniuses in G(∞) of those primes of K which are ramified over F but do not divide P . More precisely, we construct yet another level structure H1+ ⊂ H + , a funcn tion ψ1+ on CMH + , and a CM point x+ 1 ∈ CMH1+ of conductor P , such 1 that ∀γ ∈ Galab ψ1+ (γ · x+ χ0 (σ)ψ + (σγ · x+ ). K : 1)= σ∈G1 /G2

This H1+ corresponds to an OF -order R1+ ⊂ B which only differs from R+ at those finite places v = P of F which ramify in K. This part of the proof is responsible for our general assumption that N and D are relatively prime. Indeed, to establish the above formula, we need to know that for all v = P that ramify in K, the local component Hv+ = Hv = R× v of H + is a maximal order in a split quaternion algebra. It seems likely that the case where Rv is an Eichler order of level v in a split algebra could still be handled by similar methods.

Dealing with G0 /G1 We finally obtain b(x, χ0 ) =

1 c(x+ 1 ) with q |G0 /G2 |

c(y) =

σ∈G0 /G1

χ0 (σ)ψ1+ (σ · y).

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We prove that c(y) = 0 for sufficiently many y’s in CMH + (P n ), n 0, using 1 a theorem of M. Ratner on uniform distribution of unipotent orbits on p-adic Lie groups. Just as in our previous work, we show that the elements of G0 /G1 act irrationally on the relevant CM points. Slightly more precisely, we prove that for y as above, the images r(z) of vectors of the form r(z) = (σ · z)σ∈G0 /G1 ∈ CMH + (P n )G0 /G1 1

are uniformly distributed in some appropriate space, as z runs through the Galois orbit of y, and n goes to infinity. In the indefinite case, r(z) is the vector of supersingular points in characteristic which is obtained by reducing the coordinates of r(z) at some suitable place of K[P ∞ ]. In the definite case, MH itself plays the role of the supersingular locus. The uniform distribution theorem implies that the image of the Galois orbit of y tends to be large, and it easily follows that c(y) is nonzero. We have chosen to present a more general variant of the uniform distribution property alluded to above in a separate paper [5], which is quoted here in propositions 4.17 and 5.6 (for respectively the indefinite and the definite case). Although it really is the kernel of our proof, one may consider [5] as a black box while reading this paper. On the other hand, we also provide a different proof for the indefinite case, based on a proven instance of the Andr´e-Oort conjecture, rather than Ratner’s theorem. Finally, the authors would like to thank Jan Nekov´aˇr and J. Milne for their help and encouragement.

1.7 Notations For any place v of F , Fv is the completion of F at v and OF,v is its ring of integers (if v is finite). If E is a vector field over F , such as K or B, we put Ev = E ⊗F Fv . More generally, if R is a module over the ring of integers OF of F , we put Rv = R ⊗OF OF,v . We denote by A (resp. Af ) the ring of adeles (resp. finite adeles) of Q, so that A = Af × R, and Af is the restricted product of the Qv ’s with respect to the Zv ’s. We put AF = A ⊗Q F and AK = A ⊗Q K. For the finite adeles, we write F = Af ⊗Q F and = Af ⊗Q K. Thus F = O F ⊗ Q where M = M ⊗ Z denotes the profinite K completion of a finitely generated Z-module M . For any affine algebraic group G/Q, we topologize G(A) = G(Af ) × G(R) in the usual way. When G is the Weil restriction G = ResF/Q G of an algebraic group G /F , we denote by gv ∈ G (Fv ) the v-component of g ∈ G(A) = G (AF ) (or G(Af ) = G (F )), and we identify G (Fv ) with the subgroup {g ∈ G(A); ∀w = v, gw = 1} of G(A) (or G(Af )).

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We put GalF = Gal(F /F ) and GalK = Gal(F /K) where F is a fixed algebraic closure of F containing K. We denote by F ab and K ab the maximal ab abelian extensions of F and K inside F , with Galois groups Galab F and GalK . N (v) We denote by Frobv the geometric Frobenius at v (the inverse of x → x ) and normalize the reciprocity map ab recF : A× F → GalF

and

ab recK : A× K → GalK

accordingly. 2 The Galois group of K[P ∞ ]/K Fix a prime P of F with residue field F = OF /P of characteristic p and order q = |F|. We have assembled here the basic facts we need pertaining to the infinite abelian extension K[P ∞ ] = ∪n≥0 K[P n ] of K. Recall that and G(∞) = Gal(K[P ∞ ]/K) = lim G(n). ←− The first section describes G(∞) as a topological group: it is an extension of a free Zp -module of rank [FP : Qp ] by a finite group G0 , the torsion subgroup of G(∞). The second section defines a filtration G(n) = Gal(K[P n ]/K)

{1} ⊂ G2 ⊂ G1 ⊂ G0 which plays a crucial role in the proof (and statement) of our main results. Finally, the third section gives an explicit formula for a certain idempotent in the group algebra of G(n).

2.1 The structure of G(∞) Lemma 2.1 The reciprocity map induces an isomorphism of topological × /K × U and G(∞) where groups between K ×n = {λ ∈ O × , λP ∈ O× }. U = ∩O P K F,P Proof We have to show that the natural continuous map × /K × U → lim K × /K × O ×n φ:K P ←− is an isomorphism of topological groups. Put ×n /K × U O×n /O×n O× Xn = K × O P P ,P P F,P so that ker(φ) = lim Xn and coker(φ) = lim(1) Xn . Note that (OP×n )n≥0 ←− ←− × is a decreasing sequence of subgroups of OK with ∩n≥0 OP×n = OF× : since

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× × OF× has finite index in OK , OP×n = OF× and Xn = OP×n ,P /OF,P for all

n 0. It follows that lim Xn = lim(1) Xn = {1}, so that φ is indeed a group ←− ←− ×n = U for n 0. In particular, isomorphism. This also shows that K × U ∩ O P × × . Being a separated K U is a locally closed, hence closed subgroup of K × /K × , K × /K × U is also compact. Being a quotient of the compact group K continuous bijection between compact spaces, φ is an homeomorphism. It easily follows that the open subgroup Gal(K[P ∞ ]/K[1]) of G(∞) is iso× × × × × × morphic to OK,P /OK OF,P . Since OK /OF× is finite and OK,P /OF,P contains [FP :Qp ]

an open subgroup topologically isomorphic to Zp profinite groups implies that

, a classical result on

Corollary 2.2 The torsion subgroup G0 of G(∞) is finite and G(∞)/G0 is [F :Q ] topologically isomorphic to Zp P p .

2.2 A filtration of G0

Let G(∞) be the subgroup of G(∞) which is generated by the Frobeniuses of those primes of K which are not above P (these primes are unramified in K[P n ] for all n ≥ 0). In particular, G(∞) is a countable but dense subgroup of G(∞). Lemma 2.3 The reciprocity map induces topological isomorphisms × )P /(S −1 OF )× (O × )P (K K and

KP× /K × FP×

−→ G(∞)

−→ G(∞)/G(∞) .

×. Here: S = OF − P and X P = {λ ∈ X, λP = 1} for X ⊂ K × )P ) in Proof Class field theory tells us that G(∞) is the image of recK ((K G(∞). Both statements thus follow from Lemma 2.1. K ∩ K × yields an isomorphism between (K × )P /(O × )P The map λ → λO K P and the group IK of all fractional ideals of K which are relatively prime to P . × )P /(O × )P to the group P P of those ideThis bijection maps (S −1 OF )× (O F K K P als in IK which are principal and generated by an element of F × – which then necessarily belongs to (S −1 OF )× . We thus obtain a perhaps more enlightenP ing description of G(∞) : it is isomorphic to IK /PFP . The isomorphism sends the class of a prime Q P of K to its Frobenius in G(∞) . Definition 2.4 We denote by G1 ⊂ G0 the torsion subgroup of G(∞) .

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P There is an obvious finite subgroup in IK /PFP . Indeed, let IFP be the group of all fractional ideals J in K for which J = OK I for some fractional ideal I of P F relatively prime to P . Then PFP ⊂ IFP ⊂ IK and IFP /PFP is finite. In fact,

× )P F × /F × O × Pic(OF ). IFP /PFP (F× )P /(S −1 OF )× (O F F Definition 2.5 We denote by G2 Pic(OF ) the corresponding subgroup of G1 . Note that G2 is simply the image of recK (F× ) in G(∞) and the isomor× is induced by the reciprocity phism between G2 and Pic(OF ) F× /F × O F map of K. By definition, G1 /G2 is isomorphic to the torsion subgroup of × )P /(F× O × )P I P /I P . We thus obtain: (K K F K Lemma 2.6 G1 /G2 is an F2 -vector space with basis {σQ mod G2 ; Q | D } where D is the squarefree product of those primes Q = P of F which ramify in K, and σQ = FrobQ ∈ G1 with Q2 = QOK . In particular, G1 /G2 = {σD mod G2 ; D | D } where σD =

Q|D

σQ for D | D .

The following lemma is an easy consequence of the above discussion. Lemma 2.7 Let Q = P be a prime of OF which does not split in K and let Q be the unique prime of OK above Q. Then the decomposition subgroup of Q in G(∞) is finite. More precisely, it is a subgroup of G2 if Q = QOK and a subgroup of G1 not contained in G2 if Q2 = QOK .

2.3 A formula ×

Let χ0 : G0 → C be a fixed character of G0 . For n > 0, we say that a character χ : G(n) → C× is primitive if it does not factor through G(n − 1). We denote by P (χ0 , n) the set of primitive characters of G(n) inducing χ0 on G0 and let e(χ0 , n) be the sum of the orthogonal idempotents def

eχ =

1 χ(σ) · σ ∈ C[G(n)], |G(n)|

χ ∈ P (χ0 , n).

σ∈G(n)

Note that e(χ0 , n) is yet another idempotent in C[G(n)].

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Lemma 2.8 For n 0, we may identify G0 with its image G0 (n) in G(n) and # $ 1 e(χ0 , n) = · q − TrZ(n) · χ0 (σ) · σ in C[G(n)]. q |G0 | σ∈G0

def

Here, TrZ(n) =

def

σ∈Z(n)

σ with Z(n) = Gal(K[P n ]/K[P n−1 ]).

Proof We denote by G∨ the group of characters of a given G. Write e(χ0 , n) = eσ (χ0 , n) · σ ∈ C[G(n)] σ∈G(n)

and put H(n) = G(n)/G0 (n). If n is sufficiently large, (1) G0 G0 (n) is an isomorphism, (2) G(n) H(n) induces an isomorphism from Z(n) = ker (G(n) G(n − 1)) to the kernel of H(n) H(n−1), and (3) the kernel X(n) of G(n) → H(n − 1) is the direct sum of G0 (n) and Z(n) in G(n). In particular, there exists an element χ0 ∈ G(n)∨ inducing χ0 on G0 (n) G0 and 1 on Z(n), so that P (χ0 , n) = H(n)∨ χ0 − H(n − 1)∨ χ0 = (H(n)∨ − H(n − 1)∨ ) χ0 . For σ ∈ G(n), we thus obtain ⎛ |G(n)| · eσ (χ0 , n) = ⎝

χ(σ) −

χ∈H(n)∨

⎧ if ⎨ 0 = − |H(n − 1)| if ⎩ |H(n)| − |H(n − 1)| if

⎞ χ(σ)⎠ · χ0 (σ)

χ∈H(n−1)∨

⎫ σ∈ / X(n) ⎬ · χ (σ). σ ∈ X(n) G0 (n) ⎭ 0 σ ∈ G0 (n)

Since X(n) = G0 (n) ⊕ Z(n) with χ0 = χ0 on G0 (n) and 1 on Z(n), |G(n)| · e(χ0 , n) = |G(n)| · eστ (χ0 , n) · στ σ∈G0 (n) τ ∈Z(n)

=

# $ |H(n)| − |H(n − 1)| · TrZ(n) · χ0 (σ)σ.

σ∈G0

This is our formula. Indeed, |G(n)| = |G0 | |H(n)| ,

|H(n)| = |Z(n)| |H(n − 1)| ,

and |Z(n)| = |F| = q by Lemma 2.9 below. The reciprocity map induces an isomorphism between ×n−1 /K × O ×n O×n−1 /O×n−1 O×n K×O P P ,P P P ,P P

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and Z(n). For n 0, OP×n−1 = OF× is contained in OP×n ,P , so that Z(n) OP×n−1 ,P /OP×n ,P . On the other hand, for any n ≥ 1, the F-algebra OP n /P OP n is isomorphic to F[] = F[X]/X 2 F[X], and the projection OP n ,P → OP n ,P /P OP n ,P OP n /P OP n induces an isomorphism between OP×n ,P /OP×n+1 ,P and F[]× /F× {1 + α; α ∈ F}. We thus obtain: Lemma 2.9 For n 0, Z(n) F as a group.

3 Shimura Curves Let F be a totally real number field. To each finite set S of finite places of F such that |S| + [F : Q] is odd, we may attach a collection of Shimura curves over F . If K is a totally imaginary quadratic extension of F in which the primes of S do not split, these curves are provided with a systematic supply of CM points defined over the maximal abelian extension K ab of K. As explained in the introduction, our aim in this paper (for the indefinite case) is to prove the non-triviality of certain cycles supported on these points. This section provides some of the necessary background on Shimura curves, with [2] as our main reference. Further topics are discussed in Section 3.1 of [5]. To simplify the exposition, we require S to be nonempty if F = Q. This rules out precisely the case where our Shimura curves are the classical modular curves over Q. This assumption implies that our Shimura curves are complete – there are no cusps to be added, but then also no obvious way to embed the curves into their Jacobians. We note however that everything works with the obvious modifications in the non-compact case as well – see [4] and [27], as well as [5].

3.1 Shimura curves Let {τ1 , · · · , τd } = HomQ (F, R) be the set of real embeddings of F . We shall always view F as a subfield of R (or C) through τ1 . Let B be a quaternion algebra over F which ramifies precisely at S ∪ {τ2 , · · · , τd }, a finite set of even order. Let G be the reductive group over Q whose set of points on a commutative Q-algebra A is given by G(A) = (B ⊗ A)× . Let Z be the center of G. In particular, GR G1 × · · · × Gd where Bτi = B ⊗F,τi R and Gi is the algebraic group over R whose set of points on a commutative R-algebra A is given by Gi (A) = (Bτi ⊗R A)× . Fix ∈ {±1} and let X be the G(R)def conjugacy class of the morphism from S = ResC/R (Gm,C ) to GR which

Nontriviality of Rankin-Selberg L-functions maps z = x + iy ∈ S(R) = C× to - . x y , 1, · · · , 1 ∈ G1 (R) × · · · × Gd (R) G(R). −y x

143

(3.1)

We have used an isomorphism of R-algebras Bτ1 M2 (R) to identify G1 and GL2 /R; the resulting conjugacy class X does not depend upon this choice, but it does depend on , cf. Section 3.3.1 of [5] and Remark 3.1 below. It is well-known that X carries a complex structure for which the left action of G(R) is holomorphic. For every compact open subgroup H of G(Af ), the quotient of G(Af )/H × X by the diagonal left action of G(Q) is a compact Riemann surface def

an MH = G(Q)\ (G(Af )/H × X) . an The Shimura curve MH is Shimura’s canonical model for MH . It is a proper and smooth curve over F (the reflex field) whose underlying Riemann surface an MH (C) equals MH .

Remark 3.1 With notations as above, let h : S → GR be the morphism defined by (3.1). There are G(R)-equivariant diffeomorphisms X ghg−1

←− G(R)/H∞ ←− g

−→ C R −→ g · i

where H∞

= StabG(R) (h) = StabG(R) (±i) = R× SO2 (R) × G2 (R) × · · · × Gd (R)

with on CR through its first component G1 (R) GL2 (R) by # a b $G(R) acting aλ+b −1 ·λ c d · λ = cλ+d . With these conventions, the derivative of λ → gh(z)g × at λ = g · i equals z/z (for g ∈ G(R), λ ∈ C R and z ∈ C = S(R)). In other words, the above bijection between X and C R is an holomorphic diffeomorphism. This computation shows that the Shimura curves of the introduction, which are also those considered in [33] or [17], correspond to the case where = 1. On the other hand, Carayol explicitly works with the = −1 case in our main reference [2].

3.2 Connected components We denote by c

MH −→ MH → Spec(F )

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the Stein factorization of the structural morphism MH → Spec(F ), so that def

MH = Spec Γ(MH , OMH ) is a finite e´ tale F -scheme and the F -morphism c : MH → MH is proper and smooth with geometrically connected fibers. In particular, an MH (F ) π0 (MH ×F F ) π0 (MH (C)) π0 (MH ).

(3.2)

Remark 3.2 (1) In the notations of Remark 3.1, let X + be the connected component of h in X, so that X + = G(R)+ · h H where G(R)+ is the def def neutral component of G(R) and H = {λ ∈ C; ·(λ) > 0}. Put G(Q)+ = an G(R)+ ∩ G(Q). Then π0 (MH ) G(Q)+ \G(Af )/H, corresponding to the decomposition ) ) + an ←− −→ MH α Γα \H α Γα \X (3.3) −1 + [g · i] ∈ Γα \H ←− [x = ghg ] ∈ Γα \X −→ [(α, x)] where α ∈ G(Af ) runs through a set of representatives of the finite set G(Q)+ \G(Af )/H, Γα is the discrete subgroup αHα−1 ∩ G(Q)+ of G(R)+ + and Γα ⊂ PGL+ → 2 (R) is its image through the obvious map G(R) + + GL2 (R) → PGL2 (R). (2) The strong approximation theorem [29, p. 81] and the norm theorem [29, × → F× induces a bijection p. 80] imply that the reduced norm nr : B

an π0 (MH ) G(Q)+ \G(Af )/H −→ Z(Q)+ \Z(Af )/nr(H)

(3.4)

where Z(Q)+ = nr(G(Q)+ ) is the subgroup of totally positive elements in Z(Q) = F × . Using also (3.2), we obtain a left action of GalF on the RHS of (3.4). The general theory of Shimura varieties implies that the latter action factors through Galab F , where it is given by the following reciprocity law (see Lemma 3.12 in [5]): for λ ∈ F× , the element σ = recF (λ) of Galab F acts on the RHS of (3.4) as multiplication by λ . In particular, this action is transitive and MH is therefore a connected F -curve (although not a geometrically connected one).

3.3 Related group schemes The Jacobian JH of MH is the identity component of the relative Picard scheme PH of MH → Spec(F ) and the N´eron-Severi group NSH of MH is the quotient of PH by JH . By [15], JH is an abelian scheme over F while NSH is a “separable discrete” F -group scheme. The canonical isomorphism [14, V.6.1] of F -group schemes

PH −→ ResMH /F (PicMH /MH )

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145

induces an isomorphism between JH and ResMH /F (Pic0MH /MH ), so that NSH −→ ResMH /F PicMH /MH /Pic0MH /MH −→ ResMH /F (Z) where we have identified PicMH /MH /Pic0MH /MH with the constant MH group scheme Z using the degree map degH : PicMH /MH → Z. We denote by degH : PH → NSH = PH /JH the quotient map, so that degH = ResMH /F (degH ) under the above identifications. Remark 3.3 If MH (F ) = {sα } with sα : Spec(F ) → MH , ) MH ×F Spec(F ) = C , PH ×F Spec(F ) α α JH ×F Spec(F ) = J , and NS α H ×F Spec(F ) α

= =

P , α α α Zα

where Cα = c−1 (sα ), Pα = Pic(Cα ), Jα = Pic0 (Cα ) and Zα = s∗α (Z) is isomorphic to Z over F . With these identifications, degH maps (pα ) ∈ α Pα to (deg(pα )) ∈ α Z.

3.4 Hecke operators As H varies among the compact open subgroups of G(Af ), the Shimura curves {MH }H form a projective system with finite flat transition maps which is equipped with a “continuous” right action of G(Af ). Specifically, for any element g ∈ G(Af ) and for any compact open subgroups H1 and H2 of G(Af ) such that g −1 H1 g ⊂ H2 , multiplication on the right by g in G(Af ) an an defines a map MH → MH which descends to a finite flat F -morphism 1 2 [·g] = [·g]H1 ,H2 : MH1 → MH2 . We shall refer to such a map as the degeneracy map induced by g. Letting H1 and H2 vary, these degeneracy maps together define an automorphism [·g] of lim{MH }H . ←− There is a natural left action of the Hecke algebra def

TH = EndZ[G(Af )] (Z[G(Af )/H]) Z[H\G(Af )/H] on PH , JH and NSH . With the ring structure induced by its representation as an endomorphism algebra, TH is the opposite of the most frequently encountered Hecke algebra: for α ∈ G(Af ), the Hecke operator TH (α) ∈ TH

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corresponding to the double class HαH acts by TH (α) = f∗ ◦ [·α]∗ ◦ f ∗ where f and f are the obvious transition maps in the following diagram

MH

MH∩αHα−1 r f rrr r r r y rr r

[·α]

/ Mα−1 Hα∩H LLL LLLf LLL L% TH (α) / MH

The degree of TH (α) is the degree of f , namely the index of H ∩ αHα−1 in an H.1 On the level of divisors, TH (α) maps x = [g, h] ∈ MH to an TH (α)(x) = [gαi , h] ∈ DivMH ) where HαH = αi H. If α belongs to the center of G(Af ), then [·α] : MH → MH is an automorphism of MH /F and TH (α) = [·α]∗ . Definition 3.4 We denote by θM and θJ the induced left action of Z(Af ) on MH and JH : θM (α) = [·α] and θJ (α) = [·α]∗ . These actions factor through × for some OF -order R ⊂ B, Z(Q)\Z(Af )/Z(Af ) ∩ H. When H = R × and we thus obtain left actions Z(Af ) ∩ H = O F θM : Pic(OF ) → AutF MH

and

θJ : Pic(OF ) → AutF JH .

3.5 The Hodge class and the Hodge embedding Let S be a scheme. To any commutative group scheme G over S, and indeed to any presheaf of abelian groups G on the category of S-schemes, we may attach a presheaf of Q-vector spaces G ⊗ Q by the following rule: for any S-scheme def X, G ⊗ Q(X) = G(X) ⊗ Q. To distinguish between the sections of G(X) and the sections of G ⊗ Q(X), we write X → G for the former and X G for the latter, but we will refer to both kind of sections as morphisms. This construction is functorial in the sense that given two commutative group schemes G1 and G2 over S, any element α of def

Hom0S (G1 , G2 ) = HomS (G1 , G2 ) ⊗ Q defines a morphism α : G1 G2 : choose n ≥ 1 such that nα = α0 ⊗ 1 for some α0 ∈ HomS (G1 , G2 ), set α(f ) = α0 (f ) ⊗ n1 ∈ G2 (X) ⊗ Q for f ∈ G1 (X) and extend by linearity to G1 (X) ⊗ Q. The resulting morphism does 1 In general, the degree of MH → MH equals [HZ(Q) : H Z(Q)] = [H : H ] · [H ∩ Z(Q) : H ∩ Z(Q)]. But here H = H ∩ αHα−1 , so that H ∩ Z(Q) H ∩ Z(Q).

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not depend upon the choice of n and α0 and furthermore satisfies α(λ1 f1 + λ2 f2 ) = λ1 α(f1 ) + λ2 α(f2 ) in G2 ⊗ Q(X) for any λ1 , λ2 ∈ Q and f1 , f2 : X G1 . This said, the “Hodge embedding” is a morphism ιH : MH JH over F which we shall now define. To start with, consider the F -morphism (·)H : MH → PH which for any F -scheme X maps x ∈ MH (X) to the element (x) ∈ PH (X) which is represented by the effective relative Cartier divisor on MH ×F X defined by x (viewing x as a section of MH ×F X → X). Our morphism ιH is the composite of this map with a retraction PH JH of JH → PH . Defining the latter amounts to defining a section NSH PH of degH : PH NSH and since degH = ResMH /F (degH ), we may as well search for a section Z PicMH /MH of degH : PicMH /MH Z. In other word, we now want to construct an element (the Hodge class) δH ∈ PicMH /MH (MH ) ⊗ Q = Pic(MH ) ⊗ Q such that deg(δH ) = 1 in Z(MH ) ⊗ Q = Q (recall from Remark 3.2 that MH is connected). The resulting morphism ιH : MH JH will thus be given by ιH (x) = (x)H − sH ◦ degH (x)H

in JH ⊗ Q(X)

for any F -scheme X and x ∈ MH (X), with sH : NSH PH defined by Z PicMH /MH def sH = ResMH /F . n → n · δH We may now proceed to the definition of the Hodge class δH . For a compact open subgroup H of H, the ramification divisor of the transition map f = TH ,H : MH → MH is defined by def RH /H = lengthOM ,x (Ωf )x · x in Div(MH ) H

x

where x runs through the finitely many closed points of MH in the support of the sheaf of relative differentials Ωf = ΩMH /MH . The branch divisor is the flat push-out of RH /H : def

BH /H = f∗ RH /H

in Div(MH ).

When H is a normal subgroup of H, MH = MH /H and the branch divisor pulls-back to f ∗ BH /H = deg(f ) · RH /H . For H ⊂ H ⊂ H, f = TH ,H , g = TH ,H and h = f ◦ g = TH ,H , 0 → g ∗ Ωf → Ωh → Ωg → 0

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is an exact sequence of coherent sheaves on MH (this may be proven using Proposition 2.1 of [16, Chapter IV], the flatness of g and the snake lemma). This exact sequence shows that RH /H and BH /H

= =

RH /H + g ∗ RH /H f∗ BH /H + deg(g) · BH /H

in Div(MH ) in Div(MH ).

(3.5)

If H is sufficiently small, RH /H = 0 and BH /H = 0 for any H ⊂ H (see for instance [2, Corollaire 1.4.1.3]). In particular, 1 BH /H ∈ Div(MH ) ⊗ Q deg(f )

def

BH =

does not depend upon H , provided that H is sufficiently small. When H itself is sufficiently small, BH = 0. In general: Lemma 3.5 f ∗ BH = BH + RH /H in Div(MH ) ⊗ Q. Proof Let H be a sufficiently small normal subgroup of H contained in H . With notations as above, g ∗ : Div(MH ) → Div(MH ) is injective, g ∗ f ∗ BH g ∗ BH

and

= =

1 h∗ h∗ RH /H deg(h) 1 ∗ deg(g) g g∗ RH /H

= RH /H , = RH /H .

The lemma thus follows from (3.5). On the other hand, Hurwitz’s formula [16, IV, Prop. 2.3] tells us that KH = f ∗ KH + class of RH /H

in Pic(MH )

where KH is the canonical class on MH , namely the class of ΩMH /F . It follows that f ∗ (KH + BH ) = KH + BH f∗ (KH + BH ) = deg(f ) · (KH + BH )

in Pic(MH ) ⊗ Q, in Pic(MH ) ⊗ Q.

(3.6)

If H is sufficiently small, BH = 0 and deg(KH ) > 0. The above formulae therefore imply that deg(KH + BH ) > 0 for any H, and we may thus define 1

def

δH =

deg(KH + BH )

· (KH + BH ) ∈ Pic(MH ) ⊗ Q.

By construction: deg(δH ) = 1, and

f ∗ δH f∗ δH

= deg(f ) · δH = δH

in Pic(MH ) ⊗ Q in Pic(MH ) ⊗ Q.

Lemma 3.6 For any α ∈ G(Af ), TH (α)(δH ) = deg(TH (α)) · δH .

(3.7)

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149

Proof Given the definition of TH (α), this follows from (3.7) once we know that for any H and α, [·α]∗ δH = δα−1 Hα in Pic(Mα−1 Hα ) ⊗ Q. This is obvious if H is sufficiently small and the general case follows, using (3.7) again. Remark 3.7 Let dH be the smallest positive integer such that dH BH belongs 0 to Div(MH ) ⊂ Div(MH ) ⊗ Q and put δH = dH · KH + class of dH BH 0 in Pic(MH ). The proof just given shows that the relation TH (α)(δH ) = 0 deg(TH (α)) · δH already holds in Pic(MH ). With notations $ as in Remark 3.2, the restriction of BH to Γα \H equals # −1 1 − e · x where x runs through a set of representatives of Γα \H in x x H and ex is the order of its stabilizer in Γα ⊂ PGL+ 2 (R). Compare with [11, section 23]. Thus, dH is the largest common multiple of the ex ’s. Remark 3.8 With the notations of Remark 3.3, let δα ∈ Pic(Cα ) ⊗ Q be the restriction to Cα of the pull-back of δH to Pic(MH ×F Spec(F )) ⊗ Q = Pic(Cα ) ⊗ Q. α

Then deg(δα ) = 1 and the restriction of ιH to Cα maps x ∈ Cα (F ) to (x) − δα ∈ Jα (F ) ⊗ Q. In particular, the image of ιH on MH (F ) spans JH (F ) ⊗ Q over Q. The terminology Hodge Class is due to S. Zhang. In his generalization of the Gross-Zagier formulae to the case of Shimura curves, the morphism ιH : MH JH plays the role of the embedding x → (x) − (∞) of a classical modular curve into its Jacobian. This is why we refer to ιH as the Hodge “embedding”. It is a finite morphism, in the sense that some nonzero multiple nιH of ιH is a genuine finite morphism from MH to JH . More generally: Lemma 3.9 Let π : JH A be a nonzero morphism of abelian varieties over F . Then α = π ◦ ιH : MH A is finite (in the above sense). Proof We may assume that π : JH → A and α : MH → A are genuine morphisms. Since MH is a connected complete curve over F , α is either finite or constant, and it can not be constant by Remark 3.8.

3.6 Differentials and automorphic forms Let ΩH = ΩMH /F be the sheaf of differentials on MH and denote by Ωan H the an pull-back of ΩH to MH , so that Ωan is the sheaf of holomorphic 1-forms on H

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an MH . The right action of G(Af ) on the projective system {MH }H induces a C-linear left action of G(Af ) on the inductive system {Γ(Ωan H )}H of global sections of these sheaves. We want to identify lim Γ(Ωan ), together with its H −→ G(Af )-action, with a suitable space S2 of automorphic forms on G. Fix an isomorphism G(R) GL2 (R) × G2 (R) × · · · × Gd (R) as in section 3.1 and let S2 be the complex vector space of all functions φ : G(A) = G(Af ) × G(R) → C with the following properties:

P1 P2 P3 P4

φ is left G(Q)-invariant. φ is right invariant under R∗ × G2 (R) × · · · × Gd (R) ⊂ G(R). φ is right invariant under some compact open subgroup of G(Af ). For every g ∈ G(A) and θ ∈ R, cos(θ) sin(θ) φ g , 1, · · · , 1 = exp(2iθ)φ(g). − sin(θ) cos(θ)

P5 For every g ∈ G(A), the function

1 y z = x + iy → φ(g, z) = φ g × 0 y def

x 1

, 1, · · · , 1

is holomorphic on H . There is a left action of G(Af ) on S2 given by (g · φ)(x) = φ(xg). Proposition 3.10 There is a G(Af )-equivariant bijection ∼

lim Γ(Ωan H ) −→ S2 −→ H which identifies Γ(Ωan H ) with S2 .

Proof Recall from Remark 3.2 that there is a decomposition ) an MH = α Γα \H where α runs through a set of representatives of G(Q)+ \G(Af )/H and Γα is the image of Γα = αHα−1 ∩ G(Q)+ in PGL+ 2 (R). an Let ω ∈ Γ(Ωan ) be a global holomorphic 1-form on MH . The restriction H an of ω to the connected component Γα \H of MH pulls back to a Γα -invariant holomorphic form on H . The latter equals fα (z)dz for some holomorphic function fα on H such that fα |γ = fα for all γ ∈ Γα , where az + b −2 (f |γ)(z) = det(γ)(cz + d) f cz + d

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151

# $ for a function f on H and γ = ac db in GL+ 2 (R). Since # $ ) G(A) = α G(Q) · αH × G(R)+ , + we may write any element g ∈ G(A) as a product g = gQ (αh × gR ) for some α with gQ ∈ G(Q), h ∈ H and + gR ∈ G(R)+ = GL+ 2 (R) × G2 (R) × · · · × Gd (R). + + + We put φω (g) = fα |gR,1 (i) where gR,1 is the first component of gR . If −1

g also equals gQ (α h × g R ), then α = α and g Q gQ belongs to G+ (Q) ∩ −1 αHα = Γα , so that +

−1

+ + fα |g R,1 = fα |g Q gQ gR,1 = fα |gR,1 . +

It follows that φω is a well-defined complex valued function on G(A). We leave it to the reader to check that φω belongs to S2H . Conversely, any Hinvariant element φ in S2 defines an holomorphic differential 1-form ωφ on an MH : with notations as in P5, the restriction of ωφ to Γα \H pulls back to φ(α, z)dz on H . Remark 3.11 The complex cotangent space of JH at 0 is canonically H isomorphic to Γ(Ωan H ), and therefore also to S2 . With these identifications, the right action of TH = EndZ[G(Af )] (Z[G(Af )/H]) on S2H HomZ[G(Af )] (Z[G(Af )/H], S2 ) coincides with the right action induced on the cotangent space by the left action of TH on JH . Remark 3.12 When = 1, S2 is exactly the subspace of S which is defined in the introduction, given our choice of an isomorphism between G1 and GL2 (R) (cf. Remark 1.7 and Definition 1.8). This follows from the relevant properties of lowest weight vectors, much as in the classical case [6, Section 11.5].

3.7 The P -new quotient Let P be a prime of F where B is split, and consider a compact open subgroup H of G(Af ) which decomposes as H = H P RP× , where RP is an Eichler order in BP M2 (FP ) and H P is a compact open subgroup of G(Af )P = {g ∈ G(Af ); gP = 1}. Definition 3.13 The P -new quotient of JH is the largest quotient π : JH P −new JH of JH such that for any Eichler order RP ⊂ BP strictly containing × RP , π ◦ f ∗ = 0 where H = H P R P and f ∗ : JH → JH is the morphism induced by the degeneracy map f : MH → MH .

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P −new The quotient map π : JH JH induces an embedding from the complex P −new cotangent space of JH at 0 into the complex cotangent space of JH at 0. Identifying the latter space first with Γ(Ωan H ) and then with the space of Hinvariant elements in S2 (Proposition 3.10), we obtain the P -new subspace H H S2,P −new of S2 . By construction, H H S2H = S2,P −new ⊕ S2,P −old ×

H H where S2,P −old is the subspace of S2 spanned by the elements fixed by R P for some Eichler order R P ⊂ BP strictly containing RP .

3.8 CM Points Let K be a totally imaginary quadratic extension of F and put T = ResK/Q (Gm,K ). Any ring homomorphism K → B induces an embedding T → G. A morphism h : S → GR in X is said to have complex multiplication by K if it factors through the morphism TR → GR which is induced by an F -algebra homomorphism K → B. For a compact open subgroup H of an G(Af ), we say that x ∈ MH (C) is a CM point if x = [g, h] ∈ MH for some g ∈ G(Af ) and h ∈ X with complex multiplication by K. We assume that K splits B, which amounts to requiring that Kv is a field for every finite place v of F where B ramifies. Then, there exists an F -algebra homomorphism K → B, and any two such homomorphisms are conjugated by an element of B × = G(Q). We fix such an homomorphism and let T → G be the induced morphism. In each of the two connected components of X, there is exactly one morphism S → GR which factors through TR → GR . These two morphisms are permuted by the normalizer of T (Q) in G(Q), and T (Q) is their common stabilizer in G(Q). They correspond respectively to z ∈ S → (z or z¯, 1, · · · , 1) ∈ T1 × · · · × Td TR where Ki = K ⊗F,τi R, Ti = ResKi /R (Gm,Ki ) for 1 ≤ i ≤ d, and where we have chosen an extension τ1 : K → C of τ1 : F → R to identify K1 with C and T1 with S. We choose τ1 in such a way that the morphism hK : S → GR corresponding to z → (z, 1, · · · 1) belongs to the connected component of the morphism h : S → GR which is defined by (3.1). The map an g ∈ G(Af ) → [g, hK ] ∈ MH = G(Q)\ (G(Af )/H × X) an then induces a bijection between the set of K-CM points in MH and the set CMH = T (Q)\G(Af )/H of the introduction. In the sequel, we will use

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this identification without any further reference. In particular, we denote by an [g] ∈ MH = MH (C) the CM point corresponding to [g, hK ]. By Shimura’s theory, the CM points are algebraic and defined over the maximal abelian extension K ab of K. Moreover, for λ ∈ T (Af ) and g ∈ G(Af ), the action of σ = recK (λ) ∈ Galab K on x = [g] ∈ CMH is given by the following reciprocity law (viewing K as a subfield of C through τ1 ): σ · x = [λ g] ∈ CMH . We refer to Sections 3.1.2 and 3.2.1 of [5] for a more detailed discussion of CM points on Shimura curves, including a proof of the above facts.

4 The Indefinite Case To the data of F , B, K (and ), we have attached a collection of Shimura curves {MH } equipped with a systematic supply of CM points defined over the maximal abelian extension K ab of K. We now also fix a prime P of F where B is split, and restrict our attention to the CM points of P -power conductor in × for some OF -order R ⊂ B. a given Shimura curve M = MH , with H = R We assume that (H1) RP is an Eichler order in BP M2 (FP ). (H2) For any prime Q = P of F which ramifies in K, B is split at Q and RQ is a maximal order in BQ M2 (FQ ). We put J = JH , CM = CMH , ι = ιH and so on. . . We denote by CM(P n ) ⊂ M (K[P n ]) the set of CM points of conductor P n and put CM(P ∞ ) = ∪n≥0 CM(P n ) ⊂ M (K[P ∞ ]). Thanks to (H1), we have the notion of good CM points, as defined in the introduction. Recall that all CM points are good when RP is maximal; otherwise, the good CM points are those which are of type I or II, in the terminology of section 6. In this section, we study the contribution of these points to the growth of the Mordell-Weil groups of suitable quotients A of J, as one ascends the abelian extension K[P ∞ ] of K.

4.1 Statement of the main results We say that π : J A is a surjective morphism if some nonzero multiple of π is a genuine surjective morphism J → A of abelian varieties over F . We

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say that π is P -new if it factors through the P -new quotient of J, cf. Definition 3.13. We say that π is Pic(OF )-equivariant if A is endowed with an action θA of Pic(OF ) such that ∀σ ∈ Pic(OF ) :

θA (σ) ◦ π = π ◦ θJ (σ) in Hom0 (J, A),

cf. Definition 3.4. def We put C = Z[Pic(OF )]. For a character ω : Pic(OF ) → C× , we let Q{ω} be the image of the induced morphism ω∗ : C ⊗ Q → C. Then Q{ω} only depends upon the Aut(C)-conjugacy class {ω} of ω, and C ⊗ Q 0 {ω} Q{ω}. If A is endowed with an action of Pic(OF ), then End (A) is a C ⊗ Q-algebra and we write A ⊕{ω} A{ω} for the corresponding decomposition in the category Ab0F of abelian varieties over F up to isogenies. If A = A{ω} for some ω, then C acts on A through its quotient Z{ω}, the image of C in Q{ω}. Recall from section 2 that the torsion subgroup G0 of G(∞) = Gal(K[P ∞ ]/K) contains a subgroup G2 which is canonically isomorphic to Pic(OF ). For a character χ of G(∞) or G0 , we denote by Res(χ) the induced character on Pic(OF ). For a character χ of G(n), we denote by eχ ∈ C[G(n)] the idempotent of χ. We say that χ is primitive if it does not factor through G(n − 1). Theorem 4.1 Suppose that π : J A is a surjective, Pic(OF )-equivariant and P -new morphism. Fix a character χ0 of G0 such that A{ω} = 0 where ω = Res(χ0 ). Then: for any n 0 and any good CM point x ∈ CM(P n ), there exists a primitive character χ of G(n) inducing χ0 on G0 such that eχ α(x) = 0 in A ⊗ C, where α is defined in Lemma 3.9. Replacing π by π{ω} : J A A{ω} and using Lemma 4.6 below, one easily checks that Theorem 4.1 is in fact equivalent to the following variant, in which we use ω to embed Z{ω} into C. Theorem 4.2 Suppose that π : J A is a surjective, Pic(OF )-equivariant and P -new morphism. Suppose also that A = A{ω} = 0 for some character ω of Pic(OF ). Fix a character χ0 of G0 inducing ω on G2 Pic(OF ). Then: for any n 0 and any good CM point x ∈ CM(P n ), there exists a primitive character χ of G(n) inducing χ0 on G0 such that eχ α(x) = 0 in A ⊗Z{ω} C.

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Remark 4.3 In the situation of Theorem 1.10, = 1 and A = J/AnnT (C · Φ)J where C · Φ = S2 (π )H , π is an automorphic representation of G with central × with R defined by character ω : Z(Af ) Pic(OF ) → C× and H = R (1.6) – see Remarks 3.1, 3.11 and 3.12. The assumptions (H1) and (H2) are then satisfied, and the projection J → A is surjective and Pic(OF )-equivariant (with A = A{ω}). By construction, there is no Eichler order in BP strictly containing RP whose group of invertible elements fixes Φ. It thus follows from the discussion after Definition 3.13 that J → A also factors through the P -new quotient of J. Since = 1, Theorem 4.1 asserts that for any character χ0 of G0 such that χ0 · ω = 1 on Z(Af ), and any good CM point x ∈ CM(P n ) with n sufficiently large, there exists a character χ ∈ P (n, χ0 ) such that a(x, χ) = eχ−1 α(x) = 0

in A(K[P n ]) ⊗ C.

This is exactly the statement of Theorem 1.10.

4.2 An easy variant As an introduction to this circle of ideas, we will first show that a weaker variant of Theorem 4.1 can be obtained by very elementary methods, in the spirit of [22]. Thus, let π : J A be a nonzero surjective morphism, and put α = π ◦ ι : M A where ι : M J is the “Hodge embedding” of section 3.5. Then: Proposition 4.4 For all n 0 and all x ∈ CM(P n ), α(x) = 0

in A(K[P n ]) ⊗ Q.

Proof Using Lemma 3.9, we may assume that α : M → A is" a finite" morphism. In particular, there exists a positive integer d such that "α−1 (x)" ≤ d for any x ∈ A(C). On the other hand, it follows from Lemma 2.7 that the torsion subgroup of A(K[P ∞ ]) is finite, say of order t > 0. Then α maps at most dt points in CM(P ∞ ) to torsion points in A, and the proposition follows.

Corollary 4.5 There exists a character χ : G(n) → C× such that eχ α(x) = 0 in A(K[P n ]) ⊗ C.

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Suppose moreover that π is a Pic(OF )-equivariant morphism. On A(K[P n ]) ⊗ C, we then also have an action of Pic(OF ). For a character ω : Pic(OF ) → C× , let eω χ α(x) be the ω-component of eχ α(x). For any σ ∈ Aut(C), the automorphism 1 ⊗ σ of A(K[P n ]) ⊗ C maps eω χ α(x) to σ◦ω eσ◦χ α(x): if the former is nonzero, so is the latter. Lemma 4.6 eω χ α(x) = 0 unless Res(χ) = ω on G2 Pic(OF ).

Proof Write x = [g] for some g ∈ G(Af ). For λ ∈ Z(Af ), put Pic(OF ) ! [λ] = σ = recK (λ) ∈ G2 . If ρ denotes the Galois action, we find that θM (σ)(x) = [gλ] = [λg] = ρ(σ )(x)

in M (K[P n ]).

It follows that θJ (σ)(ιx) = ρ(σ )(ιx) in J(K[P n ]) ⊗ Q, θA (σ)(α(x)) = ρ(σ )(α(x)) in A(K[P n ]) ⊗ Q, ω ω and ω(σ) · eχ α(x) = χ(σ ) · eχ α(x) in A(K[P n ]) ⊗ C.

(4.1)

In particular, eω χ α(x) = 0 if ω(σ) = χ(σ ) for some σ ∈ G2 .

We thus obtain the following refinement of Proposition 4.4. Proposition 4.7 Let ω : Pic(OF ) → C× be any character such that A{ω} = 0. Then for all n 0 and all x ∈ CM(P n ), there exists a character χ : G(n) → C× inducing ω on G2 such that eχ α(x) = 0 in A(K[P n ]) ⊗ C. Proof Applying Proposition 4.4 to π{ω} : J A A{ω}, we find a character χ on G(n) such that eχ α(x) = 0 in A{ω}(K[P n ])⊗C. Lemma 4.6 then implies that Res(χ ) = σ · ω for some σ ∈ Aut(C), and we take χ = σ −1 ◦ χ . Remark 4.8 In contrast to Theorem 4.1, this proposition does not require π to be P -new, nor x to be good. It holds true without the assumptions (H1) and (H2). On the other hand, Theorem 4.1 yields a primitive character whose tame part χ0 is fixed but arbitrary, provided it coincides with ω on Pic(OF ). This seems to entail a significantly deeper assertion on the growth of the MordellWeil groups of A along K[P ∞ ]/K.

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4.3 Proof of Theorem 4.2 To prove that eχ α(x) is nonzero for some primitive character χ of G(n) inducing χ0 on G0 , it is certainly sufficient to show that the sum of these values is a nonzero element in A(K[P n ]) ⊗Z{ω} C. Provided that n is sufficiently large, Lemma 2.8 implies that this sum is equal to 1 e(χ0 , n) · α(x) = ·π χ0 (σ)σ · d(x) (4.2) q |G0 | σ∈G0

where d(x) = (q − TrZ(n) )(ιx). When x is a good CM point, d(x) may be computed using the distribution relations of section 6. Lemma 4.9 Let δ be the exponent of P in the level of the Eichler order RP ⊂ BP . If n is sufficiently large, the following relations hold in the P -new quotient J P −new ⊗ Q of J ⊗ Q. (i) If δ = 0, d(x) = q · ιx − TPl · ιx + ιx where TPl ∈ T is a certain Hecke operator, x = pru (x) belongs to CM(P n−1 ) and x = prl (x ) belongs to CM(P n−2 ). (ii) If δ = 1, d(x) = q · ιx + ιx with x = pr(x) in CM(P n−1 ). (iii) If δ ≥ 2 and x is a good CM point, d(x) = q · ιx. Proof We refer the reader to section 6 for the notations and proofs. Strictly speaking, we do not show there that x belongs to CM(P n−1 ) and x belongs to CM(P n−2 ). This however easily follows from the construction of these points. Also, Lemmas 6.6, 6.11 and 6.14 compute formulas involving the image of λ∈O× /On× recK (λ) · x in the P -new quotient of the free abelian n−1 group Z[CM]. To retrieve the above formulas, use the discussion preceding Lemma 2.9 and the compatibility of ι with the formation of the P -new quotients of Z[CM] and J. Since π : J A is P -new, these relations also hold in A ⊗ Q. In particular, for δ ≥ 2, part (iii) of the above lemma implies that Theorem 4.2 is now a consequence of the following theorem, whose proof will be given in sections 4.5-4.6. Theorem 4.10 Suppose that π : J A is a surjective, Pic(OF )-equivariant morphism such that A = A{ω} = 0 for some character ω of Pic(OF ). Fix a character χ of G0 inducing ω on G2 Pic(OF ). Then for all but finitely

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many x ∈ CM(P ∞ ), eχ · α(x) =

1 χ(σ)σ · α(x) = 0 |G0 |

in A ⊗Z{ω} C.

σ∈G0

Remark 4.11 This is the statement which is actually used in [17]. In the next subsection, we will show that Theorem 4.2 also follows from the above theorem when δ = 0 or 1, provided that we change the original P new parameterization π : J A of A to a non-optimal parameterization π + : J + A. Although this new parameterization will still satisfy to the assumptions (H1) and (H2), the proof of Theorem 4.10 only requires (H2) to hold.

4.4 Changing the level Suppose first that δ = 0. Let RP+ ⊂ RP be the Eichler order of level P 2 in BP which is constructed in section 6.5. Put H + = H P RP+ , M + = MH + , J + = JH + and so on. By Lemma 6.16, there exists degeneracy maps d0 , d1 and d2 : M + → M as well as an element ϑ ∈ C × with the property that for all x ∈ CM(P n ) with n ≥ 2, there exists a CM point x+ ∈ CM+ (P n ) such that (d0 , d1 , d2 )(x+ ) = (x, x , ϑ−1 x ). Combining this with part (1) of Lemma 4.9 (and using also the results of section 3.5, especially formula 3.7 and Lemma 3.6) we obtain: # $ d(x) = q(d0 )∗ − TPl (d1 )∗ + ϑ(d2 )∗ (ι+ x+ ) in J ⊗ Q so that π ◦ d(x) = α+ (x+ ) in A ⊗ Q, where α+ = π + ◦ ι+ with ⎛ ⎞ d0 π + = π ◦ (q, −TPl , ϑ) ◦ ⎝ d1 ⎠ : J + → J 3 → J A. d2 ∗ In particular, (4.2) becomes e(χ0 , n) · α(x) =

1 · χ0 (σ)σ · α+ (x+ ) in A ⊗Z{ω} C. q |G0 | σ∈G0

Theorem 4.2 for π thus follows from Theorem 4.10 for π + , once we know that our new parameterization π + : J + A is surjective and Pic(OF )-equivariant (these are the assumptions of Theorem 4.10). The Pic(OF )-equivariance is straightforward. Since the second and third morphisms in the definition of π +

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are surjective, it remains to show that the first one is also surjective, which amounts to showing that the induced map on the (complex) cotangent spaces at 0 is an injection. In view of Proposition 3.10, this all boils down to the following lemma. Lemma 4.12 The kernel of (d∗0 , d∗1 , d∗2 ) : (S2H )3 → S2H is trivial. +

Proof With notations as in section 6.5, the above map is given by (F0 , F1 , F2 ) → F0 + F1 + F2 where Fi (g) = d∗i Fi (g) = Fi (gbi ) with bi ∈ BP× such that bi L(0) = L(2−i). Here, L = (L(0), L(2)) is a 2-lattice in some simple left BP M2 (FP )module V FP2 such that RP = {α ∈ BP ; αL(0) ⊂ L(0)}. Put Ri = {α ∈ BP ; αL(2 − i) ⊂ L(2 − i)} so that R2 = RP and Ri× = bi R2× b−1 fixes Fi . One easily checks that R0× ∩ i R1× and R1× ∩ R2× generate R1× inside BP× . By [24, Chapter 2, section 1.4], R0× and R1× (resp. R1× and R2× ) generate the subgroup (BP× )0 of all elements in × BP× GL2 (FP ) whose reduced norm (=determinant) belongs to OF,P ⊂ FP× . Suppose that F0 + F1 + F2 ≡ 0 on G(A). Then F2 = −F0 − F1 is fixed by and R0× ∩R1× and therefore also by (BP× )0 . Being continuous, left invariant under G(Q) and right invariant under (BP× )0 H P , the function F2 : G(A) → C is then also left (and right) invariant under the kernel G1 (A) of the reduced × → F× by the strong approximation theorem [29, p. norm nr : G(A) = B 81]. For any g ∈ G(A) and θ ∈ R, we thus obtain (using the property P4 of section 3.6) cos θ sin θ F2 (g) = F2 g × , 1, · · · , 1 = e2iθ F2 (g), − sin θ cos θ R2×

so that F2 ≡ 0 on G(A). Similarly, F0 ≡ 0 on G(A). It follows that F1 ≡ 0, hence F0 ≡ F1 ≡ F2 ≡ 0 on G(A) and (d∗0 , d∗1 , d∗2 ) is indeed injective. Suppose next that δ = 1. Using now Lemma 6.17, we find two degeneracy maps d01 and d12 : M + → M , as well as an F -automorphism ϑ of M such that for all x ∈ CM(P n ) with n ≥ 2, there exists a CM point x+ ∈ CM+ (P n ) such that (x, x ) equals (d01 , ϑ−1 d12 )(x+ ) or

(d12 , ϑd01 )(x+ ).

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Theorem 4.2 (with π) thus again follows from Theorem 4.10 with ⎧ ⎫ −1 ⎨ (q, ϑ∗ ) ⎬ d01 π = π ◦ ◦ : J+ → J2 → J A or ⎩ ⎭ d12 ∗ (ϑ∗ , q) once we know that π induces an injection on the complex cotangent spaces. Since π is P -new, this now amounts to the following lemma (see section 3.7). H 2 H Lemma 4.13 The kernel of (d∗01 , d∗12 ) : (S2,P is trivial. −new ) → S2 +

Proof With notations as in section 6.5, the above map is now given by (F01 , F12 ) → F01 + F12 where F01 = F01 and F12 (g) = F12 (gb12 ) with b12 ∈ BP× such that b12 (L(0), L(1)) = (L(1), L(2)) for some 2-lattice L = (L(0), L(2)) in V such that RP× = R0× ∩ R× 1 with

Ri = {α ∈ BP ; αL(i) ⊂ L(i)}. If F01 +F12 = 0, F01 = −F12 is fixed by R0× ∩R1× and R1× ∩R2× . It is therefore × also fixed by R1 so that F01 and F12 both belong to the P -old subspace of S2H .

4.5 Geometric Galois action We now turn to the proof of Theorem 4.10. Thus, let π : J A be a surjective and Pic(OF )-equivariant morphism such that A = A{ω} = 0 for some character ω of Pic(OF ), and let χ be a fixed character of G0 inducing ω on G2 Pic(OF ). For 0 ≤ i ≤ 2, let Ci be the subring of C which is generated by the values of χ on Gi , so that C2 ⊂ C1 ⊂ C0 , C1 is finite flat over C2 , and so is C0 over C1 . Since χ induces ω on G2 , the canonical factorization of ω∗ : C → C yields an isomorphism between Z{ω} and C2 . Let Ai be the (nonzero) abelian variety over F which is defined by Ai (X) = A(X) ⊗C Ci = A(X) ⊗Z{ω} Ci for any F -scheme X (note that A2 A). Upon multiplying π by a suitable integer, we may assume that π and α are genuine morphisms. For any x ∈ CM(P n ), we may then view def a(x) = χ(σ)σ · α(x) σ∈G0

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as an element of A0 (K[P ∞ ]) = A1 (K[P ∞ ]) ⊗C1 C0 = A(K[P ∞ ]) ⊗Z{ω} C0 , and we now have to show that a(x) is a nontorsion element in this group, provided that n is sufficiently large. Using formula (4.1), which applies thanks to the Pic(OF )-equivariance of π, we immediately find that a(x) = |G2 | χ(σ)σ · α(x) σ∈R

where R ⊂ G0 is the following set of representatives for G0 /G2 . We first choose a set of representatives R ⊂ G0 of G0 /G1 containing 1, and then take R = {τ σD ; τ ∈ R and D | D }

where D ⊂ OF and the σD ’s for D | D were defined in Lemma 2.6. The next lemma will allow us to further simplify a(x). Lemma 4.14 There exists a Shimura curve M1 and a collection of degeneracy maps {dD : M1 → M ; D | D } such that for all n ≥ 0, ∀x ∈ CM(P n ), ∃x1 ∈ CM1 (P n ) s.t. ∀D | D :

σD x = dD (x1 ).

Proof Our assumption (H2) asserts that for any Q | D , RQ is a maximal order in BQ M2 (FQ ). Let ΓQ be the set of elements in RQ whose reduced norm × × is a uniformizer in OF,Q , and choose some αQ in ΓQ . Then ΓQ = RQ αQ RQ and R1,Q = RQ ∩ αQ RQ α−1 Q ⊂ BQ def

× , where R1 is the unique OF is an Eichler order of level Q. Put H1 = R 1 order in B which agrees with R outside D , and equals R1,Q at Q | D . Put M1 = MH1 , CM1 = CMH1 and so on. For D | D , put def αD = αQ ∈ G(Af ). (4.3) Q|D

Then α−1 D H1 αD ⊂ H. Let dD = [·αD ] : M1 → M be the corresponding degeneracy map. Recall also that σD = Q|D σQ for D | D , where σQ ∈ G1 is the geometric Frobenius of the unique prime Q of K above Q (so that Q2 = QOK ). Let πQ ∈ OK,Q be a local uniformizer at Q, and for D | D , put πD = Q|D πQ × , so that σD is the restriction of recK (πD ) to K[P ∞ ]. in K

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Consider now some x = [g] ∈ CM(P n ), with g ∈ G(Af ) and n ≥ 0. −1 For each Q | D , KQ ∩ gQ RQ gQ = OK,Q . In particular, πQ belongs to −1 −1 × gQ RQ gQ and gQ πQ gQ belongs to ΓQ : there exists r1,Q and r2,Q in RQ such that πQ gQ = gQ r1,Q αQ r2,Q . Put ri = Q|D ri,Q ∈ H and x1 = [gr1 ] ∈ CM1 . For D | D , we find that dD (x1 ) = [gr1 αD ] = [gr1 αD r2 ] = [πD g] = σD x.

× ∩ gr1 H1 (gr1 )−1 is obviously Finally, x1 belongs to CM1 (P n ), because K × −1 ∩ gHg away from D , and equal to K × KQ ∩ gQ r1,Q R1,Q (gQ r1,Q )−1 = × × −1 = KQ ∩ gQ r1,Q RQ (gQ r1,Q )−1 ∩ gQ r1,Q αQ RQ αQ (gQ r1,Q )−1 × −1 −1 −1 = KQ ∩ gQ RQ gQ ∩ KQ ∩ πQ gQ R× Q gQ πQ × −1 = KQ ∩ gQ RQ gQ

for Q | D . This finishes the proof of Lemma 4.14. Put J1 = Pic0 M1 and let ι1 : M1 J1 be the corresponding “Hodge embedding”. With notations as above, we find that a(x) = |G2 | b(x1 ), where for any CM point y ∈ CM1 (P ∞ ), def b(y) = χ(τ )τ · α1 (y) in A1 (K[P ∞ ]) ⊗C1 C0 . τ ∈R def

Here, α1 = π1 ◦ ι1 with π1 : J1 → A1 defined by J1 j

−→ J {D|D} −→ ((dD )∗ (j))D|D

π

−→

A{D|D} (aD )D|D

−→ A1 = A ⊗C C1 −→ D|D χ(σD )aD

Indeed, Lemma 4.14 (together with the formula (3.7)) implies that α1 (x1 ) = χ(σD )σD · α(x) in A(K[P ∞ ]) ⊗C C1 = A1 (K[P ∞ ]). D|D

We now have to show that for all x ∈ CM1 (P n ) with n 0, b(x) = torsion

in A1 (K[P ∞ ]) ⊗C1 C0 .

We will need to know that our new parameterization π1 : J1 → A1 is still surjective. As before (Lemmas 4.12 and 4.13), this amounts to the following lemma. Lemma 4.15 The kernel of

D|D

d∗D : (S2H ){D|D } → S2H1 is trivial.

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Proof We retain the notations of the proof of Lemma 4.14. The map under consideration is given by (FD )D|D → D|D αD · FD where (αD · F )(g) = F (gαD ) for any F : G(A) → C and g ∈ G(A). We show that it is injective by induction on the number of prime divisors of D . There is nothing to prove if D = OF . Otherwise, let Q be a prime divisor of ×

D . We put DQ = D /Q and H1 = R1 , where R1 is the unique OF -order in B which agrees with R1 outside Q, and equals RQ at Q. The functions F0 = αD · FD and F1 = αD · FDQ D|DQ

D|DQ

H

then belong to S2 1 , and

αD · FD = F0 + αQ · F1 .

D|D × If this function is trivial on G(A), F0 = −αQ · F1 is fixed by RQ and × −1 αQ RQ αQ . Arguing as in the proof of Lemma 4.12, we obtain F0 ≡ F1 ≡ 0 on G(A). By induction, FD ≡ 0 on G(A) for all D | D .

4.6 Chaotic Galois action We still have to show that for all but finitely many x in CM1 (P ∞ ), b(x) is a nontorsion point in A0 (K[P ∞ ]). Two proofs of this fact may be extracted from the results of [5]. These proofs are both based upon the following elementary observations: • The torsion submodule of A0 (K[P ∞ ]) is finite. This easily follows from Lemma 2.7. We thus want b(x) to land away from a given finite set, provided that x belongs to CM1 (P n ) with n 0. • The map x → b(x) may be decomposed as follows: CM1 (P ∞ ) x

Δ

−→ M1R −→ (σx)σ∈R

α

1 −→

AR 1 (aσ )σ∈R

Σ

−→ A 0 → χ(σ)aσ

In this decomposition, the second and third maps are algebraic morphisms defined over F . Moreover: Σ is surjective (this easily follows from the definitions) and α1 is finite (by Lemmas 4.15 and 3.9). In some sense, this decomposition separates the geometrical and arithmetical aspects in the definition of b(x).

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Suppose that b(x) is a torsion point in A0 (K[P ∞ ]) for infinitely many x ∈ CM1 (P ∞ ). We may then find some element a0 in A0 (C) such that E = b−1 (a0 ) is an infinite subset of CM1 (P ∞ ). Since Σ ◦ α1 : M1R → A0 is continuous for the Zariski topology, we see that (Σ ◦ α1 )−1 (a0 ) contains the Zar

Zariski closure Δ(E)

of Δ(E) in M1R (C). As explained in Remark 3.21 of

[5], a proven case of the Andr´e-Oort conjecture implies that Δ(E) a connected component of M1R (C).

Zar

contains

Zar

More precisely, let Z be any irreducible component of Δ(E) containing infinitely many points of Δ(E). The last remark of [8, Section 7.3] tells us that Z is a subvariety of Hodge type in M1R , and the list of all such subvarieties is easy to compile, following the method of [7, Section 2], see the forthcoming addendum to [5]. We find that Z is a product of curves Zi ⊂ M1Ri for a certain partition {Ri } of R. Now Proposition 3.18 of [5] implies that the partition is trivial: each Ri is a singleton, and Z is indeed a connected component of M1R . Remark 4.16 This last reference requires E to be an infinite collection of P -isogenous CM points, where two CM points x and x are said to be P isogenous if they can be represented by g and g ∈ G(Af ) with gv = gv for all v = P . Now, if a P -isogeny class contains a CM point of conductor P n for some n ≥ 0, it is actually contained in CM1 (P ∞ ) and any other P -isogeny class in CM1 (P ∞ ) also contains a point of conductor P n . Since CM1 (P n ) is finite, we thus see that CM1 (P ∞ ) is the disjoint union of finitely many P -isogeny classes, and one of them at least has infinite intersection with our infinite set E. We thus obtain a collection of connected components (Cσ )σ∈R of M1 (C) with the property that for all (xσ )σ∈R in σ∈R Cσ , χ(σ)α1 (xσ ) = a0 . σ∈R

It easily follows that α1 should then be constant on C1 . Being defined over F on the connected curve M1 (cf. Remark 3.2), α1 would then be constant on M1 , a contradiction. Second proof (using a theorem of M. Ratner) Let U be a nonempty open subscheme of Spec(OF ) such that for every closed point v ∈ U , v = P , Bv M2 (Fv ) and R1,v is maximal. Shrinking U if necessary, we may assume that M1 has a proper and smooth model M1 over U , which agrees locally with the models considered in [5], and α1 : M1 → A1

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extends uniquely to a finite morphism α1 : M1 → A1 , where A1 is the N´eron model of A1 over U . In the Stein factorization c

M1 −→ MH1 → U of the stuctural morphism M1 → U , the scheme of connected components M1 is then a finite and e´ tale cover of U , and the fibers of c are geometrically connected. For each closed point v ∈ U , we choose a place v of F ⊂ C above v, with valuation ring O(v) and residue field F(v), an algebraic closure of the residue field F(v) of v. We thus obtain the following diagram of reduction maps redv : redv :

M1 (F ) ↓c M1 (F )

←−

M1 (O(v)) −→ M1 (F(v)) ↓c ↓c ←− M1 (O(v)) −→ M1 (F(v)).

We denote by C → C(v) the induced bijection between the sets of geometrical connected components in the generic and special fibers, and we denote by Cx the connected component of x ∈ M1 (F ). In particular, Cx (v) = c−1 (redv c(x)) is the connected component of redv (x). Inside M1 (F(v)), there is a finite collection of distinguished points, namely the supersingular points as described in Section 3.1.3 of [5]. We denote by C ss (v) the set of supersingular points inside C(v). We let d > 0 be a uniform upper bound on the number of geometrical points in the fibers of α1 : M1 → A1 (such a bound does exist, thanks to the generic flatness theorem, see for instance [13, Corollaire 6.9.3]). We let t > 0 be the order of the torsion subgroup of A0 (K[P ∞ ]). One easily checks that the order of C ss (v) goes to infinity with the order of the residue field F(v) of v. Shrinking U if necessary, we may therefore assume that for all C and v, |C ss (v)| > td. Now, let v be a closed point of U which is inert in K (there are infinitely many such points). Then Lemma 3.1 of [5] states that any CM point x ∈ CM1 reduces to a supersingular point redv (x) ∈ Cxss (v) and we have the following crucial result. Proposition 4.17 For all but finitely many x in CM1 (P ∞ ), the following prop ss erty holds. For any (zσ )σ∈R in σ∈R Cσx (v), there exists some γ ∈ Galab K such that ∀σ ∈ R :

redv (γσ · x) = zσ

in M1 (F(v)).

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Proof This is a special case of Theorem 3.5 of [5], except that the latter deals with P -isogeny classes of CM points instead of the set of all CM points of P -power conductor that we consider here. However, we have already observed in Remark 4.16 that CM1 (P ∞ ) is the disjoint union of finitely many such P -isogeny classes, and the proposition follows. Corollary 4.18 For all but finitely many x in CM1 (P ∞ ), the following property holds. For any z in Cxss (v), there is a γ ∈ Galab K such that redv (γ · b(x)) − redv (b(x)) = α1 (z) − α1 (z1 ) in A0 (F(v)) = A1 (F(v)) ⊗C1 C0 , with z1 = redv (x) ∈ Cxss (v). Proof Take zσ = redv (σx) for σ = 1 in R. This finishes the proof of Theorem 4.10. Indeed, the Galois orbit of any torsion point in A0 (K[P ∞ ]) has at most t elements, while the above corollary implies that for all but finitely many x ∈ CM1 (P ∞ ), " " " " 1 " " " " ab ab "GalK · b(x)" ≥ "redv GalK · b(x) " ≥ |α1 (Cxss (v))| ≥ |Cxss (v)| > t. d

5 The definite case Suppose now that B is a definite quaternion algebra, so that B ⊗ R H[F :Q] . Let K be a totally imaginary quadratic extension of F contained in B. We put G = ResF/Q (B × ), T = ResF/Q (K × ) and Z = ResF/Q (F × ) as before.

5.1 Automorphic forms and representations We denote by S2 the space of all weight 2 automorphic forms on G, namely the space of all smooth (=locally constant) functions θ : G(Q)\G(Af ) → C. There is an admissible left action of G(Af ) on S2 , given by right translations: for g ∈ G(Af ) and x ∈ G(Q)\G(Af ), (g · θ)(x) = θ(xg). This representation is semi-simple, and S2 is the algebraic direct sum of its irreducible subrepresentations. An irreducible representation π of G(Af ) is

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automorphic if it occurs in S2 . It then occurs with multiplicity one, and we denote by S2 (π ) the corresponding subspace of S2 , so that S2 = ⊕π S2 (π ). If π is finite dimensional, it is of dimension 1 and corresponds to a smooth character χ of G(Af ). Then χ is trivial on G(Q), S2 (π ) equals C · χ, and χ factors through the reduced norm G(Af ) → Z(Af ). A function θ ∈ S2 is said to be Eisenstein if it belongs to the subspace spanned by these finite dimensional subrepresentations of S2 . Equivalently, θ is an Eisenstein function if and only if it factors through the reduced norm (because any such function spans a finite dimensional G(Af )-invariant subspace of S2 ). We say that π is cuspidal if its representation space has infinite dimension. The space of (weight 2) cuspforms S20 is the G(Af )-invariant subspace of S2 which is spanned by its irreducible cuspidal subrepresentations. Thus, θ = 0 is the only cuspform which is also Eisenstein.

5.2 The exceptional case The Jacquet-Langlands correspondence assigns, to every cuspidal representation π of G(Af ) as above, an irreducible automorphic representation π = JL(π ) of GL2 /F , of weight (2, · · · , 2). We say that (π , K) is exceptional if (π, K) is exceptional. Thus, (π , K) is exceptional if and only if π π ⊗ η, where η is the quadratic character attached to K/F . We want now to describe a simple characterization of these exceptional cases. Write π = ⊗πv , π = ⊗πv and let N be the conductor of π. For every finite place v of F not dividing N , πv πv π(μ1,v , μ2,v ) π(μ2,v , μ1,v ) for some unramified characters μi,v : Fv× → C× , i = 1, 2. These characters are uniquely determined by βi,v = μi,v (v ), where v is any local uniformizer at v, and by the strong multiplicity one theorem, the knowledge of all but finitely many of the unordered pairs {β1,v , β2,v } uniquely determines π and π . On the other hand, the representation space S(πv ) of πv contains a unique line C · φv of vectors which are fixed by the maximal compact open subgroup Hv = GL2 (OF,v ) of Gv = GL2 (Fv ) Bv× . The spherical Hecke algebra EndZ[Gv ] (Z[Gv /Hv ]) Z[Hv \Gv /Hv ] acts on C · φv , and the eigenvalues of the Hecke operators / # $ 0 / # $ 0 Tv = Hv 0v 01 Hv and Sv = Hv 0v 0v Hv

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are respectively given by av = (N v)1/2 (β1,v + β2,v )

and sv = β1,v · β2,v

where N v is the order of the residue field of v. Note that sv = ωv (v ) where ωv = μ1,v μ2,v is the local component of the central character ω of π and π . Therefore, the knowledge of ω and all but finitely many of the av ’s uniquely determines π and π . Proposition 5.1 (π, K) and (π , K) are exceptional if and only if av = 0 for all but finitely many of the v’s which are inert in K. Proof With notations as above, the v-component of π ⊗ η is equal to π(μ1,v ηv , μ2,v ηv ) where ηv is the local component of η. Therefore, π π ⊗ η if and only if {β1,v , β2,v } = {β1,v η(v), β2,v η(v)} for almost all v, where η(v) = ηv (v ) equals 1 if v splits in K, and −1 if v is inert in K. The proposition easily follows. Remark 5.2 It is well-known that the field Eπ ⊂ C generated by the av ’s and the values of ω is a number field. Moreover, for any finite place λ of Eπ with residue characteristic , there exists a unique (up to isomorphism) continuous representation ρπ,λ : GalF → GL2 (Eπ,λ ) such that for every finite place v N , ρπ,λ is unramified at v and the characteristic polynomial of ρπ,λ (Frobv ) equals X 2 − av X + N v · sv ∈ Eπ [X] ⊂ Eπ,λ [X]. See [25] and the reference therein. Put E = Eπ,λ and let V = E 2 be the representation space of ρ = ρπ,λ . If π π ⊗ η, then ρ ρ ⊗ η (viewing now η as a Galois character). In particular, there exists θ ∈ GL(V ) such that θ◦ρ=η·ρ◦θ

on GalF .

Since ρ is absolutely irreducible (this follows from the arguments in Section 2 of [25]), θ2 is a scalar in E × but θ is not. Let E be a quadratic extension of E containing a square root c of θ 2 . Put V = V ⊗E E . Then V = V+ ⊕ V− where θ = ±c on V± , and dimE V± = 1. Moreover, ∀σ ∈ GalF :

(ρ ⊗ IdE )(σ)(V± ) = V±η(σ) .

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169 ×

F It easily follows that ρ ⊗ IdE IndGal is the GalK (α), where α : GalK → E continuous character giving the action of GalK on V+ . Conversely, suppose that the base change of ρ to an algebraic closure E of × F E is isomorphic to IndGal GalK (α) for some character α : GalK → E . Then π π ⊗ η by Proposition 5.1, since

av = Tr(ρ(Frobv )) = 0 for almost all v’s that are inert in K. 5.3 CM points and Galois actions Given a compact open subgroup H of G(Af ), we say that θ ∈ S2 has level H if it is fixed by H. The space S2H of all such functions may thus be identified with the finite dimensional space of all complex valued function on the finite set def

MH = G(Q)\G(Af )/H. In particular, any such θ yields a function ψ = θ ◦ red on def

CMH = T (Q)\G(Af )/H, where red : CMH → MH is the obvious map. Also, θ is an Eisenstein function if and only if it factors through the map c : MH → NH which is induced by the reduced norm, where def

NH = Z(Q)+ \Z(Af )/nr(H) and Z(Q)+ = nr(G(Q)) is the subgroup of totally positive elements in Z(Q) = F × . We will need to consider a somewhat weaker condition. Recall from the introduction that the set CMH of CM points, is endowed with the following Galois action: for x = [g] ∈ CMH and σ = recK (λ) ∈ Galab K (with g ∈ G(Af ) and λ ∈ T (Af )), σ · x = [λg] ∈ CMH . The Galois group Galab F similarly acts on NH , and we thus obtain an action of Galab on N : for x = [z] ∈ NH and σ = recK (λ) ∈ Galab H K K (with z ∈ Z(Af ) and λ ∈ T (Af )), σ · x = [nr(λ)z] ∈ NH . By construction, the composite map red

c

CMH −→ MH −→ NH is Galab K -equivariant (and surjective).

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Definition 5.3 We say that θ : MH → C is exceptional (with respect to K) if there exists some z0 ∈ NH with the property that θ is constant on c−1 (z), for all z in Galab K · z0 . Remark 5.4 Since K is quadratic over F , there are at most two Galab K -orbits in NH . If there is just one, θ : MH → C is exceptional if and only if it is Eisenstein: this occurs for instance whenever nr(H) is the maximal compact open subgroup of Z(Af ), provided that K/F ramifies at some finite place. On the other hand, if there are two Galab K -orbits in NH , there might be exceptional θ’s which are not Eisenstein. Lemma 5.5 Let π be any cuspidal representation of G(Af ). Suppose that S2 (π ) contains a nonzero θ which is exceptional with respect to K. Then (π , K) is exceptional. Proof Let H = v Hv be a compact open subgroup of G(Af ) such that θ is right invariant under H. Since π is cuspidal, θ is not Eisenstein and the above remark shows that there must be exactly two Galab K -orbits in NH , say X and Y , with θ constant on c−1 (z) for all z in X, but θ(x1 ) = θ(x2 ) for some x1 and x2 in MH with c(x1 ) = c(x2 ) = y ∈ Y . For all but finitely many v’s, Hv = Rv× where Rv M2 (OF,v ) is a maximal order in Bv M2 (Fv ). For any such v, we know that θ|Tv = av θ where for x = [g] ∈ MH with g ∈ G(Af ), (θ|Tv )(x) =

θ(xv,i )

with xv,i = [gγv,i ] ∈ MH .

i∈Iv

# $ ) Here, Hv 0v 01 Hv = i∈Iv γv,i Hv with v a local uniformizer in Fv . Note that for x in c−1 (y), the xv,i ’s all belong to c−1 (Frobv · y). If v is inert in K, Frobv · y belongs to X and θ is constant on its fiber, say θ(x ) = θ(v, y) for all x ∈ c−1 (Frobv · y). For such v’s, we thus obtain av θ(x1 ) = |Iv | θ(v, y) = av θ(x2 ). Since θ(x1 ) = θ(x2 ) by construction, av = 0 whenever v is inert in K. The lemma now follows from Proposition 5.1.

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5.4 Main results To prove our main theorems in the definite case, we shall proceed backwards, starting from the analog of Proposition 4.17, and ending with our target result, the analog of Theorem 4.1. Thus, let R ⊂ G0 be the set of representatives for G0 /G2 which we considered in section 4.5. Recall that R = {τ σD ; τ ∈ R and D | D } where R ⊂ G0 is a set of representatives for G0 /G1 containing 1, while D × and the σD ’s for D | D were defined in Lemma 2.6. Suppose that H = R for some OF -order R ⊂ B, and consider the following maps: CMH (P ∞ ) x

/ MR H

R MH

/ (red(τ · x))τ ∈R

(aτ )τ ∈R

RED

/ NR H

C

/ (c(aτ ))τ ∈R

R If we endow NH with the diagonal Galois action, the composite R C ◦ RED : CMH (P ∞ ) → NH

becomes a G(∞)-equivariant map. In particular, for any x ∈ CMH (P ∞ ), RED(G(∞) · x) ⊂ C −1 (G(∞) · C ◦ RED(x)). The following key result is our initial input from [5]. Proposition 5.6 For all but finitely many x ∈ CMH (P ∞ ), RED(G(∞) · x) = C −1 (G(∞) · C ◦ RED(x)). Proof Given the definition of G1 , this is just a special case of Corollary 2.10 of [5] except that the latter deals with P -isogeny classes of CM points instead of the set of all CM points of P -power conductor which we consider here. Nevertheless, CMH (P ∞ ) breaks up as the disjoint union of finitely many such P -isogeny classes, cf. Remark 4.16. The proposition follows. Corollary 5.7 Let θ be any non-exceptional function on MH , and let ψ = θ ◦ red be the induced function on CMH . Let χ be any character of G0 . Then, for any CM point x ∈ CMH (P n ) with n sufficiently large, there exists some y ∈ G(∞) · x such that χ(τ )ψ(τ · y) = 0. τ ∈R

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Proof Replacing x by σ · x for some σ ∈ G(∞), we may assume that θ(p1 ) = θ(p2 ) for some p1 and p2 in c−1 (c ◦ red(x)). If n is sufficiently large, the proposition then produces y1 and y2 in G(∞) · x such that red(y1 ) = p1 , red(y2 ) = p2 and red(τ · y1 ) = red(τ · x) = red(τ · y2 ) for any τ = 1 in R. If a(y) = τ ∈R χ(τ )ψ(τ · y), we thus obtain a(y1 ) − a(y2 ) = θ(p1 ) − θ(p2 ) = 0, and one at least of a(y1 ) or a(y2 ) is nonzero. Suppose now that we are given an irreducible cuspidal representation π of G(Af ), with (unramified) central character ω. We still consider a level struc× for some OF -order R ⊂ B, but we now also require ture of the form H = R the following condition. (H2) For any prime Q = P of F which ramifies in K, B is split at Q and RQ is a maximal order in BQ M2 (FQ ). Our next result is the analog of Theorem 4.10. Proposition 5.8 Suppose that θ is a nonzero function in S2 (π )H , and let ψ be the induced function on CMH . Let χ be any character of G0 such that χ·ω = 1 n on A× F . Then, for all x ∈ CM(P ) with n sufficiently large, there exists some y ∈ G(∞) · x such that def a(y) = χ(σ)ψ(σ · y) = 0. σ∈G0

Proof Since Z(Af ) acts on S2 (π ) ! θ through ω, we find that a(y) = |G2 | χ(σ)ψ(σ · y). σ∈R

= |G2 |

χ(τ )

τ ∈R

Using Lemma 5.9 below, we obtain a(y) = |G2 |

χ(σD )ψ(σD τ · y).

D|D

χ(τ )ψ1 (τ · y1 )

τ ∈R

× where ψ1 : CMH1 → C is induced by a nonzero function θ1 of level H1 = R 1 n in S2 (π ), and y → y1 is a Galois equivariant map from CMH (P ) to

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173

CMH1 (P n ). Since θ1 is non-exceptional by Lemma 5.5, we may apply Corollary 5.7 to θ1 and x1 ∈ CMH1 (P n ), thus obtaining some y1 = γ · x1 in G(∞) · x1 such that −1 χ(τ )ψ1 (τ · y1 ) = |G2 | a(y) = 0, τ ∈R

with y = γ · x ∈ G(∞) · x. Lemma 5.9 There exists an OF -order R1 ⊂ R, a nonzero function θ1 of level × in S2 (π ), and for each n ≥ 0, a Galois equivariant map x → x1 H1 = R 1

from CMH (P n ) to CMH1 (P n ) such that χ(σD )ψ(σD · x) = ψ1 (x1 ),

(5.1)

D|D

where ψ = θ ◦ red and ψ1 = θ1 ◦ red as usual. Proof The proof is very similar to that of Lemma 4.14. We put R1 = R ∩ αD Rα−1 D ⊂ B where for any prime divisor Q of D , ΓQ is the set of elements in RQ M2 (OF,Q ) whose reduced norm (=determinant) is a uniformizer in OF,Q , αQ is a chosen element in ΓQ , and αD = Q|D αQ for any divisor D of D . We then define θ1 = χ(σD )(αD · θ). D|D

× in S2 (π ). Thus, θ1 is a function of level H1 = R 1 Consider now some x = [g] ∈ CMH (P n ), with g ∈ G(Af ) and n ≥ 0. −1 For each Q | D , we know that KQ ∩ gQ RQ gQ = OK,Q . If πQ denotes −1 a fixed generator of the maximal ideal of OK,Q , we thus find that gQ πQ gQ × × × belongs to ΓQ . Since ΓQ = RQ αQ RQ , there exists r1,Q and r2,Q in RQ such −1 that gQ πQ gQ = r1,Q αQ r2,Q . For i ∈ {1, 2}, we put ri = Q|D ri,Q and view it as an element of H ⊂ G(Af ). One easily checks, as in the proof of Lemma 4.14, that the CM point x1 = [gr1 ] in CMH1 has conductor P n , and we claim that (i) the map x → x1 is well-defined and Galois equivariant; (ii) formula (5.1) holds for all x ∈ CMH (P ∞ ). For (i), suppose that we replace g by g = λgh for some λ ∈ T (Af ) and × h ∈ H. For Q | D , let r1,Q and r2,Q be elements of RQ such that −1

g Q πQ gQ = r1,Q αQ r2,Q .

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× Since gQ = λQ gQ hQ and πQ λQ = λQ πQ in BQ , we find that −1 r1,Q αQ r2,Q = gQ πQ gQ = hQ r1,Q αQ r2,Q h−1 Q . −1

−1 In particular, r1,Q hQ r1,Q equals αQ r2,Q hQ r 2,Q α−1 Q , and thus belongs to × × × −1 R1,Q = RQ ∩ αQ RQ αQ . It follows that for r1 = Q|D r1,Q ∈ H, [g r1 ] n equals [λgr1 ] in CMH1 (P ), and this finishes the proof of (i). For (ii), we simply have to observe that for any divisor D of D , if ψD denotes the function on CMH1 which is induced by αD · θ ∈ S2 (π ),

ψD (x1 ) = θ(gr1 αD ) = θ(gr1 αD r2 ) = θ(πD g) = ψ(σD · x) where πD = Q|D πQ , so that σD = recK (πD ). To complete the proof of the lemma, it remains to show that θ1 is nonzero. This may be proved by induction, exactly as in Proposition 5.3 of [27], or Lemma 4.15 above in the indefinite case. The final step of the argument runs as follows: if ϑ+ρ(πQ )(ϑ ) = 0 for some ϑ and ϑ in S2 (π ) that are fixed by R× Q, × × −1 then ϑ = −ρ(πQ )(ϑ ) is fixed by the group spanned by RQ and πQ RQ πQ . This group contains the kernel of the reduced norm BP× → FP× , and the strong approximation theorem then implies that ϑ is Eisenstein, hence zero. Finally, suppose moreover that the following condition holds. (H1) RP is an Eichler order in BP M2 (FP ). We then have the notion of good CM points. We say that θ ∈ S2 (π ) is P -new if it is fixed by RP× , and π contains no nonzero vectors which are fixed by × RP for some Eichler order RP ⊂ BP strictly containing RP . The following is the analog of Theorem 4.1. Theorem 5.10 Suppose that θ is a nonzero function in S2 (π )H . Suppose moreover that θ is P -new, and let ψ be the induced function on CMH . Let χ0 be any character of G0 such that χ0 · ω = 1 on A× F . Then, for any good CM point x ∈ CMH (P n ) with n sufficiently large, there exists a primitive character χ of G(n) inducing χ0 on G0 such that def a(x, χ) = χ(σ)ψ(σ · x) = 0. σ∈G(n)

Proof Since a(γ · x, χ) = χ−1 (γ)a(x, χ) for any γ ∈ G(n), it suffices to show that for some y in the Galois orbit of x, the average of the a(y, χ)’s is nonzero (with χ running through the set P (n, χ0 ) of primitive characters of

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G(n) inducing χ0 on G0 ). By Lemma 2.8, this amounts to showing that χ0 (σ)ψ∗ (σ · d(y)) = 0 σ∈G0

for some y ∈ G(∞) · x, where ψ∗ : Z[CMH ] → C is the natural extension of ψ and def

d(y) = q · y − TrZ(n) (y) ∈ Z[CMH ]. Since θ is P -new, ψ∗ factors through the P -new quotient Z[CMH ]P −new of Z[CMH ]. In the latter, the image of d(y) may be computed using the distribution relations of the appendix, provided that y (or x) is a good CM point of conductor P n with n sufficiently large. We find that ψ∗ (d(y)) = ψ + (y + ) ×

where H + = R+ for some OF -order R+ ⊂ B, θ + is a function of level H + in S2 (π ), ψ + is the induced function on CMH + , and y + belongs to CMH + (P n ). Moreover, the map y → y + commutes with the action of Galab K, so that y + belongs to G(∞) · x+ ⊂ CMH1 (P n ) and ψ∗ (σ · d(y)) = ψ∗ (d(σy)) = ψ + ((σy)+ ) = ψ + (σ · y + ) for any σ ∈ G0 . We now have to show that def a(y + ) = χ0 (σ)ψ + (σ · y + ) = 0 σ∈G0

for some y + ∈ G(∞) · x+ , provided that n is sufficiently large. When δ ≥ 2, R+ = R, x+ = x and θ+ = q·θ with q = |OF /P |. Otherwise, + R is the unique OF -order in B which agrees with R outside P , and whose localization RP+ at P is the Eichler order of level P 2 constructed in section 6.5. In particular, R+ satisfies (H2). Moreover: b0 · θ0 + b1 · θ1 + b2 · θ2 if δ = 0 + θ = b01 · θ01 + b12 · θ12 if δ = 1 where the b∗ ’s are the elements of BP× defined in section 6.5, while the θ∗ ’s are the elements of S2 (π )H which are respectively given by and or

(θ0 , θ1 , θ2 ) = (q · θ, −T · θ, γ · θ) if δ = 0 (θ01 , θ12 ) = (q · θ, γ −1 · θ) (θ01 , θ12 ) = (γ · θ, q · θ) if δ = 1.

In the above formulas, T and γ are certain Hecke operators in TH , with γ ∈ T× H . The argument that we already used in Lemmas 4.12 and 4.13 shows that

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θ + = 0 in all (four) cases, and we may therefore apply Proposition 5.8 to conclude the proof of our theorem.

6 Appendix: Distribution Relations Fix a number field F , a quadratic extension K of F and a quaternion algebra B over F containing K. Let OF and OK be the ring of integers in F and × and put CMH = K × \B × /H K. Let H be a compact open subgroup of B + × × where K+ is the subgroup of K which consists of those elements which are positive at every real place of K. × × The Galois group Galab acts on CMH by σ · x = [λ b] for K K + \K × ,b∈B × ). Here, is a fixed element in σ = recK (λ) and x = [b] (λ ∈ K {±1}. We extend this action by linearity to the free abelian group Z[CMH ] generated by CMH . On the latter, we also have a Galois equivariant left action of the Hecke algebra × /H]) Z[H\B × /H]. TH = EndZ[B × ] (Z[B def

× /H acts on Z[B × /H] or Z[CMH ] by An element [α] ∈ H\B [b] →

n

[bαi ]

for HαH =

i=1

n 1

×. αi H and b ∈ B

i=1

A distribution relation is an expression relating these two actions. The aim of this section is to establish some of these relations when H = H P RP× where P is a prime of F where B is split, H P is any compact open subgroup of × )P = {b ∈ B; bP = 1} and RP ⊂ BP is an Eichler order of level P δ for (B some δ ≥ 0. More precisely, we shall relate the action of the “decomposition group at P ” to the action of the local Hecke algebra T(RP× ) = EndZ[B× ] (Z[BP× /RP× ]) Z[RP× \BP× /RP× ] ⊂ TH P

on CMH . This naturally leads us to the study of the left action of KP× and T(RP× ) on BP× /RP× . For any x = [b] ∈ BP× /RP× , the stabilizer of x in KP× equals O(x)× where O(x) = KP ∩ bRP b−1 is an OF,P -order in KP . On the other hand, any OF,P -order O ⊂ KP is equal to def

On = OF,P + P n OK,P for a unique integer n = P (O) (cf. section 6.1 below). For x as above, we def put P (x) = P (O(x)). This function on BP× /R× P obviously factors through

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KP× \BP× /RP× . Using the decomposition × × × P × × × P × P Galab K \CMH K \B /H (K ) \(B ) /H × KP \BP /RP

we thus obtain a Galois invariant fibration P : CMH → N with the property that for any x ∈ CMH with n = P (x), x is fixed by the closed subgroup recK (On× ) of Galab K . If n ≥ 1, we put def Tr(x) = recK (λ) · x ∈ Z[CMH ]. × × λ∈On−1 /On

This is, on the Galois side, the expression that we will try to compute in terms of the action of the local Hecke algebra. When δ ≥ 1 (so that RP is not a maximal order), our formulas simplify in the P -new quotient Z[CMH ]P −new of Z[CMH ]. The latter is the quotient of Z[CMH ] by the Z-submodule which is spanned by the elements of the form × and {αi } ⊂ B × is a set of representatives of R × /R× [bαi ] where b ∈ B P P P for some Eichler order RP ⊂ RP ⊂ BP of level P δ with δ < δ. We start this section with a review on the arithmetic of On . The next three sections establish the distribution relations for Tr(x) when δ = 0, δ = 1 and δ ≥ 2 respectively. The final section explains how the various points that are involved in the formulas for δ = 0 or δ = 1 may all be retrieved from a single CM point of higher level δ = 2. To fix the notation, we put F = OF /P OF,P /P OF,P and let F[] = F[X]/X 2 F[X] be the infinitesimal deformation F-algebra. We choose a local uniformizer P of F at P . We set εP = −1, 0 or 1 depending upon whether P is inert, ramifies or splits in K. We denote by P (resp. P and P ∗ ) the primes of K above P and let σP (resp. σP and σP ∗ ) be the corresponding geometric Frobeniuses.

6.1 Orders Since OK,P /OF,P is a torsionfree rank one OF,P -module, we may find an OF,P -basis (1, αP ) of OK,P . Let O be any OF,P -order in KP . The projection of O ⊂ OK,P = OF,P ⊕ OF,P αP to the second factor equals P n OF,P αP for a well-defined integer n = P (O) ≥ 0. Since OF,P ⊂ O, O = OF,P ⊕ P n OF,P αP = OF,P + P n OK,P . def

Conversely, ∀n ≥ 0, On = OF,P + P n OK,P is an OF,P -order in KP . Since any On -ideal is generated by at most two elements (it is already generated by two elements as an OF,P -module), On is a Gorenstein ring for any

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n ≥ 0 [1]. For n = 0, O0 = OK,P and the F-algebra O0 /P O0 is a degree 2 extension of F if εP = −1, is isomorphic to F[] if εP = 0 and to F2 if εP = 1. For n > 0, On is a local ring with maximal ideal P On−1 and On /P On is again isomorphic to F[]. Lemma 6.1 For any n ≥ 0, the left action of On× on P(On /P On ) factors × through On× /On+1 . Its set of fixed points is given by the following formula ⎧ ∅ if n = 0 and εP = −1, ⎪ ⎪ ⎨ × {PO /P O } if n = 0 and εP = 0, 0 0 P(On /P On )On = ∗ ⎪ {PO /P O , P O /P O } if n = 0 and εP = 1, 0 0 0 0 ⎪ ⎩ {P On−1 /P On } if n > 0. × The remaining points are permuted faithfully and transitively by On× /On+1 .

Proof This easily follows from the above discussion together with the observation that the quotient map On → On /P On induces a bijection between × On× /On+1 and (On /P On )× /F× .

6.2 The δ = 0 case Let V be a simple left BP -module, so that V FP2 as an FP -vector space. The embedding KP → BP endows V with the structure of a (left) KP -module for which V is free of rank one. Let L be the set of OF,P -lattices in V and pick L0 ∈ L such that {α ∈ BP ; αL0 ⊂ L0 } = RP . Then b → bL0 yields a bijection between BP× /RP× and L. The induced left actions of KP× and T(RP× ) on Z[L] are respectively given by (λ, L) → λL and

[RP× αRP× ](L)

=

n

bLi

i=1

)n × × for λ ∈ KP× , L = bL0 ∈ L, α ∈ BP× , R× P αRP = i=1 αi RP and Li = αi L0 . The function P on L maps a lattice L to the unique integer n = P (L) such that {x ∈ KP× ; xL ⊂ L} equals On . Lemma 6.2 The function P defines a bijection between KP× \BP× /R× P KP× \L and N. Proof Fix a KP -basis e of V . For any n ≥ 0, P (On e) = n – this shows that P is surjective. Conversely, let L be a lattice with P (L) = n. Then L is a free (rank one) On -module by [1, Proposition 7.2]. In particular, there

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179

exists an element λ ∈ KP× such that L = On λe = λOn e. This shows that

P : KP× \L → N is also injective. Definition 6.3 Let L ⊂ V be a lattice. (i) The lower (resp. upper) neighbors of L are the lattices L ⊂ L (resp. L ⊂ L ) such that L/L F (resp. L /L F). (ii) The lower (resp. upper) Hecke operator TPl (resp. TPu ) on Z[L] maps L to the sum of its lower (resp. upper) neighbors. (iii) If n = P (L) ≥ 1, the lower (resp. upper) predecessor of L is defined by def

def

prl (L) = P On−1 L (resp. pru (L) = On−1 L). Remark 6.4 TPl and TPu are the local Hecke operators corresponding to × × −1 × respectively R× RP where α is any element of RP P αRP and RP α M2 (OF,P ) whose reduced norm (= determinant) is a uniformizer in FP . Lemma 6.5 Let L be a lattice in V and put n = P (L). (i) If n = 0, there are exactly 1 + εP lower neighbors L of L for which

P (L ) = 0, namely L = PL if εP = 0 and L = PL or P ∗ L if εP = 1. (ii) If n > 0, there is a unique lower neighbor L of L for which P (L ) ≤ n, namely L = prl (L) for which P (L ) = n − 1. (iii) In both cases, the remaining lower neighbors have P = n + 1. They × are permuted faithfully and transitively by On× /On+1 and L is their common upper predecessor. Proof This is a straightforward consequence of Lemma 6.1, together with the fact already observed in the proof of Lemma 6.2 that any lattice L with n = P (L) is free of rank one over On . We leave it to the reader to formulate and prove an “upper” variant of this lemma. The function L → prl (L) (resp. pru (L)) commutes with the action of KP× , and so does the induced function on {[b] ∈ BP× /RP× , P (b) ≥ 1}. The × -equivariant function on {[b] ∈ B × /H, P (bp ) latter function extends to a K × × P P ≥ 1} with values in B /H (take the identity on (B ) /H ). Dividing by × K+ , we finally obtain Galois equivariant functions prl and pru on {x ∈ CMH , P (x) ≥ 1} with values in CMH . These functions do not depend upon the various choices that we made (V and L0 ).

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Corollary 6.6 For x ∈ CMH with P (x) = n ≥ 1, Tr(x) = TPl (x ) − x where x = pru (x), x = prl (x ) if n ≥ 2 and ⎧ (εP = −1) ⎨ 0 x = σP x (εP = 0) ⎩ (σP + σP (εP = 1) ∗ )x

if n = 1.

Note that if P (x) = 1, P (x ) = 0 and x is indeed defined over an abelian extension of K which is unramified above P .

6.3 The δ = 1 case BP× /R× P

With V as above, may now be identified with the set L1 of all pairs of lattices L = (L(0), L(1)) ∈ L2 such that L(1) ⊂ L(0) with L(0)/L(1) F. Indeed, BP× GL(V ) acts transitively on L1 and there exists some L0 = (L0 (0), L0 (1)) ∈ L1 whose stabilizer equals RP× . To each L ∈ L1 , we may now attach two integers, namely

P,0 (L) = P (L(0))

and P,1 (L) = P (L(1)).

For L ∈ L1 , P (L) = max( P,0 (L), P,1 (L)) and exactly one of the following three situations occurs (see Lemma 6.5). Definition 6.7 We say that • L is of type I if P,0 (L) = n − 1 < P,1 (L) = n. The leading vertex of L equals L(1) and if n ≥ 2, we define the predecessor of L by pr(L) = (L(0), prl L(0)) = (L(0), P On−2 L(0)). • L is of type II if P,0 (L) = n > P,1 (L) = n − 1. Then L(0) is the leading vertex and for n ≥ 2, the predecessor of L is defined by pr(L) = (pru L(1), L(1)) = (On−2 L(1), L(1)). • L is of type III if P,0 (L) = n = P,1 (L) (in which case n = 0, εP = 0 or 1 and L(1) = PL(0) or L(1) = P ∗ L(0)). As a convention, we define the leading vertex of L to be L(0). Remark 6.8 The type of L together with the integer n = P (L) almost determines the KP× -homothety class of L. Indeed, Lemma 6.2 implies that we can move the leading vertex of L to On e ({e} is a KP -basis of V ). Then L = (On−1 e, On e) if L is of type I, L = (On e, P On−1 e) if L is of type II

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and L = (On e, PO n e) or (On e, P ∗ On e) if L is of type III (in which case n = 0). Definition 6.9 The lower (resp. upper) Hecke operator TPl (resp. TPu ) on Z[L1 ] maps L ∈ L1 to the sum of all elements L ∈ L1 such that L (0) = L(0) but L (1) = L(1) (resp. L (1) = L(1) but L (0) = L(0)). The P -new quotient Z[L1 ]P −new of Z[L1 ] is the quotient of Z[L1 ] by the Z-submodule which is spanned by the elements of the form L (0)=M L or L (1)=M L with M a lattice in V . By construction, TPl ≡ TPu ≡ −1

on Z[L1 ]P −new .

Remark 6.10 For i ∈ {0, 1}, put R(i) = {b ∈ BP ; bL0 (i) ⊂ L0 (i)} so that RP = R(0) ∩ R(1). Then TPl and TPu are the local Hecke operators corresponding to respectively RP× αRP× and RP× βRP× , for any α in R(0)× − R(1)× ) and β in R(1)× − R(0)× . Also, R(0)× = RP× RP× βRP× and R(1)× = ) RP× RP× αRP× . For L ∈ L1 and λ ∈ KP× , L and λL have the same type and pr(λL) = λpr(L) (if P (L) ≥ 2). We thus obtain a Galois invariant notion of type on CMH and a Galois equivariant map x → pr(x) on {x ∈ CMH ; P (x) ≥ 2} with values in CMH . The following is then an easy consequence of Lemma 6.5. Lemma 6.11 For x ∈ CMH with P (x) ≥ 2, TPl (pr(x)), if x is of type I Tr(x) = TPu (pr(x)), if x is of type II. In the P -new quotient of Z[CMH ], these relations simplify to: Tr(x) = −pr(x). Remark 6.12 In contrast to the δ = 0 case, the above constructions do depend upon the choice of L0 . More precisely, our definition of types on CMH are sensitive to the choice of an orientation on RP : changing L0 = (L0 (0), L0 (1)) to L0 = (L0 (1), P L0 (0)) exchanges type I and type II points.

6.4 The δ ≥ 2 case BP× /RP×

We now have Lδ where Lδ is the set of all pairs of lattices L = (L(0), L(δ)) in V such that L(δ) ⊂ L(0) with L(0)/L(δ) OF /P δ . We

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refer to such pairs as δ-lattices. To each L ∈ Lδ , we may attach the sequence of intermediate lattices L(δ) L(δ − 1) · · · L(1) L(0) def

and the sequence of integers P,i (L) = P (L(i)), for 0 ≤ i ≤ δ. The function

P corresponds to

P (L) = max( P,i (L)) = max( P,0 (L), P,δ (L)). Using Lemma 6.5, one easily checks that the sequence P,i (L) satisfies the following property: there exists integers 0 ≤ i1 ≤ i2 ≤ δ such that P,i+1 (L)−

P,i (L) equals −1 for 0 ≤ i < i1 , 0 for i1 ≤ i < i2 and 1 for i2 ≤ i < δ. Moreover, P,i (L) = 0 for all i1 ≤ i ≤ i2 if i2 = i1 in which case εP = 0 or 1, and i2 − i1 ≤ 1 if εP = 0. For our purposes, we only need to distinguish between three types of δ-lattices. Definition 6.13 We say that L ∈ Lδ is of type I if P,0 (L) < P,δ (L), of type II if P,0 (L) > P,δ (L) and of type III if P,0 (L) = P,δ (L). The P -new quotient Z[Lδ ]P −new of Z[Lδ ] is the quotient of Z[Lδ ] by the Zsubmodule which is spanned by the elements of the form

(L (1),L (δ))=M L

or

(L (0),L (δ−1))=M L

with M ∈ Lδ−1 . It easily follows from Lemma 6.5 that for any L ∈ Lδ which is not of type III, Tr(L) = 0 in Z[Lδ ]P −new where Tr(L) = λ∈O× /On× λL n−1 for n = P (L). Indeed, L if L is of type I, (L (0),L (δ−1))=(L(0),L(δ−1)) Tr(L) = if L is of type II. (L (1),L (δ))=(L(1),L(δ)) L Extending the notion of types to CMH as in the previous section, we obtain: Lemma 6.14 For any x ∈ CMH which is not of type III, Tr(x) = 0 in Z[CMH ]P −new . Remark 6.15 If δ is odd, P is bounded on the set of type III points in Lδ or CMH . If δ is even, there are type III points with P = n for any n ≥ δ/2. In both cases, there are type I and type II points with P = n for any n > δ/2.

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183

6.5 Predecessors and degeneracy maps Suppose first that δ = 0 and let L0 (0) be a lattice in V such that RP = {α ∈ BP ; αL0 (0) ⊂ L0 (0)}. Choose a lattice L0 (2) ⊂ L0 (0) such that L0 = (L0 (0), L0 (2)) is a 2-lattice and let R+ P = {α ∈ BP ; αL0 (0) ⊂ L0 (0) and αL0 (2) ⊂ L0 (2)}

(6.1)

be the corresponding Eichler order (of level P 2 ). Put H + = H P (RP+ )× . To a 2-lattice L, we may attach three lattices: d0 (L) = L(2),

d1 (L) = L(1)

and d2 (L) = L(0).

Conversely, to each lattice L with n = P (L) ≥ 2, we may attach a unique 2-lattice L+ = (On−2 L, L) with the property that (d0 , d1 , d2 )(L+ ) = (L, pru L, pru ◦ pru L) = (L, L , P −1 L ) where L = pru L and L = prl L . Being KP× -equivariant, these constructions have Galois equivariant counterparts on suitable spaces of CM points. More precisely: Choose bi ∈ BP× such that bi L0 (0) = L0 (2 − i). ×. Define di : CMH + → CMH by di ([b]) = [bbi ] for b ∈ B Define ϑ : CMH → CMH by ϑ([b]) = [bP ] for b ∈ B × . Use the identifications BP× /RP× ↔ L and BP× /(RP+ )× ↔ L2 to define the KP× -equivariant map x → x+ on {[b] ∈ BP× /RP× ; P (bL0 (0)) ≥ 2} with values in BP× /(RP+ )× which corresponds to L → L+ on the level of lattices. × /H = (B × )P /H P × B × /R× (and simi• Using the decomposition B P P × /H + ), extend x → x+ to a K × -equivariant map defined on larly for B × /H with values in B × /H + (take the identity on the suitable subset of B × P (B ) ). × • Dividing out by K+ , we thus obtain a Galois equivariant map x → x+ on {x ∈ CMH ; P (x) ≥ 2} with values in CMH + . • • • •

By construction: Lemma 6.16 (δ = 0) For any x ∈ CMH with P (x) ≥ 2, (d0 , d1 , d2 )(x+ ) = (x, x , ϑ−1 x )

in CM3H

where x = pru (x) and x = prl (x ). The δ = 1 case is only slightly more difficult. Fix a 1-lattice (L0 (0), L0 (1)) whose stabilizer equals RP× and let L0 (2) be a sublattice of L0 (1) such that

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L0 = (L0 (0), L0 (2)) is a 2-lattice. Define RP+ by the same formula (6.1), so × that RP+ is again an Eichler order of level P 2 . Put H + = H P (R+ P) . To each 2-lattice L we may attach two 1-lattices, namely d01 (L) = (L(0), L(1)) and d12 (L) = (L(1), L(2)). Conversely suppose that L is a 1lattice with n = P (L) ≥ 2. If L is of type I, L+ = (On−2 L(0), L(1)) is a 2-lattice and (d01 , d12 )(L+ ) = (ϑ−1 pr(L), L) where ϑ is now the permutation of L1 which maps (L(0), L(1)) to (L(1), P L(0)). If L is of type II, L+ = (L(0), P On−2 L(1)) is a 2-lattice and (d01 , d12 )(L+ ) = (L, ϑpr(L)). These constructions are again equivariant with respect to the action of KP× , and may thus be extended to Galab K -equivariant constructions on CM points. More precisely, • Choose b01 = 1 and b12 ∈ BP× such that b12 (L0 (0), L0 (1)) = (L0 (1), L0 (2)). Define d01 and d12 : CMH + → CMH by d01 ([b]) = [bb01 ], ×. d12 ([b]) = [bb12 ] for b ∈ B × • Choose ω in BP such that ω(L0 (0), L0 (1)) = (L0 (1), P L0 (0)) and define ×. ϑ : CMH → CMH by ϑ([b]) = [bω] for b ∈ B • Proceeding as above in the δ = 0 case, extend L → L+ to a Galois equivariant function x → x+ defined on {x ∈ CMH ; P (x) ≥ 2} with values in CMH + . With these notations, we obtain: Lemma 6.17 (δ = 1) For any x ∈ CMH with P (x) ≥ 2, (d12 , ϑd01 )(x+ ) if x is of type I, (x, pr(x)) = (d01 , ϑ−1 d12 )(x+ ) if x is of type II.

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[25] R. Taylor. On Galois representations associated to Hilbert modular forms. Invent. Math., 98(2):265–280, 1989. [26] V. Vatsal. Uniform distribution of Heegner points. Invent. Math., 148:1–46, 2002. [27] V. Vatsal. Special values of anticylotomic L-functions. Duke Math J., 116(2):219– 261, 2003. [28] V. Vatsal. Special value formulae for Rankin L-functions. In Heegner points and Rankin L-series, volume 49 of Math. Sci. Res. Inst. Publ., 165–190. Cambridge Univ. Press, Cambridge, 2004. [29] M.-F. Vign´eras. Arithm´etique des alg`ebres de quaternions, Lecture Notes in Mathematics, Vol. 800, Springer-Verlag, 1980. [30] J.-L. Waldspurger. Sur les valeurs de certaines fonctions L automorphes en leur centre de symm´etrie. Compos. Math., 54:174–242, 1985. [31] S. Zhang. Gross-Zagier formula for GL2 . Asian J. Math., 5(2):183–290, 2001. [32] S. Zhang. Heights of Heegner points on Shimura curves. Ann. of Math. (2), 153(1):27–147, 2001. [33] S. Zhang. Gross-Zagier formula for GL2 II. In Heegner points and Rankin Lseries, 191–242. MSRI Publications, 2003.

A correspondence between representations of local Galois groups and Lie-type groups Fred Diamond Department of Mathematics, Brandeis University, Waltham, MA 02454, USA. Current address: Department of Mathematics, King’s College London, WC2R 2LS, UK. [email protected] a

Research supported by NSF grants DMS-9996345, 0300434

Introduction Serre conjectured in [13] that every continuous, irreducible odd representation ρ : GQ → GL2 (Fp ) arises from a modular form. Moreover he refines the conjecture by specifying an optimal weight and level for a Hecke eigenform giving rise to ρ. Viewing Serre’s conjecture as a manifestation of Langlands’ philosophy in characteristic p, this refinement can be viewed as a local-global compatibility principle, the weight of the form reflecting the behavior of ρ at p, the level reflecting the behavior at primes other than p. The equivalence between the “weak” conjecture and its refinement (for > 2) follows from work of Ribet [11] and others (see [6]). Remarkable progress has recently been made on the conjecture itself by Khare and Wintenberger; see for example Khare’s article in this volume. Serre’s conjecture is generalized in [5] to the context of Hilbert modular forms and two-dimensional representations of GK where K is a totally real number field in which p is unramified. The difficulty in formulating the refinement lies in the specification of the weight. This is handled in [5] by giving a recipe for a set Wp (ρ) of irreducible Fp -representations of GL2 (OK /p) for each prime p|p in terms of ρ|Ip ; the sets Wp (ρ) then conjecturally characterize the types of local factors at primes over p of automorphic representations giving rise to ρ. We omit the subscript p since we shall be concerned only with local behavior, so now K will denote a finite unramified extension of Qp with residue field k. The purpose of the paper is to prove that if the local Galois representation is semisimple, then W (ρ) is essentially the set of Jordan-H¨older constituents of 187

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the reduction of an irreducible characteristic zero representation of GL2 (k). Moreover, denoting this representation by α(ρ) we obtain Theorem 0.18 There is a bijection {ρ : GK −→ GL2 (Fp )}/ equivalence of ρ|ss IK α$ {irreducible Qp -representations of GL2 (k) not factoring through det}/ ∼ such that W (ρss ) contains the set of Jordan-H¨older factors of the reduction of α(ρ). Moreover, the last inclusion is typically an equality and one can explicitly describe the exceptional weights. We remark that the local Langlands correspondence also gives rise to a bijection between the sets in the theorem by taking the K-type corresponding to a tamely ramified lift of ρ. The bijection of the theorem however has a different flavor. Indeed if [k : Fp ] is odd, then irreducible ρ correspond to principal series and special representations, while reducible ρ correspond to supercuspidal ones. A generalization of Serre’s Conjecture to the setting of GLn was formulated by Ash and others in [1], [2], and Herzig’s thesis [8] pursues the idea of relating the set of Serre weights of a semi-simple ρ : GQp → GLn (Fp ) to the reduction of an irreducible characteristic zero representation of GLn (Fp ). However Herzig shows that the phenomenon described in Theorem 0.18 does not persist for n > 2; instead he defines an operator R on the irreducible mod p representations of GLn (Fp ) and shows that the regular (i.e., up to certain exceptions) Serre weights of ρ are given by applying R to the constituents of the reduction of a certain characteristic zero representation V (ρ). Herzig also show that such a relationship holds in the context of GL2 (k). Moreover, the association ρ → V (ρ) appears to be compatible with the local Langlands correspondence in the sense described above. In this light, Theorem 0.18 can be viewed as saying that Herzig’s operator R typically sends the set of irreducible constituents of the reduction of one Qp -representation of GL2 (k) to those of another. One can also view Theorem 0.18 in the context of the theory of mod p and p-adic local Langlands correspondences being developed by Breuil and others (see [3], [4], [7]). In particular, one would like a mod p local Langlands correspondence to associate a mod p representation of GL2 (K) to ρ, and local-global compatibility considerations suggest that the set of Serre weights

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comprise the constituents of its GL2 (OK )-socle. One would also like a p-adic local Langlands correspondence associating p-adic representations of GL2 (K) to suitable lifts of ρ, and satisfying some compatibility with the mod p correspondence with respect to reduction. One can thus speculate that the theorem reflects some property of the hypothetical p-adic correspondence for GL2 . The paper is organized as follows: In Section 1, we compute the semisimplification of the reduction mod p of the irreducible characteristic zero representations of GL2 (k). The main theorem is proved in Section 2, and the exceptional weights are described in Section 3 for the sake of completeness. The author is grateful to Florian Herzig, Richard Taylor and the referee for their feedback on an earlier draft.

1 A Brauer character computation In this section we compute the Jordan-H¨older constituents of the reduction mod p of the irreducible characteristic zero representations of GL2 (k). For SL2 (k), this is essentially done in [14]. See also [8] for another method of doing this calculation based on work of Jantzen. We first recall the irreducible Qp -representations of G = GL2 (k) (see for example [9, Ch. 28] or [10, XVIII, §12]). Let B denote the subgroup of upper-triangular matrices in G. For a pair of × homomorphisms χ1 , χ2 : k × → Q , we let I(χ1 , χ2 ) denote the (q + 1)dimensional representation of G induced from the character of B defined by x 0

w y

→ χ1 (x)χ2 (y).

I(χ1 , χ2 ) ∼ I(χ1 , χ2 ) if and only if {χ1 , χ2 } = {χ1 , χ2 }. If χ1 = χ2 , then I(χ1 , χ2 ) is irreducible. I(χ, χ) ∼ χ ◦ det ⊕spχ for an irreducible q-dimensional representation spχ . The remaining irreducible Q-representations of G are parametrized as follows. Let k be a quadratic extension of k, σ the non-trivial k-automorphism × of k and Nm the norm from k to k. For each homomorphism ξ : k × → Q such that ξ = ξ◦σ, there is an irreducible (q−1)-dimensional Q-representation Θ(ξ) of G, and Θ(ξ) ∼ Θ(ξ ) if and only if ξ ∈ {ξ, ξ ◦ σ}. Moreover for any × homomorphism χ : k × → Q , we have (χ ◦ det)Θ(ξ) ∼ Θ((χ ◦ Nm)ξ)). Letting i denote a k-algebra embedding k → M2 (k), the character table of G is as follows:

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Representation Conjugacy class of:

x 0

x 0 x 0 0 y

χ ◦ det

spχ

I(χ1 , χ2 )

Θ(ξ)

χ(x)2

qχ(x)2

(q + 1)χ1 (x)χ2 (x)

(q − 1)ξ(x)

χ(x)2

0

χ1 (x)χ2 (x)

−ξ(x)

χ(xy)

χ(xy)

χ1 (x)χ2 (y) + χ1 (y)χ2 (x)

0

0 x 1 x ∈ k×

i(z) ∈ k

×

−χ(zz )

σ

σ

χ(zz )

0

−ξ(z) − ξ(z σ )

Next we recall the irreducible Fp -representations of GL2 (k). Let S = k(Fp ), the set of embeddings k → Fp . For integers mτ , nτ with nτ ≥ 0 for each τ ∈ S, we have the representation mτ 2 Vm, k ⊗k Symnτ −1 k 2 ⊗k,τ Fp . n = ⊗τ ∈S det

We make the convention that Sym−1 = 0, so that the dimension of Vm, n is equal to τ ∈S nτ . If 1 ≤ nτ ≤ p for all τ , then Vm, n is irreducible; assuming further that 0 ≤ mτ ≤ p − 1 for each τ and some mτ < p − 1, then the Vm, n are inequivalent and form a complete list of the irreducible Fp -representations of GL2 (k). Recall that the semisimplification of an Fp -representation of G is determined by its Brauer character, which is a Qp -valued function on the p-regular conjugacy classes of G (see [12, 18.1, 18.2] for example). Letting ˜ denote the Teichm¨uller lift, the Brauer character of Vm, n , which we denote βm, n , is as follows:

x 0

0 y

i(z) ∈ k×

τ ∈S

τ ∈S

τ˜(xy)mτ

τ˜ (z)(q+1)mτ

0≤ν≤nτ −1

τ˜(y)ν τ˜(x)nτ −1−ν

0≤ν≤nτ −1

τ˜ (z)nτ −1+(q−1)ν

where τ denotes either extension of τ to k . If V is a finite-dimensional Qp -representation of G, then there exists a Zp lattice L ⊂ V stable under the action of G. Reducing L modulo the maximal ideal of Zp then yields an Fp -representation L of G whose Brauer character is the restriction of the character of V to the p-regular classes of G. In particular, the semisimplification of L is independent of the choice of lattice L, and we denote it V and call it the reduction of V . We now compute V for all irreducible V (i.e., the decomposition matrix of G with respect to reduction mod p). First note that any homomorphism × χ : k × → Qp can be written in the form τ τ˜aτ for some integers aτ with aτ 0 ≤ aτ ≤ p − 1, in which case χ = τ τ . Moreover, if V ∼ (χ ◦ det) ⊗ V , then V ∼ (χ ◦ det) ⊗ V , so we can replace V by such a twist in order to compute its reduction.

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We first consider the representations I(χ1 , χ2 ). Twisting by χ−1 1 ◦ det, we need only consider those of the form I(1, χ). The reduction is then given by the following proposition: Proposition 1.1 Let V = I(1, τ τ˜aτ ) with 0 ≤ aτ ≤ p − 1 for each τ ∈ S. Then V ∼ ⊕J⊂S VJ , where VJ = Vm J and nJ defined as follows: J , nJ with m mJ,τ

=

nJ,τ

=

0, aτ + δJ (τ ),

and

if τ ∈ J, if τ ∈ J,

aτ + δJ (τ ), p − aτ − δJ (τ ),

if τ ∈ J, if τ ∈ J,

where δJ is the characteristic function of J (p) = { τ ◦Frob | τ ∈ J }. Moreover the non-zero VJ are inequivalent. Proof. We need to show that the sum of the βm J , nJ coincides with the character of V on p-regular conjugacy classes. x 0 We first consider conjugacy classes of elements of the form with 0 y x, y ∈ k. Let us choose an embedding τ0 : k → Fp and index the elements of S by setting τi = τ ◦ Frobip for i ∈ Z/f Z. We then have βm, n = =

x ˜ x ˜

f −1 i=0

f −1 i=0

x 0 0 y

mi pi

f −1

y˜

i=0

mi pi

f −1

y˜

0≤ ν ≤ n−1

(2mi +ni −1)p

i

i=0

νi pi

x ˜

f −1

(˜ y /˜ x)

i=0

f −1 i=0

bi pi

(ni −1−νi )pi

,

m≤ b≤m+ n−1

where we have simply written mi for mτi , ni for nτi and w ˜ for τ˜0 (w). (We also abuse notation in viewing i as an integer when it appears as an exponent of p and as a congruence class when it appears as an index.) Taking (m, n) = (m J , nJ ) and viewing J ⊂ {0, 1, . . . , f − 1}, we have f −1 (2mi + ni − 1)pi i=0 i i = i∈J (ai − 1 + δJ (i))p + i∈J (ai − 1 + δJ (i) + p)p =

(1 − δJ (0))(q − 1) +

f −1 i=0

ai pi ,

(1.1)

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so that x ˜

f −1 i=0

(2mi +ni −1)pi

βm J , nJ

=x ˜

x 0

0 y

f −1 i=0

ai pi

=x ˜

, giving

f −1 i=0

ai pi

(˜ y /˜ x)

d

d∈BJ

where BJ =

f −1

d=

i=0

" " i " 0 ≤ bi < ai + δJ (i), bi p " ai + δJ (i) ≤ bi < p,

if i ∈ J, if i ∈ J

2 .

(1.2)

Note that if d ∈ BJ , then 0 ≤ d ≤ q − 1. Since the only dependence relation among the functions w → w ˜ d on k × for 0 ≤ d ≤ q − 1 is that w ˜0 = w ˜ q−1 , we see that VJ = 0 if and only if BJ = ∅. Moreover if VJ ∼ VJ , then either BJ = BJ or one is gotten from the other by replacing 0 by q − 1. One sees easily that the first case implies that either BJ = ∅ or J = J , and that the second is impossible. We thus conclude that the the non-zero Vm J , nJ are inequivalent. To complete the proof of the proposition, we use another description of the BJ : Lemma 1.2 Suppose that 0 ≤ d ≤ q − 1. Write d = f −1 p − 1 for each i ∈ Z/f Z. If d = i=0 ai pi , then d ∈ BJ ⇐⇒ J =

f −1

" f −1 2 f −1 " " i i j ∈ Z/f Z " bi+j+1 p < ai+j+1 p . " i=0

Furthermore, d =

i

bi+j+1 p <

i=0

i=0

ai pi ∈ BJ if and only if J = S or J = ∅.

Proof. First note that if d = f −1

bi pi with 0 ≤ bi ≤

i=0

f −1

f −1 i=0

ai+j+1 pi

bi pi =

f −1 i=0

ai pi , then

if and only if

bj−r < aj−r

i=0

where r ∈ {0, . . . , f − 1} is chosen so that bj−r = aj−r , bj−r+1 = aj−r+1 , . . . , bj = aj . Indeed if bj−r < aj−r , then bj pf −1 + bj−1 pf −2 + · · · + bj+1 <

bj pf −1 + bj−1 pf −2 + · · · + bj−r pf −1−r + pf −1−r

≤ aj pf −1 + aj−1 pf −2 + · · · + aj−r pf −1−r ≤ aj pf −1 + aj−1 pf −2 + · · · + aj+1 .

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f −1 Now suppose that d = i=0 bi pi ∈ BJ and j ∈ J. Then bj ≤ aj and if equality holds then j − 1 ∈ J, so bj−1 ≤ aj−1 . Iterating, we find that either f −1 d = i=0 ai pi and J = S, or that bj−r < aj−r , bj−r+1 = aj−r+1 , . . . , bj = aj , for some r ∈ {0, . . . , f − 1}, yielding the desired inequality. The case j ∈ J is similar. f −1 Conversely, suppose that d = i=0 ai pi and J is given by the formula in the statement of the lemma. If j ∈ J, then we have bj−r < aj−r , bj−r+1 = aj−r+1 , . . . , bj = aj , for some r ∈ {0, . . . , f − 1}, so either bj < aj , or bj = aj and the inequality for j − 1 gives j − 1 ∈ J. In either case we have bj < aj + δJ (j). Similarly we find that j ∈ J implies that aj + δJ (j) ≤ bj . f −1 Finally, it is clear that i=0 ai pi is in both B∅ and BS . Returning to the proof of Proposition 1.1, the lemma gives q−1 f −1 i d (˜ y /˜ x) = (˜ y /˜ x) i=0 ai p + (˜ y /˜ x) d J

d∈BJ

=

d=0

1+q f −1 i 1 + (˜ y /˜ x) i=0 ai p

if y˜ = x ˜, if y˜ = x ˜,

from which it follows that x 0 (q + 1) τ τ˜(x)aτ , if y = x, βm = J , nJ aτ aτ 0 y τ ˜ (x) + τ ˜ (y) , if y = x. τ τ J

Now consider conjugacy classes of elements of the form i(z) for z ∈ k × . Choosing an embedding τ0 of k extending τ0 and writing z˜ for τ˜0 (z), we have f −1 f −1 i i βm, ˜ i=0 (q+1)mi p z˜ i=0 (ni −1+(q−1)νi )p n (i(z)) = z f −1

=

z˜

i=0

0≤ ν ≤ n−1

(2mi +ni −1)pi

z˜(q−1)

f −1 i=0

bi pi

.

m≤ b≤m+ n−1

Summing over J and using (1.1) then gives βm J , nJ (i(z)) J f −1

= z˜

i=0

ai pi (q−1)(1−δJ (0))

z˜

⎛ f −1

= z˜

i=0

i

ai p

⎝

Jf −1

J

d∈BJ

(q−1)d

z˜

d∈BJ

z˜(q−1)d

+

Jf −1

d∈BJ

⎞ z˜(q−1)(1+d) ⎠

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Fred Diamond

f −1 where BJ is as in (1.2). According to Lemma 1.2, the values of d = i=0 ai pi f −1 contributing to the first sum are those with 0 ≤ d < i=0 ai pi , the values f −1 contributing to the second are those with i=0 ai pi < d ≤ q − 1, and there f −1 is one occurrence of i=0 ai pi in each. It follows that

f −1

βm ˜ J , nJ (i(z)) = z

J

i=0

ai pi

q

z˜(q−1)d = 0,

d=0

since z˜q−1 = 1, but z˜q

2

−1

= 1. This completes the proof of Proposition 1.1.

Note that when χ is trivial, so V ∼ det ⊕sp, the proposition gives V ∼ V0,1 ⊕ V0,p , the first factor being the reduction of det and the second being that of sp. If χ is non-trivial, then I(1, χ) is irreducible and its reduction is given by the proposition. Now we turn our attention to the (q − 1)-dimensional representations Θ(ξ). Choosing τ0 : k → Fp as in the proof of the theorem, we can write ξ = (˜ τ0 )n 2 for some n, determined mod (q − 1). Since ξ = ξ ◦ σ, we have that n is not divisible by q + 1 and can therefore be written in the form α + (q + 1)β with 1 ≤ α ≤ q, 0 ≤ β ≤ q − 2. Twisting by τ˜0 −β ◦ det, we can assume f −1 aτ n = α and write ξ = τ˜0 i=0 (˜ τi ) i where τi = τ0 ◦ Frobip , τi = τi |k and 0 ≤ aτi ≤ p − 1 for i = 0, . . . , f − 1. f −1 aτ Proposition 1.3 Let V = Θ τ˜0 i=0 (˜ τi ) i with 0 ≤ aτ ≤ p − 1 for each τ ∈ S. Then V ∼ ⊕J⊂S VJ , where VJ = Vm J and nJ defined as J , nJ with m follows:

mJ,τ

and

nJ,τ

⎧ δJ (τ ), if τ = τ0 ∈ J, ⎪ ⎪ ⎨ aτ + 1, if τ = τ0 ∈ J, = ⎪ 0, if τ ∈ J, τ = τ0 , ⎪ ⎩ aτ + δJ (τ ), if τ ∈ J, τ = τ0 , ⎧ aτ + 1 − δJ (τ ), if τ = τ0 ∈ J, ⎪ ⎪ ⎨ p − aτ − 1 + δJ (τ ), if τ = τ0 ∈ J, = ⎪ a + δJ (τ ), if τ ∈ J, τ = τ0 , ⎪ ⎩ τ p − aτ − δJ (τ ), if τ ∈ J, τ = τ0 ,

where δJ is the characteristic function of J (p) = { τ ◦Frob | τ ∈ J }. Moreover the non-zero VJ are inequivalent.

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Proof. Taking (m, n) = (m J , nJ ) as in (1.1) now gives f −1

(2mi + ni − 1)pi = 1 + (1 − δJ (0))(q − 1) +

i=0

ai pi ,

(1.3)

i=0

so that βm J , nJ where

f −1

x 0

0 y

⎧ ⎪ ⎪ ⎨

" " " f −1 " i " BJ = d = bi p " ⎪ ⎪ " i=0 ⎩ "

⎛

=x ˜1+

f −1 i=0

i

ai p

⎝

⎞ (˜ y /˜ x) d ⎠ ,

d∈BJ

δJ (0) ≤ b0 < a0 + 1, a0 + 1 ≤ b0 < p + δJ (0), 0 ≤ bi < ai + δJ (i), ai + δJ (i) ≤ bi < p,

if 0 ∈ J, if 0 ∈ J, if i ∈ J, i = 0, if i ∈ J, i = 0

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

.

(1.4) Note that if d ∈ BJ , then 1 ≤ d ≤ q − 1. Since there are no dependence relations among the functions w → w ˜ d on k × for such d, we see as in the proof of Proposition 1.1 that the non-zero VJ are inequivalent. f −1 Lemma 1.4 Suppose that 1 ≤ d ≤ q − 1. Write d = i=0 bi pi with 0 ≤ bi ≤ f −1 p − 1 for each i ∈ Z/f Z. If d ≤ i=0 ai pi , then " 2 j j " " i i d ∈ BJ ⇐⇒ J = j ∈ {0, . . . , f − 1} " 0 < bi p ≤ ai p . " i=0

If

f −1 i=0

i=0

ai pi < d, then

d∈

BJ

⇐⇒ J =

" j 2 j " " i i j ∈ {0, . . . , f − 1} " bi p ≤ ai p . " i=0

i=0

f −1 Proof. Write d = i=0 bi pi with δJ (0) ≤ b0 < p + δJ (0), and 0 ≤ bi < p for i = 1, . . . , f − 1. We then have (b0 , b1 , . . . , bf −1 ) = (b0 , b1 , . . . , bf −1 ) unless f − 1 ∈ J and b0 = 0, in which case (b0 , b1 , . . . , bf −1 ) = (p, p − 1, . . . , p − 1, br − 1, br+1 , . . . , bf −1 ) where r is the least positive integer such that br > 0. It follows that if f −1 ∈ J, then j j j j i i i 0< bi p ≤ ai p ⇐⇒ bi p ≤ ai pi . i=0

i=0

i=0

i=0

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Fred Diamond

If d ∈ BJ , then we have j ∈ J ⇐⇒

j

bi pi ≤

j

i=0

ai pi

(1.5)

i=0

for j = 0, . . . , f − 1 by induction on j. In particular, f − 1 is in J if and only if f −1 d ≤ i=0 ai pi , and (1.5) translates into the desired formula for BJ in either case. Suppose conversely that J is as defined in the statement of the lemma. In f −1 i particular, f − 1 is in J if and only if d ≤ i=0 ai p , so (1.5) holds for j = 0, . . . , f − 1. Therefore b0 ≤ a0 if and only if 0 ∈ J, and we deduce that j ∈ J ⇐⇒ bj < aj + δJ (j) for j = 1, . . . , f − 1 by induction. It follows that d ∈ BJ .

Returning to the proof of Proposition 1.3, the lemma gives

⎛ ⎝

⎞ (˜ y /˜ x)

d⎠

=

d∈BJ

J

q−1

d

(˜ y /˜ x) =

d=1

q − 1 if y˜ = x ˜, 0 if y˜ = x ˜,

since (˜ y /˜ x)q−1 = 1. It follows that

βm J , nJ

J

x 0 0 y

=

(q − 1)ξ(x), 0,

if y = x, if y = x.

f −1 aτ where ξ = τ˜0 i=0 (˜ τi ) i . Now for conjugacy classes of elements of the form i(z) for z ∈ k × , we have

βm J , nJ (i(z))

J −1 1+ fi=0 ai pi (q−1)(1−δJ (0))

= z˜

z˜

⎛ −1 1+ fi=0 ai pi

= z˜

⎝

Jf −1

⎛ ⎝

J

d∈BJ

⎛ ⎝

⎞ z˜(q−1)d ⎠

d∈BJ

⎞

z˜(q−1)d ⎠ +

Jf −1

⎛ ⎝

⎞⎞ z˜(q−1)(1+d) ⎠⎠

d∈BJ

where BJ is as in (1.4). According to Lemma 1.4, the values of d contributing f −1 to the first sum are those with 1 ≤ d ≤ i=0 ai pi , the values contributing to

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197

f −1 the second are those with i=0 ai pi < d ≤ q − 1. It follows that βm J , nJ (i(z)) J q −1 −1 1+ fi=0 ai pi (q−1)(1+ fi=0 ai pi ) (q−1)d = z˜ −1 − z˜ + z˜ = −˜ z

1+

f −1 i=0

ai pi

− z˜

q(1+

f −1 i=0

d=0 ai pi )

= −ξ(z) − ξ(z ). σ

This completes the proof of Proposition 1.3. 2 The correspondence In this section we construct the bijection of Theorem 0.18. Let ω0 and ω0 × denote fundamental characters IK → Fp corresponding to embeddings τ0 : k → Fp and τ0 : k → Fp chosen as in the preceding section. Thus ω0 has order q 2 − 1, and ω0 = (ω0 )q+1 . If ρ : GK → GL2 (Fp ) is a continuous representation, then ρ|ss IK is equivalent to one of the form ω0r ⊕ ω0s or

(ω0 )t

⊕

(ω0 )qt

for some r, s ∈ Z, for some t ∈ Z not divisible by q + 1,

according to whether or not ρ is reducible. In [5], the weight part of an analogue of Serre’s conjecture is formulated over totally real fields by defining a set W (ρ) of irreducible Fp -representations of GL2 (k). We recall the definition in the easiest case, when ρ|IK is semi-simple. In the case ρ ∼ ω0r ⊕ ω0s , then we define W (ρ) by the rule Vm, n ∈ W (ρ) ⇐⇒ f −1 r ≡ mi pi + i∈J ∗ ni pi mod (q − 1), i=0 for some J ∗ ⊂ S f −1 i i s ≡ m p + n p mod (q − 1), ∗ i i i=0 i∈J (where as usual, mi = mτi with τi = τ0 ◦ Frobi ). In the case ρ ∼ (ω0 )t ⊕ (ω0 )qt for some c ≡ 0Λ-mod(q +1), we let S = {0, 1, . . . , 2f −1} and define π : S → S by reduction mod f . We then define W (ρ) by f −1 t ≡ i=0 (q + 1)mi pi + i∈J ∗ ni pi mod (q 2 − 1) Vm, ∼ n ∈ W (ρ) ⇐⇒ for some J ∗ ⊂ S such that π : J ∗ → S. Let RI denote the set of equivalence classes of Qp -representations of IK as above; note that there are (q 2 − q)/2 of each type. Let RG denote the set

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of equivalence classes of representations of GL2 (k) of the form I(χ1 , χ2 ) or Θ(ξ); note that there are (q 2 − q)/2 of each of these as well. Recall that if V is a Qp representation of GL2 (k), then V denotes the semi-simplification of its reduction modulo the maximal ideal of Z p . Theorem 2.1 There is a bijection β : RG → RI such that if ρ : GK → GL2 (Fp ) restricts to β(V ) on IK and Vm, n is an irreducible subrepresentation of V , then Vm, n ∈ W (ρ). Note that Theorem 0.18 follows from Theorem 2.1 on replacing I(χ, χ) by its q-dimensional irreducible subrepresentation and setting α(ρ) = β −1 (ρ|ss IK ). Proof. We first define β for the representations considered in Propositions 1.1 and 1.3. Suppose that b0 , . . . , bf −1 are integers with 1 ≤ bi ≤ p. If f is odd, then we let f −1

β(I(1, τ˜0

i=0

(bi −1)pi

)) =

(ω0 )b0 +b2 p

2

+···+bf −1 pf −1 +b1 pf+1 +···+bf −2 p2f −2

⊕ (ω0 )b1 p+b3 p

3

+···+bf −2 pf −2 +b0 pf +b2 pf +2 +···+bf −1 p2f −1

and β(Θ((˜ τ0 )1+

f −1 i=0

(bi −1)pi

)) =

b +b p2 +···+bf −1 pf −1 ω00 2

1+b1 p+b3 p3 +···+bf −2 pf −2

⊕ ω0

.

If f is even, then we let f −1

β(I(1, τ˜0

i=0

(bi −1)pi

b +b2 p2 +···+bf −2 pf −2

)) = ω00

b p+b3 p3 +···+bf −1 pf −1

⊕ ω01

and β(Θ((˜ τ0 )1+

f −1 i=0

(bi −1)pi

)) =

2 f −2 f f +1 2f −1 (ω0 )b0 +b2 p +···+bf −2 p +p +b1 p +···+bf −1 p

⊕ (ω0 )1+b1 p+b3 p

3

+···+bf −1 pf −1 +b0 pf +b2 pf +2 +···+bf−2 p2f −2

.

There is no ambiguity in replacing b0 = · · · = bf −1 = 1 with b0 = · · · = f −1

bf −1 = p in the formula for β(I(1, τ˜0

i=0

(bi −1)pi

)) as it just exchanges the

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199

two characters in the sum. We also need to check that the exponents of ω0 are not divisible by q + 1. This follows from the fact that if f is odd, then b0 + b2 p2 + · · · + bf −1 pf −1 + b1 pf +1 + · · · + bf −2 p2f −2 ≡ b0 − b1 p + b2 p2 − . . . − bf −2 pf −2 + bf −1 pf −1 mod (q + 1)

(2.1)

and 1 ≤ b0 − b1 p + b2 p2 − . . . − bf −2 pf −2 + bf −1 pf −1 ≤ q, and that if f is even, then b0 + b2 p2 + · · · + bf −2 pf −2 + pf + b1 pf +1 + · · · + bf −1 p2f −1 ≡ −1 + b0 − b1 p + b2 p2 − . . . − bf −2 pf −2 − bf −1 pf −1 mod (q + 1), and 1 − q ≤ b0 − b1 p + b2 p2 − . . . − bf −2 pf −2 − bf −1 pf −1 ≤ 0. (2.2) × We extend β to all of RG by twisting. If χ : k × → Qp is a character, ×

then we let β(χ) denote the character IK → Fp corresponding to χ by local class field theory, i.e., if χ = τ˜0r , then β(χ) = ω0r . Any representation in RG can be written in the form (χ ◦ det) ⊗ V for some χ and some V for which we have already defined β(V ). We then let β((χ ◦ det) ⊗ V ) = β(χ) ⊗ β(V ). We need to check there is no ambiguity in the definition. If χ1 = χ2 , then I(χ1 , χ2 ) has two expressions of the above form, namely (χ2 ◦ det) ⊗ I(1, χ) and (χ2 χ ◦ det) ⊗ I(1, χ−1 ), so it suffices to check that β(I(1, χ)) = f −1

β(χ) ⊗ β(I(1, χ−1 )). If χ = τ˜0 −1

f −1

(bi −1)pi

i=0

(bi −1)pi

with each bi ∈ {1, . . . , p}, then

bi

χ = τ˜0 where = p + 1 − bi . It is then straightforward to check that replacing β(I(1, χ)) with β(χ)β(I(1, χ−1 )) simply interchanges the two characters of IK . Similarly each Θ(ξ) has two expressions as above, given explicitly by twisting the identity Θ(ξ) ∼ (χ ◦ det) ⊗ Θ(ξ ) where i=0

f −1

f −1

f −1

(b −1)pi

ξ = (˜ τ0 )1+ i=0 (bi −1)p , ξ = (˜ τ0 )1+ i=0 (bi −1)p and χ = τ˜0 i=0 i with each bi ∈ {1, . . . , p} and bi = p + 1 − bi . We find that β(Θ(ξ)) = β(χ) ⊗ β(Θ(ξ )), the characters of IK again being interchanged. Since RI and RG have the same cardinality, it suffices to show that β is surjective in order to conclude it is a bijection. Therefore it suffices to show that every representation in RI is a twist of one of the form β(V ) for some V as in Proposition 1.1 or 1.3. For representations of the form (ω0 )t ⊕ (ω0 )qt , this follows from (2.1) and (2.2). Indeed since the values of b0 − b1 p+· · ·±bf −1 pf −1 are distinct, we see that there is an exponent in every non-zero congruence class mod (q + 1). For representations of the form ω r ⊕ ω s , it suffices to note similarly that r − s mod (q − 1) arises as the difference of exponents of ω0 for some β(V ). i

i

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Fred Diamond

Suppose now that ρ : GK → GL2 (Fp ) restricts to β(V ) on IK . To prove the assertion about W (ρ), we can twist V by χ ◦ det and ρ by a character restricting to β(χ) and so assume V is as in Proposition 1.1 or 1.3. We now need to show that each non-zero Vm J , nJ is in W (ρ). f −1

(b −1)pi

Suppose first that V = I(1, τ˜0 i=0 i ) with each bi ∈ {1, . . . , p} and f is odd. Given J, we let J = { j ∈ S | j ≡ i mod f for some i ∈ J }, J0 = { j ∈ S | j is even }, J1 = { j ∈ S | j is odd } and J ∗ = (J0 ∩J )∪(J1 J ). f −1 ∼ We then have π : J ∗ → S and i=0 (q+1)mJ,i pi + i∈J ∗ nJ,i pi is congruent mod (q 2 − 1) to (bi − 1 + δJ (i))pi i∈J

+

(bi −1 + δJ (i))pi +

i∈J0 ∩J

≡

(p − bi + 1 − δJ (i))pi

i∈J1 J

(bi − 1 + δJ (i))pi +

i∈J0

≡

pi+1

i∈J1 J

bi p mod (q − 1) i

2

i∈J0

i i+1 . It follows that Vm J , nJ ∈ W (ρ). i∈J0 (δJ (i) − 1)p ≡ i∈J1 J p f −1 1+ i=0 (bi −1)pi If V = Θ((˜ τ0 ) ) and f is even, then we proceed exactly as above, but with J0 = {0, 2, . . . , f −2, f +1, f +3, . . . , 2f −1} and J1 = S f −1 (b −1)pi J0 . The remaining cases are similar, but simpler. If V = I(1, τ˜0 i=0 i ) f −1 i (resp. Θ((˜ τ0 )1+ i=0 (bi −1)p )) and f is even (resp. odd), we let J0 = { j ∈ S | j is even }, J1 = { j ∈ S | j is odd } and J ∗ = (J0 ∩ J) ∪ (J1 J). In each

since

case a calculation similar to the one above shows that Vm J , nJ ∈ W (ρ).

3 Exceptional weights Let β be the bijection of Theorem 2.1, and suppose throughout this section that ρ : GK → GL2 (Fp ) restricts to β(V ) on IK . We say that Vm, n is an exceptional weight for ρ (or V ) if it lies in the complement in W (ρ) of the set of constituents of V . In this section we characterize the exceptional weights. We first give a sufficient condition for there to be none. f −1 Theorem 3.1 Suppose V = I(˜ τ0c , τ˜0a+c ) with a = i=0 ai pi . If f is odd and 1 ≤ ai ≤ p − 2 for each i, then there are no exceptional weights for V . If f is even and a ≡ ± q−1 mod (q − 1) or 1 ≤ ai ≤ p − 2 for each i, then there are p+1

A correspondence between representations

201

no exceptional weights for V unless a ≡ ±2 q−1 mod (q − 1), in which case p+1 the only exceptional weights are those Vm, n = p. n ∈ W (ρ) with f −1 i Suppose V = Θ((˜ τ0 )1+a+c(q+1) ) with a = a p i=0 i . If f is even and 1 ≤ ai ≤ p − 2 for each i, then there are no exceptional weights for V . If f q+1 is odd and 1 + a ≡ ± p+1 mod (q + 1) or 1 ≤ ai ≤ p − 2 for each i, then q+1 there are no exceptional weights for V unless 1 + a ≡ ±2 p+1 mod (q + 1), in which case the only exceptional weights are those Vm, n = p. n ∈ W (ρ) with We remark that the special cases in the statement are precisely those where ρ is the sum of two characters whose ratio is trivial or cyclotomic on inertia. Proof. We first treat the cases where 1 ≤ ai ≤ p − 2 for each i. In this case the f Vm J , nJ of Propositions 1.1 and 1.3 are all non-zero, so V has 2 constituents. If β(V ) is irreducible, then Proposition 3.1 of [5] shows that #W (ρ) ≤ 2f , so it follows that equality holds and there are no exceptional weights in this case. So suppose that β(V ) is reducible. Propositions 3.4 and 3.5 of [5] then show that #W (ρ) ≤ 2f unless f −1

(−1)i bi pi ≡ (p + 1)

i=0

(−1)i pi mod (q − 1)

i∈J ∗

for some J ∗ ⊂ S, where each bi = ai + 1 ∈ {2, . . . , p − 1}. If f is even, then the left-hand side is strictly between 1 − q and 0 while the right hand side is between −2(q − 1) and q − 1. Setting f −1

i=0

(−1)i bi pi = c(q − 1) + (p + 1)

(−1)i pi

i∈J

for c = 0, ±1 and solving p-adically, the restriction on bi forces either J = {0, 2, . . . , f − 2} and a = (1, p − 2, 1, . . . , p − 2) or J = {1, 3, . . . , f − 1} and a = (p − 2, 1, p − 2, . . . , 1), which gives a ≡ ±2 q−1 p+1 mod (q − 1). In this case f one has #W (ρ) ≤ 2 + 1 if p > 3, so there is at most one exceptional weight. Note also that there is an element of W (ρ) of the form Vm, p , but no such factor of V since a ≡ 0 mod (q − 1). If p = 3, one has #W (ρ) ≤ 2f + 2 and two elements of W (ρ) of the form Vm, p accounting for all exceptional weights. The argument in the case of odd f is similar, but the left-hand side is strictly between 0 and q −1 while the right-hand side is between −(q −1) and 2(q −1) and we get that either J = {0, 2, . . . , f − 1} and a = (1, p − 2, 1, . . . , p − 2, 1) or J = {1, 3, . . . , f − 2} and a = (p − 2, 1, p − 2, . . . , 1, p − 2), giving q+1 1 + a ≡ ±2 p+1 mod (q + 1). We now turn our attention to the remaining cases. Suppose first that p > 2 and f is even. Twisting V and ρ, we can assume that

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Fred Diamond

a = (p − 1, 0, p − 1, 0, . . . , p − 1, 0) and 3

ρIK ∼ ω0p+p

+···+pf −1

3

⊕ ω0p+p

+···+pf −1

.

For each J ∗ ⊂ S, we explicitly describe the Vm, n such that f /2 i=0

p1+2i ≡

f −1 i=0

mi pi +

i∈J ∗

ni pi ≡

f −1

mi pi +

ni pi mod (q − 1).

i∈J ∗

i=0

Propositions 3.4 and 3.5 of [5] show that this holds for a unique n unless J ∗ = {0, 2, . . . , f − 2} or {1, 3, . . . , f − 1}. For each of these two values of J ∗ , there are two possibilities for n, namely (p, 1, p, 1, . . . , p, 1) and (1, p, 1, p, . . . , 1, p). Otherwise there is an i such that χJ ∗ (i − 1) = χJ ∗ (i) where χJ ∗ is the characteristic function of J ∗ , and n is characterized as the unique f -tuple such that • • • •

ni ∈ {0, p − 1, p} for all i; if ni−1 = 1, then ni = p; if ni−1 = p − 1 or p, then ni = p − 1 if χJ ∗ (i − 1) = χJ ∗ (i); if ni−1 = p − 1 or p, then ni = 1 if χJ ∗ (i − 1) = χJ ∗ (i). Note also that mi pi mod (q − 1) is determined by n and J ∗ . It is then straightforward to check that each such Vm, n arises as Vm J , nJ where J = {j ∈ {0, 2, . . . , f − 2} | nj = p − 1 or p} ∪ {j ∈ {1, 3, . . . , f − 1} | nj = 1}, so there are no exceptional weights. (Note that (J )∗ need not coincide with J ∗ .) The case of odd f , p > 2 is similar. We assume a = (p − 1, 0, p − 1, 0, . . . , p − 1) and 3

ρIK ∼ ω01+p+p

+···+pf −2

3

⊕ ω01+p+p

+···+pf −2

.

For each J ∗ , there is a unique possibility for n characterized exactly as in the case of f even, and we set J = { j ∈ {0, 2, . . . , f − 1} | nj = p − 1 or p } ∪ { j ∈ {1, 3, . . . , f − 2} | nj = 1 } to conclude there are no exceptional weights.

A correspondence between representations

203

Finally if p = 2, then one argues as above using Proposition 3.6 of [5], but with two changes. Firstly, we find also that there are two possibilities for n if J ∗ = ∅ or S, namely n = (1, 1, . . . , 1) or (2, 2, . . . , 2), the latter being exceptional. Secondly, to generalize the characterization of n one defines ni (x) ∈ {0, x − 1, x} and then sets n = n(p) with p = 2. We finish with a complete characterization of the exceptional weights. Theorem 3.2 Suppose that Vm, n ∈ W (ρ). r s ∗ If ρ|IK ∼ ω0 ⊕ ω0 , then Vm, n is exceptional if and only if for each J ⊂ S such that f −1

r≡

i

mi p +

f −1 i

ni p ,

s≡

i∈J ∗

i=0

mi pi +

i=0

ni pi mod (q − 1), (3.1)

i∈J ∗

we have ni = p and χJ ∗ (i − 1) = χJ ∗ (i) for some i ∈ S. If ρ|IK ∼ (ω0 )t ⊕ (ω0 )qt , then Vm, n is exceptional if and only if for each ∗ J ⊂ S such that t≡

f −1

i=0

i∈J ∗

(q + 1)mi pi +

ni pi mod (q 2 − 1) and

∼

π : J ∗ → S, (3.2)

we have ni = p and χJ ∗ (i − 1) = χJ ∗ (i) for some i ∈ S . Proof. We note first that every Vm, n as in the statement of the theorem is indeed exceptional, for if it is equivalent to Vm J , nJ for some J ⊂ S, then the proof of Theorem 2.1 provides a J ∗ such that (3.1) or (3.2) holds, but the explicit formula for nJ shows that ni < p whenever χJ ∗ (i − 1) = χJ ∗ (i). Suppose on the other hand that (3.1) or (3.2) holds for some J ∗ such that χJ ∗ (i − 1) = χJ ∗ (i) whenever ni = p. We then choose J so that J ∗ is as in the proof of Theorem 2.1 and verify that Vm, n = Vm J , nJ (except possibly in the case of reducible ρ with r = s, where the result is already immediate from Theorem 3.1). We can twist V and ρ and so assume V is as in the statements of Proposition 1.1 or 1.3. f −1

Suppose first that V = I 1, τ˜0

i=0

ai pi

with f even and 0 ≤ ai p − 1 for

each i. We then have f −1

mi pi +

i=0

ni pi ≡ b0 + b2 p2 + · · · bf −2 pf −2 mod (q − 1) (3.3)

i∈J ∗

i=0 f −1

mi pi +

i∈J ∗

ni pi ≡ b1 p + b3 p3 + · · · bf −1 pf −1 mod (q − 1) (3.4)

204

Fred Diamond

where each bi = ai + 1. Let bi = ni − δJ (i) + 1 for i ∈ J and bi = p − ni − δJ (i) + 1 for i ∈ J. The condition that χJ ∗ (i − 1) = χJ ∗ (i) whenever ni = p guarantees that 1 ≤ bi ≤ p for each i. It is then straightforward to check that f −1

i=0

i∈J ∗

(−1)i bi pi ≡

ni pi −

ni pi mod (q − 1),

i∈J ∗

f −1 which by (3.3) is congruent to i=0 bi pi . Since 1 ≤ bi , bi ≤ p for each i and f −1 i we have ruled out the case i=0 bi p ≡ 0 mod (q − 1), it follows that bi = bi for all i, and therefore that n = nJ . We then compute that f −1

mi pi

≡

i=0

≡

bi pi −

even i

ni pi

i∈J ∗

(ai + δJ (i))pi +

even i ∈ J ∗ ≡ mJ,i pi ,

(ai + δJ (i))pi

odd i ∈ J ∗

i∈S

hence Vm, n = Vm J , nJ .

f −1 a pi Suppose next that V = I 1, τ˜0 i=0 i but f is odd. We then start with

the congruence f −1

mi (q + 1)pi +

ni pi ≡ b0 + b2 p2 + · · · b2f −2 p2f −2 mod (q 2 − 1)

i∈J ∗

i=0

instead of (3.3). Defining bi and arguing as above then gives f −1

(−1)i bi pi ≡

i=0

f −1

(−1)i bi pi mod (q + 1),

i=0

so b = b and n = nJ . Similarly one finds that f −1

(q + 1)

mi pi ≡

i=0

bi pi −

ni pi

i∈J ∗ even i ∈ S ≡ (q + 1) (ai + δJ (i))pi mod (q 2 − 1) i∈J

giving Vm, n = Vm J , nJ .

A correspondence between representations

205

f −1 1 Now suppose that V = Θ (˜ τ0 )1+ i=0 ai p with f even. We then have f −1

mi (q + 1)pi +

ni pi

i∈J ∗

i=0

≡ b0 + b2 p2 + · · · + bf −2 pf −2 + pf + bf +1 pf +1 + · · · + b2f −1 p2f −1 mod (q 2 − 1). We define bi for i > 0 as above, but set b0 = n0 + δJ (0) or p − n0 + δJ (0) according to whether 0 ∈ J ∗ . Arguing as above, with special attention to the terms with i = 0, f , then gives f −1

(−1)i bi pi ≡ 1 +

ni pi −

i∈J ∗ ,0≤i
i=0

ni pi

i∈J ∗ ,0≤i
f −1

≡

(−1)i bi pi mod (q + 1),

i=0

so that b = b and n = nJ . Again computing mod (q 2 − 1), but with special attention to i = 0, f , gives f −1

f −1

mi p ≡ i

i=0

mJ,i pi mod (q − 1)

i=0

so that Vm, n = Vm J , nJ .

f −1 1 Finally suppose that V = Θ (˜ τ0 )1+ i=0 ai p with f odd. Starting with

f −1

mi pi +

i=0

i=0

ni pi

≡ b0 + b2 p2 + · · · bf −1 pf −1 mod (q − 1)

ni pi

≡ 1 + b1 p + b3 p3 + · · · bf −1 pf −1 mod (q − 1),

i∈J ∗

f −1

mi pi +

i∈J ∗

and defining bi as in the preceding case, we get f −1 i=0

(−1)i bi pi

≡

f −1 i=0

(−1)i bi pi mod (q − 1),

206

Fred Diamond

again giving b = b since r = s. Checking again that f −1

i=0

f −1

mi p ≡ i

mJ,i pi mod (q − 1)

i=0

yields Vm, n = Vm J , nJ .

Bibliography [1]

[2] [3] [4] [5] [6]

[7] [8] [9] [10] [11] [12] [13] [14]

A. Ash, W. Sinnott, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod p cohomology of GL(n, Z), Duke Math. J. 105 (2000), 1–24. A. Ash, D. Doud, D. Pollack, Galois representations with conjectural connections to arithmetic cohomology., Duke Math. J. 112 (2002), 521–579. C. Breuil, Sur quelques repr´esentations modulaires et p-adiques de GL2 (Qp ). I, Comp. Math. 138 (2003), 165–188. C. Breuil, Sur quelques repr´esentations modulaires et p-adiques de GL2 (Qp ). II, J. Inst. Math. Jussieu 2 (2003), 23–58. K. Buzzard, F. Diamond, A.F. Jarvis, On Serre’s conjecture for mod Galois representations over totally real fields, preprint. F. Diamond, The refined conjecture of Serre, in Elliptic Curves and Fermat’s Last Theorem, J.Coates and S.-T.Yau (eds.), Hong Kong 1993, Intl. Press, 2nd ed. (1997), 172–186. M. Emerton, A local-global compatibility conjecture in the p-adic Langlands programme for GL2 /Q, Pure Appl. Math. Q. 2 (2006), 279-393. F. Herzig, The weight in a Serre-type conjecture for tame n-dimensional Galois representations, Ph.D. Thesis, Harvard University, 2006. G. James, M. Liebeck, Representations and Characters of Groups, 2nd ed. Cambridge Univ. Press, 2001. S. Lang, Algebra, 3rd ed. Addison-Wesley, 1993, GTM 211, Springer, 2002. K. Ribet, On modular representations of Gal(Q/Q) arising from modular forms, Inv. Math. 100 (1990), 431–476. J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag, 1977. J.-P. Serre, Sur les repr´esentations modulaires de degr´e 2 de Gal(Q/Q), Duke Math. J. 54 (1987), 179–230. B. S. Upadhyaya, Composition factors of the principal indecomposable modules for the special linear groups SL(2, q), Jour. London Math. Soc. 17 (1978), 437-445.

Non-vanishing modulo p of Hecke L–values and application Haruzo Hida Department of Mathematics UCLA Los Angeles, Ca 90095-1555 U.S.A. [email protected] a a

The author is partially supported by the NSF grants: DMS 0244401 and DMS 0456252

Contents 1 2

3

4

5

Introduction 208 Hilbert Modular Forms 209 2.1 Abelian variety with real multiplication 209 2.2 Abelian varieties with complex multiplication 210 2.3 Geometric Hilbert modular forms 212 2.4 p–Adic Hilbert modular forms 213 2.5 Complex analytic Hilbert modular forms 214 2.6 Differential operators 216 217 2.7 Γ0 –level structure and Hecke operators 2.8 Hilbert modular Shimura varieties 218 2.9 Level structure with “Neben” character 220 2.10 Adelic Hilbert modular forms and Hecke algebras 221 Eisenstein series 227 3.1 Arithmetic Hecke characters 227 3.2 Hilbert modular Eisenstein series 228 3.3 Hecke eigenvalues 230 3.4 Values at CM points 233 Non-vanishing modulo p of L–values 236 4.1 Construction of a modular measure 236 4.2 Non-triviality of the modular measure 239 4.3 l–Adic Eisenstein measure modulo p 241 Anticyclotomic Iwasawa series 246 5.1 Adjoint square L–values as Petersson metric 248 5.2 Primitive p–Adic Rankin product 251 5.3 Comparison of p–adic L–functions 261 5.4 A case of the anticyclotomic main conjecture 265

207

208

Haruzo Hida 1 Introduction

Let F be a totally real field and M/F be a totally imaginary quadratic extension (a CM field). We fix a prime p > 2 unramified in M/Q and suppose that all prime factors of p in F split in M (M is p–ordinary). Fixing two embeddings i∞ : Q → C and ip : Q → Qp , we take a p–ordinary CM type Σ of M . Thus Σp = {ip ◦ σ} is exactly a half of the p–adic places of M . Fix a Hecke character λ of infinity type kΣ + κ(1 − c) with 0 < k ∈ Z and κ = σ∈Σ κσ σ with κσ ≥ 0. If the conductor C of λ is a product of primes split in M/F , we call λ has split conductor. Throughout this paper, we assume that λ has split conductor. We fix a prime l Cp of F . As is well known ([K] and [Sh1]), for a finite order Hecke character χ and for a power Ω of the N´eron period of an abelian scheme (over a p–adic valuation ring) of CM type (p) Σ, the L–value L (0,λχ) is (p–adically) integral (where the superscript: “(p)” Ω indicates removal of Euler factors at p). The purpose of this paper is three fold: (p)

for (1) To prove non-vanishing modulo p of Hecke L–values L (0,λχ) Ω “almost all” anticyclotomic characters χ of finite order with l–power conductor (under some mild assumptions; Theorem 4.3); − (2) To prove the divisibility: L− p (ψ)|F (ψ) in the anticyclotomic Iwasawa − algebra Λ of M for an anticyclotomic character ψ of split conductor, where L− p (ψ) is the anticyclotomic Katz p–adic L–function of the branch character ψ and F − (ψ) is the corresponding Iwasawa power series (see Theorem 5.1). − (3) To prove the equality L− p (ψ) = F (ψ) up to units under some assump√ tions if F/Q is an abelian extension (Theorem 5.8) and M = F [ D] for 0 > D ∈ Z.

Roughly speaking, F − (ψ) is the characteristic power series of the ψ-branch of the Galois group of the only Σp –ramified p–abelian extension of the anticyclotomic tower over the class field of ψ. The first topic is a generalization of the result of Washington [Wa] (see also [Si]) to Hecke L–values, and the case where λ has conductor 1 has been dealt with in [H04c] basically by the same technique. The phrase “almost all” is in the sense of [H04c] and means “Zariski densely populated characters”. If l has degree 1 over Q, we can prove a stronger non-vanishing modulo p outside a (non-specified) finite set. In [HT1] and [HT2], we have shown the divisibility in item (2) in Λ− ⊗Z Q and indicated that the full divisibility holds except for p outside an explicit finite set S of primes if one obtains the result claimed in (1). We will show that S is limited to ramified primes and even primes.

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Though the result in (2) is a direct consequence of the vanishing of the − μ–invariant of L− p (ψ) proven in [H04c] by the divisibility in Λ ⊗Z Q, we shall give another proof of this fact using the non-vanishing (1). We will actually show a stronger result (Corollary 5.6) asserting that the relative class number h(M/F ) times L− p (ψ) divides the congruence power series of the CM component of the nearly ordinary Hecke algebra (which does not directly follow from the vanishing of μ). Our method to achieve (2) is a refinement of the work [HT1] and [HT2], and this subtle process explains the length of the paper. Once the divisibility (2) is established, under the assumption of (3), if ψ √ − descends to a character of Gal(Q/Q[ D]), we can restrict L− p (ψ) and F (ψ) √ to a Zp -extension of an abelian extension of Q[ D], and applying Rubin’s identity of the restricted power series ([R] and [R1]), we conclude the identity − L− p (ψ) = F (ψ). We should mention that the stronger divisibility of the congruence power series by h(M/F )L− p (ψ) in this paper will be used to prove the equality of L− (ψ) and F − (ψ) under some mild conditions on ψ for general base fields F in our forthcoming paper [H04d]. We shall keep the notation and the assumptions introduced in this introduction throughout the paper.

2 Hilbert Modular Forms We shall recall algebro-geometric theory of Hilbert modular forms limiting ourselves to what we need later.

2.1 Abelian variety with real multiplication Let O be the integer ring of F , and put O∗ = {x ∈ F |Tr(xO) ⊂ Z} (which is the inverse different d−1 ). We fix an integral ideal N and a fractional ideal c of F prime to N. We write A for a fixed base algebra, in which N (N) and N (c) is invertible. The Hilbert modular variety M(c; N) of level N classifies triples (X, Λ, i)/S formed by • An abelian scheme π : X → S for an A–scheme S with an embedding: O → End(X/S ) making π∗ (ΩX/S ) a locally free O ⊗ OS –module of rank 1; • An O–linear polarization Λ : X t = Pic0X/S ∼ = X ⊗ c; • A closed O–linear immersion i = iN : (Gm ⊗ O∗ )[N] → X. By Λ, we identify the O–module of symmetric O–linear homomorphisms with c. Then we require that the (multiplicative) monoid of symmetric O–linear isogenies induced locally by ample invertible sheaves be identified with the set of

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totally positive elements c+ ⊂ c. Thus M(c; N)/A is the coarse moduli scheme of the following functor from the category of A–schemes into the category SET S: / 0 P(S) = (X, Λ, i)/S , where [ ] = { }/ ∼ = is the set of isomorphism classes of the objects inside the brackets, and we call (X, Λ, i) ∼ = (X , Λ , i ) if we have an O–linear isomor such that Λ = φ ◦ Λ ◦ φt and φ ◦ i = i . The scheme phism φ : X/S → X/S M is a fine moduli if N is sufficiently deep. In [K] and [HT1], the moduli M is described as an algebraic space, but it is actually a quasi-projective scheme (e.g. [C], [H04a] Lectures 5 and 6 and [PAF] Chapter 4).

2.2 Abelian varieties with complex multiplication p for the p–adic We write | · |p for the p–adic absolute value of Qp and Q completion of Qp under | · |p . Recall the p–ordinary CM type (M, Σ), and let R be the integer ring of M . Thus Σ%Σc for the generator c of Gal(M/F ) gives the set of all embeddings of M into Q. For each σ ∈ (Σ ∪ Σc), ip σ induces a p–adic place pσ giving rise to the p–adic absolute value |x|pσ = |ip (σ(x))|p . We write Σp = {pσ |σ ∈ Σ} and Σp c = {pσc |σ ∈ Σ}. By ordinarity, we have Σp ∩ Σp c = ∅. For each O–lattice a ⊂ M whose p–adic completion ap is identical to Rp = R ⊗Z Zp , we consider the complex torus X(a)(C) = CΣ /Σ(a), where Σ(a) = {(i∞ (σ(a)))σ∈Σ |a ∈ a}. By a theorem in [ACM] 12.4, this complex torus is algebraizable to an abelian variety X(a) of CM type (M, Σ) over a number field. Let F be an algebraic closure of the finite field Fp of p–elements. We write p unramified over W for the p–adically closed discrete valuation ring inside Q Zp with residue field F. Thus W is isomorphic to the ring of Witt vectors with coefficients in F. Let W = i−1 p (W ), which is a strict henselization of p , which Z(p) = Q∩Zp . In general, we write W for a finite extension of W in Q is a complete discrete valuation ring. We suppose that p is unramified in M/Q. Then the main theorem of complex multiplication ([ACM] 18.6) combined with the criterion of good reduction over W [ST] tells us that X(a) is actually defined over the field of fractions K of W and extends to an abelian scheme over W (still written as X(a)/W ). All endomorphisms of X(a)/W are defined over W. We write θ : M → End(X(a)) ⊗Z Q for the embedding of M taking α ∈ M to the complex multiplication by Σ(α) on X(a)(C) = CΣ /Σ(a). Let R(a) = {α ∈ R|αa ⊂ a}. Then R(a) is an order of M over O. Recall the prime l p of F in the introduction. The order R(a) is determined by its

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conductor ideal which we assume to be an l–power le . In other words, R(a) = Re := O + le R. The following three conditions for a fractional Re –ideal a are equivalent (cf. [IAT] Proposition 4.11 and (5.4.2) and [CRT] Theorem 11.3): (I1) a is Re –projective; (I2) a is locally principal; (I3) a is a proper Re –ideal (that is, Re = R(a)). Thus Cle := Pic(Re ) is the group of Re –projective fractional ideals modulo principal ideals. The group Cle is finite and called the ring class group modulo le . We choose and fix a differential ω = ω(R) on X(R)/W so that H 0 (X(R), ΩX(R)/W ) = (W ⊗Z O)ω. If ap = Rp , X(R ∩ a) is an e´ tale covering of both X(a) and X(R); so, ω(R) induces a differential ω(a) first by pull-back to X(R∩a) and then by pull-back inverse from X(R ∩ a) to X(a). As long as the projection π : X(R ∩ a) X(a) is e´ tale, the pull-back inverse (π ∗ )−1 : ΩX(R∩a)/W → ΩX(a)/W is a surjective isomorphism. We thus have H 0 (X(a), ΩX(R)/W ) = (W ⊗Z O)ω(a). We choose a totally imaginary δ ∈ M with Im(i∞ (σ(δ))) > 0 for all σ ∈ Σ such that (a, b) → (c(a)b − ac(b))/2δ gives the identification R ∧ R ∼ = d−1 c−1 . We assume that c is prime to p (( ) = l ∩ Z). This Riemann form: R∧R ∼ = c∗ = d−1 c−1 gives rise to a c–polarization Λ = Λ(R) : X(R)t ∼ = X(R) ⊗ c, which is again defined over W. Here d is the different of F/Q, and c∗ = {x ∈ F |TrF/Q (xc) ⊂ Z}. Since we have Re ∧ Re = le (O ∧ R) + l2e (R ∧ R), the pairing induces Re ∧ Re ∼ = (cl−e )∗ , and this pairing induces a cl−e NM/F (a)−1 –polarization Λ(a) on X(a) for a proper Re –ideal a. We choose a local generator a of al . Multiplication by a induces an isomorphism Re,l ∼ = al . Since X(Re )/W has a subgroup C(Re ) = R/(O + le R) ⊂ X(Re ) isomorphic e´ tale-locally to O/le . This subgroup C(Re ) is sent by multiplication by a to C(a) ⊂ X(a)/W , giving rise to a Γ0 (le )–level structure C(a) on X(a). For our later use, we choose ideals F and Fc of R prime to c so that F ⊂ Fcc and F + Fc = R. The product C = FFc shall be later the conductor of the Hecke character we study. We put f = F ∩ O and f = Fc ∩ O; so, f ⊂ f . We shall define a level f2 –structure on X(a): supposing that a is prime to f, we have af ∼ = Rf = RF × RFc , which induces a canonical identification i(a) : f∗ /O∗ = f−1 /O ∼ = F−1 /R ∼ = F−1 aF /aF ⊂ X(a)[f].

(2.1)

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This level structure induces i (a) : X(a)[f] f−1 /O by the duality under Λ. In this way, we get many sextuples: (X(a), Λ(a), i(a), i (a), C(a)[l], ω(a)) ∈ M(cl−e (aac )−1 ; f2 , Γ0 (l))(W) (2.2) as long as le is prime to p, where C(a)[l] = {x ∈ C(a)|lx = 0}. A precise definition of the moduli scheme of Γ0 –type: M(cl−e (aac )−1 ; f2 , Γ0 (l)) classifying such sextuples will be given in 2.7. The point x(a) = (X(a), Λ(a), i(a), i (a)) of the moduli scheme M(c(aac )−1 ; f2 ) is called a CM point associated to X(a).

2.3 Geometric Hilbert modular forms We return to the functor P in 2.1. We could insist on freeness of the differentials π∗ (ΩX/S ), and for ω with π∗ (ΩX/S ) = (OS ⊗Z O)ω, we consider the functor classifying quadruples (X, Λ, i, ω): 0 / Q(S) = (X, Λ, i, ω)/S . Let T = ResO/Z Gm . We let a ∈ T (S) = H 0 (S, (OS ⊗Z O)× ) act on Q(S) by (X, Λ, i, ω) → (X, Λ, i, aω). By this action, Q is a T –torsor over P; so, Q is representable by an A–scheme M = M(c; N) affine over M = M(c; N)/A . For each character k ∈ X ∗ (T ) = Homgp−sch (T, Gm ), if F = Q, the k −1 – eigenspace of H 0 (M/A , OM/A ) is by definition the space of modular forms of weight k integral over A. We write Gk (c, N; A) for this space of A–integral modular forms, which is an A–module of finite type. When F = Q, as is well known, we need to take the subsheaf of sections with logarithmic growth towards cusps (the condition (G0) below). Thus f ∈ Gk (c, N; A) is a functorial rule assigning a value in B to each isomorphism class of (X, Λ, i, ω)/B (defined over an A–algebra B) satisfying the following three conditions: (G1) f (X, Λ, i, ω) ∈ B if (X, Λ, i, ω) is defined over B; (G2) f ((X, Λ, i, ω) ⊗B B ) = ρ(f (X, Λ, i, ω)) for each morphism ρ : B/A → B/A ; (G3) f (X, Λ, i, aω) = k(a)−1 f (X, Λ, i, ω) for a ∈ T (B). By abusing the language, we pretend f to be a function of isomorphism classes of test objects (X, Λ, i, ω)/B hereafter. The sheaf of k −1 –eigenspace OM [k −1 ] under the action of T is an invertible sheaf on M/A . We write this sheaf as ω k (imposing (G0) when F = Q). Then we have Gk (c, N; A) = H 0 (M(c; N)/A , ωk/A )

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as long as M(c; N) is a fine moduli space. Writing X = (X, λ, i, ω) for the universal abelian scheme over M, s = f (X)ω k gives rise to the section of ω k . Conversely, for any section s ∈ H 0 (M(c; N), ω k ), taking a unique morphism φ : Spec(B) → M such that φ∗ X = X for X = (X, Λ, i, ω)/B , we can define f ∈ Gk by φ∗ s = f (X)ω k . We suppose that the fractional ideal c is prime to Np, and take two ideals a and b prime to Np such that ab−1 = c. To this pair (a, b), we can attach the Tate AVRM T atea,b (q) defined over the completed group ring Z((ab)) made ξ of formal series f (q) = ξ−∞ a(ξ)q (a(ξ) ∈ Z). Here ξ runs over all elements in ab, and there exists a positive integer n (dependent on f ) such that a(ξ) = 0 if σ(ξ) < −n for some σ ∈ I. We write A[[(ab)≥0 ]] for the subring of A[[ab]] made of formal series f with a(ξ) = 0 for all ξ with σ(ξ) < 0 for at least one embedding σ : F → R. Actually, we skipped a step of introducing the toroidal compactification of M whose (completed) stalk at the cusp corresponding to (a, b) actually carries T atea,b (q). However to make exposition short, we ignore this technically important point, referring the reader to the treatment in [K] Chapter I, [C], [DiT], [Di], [HT1] Section 1 and [PAF] 4.1.4. The scheme T ate(q) can be extended to a semi-abelian scheme over Z[[(ab)≥0 ]] adding the fiber Gm ⊗ a∗ . Since a is prime to p, ap = Op . Thus if A is a Zp –algebra, we have a canonical isomorphism: Lie(T atea,b (q) mod A) = Lie(Gm ⊗ a∗ ) ∼ = A ⊗Z a ∗ ∼ = A ⊗Z O∗ . By Grothendieck-Serre duality, we have ΩT atea,b (q)/A[[(ab)≥0 ]] ∼ = A[[(ab)≥0 ]]. Indeed we have a canonical generator ωcan of ΩT ate(q) induced by dt t ⊗ 1 on ∗ ∗ ∗ Gm ⊗ a . We have a canonical inclusion (Gm ⊗ O )[N] = (Gm ⊗ a )[N] into Gm ⊗a∗ , which induces a canonical closed immersion ican : (Gm ⊗O∗ )[N] → T ate(q). As described in [K] (1.1.14) and [HT1] page 204, T atea,b (q) has a canonical c–polarization Λcan . Thus we can evaluate f ∈ Gk (c, N; A) at (T atea,b (q), Λcan , ican , ωcan ). The value f (q) = fa,b (q) actually falls in A[[(ab)≥0 ]] (if F = Q : Koecher principle) and is called the q–expansion at the cusp (a, b). When F = Q, we impose f to have values in the power series ring A[[(ab)≥0 ]] when we define modular forms: (G0) fa,b (q) ∈ A[[(ab)≥0 ]] for all (a, b).

2.4 p–Adic Hilbert modular forms Suppose that A = limn A/pn A and that N is prime to p. We can think of a ←− functor / 0 P(A) = (X, Λ, ip , iN )/S

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similar to P that is defined over the category of p–adic A–algebras B = limn B/pn B. An important point is that we consider an isomorphism of ind←− group schemes ip : μp∞ ⊗Z O∗ → X[p∞ ] (in place of a differential ω), which m ⊗ O∗ ∼ for the formal completion V at the characteristic induces G = X p–fiber of a scheme V over A. It is a theorem (due to Deligne-Ribet and Katz) that this functor is representable by the formal Igusa tower over the formal completion M(c; N) of M(c; N) along the ordinary locus of the modulo p fiber (e.g., [PAF] 4.1.9). A p–adic modular form f/A for a p–adic ring A is a function (strictly speaking, a functorial rule) of isomorphism classes of (X, Λ, ip , iN )/B satisfying the following three conditions: (P1) f (X, Λ, ip , iN ) ∈ B if (X, Λ, ip , iN ) is defined over B; (P2) f ((X, Λ, ip , iN ) ⊗B B ) = ρ(f (X, Λ, ip , iN )) for each continuous A–algebra homomorphism ρ : B → B ; (P3) fa,b (q) ∈ A[[(ab)≥0 ]] for all (a, b) prime to Np. We write V (c, N; A) for the space of p–adic modular forms satisfying (P1-3). This V (c, N; A) is a p–adically complete A–algebra. We have the q–expansion principle valid both for classical modular forms and p–adic modular forms f , (q-exp)

f is uniquely determined by the q–expansion: f → fa,b (q) ∈ A[[(ab)≥0 ]].

This follows from the irreducibility of (the Hilbert modular version of) the Igusa tower proven in [DeR] (see also [PAF] 4.2.4). m ⊗ O∗ has a canonical invariant differential dt , we have ωp = Since G t dt ip,∗ ( t ) on X. This allows us to regard f ∈ Gk (c, N; A) a p–adic modular form by f (X, Λ, ip , iN ) := f (X, Λ, iN , ωp ). By (q-exp), this gives an injection of Gk (c, N; A) into the space of p–adic modular forms V (c, N; A) (for a p–adic ring A) preserving q–expansions.

2.5 Complex analytic Hilbert modular forms Over C, the category of test objects (X, Λ, i, ω) is equivalent to the category of triples (L, Λ, i) made of the following data (by the theory of theta functions): L is an O–lattice in O ⊗Z C = CI , an alternating pairing Λ : L ∧O L ∼ = c∗ and ∗ ∗ i : N /O → F L/L. The alternating form Λ is supposed to be positive in the sense that Λ(u, v)/ Im(uv c ) is totally positive definite. The differential ω can

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be recovered by ι : X(C) = CI /L so that ω = ι∗ du where u = (uσ )σ∈I is the variable on CI . Conversely " " LX = ω ∈ O ⊗Z C"γ ∈ H1 (X(C), Z) γ

is a lattice in CI , and the polarization Λ : X t ∼ = X ⊗ c induces L ∧ L ∼ = c∗ . Using this equivalence, we can relate our geometric definition of Hilbert modular forms with the classical analytic definition. Define Z by the product of I copies of the upper half complex plane H. We regard Z ⊂ F ⊗Q C = CI made up of z = (zσ )σ∈I with totally positive imaginary part. For each z ∈ Z, we define √ Lz = 2π −1(bz + a∗ ), √ √ Λz (2π −1(az + b), 2π −1(cz + d)) = −(ad − bc) ∈ c∗ √ with iz : N∗ /O ∗ → CI /Lz given by iz (a mod O∗ ) = (2π −1a mod Lz ). Consider the following congruence subgroup Γ11 (N; a, b) given by " ( '# $ a b ∈ SL (F )" d ∈ O, b ∈ (ab)∗ , c ∈ Nabd and d − 1 ∈ N . "a, 2 c d We let g = (gσ ) ∈ SL2 (F ⊗Q R) = SL2 (R)I act on Z by linear fractional transformation of gσ on each component zσ . It is easy to verify (Lz , Λz , iz ) ∼ = (Lw , Λw , iw ) ⇐⇒ w = γ(z) for γ ∈ Γ11 (N; a, b). The set of pairs (a, b) with ab−1 = c is in bijection with the set of cusps (unramified over ∞) of Γ11 (N; a, b). Two cusps are equivalent if they transform each other by an element in Γ11 (N; a, b). The standard choice of the cusp is (O, c−1 ), which we call the infinity cusp of M(c; N). Write Γ11 (c; N) = Γ11 (N; O, c−1 ). For each ideal t, (t, tc−1 ) gives another cusp. The two cusps (t, tc−1 ) and (s, sc−1 ) are equivalent under Γ11 (c; N) if t = αs for an element × α ∈ F × with α ≡ 1 mod N in FN . We have M(c; N)(C) ∼ = Γ11 (c; N)\Z, canonically. Let G = ResO/Z GL(2). Take the following open compact subgroup of G(A(∞) ): '# $ ( " a b ∈ G(Z) "c ∈ NO and a ≡ d ≡ 1 mod NO , U11 (N) = c d −1

and put K = K11 (N) = ( d0 10 ) U11 (N) ( d0 10 ) for an idele d with dO = d and = c and c(c) = 1, we see that d(d) = 1. Then taking an idele c with cO −1 Γ11 (c; N) ⊂ ( 0c 10 ) K ( 0c 10 ) ∩ G(Q)+ ⊂ O× Γ11 (c; N)

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for G(Q)+ made up of all elements in G(Q) with totally positive determinant. Choosing a complete representative set {c} ⊂ FA× for the strict ray class group ClF+ (N) modulo N, we find by the approximation theorem that 3 G(A) = G(Q) ( 0c 10 ) K · G(R)+ + c∈ClF (N)

for the identity connected component G(R)+ of the Lie group G(R). This shows 3 G(Q)\G(A)/KCi ∼ M(c; N)(C), (2.3) = G(Q)+ \G(A)+ /KCi ∼ = + c∈ClF (N)

where G(A)+ = G(A(∞) )G(R)+ and Ci is the stabilizer in G(R)+ of i = √ √ ( −1 . . . , −1) ∈ Z. By (2.3), a ClF+ (N)–tuple (fc )c with fc ∈ Gk (c, N; C) can be viewed as a single automorphic form defined on G(A). Recall the identification of X ∗ (T ) with Z[I] so that k(x) = σ σ(x)kσ . Regarding f ∈ Gk (c, N; C) as a holomorphic function of z ∈ Z by f (z) = f (Lz , Λz , iz ), it satisfies the following automorphic property: # $ f (γ(z)) = f (z) (cσ zσ + dσ )kσ for all γ = ac db ∈ Γ11 (c; N). (2.4) σ

The holomorphy of f is a consequence of the functoriality (G2). The function f has the Fourier expansion a(ξ)eF (ξz) f (z) = ξ∈(ab)≥0

√ at the cusp corresponding to (a, b). Here eF (ξz) = exp(2π −1 σ ξ σ zσ ). This Fourier expansion gives the q–expansion fa,b (q) substituting q ξ for eF (ξz).

2.6 Differential operators Shimura studied the effect on modular forms of the following differential operators on Z indexed by k ∈ Z[I]: # $ ∂ 1 kσ σ √ √ δkσσ +2rσ −2 · · · δkσσ , and δkr = + δk = 2π −1 ∂zσ 2yσ −1 σ (2.5) where r ∈ Z[I] with rσ ≥ 0. An important point is that the differential operator preserves rationality property at CM points of (arithmetic) modular forms, although it does not preserve holomorphy (see [AAF] III and [Sh1]). We shall describe the rationality. The complex uniformization ι : X(a)(C) ∼ = CΣ /Σ(a)

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induces a canonical base ω∞ = ι∗ du of ΩX(a)/C over R ⊗Z R, where u = (uσ )σ∈Σ is the standard variable on CΣ . Define a period Ω∞ ∈ CΣ = O ⊗Z C by ω(R) = ω(a) = Ω∞ ω∞ . Here the first identity follows from the fact that ω(a) is induced by ω(R) on X(R). We suppose that a is prime to p. Here is the rationality result of Shimura for f ∈ Gk (c, f2 ; W): (δkr f )(x(a), ω∞ ) = (δkr f )(x(a), ω(a)) ∈ Q. (S) Ωk+2r ∞ Katz interpreted the differential operator in terms of the Gauss-Manin connection of the universal AVRM over M and gave a purely algebro-geometric definition of the operator (see [K] Chapter II and [HT1] Section 1). Using this algebraization of δkr , he extended the operator to geometric modular forms and p–adic modular forms. We write his operator corresponding to δ∗k as dk : V (c, N; A) → V (c, N; A). The level p–structure ip (a) : (Gm ⊗ O ∗ )[p∞ ] ∼ = ∞ MΣ /aΣ → X(a)[p ] (aΣ = P∈Σp aP = RΣ ) induces an isomorphism m ⊗ O∗ ∼ /W at the origin. ιp : G for the p–adic formal group X(a) = X(a) Then ω(R) = ω(a) = Ωp ωp (Ωp ∈ O ⊗Z W = W Σ ) for ωp = ιp,∗ dt t . An important formula given in [K] (2.6.7) is: for f ∈ Gk (c, f2 ; W), (dr f )(x(a), ωp ) = (dr f )(x(a), ω(a)) = (δkr f )(x(a), ω(a)) ∈ W. Ωk+2r p The effect of dr on q–expansion of a modular form is given by dr a(ξ)q ξ = a(ξ)ξ r q ξ . ξ

(K)

(2.6)

ξ

See [K] (2.6.27) for this formula.

2.7 Γ0 –level structure and Hecke operators We now assume that the base algebra A is a W–algebra. Choose a prime q of F . We are going to define Hecke operators U (qn ) and T (1, qn ) assuming for simplicity that q pN, though we may extend the definition for arbitrary q (see [PAF] 4.1.10). Then X[qr ] is an e´ tale group over B if X is an abelian scheme over an A–algebra B. We call a subgroup C ⊂ X cyclic of order qr if C∼ = O/qr over an e´ tale faithfully flat extension of B. We can think of quintuples (X, Λ, i, C, ω)/S adding an additional information C of a cyclic subgroup scheme C ⊂ X cyclic of order qr . We define the space of classical modular forms Gk (c, N, Γ0 (qr ); A) (resp. the space V (c, N, Γ0 (qr ); A) of p–adic modular forms) of level (N, Γ0 (qr )) by (G14) (resp. (P1-3)) replacing test objects (X, Λ, i, ω) (resp. (X, Λ, iN , ip )) by (X, Λ, i, C, ω) (resp. (X, Λ, iN , C, ip )).

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Our Hecke operators are defined on the space of level (N, Γ0 (qr )). The operator U (qn ) is defined only when r > 0 and T (1, qn ) is defined only when r = 0. For a cyclic subgroup C of X/B of order qn , we can define the quotient abelian scheme X/C with projection π : X → X/C . The polarization Λ and the differential ω induce a polarization π∗ Λ and a differential (π ∗ )−1 ω on X/C . If C ∩ C = {0} (in this case, we say that C and C are disjoint), π(C) gives rise to the level Γ0 (qr )–structure on X/C . Then we define for f ∈ Gk (cqn ; N, Γ0 (qr ); A), 1 f |U (qn )(X, Λ, C, i, ω) = f (X/C , π∗ Λ, π ◦ i, π(C), (π∗ )−1 ω), N (qn ) C (2.7) where C runs over all e´ tale cyclic subgroups of order qn disjoint from C. Since π∗ Λ = π ◦ Λ ◦ πt is a cqn –polarization, the modular form f has to be defined for abelian varieties with cqn –polarization. Since q N, forgetting the Γ0 (qn )–structure, we define for f ∈ Gk (cqn ; N; A) 1 f (X/C , π∗ Λ, π ◦ i, (π ∗ )−1 ω), (2.8) f |T (1, qn )(X, Λ, i, ω) = N (qn ) C

where C runs over all e´ tale cyclic subgroups of order qn . We can check that f |U (qn ) and f |T (1, qn ) belong to V (c, N, Γ0 (qr ); A) and also stay in Gk (c, N, Γ0 (qr ); A) if f ∈ Gk (cq, N, Γ0 (qr ); A). We have U (qn ) = U (q)n .

2.8 Hilbert modular Shimura varieties We extend the level structure i limited to N–torsion points to far bigger structure η (p) including all prime-to–p torsion points. Since the prime-to–p torsion on an abelian scheme X/S is unramified at p (see [ACM] 11.1 and [ST]), the extended level structure η (p) is still defined over S if S is a W–scheme. Triples (p) (X, Λ, η(p) )/S for W–schemes S are classified by an integral model Sh/W (cf. [Ko]) of the Shimura variety Sh/Q associated to the algebraic Q–group G = ResF/Q GL(2) (in the sense of Deligne [De] 4.22 interpreting Shimura’s original definition in [Sh] as a moduli of abelian schemes up to isogenies). Here the classification is up to prime-to–p isogenies, and Λ is an equivalence class of polarizations up to prime-to–p O–linear isogenies. To give a description of the functor represented by Sh(p) , we introduce some (p) more notations. We consider the fiber category AF over schemes defined by (Object) abelian schemes X with real multiplication by O; (Morphism) HomA(p) (X, Y ) = Hom(X, Y ) ⊗Z Z(p) , F

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where Z(p) is the localization of Z at the prime ideal (p), that is, ( 'a" "bZ + pZ = Z, a, b ∈ Z . Z(p) = b Isomorphisms in this category are isogenies with degree prime to p (called “prime-to–p isogenies”), and hence the degree of polarization Λ is supposed to be also prime to p. Two polarizations are equivalent if Λ = cΛ = Λ ◦ i(c) for a totally positive c prime to p. We fix an O–lattice L ⊂ V = F 2 with O– hermitian alternating pairing ·, · inducing a self duality on Lp = L⊗Z Zp . We consider the following condition on an AVRM X/S with θ : O → End(X/S ): (det) the characteristic polynomial of θ(a) (a ∈ O) on Lie(X) over OS is given by σ∈I (T − σ(a)), where I is the set of embeddings of F into Q. This condition is equivalent to the local freeness of π∗ ΩX/S over OS ⊗Z O for π : X → S. For an open-compact subgroup K of G(A(∞) ) maximal at p (i.e. K = GL2 (Op ) × K (p) ), we consider the following functor from Z(p) –schemes into SET S: 4 5 (p) PK (S) = (X, Λ, η (p) )/S with (det) . (2.9) Here η (p) : L⊗Z A(p∞) ∼ = V (p) (X) = T (X)⊗Z A(p∞) is an equivalence class (p) of η modulo multiplication η(p) → η (p) ◦ k by k ∈ K (p) for the Tate module T (X) = limN X[N] (in the sheafified sense that η (p) ≡ (η )(p) mod K ←− e´ tale-locally), and a Λ ∈ Λ induces the self-duality on Lp . As long as K (p) is (p) sufficiently small (for K maximal at p), PK is representable over any Z(p) – algebra A (e.g. [H04a], [H04b] Section 3.1 and [PAF] 4.2.1) by a scheme ShK/A = Sh/K, which is smooth by the unramifiedness of p in F/Q. We (p) let g ∈ G(A(p∞) ) act on Sh/Z(p) by x = (X, Λ, η) → g(x) = (X, Λ, η ◦ g), which gives a right action of G(A) on Sh(p) through the projection G(A) G(A(p∞) ). 1 (c; N) for By the universality, we have a morphism M(c; N) → Sh(p) /Γ 1 the open compact subgroup: # # $ $ 11 (c; N) = ( c 0 ) K11 (N) ( c 0 )−1 = cd−1 0 U11 (N) cd−1 0 −1 Γ 01 01 0 1 0 1 maximal at p. The image of M(c; N) gives a geometrically irreducible com 1 (c; N). If N is sufficiently deep, by the universality of ponent of Sh(p) /Γ 1

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1 (c; N). By the M(c; N), we can identify M(c; N) with its image in Sh(p) /Γ 1 action on the polarization Λ → αΛ for a suitable totally positive α ∈ F , we can bring M(c; N) into M(αc; N); so, the image of limN M(c; N) in Sh(p) ←− only depends on the strict ideal class of c. For each x = (X, Λ, i) ∈ M(c; N)(S) for a W–scheme S, choosing η (p) 1 (c; N) = i, we get a point x = (X, Λ, η(p) ) ∈ Sh(p) (S) so that η (p) mod Γ 1 projecting down to x = (X, Λ, i). Each element g ∈ G(A) with totally positive determinant in F × acts on x = (X, Λ, η (p) ) ∈ Sh(p) by x → g(x) = (X, det(g)Λ, η (p) ◦ g). This action is geometric preserving the base scheme Spec(W) and is compatible with the action of G(A(p∞) ) given as above (see [PAF] 4.2.2), because Λ = det(g)Λ. Then we can think of the projection of g(x) in M(c; N). By abusing the notation slightly, if the lift η (p) of i is clear in the context, we write g(x) ∈ M(c; N) for the image of g(x) ∈ Sh(p) . If the action of g is induced by a prime-to–p isogeny α : X → g(X), we write g(x, ω) = (g(x), α∗ ω) for (x, ω) ∈ M(c; N) if there is no ambiguity of α. When det(g) is not rational, the action of g is often non-trivial on Spec(W); see [Sh] II, [Sh1] and [PAF] 4.2.2.

2.9 Level structure with “Neben” character In order to make a good link between classical modular forms and adelic automorphic forms (which we will describe in the following subsection), we would like to introduce “Neben” characters. We fix two integral ideals N ⊂ n ⊂ O. We think of the following level structure on an AVRM X: i : (Gm ⊗ O∗ )[N] → X[N] and i : X[n] O/n

(2.10) i

→ with Im(i) ×X[N] X[n] = Ker(i ), where the sequence (Gm ⊗ O∗ )[N] − i

→ O/n is required to induce an isomorphism X[N] − (Gm ⊗ O∗ )[N] ⊗O O/n ∼ = (Gm ⊗ O ∗ )[n] under the polarization Λ. When N = n, this is exactly a Γ11 (N)–level structure. We fix two characters 1 : (O/n)× → A× and 2 : (O/N)× → A× , and we insist for f ∈ Gk (c, N; A) on the version of (G0-3) for quintuples (X, Λ, i · d, a · i , ω) and the equivariancy: f (X, Λ, i · d, a · i , ω) = 1 (a)2 (d)f (X, Λ, i, i , ω) for a, d ∈ (O/N)× . (Neben) Here Λ is the polarization class modulo multiple of totally positive numbers in F prime to n. We write Gk (c, Γ0 (N), ; A) ( = (1 , 2 )) for the A–module of geometric modular forms satisfying these conditions.

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2.10 Adelic Hilbert modular forms and Hecke algebras Let us interpret what we have said so far in automorphic language and give a definition of the adelic Hilbert modular forms and their Hecke algebra of level N (cf. [H96] Sections 2.2-4 and [PAF] Sections 4.2.8–4.2.12). We first recall formal Hecke rings of double cosets. For that, we fix a prime element q of Oq for every prime ideal q of O. We consider the following open compact subgroup of G(A(∞) ): ( '# $ " a b ∈ GL (O) , "c ≡ 0 mod NO U0 (N) = 2 c d ( '# $ (2.11) " a b ∈ U (N)"a ≡ d ≡ 1 mod NO , U11 (N) = 0 c d and Z = Z . Then we introduce the following semi = O ⊗Z Z where O group '# $ ( " a b ∈ G(A(∞) ) ∩ M (O) dN ∈ O × , "c ≡ 0 mod NO, Δ0 (N) = 2 N c d (2.12) where dN is the projection of d ∈ O to q|N Oq for prime ideals q. Writing T0 for the maximal diagonal torus of GL(2)/O and putting ' ( " "dN = 1 , (2.13) D0 = diag[a, d] = ( a0 d0 ) ∈ T0 (FA(∞) ) ∩ M2 (O) we have (e.g. [MFG] 3.1.6 and [PAF] Section 5.1) (2.14) Δ0 (N) = U0 (N)D0 U0 (N). In this section, writing pα = p|p pα(p) with α = (α(p)), the group U is assumed to be a subgroup of U0 (Npα ) with U ⊃ U11 (Npα ) for some multiexponent α (though we do not assume that N is prime to p). Formal finite linear combinations δ cδ U δU of double cosets of U in Δ0 (Npα ) form a ring R(U, Δ0 (Npα )) under convolution product (see [IAT] Chapter 3 or [MFG] 3.1.6). The algebra is commutative and is isomorphic to the polynomial ring over the group algebra Z[U0 (Npα )/U ] with variables# {T (q), $ T (q, q)}q for primes q, T (q) corresponding to the double coset U 0q 10 U and T (q, q) (for primes q Npα ) corresponding to U0 q U . Here we have chosen a prime element q in Oq . The group element u ∈ U0 (Npα )/U in Z[U0 (Npα )/U ] corresponds to the double coset U uU (cf. [H95] Section 2). The double coset ring R(U, Δ0 (Npα )) naturally acts on the space of modular forms on U whose definition we now recall. Recall that T0 is the diagonal torus of GL(2)/O ; so, T0 = G2m/O . Since T0 (O/N ) is canonically a quotient of U0 (N ) for an ideal N , a character : T0 (O/N ) → C× can be considered as a character of U0 (N ). Writing (( a0 d0 )) = 1 (a)2 (d), if − = −1 1 2

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factors through O/N for N|N , then we can extend the character # a b $ of U0 (N ) − to U0 (N) by putting (u) = 1 (det(u)) (d) for u = c d ∈ U0 (N). In this sense, we hereafter assume that is defined modulo N and regard as a character of U0 (N). We choose a Hecke character + : FA× /F × → C× with infinity type (1 − [κ])I (for an integer [κ]) such that + (z) = 1 (z)2 (z) for × . We also write t for the restriction of + to the maximal torsion z ∈ O + subgroup ΔF (N) of ClF+ (Np∞ ) (the strict ray class group modulo Np∞ : limn ClF+ (Npn )). ←− Writing I for the set of all embeddings of F into Q and T 2 for ResO/Z T0 (the diagonal torus of G), the group of geometric characters X ∗ (T 2 ) is isomorphic to Z[I]2 so that (m, n) ∈ Z[I]2 send diag[x, y] ∈ T 2 to xm y n = mσ σ(y)nσ ). Taking κ = (κ1 , κ2 ) ∈ Z[I]2 , we assume [κ]I = σ∈I (σ(x) κ1 + κ2 , and we associate with κ a factor of automorphy:

Jκ (g, τ ) = det(g∞ )κ2 −I j(g∞ , τ )κ1 −κ2 +I for g ∈ G(A) and τ ∈ Z. (2.15) We define Sκ (U, ; C) by the space of functions f : G(A) → C satisfying the following three conditions (e.g. [H96] Section 2.2 and [PAF] Section 4.3.1): (S1) f (αxuδ) = (u)t+ (z)f (x)Jκ (u, i)−1 for all α ∈ G(Q) and all u ∈ U · Ci and z ∈ ΔF (N) (ΔF (N) is the maximal torsion subgroup of ClF+ (Np∞ )); (S2) Choose u ∈ G(R) with u(i) = τ for τ ∈ Z, and put fx (τ ) = f (xu)Jκ (u, i) for each x ∈ G(A(∞) ) (which only depends on τ ). Then fx is a holomorphic function on Z for all x; (S3) fx (τ ) for each x is rapidly decreasing as ησ → ∞ (τ = ξ + iη) for all σ ∈ I uniformly. If we replace the word “rapidly decreasing” in (S3) by “slowly increasing”, we get the definition of the space Gκ (U, ; C). It is easy to check (e.g. [MFG] 3.1.5) that the function fx in (S2) satisfies the classical automorphy condition: f (γ(τ )) = (x−1 γx)f (τ )Jκ (γ, τ ) for all γ ∈ Γx (U ), −1

(2.16)

where Γx (U ) = xU x G(R) ∩G(Q). Also by (S3), fx is rapidly decreasing towards all cusps of Γx (e.g. [MFG] (3.22)); so, it is a cusp form. Imposing that f have the central character + in place of the action of ΔF (N) in (S1), we define the subspace Sκ (N, + ; C) of Sκ (U0 (N), ; C). The symbols κ = (κ1 , κ2 ) and (ε1 , ε2 ) here correspond to (κ2 , κ1 ) and (ε2 , ε1 ) in [PAF] Section 4.2.6 (page 171) because of a different notational convention in [PAF]. If we restrict f as above to SL2 (FA ), the determinant factor det(g)κ2 in the factor of automorphy disappears, and the automorphy factor becomes only +

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dependent on k = κ1 − κ2 + I ∈ Z[I]; so, the classical modular form in Gk has single digit weight k ∈ Z[I]. Via (2.3), we have an embedding of Sκ (U0 (N ), ; C) into Gk (Γ0 (N ), ; C) = [c]∈Cl+ Gk (c, Γ0 (N ), ; C) (c F

+ running over a complete representative set for the strict ideal class # group $ClF ) −1 bringing f into (fc )[c] for fc = fx (as in (S3)) with x = cd0 10 (for = d). The cusp form fc is determined by the restriction d ∈ FA× with dO of f to x · SL2 (FA ). If we vary the weight κ keeping k = κ1 − κ2 + I, the image of Sκ in Gk (Γ0 (N ), ; C) transforms accordingly. By this identification, the Hecke operator T (q) for non-principal q makes sense as an operator acting on a single space Gκ (U, ; C), and its action depends on the choice of κ. In other words, we have the double digit weight κ = (κ1 , κ2 ) for adelic modular forms in order to specify the central action of G(A). For a given f ∈ Sκ (U, ; C) and a Hecke character λ : FA× /F × → C× , the tensor product (f ⊗ λ)(x) = f (x)λ(det(x)) gives rise to a different modular form in Sκλ (U, λ ; C) for weight κλ and character λ dependent on λ, although the two modular forms have the same restriction to SL2 (FA ). We identify I with σ σ in Z[I]. It is known that Gκ = 0 unless κ1 + κ2 = [κ1 + κ2 ]I for [κ1 + κ2 ] ∈ Z, because I − (κ1 + κ2 ) is the infinity type of the central character of automorphic representations generated by Gκ . We write simply [κ] for [κ1 + κ2 ] ∈ Z assuming Gκ = 0. The SL(2)–weight of the central character of an irreducible automorphic representation π generated by f ∈ Gκ (U, ; C) is given by k (which specifies the infinity type of π∞ as a discrete series representation of SL2 (FR )). There is a geometric meaning of the weight κ: the Hodge weight of the motive attached to π (cf. [BR]) is given by {(κ1,σ , κ2,σ ), (κ2,σ , κ1,σ )}σ , and thus, the requirement κ1 − κ2 ≥ I is the regularity assumption for the motive (and is equivalent to the classical weight k ≥ 2I condition). Choose a prime element q of Oq for each prime q of F . We extend × → C× to F ×(∞) → C× just by putting − (m ) = 1 for m ∈ Z. − : O q A This is possible because Fq× = Oq× × qZ for qZ = {qm |m ∈ Z}. Similarly,#we extend 1 to FA×(∞) . Then we define (u) = 1 (det(u))− (aN ) for $ a b u = c d ∈ Δ0 (N).6 Let U "be the 7unipotent algebraic subgroup of GL(2)/O defined by U(A) = ( 10 a1 ) "a ∈ A . For each U yU ∈ R(U, Δ0 (Npα )), we 8 decompose U yU = t∈D0 ,u∈U (O) utU for finitely many u and t (see [IAT] Chapter 3 or [MFG] 3.1.6) and define (t)−1 f (xut). (2.17) f |[U yU ](x) = t,u

We check that this operator preserves the spaces of automorphic forms: Gκ (N, ; C) and Sκ (N, ; C). This action for y with yN = 1 is independent

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of the choice of the extension of to T0 (FA ). When yN = 1, we may assume that yN ∈ D0 ⊂ T0 (FA ), and in this case, t can be chosen so that tN = yN (so tN is independent of single right cosets in the double coset). If we extend (∞) to T0 (FA ) by choosing another prime element q and write the extension as , then we have (tN )[U yU ] = (tN )[U yU ] , where the operator on the right-hand-side is defined with respect to . Thus the sole difference is the root of unity (tN )/ (tN ) ∈ Im(|T0 (O/N) ). Since it depends on / the # choice $ 0of q , we make the choice once and for all, and write T (q) for U 0q 10 U (if q|N). By linearity, these action of double cosets extends to the ring action of the double coset ring R(U, Δ0 (Npα )). To introduce rationality of modular forms, we recall Fourier expansion of adelic modular forms (cf. [H96] Sections 2.3-4). Recall the embedding i∞ : Q → C, and identify Q with the image of i∞ . Recall also the differental idele = dO. Each member f of Sκ (U, ; C) has its d ∈ FA× with d(d) = 1 and dO Fourier expansion: a(ξyd, f )(ξy∞ )−κ2 eF (iξy∞ )eF (ξx), (2.18) f ( y0 x1 ) = |y|A 0ξ∈F

where eF : FA /F → C× is the additive character which has eF (x∞ ) = exp(2πi σ∈I xσ ) for x∞ = (xσ )σ ∈ RI = F ⊗Q R. Here y → a(y, f ) is a function defined on y ∈ FA× only depending on its finite part y (∞) . The × F∞ ) ∩ F × of integral ideles. function a(y, f ) is supported by the set (O A Let F [κ] be the field fixed by {σ ∈ Gal(Q/F )|κσ = κ}, over which the character κ ∈ X ∗ (T 2 ) is rational. Write O[κ] for the integer ring of F [κ]. We also define O[κ, ] for the integer ring of the field F [κ, ] generated by the values of over F [κ]. For any F [κ, ]–algebra A inside C, we define " 7 6 Sκ (U, ; A) = f ∈ Sκ (U, ; C)"a(y, f ) ∈ A as long as y is integral . (2.19) As we have seen, we can interpret Sκ (U, ; A) as the space of A–rational global sections of a line bundle of a variety defined over A; so, we have, by the flat base-change theorem (e.g. [GME] Lemma 1.10.2), Sκ (N, ; A) ⊗A C = Sκ (N, ; C).

(2.20)

The Hecke operators preserve A–rational modular forms (e.g., [PAF] 4.2.9). We define the Hecke algebra hκ (U, ; A) ⊂ EndA (Sκ (U, ; A)) by the A–subalgebra generated by the Hecke operators of R(U, Δ0 (Npα )).

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For any Qp –algebras A, we define Sκ (U, ; A) = Sκ (U, ; Q) ⊗Q,ip A.

(2.21)

By linearity, y → a(y, f ) extends to a function on FA× × Sκ (U, ; A) with values in A. We define the q–expansion coefficients (at p) of f ∈ Sκ (U, ; A) by ap (y, f ) = yp−κ2 a(y, f ) and a0,p (y, f ) = N (yd−1 )[κ2 ] a0 (y, f ),

(2.22)

×

where N : FA× /F × → Qp is the character given by N (y) = yp−I |y (∞) |−1 A . Here we note that a0 (y, f ) = 0 if κ2 ∈ ZI. Thus, if a0 (y, f ) = 0, [κ2 ] ∈ Z is well defined. The formal q–expansion of an A–rational f has values in the space of functions on FA×(∞) with values in the formal monoid algebra A[[q ξ ]]ξ∈F+ of the multiplicative semi-group F+ made up of totally positive elements, which is given by ⎧ ⎫ ⎨ ⎬ (2.23) f (y) = N (y)−1 a0,p (yd, f ) + ap (ξyd, f )q ξ . ⎩ ⎭ ξ0

p We now define for any p–adically complete O[κ, ]–algebra A in Q ' ( " p )"ap (y, f ) ∈ A for integral y . (2.24) Sκ (U, ; A) = f ∈ Sκ (U, ; Q As we have already seen, these spaces have geometric meaning as the space of A–integral global sections of a line bundle defined over A of the Hilbert modular variety of level U (see [PAF] Section 4.2.6), and the q–expansion above for a fixed y = y (∞) gives rise to the geometric at the # q–expansion $ infinity cusp of the classical modular form fx for x = y0 10 (see [H91] (1.5) and [PAF] (4.63)). We have chosen a complete representative set {ci }i=1,...,h in finite ideles × × F∞+ for the strict idele class group F × \FA× /O , where h is the strict class −1

number of F . Let ci = ci O. Write ti = ci d0 10 and consider fi = fti as defined in (S2). The collection (fi )i=1,...,h determines f , because of the approximation theorem. Then f (ci d−1 ) gives the q–expansion of fi at the Tate abelian variety with ci –polarization Tatec−1 ,O (q) (ci = ci O). By (q–exp), the i q–expansion f (y) determines f uniquely. We write T (y) for the # Hecke $ operator acting on Sκ (U, ; A) corresponding to the double coset U y0 10 U for an integral idele y. We renormalize T (y) to have a p–integral operator T(y): T(y) = yp−κ2 T (y). Since this only affects T (y) with yp = 1, T(q) = T (q ) = T (q) if q p. However T(p) = T (p)

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for primes p|p. The renormalization is optimal to have the stability of the A– integral spaces under Hecke operators. We define q = N (q)T (q, q) for q Npα , which is equal to the central action of a prime element q of Oq times N (q) = |q |−1 A . We have the following formula of the action of T (q) and T (q, q) (e.g., [PAF] Section 4.2.10): ap (yq , f ) + ap (yq−1 , f |q) if q is outside n ap (y, f |T(q)) = ap (yq , f ) otherwise, (2.25) 1 where the level n of U is the ideal maximal under the condition: U1 (n) ⊂ U ⊂ U0 (N). Thus T(q ) = U (q) (up to p–adic units) when q is a factor of the level of U (even when q|p; see [PAF] (4.65–66)). Writing the level of U as Npα , we assume either p|Npα or [κ] ≥ 0,

(2.26)

since T(q) and q preserve the space Sκ (U, ; A) under this condition (see [PAF] Theorem 4.28). We then define the Hecke algebra hκ (U, ; A) (resp. hκ (N, + ; A)) with coefficients in A by the A–subalgebra of the A–linear endomorphism algebra EndA (Sκ (U, ; A)) (resp. EndA (Sκ (N, + ; A))) generated by the action of the finite group U0 (Npα )/U , T(q) and q for all q. We have canonical projections: R(U11 (Npα ), Δ0 (Npα )) R(U, Δ0 (Npα )) R(U0 (Npβ ), Δ0 (Npβ )) for all α ≥ β (⇔ α(p) ≥ β(p) for all p|p) taking canonical generators to the corresponding ones, which are compatible with inclusions Sκ (U0 (Npβ ), ; A) → Sκ (U, ; A) → Sκ (U11 (Npα ), ; A). We get a projective system of Hecke algebras {hκ (U, ; A)}U (U running through open subgroups of U0 (Np) containing U11 (Np∞ )), whose projective limit (when κ1 − κ2 ≥ I) gives rise to the universal Hecke algebra h(N, ; A) for a complete p–adic algebra A. This algebra is known to be independent of κ (as long as κ1 − κ2 ≥ I) and has canonical generators T(y) over A[[G]] (for G = (Op × (O/N(p) ))× × ClF+ (Np∞ )), where N(p) is the prime-to–p part of N. Here note that the operator q is included in the action of G, (U, ; A), hn.ord (Npα , + ; A) and because q ∈ ClF+ (Np∞ ). We write hn.ord κ κ n.ord n.ord h = h (N, ; A) for the image of the (nearly) ordinary projector e = limn T(p)n! . The algebra hn.ord is by definition the universal nearly ordinary Hecke algebra over A[[G]] of level N with “Neben character” . We also note here that this algebra hn.ord (N, ; A) is exactly the one h(ψ + , ψ )

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employed in [HT1] page 240 (when specialized to the CM component there) if A is a complete p–adic valuation ring. Let ΛA = A[[Γ]] for the maximal torsion-free quotient Γ of G. We fix a splitting G = Γ × Gtor for a finite group Gtor . If A is a complete p–adic valuation ring, then hn.ord (N, ; A) is a torsion-free ΛA –algebra of finite rank and is ΛA –free under some mild conditions on N and ([PAF] 4.2.12). Take a point P ∈ Spf(Λ)(A) = Homcont (Γ, A× ). Regarding P as a character of G, we call P arithmetic if it is given locally by an algebraic character κ(P ) ∈ X ∗ (T 2 ) with κ1 (P ) − κ2 (P ) ≥ I. Thus if P is arithmetic, P = P κ(P )−1 is a character of T 2 (O/pα N) for some multi-exponent α ≥ 0. Similarly, the restriction of P to ClF+ (Np∞ ) is a p–adic Hecke character P + induced by an arithmetic Hecke character of infinity type (1 − [κ(P )])I. As long as P is arithmetic, we have a canonical specialization morphism: α hn.ord (N, ; A) ⊗ΛA ,P A hn.ord κ(P ) (Np , P + ; A),

which is an isogeny (surjective and of finite kernel) and is an isomorphism if hn.ord is ΛA –free. The specialization morphism takes the generators T(y) to T(y).

3 Eisenstein series We shall study the q–expansion, Hecke eigenvalues and special values at CM points of an Eisenstein series defined on M(c; N).

3.1 Arithmetic Hecke characters Recall the CM type Σ ordinary at p and the prime ideal l of O introduced in the introduction. We sometimes regard Σ as a character of TM = ResM/Q Gm sending x ∈ M × to xΣ = σ∈Σ σ(x). More generally, each integral linear combination κ = σ∈ΣΣc κσ σ is regarded as a character of TM by x → κσ an arithmetic Hecke character λ of infinity type kΣ + σ σ(x) . We fix κ(1 − c) for κ = σ∈Σ κσ σ ∈ Z[Σ] and an integer k. This implies, regarding kΣ+κ(1−c) for x∞ ∈ TM (R). λ as an idele character of TM (A), λ(x∞ ) = x∞ We assume the following three conditions: (crt) k > 0 and κ ≥ 0, where we write κ ≥ 0 if κσ ≥ 0 for all σ. (opl) The conductor C of λ is prime to p and ( ) = l ∩ Z. (spt) The ideal C is a product of primes split over F .

228

Haruzo Hida 3.2 Hilbert modular Eisenstein series

We shall define an Eisenstein series whose special values at CM points interpolate the values L(0, λχ) for anticyclotomic characters χ of finite order. We split the conductor C in the following way: C = FFc with F + Fc = R and F ⊂ Fcc . This is possible by (spt). We then define f = F ∩ O and f = Fc ∩ O. Then f ⊂ f . Here X = O/f ∼ = R/F and Y = O/f ∼ = R/Fc . Let φ : X × Y → C be a function such that φ(ε−1 x, εy) = N (ε)k φ(x, y) for all ε ∈ O × with the integer k as above. We put X ∗ = f∗ /O ∗ ; so, X ∗ is naturally the Pontryagin dual module of X under the pairing (x∗ , x) = eF (x∗ x) = (Tr(x∗ x)), where (x) = exp(2πix) for x ∈ C. We define the partial Fourier transform P φ : X ∗ × Y → C of φ by φ(a, y)eF (ax), (3.1) P φ(x, y) = N (f)−1 a∈X

where eF is the restriction of the standard additive character of the adele ring FA to the local component Ff at f. A function φ as above can be interpreted as a function of (L, Λ, i, i ) in 2.5. Here i : X ∗ → f−1 L/L is the level f–structure. We define an Of –submodule P V (L) ⊂ L ⊗O Ff specified by the following conditions: P V (L) ⊃ L ⊗O Of , P V (L)/Lf = Im(i) (Lf = L ⊗O Of ).

(PV)

By definition, we may regard i−1 : P V (L) P V (L)/ (L ⊗O Of ) ∼ = f∗ /O∗ . By Pontryagin duality under Tr ◦ λ, the dual map of i gives rise to i : P V (L) O/f. Taking a lift i : (f2 )∗ /O∗ → P V (L)/fLf with i mod Lf = i, we have an exact sequence: i

i

→ P V (L)/fLf − → O/f → 0. 0 → (f2 )∗ /O ∗ − This sequence is kept under α ∈ Aut(L) with unipotent reduction modulo f2 , and hence, the pair (i, i ) gives a level Γ11 (f2 )–structure: Once we have chosen a generator f of f in Of , by the commutativity of the following diagram: i

i

(f2 )∗ /O∗ −−−−→ P V (L)/fLf −−−−→ O/f ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ f :∩ f :∩ : (f2 )∗ /O∗ −−−−→

X[f2 ]

(3.2)

−−−−→ O/f2 ,

giving (i, i ) is equivalent to having the bottom sequence of maps in the above diagram. This explains why the pair (i, i ) gives rise to a level Γ11 (f2 )–structure;

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229

strictly speaking, the exact level group is given by: Γ11,0 (f2 ) =

'# $ a b c d

" ( " ∈ SL2 (Of )"a ≡ d ≡ 1 mod f, c ≡ 0 mod f2 . (3.3)

We regard P φ as a function of L ⊗O F supported on (f−2 L) ∩ P V (L) by P φ(w) =

P φ(i−1 (w), i (w))

if (w mod L) ∈ Im(i),

0

otherwise.

(3.4)

For each w = (wσ ) ∈ F ⊗Q C = CI , the norm map N (w) = σ∈I wσ is well defined. For any positive integer k > 0, we can now define the Eisenstein series Ek . Writing L = (L, λ, i) for simplicity, we define the value Ek (L; φ, c) by " P φ(w) {(−1)k Γ(k + s)}[F :Q] " % . k |N (w)|2s "s=0 N (w) |DF | w∈f−1 L/O × (3.5) Here “ ” indicates that we are excluding w = 0 from the summation. As shown by Hecke, this type of series is convergent when the real part of s is sufficiently large and can be continued to a meromorphic function well defined at s = 0 (as long as either k ≥ 2 or φ(a, 0) = 0 for all a). The weight of the Eisenstein series is the parallel weight kI = σ kσ. If either k ≥ 2 or φ(a, 0) = 0 for all a, the function Ek (c, φ) gives an element in GkI (c, f2 ; C), whose q–expansion at the cusp (a, b) computed in [HT1] Section 2 is given by Ek (L; φ, c) =

N (a)−1 Ek (φ, c)a,b (q) = 2−[F :Q] L(1 − k; φ, a) N (a) N (a)k−1 q ξ , (3.6) + φ(a, b) |N (a)| × 0ξ∈ab (a,b)∈(a×b)/O ab=ξ

where L(s; φ, a) is the partial L–function given by the Dirichlet series: ξ∈(a−{0})/O×

φ(ξ, 0)

N (ξ) |N (ξ)|

k

|N (ξ)|−s .

If φ(x, y) = φX (x)φY (y) for two functions φX : X → C and φY : Y → C with φY factoring through O/f , then we can check easily that Ek (φ) ∈ GkI (c, ff ; C).

230

Haruzo Hida 3.3 Hecke eigenvalues

× × We take a Hecke character λ as in 3.1. Then the restriction λ−1 C : RF × RFc → × W induces a locally constant function ψ : (O/f) × (O/f ) → W supported on (O/f)× ×(O/f )× , because λC factor through (R/C)× which is canonically isomorphic to (O/f)× × (O/f )× . Since λ is trivial on M × , ψ satisfies

ψ(εx, εy) = εkΣ+κ(1−c) ψ(x, y) = N (ε)k ψ(x, y) for any unit ε ∈ O× . We regard the local uniformizer q ∈ Oq as an idele. For each ideal A of F , e(q) decomposing A = q qe(q) for primes q, we define e(A) = q q ∈ FA× . We then define a partial Fourier transform ψ ◦ : X × Y → W by ψ(u, b)eF (−ua−e(f) ). (3.7) ψ ◦ (a, b) = u∈O/f

By the Fourier inversion formula, we have P ψ ◦ (x, y) = ψ( e(f) x, y).

(3.8)

From this and the definition of Ek (L) = Ek (L; ψ◦ , c), we find Ek (X, Λ, i ◦ x, i ◦ y, aω) = N (a)−k λF (x)λ−1 Fc (y)Ek (X, Λ, i, i , ω) (3.9)

for x ∈ (O/f)× = (R/F)× and y ∈ (O/f )× = (R/Fc )× . Because of this, Ek (ψ ◦ , c) actually belongs to GkI (c, Γ0 (ff ), λ ; C) for λ,1 = λFc and λ,2 = λF identifying OF = RF and OFc = RFc . Recall GkI (Γ0 (N), ; C) = c∈Cl+ GkI (c, Γ0 (N), ; C). Via this decomposition we F

extend each ClF+ –tuple (fc )c in GkI (Γ0 (ff ), λ ; C) to an automorphic form f ∈ GkI (ff , λ+ ; C) as follows: (i) f (zx) = λ(z)|z|A f (x) for z in the center FA× ⊂ G(A) (so λ+ (z) = λ(z)|z|A ); (ii) f (xu) = λ (u)f (x) for u ∈ U0 (ff ); (iii) fx = fc if x = ( 0c 10 ) for an idele c with cO = c and c(c) = 1. We now compute the effect of the operator q (defined above (2.25)) on Ek for a fractional ideal q prime to the level f. A geometric interpretation of the operator q is discussed, for example, in [H04b] (5.3) (or [PAF] 4.1.9), and has the following effect on an AVRM X: X → X ⊗O q. The level structure i is intact under this process. The c–polarization Λ induces a cq−2 –polarization on X ⊗O q. On the lattice side, q brings L to qL.

Hecke L–values

231

To simplify our notation, we write t(w; s) =

e(f) −1 i (w))λ−1 λ−1 F ( Fc (i (w)) N (w)k |N (w)|2s

for each term of the Eisenstein series and c(s) for the Gamma factor in front of the summation, where D = N (d) is the discriminant of F . First we compute the effect of the operator q when q = (ξ) for ξ ∈ F naively as follows: Ek ((qL, Λ, i); ψ ◦ , cq−2 ) = c(s) q=(ξ)

= c(s)

t(w; s)|s=0

w∈f−1 qL/O× −k t(ξw; s)|s=0 = λF (ξ)−1 λ−1 Ek (L; ψ◦ , c). Fc (ξ)N (ξ)

w∈f−1 L/O ×

(3.10) −1 Here we agree to put λ−1 F (x) = 0 if xRF = RF and λFc (y) = 0 if yRFc = RFc . The result of the above naive calculation of the eigenvalue of q shows that our way of extending the Eisenstein series (Ek (ψ ◦ ; c))c to an adelic automorphic form G(A) is correct (and canonical): This claim follows from −k = λ(q) = λ(q )|q |A N (q), λF (ξ)−1 λ−1 Fc (ξ)N (ξ)

because the operator q on GkI (U, ; C) is defined (above (2.25)) to be the central action of q ∈ FA× (that is, multiplication by λ(q )|q |A ) times N (q). We obtain λFc (ξ 2 )Ek |q−1 (L; ψ◦ , c) = Ek ((q−1 L, Λ, i); ψ ◦ , cq2 ) = λF (ξ)λFc (ξ)N (ξ)k Ek (L; ψ ◦ , c).

(3.11)

The factor λFc (ξ 2 ) in the left-hand-side comes from the fact that i with respect to the c–polarization ξ 2 Λ of q−1 L is the multiple by ξ 2 of i with respect to cq2 –polarization Λ of q−1 L. We now compute the effect of the Hecke operator T (1, q) = T (q) for a prime q f. Here we write L for an O-lattice with L /L ∼ = O/q. Then L ∧ ∗ L = (qc) ; so, Λ induces a qc–polarization on L , and similarly it induces q2 c–polarization on q−1 L. By (2.8), Ek |T (q) is the sum of the terms t( ; s) with multiplicity extended over q−1 f−1 L. The multiplicity for each ∈ f−1 L is N (q) + 1 and only once for ∈ q−1 f−1 L − f−1 L (thus, N (q) times for

232

Haruzo Hida

∈ f−1 L and once for ∈ q−1 f−1 L). This shows c(0)−1 N (q)Ek |T (q)(L; ψ ◦ , qc) ⎫ ⎧ ⎬" ⎨ " = t(w; s) + t(w; s) " ⎭ ⎩ s=0 L w∈f−1 L /O × w∈f−1 q−1 L /O× 6 7 = c(0)−1 N (q)Ek (L; ψ◦ , c) + Ek |q−1 (L; ψ◦ , cq2 ) . In short, we have Ek (ψ ◦ , qc)|T (q) = Ek (ψ ◦ , c) + N (q)−1 Ek (ψ ◦ , cq2 )|q−1 .

(3.12)

Suppose that q is principal generated by a totally positive ξ ∈ F . Substituting ξ −1 Λ for Λ, i will be transformed into ξ −1 i , and we have λFc (ξ)Ek (ψ ◦ , c)|T (q) = Ek (ψ ◦ , qc)|T (q) We combine this with (3.11) assuming q = (ξ) with 0 ' ξ ∈ F : k−1 Ek (ψ ◦ , c)|T (q)(L) = (λ−1 )Ek (ψ ◦ , c), Fc (ξ) + λF (ξ)N (q)

(3.13)

−1 which also follows from (3.6) noting that ψ◦ (a, b) = G(λ−1 F )λF (a)λFc (b) for −1 the Gauss sum G(λF ). We now look into the operator [q] for a prime q outside the level f. This operator brings a level Γ0 (q)–test object (X, C, i) with level f structure i outside q to (X/C, i), where the level f–structure i is intact under the quotient map: X → X/C. On the lattice side, taking the lattice LC with LC /L = C, it is defined as follows:

f |[q](L, C, Λ, i) = N (q)−1 f (LC , Λ, i).

(3.14)

The above operator is useful to relate U (q) and T (q). By definition, f |U (q)(L, Λ, C, i) = N (q)−1 f (L , Λ, C , i) L ,L =LC

for C = LC + L /L = q−1 L/L . Thus we have U (q) = T (q) − [q].

(3.15)

A similar computation yields: [q] ◦ U (q) = N (q)−1 q−1 .

(3.16)

Lemma 3.1 Let q be a prime outside f. Suppose that qh = (ξ) for a totally positive ξ ∈ F . Let Ek (ψ, c) = Ek (ψ ◦ , c) − Ek (ψ ◦ , cq)|[q] and Ek (ψ, c) = Ek (ψ ◦ , c) − N (q)Ek (ψ ◦ , cq−1 )|q|[q]. Then we have

Hecke L–values (1) (2) (3) (4)

233

Ek (ψ, c)|U (q) = Ek (ψ ◦ , q−1 c) − Ek (ψ ◦ , c)|[q], Ek (ψ, c)|U (qh ) = λ−1 Fc (ξ)Ek (ψ, c), Ek (ψ, c)|U (q)=(Ek (ψ ◦ , qc)−N (q)Ek (ψ ◦ , c)|q|[q])|(N (q)−1 q−1 ) Ek (ψ, c)|U (qh ) = λF (ξ)N (q)h(k−1) Ek (ψ, c).

Proof We prove (1) and (3), because (2) and (4) follow by iteration of these formulas combined with the fact: λFc (ξ)Ek (ψ ◦ , c) = Ek (ψ ◦ , ξc) for a totally positive ξ ∈ F . Since (3) can be proven similarly, we describe computation to get (1), writing Ek (c) = Ek (ψ ◦ , c): Ek (ψ; c)|U (q) = Ek (c)|U (q) − Ek (cq)|[q]|U (q) (3.16)

= Ek (c)|U (q) − N (q)−1 Ek (cq)|q−1

(3.15)

= Ek (c)|T (q) − Ek (cq)|[q] − N (q)−1 Ek (cq)|q−1

(3.12)

= Ek (cq−1 ) + N (q)−1 Ek (cq)|q−1 − Ek (c)|[q] − N (q)−1 Ek (cq)|q−1 = Ek (cq−1 ) − Ek (c)|[q].

Remark 3.2 As follows from the formulas in [H96] 2.4 (T1) and [H91] Section 7.G, the Hecke operator T (q) and U (q) commutes with the Katz differential operator as long as q p. Thus for E(λ, c) = dκ Ek (ψ, c) and E (λ, c) = dκ Ek (ψ, c), we have under the notation of Lemma 3.1 E (λ, c)|U (qh ) =λ−1 Fc (ξ)E (λ, c),

E(λ, c)|U (qh ) =λF (ξ)N (q)h(k−1) E(λ, c).

(3.17)

3.4 Values at CM points We take a proper Rn+1 –ideal a for n > 0, and regard it as a lattice in CΣ by a → (aσ )σ∈Σ . Then Λ(a) induces a polarization of a ⊂ CΣ . We suppose that a is prime to C (the conductor of λ). For a p–adic modular form f of the form dκ g for classical g ∈ GkI (c, Γ1,0 (f2 ); W), we have by (K) in 2.6 f (x(a), ω∞ ) f (x(a), ωp ) = f (x(a), ω(a)) = . kΣ+2κ Ωp ΩkΣ+2κ ∞ Here x(a) is the test object: x(a) = (X(a), Λ(a), i(a), i (a))/W . π κ ΓΣ (kΣ+κ) √ We write c0 = (−1)k[F :Q] Im(δ) . Here ΓΣ (s) = σ∈Σ Γ(sσ ), κ DΩkΣ+2κ ∞ Ωs∞ = σ Ωsσσ , Im(δ)s = σ Im(δ s )sσ , and so on, for s = σ sσ σ. By

234

Haruzo Hida

definition (see [H04c] 4.2), we find, for e = [R× : O× ], κ (c0 e)−1 δkI Ek (c)(x(a), ω(a)) (∞) " λ−1 )" e(F) C (w)λ(w = λ−1 ( ) " C s NM/Q (w) s=0 −1 × w∈F

a/R

e(F) = λ−1 )λ(a)NM/Q (Fa−1 )s C (

wFa−1 ⊂Rn+1

λ(wFa−1 ) "" " NM/Q (wFa−1 )s s=0

e(F) = λ−1 )λ(a)Ln+1 C ( [Fa−1 ] (0, λ),

(3.18) where for an ideal class [A] ∈ Cln+1 represented by a proper Rn+1 –ideal A, λ(b)NM/Q (b)−s Ln+1 [A] (s, λ) = b∈[A]

is the partial L–function of the class [A] for b running over all Rn+1 –proper integral ideals prime to C in the class [A]. In the second line of (3.18), we regard λ as an idele character and in the other lines as an ideal character. For = aR and aC = 1, we have λ(a(∞) ) = λ(a). an idele a with aR κ We put E(λ, c) = d Ek (ψ, c) and E (λ, c) = dκ Ek (ψ, c) as in Remark 3.2. We want to evaluate E(λ, c) and E (λ, c) at x = (x(a), ω(a)). Here c is the polarization ideal of Λ(a); so, if confusion is unlikely, we often omit the reference to c (which is determined by a). Thus we write, for example, E(λ) and E (λ) for E(λ, c) and E (λ, c). Then by definition and (K) in 2.6, we have for x = (x(a), ω(a)) κ κ Ek (ψ ◦ , c)(x) − N (q)−1 δkI Ek (ψ ◦ , cq)(x(aRn ), ω(aRn )) E (λ)(x) = δkI κ κ E(λ)(x) = δkI Ek (ψ ◦ , c)(x) − δkI Ek (ψ ◦ , cq−1 )(x(qaRn ), ω(aRn ))

(3.19) because C(a) = aRn /a and hence [q](x(a)) = x(aRn ). To simplify notation, write φ([a]) = λ(a)−1 φ(x(a), ω(a)). By (3.9), for φ = E(λ) and E (λ), the value φ([a]) only depends on the ideal class [a] but not the individual a. The formula (3.19) combined with (3.18) shows, for a proper Rn+1 –ideal a, −1 n e−1 λC (e(F) )E (λ)([a]) = c0 Ln+1 L[Fa−1 Rn ] (0, λ) [Fa−1 ] (0, λ) − N (q) n (0, λ) − λ(q)L e−1 λC (e(F) )E(λ)([a]) = c0 Ln+1 −1 a−1 R ] (0, λ) −1 [Fq [Fa ] n (3.20)

Hecke L–values where e = [R× : O× ]. Now we define Ln (s, λ) = λ(a)NM/Q (a)−s ,

235

(3.21)

a

where a runs over all proper ideals in Rn prime to C and NM/Q (a) = [Rn : a]. ×

For each primitive character χ : Clf → Q , we pick n + 1 = mh so that (m − 1)h ≤ f ≤ n + 1, where qh = (ξ) for a totally positive ξ ∈ F . Then we have χ(a)E (λ)([a]) e−1 λC (e(F) ) [a]∈Cln+1

# $ = c0 χ(F) Ln+1 (0, λχ−1 ) − Ln (0, λχ−1 ) χ(a)E(λ)([a]) e−1 λC (e(F) ) [a]∈Cln+1

# $ = c0 χ(F) Ln+1 (0, λχ−1 ) − λχ−1 (q)N (q)Ln (0, λχ−1 ) . (3.22)

As computed in [H04c] 4.1 and [LAP] V.3.2, if k ≥ f then the Euler q–factor of Lk (s, χ−1 λ) is given by

k−f

(χ−1 λ(q))j N (q)j−2sj if f > 0,

j=0 k−1

(χ−1 λ(q))j N (q)j−2sj

j=0

+ N (q) −

M/F q

(χ−1 λ(q))k N (q)k−1−2ks L0q (s, χ−1 λ) if f = 0, (3.23)

where

M/F q

is 1, −1 or 0 according as q splits, remains prime or ramifies

in M/F , and L0q (s, χ−1 λ) is the q–Euler factor of the primitive L–function L(s, χ−1 λ). We define a possibly imprimitive L–function L(q) (s, χ−1 λ) = Lq (s, χ−1 λ)L0 (s, χ−1 λ) removing the q–Euler factor. Combining all these formulas, we find χ(a)E(λ)([a]) = c0 L(q) (0, χ−1 λ), e−1 λχ−1 (e(F) ) [a]∈Cln+1

(3.24)

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Haruzo Hida

e−1 λχ−1 (e(F) )

χ(a)E (λ)([a])

[a]∈Cln+1

⎧ (q) ⎪ (0, χ−1 λ) ⎪ ⎨c0 L = c0 M/F Lq (1, χ−1 λ)L(0, χ−1 λ) q ⎪ ⎪ ⎩ −c0 χ−1 λ(Q)Lq (1, χ−1 λ)L(0, χ−1 λ)

if f > 0, if f = 0 and

M/F q

= 0,

if q = Q2 in R and f = 0. (3.25)

All these values are algebraic in Q and integral over W.

4 Non-vanishing modulo p of L–values We construct an F–valued measure (F = Fp as in 2.2) over the anti-cyclotomic class group Cl∞ = limn Cln modulo l∞ whose integral against a character χ ←− is the Hecke L–value L(0, χ−1 λ) (up to a period). The idea is to translate the Hecke relation of the Eisenstein series into a distribution relation on the profinite group Cl∞ . At the end, we relate the non-triviality of the measure to the q–expansion of the Eisenstein series by the density of {x(a)}a (see [H04c]).

4.1 Construction of a modular measure We choose a complete representative set {c}[c]∈Cl+ of the strict ideal class F

group ClF+ made up of ideals c prime to pfl. For each proper Rn –ideal a, the polarization ideal c(a) of x(a) is equivalent to one of the representatives c (so [c] = [c(a)]). Writing c0 for c(R), we have c(a) = c0 l−n (aac )−1 . Take a modular form g in GkI (Γ0 (ff l), λ ; W ). Thus g = (g[c] ) is an h– tuple of modular forms for h = |ClF+ |. Put f = (f[c] )c for f[c] = dκ g[c] for the differential operator dκ = σ dκσσ in 2.6. We write f (x(a)) for the value of f[c(a)] (x(a)). Similarly, we write f (X, Λ, i, ω) for f[c] (X, Λ, i, ω) for the ideal class c determined by Λ. The Hecke operator U (l) takes the space V (c, Γ0 (ff l), λ ; W ) into V (cl−1 , Γ0 (ff l), λ ; W ). Choosing cl in the representative set equivalent to the ideal cl−1 , we have a canonical isomorphism V (cl, Γ0 (ff l), λ ; W ) ∼ = V (cl , Γ0 (ff l), λ ; W ) sending f to f given by f (X, ξΛ, i, i , ω) = f (X, Λ, i, i , ω) for totally positive ξ ∈ F with ξcl = l−1 c. This map is independent of the choice of ξ. Since the image of M(c; N) in Sh(p) depends only on N and the strict ideal class of c as explained in 2.8, the Hecke operator U (l) is induced from the algebraic on the Shimura variety associated to the # correspondence $ double coset U 0l 10 U . So we regard U (l) as an operator acting on h–tuple

Hecke L–values

237

of p–adic modular forms in V (Γ0 (ff l), λ ; W ) = c V (c, Γ0 (ff l), λ ; W ) inducing permutation c → cl on the polarization ideals. Suppose that g|U (l) = ag with a ∈ W × ; so, f |U (l) = af (see Remark 3.2). The Eisenstein series (E(λ, c))c satisfies this condition by Lemma 3.1. The operator U (lh ) (h = |ClF+ |) takes V (c, Γ0 (ff l), λ ; W ) into itself. Thus fc |U (lh ) = ah fc . identify T (X(R) ) = R = R ⊗Z Z, Choosing a base w = (w1 , w2 ) of R /Q ! (a, b) → aw1 + bw2 ∈ T (X(R)). This gives a level structure 2 by O with O (p) 2 η (R) : F ⊗Q A(p∞) ∼ = V (p) (X(R)) defined over W. Choose the base w satisfying the following two conditions: (B1) w2,l = 1 and Rl = Ol [w1,l ]; (B2) By using the splitting: Rf = RF ×RFc , w1,f = (1, 0) and w2,f = (0, 1). Let a be a proper Rn –ideal (for Rn = O+ln R) prime to f. Recall the generator n = l of lOl . Regarding ∈ FA× , wn = (n w1 , w2 ) is a base of R (p) 2 (p∞) ∼ (p) and gives a level structure η (Rn ) : F ⊗Q A = V (X(Rn )). We choose a complete representative set A = {a1 , . . . , aH } ⊂ MA× so that MA× = 8H × × × × j=1 M aj Rn M∞ . Then aRn = αaj Rn for α ∈ M for some index j. We then define η (p) (a) = αaj η (p) (Rn ). The small ambiguity of the choice of α does not cause any trouble. Write x0 (a) = (X(a), Λ(a), i(a), i (a), C(a), ω(a)). This is a test object of level Γ11,0 (f2 ) ∩ Γ0 (l) (see (3.3) for Γ11,0 (f2 )). We pick a subgroup C ⊂ X(Rn ) such that C ∼ = O/lm (m > 0) but C ∩ C(Rn ) = {0}. Then we define x0 (Rn )/C by X(Rn ) C + C(Rn )[l] , π∗ Λ(Rn ), π ◦ i(Rn ), π −1 ◦ i (Rn ), , (π ∗ )−1 ω(Rn ) C C for the projection map π : X(Rn ) X(Rn )/C. We can write x0 (Rn )/C = x0 (a) ∈ M(cl−n−m , f2 , Γ0 (l))(W) for a proper Rn+m –ideal a ⊃ Rn with (aac ) = l−2m , and for u ∈ Ol× we have u 1 m l (4.1) x0 (a) = x0 (Rn )/C = (x0 (Rm+n )). 0

1

See Section 2.8 in the text for the action of g =

1

u m l

0

1

on the point

x0 (Rm+n ), and see [H04c] Section 3.1 for details of the computation leading to (4.1). Let TM = ResM/Q Gm . For each proper Rn –ideal a, we have an embedding ρa : TM (A(p∞) ) → G(A(p∞) ) given by αη (p) (a) = η(p) (a) ◦ ρa (α). Since

238

Haruzo Hida

det(ρa (α)) = ααc 0, α ∈ TM (Z(p) ) acts on Sh(p) through ρa (α) ∈ G(A). We have ρa (α)(x(a)) = (X(a), (ααc )Λ(a), η (p) (a)ρa (α)) = (X(αa), Λ(αa), η (p) (αa)) for the prime-to–p isogeny α ∈ EndO (X(a)) = R(p) . Thus TM (Z(p) ) acts on Sh(p) fixing the point x(a). We find ρ(α)∗ ω(a) = αω(a), and by (B2), we have g(x(αa), αω(a)) = g(ρ(α)(x(a), ω(a))) = α−kΣ λF (α)λFc (α)g(x(a), ω(a)). From this, we conclude f (x(αa), αω(a)) = f (ρ(α)(x(a), ω(a))) = α−kΣ−κ(1−c) λF (α)λFc (α)f (x(a), ω(a)), because the effect of the differential operator d is identical with that of δ at the CM point x(a) by (K). By our choice of the Hecke character λ, we find λ(αa) = α−kΣ−κ(1−c) λF (α)λFc (α)λ(a). If a and α is prime to Cp, then the value α−kΣ−κ(1−c) λC (α) is determined independently of the choice of α for a given ideal αa, and the value λ(a)−1 f (x(a), ω(a)) is independent of the representative set A = {aj } for Cln . Defining f ([a]) = λ(a)−1 f (x(a), ω(a)) for a proper Rn –ideal a prime to Cp, (4.2) we find that f ([a]) only on the proper ideal class [a] ∈ Cln . depends 1 u l (x(a)), where lh = () for an element ∈ F . We write x(au ) = 0 1

Then au depends only on u mod lh , and {au }u mod lh gives a complete representative set for proper Rn+h –ideal classes which project down to the ideal class [a] ∈ Cln . Since au Rn = −1 a, we find λ(au ) = λ(l)−h λ(a). Then we have 1 ah f ([a]) = λ(a)−1 f |U (lh )(x(a)) = f ([au ]), λ(l)h N (l)h h u mod l

and we may define a measure ϕf on Cl∞ with values in F by φdϕf = b−m φ(a−1 )f ([a]) (for b = ah λ(l)h N (l)h ). Cl∞

a∈Clmh

(4.3)

Hecke L–values

239

4.2 Non-triviality of the modular measure The non-triviality of the measure ϕf can be proven in exactly the same manner as in [H04c] Theorems 3.2 and 3.3. To recall the result in [H04c], we need to describe some functorial action on p–adic modular forms, commuting with U (lh ). Let q be a prime ideal of F . For a test object (X, Λ, η) of level Γ0 (Nq), η induces a subgroup C ∼ = O/q in X. Then we can construct canonically [q](X, Λ, η) = (X , Λ, η ) with X = X/C (see [H04b] Subsection 5.3). If q splits into QQ in M/F , choosing ηq induced by X(a)[q∞ ] ∼ = MQ /RQ × MQ /RQ ∼ = Fq /Oq × Fq /Oq , we always have a canonical level q–structure on X(a) induced by the choice of the factor Q. Then [q](X(a)) = X(aQ−1 n ) for Qn = Q ∩ Rn for a proper Rn –ideal a. When q ramifies in M/F as q = Q2 , X(a) has a subgroup C = X(a)[Qn ] isomorphic to O/q; so, we can still define [q](X(a)) = X(aQ−1 n ). The effect of [q] on the q–expansion at the infinity cusp (O, c−1 ) is computed in [H04b] (5.12) and is given by −1 the q–expansion of f atthe cusp (q, c ). The operator [q] corresponds to the 1 0 action of g = 0 q−1 ∈ GL2 (Fq ). Although the action of [q] changes the polarization ideal by c → cq, as in the case of Hecke operator, we regard it as a linear map well defined on V (Γ0 (ff l), λ ; W ) into V (Γ0 (ff lq), λ ; W ) (inducing the permutation c → cq ) For ideals A in F , we can think of the association X → X ⊗O A for each AVRM X. There are a natural polarization and a level structure on X ⊗ A induced by those of X. Writing (X, Λ, η) ⊗ A for the triple made out of (X, Λ, η) after tensoring A, we define f |A(X, Λ, η) = f ((X, Λ, η) ⊗ A). For X(a), we have A(X(a)) = X(Aa). The effect of the operator A on the Fourier expansion at (O, c−1 ) is given by that at (A−1 , Ac) (see [H04b] (5.11) or [PAF] (4.53)). The operator A induces an automorphism of V (Γ0 (ff l), λ ; W ). By q–expansion principle, f → f |[q] and f → f |A are injective on the space of (p–adic) modular forms, since the effect on the q-expansion at one cusp of the operation is described by the q-expansion of the same form at another cusp. We fix a decomposition Cl∞ = Γf × Δ for a finite group Δ and a torsionfree subgroup Γf . Since each fractional R–ideal A prime to l defines a class [A] in Cl∞ , we can embed the ideal group of fractional ideals prime to l into Cl∞ . alg alg We write Cl∞ for its image. Then Δalg = Δ ∩ Cl∞ is represented by prime ideals of M non-split over F . We choose a complete representative set for Δalg as {sR−1 |s ∈ S, r ∈ R}, where S contains O and ideals s of F outside p C, R is made of square-free product of primes in F ramifying in M/F , and R is a unique ideal in M with R2 = r. The set S is a complete representative set for the image ClF0 of ClF in Cl0 and {R|r ∈ R} is a complete representative set

240

Haruzo Hida

for 2–torsion elements in the quotient Cl0 /ClF0 . We fix a character ν : Δ → F× , and define " −1 −1 fν = λν (R) νλ (s)f |s "[r]. (4.4) r∈R

s∈S

Choose a complete representative set Q for Cl∞ /Γf Δalg made of primes (p) of M split over F outside plC. We choose ηn out of the base (w1 , w2 ) n so that at q = Q ∩ F , w1 = (1, 0) ∈ RQ × RQc = Rq and of R w2 = (0, 1) ∈ RQ × RQc = Rq . Since all operators s, [q] and [r] involved in this definition commutes with U (l), fν |[q] is still an eigenform of U (l) with the same eigenvalue as f . Thus in particular, we have a measure ϕfν . We define another measure ϕνf on Γf by φdϕνf = λν −1 (Q) φ|Qdϕfν |[q] , Γf

Γf

Q∈Q

where φ|Q(y) = φ(y[Q]−1 f ) for the projection [Q]f in Γf of the class [Q] ∈ Cl∞ . Lemma 4.1 If χ : Cl∞ → F× is a character inducing ν on Δ, we have ν χdϕf = χdϕf . Γf

Cl∞

Proof Write Γf,n for the image of Γf in Cln . For a proper Rn –ideal a, by the above definition of these operators, f |s|[r]|[q]([a]) = λ(a)−1 f (x(Q−1 R−1 a), ω(Q−1 R−1 a)). For sufficiently large n, χ factors through Cln . Since χ = ν on Δ, we have χdϕνf = λχ−1 (QRs−1 a)f |s|[r]|[q]([a]) Γf

Q∈Q s∈S r∈R a∈Γf,n

= because Cl∞ =

8

χ(QRs−1 a)f ([Q−1 R−1 sa]) =

a,Q,s,r

χdϕf , Cl∞

−1 −1 R s]Γf . Q,s,R [Q

m/Z ) ∼ We identify Hom(Γf , F× ) ∼ = Hom(Γf , μ∞ ) with Hom(Γf , G = m over Z . Choosing a basis β = for the formal multiplicative group G {γ1 , . . . , γd } of Γf over Z (so, Zβ = j Zγj ⊂ Γf ) is to choose a multiplicative group Gβm = Hom(Zβ , Gm ) over Z whose formal completion along m/Z ). Thus we may regard the identity of Gβm (F ) giving rise to Hom(Γf , G d G m

Hecke L–values

241

Hom(Γf , μ∞ ) as a subset of Hom(Zβ , Gm ) ∼ = Gβm . We call a subset X of characters of Γf Zariski-dense if it is Zariski-dense as a subset of the algebraic group Gβm/Q (for any choice of β). Then we quote the following result ([H04c] Theorems 3.2 and 3.3): Theorem 4.2 Suppose that p is unramified in M/Q and Σ is ordinary for p. Let f = 0 be an eigenform defined over F of U (l) of level (Γ0 (ff l), λ ) with non-zero eigenvalue. Fix a character ν : Δ → F× , and define fν as in (4.4). If f satisfies the following two conditions: (H1) There exists a strict ideal class c ∈ ClF with the following two properties: (a) the polarization ideal c(Q−1 R−1 s) is in c for some (Q, R, s) ∈ Q × S × R; (b) for any given integer r > 0, the N (l)r modular forms fψ,c | ( 10 u1 ) for u ∈ l−r /O are linearly independent over F, (H2) λ and f are rational over a finite field,

; then the set of characters χ : Γf → F× with non-vanishing Cl∞ νχdϕf = 0 is Zariski dense. If rankZ Γf = 1, under the same assumptions, the nonvanishing holds except for finitely many characters of Γf . Here νχ is the character of Cl∞ = Γf × Δ given by νχ(γ, δ) = ν(δ)χ(γ) for γ ∈ Γf and δ ∈ Δ. 4.3 l–Adic Eisenstein measure modulo p We apply Theorem 4.2 to the Eisenstein series E(λ) in (3.17) for the Hecke character λ fixed in 3.1. Choosing a generator π of mW , the exact kI 0 kI → ω kI sequence ω kI /W − /W ω /F induces a reduction map: H (M, ω /W ) → ◦ H 0 (M, ω kI /F ). We write Ek (ψ , c) mod Λ-modmW for the image of the Eisen◦ stein series Ek (ψ , c). Then we put f = (dκ (Ek (ψ ◦ , c)) mod mW )c ∈ V (Γ0 (ff l), λ ; F). By definition, the q–expansion of f[c] is the reduction modulo mW of the q–expansion of E(λ, c) of characteristic 0. We fix a character ν : Δ → F× as in the previous section and write ϕ = ϕf and ϕν = ϕνf . By (3.24) combined with Lemma 4.1, we have, for a character χ : CL∞ → F× with χ|Δ = ν, π κ ΓΣ (kΣ + κ)L(l) (0, χ−1 λ) χdϕν = χdϕ = Cχ(F) mod mW , ΩkΣ+2κ Γf Cl∞ ∞ (4.5)

242

Haruzo Hida

where C is a non-zero constant given by the class modulo mW of (−1)k[F :Q] (R× : O × )λ−1 (e(F) ) √ . Im(δ)κ D The non-vanishing of C follows from the unramifiedness of p in M/Q and that F is prime to p. Theorem 4.3 Let p be an odd prime unramified in M/Q. Let λ be a Hecke character of M of conductor C and of infinity type kΣ + κ(1 − c) with 0 < k ∈ Z and 0 ≤ κ ∈ Z[Σ] for a CM type Σ that is ordinary with respect × to p. Suppose (spt) and (opl) in 3.1. Fix a character ν : Δ → Q . Then π κ ΓΣ (kΣ+κ)L(l) (0,ν −1 χ−1 λ) ∈ W for all characters χ : Cl∞ → μ∞ (Q) facΩkΣ+2κ ∞ toring through Γf . Moreover, for Zariski densely populated character χ in Hom(Γf , μ∞ ), we have π κ ΓΣ (kΣ + κ)L(l) (0, ν −1 χ−1 λ) ≡ 0 mod mW , ΩkΣ+2κ ∞ unless the following three conditions are satisfied by ν and λ simultaneously: (M1) M/F is unramified everywhere; (M2) The strict ideal class (in F ) of the polarization ideal c0 of X(R) is not M/F a norm class of an ideal class of M (⇔ = −1); c0 (M3) The ideal character a → (λν −1 N (a) mod mW ) ∈ F× of F is equal to the character M/F of M/F . · If l is a split prime of degree 1 over Q, under the same assumptions, the non-vanishing holds except for finitely many characters of Γf . If (M1-3) are satisfied, the L–value as above vanishes modulo m for all anticyclotomic characters χ. See [H04b] 5.4 for an example of (M, c0 , Σ) satisfying (M1-3). Proof By Theorem 4.2, we need to verify the condition (H1-2) for E(λ). The rationality (H2) follows from the rationality of Ek (ψ ◦ , c) and the differential operator d described in 2.6. For a given q–expansion h(q) = ξ a(ξ, h)q ξ ∈ −1 F[[c−1 ), we know that, for u ∈ Ol ⊂ FA , ≥0 ]] at the infinity cusp (O, c a(ξ, h|αu ) = eF (uξ)a(ξ, h) for αu = ( 10 u1 ). The condition (H1) for h concerns the linear independence of h|αu for u ∈ l−r Ol /Ol . For any function φ : c−1 /lr c−1 = O/lr → F, we write

Hecke L–values h|φ =

ξ

243

φ(ξ)a(ξ, h)q ξ . By definition, we have

h|Rφ =

φ(u)h|αu = h|φ∗

u∈O/lr

for the Fourier transform φ∗ (v) = u φ(u)eF (uv). For the characteristic function χv of v ∈ c−1 /lr c, we compute its Fourier transform χ∗v (u) = eF (au)χv (a) = eF (vu). a∈O/lr

Since the Fourier transform of the finite group O/lr is an automorphism (by the inversion formula), the linear independence of {h|αu = h|χ∗u }u is equivalent to the linear independence of {h|χu }u . We recall that fν is a tuple (fν,[c] )c ∈ V (Γ01 (f2 ), Γ0 (l); W ). Thus we need to prove: there exists c such that for a given congruence class u ∈ c−1 /lr c−1 a(ξ, fν,[c] ) ≡ 0 mod mW for at least one ξ ∈ u.

(4.6)

Since a(ξ, dκ h) = ξ κ a(ξ, h) ((2.6)), (4.6) is achieved if ) ≡ 0 mod mW for at least one ξ ∈ u prime to p a(ξ, fν,[c]

(4.7)

holds for f = (Ek (ψ ◦ , c) − N (l)Ek (ψ ◦ , cl−1 )|l|[l])c , because l p. Up to a non-zero constant, ψ ◦ (a, b) in (3.7) is equal to φ(a, b) = × λF (a)λ−1 Fc (b) for (a, b) ∈ (O/f) . Thus we are going to prove, for a well chosen c, ) ≡ 0 mod mW for at least one ξ ∈ u prime to p, a(ξ, fν,[c]

(4.8)

where f[c] = Ek (φ, c) − N (l)Ek (φ, cl−1 )|l|[l]. Recall (4.4):

fν

=

λν

−1

(R)

r∈R

−1

νλ

" (s)f |s "[r].

(4.9)

s∈S

As computed in [H04b] (5.11) and (5.12), we have N (s−1 r)−1 Ek (φ, c)|s|[r]O,c−1 (q) = 2−[F :Q] L(1 − k; φ, s−1 r) N (a) qξ φ(a, b) N (a)k−1 . (4.10) + |N (a)| −1 −1 −1 × 0ξ∈c

r

(a,b)∈(s

r×c ab=ξ

s)/O

244

Haruzo Hida k

N (a) −1 a(ξ, fa,b ) is given by Thus we have, writing t(a, b) = φ(a, b) |N (a)| , N (a)

t(a, b) −

(a,b)∈(a×b)/O × ab=ξ

t(a, b)

(a,b)∈(ar×lb)/O× ab=ξ

=

t(a, b).

(a,b)∈(a×(b−lb))/O ab=ξ

(4.11)

×

We have the freedom of moving around the polarization ideal class [c] in the coset NM/F (ClM )[c0 ] for the polarization ideal class [c0 ] of x(R). We first look into a single class [c]. We choose c−1 to be a prime q prime to pfl (this is possible by changing c−1 in its strict ideal class and choosing δ ∈ M suitably). We take a class 0 ' ξ ∈ u for u ∈ O/lr so that (ξ) = qnle for an integral ideal n plC prime to the relative discriminant D(M/F ) and 0 ≤ e ≤ r. Since we have a freedom of choosing ξ modulo lr , the ideal n moves around freely in a given ray class modulo lr−e . We pick a pair (a, b) ∈ F 2 with ab = ξ with a ∈ s−1 and b ∈ qs. Then (a) = s−1 lα x for an integral ideal x prime to l and (b) = sqle−α x for an integral ideal x prime to l. Since (ab) = qnle , we find that xx = n. By (4.11), b has to be prime to l; so, we find α = e. Since xx = n and hence r = O because n is prime to D(M/F ). Thus for each factor x of n, we could have two possible pairs (ax , bx ) with ax bx = ξ such that e −1 −1 ((ax ) = s−1 ) x l x, (bx ) = (ξax ) = sx qnx

for sx ∈ S representing the ideal class of the ideal le x. We put ψ = ν −1 λ. We then write down the q–expansion coefficient of q ξ at the cusp (O, q) (see [H04c] (4.30)): G(ψf )−1 a(ξ, fν ) = ψF−1 (ξ)ψ(nle )−1 N (nle )−1 c

1 − (ψ(y)N (y))e(y)+1 y|n

1 − ψ(y)N (y)

where n = y|n ye(y) is the prime factorization of n. We define, for the valuation v of W (normalized so that v(p) = 1) ⎞ ⎛ 1 − (ψ(y)N (y))e(y)+1 ⎠, μC (ψ) = Infn v ⎝ 1 − ψ(y)N (y)

,

(4.12)

(4.13)

y|n

where n runs over a ray class C modulo lr−e made of all integral ideals prime to Dl of the form q−1 ξl−e , 0 ' ξ ∈ u. Thus if μC (ψ) = 0, we get the desired non-vanishing. Since μC (ψ) only depends on the class C, we may assume

Hecke L–values

245

(and will assume) that e = 0 without losing generality; thus ξ is prime to l, and C is the class of u[q−1 ]. Suppose that n is a prime y. Then by (4.12), we have (ξ)(1 + (ψ(y)N (y))−1 ). G(ψf )−1 a(ξ, fν ) = ψF−1 c If ψ(y)N (y) ≡ −1 mod mW for all prime ideals y in the ray class C modulo lr , the character a → (ψ(a)N (a) mod mW ) is of conductor lr . We write ψ for the character: a → (ψ(a)N (a) mod mW ) of the ideal group of F with values r . Since ν is anticyclotomic, in F× . This character therefore has conductor C|l × its restriction to FA has conductor 1. Since λ has conductor C prime to l, the conductor of ψ is a factor of the conductor of λ mod mW , which is a factor of = 1. pC. Thus C|pC. Since l pC, we find that C are going to show that if μC (ψ) > 0, M/F is unramified and ψ ≡ We mod mW . We now choose two prime ideals y and y so that qyy = (ξ) with ξ ∈ u. Then by (4.12), we have 1 1 1 + . (ξ) 1 + G(ψf )−1 a(ξ, fν ) = ψF−1 c ψ(y)N (y) ψ(y )N (y ) (4.14) Since ψ(yy ) = ψ(u[q−1 ]) = ψ(C) = −1, we find that if a(ξ, fν ) ≡ 0 mod mW , M/F

−1 = ψ(y/y ) = ψ(l−1 )ψ(y2 ) = −ψ(y2 ). Since we can choose y arbitrary, we find that ψ is quadratic. Thus μC (ψ) > 0 if and only if ψ(c) = −1, which is independent of the choice of u. Since we only need to show the existence of c with ψ(c) = 1, we can vary the strict ideal class [c] in [c0 ]NM/F (ClM ). By class field theory, assuming that ψ has conductor 1, we have ψ(c) = −1 for all [c] ∈ [c0 ]NM/F (ClM ) M/F if and only if ψ(c0 ) = −1 and ψ(a) = for all a ∈ ClF . a

(4.15)

If M/F is unramified, by definition, 2δc∗ = 2δd−1 c−1 = R. Taking squares, we find that (dc)2 = 4δ 2 ' 0. Thus 1 = ψ(d−2 c−2 ) = (−1)[F :Q] , and this never happens when [F : Q] is odd. Thus (4.15) is equivalent to the three conditions (M1-3). The conditions (M1) and (M3) combined is equivalent to ψ ∗ ≡ ψ mod mW , where the dual character ψ ∗ is defined by ψ ∗ (x) = ψ(x−c )N (x)−1 . Then the vanishing of L(0, χ−1 ν −1 λ) ≡ 0 for all anti-cyclotomic χν follows from the functional equation of the p-adic Katz measure interpolating the p–adic Hecke L–values. This finishes the proof.

246

Haruzo Hida 5 Anticyclotomic Iwasawa series

We fix a conductor C satisfying (spt) and (opl) in 3.1. We consider Z = Z(C) = limn ClM (Cpn ) for the ray class group ClM (x) of M modulo x. ←− We split Z(C) = ΔC × ΓC for a finite group Δ = ΔC and a torsion-free subgroup ΓC . Since the projection: Z(C) Z(1) induces an isomorphism ΓC = Z(C)/ΔC ∼ = Z(1)/Δ1 = Γ1 , we identify ΓC with Γ1 and write it as Γ, which has a natural action of Gal(M/F ). We define Γ+ = H 0 (Gal(M/F ), Γ) and Γ− = Γ/Γ+ . Write π− : Z → Γ− and πΔ : Z → Δ for the two projec× tions. Take a character ϕ : Δ → Q , and regard it as a character of Z through the projection: Z Δ. The Katz measure μC on Z(C) associated to the p– adic CM type Σp as in [HT1] Theorem II induces the anticyclotomic ϕ–branch μ− ϕ by

φdμ− ϕ = Γ−

φ(π− (z))ϕ(πΔ (z))dμC (z). Z(C)

− We write L− p (ϕ) for this measure dμϕ regarding it as an element of the algebra − Λ = W [[Γ− ]] made up of measures with values in W . [F :Q] − of M on We look into the arithmetic of the unique Zp –extension M∞ −1 −1 − which we have cσc = σ for all σ ∈ Gal(M∞ /M ) for complex con− /M is called the anticyclotomic tower over jugation c. The extension M∞ ∞ M . Writing M (Cp ) for the ray class field over M modulo Cp∞ , we identify Z(C) with Gal(M (Cp∞ )/M ) via the Artin reciprocity law. Then one has − − Gal(M (Cp∞ )/M∞ ) = Γ+ × ΔC and Gal(M∞ /M ) = Γ− . We then define ∞ MΔ by the fixed field of ΓC in M (Cp ); so, Gal(MΔ /M ) = Δ. Since ϕ is a − − character of Δ, ϕ factors through Gal(M∞ MΔ /M ). Let L∞ /M∞ MΔ be the maximal p–abelian extension unramified outside Σp . Each γ ∈ Gal(L∞ /M ) − acts on the normal subgroup X = Gal(L∞ /M∞ MΔ ) continuously by conjugation, and by the commutativity of X, this action factors through − /M ). Then we look into the Γ− –module: X[ϕ] = X ⊗ΔC ,ϕ W . Gal(MΔ M∞ As is well known, X[ϕ] is a Λ− –module of finite type, and in many cases, it is torsion by a result of Fujiwara (cf. [Fu], [H00] Corollary 5.4 and [HMI] Section 5.3) generalizing the fundamental work of Wiles [Wi] and Taylor-Wiles [TW]. If one assumes the Σ–Leopoldt conjecture for abelian extensions of M , we know that X[ϕ] is a torsion module over Λ− unconditionally (see [HT2] Theorem 1.2.2). If X[ϕ] is a torsion Λ− –module, we can think of the characteristic element F − (ϕ) ∈ Λ− of the module X[ϕ]. If X[ϕ] is not of torsion over Λ− , we simply put F − (ϕ) = 0. A character ϕ of Δ is called anticyclotomic if ϕ(cσc−1 ) = ϕ−1 .

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We are going to prove in this section the following theorem: Theorem 5.1 Let ψ be an anticyclotomic character of Δ. Suppose (spt) and (opl) in 3.1 for the conductor C(ψ) of ψ. If p is odd and unramified in F/Q, − then the anticyclotomic p–adic Hecke L–function L− p (ψ) is a factor of F (ψ) − in Λ . Regarding ϕ as a Galois character, we define ϕ− (σ) = ϕ(cσc−1 σ −1 ) for σ ∈ Gal(M /M ). Then ϕ− is anticyclotomic. By enlarging C if necessary, we can find a character ϕ such that ψ = ϕ− for any given anticyclotomic ψ (e.g. [GME] page 339 or [HMI] Lemma 5.31). Thus we may always assume that ψ = ϕ− . It is proven in [HT1] and [HT2] that Lp (ϕ− ) is a factor of F − (ϕ− ) in − Λ− ⊗Z Q. Thus the improvement concerns the p–factor of L− p (ϕ ), which has been shown to be trivial in [H04b]. The main point of this paper is to give another proof of this fact reducing it to Theorem 4.3. The new proof actually gives a stronger result: Corollary 5.6, which is used in our paper [H04d] to − prove the identity L− p (ψ) = F (ψ) under suitable assumptions on ψ. The proof is a refinement of the argument in [HT1] and [HT2]. We first deduce a refinement of the result in [HT1] Section 7 using a unique Hecke eigenform (in a given automorphic representation) of minimal level. The minimal level is possibly a proper factor of the conductor of the automorphic representation. Then we proceed in the same manner as in [HT1] and [HT2]. Here we describe how to reduce Theorem 5.1 to Corollary 5.6. Since the result is known for F = Q by the works of Rubin and Tilouine, we may assume that F = Q. Put Λ = W [[Γ]]. By definition, for the universal Galois character ψ : Gal(M (Cp∞ )/M ) → Λ× sending δ ∈ ΔC to ψ(δ) and γ ∈ Γ to the group element γ ∈ Γ ⊂ Λ, the Pontryagin dual of the adjoint Selmer group Sel(Ad(IndF M ψ)) defined in [MFG] 5.2 is isomorphic to the direct sum ClM ⊗Z Λ of X[ψ]⊗Λ− Λ and ClF ⊗Z Λ . Thus the characteristic power series of the Selmer group is given by (h(M )/h(F ))F − (ψ). To relate this power series (h(M )/h(F ))F − (ψ) to congruence among automorphic forms, we identify Of ∼ = RF ∼ = RFc . Recall the maximal diagonal torus T0 ⊂ GL(2)/O . Thus ψ restricted to (RF × RFc )× gives rise to the character ψ of T02 (Of ). We then extend ψ to acharacter ψF of

T02 (Of × OD(M/F ) ) by ψF (xf , yf , x , y ) = ψ(xf , yf ) −

M/F y

. Then we define

the level ideal N by (C(ψ ) ∩ F )D(M/F ) and consider the Hecke algebra hn.ord = hn.ord (N, ψF ; W ). It is easy to see that there is a unique W [[Γ]]– algebra homomorphism λ : hn.ord → Λ such that the associated Galois representation ρλ ([H96] 2.8) is IndF M ψ. Here Γ is the maximal torsion-free

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quotient of G introduced in 2.10. Note that the restriction of ρλ tothe decom ψF,1 0 position group Dq at a prime q|N is the diagonal representation 0 ψF,2 with values in GL2 (W ), which we write ρq . We write H(ψ) for the congruence power series H(λ) of λ (see [H96] Section 2.9, where H(λ) is written as η(λ)). Writing T for the local ring of hn.ord through which λ factors, the divisibility: H(ψ)|(h(M )/h(F ))F − (ψ) follows from the surjectivity onto T of the natural morphism from the universal nearly ordinary deformation ring Rn.ord of IndM F ψ mod mW (without deforming ρq for each q|N and the restriction of the determinant character to Δ(N)). See [HT2] Section 6.2 for details of this implication. The surjectivity is obvious from our construction of hn.ord (N, ψF ; W ) because it is generated by Tr(ρλ (F robq )) for primes q outside pN and by the diagonal entries of ρλ restricted to Dq for q|pN. Thus we need to prove (h(M )/h(F ))L− p (ψ)|H(ψ), which is the statement of Corollary 5.6. This corollary will be proven in the rest of this section. As a final remark, if we write Tχ for the quotient of T which parametrizes all p–adic modular Galois representations congruent to IndF M ψ with a given W W [[Γ+ ]] = Tχ [[Γ+ ]] for the determinant character χ, we have T ∼ = Tχ ⊗ maximal torsion-free quotient Γ+ of ClF+ (Np∞ ) (cf. [MFG] Theorem 5.44). This implies H(ψ) ∈ W [[Γ− ]].

5.1 Adjoint square L–values as Petersson metric We now set G := ResO/Z GL(2). Let π be a cuspidal automorphic representation of G(A) which is everywhere principal at finite places and holomorphic discrete series at all archimedean places. Since π is associated to holomorphic automorphic forms on G(A), π is rational over the Hecke field generated by eigenvalues of the primitive Hecke eigenform in π. We have π = π (∞) ⊗ π∞ for representations π (∞) of G(A(∞) ) and π∞ of G(R). We further decompose π (∞) = ⊗q π(1,q , 2,q ) for the principal series representation π(1,q , 2,q ) of GL2 (Fq ) with two char× acters 1,q , 2,q : Fq× → Q . By the rationality of π, these characters have values in Q. The central character of π(∞) is given by + = q (1,q 2,q ), which is a Hecke character of F . However 1 = q 1,q and 2 = q 2,q are just characters of FA×(∞) and may not be Hecke characters. In the space of automorphic forms in π, there is a unique normalized Hecke eigenform f = fπ of minimal level satisfying the following conditions (see [H89] Corollary 2.2): (L1) The level N is the conductor of − = 2 −1 1 .

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249

# $ (L2) Note that π : ac db → 1 (ad − bc)− (d) is a character of U0 (N) whose restriction to#U0 (C(π)) for the conductor C(π) of π induces the $ “Neben” character ac db → 1 (a)2 (d). Then f : G(Q)\G(A) → C satisfies f (xu) = π (u)f (x). (L3) The cusp form f corresponds to holomorphic cusp forms of weight κ = (κ1 , κ2 ) ∈ Z[I]2 . In short, fπ is a cusp form in Sκ (N, + ; C). It is easy to see that Π = π ⊗ −1 2 has conductor N and that v ⊗ 2 is a constant multiple of f for the new vector v of Π (note here that Π may not be automorphic, but Π is an admissible irreducible representation of G(A); so, the theory of new vectors still applies). Since the conductor C(π) of π is given by the product of the conductors of 1 and 2 , the minimal level N is a factor of the conductor C(π) and is often a proper divisor of C(π). By (L2), the Fourier coefficients a(y, f ) satisfy a(uy, f ) = 1 (u)a(y, f ) for In particular, the function: y → a(y, f )a(y, f ) only × (O = O ⊗Z Z). u∈O depends on the fractional ideal yO. Thus writing a(a, f )a(a, f ) for the ideal a = yO, we defined in [H91] the self Rankin product by a(a, f )a(a, f )N (a)−s , D(s − [κ] − 1, f, f ) = a⊂O

where N (a) = [O : a] = |O/a|. We have a shift: s → s − [κ] − 1, because in order to normalize the L–function, we used in [H91] (4.6) the unitarization ([κ]−1)/2 π u = π ⊗ | · |A in place of π to define the Rankin product. The weight u κ of the unitarization satisfies [κu ] = 1 and κu ≡ κ mod QI. Note that (cf. [H91] (4.2b)) fπu (x) := fπu (x) = D −([κ]+1)/2 fπ (x)| det(x)|A

([κ]−1)/2

.

(5.1)

We are going to define Petersson metric on the space of cusp forms satisfying (L1-3). For that, we write X0 = X0 (N) = G(Q)+ \G(A)+ /U0 (N)FA× SO2 (FR ). We define the inner product (f, g) by [κ]−1 f (x)g(x)| det(x)|A dx (f, g)N = X0 (N)

(5.2)

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Haruzo Hida

with respect to the invariant measure dx on X0 as in [H91] page 342. In exactly the same manner as in [H91] (4.9), we obtain Ds (4π)−I(s+1)−(κ1 −κ2 ) ΓF ((s + 1)I + (κ1 − κ2 ))ζF (2s + 2)D(s, f, f ) (N)

= N (N)−1 D −[κ]−2 (f, f E0,0 (x, 1, 1; s + 1))N , (N) where D is the discriminant N (d) of F , ζF (s) = ζF (s) q|N (1 − N (q)−s ) for the Dedekind zeta function ζF (s) of F and Ek,w (x, 1, 1; s) (k = κ1 −κ2 +I and w = I − κ2 ) is the Eisenstein series of level N defined above (4.8e) of [H91] for the identity characters (1, 1) in place of (χ−1 ψ −1 , θ) there. (N) By the residue formula at s = 1 of ζF (2s)E0,0 (x, 1, 1; s) (e.g. (RES2) in [H99] page 173), we find (4π)−I−(κ1 −κ2 ) ΓF (I + (κ1 − κ2 ))Ress=0 ζF (2s + 2)D(s, f, f ) 2[F :Q]−1 π [F :Q] R∞ h(F ) √ = D−[κ]−2 N (N)−1 (f, f )N , (1 − N (q)−1 ) w D q|N (N)

(5.3) where w = 2 is the number of roots of unity in F , h(F ) is the class number of F and R∞ is the regulator of F . Since f corresponds to v⊗2 for the new vector v ∈ Π of the principal series representation Π(∞) of minimal level in its twist class {Π ⊗ η} (η running over all finite order characters of FA×(∞) ), by making product f · f , the effect of tensoring 2 disappears. Thus we may compute the Euler factor of D(s, f, f ) as if f were a new vector of the minimal level representation (which has the “Neben” character with conductor exactly equal to that of Π). Then for each (N) prime factor q|N, the Euler q–factor of ζF (2s + 2)D(s, f, f ) is given by ∞ −1 a(qν , f )a(qν , f )N (q)−νs = 1 − N (q)[κ]−s , ν=0

because a(q, f )a(q, f ) = N (q)[κ] by [H88] Lemma 12.2. Thus the zeta func(N) tion ζF (2s + 2)D(s, f, f ) has the single Euler factor (1 − N (q)−s−1 )−1 at q|N, and the zeta function ζF (s + 1)L(s + 1, Ad(f )) has its square (1 − N (q)−s−1 )−2 at q|N, because L(s + 1, Ad(f )) contributes one more factor (1 − N (q)−s−1 )−1 . The Euler factors outside N are the same by the standard computation. Therefore, the left-hand-side of (5.3) is given by (N)

ζF (2s + 2)D(s, f, f ) ⎛ ⎞ = ⎝ (1 − N (q)−s−1 )⎠ ζF (s + 1)L(s + 1, Ad(f )) q|N

(5.4)

Hecke L–values

251

By comparing the residue at s = 0 of (5.4) with (5.3) (in view of (5.1)), we get (fπu , fπu )N = D −[κ]−1 (fπ , fπ )N = DΓF ((κ1 − κ2 ) + I)N (N)2−2((κ1 −κ2 )+I)+1 π −((κ1 −κ2 )+2I) L(1, Ad(f )) (5.5) for the primitive adjoint square L–function L(s, Ad(f )) (e.g. [H99] 2.3). sσ for s ;= Here we have written xs = σx σ sσ σ ∈ C[I], and ∞ ΓF (s) = σ Γ(sσ ) for the Γ–function Γ(s) = 0 e−t ts−1 dt. This formula is consistent with the one given in [HT1] Theorem 7.1 (but is much simpler).

5.2 Primitive p–Adic Rankin product Let N and J be integral ideals of F prime to p. We shall use the notation introduced in 2.10. Thus, for a p–adically complete valuation ring W ⊂ p , hn.ord (N, ψ; W ) and hn.ord (J, χ; W ) are the universal nearly ordinary Q Hecke algebra with level (N, ψ) and (J, χ) respectively. The character ψ = (p) t ) is made of the characters of ψj of T0 (Op × (O/N )) (for (ψ1 , ψ2 , ψ+ t to ΔF (N) (the an ideal N ⊂ N) of finite order and for the restriction ψ+ + ∞ torsion part of ClF (N p )) of a Hecke character ψ+ extending ψ1 ψ2 . Similarly we regard χ as a character of G(J ) for an ideal J ⊂ J); so, ψ − = ψ1−1 ψ2 and χ− are well defined (finite order) character of T0 (Op × (O/N)) and T0 (Op × (O/J)) respectively. In particular we have C(p) (ψ − )|N and C(p) (χ− )|J, where C(p) (ψ − ) is the prime-to–p part of the conductor C(ψ − ) of ψ− . We assume that C(p) (ψ − ) = N, and C(p) (χ− ) = J.

(5.6)

For the moment, we also assume for simplicity that × × ψq− = χ− q on Oq for q|JN and ψ1 = χ1 on O .

(5.7)

Let λ : hn.ord (N, ψ; W ) → Λ and ϕ : hn.ord (J, χ; W ) → Λ be Λ–algebra homomorphisms for integral domains Λ and Λ finite torsionfree over Λ. For each arithmetic point P ∈ Spf(Λ)(Qp ), we let fP ∈ Sκ(P ) (U0 (Npα ), ψP ; Qp ) denote the normalized Hecke eigenform of minimal level belonging to λ. In other words, for λP = P ◦ λ : hn.ord → Qp , we have a(y, fP ) = λP (T (y)) for all integral ideles y with yp = 1. In the automorphic representation generated by fP , we can find a unique automorphic form fPord with a(y, fPord ) = λ(T (y)) for all y, which we call the (nearly) ordinary projection of fP . Similarly, using ϕ, we define gQ ∈ Sκ(Q) (U0 (Jpβ ), χQ ; Qp )

252

Haruzo Hida

for each arithmetic point Q ∈ Spf(Λ )(Qp ). Recall that we have two charac associated to ψP . Recall ψP = (ψP,1 , ψP,2 , ψP + ) : ters (ψP,1 , ψP,2 ) of T0 (O) 2 × (F × /F × ) → C× . The central character ψP + of fP coincides with T0 (O) A × and has infinity type (1 − [κ(P )])I. We suppose ψP,1 ψP,2 on O The character ψP,1 χ−1 Q,1 is induced by a global finite order character θ. (5.8) This condition combined with (5.6) implies that θ is unramified outside p. As seen in [H91] 7.F, we can find an automorphic form gQ |θ−1 on G(A) whose Fourier coefficients are given by a(y, gQ |θ−1 ) = a(y, gQ )θ −1 (yO), where θ(a) = 0 if a is not prime to C(θ). The above condition implies, as explained in the previous subsection, y → a(y, fP )a(y, gQ |θ−1 )θ(y) factors through the ideal group of F . Note that a(y, fP )a(y, gQ |θ −1 )θ(y) = a(y, fP )a(y, gQ ) as long as yp is a unit. We thus write a(a, fP )a(a, gQ |θ−1 )θ(a) for the above product when yO = a and define D(s −

[κ(P )] + [κ(Q)] − 1, fP , gQ |θ−1 , θ −1 ) 2 = a(a, fP )a(a, gQ |θ −1 )θ(a)N (a)−s . a

Hereafter we write κ = κ(P ) and κ = κ(Q) if confusion is unlikely. (x) = gQ |θ−1 (x)θ(det(x)), Note that for gQ D(s, fP , gQ ) := D(s, fP , gQ , 1) = D(s, fP , gQ |θ −1 , θ−1 ).

Though the introduction of the character θ further complicates our notation, , since the local component we can do away with it just replacing gQ by gQ π(χQ,1,q , χQ,2,q ) of the automorphic representation generated by gQ satisfies χ1,Q = ψ1,P , and hence without losing much generality, we may assume a slightly stronger condition: × ψP,1 = χQ,1 on O

(5.9)

in our computation. For each holomorphic Hecke eigenform f , we write M (f ) for the rank <(f ) for its dual, ρf for the p–adic 2 motive attached to f (see [BR]), M Galois representation of M (f ) and ρf for the contragredient of ρf . Here p is the p–adic place of the Hecke field of f induced by ip : Q → Qp . Thus

Hecke L–values

253

L(s, M (f )) coincides with the standard L–function of the automorphic representation generated by f , and the Hodge weight of M (fP ) is given by {(κ1,σ , κ2,σ ), (κ1,σ , κ2,σ )}σ for each embedding σ : F → C. We have det(ρfP (F robq )) = ψPu (q)N (q)[κ] (p = q; see [MFG] 5.6.1). Lemma 5.2 Suppose (5.6) and (5.8). For primes q p, the Euler q–factor of L(NJ) (2s − [κ] − [κ ], ψPu χ−u Q )D(s −

[κ] + [κ ] ) − 1, fP , gQ 2

<(gQ )) given by is equal to the Euler q–factor of Lq (s, M (fP ) ⊗ M " # $−1 det 1 − (ρfP ⊗ ρgQ )(F robq )"V I N (q)−s ) , where V is the space of the p–adic Galois representation of the tensor product: ρfP ⊗ ρgQ and V I = H 0 (I, V ) for the inertia group I ⊂ Gal(Q/F ) at q. Proof As already explained, we may assume (5.9) instead of (5.8). By abusing the notation, we write π(ψP,1,q , ψP,2,q ) (resp. π(χQ,1,q , χQ,2,q )) for the q–factor of the representation generated by fP (resp. gQ ). By the work of Carayol, R. Taylor and Blasius-Rogawski combined with a recent work of Blasius [B], the restriction of ρfP to the Decomposition group at q is isomorphic to diag[ψP,1,q , ψP,2,q ] (regarding ψi,P,q as Galois characters by local class field theory). The same fact is true for gQ . If q|JN, then V I is one dimensional on which F robq acts by ψP,1,q (q )χQ,1,q (q ) = a(q, fP )a(q, gQ ) × and because ψi,P,q χj,Q,q is ramified unless i = j = 1 (⇔ ψ1 = χ1 on O −1 − − − ψq = ψq ). If q JN, both π(ψP,1,q , ψP,2,q ) ⊗ ψP,1,q = π(1, ψP,q ) and − π(χQ,1,q , χQ,2,q ) ⊗ χ−1 Q,1,q = π(1, χQ,q ) are unramified principal series. By ψP,1,q = χQ,1,q :(5.8), we have an identity: −1 c ∼ c ⊗ χQ,1,q ) ) ⊗ (ρgQ ρfP ⊗ ρgQ = (ρfP ⊗ ψP,1,q

on the inertia group, which is unramified. Therefore V is unramified at q. At the same time, the L–function has full Euler factor at q JN. would like to compute fP |τ (x) := ψPu (det(x))−1 fP (xτ ) for τ (N ) = # 0We $ −1 ∈ G(A(∞) ) for an idele N = N (P ) with N (pN) = 1 and N O = N 0 − C(ψP ) (whose prime-to–p factor is N). We continue to abuse notation and −1 − − write π(ψP,1,q , ψP,2,q ) ⊗ ψP,1,q as π(1, ψP,q ) (thus ψP,q is the character of − − u − − ) = ψP,q /|ψP,q | Fq× inducing the original ψP,q on Oq× ). We write (ψP,q − − (which is a unitary character). In the Whittaker model V (1, ψP,q ) of π(1, ψP,q ) (realized in the space of functions on GL2 (Fq )), we have a unique function

254

Haruzo Hida

φq on GL2 (Fq ) whose Mellin transform gives rise to the local L–function of π(1, ψq− ). In particular, we have (cf. [H91] (4.10b)) φq |τq (x) := (ψP− )u (det(x))−1 φq (xτq ) = W (φq )|ψP− (Nq )|1/2 φq (x), where φq is the complex conjugate of φq belonging to the space repre−

sentation V (1, ψ P,q ) and W (φq ) is the epsilon factor of the representation − π(1, ψP,q ) as in [H91] (4.10c) (so, |W (φq )| = 1). Then φq ⊗ ψP,1,q (x) := ψP,1,q (det(x))φq (x) is in V (ψP,1,q , ψP,2,q ) and gives rise to the q–component of the global Whittaker model of the representation π generated by fP . The above formula then implies (φq ⊗ ψP,1,q )|τq := ψPu (det(x))−1 (φq ⊗ ψP,1,q )(xτq ) (1−[κ])/2

u = ψP,1,q (Nq )|Nq |q

W (φq )(φq ⊗ ψ P,1,q )(x). Define the root number Wq (fP ) = W (φq ) and W (fP ) = q Wq (fP ). Here note that Wq (fP ) = 1 if the prime q is outside C(ψP,1 )C(ψP,2 )D. We conclude from the above computation the following formula: u (N )|N |A fP |τ (x) := ψPu (det(x))−1 fP (xτ ) = W (fP )ψP,1

(1−[κ])/2 c fP (x),

(5.10) where fPc is determined by a(y, fPc ) = a(y, fP ) for all y ∈ FA× . This shows + u W (fP )W (fPc ) = ψP,∞ (−1) = ψP,∞ (−1).

(5.11)

Using the formula (5.10) instead of [H91] (4.10b), we can prove in exactly the same manner as in [H91] Theorem 5.2 the following result: Theorem 5.3 Suppose (5.6) and (5.7). There exists a unique element D in the W Λ satisfying the following interpolation property: field of fractions of Λ⊗ Let (P, Q) ∈ Spf(Λ) × Spf(Λ ) be an arithmetic point such that × . (W) κ1 (P ) − κ1 (Q) > 0 ≥ κ2 (P ) − κ2 (Q) and ψP,1 = χQ,1 on O Then D is finite at (P, Q) and we have D(P, Q) = W (P, Q)C(P, Q)S(P )−1 E(P, Q)

<(gQ )) L(p) (1, M (fP ) ⊗ M , (fP , fP )

where, writing k(P ) = κ1 (P ) − κ2 (P ) + I, W (P, Q) = −1 −1 u (−1)k(Q) N (J)([κ(Q)]+1)/2 χQ (dp )G(χQ,1,p ψP,1,p )G(χQ,2,p ψP,1,p ) · − −1 (−1)k(P ) N (N)([κ(P )]−1)/2 ψPu (dp )G((ψP,p ) ) p|p

Hecke L–values

255

C(P, Q) = 2([κ(P )]−[κ(Q)])I−2k(P ) π 2κ2 (P )−([κ(Q)]+1)I × ΓF (κ1 (Q) − κ2 (P ) + I)ΓF (κ2 (Q) − κ2 (P ) + I), S(P ) = # $ $# ψP− (p ) − 1 1 − ψP− (p )|p |p − pCp (ψP )

(ψP− (p )|p |p )δ(p) ,

− p|Cp (ψP )

E(P, Q) = pCp (χ− Q)

−1 −1 (1 − χQ,1 ψP,1 (p ))(1 − χQ,2 ψP,1 (p )) −1 (1 − χ−1 Q,1 ψP,1 (p )|p |p )(1 − χQ,2 ψP,1 (p )|p |p )

×

p|Cp (χ− Q)

−1 χQ,2 ψP,1 (p

γ(p)

−1 )(1 − χQ,1 ψP,1 (p ))

(1 − χ−1 Q,1 ψP,1 (p )|p |p )

.

− Here Cp (ψP− ) := C(ψP− ) + (p) = p|p pδ(p) and Cp (χ− P ) := C(χQ ) + (p) = γ(p) . Moreover for the congruence power series H(λ) of λ, H(λ)D ∈ p|p p W Λ . Λ⊗ The expression of p–Euler factors and root numbers is simpler than the one given in [H91] Theorem 5.1, because automorphic representations of fP and gQ are everywhere principal at finite places (by (5.6)). The shape of the constant W (P, Q) appears to be slightly different from [H91] Theorem 5.2. Firstly the present factor (−1)k(P )+k(Q) is written as (χQ+ ψP + )∞ (−1) in −1 [H91]. Secondly, in [H91], it is assumed that χ−1 Q,1 and ψP,1 are both induced by a global character ψP and ψP unramified outside p. Thus the factor )(−1) appears there. This factor is equal to (χQ,1,p ψP,1,p )(−1) = (χQ,∞ ψP,∞ θp (−1), which is trivial because of the condition (W). We do not need to assume the individual extensibility of χQ,1 and ψP,1 . This extensibility is assumed in order to have a global Hecke eigenform fP◦ = fPu ⊗ ψP . However this assumption is redundant, because all computation we have done in [H91] can be done locally using the local Whittaker model. Also C(P, Q) in the above theorem is slightly different from the one in [H91] Theorem 5.2, because (fP , fP ) = D[κ(P )]+1 (fP◦ , fP◦ ) for fP◦ appearing in the formula of [H91] Theorem 5.2. Proof We start with a slightly more general situation. We shall use the symbol introduced in [H91]. Suppose C(ψ − )|N and C(χ− )|J, and take normalized Hecke eigenforms f ∈ Sκ (N, ψ+ ; C) and g ∈ Sκ (J, χ+ ; C). Suppose

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c f ). Then ψ1 = χ1 . We define f c ∈ S#κ (N, ψ $ + ; C) by a(y, f ) = a(y, −1 0 c −1 c g c )(w). Then f (w) = f (ε wε) for ε = . We put Φ(w) = (f 0 1 we see Φ(wu) = ψ(u)χ−1 (u)Φ(w) for u ∈ U = U0 (N ) ∩ U0 (J ). Since ψ1 =# χ1$, we find that ψ(u)χ(u)−1 = ψ − (χ− )−1 (d) = ψ u (χu )−1 (d) if c u = ac db . We write simply ω for the central character of f g c , which is the −[κ ]−[κ]

u Hecke character ψ+ (χu+ )−1 | · |A

. Then we have Φ(zw) = ω(z)Φ(w),

−1

and Φ(wu∞ ) = J#κ (u$∞ , i) Jκ (u∞ , i)−1 Φ(w). We then define ω ∗ (w) = ω(dN J ) for w = ac db ∈ B(A)U · G(R)+ . Here B is the algebraic subgroup of G made of matrices of the form ( y0 x1 ). We extend ω ∗ : G(A) → C outside B(A)U · G(R)+ just by 0. Similarly we define η : G(A) → C by |y|A if g = ( y0 x1 ) zu with z ∈ FA× and u ∈ U · SO2 (FR ) η(w) = 0 otherwise. For each Q–subalgebra A ⊂ A, we write B(A)+ = B(A)∩G(A(∞ )×G(R)+ . Note that Φ(w)ω ∗ (w)η(w)s−1 for s ∈ C is left invariant under B(Q)+ . Then we compute Z(s, f, g) = Φ(w)ω ∗ (w)η(w)s−1 dϕB (w) B(Q)+ \B(A)+

for the measure have

ϕB ( y0 x1 )

Z(s, f, g) = × FA+

=D dy→y

1 2

FA /F

FA× 1

= D s+ 2

× = |y|−1 A dx ⊗ d y defined in [H91] page 340. We

× Φ ( y0 x1 ) dx|y|s−1 A d y

√ −(κ +κ ) a(dy, f )a(dy, g)eF (2 −1y∞ )y∞ 2 2 |y|sA d× y

FA×

√ −(κ +κ ) a(y, f )a(y, g)eF (2 −1y∞ )y∞ 2 2 |y|sA d× y

=D s+ 2 (4π)−sI+κ2 +κ2 ΓF (sI − κ2 − κ2 )D(s − 1

[κ] + [κ ] − 1, f, g). 2

Define C∞+ ⊂ G(R)+ by the stabilizer in G(R)+ of i ∈ Z. We now choose an invariant measure ϕU on X(U ) = G(Q)+ \G(A)+ /U C∞+ so that φ(γw)dϕU (w) = φ(b)dϕB (b) X(U ) γ∈O × B(Q) \G(Q) + +

B(Q)+ \B(A)+

whenever φ is supported on B(A(∞) )u · G(R)+ and the two integrals are absolutely convergent. There exists a unique invariant measure ϕU as seen

Hecke L–values

257

in [H91] page 342 (where the measure is written as μU ). On B(A)+ , Φ(w) = c f g c (w)Jκ (w, i)Jκ (w, i) and the right-hand-side is left C∞+ invariant (cf. (S2) in 2.10). Then by the definition of ϕU , we have B(Q)+ \B(A)+

Φω ∗ η s−1 dϕB =

c

f (w)g c (w)E(w, s − 1)dϕU (w), X(U )

(5.12) where

E(w, s) =

ω ∗ (γw)η(γw)s Jκ (γw, i)Jκ (γw, i).

γ∈O× B(Q)+ \G(Q)+

× . By definition, Note that E(zw, s) = (ψ+ −u χu+ )(z)E(w, s) for z ∈ O E(αx) = E(x) for α ∈ G(Q)+ ; in particular, it is invariant under α ∈ F × . For z ∈ FR× , one has E(zw, s) = N (z)[κ]+[κ ]−2 E(w, s). It follows that 1−([κ]+[κ])/2 −u u E(w, s) has eigenvalue ψ+ χ+ (z) under the central | det(w)|A × × × action of z ∈ F O FR . The averaged Eisenstein series: E(w, s) =

1−([κ]+[κ ])/2

u −u ψ+ χ+ (a)| det(aw)|A

a∈ClF 1−([κ]+[κ ])/2

= | det(w)|A

E(aw, s)

ψ+ χ+ (a)E(aw, s)

a∈ClF −u u χ+ (z)E(w, s), where a runs over complete represatisfies E(zw, s) = ψ+ × × × × sentative set for FA /F O FR and ψ+ is the central character of f and χ+ is the central character of g c . Defining the P GL2 modular variety X(U ) = X(U )/FA× , by averaging (5.12), we find 1 [κ] + [κ ] Ds+ 2 (4π)−sI+κ2 +κ2 ΓF (sI − κ2 − κ2 )D(s − − 1, f, g) 2 c ([κ]+[κ ])/2−1 = f (w)g c (w)E(w, s − 1)| det(w)|A dϕU (w).

X(U )

(5.13) Writing U = U0 (L) and writing r = κ2 − κ2 , we define an Eisenstein series Ek−k ,r (ω u , 1; s) by [κ ]−[κ] √ [κ] + [κ ] N (L)−1 D| det(w)|A 2 L(L) (2s, ω u )E(w, s + − 1), 2

258

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where k = κ1 − κ2 and k = κ1 − κ2 . The ideal L is given by N ∩ J. Then, ] changing variable s − [κ]+[κ − 1 → s, we can rewrite (5.13) as 2 k + k (L) )L (2s + 2, ψ u χ−u )D(s, f, g) 2 = N (L)−1 D −(3+[κ]+[κ ])/2 (f c , g c Ek−k ,r (ω u , 1; s + 1))L

Ds+ 2 (4π)−sI− 1

= N (L)−s−

k+k 2

[κ]+[κ ] 2

ΓF (sI +

D −(3+[κ]+[κ ])/2 (f c |τ, (g c |τ )Gk−k ,r (ω u , 1; s + 1))L , (5.14) [κ ]−[κ]

where Gk−k ,r (ω u , 1; s) = N (L)s−1+ 2 Ek−k ,r (ω u , 1; s)|τ for τ of level L, and [κ]−1 (φ, ϕ)L = φ(w)ϕ(w)| det(w)|A dϕU (w) X(U )

is the normalized Petersson inner product on Sκ (U, ψ; C). This formula is equivalent to the formula in [H91] (4.9) (although we have more general forms f and g with character ψ and χ not considered in [H91]). In [H91] (4.9), k is written as κ and r is written as w − ω. Let E be the Eisenstein measure of level L = N ∩ J defined in [H91] Section 8, where L is written as L. We take an idele L with LO = L and L(L) = 1. Similarly we take ideles J and N replacing in the above formula L by J and N, respective;y, and L by the corresponding J and N , respectively. The algebra homomorphism ϕ : hn.ord (J, χ; W ) → Λ induces, by the ∗ W –duality, ϕ∗ : Λ → Sn.ord (J, χ; W ), where Sn.ord (J, χ; W ) is a subspace of p–adic modular forms of level (J, χ) (see [H96] 2.6). We then consider the convolution as in [H91] Section 9 (page 382): D =λ∗ϕ=

E ∗λ ([L/J] ◦ ϕ∗ ) , for E ∗λ ([L/J] ◦ ϕ∗ ) ∈ Λ⊗Λ H(λ) ⊗ 1

where [L/J] is the operator defined in [H91] Section 7.B and all the ingredient of the above formula is as in [H91] page 383. An important point here is that ) defined we use the congruence power series H(λ) ∈ Λ (so H(λ) ⊗ 1 ∈ Λ⊗Λ n.ord u with respect to h (N, ψ; W ) instead of h(ψ , ψ1 ) considered in [H91] page 379 (so, H(λ) is actually a factor of H in [H91] page 379, which is an improvement). We write the minimal level of fPord as Npα for pα = p|p pα(p) . Then α(p) . The integer α(p) is given by the exponent we define α = p|p p − of p in C(ψP ) or 1 whichever larger. We now compute D(P, Q). We shall give the argument only when j = [κ(P )] − [κ(Q)] ≥ 1, since the other case can be treated in the same manner as in [H91] Case II (page 387).

Hecke L–values

259

j u α = (·, ·)α and put ψ , 1; 1 − Put G = GjI,0 χ−u P Q 2 . We write (·, ·)Np m = Lpα . As before, m = Lα satisfies mO = m and m(m) = 1. Put r(P, Q) = κ2 (Q) − κ2 (P ), which is non-negative by the weight condition (W) in the theorem. Then in exactly the same manner as in [H91] Section 10 √ (page 386), we find, for c = (2 −1)j[F :Q] π [F :Q] , (−1)k(Q) N (J/L)−1 N (Lpα )[κ(Q)]−1 c((fPord )c |τ (N α ), fPord )α D(P, Q) r(P,Q)

ord = ((fPord )c |τ (m), (gQ |τ (Jα )|τ (m)) · (δjI

G))m . (5.15)

By [H91] Corollary 6.3, we have, for r = r(P, Q), j −u u r −r . δjI G = ΓF (r + I)(−4π) GjI+2r,r χQ ψP , 1; 1 − 2 Then by (5.14), we get c(−1)k(Q) N (Jpα )−1 C(P, Q)−1 (fPord |τ (N α ), fPord )α D(P, Q) = L(m) (2 − [κ(P )] + [κ(Q)], ωP,Q ) × D(

[κ(Q)] − [κ(P )] ord ord , fP , (gQ |τ (J α ))c ), (5.16) 2

where ωP,Q = χ−1 Q+ ψP + for the central characters χQ+ of gQ and ψP + of fP . Now we compute the Petersson inner product (fPord |τ (N α ), fPord )α in terms of (fP , fP ). Note that for f, g ∈ Sκ (U0 (N ), ; C) ([κ]−1)/2

(f u )|τ (N ) = |N |A

(f |τ (N ))u and (f, g)N = D [κ]+1 (f u , g u ). (5.17) The computation we have done in [H91] page 357 in the proof of Lemma 5.3 (vi) is valid without any change for each p|p, since at p–adic places, fP in [H91] has the Neben type we introduced in this paper also for places outside p. The difference is that we compute the inner product in terms of (fP , fP ) not (fP◦ , fP◦ ) as in [H91] Lemma 5.3 (vi), where fP◦ is the primitive form associated u u assuming that ψP,1 lifts to a global finite order character (the to fPu ⊗ ψP,1 −u character ψP,1 is written as ψ in the proof of Lemma 5.3 (vi) of [H91]). Note −1 by definition and hence (fP◦ , fP◦ ) = (fPu , fPu ), because here fP◦ = fP ⊗ ψP,1 tensoring a unitary character to a function does not alter the hermitian inner product. Thus we find ord α ord (fPord,u |τ (N α ), fPord,u ) α ([κ(P )]−1)/2 (fP |τ (N ), fP ) = |N . | A (fP◦ , fP◦ ) (fP , fP ) (5.18) A key point of the proof of Lemma 5.3 (vi) is the formula writing down −u u in terms of fP◦ . Even without assuming the liftability of ψP,1 fPord,u ⊗ ψP,1

260

Haruzo Hida

to a global character, the same formula is valid for fPord and fP before tensor−1 ing ψP,1 (by computation using local Whittaker model). We thus have fPord = fP |R for a product R = p|p Rp of local operators Rp given as follows: If the prime p is a factor of C(ψP− ), then Rp is the identity operator. If p is prime to − ) is spherical), then f |Rp = f − ψP,2 (p )f |[p ], where C(ψP− ) (⇔ π(1, ψP,p −1 f |[p ](x) = |p |p f |g with f |g(x) = f (xg) for g = p 0 . Writing U for 0

1

the level group of fP and U = U ∩ U0 (p), we note f |T (p) = TrU/U (f |g −1 ). This shows [κ(P )]

(fP |[p ], fP )U = |p |p

[κ(P )]

= |p |p

(fP , fP |g −1 )U

(fP , TrU/U (fP |g −1 ))U = (a + b)(fP , fP )U ,

[κ(P )]

[κ(P )]

ψP,1 (p ), b = |p |p where a = |p |p Petersson metric on X(U ). Similarly we have,

ψP,2 (p ), and (·, ·)U is the

(fP , fP |[p ])U = (a + b)(fP , fP )U [κ(P )]

and (fP |[p ], fP |[p ])U = |p |p

(|p |p + 1)(fP , fP )U .

By (5.10) and by (5.18), we conclude from [H91] Lemma 5.3 (vi) ((fPord )c |τ (N α ), fPord )α (fP , fP ) (−1)k(P ) ψPu (dp )W (fP )S(P ) −1 G(ψP,2,p ψP,1,p ) × ψP,2 (α )

(1−[κ(P )])/2

= |N |A

(5.19)

− p|((p)+C(ψP ))

for p running over the prime factors of p. We now compute the extra Euler factors: E(P, Q) and W (P, Q). Again the computation is the same as in [H91] Lemma 5.3 (iii)-(v), because the level structure and the Neben character at p–adic places are the same as in [H91] for fP and gQ and these factors only depend on p–adic places. Then we get the Euler p–factor E(P, Q) and W (P, Q) as in the theorem from [H91] lemma 5.3.

Remark 5.4 We assumed the condition (5.7) to make the proof of the theorem simpler. We now remove this condition. Let E be the set of all prime factors − × q of IN such that χ− q = ψq on Oq . Thus we assume that E = ∅. Then in the proof of Lemma 5.2, the inertia group at q ∈ E fixes a two-dimensional −1 c , one corresponding to ψ1,q ⊗χ subspace of ρfP ⊗ρgQ 1,Q and the other coming −1 from ψ2,q ⊗ χ2,Q . The Euler factor corresponding to the latter does not appear

Hecke L–values

261

in the Rankin product process; so, we get an imprimitive L–function, whose missing Euler factors are −1 (1 − ψ2,q χ−1 . E (P, Q)−1 = 2,Q (q )) q∈E

Thus the final result is identical to Theorem 5.3 if we multiply E(P, Q) by E (P, Q) in the statement of the theorem. In our application, λ and ϕ will be ϕ (Λ–adic) automorphic inductions of Λ–adic characters λ, : Gal(M /M ) → × − and ϕ Λ for an ordinary CM field M/F . If the prime-to–p conductors of λ − are made of primes split in M/F , E is the set of primes ramifying in M/F . ⊗ϕ −1 (F robq )) Then E (P, Q) is the specialization of E = q∈E (1 − λ is not divisible by the prime element of W (that at (P, Q), and E ∈ Λ⊗Λ and ϕ is, the μ–invariant of E vanishes). Actually, we can choose λ so that −1 ϕ λ (F robq ) ≡ 1 mod mΛ , and under this choice, we may assume that E ∈ ×. (Λ⊗Λ)

5.3 Comparison of p–adic L–functions For each character ψ : ΔC → W × , we have the extension ψ : Gal(Q/M ) → Λ sending (γ, δ) ∈ Γ × ΔC to ψ(δ)γ for the group element γ ∈ Γ inside the group algebra Λ. Regarding ψ as a character of Gal(Q/M ), the induced representation IndF M ψ is modular nearly ordinary at p, and hence, by the universality of the nearly p–ordinary Hecke algebra hn.ord (W ) defined in [H96] U → Λ such that 2.5, we have a unique algebra homomorphism λ : hn.ord U Hecke ∼ ψ λ ◦ ρ IndF for the universal nearly ordinary modular Galois rep= M resentation ρHecke with coefficients in h, where U = U11 (N) with N = DM/F c(ψ) for the relative discriminant DM/F of M/F and c = C(ψ) ∩ F for the conductor C(ψ) of ψ. Thus for each arithmetic point P ∈ Spf(Λ)(Qp ) (in the sense of [H96] 2.7), we have a classical Hecke eigenform θ(ψP ) of weight κ(P ). We suppose that the conductor C(ψ − ) of ψ − consists of primes of M split over F . Then the automorphic representation π(ψP ) of weight κ(P ) is everywhere principal at finite places. By [H96]7.1, the Hecke character ψP has infinity type (κ1 (P )σ|F σ + κ2 (P )σ|F σc). ∞(ψP ) = − σ∈Σ

In the automorphic representation π(ψP ) generated by right translation of θ(ψP ), we have a unique normalized Hecke eigenform f (ψP ) of minimal level.

262

Haruzo Hida

The prime-to–p level of the cusp form f (ψP ) is equal to N(ψ) = NM/F (C(ψ − ))DM/F , and it satisfies (L1-3) in 5.1 for = P given by ψP |Gal(M L /ML ) 1,q = an extension of ψP |Gal(M L /ML ) to Gal(M L /Fq ) ψP |Gal(M /M ) L L 2,q = another extension of ψP | to Gal(M L /Fq ) Gal(M L /ML )

if q = LL, otherwise, if q = LL, otherwise, (5.20)

where L and L are distinct primes in M . Write t+ for the restriction of + = 1 2 to ΔF (N ), which is independent of P (because it factors through the torsion part of ClF+ (N p∞ )). Since {f (ψP )}P is again a p–adic analytic family of cusp forms, they are induced by a new algebra homomorphism λψ : h = hn.ord (N, ; W ) → Λ. Since λψ is of minimal level, the congruence module C0 (λψ ; Λ) is a well defined Λ– module of the form Λ/H(ψ)Λ (see [H96] 2.9). Actually we can choose H(ψ) in Λ− = W [[Γ− ]] (see [GME] Theorem 5.44). The element H(ψ) is called the congruence power series of λψ (identifying Λ− with a power series ring over W of [F : Q] variables) By Theorem 5.3 and Remark 5.4, we have the (imprimitive) p–adic Rankin × as in product D = λψ ∗ λϕ with missing Euler factor E ∈ (Λ⊗Λ) Remark 5.4 for two characters ψ : ΔC → W × and ϕ : ΔC → W × . Writing R W Λ, we have D = H(ψ) . R = D · H(ψ) ∈ Λ⊗ We define two p–adic L–functions Lp (ψ −1 ϕ) and Lp (ψ −1 ϕc ) by Lp (ψ −1 ϕ)(P, Q) = E (P, Q)Lp (ψP−1 ϕ Q ) and Q,c ) Lp (ψ −1 ϕc )(P, Q) = Lp (ψP−1 ϕ for the Katz p–adic L–function Lp , where χc (σ) = χ(cσc−1 ). We follow the argument in [H91], [HT1] and [H96] to show the following identity of p–adic L–functions, which is a more precise version of [HT1] Theorem 8.1 without the redundant factor written as Δ(M/F ; C) there: Theorem 5.5 Let ψ and ϕ be two characters of ΔC with values in W × . Suppose the following three conditions: (i) C(ψ − )C(ϕ− ) is prime to any inert or ramified prime of M ; (ii) At each inert or ramified prime factor q of C(ψ)C(ϕ), ψq = ϕq on Rq× ;

Hecke L–values

263

(iii) For each split prime q|C(ψ)C(ϕ), we have a choice of one prime factor × . Q|q so that ψQ = ϕQ on RQ Then we have, for a power series L ∈ Λ⊗Λ, Lp (ψ −1 ϕ)Lp (ψ −1 ϕc ) L = . − H(ψ) (h(M )/h(F ))L− p (ψ ) The power series L is equal to (λψ ∗ λϕ ) · H(ψ) up to units in Λ⊗Λ. The improvement over [HT1] Theorem 8.1 is that our identity is exact without missing Euler factors, but we need to have the additional assumptions (i)–(iii) to ensure the matching condition (5.6) for the automorphic induction of ψP and ϕ Q . The proof of the theorem is identical to the one given in [HT1] Section 10, since the factors C(P, Q), W (P, Q), S(P ) and E(P, Q) appearing in Theorem 5.3 are identical to those appearing in [H91] Theorem 5.2 except for the power of the discriminant D, which is compensated by the difference of (fP◦ , fP◦ ) appearing in [H91] Theorem 5.2 from (fP , fP ) in Theorem 5.3. Since we do not lose any Euler factors in (5.5) (thanks to our minimal level structure and the assumption (5.6) assuring principality everywhere), we are able to remove the missing Euler factor denoted Δ(1) in [HT1] (0.6b). If M/F is ramified at some finite place, by Theorem 4.3 and Remark 5.4, we can always choose a pair (l, ϕ) of a prime ideal l and a character ϕ of l–power conductor so that (i) l is a split prime of F of degree 1; (ii) Lp (ψ −1 ϕ)Lp (ψ −1 ϕc ) is a unit in Λ⊗Λ. Even if M/F is unramified, we shall show that such choice is possible: Writing Ξ for the set of primes q as specified in the condition (3) in the theorem. Then we choose a prime L split in M/Q so that LL is outside Ξ. We then take a finite order character ϕ : MA× /M × → W × of conductor Lm so that ϕL is trivialbut the reduction modulo mW of ψ −1 ϕϕ N and ψ −1 ϕc ϕc N is not equal to M/F . Then again by Theorem 4.3, we find (infinitely many) ϕ with · unit power series Lp (ψ −1 ϕϕ )Lp (ψ −1 (ϕϕ )c ). This implies the following corollary:

Corollary 5.6 Suppose that C(ψ − ) is prime to any " inert or ramified prime of − " H(ψ). (ψ ) M . Then in Λ− , we have (h(M )/h(F ))L− p As a byproduct of the proof of Theorem 5.5, we can express the p–adic L–value Lp (ψP− ) (up to units in W and the period in C× ) by the Petersson

264

Haruzo Hida

metric of the normalized Hecke eigenform fP = f (ψP ) of minimal level (in the automorphic induction πP of ψP ). We shall describe this fact. We are going to express the value L(1, Ad(fP )) in terms of the L–values of the Hecke character ψP . The representation πP is everywhere principal outside archimedean places if and only if ψ − has split conductor. Then M/F L(s, Ad(fP )) = L(1, )L(0, (ψP− )∗ ). · The infinity type of ψP− is (κ1 (P ) − κ2 (P )) + (κ2 (P ) − κ1 (P ))c = (k(P ) − I)(1−c), where we identify Z[Σ] with Z[I] sending σ to σ|F . Thus the infinity type of (ψP− )∗ is given by k(P ) − k(P )c + 2Σc = 2Σ + (k(P ) − 2I)(1 − c). Thus we have Lp ((ψ− )∗ ) = N (c)ψ− (c)W (ψ− )−1 Lp (ψ− ) P

P

P

= CWp (ψP− )E(P )

P

πk(P )−I ΓΣ (k(P ) − I)L(0, ψP− ) 2(k(P )−I)

= C Wp ((ψ− )∗P )E(P )

Ω∞ k(P )−2I π ΓΣ (k(P ))L(0, (ψ − )∗P ) 2(k(P )−I)

,

Ω∞

(5.21) where C and C are constants in W × and E(P ) = (1 − ψP− (Pc ))(1 − (ψP− )∗ (Pc )) . P∈Σp − Since we have, for the conductor Pe(P) of ψP,P ,

(ψP− )∗ (P

−e(P)

) = N (P)e(P) ψP− (P

−e(P)

),

− by definition, W ((ψP− )∗ ) is the product over P ∈ Σ of G(2δP , ψP,P ), and hence we have Wp ((ψ− )∗ ) = N (Pe(Σ) )Wp (ψ− ), P

P

is the Σp –part of the conductor of ψP− . Thus we have, for where P h(M/F ) = h(M )/h(F ), e(Σ)

π Lp ((ψP− )∗ ) = C Wp (ψP− )N (Pe(Σ) )E(P ) = C Wp (ψP− )E(P )

k(P )−2I

ΓΣ (k(P ))L(0, (ψ− )∗P ) 2(k(P )−I)

Ω∞ 2(κ1 (P )−κ2 (P )) (f (ψP ), f (ψP )) π 2(κ1 (P )−κ2 (P ))

,

h(M/F )Ω∞

(5.22) where C and C are constants in W × .

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265

We suppose that the conductor C(ψP ) is prime to p. Then Wp (ψP− ) = 1 if p is unramified in F/Q (and even if p ramifies in F/Q, it is a unit in W ). Let h = h(M ), and choose a global generator of Ph . Thus Σ ≡ 0 mod mW and Σc ≡ 0 mod mW . By the arithmeticity of the point P , we have k = k(P ) ≥ 2I. Then we have, up to p–adic unit, ψP− (Pc )h = ψP− () = (k−I)−(k−I)c (ψP− )∗ (Pc )h = (ψP− )∗ () = (k−2I)−kc . Thus ψP− (Pc ) ∈ mW if k ≥ 2I and (ψP− )∗ (Pc ) ∈ mW if k ≥ 3I. For each σ ∈ Σ, we write Pσ for the place in Σp induced by ip ◦ σ. Thus we obtain Proposition 5.7 Suppose that either k(P )σ = κ1 (P )σ − κ2 (P )σ + 1 ≥ 3 or (ψP− )∗ (Pcσ ) ≡ 1 mod mW for all σ ∈ I and that ψP has split conductor. Write h(M/F ) for h(M )/h(F ). Then up to units in W , we have h(M/F )Lp ((ψP− )∗ ) = h(M/F )Lp (ψP− ) =

π 2(k(P )−I) Wp (ψP− )(f (ψP ), f (ψP ))N 2(k(P )−I)

,

Ω∞

where f (ψP ) is the normalized Hecke eigenform of minimal level N (necessarily prime to p) of the automorphic induction of ψP .

5.4 A case of the anticyclotomic main conjecture Here we describe briefly an example of a case where the divisibility: − − − L− p (ψ)|F (ψ) implies the equality Lp (ψ) = F (ψ) (up to units), relying on the proof by Rubin of the one variable main conjecture over an imaginary quadratic field in [R] and [R1]. Thus we need to suppose √ (ab) F/Q is abelian, p [F : Q], and M = F [ D] with 0 > D ∈ Z, in order to reduce √ our case to the imaginary quadratic case treated by Rubin. Write E = Q[ D] and suppose that D is the discriminant of E/Q. We have p D since p is supposed to be unramified in M/Q. We suppose also (sp) The prime ideal (p) splits into pp in E/Q. √ √ Then we take Σ = {σ : M → Q|σ( D) = D}. By (sp), Σ is an ordinary CM type. We fix a conductor ideal c (prime to p) of E satisfying (opl) and (spt) for E/Q (in place of M/F ). We then put C = cR. We consider ZE = ZE (c) = limn ClE (cpn ). We split ZE (c) = Δc × Γc for a finite group Δ = Δc and ←−

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a torsion-free subgroup Γc . As before, we identify Γc and Γ1 for E and write + 0 it as ΓE . We then write ΓE,− = ΓE /Γ+ E for ΓE = H (Gal(E/Q), ΓE ). − We consider the anticyclotomic Zp –extension E∞ of E on which we have − cσc−1 = σ−1 for all σ ∈ Gal(E∞ /E) for complex conjugation c. Writ∞ ∞ ing E(cp ) (inside M (Cp )) for the ray class field over E modulo cp∞ , we identify ZE (c) with Gal(E(cp∞ )/E) via the Artin reciprocity law. Then − − ) = Γ+ Gal(E(cp∞ )F/E∞ E × Δc and Gal(E∞ /E) = ΓE,− . We then define EΔ by the fixed field of Γc in the composite F · E(cp∞ ); so, Gal(EΔ /E) = Δc and EΔ ⊃ F . We have MΔ ⊃ EΔ . Thus we have the restriction maps ResZ : Z(C) = Gal(M (Cp∞ )/M ) → Gal(E(cp∞ )/M ) = ZE (c), − − ResΓ : Γ− = Gal(M∞ /M ) → Gal(E∞ /E) = ΓE,− and ResΔ : ΔC = Gal(MΔ /M ) → Gal(EΔ /E) = Δc . We suppose (res) There exists an anticyclotomic character ψE of Δc such that ψ = ψE ◦ ResΔ . − Let LE ∞ /E∞ EΔ be the maximal p–abelian extension unramified outside p. E − Each γ ∈ Gal(LE ∞ /E) acts on the normal subgroup XE = Gal(L∞ /E∞ EΔ ) continuously by conjugation, and by the commutativity of XE , this action − factors through Gal(EΔ E∞ /E). Then we look into the Λ− E –module: XE [ψE χ] = XE ⊗Δc ,ψχ W for a character χ of Gal(F/Q), where Λ− E = W [[ΓE,− ]]. The projection ResΓ induces a W -algebra homomorphism Λ− → Λ− E whose kernel we write as a.

Theorem 5.8 Let the notation be as above. Suppose that ψ has order prime to − − p in addition to (ab), (sp) and (res). Then L− p (ψ) = F (ψ) up to units in Λ . Proof We shall use functoriality of the Fitting ideal FA (H) of an A-module H with finite presentation over a commutative ring A with identity (see [MW] Appendix for the definition and the functoriality Fitting ideals listed below): (i) If I ⊂ A is an ideal, we have FA/I (H/IH) = FA (H) ⊗A A/I; (ii) If A is a noetherian normal integral domain and H is a torsion Amodule of finite type, the characteristic ideal charA (H) is the reflexive closure of FA (H). In particular, we have charA (H) ⊃ FA (H). By definition, we have H0 (Ker(ResΓ ), X[ψ]) = X[ψ]/aX[ψ] ∼ =

XE [ψE χ]

χ

for χ running all characters of Gal(F/Q). By (i) above, we have FΛ− (XE [ψχ]) = FΛ− (X[ψ]) ⊗Λ− Λ− E. E

χ

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Since charΛ− (X[ψ]) = F − (ψ), by Theorem 5.1, charΛ− (X[ψ]) ⊂ L− p (ψ). Thus by (ii), we obtain − charΛ− (XE [ψχ]) ⊂ L− p (ψ)ΛE , E

χ

− − − − where L− p (ψ)ΛE is the ideal of ΛE generated by the image of Lp (ψ) ∈ Λ − in ΛE . Write RE for the integer ring of E, and let X(RE )/W be the elliptic curve with complex multiplication whose complex points give the torus C/RE . Since X(R) = X(RE ) ⊗RE R for our choice of CM type Σ, the complex and p-adic periods of X(R) and X(RE ) are identical. Thus by the factorization of Hecke L-functions, we have − − L− L− p (ψ)ΛE = p (ψE χ)ΛE . χ − /E, we Then by Rubin [R] Theorem 4.1 (i) applied to the Zp -extension E∞ find that − charΛ− (XE [ψE χ]) = L− p (ψE χ)ΛE . E

Thus (F

−

− (ψ)/L− p (ψ))ΛE

− − − − = Λ− E , and hence F (ψ)Λ = Lp (ψ)Λ .

Bibliography Books [AAF] G. Shimura, Arithmeticity in the Theory of Automorphic Forms, Mathematical Surveys and Monographs 82, AMS, 2000 [ACM] G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton University Press, 1998 [CRT] H. Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8, Cambridge Univ. Press, 1986 [GME] H. Hida, Geometric Modular Forms and Elliptic Curves, 2000, World Scientific Publishing Co., Singapore (a list of errata downloadable at www.math. ucla.edu/˜hida) [HMI] H. Hida, Hilbert Modular Forms and Iwasawa Theory, Oxford University Press, 2006 (a list of errata downloadable at www.math.ucla. edu/˜hida) [IAT] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press and Iwanami Shoten, 1971, Princeton-Tokyo [LAP] H. Yoshida, Lectures on Absolute CM period, AMS mathematical surveys and monographs 106, 2003, AMS [LFE] H. Hida, Elementary Theory of L–functions and Eisenstein Series, LMSST 26, Cambridge University Press, Cambridge, 1993

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[MFG] H. Hida, Modular Forms and Galois Cohomology, Cambridge studies in advanced mathematics 69, Cambridge University Press, Cambridge, 2000 (a list of errata downloadable at www.math.ucla.edu/˜hida) [PAF] H. Hida, p–Adic Automorphic Forms on Shimura Varieties, Springer Monographs in Mathematics. Springer, New York, 2004 (a list of errata downloadable at www.math.ucla.edu/˜hida) Articles [B] [BR] [C] [De] [DeR] [Di]

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D. Blasius, Hilbert modular forms and Ramanujan conjecture, Aspects of Mathematics E37 (2006), 35–56 D. Blasius and J. D. Rogawski, Motives for Hilbert modular forms, Inventiones Math. 114 (1993), 55–87 C.-L. Chai, Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces, Ann. of Math. 131 (1990), 541–554 P. Deligne, Travaux de Shimura, Sem. Bourbaki, Exp. 389, Lecture notes in Math. 244 (1971), 123–165 P. Deligne and K. A. Ribet, Values of abelian L–functions at negative integers over totally real fields, Inventiones Math. 59 (1980), 227–286 M. Dimitrov, Compactifications arithm´etiques des vari´et´es de Hilbert et formes modulaires de Hilbert pour Γ1 (c, n), in Geometric aspects of Dwork theory, Vol. I, II, 527–554, Walter de Gruyter GmbH & Co. KG, Berlin, 2004 M. Dimitrov and J. Tilouine, Vari´et´e de Hilbert et arithm´etique des formes modulaires de Hilbert pour Γ1 (c, N ), in Geometric Aspects of Dwork’s Theory, Vol. I, II, 555–614, Walter de Gruyter GmbH & Co. KG, Berlin, 2004 Walter de Gruyter, Berlin, 2004. K. Fujiwara, Deformation rings and Hecke algebras in totally real case, preprint, 1999 (ArXiv. math. NT/0602606) H. Hida, On p-adic Hecke algebras for GL2 over totally real fields, Ann. of Math. 128 (1988), 295–384 H. Hida, Nearly ordinary Hecke algebras and Galois representations of several variables, Proc. JAMI Inaugural Conference, Supplement to Amer. J. Math. (1989), 115–134 H. Hida, On p–adic L–functions of GL(2) × GL(2) over totally real fields, Ann. Inst. Fourier 41 (1991), 311–391 H. Hida, Control theorems of p–nearly ordinary cohomology groups for SL(n), Bull. Soc. math. France, 123 (1995), 425–475 H. Hida, On the search of genuine p–adic modular L–functions for GL(n), M´emoire SMF 67, 1996 H. Hida, Non-critical values of adjoint L-functions for SL(2), Proc. Symp. Pure Math., 1998 H. Hida, Adjoint Selmer group as Iwasawa modules, Israel J. Math. 120 (2000), 361–427 H. Hida, Control theorems of coherent sheaves on Shimura varieties of PEL– type, Journal of the Inst. of Math. Jussieu, 2002 1, 1–76 H. Hida, p–Adic automorphic forms on reductive groups, Ast´erisque 298 (2005), 147–254 (preprint downloadable at www.math.ucla. edu/˜hida)

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[H04b] H. Hida, The Iwasawa μ–invariant of p–adic Hecke L–functions, preprint, 2004 (preprint downloadable at www.math.ucla.edu/˜hida) [H04c] H. Hida, Non-vanishing modulo p of Hecke L–values, in: “Geometric Aspects of Dwork’s Theory, II” (edited by Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, and Francois Loeser), Walter de Gruyter, 2004, pp. 735–784 (preprint downloadable at www.math.ucla. edu/˜hida) [H04d] H. Hida, Anticyclotomic main conjectures, Documenta Math. Extra volume Coates (2006), 465–532 [HT1] H. Hida and J. Tilouine, Anticyclotomic Katz p–adic L–functions and congruence modules, Ann. Sci. Ec. Norm. Sup. 4-th series 26 (1993), 189–259 [HT2] H. Hida and J. Tilouine, On the anticyclotomic main conjecture for CM fields, Inventiones Math. 117 (1994), 89–147 [K] N. M. Katz, p-adic L-functions for CM fields, Inventiones Math. 49 (1978), 199–297 [Ko] R. Kottwitz, Points on Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373–444 [MW] B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Inventiones Math. 76 (1984), 179–330 [R] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Inventiones Math. 103 (1991), 25–68 [R1] K. Rubin, More“main conjectures” for imaginary quadratic fields. Elliptic curves and related topics, 23–28, CRM Proc. Lecture Notes, 4, Amer. Math. Soc., Providence, RI, 1994. [ST] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968), 452–517 [Sh] G. Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. 91 (1970), 144-222; II, 92 (1970), 528-549 [Sh1] G. Shimura, On some arithmetic properties of modular forms of one and several variables, Ann. of Math. 102 (1975), 491–515 [Si] W. Sinnott, On a theorem of L. Washington, Ast´erisque 147-148 (1987), 209–224 [TW] R. Taylor and A. Wiles, Ring theoretic properties of certain Hecke modules, Ann. of Math. 141 (1995), 553–572 [Wa] L. Washington, The non-p–part of the class number in a cyclotomic Zp – extension, Inventiones Math. 49 (1978), 87–97 [Wi] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141 (1995), 443–551

Serre’s modularity conjecture: a survey of the level one case Chandrashekhar Khare Department of Mathematics 155 South 1400 East, Room 233 Salt Lake City, UT 84112-0090 U.S.A. [email protected]

Abstract We give a survey of recent work on Serre’s conjecture. In particular we report on joint work with Wintenberger [32] which proved the conjecture for finitely many weights and levels and set up a strategy to prove Serre’s conjecture, and the author’s subsequent proof in [33] of the level 1 case of the conjecture using “weight cycles”. The article gives an informal survey of the ideas of the proof.

1 Introduction We recall the statement of Serre’s conjecture, state the main result of [33], and give a synopsis of earlier work on the conjecture.

1.1 Statement of Serre’s conjecture and the result ¯ Let ρ¯ : Gal(Q/Q) → GL2 (F) be a continuous, absolutely irreducible, two-dimensional, odd (detρ(c) = −1 for c a complex conjugation), mod p representation, with F a finite field of characteristic p. We say that such a representation is of Serre-type, or S-type, for short. We denote by N (ρ) the (prime to p) Artin conductor of ρ, and k(ρ) the weight of ρ as defined in [55]. The invariant N (ρ) is made out of (ρ|I )=p , and is divisible exactly by the primes ramified in ρ that are = p, while k(ρ) is such that 2 ≤ k(ρ) ≤ p2 − 1 if p = 2 (2 ≤ k(ρ) ≤ 4 if p = 2), and is made from information of ρ|Ip . It is an important feature of the weight k(ρ), for p > 2, that if χp is the mod p cyclotomic character, then for some i ∈ Z, 2 ≤ k(ρ ⊗ χp i ) ≤ p + 1. Serre has conjectured in [55] that such a ρ¯ arises (with respect to some fixed embedding ι : Q → Qp ) from a newform f of weight k(ρ) and level 270

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N (ρ). We fix embeddings ι : Q → Qp for all primes p hereafter, and when we say (a place above) p, we will mean the place induced by this embedding. By arises from we mean that the reduction of an integral model of the p-adic representation ρf associated to f , which is valued in GL2 (O) for O the ring of integers of some finite extension of Qp , modulo the maximal ideal of O is isomorphic to ρ: ρf

GQ −−−−→ GL2 (O) ⏐ = ⏐ = : = ρ

GQ −−−−→ GL2 (F). We have contented ourselves here with a schematic statement of the conjecture, referring to Serre’s original article [55] for a beautiful, more extensive, account of his conjecture and its many consequences. It is convenient to split the conjecture into 2 parts: 1. Qualitative form: In this form, one only asks that ρ arise from a newform of unspecified level and weight. 2. Refined form: In this form, one asks that ρ arise from a newform f ∈ Sk(ρ) (Γ1 (N (ρ))). A large body of difficult, important work of a large number of people, Ribet, Mazur, Carayol, Gross, Coleman-Voloch, Edixhoven, Diamond et al., see [45] and [23], proves that (for p > 2) the qualitative form implies the refined form. We focus only on the qualitative form of the conjecture. In fact the conjecture, especially in its qualitative form, is much older and dates from the early 1970’s. The qualitative form of the level 1 (i.e., N (ρ) = 1) conjecture/question was officially formulated in an article that Serre wrote for the Journ´ees Arithm´etiques de Bordeaux in 1975: see article 104 of [54]. Perhaps the restriction to level 1 is merely for simplicity. The fact that it is only in a qualitative form, in the weight aspect, is for a more substantial reason. The definition of the weight k(ρ) in [55] is very delicate and probably came later, motivated by resuts of Deligne and Fontaine that describe ρ|Ip when ρ arises from Sk (Γ1 (N )) with 2 ≤ k ≤ p + 1 and (N, p) = 1, and Tate’s analysis of Θ-cycles. The level 1 conjecture was proved in [33]. Theorem 1.1 A ρ of S-type with N (ρ) = 1 arises from Sk(ρ) (SL2 (Z)). This built on the ideas introduced in [32]. In this paper we survey some of the ideas which are used in the proof of Theorem 1.1.

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One of the main techniques of [32] and [33] is lifting theorems. These may be motivated as follows. The conjecture in its qualitative form predicts that given an S-type ρ (which we assume is such that 2 ≤ k(ρ) ≤ p + 1), there is a lifting ρ ρ

GQ −−−−→ GL2 (O) ⏐ = ⏐ = : = ρ

GQ −−−−→ GL2 (F) which is finitely ramified and potentially semistable at p. This was proved by Ramakrishna, under some mild conditions, in [43] (see also [63] for some improvements): he also proved that the lift ρ can be chosen so that it is crystalline at p of weight k(ρ) (i.e., of Hodge-Tate weights (k(ρ) − 1, 0)), but his method cannot control precisely the set of primes at which ρ can be ramified. The conjecture in its refined form predicts that given a ρ there is a lifting ρ thas has the same conductor outside p as ρ, and is crystalline at p of weight k(ρ). Proving the existence of such a minimal lifting in many cases is one of the key steps of [32]. This uses crucially a result of B¨ockle in [2], and a result of Taylor that we state below as Theorem 1.2. The two methods for producing liftings are quite different. This is reflected in the kinds of lifts produced. Ramakrishna’s lifts will in general be ramified at more primes than ρ while having the best possible field of definition, i.e., the fraction field of W (F) the Witt vectors of F. The lifts produced in [32] are minimally ramified, but there is little control of the field of definition. This lack of control is not of relevance for the methods below.

1.3 Taylor’s potential version of Serre’s conjecture and some earlier proven cases of the conjecture Taylor had devised (see [63], and [61], [65]) a strategy to use the Galoistheoretic lifting theorem of Ramakrishna, together with modularity lifting results pioneered by Wiles, to prove Serre’s conjecture modulo some open question in Diophantine geometry (see Question 5.5 of [64] and also [62]). The strategy of Taylor in part relied on an observation of the author (see [35]) about using a Galois-theoretic lifting result to prove descent results for mod p Hilbert modular forms.

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In [61] and [65], Taylor proved, using modularity lifting results (over totally real fields), and a result in Diophantine geometry (over “large” fields) of Moret-Bailly, Rumley and Pop, the following potential version of Serre’s conjecture: Theorem 1.2 Assume ρ is of S-type in odd residue characteristic, and im(ρ) is not solvable. Then there is a totally real field F that is Galois over Q and unramified at p, and even split above p if ρ|Dp is irreducible, such that ρ|GF arises from a cuspidal automorphic representation π of GL2 (AF ) that is unramified at all finite places, and is discrete series of weight k(ρ) at the infinite places. Taylor’s question in essence was if F in the theorem could be chosen so that F/Q was a solvable extension. In some cases Taylor in [63], and following Taylor’s method, Manoharmayum and Ellenberg, see [40] and [27], were able to control F and thus prove some (non-solvable) cases of Serre’s conjecture when the image was contained in GL2 (F5 ), GL2 (F7 ) or GL2 (F9 ). The case when the image is contained in GL2 (F4 ) had been addressed by Shepherd-Baron and Taylor, by another method, in [56]. In [61] and [65], Taylor used his potential modularity result Theorem 1.2 to prove some cases of the Fontaine-Mazur conjecture, see [28]. For this control of F of Theorem 1.2, beyond the local properties say at infinity and p, was not needed. For instance, the conjectures in [28] predict that geometric p-adic representations ρ : GQ → GL2 (O) (i.e., finitely ramified and potentially semistable at p) can be propagated into compatible systems. Taylor’s results allow this to be verified in many cases, and this is a crucial input into a result of Dieulefait recalled below, see [24], and also into the method of [32] and [33] (see Section 2 below).

1.4 Measures of the complexity of ρ The difficulty in proving the modularity of a ρ¯ can be either measured in terms of the ramification properties of ρ, for example N (ρ), k(ρ), or in terms of im(ρ). The results proven by Tate and Serre (in [60] and page 710 of [54]) early on were for the cases of residue characteristic p = 2, 3 and N (ρ) = 1, i.e., cases which were more tractable in terms of the first measure. The results of Langlands, and Tunnell, implied Serre’s conjecture for p > 2 when im(ρ) was solvable, i.e., cases which were more tractable in terms of the second measure. The results in [63], [40] and [27] also proved cases which were more

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tractable as per the second measure. In [32] the “complexity” of ρ is again, like in the results of Tate and Serre, measured in terms of the ramification properties of ρ.

1.5 The function field case Lifting theorems were used by B¨ockle and the author, [3], when proving in many cases an (n-dimensional) analog of Serre’s conjecture for function fields. This has also been proven by D. Gaitsgory in [31] using different methods. The idea of [3] was to lift the given residual representation to a characteristic 0 representation using the techniques of Ramakrishna, which sufficed because of Lafforgue’s result, [39]. For 2-dimensional representations this was established by A. J. de Jong in [19], and his use of base change to prove properties of deformation rings influenced the lifting theorems proved in [4] and [32].

1.6 Modularity lifting theorems The principal technical tool of the work of Taylor described above, and the papers [32] and [33] are the methods developed by Wiles, and Taylor and Wiles, see [67] and [66], to prove a relative version of the Fontaine-Mazur conjecture, i.e., modularity lifting theorems. Given a ρ : GQ → GL2 (O) that is finitely ramified, and potentially semistable at p (i.e., geometric) with Hodge-Tate weights (∗, 0) with the integer ∗ > 0, then a particular case of Fontaine-Mazur conjecture predicts that ρ arises from a modular form. Wiles, and Taylor, developed techniques in [67], [66] to show a relative version of this wherein one assumes that residually the representation arises from a newform, and then lifts this modularity property to ρ. Most subsequent work on Serre’s conjecture has used this technique, that assumes mod p modularity results to prove modularity of certain p-adic lifts, to prove mod p modularity in certain instances. This almost gives an appearance of circularity to the method. What saves it from this charge is that p should be treated as a variable in the statement. The kind of lifting statement invoked is that, under appropriate conditions, if one residual representation arising from a compatible system of λ-adic representations of GQ is modular, then so is the corresponding -adic representation. Hence so is the compatible system, and hence so are all the residual representations that arise from the system.

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1.7 Linked compatible systems and congruences between Galois representations The use of Theorem 1.2 in both [32] and [33] is indirect: it is used only to draw some Galois theoretic consequences like existence of lifts of various kinds. Then the various lifts of ρ can be inserted into compatible systems using Taylor’s work. These are purely Galois theoretic consequences which pure algebraic number theory seems unable to prove. The following definition is crucial for us: Definition 1.3 We say that two E-rational, (weakly) compatible systems of representations of GQ are linked if for some finite place λ of a number field E the semisimplifications of the corresponding residual mod λ representations arising from the two systems are isomorphic up to a twist by a (one-dimensional) character of GQ . (The twist will in practise always be by a power of the mod λ cyclotomic character.) The magic the definition works below is partly because being linked is not a transitive property. The role that two linked compatible systems played in Wiles proof of Fermat’s Last Theorem (the 3 - 5 switch) is well-known. The idea of using a sequence of linked compatible systems, by using the liftings of Ramakrishna and the result of Taylor that put these into compatible systems, had always been a tempting one, and had surely been considered by many people. But it was not clear perhaps how to profit by proliferating compatible systems. In [33], in the course of proving the level one case, it is shown how Serre’s conjecture in a given residue characteristic p is influenced by the conjecture in other residue characteristics. This influence is mediated through linked compatible systems and distribution properties of primes of the type proved by Chebyshev in the 1850’s (we will in fact use better estimates proved in [48], although this is most likely not essential) which ensure their proximity. An important starting point for the proof of Theorem 1.1 is provided by the results proven by Fontaine [29], and subsequently by Brumer, Kramer and Schoof [11], [52] that classify abelian varieties over Q with some prescribed reduction properties. The lifting results of [32] and [33] develop to a certain extent the theory of congruences between Galois representations to parallel the corresponding results known for modular forms in the work of Ribet, see [45] and [46], and subsequent work in [12], [21] and [34]. The idea of recreating in the theory of deformations of Galois representations, the theory of congruences between modular forms, already occurs in work of Boston, see [5].

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We draw the attention of the reader to Ribet’s notes available at http://math.berkeley.edu/ribet/cms.pdf which give a nice survey of some of these ideas.

1.8 Notation For F a field, Q ⊂ F ⊂ Q, we write GF for the Galois group of Q/F . For λ a prime/place of F , we mean by Dλ (resp., Iλ ) a decomposition (resp., inertia) subgroup of GF at λ. For each place p of Q, we fix embeddings ιp of Q in its completions Qp . Denote by χp the p-adic cyclotomic character, and ωp the Teichm¨uller lift of the mod p cyclotomic character χp (the latter being the reduction mod p of χp ). By abuse of notation we also denote by ωp the -adic character ι ι−1 p (ωp ) for any prime : this should not cause confusion as from the context it will be clear where the character is valued. For a number field F we denote the restriction of a character of GQ to GF by the same symbol. Mod p and p-adic Galois representations arising from newforms are said to be modular, another standard bit of terminology.

2 Compatible systems and a result of Dieulefait We state a result of Dieulefait in [24] (this was also observed later, independently, by Wintenberger in [68]), which was a precursor to the work in [32]. Theorem 2.1 Let ρ : GQ → GL2 (O) be a representation that arises from a p-divisible group over Z, i.e., ρ is unramified outside p and Barsotti-Tate at p. Then ρ is reducible. The main ingredient of the proof, both in [24] and [68], is the use of compatible systems. Let E and F be number fields. We recall that a E-rational, 2-dimensional, strictly compatible system of representations (ρλ ) of GF consists of the data: (i) for each finite place λ of E, ρλ : GF → GL2 (Eλ ) is a continuous, semisimple representation of GF , (ii) for all finite places q of F , if λ ia a place of E of residue characteristic different from q, the Frobenius semisimplification of the Weil-Deligne parameter of ρλ |Dq is independent of λ, and for almost all primes q this parameter is unramified. The results of Taylor in [61] and [65] imply that ρ is part of an E-rational, strictly compatible system (ρλ ) of 2-dimensional representations of GQ , with

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E a number field, so that for each place λ above an odd prime the representation ρλ is unramified outside . Dieulefait checks that for each λ above an odd prime the representation ρλ is Barsotti-Tate at . We recall the arguments of Taylor (see proof of Theorem 6.6 of [65] and 5.3.3 of [62]) and Dieulefait. Strictly compatible systems: We may assume that p ≥ 5 as otherwise the result is proved by Fontaine in [29]. We may assume that im(ρ) is not solvable, as otherwise we are done by Theorem 6.2 below, in conjunction with the consequence of the LanglandsTunnell theorem spelled out in Theorem 4 of [42], for instance. By Theorem 1.2, and the modularity lifting theorems of Skinner-Wiles in [59], there is a totally real field F , Galois over Q and that is unramified at p, such that ρ|GF arises from a holomorphic, cuspidal automorphic representation π of GL2 (AF ) with respect to the embedding ιp , and with AF the adeles of F . The strictly compatible system corresponding to π is such that each member is irreducible. The irreducibility is a standard consequence of the fact that ρ|GF is irreducible (as im(ρ) is not solvable: see Lemma 2.6 of [32] for instance) and Hodge-Tate. Let G := Gal(F/Q). Using Brauer’s Theorem we get subextensions Fi of F such that Gi = Gal(F/Fi ) is solvable, characters χi of Gi (that we may also regard as characters of GFi ) with values in Q (that we embed in Qp using ιp ), and integers ni such that 1G = Gi ni IndG Gi χi . Using results on base change in [38] and [1] we also get holomorphic cuspidal automorphic representations πi of GL2 (AFi ) such that if ρπi ,ιp is the representation of GFi corresponding GQ to πi w.r.t. ιp , then ρπi ,ιp = ρ|GFi . Thus ρ = Gi ni IndGF χi ⊗ ρπi ,ιp . i

Now for any prime and any embedding ι : Q → Q , we define the virtual G ¯ representation ρι = Gi ni IndGQF χi ⊗ ρπi ,ι of Gal(Q/Q) with the χi ’s now i regarded as -adic characters via the embedding ι. We check that ρι is a true representation by computing its inner product in the Grothendieck group of Q -valued (continuous, linear) representations of GQ . We claim that this is G independent of ι. This is because, as the IndGQF χi ⊗ ρπi ,ι are semisimple, i

the value of the inner product is the dimension of End(ρι ) as a Q -vector space. Using Mackey’s formula, and the fact that the compatible system of representations of GF arising from π is irreducible, we see that this dimension is independent of ι. As for ι = ιp this dimension is 1 we see that (±1)ρι is an irreducible representation, and then using the trace of the identity we see that ρι is a true representation of dimension 2. The representations ρι together constitute the strictly compatible system we seek (see proof of Theorem 6.6 of [65]). Note that ρλ for λ above = p is not ramified at p. This is easily

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deduced using that F as above is unramified at p. (The details of the proof were explained to us by Wintenberger.) Dieulefait’s refinement: Now we recall the argument in [24] to check that for each λ above an odd prime the representation ρλ is crystalline at . Consider a decomposition group D in G of an odd prime . The corresponding fixed field F has a prime λ above that is split. By [1], ρ|F arises from a cuspial automorphic representation π of GL2 (AF ), and then as ρ|F is unramified at λ we see that π is unramified at λ (because of results of Carayol and Taylor in the “ = p case”). This gives that ρλ |GF is Barsotti-Tate at λ by a result of Breuil in [7] (in the “ = p case”) which suffices as the prime λ is split. After this, to prove Theorem 2.1 we proceed as in [68]. Looking at ρλ for λ above 7, Fontaine’s work in [29] implies that ρλ is reducible and hence that ρ is reducible. (In [24] modularity lifting results together with results of Tate and Serre at p = 3 are used instead.)

3 The minimal lifting result of [32] and a sketch of its proof We refer to [20], see also Section 2 of [32], for the definition of minimal lifting ρ of ρ. Away from p, this is just a little more than requiring an equality of conductors N (ρ) = N (ρ). At p the condition is that ρ be crystalline of weight k(ρ) (which in the case k(ρ) = p + 1 is taken to be an ordinarity condition), or ρ could also be semistable of weight 2 in the k(ρ) = p+1 case. (The definition of minimal lifts makes sense even for representations valued in (abstract) local W (F)-algebras with residue field F.) In [32] a new lifting technique to prove the existence of minimal lifts was introduced which used as an essential ingredient Taylor’s result that proved a potential version of Serre’s conjecture (see Theorem 1.2). Theorem 3.1 Let p be a prime > 2. Let ρ : GQ → GL2 (F) be a S-type representation with non-solvable image. We suppose that 2 ≤ k(¯ ρ) ≤ p + 1 and k(¯ ρ) = p. (i) If k(ρ) = p + 1, then ρ¯ has a lift ρ which is minimally ramified at every . (ii) In the case k(ρ) = p + 1, there is a lifting ρ that is minimal at all primes

= p and which is crystalline of weight p + 1 at p. There is also a lifting ρ that is minimal at all primes = p and is semi-stable of weight 2 at p. We give the grandes lignes of the proof referring to [32] for more details. The idea of the proof is rather simple. The first remark is that it will suffice to prove the flatness over Zp of the corresponding minimal deformation ring RQ

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which parametrises all minimal liftings. Using obstruction theory, and using Poitou-Tate exact sequences as in the work of [67], B¨ockle in [2] has proven that the latter has a presentation W [[X1 , · · · , Xr ]]/(f1 , · · · , fs ) where s ≤ r. Thus it will suffice to prove that RQ /(p) is a finite ring. An easy argument in [32] (see Lemma 2.4 of loc. cit.) shows that this will be implied by showing that an analogous ring RF /(p), the universal mod p minimal deformation ring for ρ|GF for some totally real number field F (ρ|GF remains irreducible: see Lemma 2.6 of [32]) unramified at p, is finite. (We recall the argument: Assuming RF /(p) is finite we see that the image of the universal mod p Galois deformation ρ¯univ : GQ → GL2 (RQ /(p)) is finite. As the representation ρ¯univ is absolutely irreducible, a theorem of Carayol implies that the Noetherian ring RQ /(p) is generated by the traces of ρ¯univ . As ρ¯univ has finite image, for each prime ideal ℘ of RQ /(p), the images of these traces in the quotient by ℘ are sums of roots of unity, and there is a finite number of them. Thus we see that each of these quotients is a finite extension of F. It follows that the noetherian ring RQ /(p) is of dimension 0, and so is finite.) This will be known by the type of modularity lifting results proved in [59] and [30] if we know that ρ|GF is modular. The existence of a F with this property is precisely the content of Taylor’s potential modularity result Theorem 1.2. Remarks: It is important to note the crucial use of deformation rings (introduced by Mazur: see [41] for an account) in the above “existence” proof of minimal lifts.

4 The proof in [32] of Serre’s conjecture for low levels and low weights We present one of the main results of [32]. Theorem 4.1 Serre’s conjecture is true for residue characteristic p ≥ 3 for ρ such that N (ρ) = 1 and k(ρ) = 2, 4, 6, 8, 12, 14, or when k(ρ) = 2 and N (ρ) = 1, 2, 3, 5, 7, 11, 13, and det(ρ) = χp . We sketch a proof in some cases. The case of N (ρ) = 1, k(ρ) = 2 follows immediately from the result of Dieulefait in [24] (see Theorem 2.1) and the minimal lifting result in Theorem 3.1. k(¯ ρ) = 6, N (ρ) = 1 (see Figure 1): Suppose we have an irreducible ρ¯ with N (ρ) = 1, k(¯ ρ) = 6 as in the theorem (and we may assume p > 3 by Serre’s result for p = 3). We use Theorem 3.1 to get a compatible system

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weight 2 semistable at 5

linked mod 5

weight 6 crystalline compatible system

Fig. 1. Two linked compatible systems: the case of weight 6

(ρλ ) of weight 6, i.e., Hodge-Tate of weights (5, 0), and with good reduction everywhere. Consider a prime λ above 5 and assume the representation arising from the compatible system ρ¯5 has solvable image. The image has then to be reducible by known cases of Serre’s conjecture. Then by results in [6], and as the residual representation is globally and hence of course locally reducible at 5, we see that ρλ is ordinary at 5, and hence by Skinner-Wiles [58] (see Theorem 6.2 (1) below) corresponds to a cusp form of level 1 and weight 6 of which there are none. Otherwise, using Theorem 3.1, we get a minimal lift ρ of ρ¯5 that is unramified outside 5, and semistable (and not crystalline) of weight 2 at 5. Such a ρ by results of [61] arises from an abelian variety A over Q that is semistable and with good reduction outside 5. But by results of Brumer-Kramer [11] (see also Schoof [52]) such an A does not exist.

4.1 Even representations Theorem 4.1 proves inter alia that for p ≤ 7 there are no odd, irreducible representations ρ¯ : GQ → GL2 (Fp ) unramified outside p, as predicted by Serre’s conjectures. This, together with the results of Moon-Taguchi in [42] which handle the case of even representations following the method of [60], proves that for p ≤ 7 there are no irreducible representations ρ¯ : GQ → GL2 (Fp ) unramified outside p.

4.2 Explanation of Figures 1 and 4 The conceit of Figures 1 and 4 that depict linked compatibles systems is that each curve represents a compatible system and they intersect “at” a residual representation which arises (up to twist) from the 2 compatible systems.

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5 A variation on Wiles’ proof of Fermat’s last theorem Recall that in Section 4 of Serre’s paper [55] it is shown how the Frey construction of a semistable elliptic curve Eap ,bp ,cp over Q, associated to a Fermat triple (a, b, c), i.e., ap +bp +cp = 0, a, b, c coprime and abc = 0, and where we may assume a is −1 mod 4, b even, and p a prime > 3, leads to a S-type representation ρ (the irreducibility is a consequence of a theorem of Mazur) with k(ρ) = N (ρ) = 2. Wiles proved Fermat’s Last Theorem in [67] by showing that E is modular and hence ρ is modular and thus by Ribet’s level-lowering results, ρ arises from S2 (Γ0 (2)) which gave a contradiction as the latter space is empty.(Note that for the ρ considered here, im(ρ) is never solvable: see Proposition 21 of article 94 of [54] and also [47].) It is remarked in [32] that one may proceed differently. We expand on that remark. A possible way to prove Fermat’s Last Theorem was to show that ρ arises from a semistable abelian variety A over Q with good reduction outside 2 (note that E has conductor the radical of abc). This would give a contradiction as by the results of [11] such an A does not exist. The work in [32] now enables one to give such a proof. The minimal lifting result of [32] produces a lift of ρ that is Barsotti-Tate at p and has semistable reduction at 2, and is unramified everywhere else. Taylor’s results towards the Fontaine-Mazur conjecture prove that such a ρ arises from A which gives a contradiction. This proof while different in appearance from that of Wiles, uses all the techniques he developed in his original proof. The “simplifications” in this slightly different approach are: (i) We do not need to use the results of Langlands-Tunnell, that prove Serre’s conjecture for a ρ with solvable image, that Wiles had needed; (ii) This altered proof does not make use of the most difficult of the levellowering results which is due to Ribet, but instead uses the level lowering up to base change results of Skinner-Wiles [57].

6 Lifting results The results stated in this section are of a technical nature, but are crucial to the proof of Theorem 1.1. We refer to the original papers for the proofs.

6.1 Three Galois-theoretic lifting results Consider a S-type representation ρ with 2 ≤ k(ρ) ≤ p + 1, k(ρ) = p, p > 2 and assume that the image of ρ is not solvable.

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When we say that for some number field E, a E-rational compatible system of 2-dimensional representations of GQ lifts ρ we mean that for the place λ of E fixed by ιp , the residual representation arising from ρλ is isomorphic to ρ. The following theorem is contained in [32] and [33]. The essential idea for constructing the lifts is contained in the sketch of the proof of the minimal liftings given in Section 3, and that for constructing compatible systems of representations of GQ in Section 2. If the Weil-Deligne parameter at a prime q is (τ, N ), with τ a 2-dimensional complex representation of the Weil group and N a nilpotent matrix in M2 (C), by the inertial Weil-Deligne parameter we mean (τ |Iq , N ). In the theorem below we spell out more concretely what the property of strict compatibility means for ρλ |Dq , when the corresponding parameter is ramified, in the instances where we use this property. Theorem 6.1 1. Assume N (ρ) = 1. Then ρ lifts to a E-rational strictly compatible system (ρλ ) such that for all odd primes , and λ a prime above , ρλ is crystalline at of weight k(ρ) and unramified outside . 2. Assume N (ρ) = 1, and that ρ is ordinary at p, and thus ρ¯|Ip is of the form χp k(ρ)−1 ∗ . 0 1 (i) Then ρ lifts to an E-rational strictly compatible system (ρλ ) such that k(ρ)−2 ⊕ 1, 0) if for all λ the inertial Weil-Deligne parameter of ρλ at p is (ωp k(ρ) = p + 1 and otherwise is of the form (id, N ) with N a non-zero nilpotent matrix ∈ GL2 (C). (ii) For all primes = p, > 2, and λ above , ρλ is unramified outside { , p} and is crystalline of weight 2 at , and ρλ |Ip is of the form k−2 ∗ ωp , 0 1 and unramified if k(ρ) = 2. (iii) For the fixed place λ above p, ρλ is unramified outside p, and if k(ρ) = p + 1, then ρλ |Ip is of the form k−2 ωp χp ∗ , 0 1 and Barsotti-Tate over Qp (μp ) (and Barsotti-Tate over Qp if k(ρ) = 2). If k(ρ) = p + 1 then ρλ |Ip is of the form χp ∗ . 0 1

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3. Assume k(ρ) = 2 and N (ρ) = q an odd prime. Then ρ|Iq is of the form χ ∗ , 0 1 with χ a character of Iq that factors through its quotient (Z/qZ)∗ . Let χ be ∗ any non-trivial Zp -valued character of Iq that factors though (Z/qZ)∗ and reduces to χ. Thus we may write χ as ωqi for some 0 ≤ i ≤ q − 2. (i) Then there is an E-rational strictly compatible system (ρλ ) that lifts ρ such that for λ a prime above p fixed by ιp , ρλ is unramified outside {p, q}, is Barsotti-Tate at p, and ρλ |Iq is of the form χ ∗ . 0 1 (ii) For the place λ of E fixed above q, ρλ is unramified outside q, and either semistable of weight 2 at q or Barsotti-Tate over Q(μq ). (iii) For the place λ of E fixed above q, the residual representation ρ¯λ |Iq is of one of the following 3 forms (a) χq i+1 ∗ , 0 1 (b)

χq 0

∗ χq i

ψqi+1 0

0

,

or (c)

ψq

i+1

,

where ψq , ψq are the fundamental character of level 2 of Iq . Thus in particular the Serre weight of some twist of ρ¯λ by a power of χq is either i + 2 or q + 1 − i. The last part of the theorem uses the results of Breuil, Mezard and Savitt (see [10] and [50]). The theorem can be motivated by results on modular forms. The first, which is contained in Theorem 3.1 (1), may be motivated by recalling that if ρ is modular, then it arises from Sk (ρ)(Γ1 (N )) (see [45]: this is the level and weight optimisation result). The second and third parts may be motivated by recalling that a mod p representation that arises from S2 (Γ0 (p), χp i ), arises also from either

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Si+2 (SL2 (Z)) or Sp+1−i (SL2 (Z))(χp −i ) where the spaces are suitable spaces of mod p modular forms. The third may be motivated by Carayol’s result that if ρ arises from S2 (Γ0 (q), ), then it also arises from S2 (Γ0 (q), ) for any congruent to modulo the place above p we have fixed.

6.2 Three modularity lifting results Consider a 2-dimensional mod p > 2 representation ρ of GQ which is odd and with 2 ≤ k(ρ) ≤ p + 1 with p > 2. We do not assume that ρ is irreducible, but we do assume that ρ is modular, which in the reducible case simply means odd. The following theorem is the work of many people, Wiles, Taylor, Breuil, Conrad, Diamond, Flach, Fujiwara, Guo, Kisin, Savitt, Skinner et al. (see [67], [66], [30], [16],[17], [58], [59], [49], [37]) and is absolutely vital to us. Theorem 6.2 1. Let ρ be a lift of ρ to a p-adic representation that is unramified outside p and crystalline of weight k, with 2 ≤ k ≤ p + 1, at p. Then ρ is modular. 2. Let ρ be a lift of ρ to a p-adic representation that is unramified outside p and either semistable of weight 2, or Barsotti-Tate over Qp (μp ). Then ρ is modular. 3. Let ρ be a lift of ρ to a p-adic representation that is unramified outside a finite set of primes and is Barsotti-Tate at p. Then ρ is modular.

The theorem crucially uses developments in the modularity lifting technology that build on [67] and [66]. In particular the results of Skinner-Wiles in the residually degenerate cases are crucial to us, see [58] and [59]. The simplifications of the original method of [67] and [66], due independently to Diamond and Fujiwara in one aspect, see [22] and [30], and due to Kisin in another aspect, see Proposition 3.3.1 of [37], are again vital. The fact that we do not have the usual condition that ρ is non-degenerate, i.e., ρ|Q(μp ) is irreducible, is because when ρ|Q(μp ) is reducible in the cases envisaged in the theorem, the lift ρ is up to twist by a power of ωp ordinary at p, and in this case as our representations are also distinguished at p, results of Skinner-Wiles in [58] and [59] apply.

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q||N (¯ ρ) k

p–plane

k

q–plane (1, q + 1)

(q, 2) N

N

Fig. 2. Killing ramification at the expense of weight increase

7 The reductions in [32] of Serre’s conjectures to modularity lifting results In [32] it was observed that the lifting technique introduced there, together with the use of compatible systems, would yield Serre’s conjecture in two steps if modularity lifting results were proved in sufficient generality: 1. Reduction to level 1: Here the key observation is that a Serre-type mod p representation with weights between 2 and p + 1 always lifts to a geometric p-adic representation of GQ which is minimally ramified away from p and crystalline at p of weight between 2 and p + 1. This together with the fact that such a lift can be made part of a compatible system explains this procedure. We will explain the idea pictorially (see Figure 2, which depicts the generic case: sometimes the point plotted in the q-plane could have co-ordinates (1, 2)) to show how the conjecture for a S-type semistable ρ when the conductor of ρ is a prime q and the weight is 2 follows from the level 1 case if one knew an appropriate lifting theorem (and one does in this case!). This idea when combined with Theorem 1.1 results in Corollary 10.1 below. 2. The level 1 case using induction on the residue characteristic: Given a S-type ρ of level 1 in residue characteristic pn , the nth prime, we use Theorem 6.1 (1) to lift it to a compatible system (ρλ ), and extract from this ρpn−1 , and consider the residual representation ρ¯pn−1 . By induction this would be modular (the starting point of the induction would use the results of Tate and Serre), and if one knew an appropriate lifting theorem one would be done. The lifting theorem needed would be to show that a p-adic representation ρ unramified outside p is modular, if residually it is modular (which includes the reducible, odd case), and it is crystalline at p of weight ≤ 2p: here we are using

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the Bertrand postulate, which was proved by Chebyshev, to restrict attention to these weights. Such modularity lifting theorems especially in degenerate cases seem hard as explained later. While this approach has been superseded by the different inductive step in [33], it contained implicitly within it the following result (see Section 5 of [33]) that was used in the induction of [33] (see Figure 3): Theorem 7.1 (i) Given an odd prime p, if all 2-dimensional, mod p, odd, irreducible representations ρ of a given weight k(ρ) = k ≤ p + 1, and unramified outside p, are known to be modular, then for any prime q ≥ k − 1, all 2-dimensional, mod q, odd, irreducible representations ρ of weight k(ρ ) = k, and unramified outside q, are modular. (ii) If the level 1 case of Serre’s conjecture is known for a prime p > 2, then for any prime q the level 1 case of Serre’s conjectures is known for all 2-dimensional, mod q, odd, irreducible representations ρ of weight k(ρ) ≤ p + 1. Thus in particular it suffices to prove Serre’s conjecture for infinitely many primes (and the argument in the level 1 case partly works because there are infinitely many non-Fermat primes!).

k

p–plane

k

(1, k)

q–plane

(1, k)

N

N

Fig. 3. Compatible systems

7.1 Explanation of Figures 2, 3 and 5 Figures 2,3 and 5 are supposed to express the following idea. Serre’s conjecture for each prime p is imagined as happening on a p-plane with the axes recording the level and weight invariants (the weight axis is not semi-infinite, and stops at the ordinate p2 − 1). For instance in Figure 3, plotting a point means that one

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knows Serre’s conjecture in residue characteristic p for all S-type representations with invariants the co-ordinates of the point (a posteriori there are only finitely many isomorphism classes of such representations). Drawing an arrow between a p-plane and a q-plane describes a “move” made using compatible systems (we owe this terminology to Mazur).

8 The proof of the level 1 case: a sketch We use as an input in the proof of the full level 1 case, the fact that by Theorem 4.1 we know the level 1 case in weights 2, 4 and 6 in all residue characteristics.

8.1 The weight 8 case: a trailer for the general case Consider the weight 8 case (see Figure 4). By Theorem 7.1 it’s enough to prove that an irreducible 2-dimensional, mod 7 representation ρ¯ of level 1 and weight 8 is modular. (If the image is solvable we are done.) Such a representation is ordinary at 7 and in fact ρ|I7 is of the form χ7 ∗ , 0 1 and is tr`es ramifi´ee. By Theorem 6.1 (2) lift ρ to a strictly compatible system (ρλ ) such that the 7-adic representation ρ7 (corresponding to our fixed place of Q above 7) lifts ρ and is weight 2, semistable at 7 that is unramified outside 7. Thus ρ7 |I7 is of the form χ7 ∗ . 0 1 We now extract the 3-adic representation ρ3 determined by ι3 from (ρλ ), and consider a corresponding residual mod 3 representation ρ . Note that k(ρ ) = 2, and ρ is unramified outside 3 and 7. If ρ has solvable image we are done, as in that case we have a representation to which we can apply modularity lifting results of Theorem 6.2 (3) to conclude that the compatible system ρλ is modular and hence so is ρ. Similarly if ρ is unramified at 7 we are again done as we know by page 710 of [54] that the residual mod 3 representation is then reducible. Note that ρ |I7 is of the form 1 ∗ . 0 1

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weight 8

linked mod 3

weight 2 level 7

linked mod 7

Fig. 4. Three linked compatible systems: the weight 8 case

Using Theorem 6.1 (3) get a 3-adic lift ρ of ρ with nebentype ω72 at 7 (ω74 would also work). Thus ρ |I7 is of the form

ω72 0

∗ 1

.

Note that the behavior of the lifts ρ3 and ρ of ρ when restricted to I7 are quite different: ρ3 (I3 ) has infinite image, while ρ (I3 ) has image of order 3. Using Theorem 6.1 (3) get a compatible system (ρλ ) with ρ the member of this compatible system at the place corresponding to ι3 , and consider a residual representation ρ¯7 arising from this system at a place λ above 7 fixed by ι7 . If ρ¯7 has solvable image, and hence known to be modular, we are done by applying Theorem 6.2 (2). Now assume that ρ¯7 has non-solvable image. We get a residual representation whose Serre weight (up to twisting) is either 4 or 7 + 3 − 4 = 6 by Theorem 6.1 (3). But we know the modularity of level 1, S-type representations of weights 4 and 6. Now we again use Theorem 6.2 (2) and conclude that (ρλ ) is modular. Hence so is the residual representation ρ , and hence by another application of modularity lifting theorem we conclude that the first compatible system (ρλ ) is modular (as the compatible systems (ρλ ) and (ρλ ) are linked at the place above 3 fixed by ι3 ), and hence so is ρ (which in this case means that it does not exist!).

8.2 Estimates on distribution of primes The proof of the general case is very similar except that one uses some estimates on prime numbers of the type proven by Chebyshev. In fact, we will use the better estimates proved in [48]. (The estimates quoted in [33] were incorrect: we thank Dietrich Burde for bringing to light this error.)

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If π(x) is the prime counting function, then in Theorem 1 of [48] various estimates of the form x x ) < π(x) < B( ) A( log(x) log(x) are proven with varying values of A, B, for x > x0 where x0 depends on A, B. From such a bound we easily deduce that if we fix a real number a > C, with C = B A ,aand denote by pn the nth prime then pn+1 ≤ apn for pn+1 > max(ax0 , a a−C ), . In the arguments below we need to check, that for each non-Fermat prime P > 7, there is a non-Fermat prime p < P (for example p the largest nonFermat prime < P ) and an odd prime power divisor r ||(P − 1) so that P 2m + 1 m 1 ≤ −( )( ) p m+1 m+1 p

(8.1)

where we have set r = 2m + 1 with m ≥ 1. For non-Fermat primes 7 < P ≤ 31 we see this by inspection. For P > 31 we in fact prove the stronger 1 ) = 1.46. estimate Pp ≤ 32 − ( 30 In loc. cit. it is proven that for x > 17, x x < π(x) < 1.25506( ). log(x) log(x) From this we deduce the estimate that pn+1 ≤ (1.46)pn when pn+1 > 31. Consider the only Fermat primes 257 and 65, 537 (the latter being the largest known Fermat prime!) between 17 and 200, 000. For P = 263, the next prime after 257, p = 251 satisfies the required estimate, and for P = 65, 539, the next prime after 65, 537, p = 65, 521 satisfies the required estimate. 1 ) for 31 < P < 200, 000. From all this we deduce the estimate Pp ≤ 32 − ( 30 In loc. cit. it is also proven that π(x) <

x 3 (1 + ), log(x) 2log(x)

for x > 1, which yields the inequalities x x < π(x) < 1.130289( ), log(x) log(x) for x > 100, 000. Using a = 1.2 in the estimates at the beginning of the section, and noting that (1.2)2 < 1.46, and using that there are no successive 1 Fermat primes after the pair 3, 5, we deduce that the estimate Pp ≤ 32 − ( 30 ) is true for all P > 100, 000, and thus the truth of (8.1) for all P > 7. (As will be clear to the discerning reader, this is not necessarily the most efficient way to check this!)

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Chandrashekhar Khare r ||Pn − 1

Pn = n’th non Fermat prime

k

k

Pn –plane

–plane

(1, k(ρ)) ¯ (2, Pn )

N

N Pn –plane

k

Pn−1 –plane

k

(1, k(ρ¯ ))

(1, k(ρ¯ ))

k(ρ¯ ) ≤ Pn−1 + 1

N

N

Fig. 5. Weight reduction

8.3 The general case Now we give the general argument (see Figure 5). Rather than present the argument as an induction as we did in [33], we unravel the induction there, and lay out the skein of linked compatible systems that makes the argument work. Consider a S-type representation ρ of residue characteristic P > 7 and such that 2 < k(ρ) ≤ P + 1. We may assume that P is a non-Fermat prime by Theorem 7.1. By the estimates of Section 8.2, there is an odd prime power divisor r := 2m + 1 of P − 1 and a non-Fermat prime p < P which satisfies the bound of (8.1). We may assume that the image is non-solvable. Besides the basic idea presented in the weight 8 case, and the use of the Chebyshev-type estimates, there is one extra argument that uses the simple fact that knowing modularity for a representation ρ is equivalent to knowing it for a twist of the representation by a power of χP . This allows one to accomplish the first reduction of weight rather easily in the case when ρ at P is irreducible, as in this case we see that a twist of ρ has weight P + 3 − k (when k > 2). Now we claim that at least one of k or P + 3 − k is ≤ p + 1: otherwise we get

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that P > 2p − 1. Using (8.1) this will give a contradiction if 2p − 1 ≥

m 2m + 1 p−( ), m+1 m+1

p 1 ≥ m+1 , which is always true. Thus in the case when ρ is i.e., if m+1 irreducible at P we may reduce to the case when k(ρ) ≤ p + 1. Otherwise we use Theorem 6.1 (2) to produce a weight 2 compatible lifting (ρλ ), consider ρ (and stop if this either has solvable image or is unramified at P ) and then choose a suitable lifting (ρλ ) using Theorem 6.1 (3) with the suitability dictated by the estimate (8.1). Namely, we choose the -adic lifting ρ of ρ so that it is unramified outside , P , is Barsotti-Tate at and at P , ρ |IP is of the form:

ωPi 0

∗ 1

,

m m+1 (P − 1), ( 2m+1 )(P − 1)]. Then we stop if the residual mod with i ∈ ( 2m+1 P representation arising from (ρλ ), say ρ , has solvable image. Otherwise we know by our choice of i (and Theorem 6.2 (3)) that ρ¯ is such that, after twisting by a power of χP , k(ρ ) ≤ p + 1. (This is where the Chebyshev-type etimate is used: see [33] for more details. Note that it is essential to choose i to be in the “middle” of its possible range of values because of the dichotomy for the weight of ρ in Theorem 6.1(3).) After rechristening ρ again as ρ, and conflating now the locally at P reducible and irreducible cases, use Theorem 6.1 (1) to lift ρ to a compatible system (ρλ ), and consider ρp . We repeat this procedure and in the end conclude modularity either: (i) because we got lucky early and we hit a residual representation that was considered in the procedure which happened to have solvable image, or (ii) at 3p the end of constructing (very roughly log p ) compatible systems, we know the modularity of the last system by the proven level 1 case of Serre’s conjecture mod 7. The fact that we are able to pass the modularity property through the linkages is because of the modularity lifting theorems in Theorem 6.2: it requires some thought to convince oneself about this, but we leave this as an exercise to the interested reader.

Remark: It seems possible to avoid using the fact that Serre’s conjecture is known when the mod p representation has solvable image (p > 2), in all but the dihedral case (that case being known to Hecke).

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Chandrashekhar Khare 9 Some observations about why the proof works

We make some heuristic remarks about the proof. 1. In the first remark we try to explain why the approach in [33] was able to sidestep the technical difficulties that the initially proposed inductive approach to the level 1 conjecture in [32], sketched in Section 7 of this paper, encountered. The main innovation of [33] was to prove the level 1 case by a new method of carrying out the inductive step that used known modularity lifting results. Besides the fact that this method works, there is a “philosophical” point to be made here which might explain why the modularity lifting results that were needed by the method were indeed available. The method of [33] introduces the mildest possible singularities in the process to prove Serre’s conjecture. Here we define singularities to be points in the proof (linkages) when one encounters a residually degenerate representation. The singularity is mild when the corresponding modularity lifting theorem needed is the one proved in the work of Skinner and Wiles in [57] and [58], i.e., when the lift is (up to twist) ordinary at the residue characteristic (and which is the simplest kind of behavior at p from the view-point of p-adic Hodge theory). When the lift is not ordinary in these globally residually degenerate cases, no modularity lifting theorems seem to be known. This seems technically like a very hard problem, although the author is no expert. One would imagine that encountering a singularity makes some d´evissage possible resulting in “reducing” the conjecture to a known case. While this is in a sense true, the corresponding modularity lifting theorem required to implement this intution is more involved and contained in [58], [59]. The method presented here uses modularity lifting theorems detailed in Theorem 6.2 when the geometric liftings that need to be proved modular are at p either crystalline of weight ≤ p + 1, semistable of weight 2 or Barsotti-Tate over Qp (μp ). If the weight of a crystalline representation ρ : GQp → GL2 (O) is bigger than p + 1 (the original method proposed in [32] to prove the level 1 conjecture involved liftings that were of these kind), or it is semistable of weight > 2, or Barsotti-Tate only over a wildly ramified extension or even over a tamely ramified extension of ramification index not dividing p − 1, it is not true that if ρ is reducible, then so is ρ. Thus the method just about avoids having to deal with anything beyond mild singularities. 2. As there are no irreducible, level 1, S-type representations of weights up to 6, and this is used crucially in the proof of the level 1 conjecture, it

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is important to note how crucial the modularity lifting results of Skinner and Wiles of [58] in the reducible case are to us. 3. The difference between the approach envisaged in [32] for the level 1 case, and that of [33], may also be brought out by pointing out that the former sought to prove the level 1 conjecture in residue characteristic p by using a p , while in sequence of linked compatible systems of cardinality roughly log(p) 3p linked compatible systems are used. The interpolation the latter roughly log(p) of more compatible systems in the proof ameliorates the technical difficulties that were encountered in [32].

10 A higher level result The following result follows from the level 1 result together with the method of killing ramification: Corollary 10.1 If ρ¯ is an irreducible, odd, 2-dimensional, mod p representation of GQ with k(ρ) = 2, N (ρ) = q, with q prime, and p > 2, then it arises from S2 (Γ1 (q)). Characterisation of simple factors of J0 (p): Theorem 1.1 also implies that a simple abelian variety A over Q is a factor of J0 (p) for a prime p, up to isogeny, if and only if it is a simple GL2 -type semistable abelian variety over Q with good reduction outside p (one direction is classical needless to say).

11 Finite flat group schemes of type (p, · · · , p) over Z The weight 2, level 1 case of Serre’s conjecture proven in [32] can be rephrased as follows and where we use the terminology of [44]: Theorem 11.1 For a finite field F of characteristic p, there are no irreducible finite, flat F-vector space group schemes of rank 2 over Z. The following problem belongs to the folklore of the subject: Problem 11.2 Prove that for an odd prime p all finite, flat group schemes of type (p, · · · , p) over Z are direct sums of μp ’s and Z/pZ’s. This has been proved by Fontaine in [29] for p = 3, 5, 7, 11, 13, 17. Perhaps one could be less ambitious and pose instead:

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Problem 11.3 For a finite field F of characteristic p, prove that there are no irreducible finite flat, F-vector space group schemes over Z of rank n > 1 which are isomorphic to their Cartier duals.

12 Perspectives: the Fontaine-Mazur conjecture Serre’s conjecture, when combined with modularity lifting results, allows one to prove more cases of the Fontaine-Mazur conjecture (in its restricted version for 2-dimensional representations of GQ ) beyond the cases when the image of the representation is pro-solvable. The work in [33], when combined with modularity lifting theorems, implies that if ρ : GQ → GL2 (O) is an irreducible p-adic representation unramified outside p > 2 and at p crystalline of Hodge-Tate weights (k −1, 0) with k even and either 2 ≤ k ≤ p + 1 or ρ is ordinary at p, then ρ arises from Sk (SL2 (Z)). This also has bearing on some of the finiteness conjectures in [28]. Even after Serre’s conjecture is proved completely to prove the (restricted version of) Fontaine-Mazur conjecture one will need new modularity lifting results for lifts whose behaviour at p is complicated. Work of Berger-Breuil, see [8], and forthcoming work of Kisin, addresses some of these cases. While we have commented on how Serre’s conjecture implies modularity of certain p-adic representations, it is perhaps worth noting that in [36] it has been shown that Serre’s conjecture also implies Artin’s conjecture for 2-dimensional odd, irreducible, complex representations of GQ . The observation of [36] uses the modification in [26] of the definition of k(ρ) which allows it to take the value 1, and the results of Coleman-Voloch about the weight using this modified definition. The observation of [36] again relies on using compatible systems, but only the “constant” systems that arise from Artin representations. As almost all of this case of Artin’s conjecture is known (see [18]), the result of [36] just offers a different perspective. The Fontaine-Mazur conjectures are related to (restricted cases of) the Langlands conjectures that seek to establish intimate relationships between motives, automorphic forms and Galois representations. Thus results and methods that prove residual automorphy of Galois representations, and modularity lifting methods, will help make progress in the Langlands program in “arithmetic cases” (see [15] and [62]) in the direction of showing that certain motives are automorphic, or that certain Galois representations are motivic/automorphic. (The association of Galois representations to automorphic forms of arithmetic type is a separate problem, and in a sense precedes the others.)

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The case for 2-dimensional odd representations of GQ , which one may hope now will be understood in a not too distant future, is just a first step in this direction. For instance it will be of interest to see if the methods presented here can be generalised to prove the analog of Serre’s conjectures for totally real fields (see [9]). At first one might restrict attention to a few fields over which abelian varieties with good reduction everywhere have been classified. Even for these, while the ideas of [32] and [33] presented in this expository article will probably be useful, there are interesting technical complications to be overcome. We make a list of (not necessarily totally real) number fields over which abelian varieties with good reduction eveywhere Fontaine √ have been classified. √ proved that there are none such over Q, Q( 5), Q(i) and Q( −3). Schoof proved in [51] that there are none such over the cyclotomic fields Q(ζf ) for f = 5, 7, 8, 9, 12. Schoof has also informed us that in unpublished work he has proven a √ similar result over Q( f ) if discriminant f is equal to 5, 8, 12, 13, 17, 21 (the “first” six real quadratic fields). In all cases the corresponding space of quadratic nebentypus weight 2 cuspforms for Γ1 (f ) is zero, and for other conductors f this space is non-zero: thus the result is best possible. For f = 24 (and assuming the GRH for f = 28, 29, 33), Schoof in [53] has √ shown that any abelian variety over Q( f ) with good reduction everywhere is isogenous to a power of the elliptic curve associated by Shimura to the space of (quadratic) nebentypus weight 2 cuspforms for Γ1 (f ) (the spaces in these cases are 2-dimensional). The “weight cycle” idea of [33] and killing ramification idea of [32] give a method to prove Serre’s conjecture once one knows (a proof of) Fontaine’s result that there are no abelian varieties over Z. Thus the real quadratic fields over point” to launch the methods of [32] and [33] are √ has a√“starting √ √ √which one are no abelian Q( 3), Q( 5), Q( 13), Q( 17), Q( 21) over√which there √ √ 6) (and Q( 7), Q( 29), varieties with good reduction everywhere, and Q( √ Q( 33) under the GRH) over which all such simple abelian varieties are classified up to isogeny (and known to arise from Hilbert modular forms that are in this case base changed from Q). For the case of a general real quadratic field, it might be better to use some instances of functoriality and work with the group GSp4 /Q . Acknowledgements: I thank Najmuddin Fakhruddin for his help with the writing of this article. I would like to thank Dietrich Burde, Najmuddin Fakhruddin, Mark Kisin, Ravi Ramakrishna, Jean-Pierre Serre, Ren´e Schoof and Jean-Pierre Wintenberger for helpful correspondence during the writing

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of this expository paper. I would like to especially thank Barry Mazur for his generous, detailed comments. The presentation of the results of [33] given here follows the lines of the talks the author gave at the conference “Repr´esentations galoisiennes” in Strasbourg in July 2005. I would like to thank the organisers, Jean-Francois Boutot, Jacques Tilouine and Jean-Pierre Wintenberger, for their invitation to speak at the conference.

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Two p-adic L-functions and rational points on elliptic curves with supersingular reduction Masato Kurihara Department of Mathematics Keio University 3-14-1 Hiyoshi, Kohoku-ku Yokohama, 223-8522, Japan [email protected]

Robert Pollack Department of Mathematics and Statistics Boston University Boston, MA 02215, USA [email protected]

Introduction Let E be an elliptic curve over Q. We assume that E has good supersingular reduction at a prime p, and for simplicity, assume p is odd and ap = p + 1 − #E(Fp ) is zero. Then, as the second author showed, the p-adic L-function √ Lp,α (E) of E corresponding to α = ± −p (by Amice-V´elu and Vishik) can be written as − Lp,α (E) = f log+ p +g logp α

by using two Iwasawa functions f and g ∈ Zp [[Gal(Q∞ /Q)]] ([20] Theorem 5.1). Here log± p is the ±-log function and Q∞ /Q is the cyclotomic Zp -extension (precisely, see §1.3). In Iwasawa theory for elliptic curves, the case when p is a supersingular prime is usually regarded to be more complicated than the ordinary case, but the fact that we have two nice Iwasawa functions f and g gives us some advantage in several cases. The aim of this paper is to give such examples. 0.1. Our first application is related to the weak Birch and Swinnerton-Dyer conjecture. Let L(E, s) be the L-function of E. The so called weak Birch and Swinnerton-Dyer conjecture is the statement Conjecture (Weak BSD) L(E, 1) = 0 ⇐⇒ rankE(Q) > 0.

(0.1)

We know by Kolyvagin that the right hand side implies the left hand side, but the converse is still a very difficult conjecture. For a prime number p, 300

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let Sel(E/Q)p∞ be the Selmer group of E over Q with respect to the p-power torsion points E[p∞ ]. Hence Sel(E/Q)p∞ sits in an exact sequence 0 −→ E(Q) ⊗ Qp /Zp −→ Sel(E/Q)p∞ −→ X(E/Q)[p∞ ] −→ 0 where X(E/Q)[p∞ ] is the p-primary component of the Tate-Shafarevich group of E over Q. In this paper, we are interested in the following conjecture. Conjecture 0.1 L(E, 1) = 0 ⇐⇒ #Sel(E/Q)p∞ = ∞ This is equivalent to the weak Birch and Swinnerton-Dyer conjecture if we assume #X(E/Q)[p∞ ] < ∞. (Of course, the problem is the implication from the left hand side to the right hand side.) We note that Conjecture 0.1 is obtained as a corollary of the main conjecture in Iwasawa theory for E over the cyclotomic Zp -extension Q∞ /Q. We also remark that if the sign of the functional equation is −1, Conjecture 0.1 was proved by Skinner-Urban [24] and Nekov´aˇr [16] in the case when p is ordinary, and by Byoung-du Kim [10] in the case when p is supersingular. In this paper, we will give a simple condition which can be checked numerically and which implies Conjecture 0.1. Suppose that p is an odd supersingular prime with ap = 0. We identify Zp [[Gal(Q∞ /Q)]] with Zp [[T ]] by the usual correspondence between a generator γ of Gal(Q∞ /Q) and 1 + T . When we regard the above two Iwasawa functions f , g as elements of Zp [[T ]], we denote them by f (T ), g(T ). The interpolation property of f (T ) and g(T ) tells us that f (0) = (p − 1)L(E, 1)/ΩE and g(0) = 2L(E, 1)/ΩE where ΩE is the N´eron period. Hence, if L(E, 1) = 0, we have f (T ) "" p−1 = . " g(T ) T =0 2 We conjecture that the converse is also true, namely Conjecture 0.2 L(E, 1) = 0 ⇐⇒

f (T ) "" p−1 = " g(T ) T =0 2

(Again, the problem is the implication from the left hand side to the right hand side.) Our first theorem says that Conjecture 0.2 implies Conjecture 0.1, namely

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Theorem 0.3 Assume that (f /g)(0) =

"

f (T ) " g(T ) " T =0

does not equal (p − 1)/2.

Then, Sel(E/Q)p∞ is infinite. This result in a different terminology was essentially obtained by PerrinRiou (cf. [19] Proposition 4.10, see also §1.6 in this paper), but we will prove this theorem in §1.4 by a different and simple method, using a recent formulation of Iwasawa theory of an elliptic curve with supersingular reduction. In §1, we also review the recent formulation of such supersingular Iwasawa theory. Conversely, assuming condition (∗)0 which will be introduced in §1.4 and which should always be true, we will show in §1.4 that the weak BSD conjecture (0.1) implies Conjecture 0.2, namely Theorem 0.4 Assume condition (∗)0 in §1.4, and rankE(Q) > 0. Then, (f /g)(0) = (p − 1)/2 holds. Combining Theorems 0.3 and 0.4, we get Corollary 0.5 Assume (∗)0 in §1.4 and #X(E/Q)[p∞ ] < ∞. Then, we have f (T ) "" p−1 rankE(Q) > 0 ⇐⇒ = . " g(T ) T =0 2 Note that the left hand side is algebraic information and the right hand side is p-adic analytic information. The weak BSD conjecture (0.1) is usually regarded to be a typical relation between algebraic and analytic information. The above Corollary 0.5 also gives such a relation, but in a different form. (Concerning the meaning of (f /g)(0) = p−1 2 , see also §1.6.) We note that f (T ) and g(T ) can be computed numerically and the condition (f /g)(0) = (p − 1)/2 can be checked numerically. For example, for N = 17 and 32, we considered the quadratic twist E = X0 (N )d of the elliptic curve X0 (N ) by the Dirichlet character χd of conductor d > 0. For p = 3 (which is a supersingular prime in both cases with ap = 0), we checked the condition (f /g)(0) = (p − 1)/2 for all Ed such that L(Ed , 1) = 0 with 0 < d < 500 and d prime to 3N . We did this by computing (f /g)(0) − (p − 1)/2 mod 3n ; the biggest n we needed was 7 for E = X0 (32)d with d = 485. For N = 17 and d = 76, 104, 145, 157, 185, ... (resp. N = 32 and d = 41, 65, 137, 145, 161, ...) ordT (f (T )) = ordT (g(T )) = 2, so the corank of Sel(E/Q)p∞ should be 2. Our computation together with Theorem 0.3 implies that this corank is ≥ 1. We give here one simple condition which implies (f /g)(0) = (p − 1)/2. Put r = min{ordT f (T ), ordT g(T )}, and set f ∗ (T ) = T −r f (T ) and

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g ∗ (T ) = T −r g(T ). If ordp (f ∗ (0)) = ordp (g ∗ (0)), then (f ∗ /g ∗ )(0) is not in Z× p and hence cannot be (p − 1)/2. Therefore, by Theorem 0.3 we have Corollary 0.6 If ordp (f ∗ (0)) = ordp (g ∗ (0)), then Sel(E/Q)p∞ is infinite. In particular, if one of f ∗ (T ) or g ∗ (T ) has λ-invariant zero and the other has non-zero λ-invariant, then Sel(E/Q)p∞ is infinite. For example, we again consider E = X0 (17)d with d > 0 and p = 3. Then, the condition on the λ-invariants in Corollary 0.6 is satisfied by many examples, namely for d = 29, 37, 40, 41, 44, 56, 65, ... with r = 1, and for d = 145, 157, 185, 293, 409, ... with r = 2. Corollary 0.6 implies a small result on the Main Conjecture (Proposition 1.5, see §1.5). The case r = 1 will be treated in detail in §2. 0.2. In 0.1, we explained that the computation of the value (f /g)(0)− (p − 1)/2, or f (r) (0) − p−1 g (r) (0), yields information on the Selmer group 2 Sel(E/Q)p∞ where r = min{ordT f (T ), ordT g(T )}. It is natural to ask what this value means. In the case r = 1, we can interpret this value very explicitly by using the p-adic Birch and Swinnerton-Dyer conjecture (cf. Bernardi and Perrin-Riou [1] and Colmez [4]). In this case, for a generator P of E(Q)/E(Q)tor , the p-adic Birch and Swinnerton-Dyer conjecture predicts that logEˆ (P ) is related to the quantity ˆ f (0) − p−1 ˆ is the logarithm of the formal group E (see 2 g (0) where logE §2.6 (2.5)). Using this formula for logEˆ (P ), we can find P numerically. More precisely, we compute ⎛> # f (0) − exp ˆ ⎝ − E

⎞ 2p log(κ(γ))[ϕ(ωE ), ωE ] #E(Q)tor ⎠ · Tam(E) p+1 $

p−1 g (0) 2

(0.2) which is a point on E(Qp ), and which would produce a point on E(Q) with a slight modification (see §2.7). Namely, we can construct a rational point of infinite order p-adically in the case r = 1 as Rubin did in his paper [22] §3 for a CM elliptic curve. Note here that we have an advantage in the supersingular case in that the two p-adic L-functions together encode logEˆ (P ) (and not just the p-adic height of a point). We did the above computation for quadratic twists of the curve X0 (17) with p = 3. We found a rational point on X0 (17)d by this method for all d such that 0 < d < 250 except for d = 197, gcd(d, 3 · 17) = 1, and the rank

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of X0 (17)d is 1. For example, for the curve X0 (17)193 : y 2 + xy + y = x3 − x2 − 25609x − 99966422 we found the rational point 915394662845247271 −878088421712236204458830141 , 25061097283236 125458509476191439016 by this method, namely 3-adically. To get this rational point, we had to compute the value (0.2) modulo 380 (to 80 3-adic digits) to recognize that the modified point constructed from the value (0.2) is a point on E(Q). For the curve X0 (32) and p = 3 we did a similar computation and found rational points on X0 (32)d for all d with 0 < d < 150, gcd(d, 6), and the rank of X0 (32)d is 1. In §4, there are tables listing points for both of these curves. To compute f (0) and g (0) to high accuracy, the usual definition of the padic L-function (namely, the computation of the Riemann sums to approximate f (0) and g (0)) is not at all suitable. We use the theory of overconvergent modular symbols as in [21] and [6]. We will explain in detail in §2 this theory and the method to compute a rational point in practice. 0.3. Next, we study a certain important subgroup of the Selmer group over Q∞ . This is related to studying common divisors of f (T ) and g(T ). For any algebraic extension F of Q, we define the fine Selmer group Sel0 (E/F ) by Sel0 (E/F ) = Ker(Sel(E/F )p∞ −→ H 1 (Fv , E[p∞ ])) v|p

where Sel(E/F )p∞ is the Selmer group of E over F with respect to E[p∞ ], and v ranges over primes of F lying over p. (The name “fine Selmer group” is due to J. Coates.) Our interest in this subsection is in Sel0 (E/Q∞ ). Let Qn be the intermediate field of Q∞ /Q with degree pn . We put e0 = rankE(Q) and Φ0 (T ) = T . For n ≥ 1, we define en =

rankE(Qn ) − rankE(Qn−1 ) pn−1 (p − 1) n

which is a non-negative integer, ωn (T ) = (1 + T )p − 1, and Φn (T ) = ωn (T )/ωn−1 (T ). The Pontryagin dual Sel0 (E/Q∞ )∨ of the fine Selmer group over Q∞ is a finitely generated torsion Zp [[Gal(Q∞ /Q)]]-module by Kato [9]. Concerning the characteristic ideal, Greenberg raised the following problem (conjecture) (see §3.1)

Two p-adic L-functions and rational points Problem 0.7 char(Sel0 (E/Q∞ )∨ ) = (

305

Φenn −1 ).

en ≥1 n≥0

We remark that this “conjecture” has the same flavor as his famous conjecture on the vanishing of the λ-invariants for class groups of totally real fields. Let Sel± (E/Q∞ ) be Kobayashi’s ±-Selmer groups ([12], or see 1.5). By definition we have Sel0 (E/Q∞ ) ⊂ Sel± (E/Q∞ ). In this subsection, we assume the μ-invariant of Sel± (E/Q∞ ) vanishes. Using Kato’s result [9] we know g(T ) ∈ char(Sel+ (E/Q∞ )∨ ), and f (T ) ∈ char(Sel− (E/Q∞ )∨ ) (cf. Kobayashi [12] Theorem 1.3, see also 1.5). Hence, any generator of char(Sel0 (E/Q∞ )∨ ) divides both f (T ) and g(T ). Thus, in the supersingular case, we can check this conjecture (Problem 0.7) numerically in many cases, by computing f (T ) and g(T ). For example, suppose that rankE(Q) = e0 and min{λ(f (T )), λ(g(T ))} = e0 . Then, we can show that the above “conjecture” is true and, moreover, char(Sel0 (E/Q∞ )∨ ) = (T e0 ) where e0 = max{0, e0 − 1} (see Proposition 3.1). For E = X0 (17)d and p = 3, the condition of Proposition 3.1 is satisfied for all d such that 0 < d < 250 except for d = 104, 193, 233. For these exceptional values, we also checked Problem 0.7 holds (see §3.3). In §3.2 we raise a question on the greatest common divisor of f (T ) and g(T ) (Problem 3.2), and study a relation with the above Greenberg “conjecture” (see Propositions 3.3 and 3.4). We would like to heartily thank R. Greenberg for fruitful discussions on all subjects in this paper, and for his hospitality when both of us were invited to the University of Washington in May 2004. Furthermore, we learned to use Wingberg’s result from him when we studied the problem in Proposition 3.4 (1). We would also like to express our hearty thanks to G. Stevens for his helpful suggestion when we studied the example (the case d = 193) in §3.3.

1 Iwasawa theory of an elliptic curve with supersingular reduction 1.1. ±-Coleman homomorphisms. Kobayashi defined in [12] §8 ±-Coleman homomorphisms. We will give here a slightly different construction of these homomorphisms using the results of the first author in [13]. Suppose that E has good supersingular reduction at an odd prime p with ap = p + 1 − #E(Fp ) = 0. We denote by T = Tp (E) the Tate module, and set V = T ⊗ Qp . For n ≥ 0, let Qp,n denote the intermediate field

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of the cyclotomic Zp -extension Qp,∞ /Qp of the p-adic field Qp such that [Qp,n : Qp ] = pn . We set Λ = Zp [[Gal(Q∞ /Q)]] = Zp [[Gal(Qp,∞ /Qp )]], and identify Λ with Zp [[T ]] by identifying γ with 1 + T . We put H1loc = lim H 1 (Qp,n , T ) ←

where the limit is taken with respect to the corestriction maps. We will define two Λ-homomorphisms Col± : H1loc −→ Λ. Let D = DdR (V ) be the Dieudonn´e module which is a two dimensional Qp -vector space. Let ωE be the N´eron differential which we regard as an element of D. Since D is isomorphic to the crystalline cohomology space 1 Hcris (E mod p/Qp ), the Frobenius operator ϕ acts on D and satisfies ϕ−2 − ap ϕ−1 + p = ϕ−2 + p = 0. We take a generator (ζpn ) of Zp (1); namely, ζpn is a primitive pn -th root of unity, and ζppn+1 = ζpn for any n ≥ 1. For n ≥ 1 and x ∈ D, put γn (x) =

n−1

ϕi−n (x) ⊗ ζpn−i + (1 − ϕ)−1 (x) ∈ D ⊗ Qp (μpn ).

i=0

Putting Gn+1 = Gal(Qp (μpn+1 )/Qp ), we define Pn : H 1 (Qp (μpn+1 ), T ) −→ Qp [Gn+1 ] by setting Pn (z) := 1 TrQp (μpn+1 )/Qp ([γn+1 (ϕn+2 (ωE ))σ , exp∗ (z)])σ. [ϕ(ωE ), ωE ] σ∈Gn+1

Here exp∗ : H 1 (Qp (μpn+1 ), T ) −→ D ⊗ Qp (μpn+1 ) is the dual exponential map of Bloch and Kato (which is the dual of the exponential map: D ⊗ Qp (μpn+1 ) −→ H 1 (Qp (μpn+1 ), V )), and [x, y] ∈ D ⊗ Qp (μpn+1 ) is the cup product of the de Rham cohomology for x, y ∈ D ⊗ Qp (μpn+1 ). In this, [ϕ(ωE ), ωE ] ∈ Zp plays the role of a p-adic period. By Proposition 3.6 in [13], we have Pn (z) ∈ Zp [Gn+1 ] (note that we slightly changed the notation γn , Pn from [13]).

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Put Gn = Gal(Qp,n /Qp ), and let i : H 1 (Qp,n , T ) −→ H 1 (Qp (μpn+1 ), T ) and π : Zp [Gn+1 ] −→ Zp [Gn ] be the natural maps. We define Pn : H 1 (Qp,n , T ) −→ Zp [Gn ] by Pn (z) =

1 π ◦ Pn (i(z)). p−1

These elements satisfy a distribution property; namely, we have πn,n−1 Pn (z) = −νn−2,n−1 Pn−2 (Nn,n−2 (z)) where πn,n−1 : Zp [Gn ] −→ Zp [Gn−1 ] is the natural projection, νn−2,n−1 : Zp [Gn−2 ] −→ Zp [Gn−1 ] is the norm map such that σ → Στ (for σ ∈ Gn−2 , τ ranges over elements in Gn−1 such that πn−1,n−2 (τ ) = σ), and Nn,n−2 : H 1 (Qp,n , T ) −→ H 1 (Qp,n−2 , T ) is the corestriction map. This relation can be proved by showing ψ(πn,n−1 Pn (z)) = ψ(−νn−2,n−1 Pn−2 (Nn,n−2 (z))) for any character ψ of Gn−1 (cf. the proof of Lemma 7.2 in [13]). Our identification of γ with 1 + T , gives an identification of Zp [Gn ] with n m Zp [T ]/((1+T )p −1). Set ωm = (1+T )p −1, and Φm = ωm /ωm−1 which is the pm -th cyclotomic polynomial evaluated at 1 + T . The above distribution relation implies that Φn−1 divides Pn (z). By induction on n, we can show that Φn−1 Φn−3 · ... · Φ1 divides Pn (z) if n is even, and Φn−1 Φn−3 · ... · Φ2 divides Pn (z) if n is odd. Put ωn+ = Φm , ωn− = Φm . 2≤m≤n,2|m

1≤m≤n,2m

Suppose that z is an element in H1loc , and zn ∈ H 1 (Qp,n , T ) is its image. Suppose at first n is odd, and write Pn (zn ) = ωn+ hn (T ) with hn (T ) ∈ Zp [T ]/(ωn ). Then, hn (T ) is uniquely determined in Zp [T ]/(T ωn− ) because ωn+ ωn− T = ωn ; so we regard hn (T ) as an element in this ring. By the above distribution property, we know that ((−1)(n+1)/2 hn (T ))n:odd≥1 is a projective − system with respect to the natural maps Zp [T ]/(T ωn+2 ) −→ Zp [T ]/(T ωn− ). − Hence, it defines an element h(T ) ∈ lim Zp [T ]/(T ωn ) = Zp [[T ]] = Λ. We ←

define Col+ (z) = h(T ). Next, suppose n is even. We write Pn (zn ) = ωn− kn (T ) with kn (T ) ∈ Zp [T ]/(T ωn+ ). By the same method as above, the distribution property implies that ((−1)(n+2)/2 kn (T ))n:even≥1 is a projective system, so it defines k(T ) ∈ lim Zp [T ]/(T ωn+ ) = Zp [[T ]] = Λ. We define Col− (z) = k(T ). Thus, we have ←

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obtained two power series from z ∈ H1loc . We define Col : H1loc −→ Λ ⊕ Λ by Col(z) = (Col+ (z), Col− (z)) = (h(T ), k(T )). The next lemma will be useful in what follows. Lemma 1.1 Suppose z ∈ H1loc , Col+ (z) = h(T ), and Col− (z) = k(T ). Then, h(0) =

p(p − 1) exp∗ (z0 ) 2p exp∗ (z0 ) and k(0) = p+1 ωE p + 1 ωE

where z0 is the image of z in H 1 (Qp , T ), and exp∗ (z0 )/ωE is the element a ∈ Qp such that exp∗ (z0 ) = aωE . We note that exp∗ (z0 )/ωE is known to be in p−1 Zp (cf. [23] Proposition 5.2), hence the right hand side of the above formula is in Zp . Proof. This follows from the construction of Col± (z) and Lemma 3.5 in [13] (cf. the proof of Lemma 7.2 in [13], pg. 220). 1.2. An exact sequence. We have defined Col = Col+ ⊕ Col− : H1loc −→ Λ ⊕ Λ. This homomorphism induces Proposition 1.2 We have an exact sequence Col

ρ

0 −→ H1loc −→ Λ ⊕ Λ −→ Zp −→ 0 where ρ is the map defined by ρ(h(T ), k(T )) = h(0) −

p−1 2 k(0).

Proof. First of all, we note that H1loc is a free Λ-module of rank 2. In fact, since H 0 (Qp,∞ , E[p]) = 0, it follows that

H 1 (Qp , E[p∞ ]) −→ H 1 (Qp,∞ , E[p∞ ])Gal(Qp,∞ /Qp ) is bijective. Taking the dual, we get an isomorphism (H1loc )Gal(Qp,∞ /Qp ) H 1 (Qp , T ). Since H 0 (Qp , E[p]) = H 2 (Qp , E[p]) = 0, H 1 (Qp , T ) is a free Zp -module of rank 2, and so H1loc is a free Λ-module of rank 2. By Lemma 1.1, we know ρ ◦ Col = 0. Hence, to prove Proposition 1.2, it suffices to show that the cokernel of Col is isomorphic to Zp . Kobayashi defined two subgroups E ± (Qp,n ) of E(Qp,n ) ⊗ Zp ([12] Definition 8.16). We will explain these subgroups in a slightly different way (this idea is due to R. Greenberg). Since E(Qp,n ) ⊗ Qp is the regular representation

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309

n of Gn , it decomposes into i=0 Vi where the Vi ’s are irreducible representations such that dimQp V0 = 1, and dimQp Vi = pi−1 (p − 1) for i > 0. Then E + (Qp,n ) (resp. E − (Qp,n )) is defined to be the subgroup consisting of all points P ∈ E(Qp,n ) ⊗ Zp such that the image of P in Vi is zero for every odd i (resp. for every positive even i). We define E ± (Qp,∞ ) as the direct limit of E ± (Qp,n ). By definition, the sequence 0 −→ E(Qp ) ⊗ Qp /Zp −→ E + (Qp,∞ ) ⊗ Qp /Zp ⊕ E − (Qp,∞ ) ⊗ Qp /Zp −→ E(Qp,∞ ) ⊗ Qp /Zp −→ 0 is exact. Since p is supersingular, E(Qp,∞ ) ⊗ Qp /Zp = H 1 (Qp,∞ , E[p∞ ]). We also know that E ± (Qp,∞ ) ⊗ Qp /Zp is the exact annihilator of the kernel of Col± with respect to the cup product ([12] Proposition 8.18). Hence, taking the dual of the above exact sequence, we get Coker(Col) (E(Qp ) ⊗ Qp /Zp )∨ Zp , which completes the proof. 1.3. p-adic L-functions and Kato’s zeta elements. We now consider global cohomology groups. For n ≥ 0, let Qn denote the intermediate field of Q∞ /Q with degree pn . We define 1 H1glob = lim H 1 (Qn , T ) = lim Het (OQn [1/S], T ) ←

←

where the limit is taken with respect to the corestriction maps, and S is the product of the primes of bad reduction and p. The image of z ∈ H1glob in H1loc we continue to denote by z. In our situation, H1glob was proved to be a free Λ-module of rank 1 (Kato [9] Theorem 12.4). Kato constructed an element zK = ((zK )n )n≥0 ∈ H1glob with the following properties [9]. For a faithful character ψ of Gn = Gal(Qn /Q) with n > 0,

ψ(σ) exp∗ (σ(zK )n ) = ωE

σ∈Gn

L(E, ψ, 1) , ΩE

and exp∗ ((zK )0 ) = ωE (1 − ap p−1 + p−1 )

L(E, 1) ΩE

where exp∗ : H 1 (Qp,n , T ) −→ D ⊗ Qp,n is the dual exponential map. Suppose that θQn ∈ Zp [Gn ] is the modular element of Mazur and Tate [15], which satisfies the distribution property πn,n−1 θQn = −νn−2,n−1 θQn−2 , and the property that for a faithful character ψ of Gn = Gal(Qn /Q) with n > 0, ψ(θQn ) = τ (ψ)

L(E, ψ −1 , 1) ΩE

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where τ (ψ) is the Gauss sum, and ψ : Zp [Gn ] −→ Zp [Image ψ] is the ring homomorphism induced by ψ. Let α and β be two roots of t2 +p = 0. We have two p-adic L-functions Lp,α √ and Lp,β by Amice-V´elu and Vishik, which are in H∞ ⊗ Qp ( −p) where ∞ 2 n −h H∞ = an T ∈ Qp [[T ]]; lim |an |p n = 0 for some h ∈ Z>0 . n→∞

n=0

As the second author proved in [20], there are two Iwasawa functions f (T ) and g(T ) in Zp [[T ]] such that Lp,α (T ) = f (T ) log+ (T ) + αg(T ) log− (T )

(1.1)

Lp,β (T ) = f (T ) log+ (T ) + βg(T ) log− (T )

(1.2)

and where log+ (T ) = p−1 Πn>0 1p Φ2n (T ) and log− (T ) = p−1 Πn>0 p1 Φ2n−1 (T ). Let Pn be as in §1.1, and zK = ((zK )n ) be the zeta element of Kato. By Lemma 7.2 in [13] by the first author, we have Pn ((zK )n ) = θQn

(1.3)

(we note that we need no assumption (for example on L(E, 1)) to get (1.3)). Since we know that Lp,α is also obtained as the limit of α−n−1 (θQn − α−1 νn−1,n θQn−1 ), using (1.3), we obtain Lp,α (T ) = Col+ (zK ) log+ (T ) + αCol− (zK ) log− (T ). Comparing this formula with (1.1), we have proved Theorem 1.3 (Kobayashi [12] Theorem 6.3) Let zK be Kato’s zeta element, and f , g be the Iwasawa functions as in (1.1). Then, we have Col+ (zK ) = f (T ) and Col− (zK ) = g(T ). 1.4. Proofs of Theorems 0.3 and 0.4. We begin by proving Theorem 0.3. For a non-zero element z in H1glob , we write Col(z) = (hz (T ), kz (T )). Since H1glob is free of rank 1 over Λ ([9] Theorem 12.4), hz (T )/kz (T ) does not depend on the choice of z ∈ H1glob . So choosing z = zK , we have by Theorem 1.3, that hz (T )/kz (T ) = f (T )/g(T ) for all non-zero z ∈ H1glob . Let ξ be a generator of H1glob . Suppose that hξ (0) = 0. By Proposition 1.2, ρ(Col(ξ)) = 0, so we get kξ (0) = 0 and hξ (0)/kξ (0) = (f /g)(0) = (p − 1)/2. This contradicts our assumption. Hence, hξ (0) = 0. This implies

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that the image ξQp of ξ in H 1 (Qp , T ) satisfies exp∗ (ξQp ) = 0 by Lemma 1.1, so ξQp is in E(Qp ) ⊗ Zp . Hence, the image ξQ of ξ in H 1 (Q, T ) is in Sel(E/Q, T ) which is the Selmer group of E/Q with respect to T . Since H1glob = H 1 (Q, Λ ⊗ T ), from an exact sequence 0 → Λ ⊗ T → Λ ⊗ T → T → 0, the natural map (H1glob )Gal(Q∞ /Q) −→ H 1 (Q, T ) is injective ((H1glob )Gal(Q∞ /Q) is the Gal(Q∞ /Q)-coinvariants of H1glob ). Thus, ξQ ∈ Sel(E/Q, T ) is of infinite order. Therefore, #Sel(E/Q, T ) = ∞, which implies #Sel(E/Q)p∞ = ∞ and establishes Theorem 0.3. Next, we introduce condition (∗)0 . We consider the composite H1glob −→ H (Q, T ) −→ H 1 (Qp , T ) of natural maps, and the property 1

(∗)0

H1glob −→ H 1 (Qp , T )

is not the zero map.

This property (∗)0 should always be true. In fact, it is a consequence of the p-adic Birch and Swinnerton-Dyer conjecture. More precisely, it follows from a conjecture that a certain p-adic height pairing is non-degenerate (see PerrinRiou [17] pg. 979 Conjecture 3.3.7 B and Remarque iii)). We now prove Theorem 0.4. As we saw in the proof of Proposition 1.2, H1loc is a free Λ-module of rank 2. The p-adic rational points E(Qp ) ⊗ Zp is a direct summand of H 1 (Qp , T ); hence, we can take a basis e1 , e2 of H1loc such that the image e01 of e1 in H 1 (Qp , T ) is not in E(Qp ) ⊗ Zp , and the image e02 of e2 in H 1 (Qp , T ) generates E(Qp ) ⊗ Zp . Since e01 is not in E(Qp ) ⊗ Zp , exp∗ (e01 ) = 0, and, by Lemma 1.1, Col+ (e1 )(0) = 0, and Col− (e1 )(0) = 0. Since e02 is in E(Qp ) ⊗ Zp , by Lemma 1.1, Col+ (e2 )(0) = Col− (e2 )(0) = 0. We also have that Col+ (e2 ) (0) = 0 and Col− (e2 ) (0) = 0. This follows from the fact that the determinant of the Λ-homomorphism Col : H1loc −→ Λ ⊕ Λ is T modulo units by Proposition 1.2. We also note that Col+ (e2 ) (0) − p−1 − + − 2 Col (e2 ) (0) = 0. Indeed, since (Col (e1 )/Col (e1 ))(0) = (p − 1)/2, + − if we had (Col (e2 )/Col (e2 ))(0) = (p − 1)/2, we would have Image(Col) ∩ T (Λ ⊕ Λ) ⊂ ((p − 1)/2, 1)T Λ + T 2 (Λ ⊕ Λ), which contradicts Proposition 1.2. Now we assume (∗)0 and rankE(Q) > 0. Let ξ, hξ (T ), kξ (T ), ... be as in the proof of Theorem 0.3. We write ξ = a(T )e1 + b(T )e2 with a(T ), b(T ) ∈ Λ. Since rankE(Q) > 0, E(Q) ⊗ Qp −→ E(Qp ) ⊗ Qp is surjective. So the image of H 1 (Q, T ) −→ H 1 (Qp , T ) is in E(Qp ) ⊗ Zp by Lemma 1.4 below which we will prove later. Therefore, a(0) = 0. Hence, (∗)0 implies

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that b(0) = 0. Thus, we get hξ (0) − Hence,

p−1 p−1 kξ (0) = b(0) Col+ (e2 ) (0) − Col− (e2 ) (0) = 0. 2 2 hξ (0) f (T ) "" p−1 = = , " g(T ) T =0 kξ (0) 2

which completes the proof. Our last task in this subsection is to prove the following well-known property. Lemma 1.4 . Let V = T ⊗ Q. The image of H 1 (Q, V ) −→ H 1 (Qp , V ) is a one dimensional Qp -vector space. Proof of Lemma 1.4. We first note that H 1 (Q, V ) = H 1 (Z[1/S], V ) where S is the product of the primes of bad reduction and p. Since V is self-dual, by the Tate-Poitou duality we have an exact sequence i∨

H 1 (Q, V ) −→ H 1 (Qp , V ) −→ H 1 (Q, V )∨ i

where i∨ : H 1 (Qp , V ) = H 1 (Qp , V )∨ −→ H 1 (Q, V )∨ is obtained as the dual of i : H 1 (Q, V ) −→ H 1 (Qp , V ). This shows that dim(Image(i)) = dim(Image(i∨ )) = dim(Coker(i)). Thus, we obtain dim(Image(i)) = 1 from dim(H 1 (Qp , V )) = 2. 1.5. Main Conjecture. We review the main conjectures in our case. We define Sel± (E/Q∞ ) to be the Selmer group which is defined by replacing the local condition at p with E ± (Qp,∞ ) ⊗ Qp /Zp (see the proof of Proposition 1.2). Then, the main conjecture is formulated as char(Sel+ (E/Q∞ )∨ ) = (g(T )) and char(Sel− (E/Q∞ )∨ ) = (f (T )) where (∗)∨ is the Pontryagin dual (Kobayashi [12]). These two conjectures are equivalent to each other ([12] Theorem 7.4) and, furthermore, each of them is equivalent to char(Sel0 (E/Q∞ )∨ ) = char(H1glob / < zK >)

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where Sel0 (E/Q∞ ) is defined as in 0.3. The inclusion ⊃ is proved by using Kato’s result [9] up to μ-invariants (Kobayashi [12] Theorem 1.3). We give here a corollary of Corollary 0.6 in §0.1. Proposition 1.5 Assume that L(E, 1) = 0, μ(f (T )) = μ(g(T )) = 0, and 1 = min{λ(f (T )), λ(g(T ))} < max{λ(f (T )), λ(g(T ))}. Then, the Iwasawa main conjecture for E is true; namely, char(Sel+ (E/Q∞ )∨ ) = (g(T )) and char(Sel− (E/Q∞ )∨ ) = (f (T )). Proof. Corollary 0.6 implies that Sel(E/Q)p∞ is infinite. Hence, by the control theorem for Sel± (E/Q∞ ) ([12] Theorem 9.3), T divides the characteristic power series of Sel± (E/Q∞ )∨ , and thus, T divides f (T ) and g(T ). Then, by our assumption, one of (f (T )) or (g(T )) equals (T ). Hence, the main conjecture for one of Sel+ (E/Q∞ ) or Sel− (E/Q∞ ) holds. This implies that both statements are true ([12] Theorem 7.4). As an example, we consider the quadratic twist E = X0 (17)d as in §0.1. Then, the condition in Proposition 1.5 is satisfied by X0 (17)d for d = 29, 37, 40, 41, 44, 56, 65, . . . . For example, when d = 37, we have char(Sel+ (E/Q∞ )∨ ) = (T ) and char(Sel− (E/Q∞ )∨ ) = ((1 + T )3 − 1). Hence, if we assume the finiteness of X(E/Q(cos 2π/9))[3∞ ], we know that rankE(Q) = 1 and rankE(Q(cos 2π/9)) = 3. Note that we get this conclusion just from analytic information (the computation of modular symbols). 1.6. Remark. We consider in this subsection a more general case. We assume that E has good reduction at an odd prime p. If p is ordinary, we assume that E does not have complex multiplication. Let α, β be the two roots of x2 − ap x + p = 0 in an algebraic closure of Qp where ap = p + 1 − #E(Fp ). If p is ordinary, we take α to be a unit in Zp as usual. If p is a supersingular prime, we have two p-adic L-functions Lp,α and Lp,β by Amice-V´elu and Vishik. If p is ordinary, Lp,α is the p-adic L-function of Mazur and SwinnertonDyer. The other function Lp,β is defined in the following way. Since E does not have complex multiplication, if p is ordinary, we can write ωE = eα + eβ in D ⊗ Qp (α) where eα (resp. eβ ) is an eigenvector of the Frobenius operator ϕ corresponding to the eigenvalue α−1 (resp. β −1 ). Perrin-Riou constructed a map which interpolates the dual exponential map (cf. [18] Theorem 3.2.3,

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[9] Theorem 16.4) H1loc ⊗Λ H∞ −→ D ⊗Qp H∞ . The image of Kato’s element zK can be written as Lp,α eα + Lp,β eβ if p is supersingular, and the eα component of the image of zK is Lp,α in the ordinary case. We simply define Lp,β by zK → Lp,α eα + Lp,β eβ in the ordinary case. Set r = min{ordT =0 Lp,α , ordT =0 Lp,β }. We conjecture that if r > 0, (r)

(r) Lp,β (0) Lp,α (0) 1 2 = (1 − α ) (1 − β1 )2

(1.4)

which is equivalent to Conjecture 0.2 in the case ap = 0. Note that if r = 0, we have Lp,α (0) Lp,β (0) L(E, 1) . 1 2 = 1 2 = ΩE (1 − α ) (1 − β ) So the above conjecture (1.4) asserts that Lp,α (0)/(1 − α−1 )2 = Lp,β (0)/ (1 − β −1 )2 if and only if r = 0. We remark that the p-adic Birch and Swinnerton-Dyer conjecture would imply r = ordT =0 Lp,α = ordT =0 Lp,β , but if ordT =0 Lp,α = ordT =0 Lp,β , then Conjecture (1.4) would follow automatically. The p-adic Birch and Swinnerton-Dyer conjecture predicts (r)

d ds

(r)

r 2 " 1 #X(E/Q) Tam(E/Q) Lp,α (κ(γ)s−1 − 1)"s=1 = 1 − Rp,α α (#E(Q)tor )2

(cf. Colmez [4]) where κ : Gal(Q∞ /Q) −→ Z× p is the cyclotomic character, and Rp,α (resp. Rp,β ) is the p-adic α-regulator (β-regulator) of E. Hence, if we admit this conjecture, Conjecture (1.4) means that Rp,α = Rp,β . (r) (r) We now establish that Lp,α (0)/(1 − α−1 )2 = Lp,β (0)/(1 − β −1 )2 implies that #Sel(E/Q)p∞ = ∞. Namely, Conjecture (1.4) implies Conjecture 0.1. We know (H1glob )Q = H1glob ⊗Q is a free ΛQ = Λ⊗Q-module of rank 1 (Kato [9] Theorem 12.4). Suppose that T s divides Kato’s element zK and T s+1 does not divide zK in (H1glob )Q . We denote by ξi the image of zK /T i in H 1 (Q, V ) for i = 1, ..., s. Clearly, ξ1 = ... = ξs−1 = 0, and ξs = 0 because the natural map (H1glob )Gal(Q∞ /Q) ⊗ Q −→ H 1 (Q, V ) is injective ((H1glob )Gal(Q∞ /Q) is the Gal(Q∞ /Q)-coinvariants of H1glob ). We assume Lp,α (0)/(1−α−1 )2 = (r)

Lp,β (0)/(1 − β −1 )2 , which implies that L(E, 1) = 0. We will show that the image of ξs in H 1 (Qp , V ) is in Hf1 (Qp , V ) = E(Qp ) ⊗ Qp . Put D 0 = Qp ωE (r)

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and −1 −2 L(i) = (1 − α−1 )−2 L(i) ) Lp,β (0)eβ ∈ D. p,α (0)eα + (1 − β (i)

We know that the image of ξi by exp∗ is L(i) times a non-zero element, and if ξi is in Hf1 (Qp , V ), the image of ξi by log : Hf1 (Qp , V ) −→ D/D0 is L(i + 1) times a non-zero element modulo D0 by Perrin-Riou’s formulas [17] Propositions 2.1.4 and 2.2.2. (Note that Conjecture R´ec(V) in [17] was proved by Colmez [3].) Suppose that r ≤ s. By our assumption, L(r) is not in D 0 , hence log(ξr−1 ) is not in D0 . Thus, ξr−1 is a non-zero element in H 1 (Qp , V ). But this is a contradiction because ξr−1 = 0 in H 1 (Q, V ) by the definition of s. Thus, we have r > s. This implies that L(s) = 0, and hence, exp∗ (ξs ) = 0. So ξs is in Hf1 (Qp , V ). Therefore, ξs is in the Selmer group Sel(E/Q, V ) with respect to V , and #Sel(E/Q)p∞ = ∞. This can be also obtained from Proposition 4.10 in Perrin-Riou [19]. Next, we will show that rankE(Q) > 0 and (∗)0 imply (1.4). These conditions imply that ξs is in Hf1 (Qp , V ) (by Lemma 1.4), and also is non-zero. Therefore, L(s + 1) is not in D0 . This shows that r = s + 1 and (1.4) holds. 2 Constructing a rational point in E(Q) We saw in the previous section that the value (f /g)(0)−(p−1)/2 is important to understand the Selmer group. In this section, we consider the case r = ordT =0 f (T ) = ordT =0 g(T ) = 1. We will see that the computation of the value f (0) − p−1 2 g (0) helps to produce a rational point in E(Q) numerically. To do this, we have to compute the value f (0) − p−1 g (0) to high accuracy, 2 which we do using the theory of overconvergent modular symbols. 2.1. Overconvergent modular symbols. Let Δ0 denote the space of degree zero divisors on P1 (Q) which naturally are a left GL2 (Q)-module under linear # a b $ fractional transformations. Let Σ0 (p) be the semigroup of matrices c d ∈ M2 (Zp ) such that p divides c, gcd(a, p) = 1 and ad − bc = 0. If V is some right Zp [Σ0 (p)]-module, then the space Hom(Δ0 , V ) is naturally a right Σ0 (p)-module by " " (ϕ"γ)(D) = ϕ(γD)"γ. For a congruence subgroup Γ ⊂ Γ0 (p) ⊂ SL2 (Z), we set " 6 7 SymbΓ (V ) = ϕ ∈ Hom(Δ0 , V ) : ϕ"γ = ϕ , the subspace of Γ-invariant maps which we refer to as the space of V -valued modular symbols of level Γ.

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We note that this space is naturally a Hecke module.#For instance, Up is $ p−1 # $ 0 defined by a=1 10 ap . Also, the action of the matrix −1 decomposes 0 1 SymbΓ (V ) into plus/minus subspaces SymbΓ (V )± . If we take V = Qp , SymbΓ (Qp ) is the classical space of modular symbols of level Γ over Qp . By Eichler-Shimura theory, each eigenform f over Qp ± of level Γ gives rise to an eigensymbol ϕ± f in SymbΓ (Qp ) with the same Hecke-eigenvalues as f . Let D(Zp ) denote the space of (locally analytic) distributions on Zp . Then D(Zp ) inherits a right Σ0 (p)-action defined by: b + dx (μ|γ)(f (x)) = μ f . a + cx Then SymbΓ (D(Zp )) is Steven’s space of overconvergent modular symbols. This space admits a Hecke-equivariant map to the space of classical modular symbols ρ : SymbΓ (D(Zp )) −→ SymbΓ (Qp ) by taking total measure. That is, ρ(Φ)(D) = Φ(D)(1Zp ). We refer to this map as the specialization map. Theorem 2.1 (Stevens) The operator Up is completely continuous on SymbΓ (D(Zp )). Moreover, the Hecke-equivariant map ∼

ρ : SymbΓ (D(Zp ))(<1) −→ SymbΓ (Qp )(<1) is an isomorphism. Here the superscript (< 1) refers to the subspace where Up acts with slope less than 1. See [26] Theorem 7.1 for a proof of this theorem. 2.2. Connection to p-adic L-functions. Now consider an elliptic curve E/Q of level N with good supersingular reduction at p. By the Modularity theorem, there is some modular form f = fE on Γ0 (N ) corresponding to E. If α is a root of x2 − ap x + p = 0, let fα (τ ) = f (τ ) + αf (pτ ) denote the p-stabilization of f to level Γ0 (N p). By Eichler-Shimura theory, there exists + a Hecke-eigensymbol ϕf, α = ϕ+ f, α ∈ SymbΓ (Qp ) ⊗ Qp (α) with the same Hecke-eigenvalues as fα . Explicitly, we have r −r ϕ+ ({r} − {s}) = πi f + fα Ω−1 α E f, α s

where ΩE is the N´eron period of E/Q.

−s

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By Stevens’ comparison theorem (Theorem 2.1), there exists a unique over+ convergent Hecke-eigensymbol Φα = Φ+ α ∈ " SymbΓ (D(Zp )) ⊗ Qp (α) such " that ρ(Φα ) = ϕf, α . (Note that since ϕf, α Up = α · ϕf, α , the symbol ϕf, α has slope 1/2 and the theorem applies.) The overconvergent symbol Φα is intimately connected to the p-adic L-function of E. Indeed, Φα ({0} − {∞}) = Lp,α (E),

(2.1)

the p-adic L-function of E viewed as a (locally analytic) distribution on Z× p. " To verify this, one uses the fact that Φα lifts ϕf, α , that Φα "Up = α · Φα and that, by definition, ' ( a Lp,α (E)(1a+pn Zp ) = α1n · ϕf, α − {∞} . n p See [26] Theorem 8.3. 2.3. Computing p-adic L-functions. As in [21] and [6], one can use the theory of overconvergent modular symbols to very efficiently compute the p- adic L-function of an elliptic curve. Indeed, by (2.1), it suffices to compute the corresponding overconvergent modular symbol Φα . To do this, we first lift ϕf, α to any overconvergent symbol Φ (not necessarily a Hecke-eigensymbol). Then, using Theorem 2.1, one can verify that " the sequence {α−n Φ"Upn } converges to Φα . Thus, as long as we can efficiently compute Up on spaces of overconvergent modular symbols, we can form approximations to the symbol Φα . To actually perform such a computation, one must work modulo various powers of p and thus we must make a careful look at the denominators that are present. If O denotes the ring of integers of Qp (α), then ϕf, α ∈ 1 1 α SymbΓ (Zp ). (The factor of α comes about from the p-stabilization of f from level N to level N p.) Let D 0 (Zp ) denote the set of distributions whose moments are all integral; that is, D 0 (Zp ) = {μ ∈ D(Zp ) : μ(xj ) ∈ Zp }. It is then possible to lift ϕf, α to a" symbol Φ in α12 SymbΓ (D 0 (Zp )) ⊗ O. As mentioned above, {α−n Φ"Upn } converges to Φα . However, the space SymbΓ (D0 (Zp )) ⊗ O is not preserved by the operator α1 Up . However, the subspace X := " 6 7 Φ ∈ SymbΓ (D0 (Zp )) ⊗ O : ρ(Φ)"Up = αρ(Φ), ρ(Φ) ∈ αSymbΓ (O)

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is preserved by α1 Up . (Note that the two conditions defining this set are clearly preserved by this operator. The key point here is that overconvergent symbols with integral moments satisfying these two conditions will still have integral moments after applying α1 Up .) Thus, to form our desired overconvergent symbol using only symbols with integral moments, we begin with α2 ϕf, α ∈ αSymb"Γ (O) and lift this symbol to an overconvergent symbol Φ in X. Then α1n Φ "Upn is in X for all n and converges to α2 Φα . To perform this computation on a computer, we need a method of approximating an overconvergent modular symbol with a finite amount of data. Furthermore, we must ensure that our approximations are stable under the action of Σ0 (p) so that Hecke operators can be computed. A method of approximating distributions using “finite approximation modules” is given in [21] and [6] Section 2.4 which we will now describe. Consider the set F(M ) = O/pM × O/pM −1 × · · · × O/p. We then have a map −1 D0 (Zp ) ⊗ O −→ F(M ) given by μ → {μ(xj ) mod pM −j }M j=0 .

Moreover, one can check that the kernel of this map is stable under the action of Σ0 (p). This allows us to give F (M ) the structure of a Σ0 (p)-module. As F (M ) is a finite set, the space SymbΓ (F (M )) can be represented on a computer. Indeed, there is a finite set of divisors such that any modular symbol of level Γ is uniquely determined by its values on these divisors. Thus, any element of SymbΓ (F (M )) can be represented by a finite list of elements in F (M ). Moreover, since SymbΓ (D0 (Zp )) ⊗ O ∼ = lim SymbΓ (F(M )), ←

these spaces provide a natural setting to perform computations with overconvergent modular symbols. To compute the p-adic L-function of E, we fix some large integer M and consider the image of α2 ϕf, α in SymbΓ (O/pM ). One can explicitly lift this symbol to a symbol Φ in SymbΓ (F (M )). (See [21] for explicit formulas related to such liftings.) In the"ordinary case, one then proceeds by simply computing the sequence {α−n Φ "Upn } which will eventually stabilize to the image of α2 Φα in SymbΓ (F(M )).

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However, in the supersingular case, α is not a" unit and thus division by α causes a loss of accuracy. The symbol α−2n Φ "Up2n is naturally a symbol in F (M − n) and is congruent to α2 Φα modulo pn . By choosing M and n appropriately, one can then produce a Up -eigensymbol to any given desired accuracy. 2.4. Twists. Let χd denote the quadratic character of conductor d and let Ed be the quadratic twist of E by χd . Let α∗ = χd (p)α and let Φα be the overconvergent eigensymbol whose special value at {0} − {∞} equals Lp,α (E). If gcd(d, N p) = 1, then

Φα∗ ,d :=

|d| a=1

" " 1 χd (a) · Φα " 0

a d

1

is a Up -eigensymbol of level Γ0 (d2 N p) with eigenvalue α∗ . Moreover, if d > 0, then Φα∗ ,d ({0} − {∞}) = Lp,α∗ (Ed ). In particular, once we have constructed an eigensymbol that computes Lp,α (E), we can use this symbol to compute the p-adic L-function of all real quadratic twists of E. (To compute imaginary quadratic twists one needs to instead use the overconvergent symbol that corresponds to ϕ− f,α .) 2.5. Computing derivatives of p-adic L-functions. Once the image of Φα has been computed in SymbΓ (F(M )) for some M , by evaluating at D = {0} − {∞} we can recover the first M moments of the p-adic L-function modulo certain powers of p. As in [6] pg. 17, we can also recover (x − {a})j dLp,α (E) mod pM a+pZp

for 0 ≤ j ≤ M − 1 where {a} denotes the Teichm¨uller lift of a. Using these values one can compute the derivative of this p-adic L-function. Indeed, if we let Lp,α (E, s) be the function of Lp,α (E) in the s-variable, then " " Lp,α (E, s)"

s=1

= =

Z× p

logp (x) dLp,α (E)

p−1 a=1

logp (x/{a}) dLp,α (E) a+pZp

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=

p−1 a=1

=

∞ (−1)j+1

a+pZp j=1

p−1 ∞ a=1 j=1

j

(−1)j+1 {a}−j j

(x/{a} − 1)j dLp,α (E)

(x − {a})j dLp,α (E). a+pZp

;

Since a+pZp (x − {a})j dLp,α (E) is divisible by pj , by calculating the above expression for j between 0 and M − 1, one obtains an approximation to Lp,α (E, 1) which is correct modulo pM −r where pr ≤ M < pr+1 . Let Lp,α (T ) be the p-adic L-function in the T -variable as in §1. Then Lp,α (κ(γ)s−1 − 1) = Lp,α (E, s), and Lp,α (T ) = f (T ) log+ (T )+ g(T ) log− (T )α. Hence, " " " " Lp,α (E, s)" = Lp,α (T )" · log(κ(γ)) s=1 T =0 # $ = f (0) log+ (0) + g (0) log− (0)α · log(κ(γ)) 1 = · (f (0) + g (0)α) · log(κ(γ)). (2.2) p So from the computation of the derivative of Lp,α (E, s), we can easily compute the values of f (0) and g (0). 2.6. p-adic Birch and Swinnerton-Dyer conjecture. We now give a precise statement of the p-adic Birch and Swinnerton-Dyer conjecture as in Bernardi and Perrin-Riou [1]. As before, E/Q is an elliptic curve for which p is a good supersingular prime with ap = 0. If r is the order of Lp,α (E/Q, s) at s = 1, then this conjecture asserts that r = rankE(Q) and 2 " 1 (r) 1 Lp,α (E/Q, s)"s=1 = 1 − Cp (E/Q) · Rp,α (E/Q) (2.3) r! α where Rp,α (E/Q) is the p-adic α-regulator of E/Q, and Cp (E/Q) =

#X(E/Q) · Tam(E/Q) (#E(Q)tor )2

(cf. also [4], [19]). In the case r = 1, we have 1 hωE (P ) Rp,α (E/Q) = − · hϕ(ωE ) (P ) + α [ϕ(ωE ), ωE ] p

(2.4)

where P is some generator of E(Q)/E(Q)tor , and ωE , ϕ, [ϕ(ωE ), ωE ] are as in §1. Here, for each ν ∈ D = DdR (Vp (E)), there is a p-adic height function hν : E(Q) −→ Qp attached to ν. If ν = ωE and if P is in the kernel of

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ˆ p ) −→ reduction modulo p, then hωE (P ) = − logEˆ (P )2 where logEˆ : E(Q ˆ Qp is the logarithm of the formal group E/Qp attached to E. Assuming the p-adic Birch and Swinnerton-Dyer conjecture (with r = 1), equations (2.2), (2.3) and (2.4) yield −2 1 1− (f (0) + g (0)α) log(κ(γ)) α p(p − 1) − 2pα = (f (0) + g (0)α) log(κ(γ)) (p + 1)2 p hω (P ) =− · hϕ(ωE ) (P ) + E α Cp (E/Q). [ϕ(ωE ), ωE ] p Equating α-coefficients of both sides gives 2p log(κ(γ)) p−1 hω (P ) · Cp (E/Q) · f (0) − g (0) = E . (p + 1)2 2 [ϕ(ωE ), ωE ] ˆ p ) = E 1 (Qp ) is the kernel of the reduction map E(Qp ) −→ Since E(Q ˆ p )) = #E(Fp ) = p + 1, and (p + 1)P is in E(Q ˆ p ). E(Fp ), (E(Qp ) : E(Q The above formula can be written as 2

logEˆ ((p + 1)P ) = −

f (0) − p−1 2 g (0) 2p log(κ(γ))[ϕ(ωE ), ωE ]. Cp (E/Q)

(2.5)

Note that the fact that we have two p-adic L-functions which are computable, gives us the advantage that logEˆ ((p + 1)P ) is expressed by computable terms. We apply this formula to a quadratic twist of E, and have a twisted version of this equation. Let Lp,α∗ (Ed , T ) be the function in the T -variable corresponding to Lp,α∗ (Ed ) defined in 2.4. We assume that the rank of Ed (Q) equals 1 and Pd is a generator of the Mordell-Weil group modulo torsion. Also, we define fd (T ) and gd (T ) by Lp,α∗ (Ed , T ) = fd (T ) log+ (T ) + gd (T ) log− (T )α∗ . Then, we have a twisted equation # $ ηd2 fd (0) − p−1 gd (0) 2 2 logEˆd ((p + 1)Pd ) = − 2p log(κ(γ))[ϕ(ωE ), ωE ]. d · Cp (Ed /Q) (2.6) ηd √ Here, ηd is defined by ωEd = d ωE . (Note that ηd is written down explicitly in [1] pg. 230. In most cases, ηd = 1.) 2.7. Computing rational points. Using equation (2.5) we can attempt to use the p-adic L-function to p-adically compute global points on E(Q). Suppose

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that we do not know X(E/Q). We first compute zp (E) = ⎛> # f (0) − exp ˆ ⎝ −

⎞ 2p log(κ(γ))[ϕ(ωE ), ωE ] #E(Q)tor ⎠ · Tam(E) p+1 $

p−1 2 g (0)

E

(2.7) ˆ The quantity where expEˆ is the inverse of logEˆ (the formal exponential of E). in the parentheses should be in pZp because of the p-adic Birch and Swinnerton -Dyer conjecture (cf. (2.5)), so the right hand side should converge. If x ˆ(t), yˆ(t) ˆ p , then represent the formal x and y coordinate functions of E/Q P˜ := (ˆ x(zp (E)), yˆ(zp (E))) ˆ p ). The p-adic Birch and Swinnerton-Dyer conjecture (2.5) is a point in E(Q predicts that there is a global point P ∈ E(Q) such that (p + 1)P = (p + 1)P˜ in E(Qp ). % (The point P should be #X(E/Q)P in the terminology of (2.5)). Thus, there is a point Q ∈ E(Qp ) of order dividing p + 1 such that P = P˜ + Q. So we can proceed in the following way. We first compute p-adically all Q of order dividing p + 1 in E(Qp ). Then, for each such Q, we check to see if P˜ + Q appears to be a global point. 2.8. Computing in practice. We explain our method in the case of computing points on a quadratic twist of E. Considering (2.6), we define zp (E, d) := ⎛> # f (0) − exp ˆ ⎝ − d E

where τd :=

ηd #Ed (Q)tor , p+1

⎞ · 2p log(κ(γ))[ϕ(ωE ), ωE ] · τd ⎠ d Tam(Ed )

p−1 gd (0) 2

$

(2.8)

and also

P˜d := (ˆ x(zp (E, d)), yˆ(zp (E, d))). To carry out this computation, we need to first get a good p-adic approximation of zp (E, d) and thus a p-adic approximation of P˜d . Then, we compute p-adic approximations of the (p + 1)-torsion in Ed (Qp ). To find a global point, we translate P˜d around by these torsion points with the hope of finding some rational point that is very close p-adically to one of these translates.

Two p-adic L-functions and rational points

323

Fortunately, if we find a candidate global point, we can simply go back to the equation of Ed to see if the point actually sits on our curve. The key terms that need to be computed in order to determine P˜d are fd (0), gd (0), expEˆ (t), x ˆ(t), yˆ(t), Tam(Ed /Q), #Ed (Q)tor , and [ϕ(ωE ), ωE ]. The most difficult of these terms to compute are fd (0) and gd (0). We cannot use Riemann sums to approximate fd (0) and gd (0), since in practice we will need a fairly high level of p-adic accuracy in order to recognize global points. (To get n digits of p-adic accuracy, one needs to sum together approximately pn modular symbols which becomes implausible for large n.) Instead, we use the theory of overconvergent modular symbols explained above to compute fd (0) and gd (0) to high accuracy. ˆ p is For the remaining terms, computing invariants of the formal group E/Q standard. (We used the package [25].) The arithmetic invariants (Tam(Ed /Q) and #Ed (Q)tor ) of the elliptic curve Ed are easy to compute. (We used the intrinsic functions of MAGMA [14].) Lastly, an algorithm to compute [ϕ(ωE ), ωE ] is outlined in [1] pg. 232. However, we sidestepped this issue in the following way. For one particular twist Ed with X(Ed /Q) = 0, we found a generator Pd of Ed (Q)/Ed (Q)tor directly using mwrank [5]. Then, using (2.5), we can determine what the value of [ϕ(ωE ), ωE ] should be to high accuracy since every other expression in (2.5) is computable to high accuracy. (We actually repeated this computation for several different twists to make sure that the predicted value of [ϕ(ωE ), ωE ] was always the same.) Lastly, to recognize the coordinates of P˜d as rational numbers, we used the method of rational reconstruction as explained in [11] and, in practice, we used the recognition function in [7]. 2.9. The computations. We performed the above described computations for the curves X0 (17) and X0 (32) and the prime p = 3. (These computations were done on William Stein’s meccah cluster.) For X0 (17) (resp. X0 (32)) we computed the associated overconvergent symbol Φα modulo 3200 (resp. 3100 ). Because of the presence of the square root in (2.8), we then were only able to compute P˜d to 100 (resp. 50) 3-adic digits for X0 (17) (resp. X0 (32)). For X0 (17) (resp. X0 (32)), we could find a global point on all quadratic twists 0 < d < 250 except for d = 197 (resp. 0 < d < 150) with gcd(d, 3N ) = 1. (For X0 (17)197 one could find a point via several different methods, but our method would require a more accurate computation of overconvergent modular symbols.) We made a table of these points in §4.

324

Masato Kurihara and Robert Pollack

Another interesting example whose d is not in the above mentioned range is Ed = X0 (17)d with d = 328. We found a global point P = (28069/25, 3626247/125) on the curve E328 : y 2 = x3 − 73964x − 490717520 by the method which we described above. We also computed 1 “ ”zp (E, d) := 2 ⎛ > # f (0) − 1 expEˆ ⎝ − d 2

⎞ · 2p log(κ(γ))[ϕ(ωE ), ωE ] · τd ⎠ . d Tam(Ed ) $

p−1 gd (0) 2

From this local point, we produced a global point P = (1398, −46240) on the curve E328 . This reflects well the fact that #X(E/Q) = 4 in this case. The point P = (1398, −46240) should be a generator of E(Q) modulo torsion.

3 The structure of fine Selmer groups and the gcd of f (T ) and g(T ) In this section, we will study the problem mentioned in §0.3, and the greatest common divisor of f (T ) and g(T ). 3.1. Greenberg’s “conjecture” (problem). Suppose that Sel0 (E/Q∞ ), en , f (T ), g(T ), ... are as in §0.3. As we explained in §0.3, we are interested in the following problem (conjecture) by Greenberg. Problem 0.7 char(Sel0 (E/Q∞ )∨ ) = (

Φenn −1 ).

en ≥1 n≥0

First of all, since E(Qp,n ) ⊗ Qp is a one dimensional regular representation of Gn = Gal(Qn /Q), by the definition of en , Ker(E(Qn ) ⊗ Qp −→ E(Qp,n )⊗Qp ) contains (Qp [T ]/Φn (T ))en −1 if en ≥ 1. Hence, Sel0 (E/Qn ) contains ((Zp [T ]/Φn (T )) ⊗ Qp /Zp )en −1 , and we always have char(Sel0 (E/Q∞ )∨ ) ⊂ ( Φenn −1 ). en ≥1 n≥0

Two p-adic L-functions and rational points

325

Coates and Sujatha conjectured in [2] that μ(Sel0 (E/Q∞ )∨ ) = 0. In fact, ∨ they showed that μclass Q(E[p]) = 0 implies μ(Sel0 (E/Q∞ ) ) = 0 ([2] Theorem 3.4) where μclass Q(E[p]) is the classical Iwasawa μ-invariant for the class group of cyclotomic Zp -extension of Q(E[p]) which is the field obtained by adjoining all p-torsion points of E. The proof of this fact can be described simply and slightly differently from [2], so we give here the proof. Put F = Q(E[p]), and denote by F∞ /F the cyclotomic Zp -extension. By Iwasawa [8] Theorem 2, 2 μclass = 0 implies H 2 (OF∞ [1/S], Z/pZ) = Het (Spec OF∞ [1/S], Z/pZ) = F 0 (S is the product of the primes of bad reduction and p). Hence, assuming μclass = 0, we have H 2 (OF∞ [1/S], E[p]) = 0. Since the p-cohomological F dimension of Spec OF∞ [1/S] is 2, the corestriction map H 2 (OF∞ [1/S], E[p]) −→ H 2 (OQ∞ [1/S], E[p]) is surjective, so we get H 2 (OQ∞ [1/S], E[p]) = 0, which implies μ(Sel0 (E/Q∞ )∨ ) = 0. In the following, we assume p is supersingular and ap = 0. As we explained in §0.3, a generator of char(Sel0 (E/Q∞ )∨ ) divides both f (T ) and g(T ) (at least up to μ-invariants). Using this fact, we will prove what we mentioned in §0.3. Proposition 3.1 Assume that rankE(Q) = e0 , min{λ(f (T )), λ(g(T ))} = e0 , and μ(Sel0 (E/Q∞ )∨ ) = 0. Then, char(Sel0 (E/Q∞ )∨ ) = (T e0 ) where e0 = max{0, e0 − 1}, and Problem 0.7 holds. Proof. Suppose, for example, λ(f (T )) = e0 . Since one divisibility of the main conjecture was proved (cf. 1.5), this assumption implies (f (T )) = (T e0 ) as ideals of Λ. If e0 = 0, then f (T ) is a unit. Thus, Sel0 (E/Q∞ ) is finite and we get the conclusion. Hence, we may assume e0 > 0. Then, Sel− (E/Q∞ )∨ ∼ (Λ/(T ))e0 (pseudo-isomorphic), and by the control theorem for Sel− (E/Q∞ ) ([12] Theorem 9.3), Sel(E/Q)p∞ is of corank e0 . Since E(Q) ⊗ Qp −→ E(Qp ) ⊗ Qp is surjective (because e0 > 0), Sel0 (E/Q) is of corank e0 − 1. By the control theorem for Sel0 (E/Q∞ ) (cf. [13] Remark 4.4), we get Sel0 (E/Q∞ )∨ ∼ (Λ/(T ))e0 −1 , which implies the conclusion. 3.2. The gcd of f (T ) and g(T ). We use the convention that we always express the greatest common divisor of elements in Λ of the form pμ h(T ) where h(T ) is a distinguished polynomial. Concerning the greatest common divisor of f (T ) and g(T ), we propose

326

Masato Kurihara and Robert Pollack

Problem 3.2 gcd(f (T ), g(T )) = T e0

Φenn −1 .

en ≥1 n≥1

Proposition 3.3 Assume μ(Sel0 (E/Q∞ )∨ ) = 0. Then, Problem 3.2 implies Problem 0.7. We note that if we assume Problem 3.2, we have min{μ(f (T )), μ(g(T ))} = 0, hence if the Galois representation on the p-torsion points of E is surjective, μ(Sel0 (E/Q∞ )∨ ) = 0 holds by [9] Theorem 13.4. Proof of Proposition 3.3. As we explained in §0.3, Sel0 (E/Q∞ ) ⊂ Sel± (E/Q∞ ) which implies gcd(f (T ), g(T )) ∈ char(Sel0 (E/Q∞ )∨ ). So if Problem 3.2 is true, we have (T e0 Φenn −1 ) ⊂ char(Sel0 (E/Q∞ )∨ ) ⊂ ( Φenn −1 ). en ≥1 n≥1

en ≥1 n≥0

The rest of the proof is the same as that of Proposition 3.1. We may assume e0 > 0. By the control theorem for Sel± (E/Q∞ ) ([12] Theorem 9.3) and our assumption that Problem 3.2 is true, Sel(E/Q)p∞ is of corank e0 . Hence, Sel0 (E/Q) is of corank e0 − 1. Hence, if F is a generator of char(Sel0 (E/Q∞ )∨ ), by the control theorem for Sel0 (E/Q∞ ) (cf. [13] Remark 4.4), we have ordT (F ) = e0 −1. This implies that Problem 0.7 is true. Next, we will assume Problem 0.7 and deduce Problem 3.2 under certain assumptions. Let H1glob and H1loc be as in §1. We consider the natural map H1glob −→ H 1 (Qn , T ) −→ H 1 (Qp,n , T ) −→ H 1 (Qp,n , T )/(Φn ), and assume (∗)n

H1glob −→ H 1 (Qp,n , T )/(Φn )

is not the zero map

for all n ≥ 0. In particular, condition (∗)0 coincides with the condition we considered in §1.4. Proposition 3.4 We assume (∗)n for all n ≥ 0 and Problem 0.7. Then, (1) The cokernel of the natural map H1glob −→ H1loc is pseudo-isomorphic to Λ. (2) If we also assume the Main Conjecture for E (cf. 1.6) and that the pprimary component X(E/Q)[p∞ ] of the Tate-Shafarevich group of E/Q is finite, then Problem 3.2 holds.

Two p-adic L-functions and rational points

327

Proof. (1) First of all, H1glob is isomorphic to Λ ([9] Theorem 12.4), and as we saw in §1.4, H1loc is isomorphic to Λ ⊕ Λ. If we denote by (a, b) the image of a generator of H1glob Λ in H1loc Λ ⊕ Λ, Φn does not divide the gcd of a and b. Indeed, if Φn divided the gcd of a and b, the map H1glob −→ H 1 (Qp,n , T )/(Φn ) would be the zero map, which contradicts (∗)n . On the other hand, we have an exact sequence ∨ 0 −→ H1loc /H1glob −→ Sel(E/Q∞ )∨ p∞ −→ Sel0 (E/Q∞ ) −→ 0;

hence, taking the Λ-torsion parts, we get an exact sequence 0 −→ (H1loc /H1glob )Λ−tor −→ (Sel(E/Q∞ )∨ p∞ )Λ−tor −→ (Sel0 (E/Q∞ )∨ )Λ−tor . We write char((H1loc /H1glob )Λ−tor ) = ((T )). By Wingberg [27] Corollary 2.5, we have ∨ char((Sel(E/Q∞ )∨ p∞ )Λ−tor ) = char((Sel0 (E/Q∞ ) )Λ−tor ).

Hence, by Problem 0.7, any irreducible factor of (T ) is of the form Φn . But Φn does not divide the gcd of a and b, so does not divide (T ). Therefore, char((H1loc /H1glob )Λ−tor ) = (1), and (H1loc /H1glob )Λ−tor is pseudo-null. This implies H1loc /H1glob ∼ Λ. (2) Put h(T ) = Πn≥0,en ≥1 Φenn −1 . By the Main Conjecture and Problem 0.7, we have char(H1glob / < zK >) = char(Sel0 (E/Q∞ )∨ ) = (h(T )). Hence, we can write zK = h(T )ξ where ξ is a generator of H1glob . By Theorem 1.3, we have f (T ) = Col+ (ξ)h(T ) and g(T ) = Col− (ξ)h(T ). We take a = a(T ), b = b(T ), e1 , e2 as in the proof of Theorem 0.4 in §1.4, namely, ξ = ae1 + be2 with a, b ∈ Λ. By the proof of Proposition 3.4 (1), a is prime to b. Hence, by Proposition 1.2, the greatest common divisor of Col+ (ξ) and Col− (ξ) is T or 1. Suppose at first that e0 = 0. Since X(E/Q)[p∞ ] is finite, Sel(E/Q)p∞ is finite. Hence, by the control theorem for Sel± (E/Q∞ ), char(Sel± (E/Q∞ )∨ ) ⊂ (T ). Hence, by the Main Conjecture, f (0) = 0 and g(0) = 0. Thus, Col± (ξ)(0) = 0. Therefore, the gcd of Col+ (ξ) and Col− (ξ) is 1, and the gcd of f (T ) and g(T ) is h(T ). Next, suppose e0 > 0. Then, E(Q) ⊗ Qp −→ E(Qp ) ⊗ Qp is surjective, so the image of H 1 (Q, T ) −→ H 1 (Qp , T ) is in E(Qp ) ⊗ Zp by Lemma 1.4. In particular, the image of ξ is in E(Qp ) ⊗ Zp . Recall that ξ = a(T )e1 + b(T )e2 .

328

Masato Kurihara and Robert Pollack

Since ξ ∈ E(Qp ) ⊗ Zp , we get a(0) = 0. It follows from T | Col± (e2 ) (cf. 1.4) that T divides Col± (ξ). Thus, the gcd of Col+ (ξ) and Col− (ξ) is T , and the gcd of f (T ) and g(T ) is T h(T ), which completes the proof. 3.3. Examples. Let E = X0 (17) and consider the quadratic twist Ed . The condition of Proposition 3.1 is satisfied for all d such that 0 < d < 250 except for d = 104, 193, 233. For d = 104, we have r = 2 and λ(fd (T )) = λ(gd (T )) = 4. We know that T 2 divides both fd (T ) and gd (T ), and a computer computation shows that fd (T ) has two addition zeroes of slope 1, and gd (T ) has two addition zeroes of slope 1/2. Hence, the greatest common divisor of fd (T ) and gd (T ) is T 2 as predicted by Problem 3.2. The computer computations also show that μ(fd (T )) = μ(gd (T )) = 0. Since the Galois representation on the 3-torsion of Ed is surjective, [9] Theorem 13.4 yields that μ(Sel0 (E/Q∞ )∨ ) = 0. Therefore, Proposition 3.3 applies and we have char(Sel0 (E/Q∞ )∨ ) = (T ). For d = 233, we have r = 1 and min{λ(fd (T )), λ(gd (T ))} = 3. In this case, a similar computation can be done and, simply by computing slopes of the zeroes, we can conclude that the gcd of fd (T ) and gd (T ) is T . Again, the relevant μ-invariants are zero and thus Proposition 3.3 yields that Sel0 (E/Q∞ ) is finite. Finally, we consider the case d = 193. We have that λ(fd (T )) = λ(gd (T )) = 7. However, in this case, both power series have 6 roots of valuation 1/6 and a simple zero at 0. Thus, simply by looking at slopes of roots, we cannot conclude that their greatest common divisor is T . To analyze this situation more carefully, we set A(T ) =

fd (T ) gd (T ) log− (T ) · log+ (T ) and B(T ) = · fd (0)T gd (0)T Φ1 (T )

which are both convergent power series on the open unit disc. We divide here by Φ1 (T ) so that every root of both A(T ) and B(T ) has valuation at most 1/6. Let π be some 6-th root of p in Qp and set A (T ) = A(πT ) and B (T ) = B(πT ). Then both A and B are convergent power series on the closed unit disc and are thus in the Tate algebra 2 ∞ n OT = f (T ) = an T : |an |p → 0 n=0

where O = Zp [π].

Two p-adic L-functions and rational points

329

We consider the image of A and B in OT /π m ∼ = (O/π m O)[T ] for various m with the hopes of noticing that the image of these power series do not share a common root in this small polynomial ring. For m = 1, by a computer computation, we have A (T ) ≡ 1 + 2T 6 and B (T ) ≡ 1 + 2T 6 , from which we deduce nothing. For m = 2, we have A (T ) ≡ 1 + 2T 6 and B (T ) ≡ 1 + 2πT + 2T 6 + πT 7 . From this, we again deduce nothing as A (T ) · (1 + 2πT ) ≡ B (T ). For m = 3 though, we have A (T ) ≡ 1+π 2 T 2 +2T 6 and B (T ) ≡ 1+2πT +π 2 T 2 +2T 6 +πT 7 +2π 2 T 8 . Now, one computes that B (T ) ≡ A (T ) · (1 + 2πT + π 2 T 2 ) + 2π 2 T 2 . If gcd(fd , gd ) = T , then there exists some common root α in OQp with valuation 1/6. If we write α = πu with u a p-adic unit, then A (u) = B (u) = 0. But the above identity then forces that 2π 2 u2 is divisible by π 3 , which is impossible! Thus, gcd(fd , gd ) = T and Sel0 (E/Q∞ ) is finite.

4 Tables In this section, we present two tables listing the rational points we constructed 3-adically on quadratic twists of X0 (17) and X0 (32). For each curve X0 (N ), we considered d such that d > 0, gcd(d, 3N ) = 1, and rankX0 (N )d = 1. In each case, we used the curve’s globally minimal Weierstrass equation. For the curve X0 (17)d , we included this equation in the table (by listing the coefficients of y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 ). For the curve X0 (32)d , the globally minimal Weierstrass equation is simply y 2 = x3 + 4d2 x. These computations were done for the curve X0 (17) (resp. X0 (32)) using an overconvergent modular symbol that was accurate mod 3200 (resp. mod 3100 ).

330

Masato Kurihara and Robert Pollack Table of global points Pd found on X0 (17)d

d

Pd (14, −32)

5

9895 −878410 ) 81 , 729 139064 −46339707 ( 1225 , 42875 )

(

28 29 (

37 40

150455134 1539419885296 ) 974169 , 961504803

(190, −2400)

(

41

49351 −2413090 ) 441 , 9261 19244 −1480800 ( 121 , 1331 ) 3952 235937 ( 9 , 27 )

(

44 56 61

88

109

23663936531 3630080811299531 , ) 729316 622835864 2195359 −3243299390 ( 1089 , ) 35937 673327635832141 −17605988238521415099109 ( 308624691600 , ) 171453361171464000

(

113 124 133 (

173 181 184

209 232 233 241 248

(

[1, −1, 1, −1156, −958144] [0, 0, 0, −1331, −1184590] [0, 0, 0, −2156, −2442160] [1, −1, 0, −2558, −3155815]

[1, −1, 1, −3664, −5408852]

3242226594695 −5838006309203270250 , ) 62457409 493600903327 123907127 1373877614721 ( 7396 , ) 636056 117267845674060 −1272481700834989645855 ( 1957797009 , ) 86626644257223

97

[1,-1,0,-941,-704158] [0, 0, 0, −1100, −890000]

[1, −1, 1, −2905, −3818278]

(

92

[1,-1,0,-578,-339015]

(959, 29095) (

73

[1, −1, 0, −17, −1734] [0, 0, 0, −539, −305270]

111991 37126789 , ) 36 216 2140023710080 1453764842693104220 ( 8942160969 , 845597567711547 )

65

193 197

1067 −34691 ) 4 , 8

[a1 , a2 , a3 , a4 , a6 ]

9693514595778788 −653587463274626644218393 ) 18312896333449 , 78367421114819312293 56621266 402821517014 ( 50625 , 11390625 ) 33961175037484 −5571288550874575360 ( 73808392329 , 20052042602765733 )

915394662845247271 −878088421712236204458830141 , ) 25061097283236 125458509476191439016

not found

(

472269 303288257 ) 400 , 8000

1106304238 −32569057816800 ( 1117249 , ) 1180932193 847800975918361973 −738030759904517944421609807 ( 716449376631184 , ) 19176893823953703981248 1313533 −1347623317 ( 1296 , ) 46656 3841589315420 −7526764618576173000 ( , ) 8216728195339 407192041

[0, 0, 0, −5324, −9476720] [0, 0, 0, −5819, −10828630] [1, −1, 1, −6469, −12690242] [1, −1, 0, −8168, −18006955] [1, −1, 1, −8779, −20063092] [0, 0, 0, −10571, −26513990] [1, −1, 0, −12161, −32713318] [1, −1, 0, −20576, −71997483] [1, −1, 0, −22523, −82454830] [0, 0, 0, −23276, −86629040] [1, −1, 1, −25609, −99966422] [1, −1, 0, −26681, −106311798] [1, −1, 1, −30031, −126947224] [0, 0, 0, −37004, −173649680] [1, −1, 1, −37324, −175895512] [1, −1, 1, −39931, −194643044] [0, 0, 0, −42284, −212111920]

Two p-adic L-functions and rational points

331

Table of global points Pd found on X0 (32)d d

Pd

5

(5, 25) 13 9 , 1421 ( 25 ,

(

13 29

845 27 ) 76531 125 )

37

61

198505 9261 ) 750533 1987241095 ( 20449 , 2924207 ) 102541 67889645 ( 1521 , 59319 )

77

( 719104 ,

( 441 ,

37 53

275625 58139738475 609800192 )

(765, 21675)

85 101 109 133 149

(

42672500 279031013300 9409

(

,

912673

5341 415835

,

)

)

9 27 314109807025 338319926884539145 ( 1937936484 , 85311839898648 ) 43061 9435425 ( 49 , ) 343

Bibliography [1]

[2] [3] [4] [5] [6]

[7]

Bernardi, D. et Perrin-Riou B., Variante p-adique de la conjecture de Birch et Swinnerton-Dyer (le cas supersingulier), C. R. Acad. Sci. Paris, 317 S´er. I (1993), 227-232. Coates, J. and Sujatha, R., Fine Selmer groups of elliptic curves over p-adic Lie extensions, Math. Annalen 331 (2005) 809-839. Colmez, P., Th´eorie d’Iwasawa des representations de de Rham d’un corps local, Ann. of Math. 148 (1998), 485-571. Colmez, P., La conjecture de Birch et Swinnerton-Dyer p-adique, S´eminaire Bourbaki 919, Ast´erisque 294 (2004), 251-319. Cremona, J., mwrank, http://www.maths.nott.ac.uk/personal/ jec/ftp/progs/mwrank.info. Darmon, H. and Pollack, R., The efficient calculation of Stark-Heegner points via overconvergent modular symbols, Israel Journal of Mathematics, 153 (2006), 319-354. Darmon, H. and Pollack, R., Stark-Heegner point computational package, http://www.math.mcgill.ca/darmon/programs/shp/shp.html.

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Masato Kurihara and Robert Pollack Iwasawa, K., Riemann-Hurwitz formula and p-adic Galois representations for number fields, Tˆohoku Math. J. 33 (1981), 263-288. Kato, K., p-adic Hodge theory and values of zeta functions of modular forms, in Cohomologies p-adiques et applications arithm´etiques III, Ast´erisque 295 (2004), 117-290. Kim, B.-D., The parity conjecture for elliptic curves at supersingular reduction primes, Compos. Math. 143 (2007), 47-72. Knuth, D. E., The art of computer programming, Vol 2., 3rd ed, Addison-Wesley, Reading, Mass. 1997. Kobayashi, S., Iwasawa theory for elliptic curves at supersingular primes, Invent. math. 152 (2003), 1-36. Kurihara, M., On the Tate-Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I, Invent. math. 149 (2002), 195-224. The MAGMA computational algebra system, http://magma.maths. usyd.edu.au. Mazur, B. and Tate, J., Refined conjectures of the “Birch and Swinnerton-Dyer type”, Duke Math. J. 54 No 2 (1987), 711-750. Nekov´aˇr, J., On the parity of ranks of Selmer groups II, C. R. Acad. Sci. Paris S´er. I 332 (2001), 99-104. Perrin-Riou, B., Fonctions L p-adiques d’une courbe elliptique et points rationnels, Ann. Inst. Fourier 43, 4 (1993), 945-995. Perrin-Riou, B., Th´eorie d’Iwasawa des representations p-adiques sur un corps local, Invent. math. 115 (1994), 81-149. Perrin-Riou, B., Arithm´etique des courbes elliptiques a` reduction supersinguli`ere en p, Experimental Mathematics 12 (2003), 155-186. Pollack, R., On the p-adic L-functions of a modular form at a supersingular prime, Duke Math. J. 118 (2003), 523-558. Pollack, R. and Stevens, G., Explicit computations with overconvergent modular symbols, preprint. Rubin, K., p-adic variants of the Birch and Swinnerton-Dyer conjecture with complex multiplication, Contemporary Math. 165 (1994), 71-80. Rubin, K., Euler systems and modular elliptic curves, in Galois representations in Arithmetic Algebraic Geometry, London Math. Soc., Lecture Note Series 254 (1998), 351-367. Skinner, C. and Urban E., Sur les d´eformations p-adiques des formes de SaitoKurokawa, C. R. Acad. Sci. Paris, S´er. I 335 (2002), 581-586. Stein, W., Computing p-adic heights, http://www.williamstein.org/ talks/harvard-talk-2004-12-08/. Stevens, G., Rigid analytic modular symbols, preprint http://math.bu. edu/people/ghs/research.d Wingberg, K., Duality theorems for abelian varieties over Zp -extensions, in Algebraic Number Theory - in honor of K. Iwasawa, Advanced Studies in Pure Mathematics 17 (1989), 471-492.

From the Birch and Swinnerton-Dyer Conjecture to non-commutative Iwasawa theory via the Equivariant Tamagawa Number Conjecture - a survey Otmar Venjakob Universit¨at Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 69120 Heidelberg, Germany. [email protected]

Introduction This paper aims to give a survey on Fukaya and Kato’s article [23] which establishes the relation between the Equivariant Tamagawa Number Conjecture (ETNC) of Burns and Flach [9] and the noncommutative Iwasawa Main Conjecture (MC) (with p-adic L-function) as formulated by Coates, Fukaya, Kato, Sujatha and the author [14]. Moreover, we compare their approach with that of Huber and Kings [24] who formulate an Iwasawa Main Conjecture (without p-adic L-functions). We do not discuss these conjectures in full generality here, in fact we are mainly interested in the case of an abelian variety defined over Q. Nevertheless we formulate the conjectures for general motives over Q as far as possible. We follow closely the approach of Fukaya and Kato but our notation is sometimes inspired by [9, 24]. In particular, this article does not contain any new result, but hopefully serves as introduction to the original articles. See [47] for a more down to earth introduction to the GL2 Main Conjecture for an elliptic curve without complex multiplication. There we had pointed out that the Iwasawa main conjecture for an elliptic curve is morally the same as the (refined) Birch and Swinnerton Dyer (BSD) Conjecture for a whole tower of number fields. The work of Fukaya and Kato makes this statement precise as we are going to explain in these notes. For the convenience of the reader we have given some of the proofs here which had been left as an exercise in [23] whenever we had the feeling that the presentation of the material becomes more transparent thereby. Since the whole paper bears an expository style we omit a lengthy introduction and just state briefly the content of the different sections: In section 1 we recall the fundamental formalism of (non-commutative) determinants which were introduced first by Burns and Flach to formulate 333

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equivariant versions of the TNC. In section 2 we briefly discuss the setting of (realisations of) motives as they are used to formulate the conjectures concerning their L-functions, which are defined in section 3. There, also the absolute version of the TNC is discussed, which predicts the order of vanishing of the L-function at s = 0, the rationality and finally the precise value of the leading coefficient at s = 0 up to the period and regulator. In subsection 3.1 we sketch how one retrieves the BSD conjecture in its classical formulation if one applies the TNC to the motive h1 (A)(1) of an abelian variety A/Q. Though well known to the experts this is not very explicit in the literature. In section 4 we consider a p-adic Lie extension of Q with Galois group G. In this context the TNC is extended to an equivariant version using the absolute version for all twists of the motive by certain representations of G. The compatibility of the ETNC with respect to Artin-Verdier/Poitou-Tate duality and the functional equation of the L-function is studied in section 5. A refinement leads to the formulation of the local -conjecture in subsection 5.1. In order to involve p-adic L-functions one has to introduce Selmer groups or better complexes. The necessary modifications of the L-function and the Galois cohomology - in a way that respects the functional equation - are described in section 6. From this the MC in the form of [14] is derived in subsection 6.2 after a short interlude concerning the new “localized K1 ”. In the Appendix we collect basic facts about Galois cohomology on the level of complexes. Acknowledgements: I am very grateful to Takako Fukaya and Kazuya Kato for providing me with updated copies of their work and for answering many questions. I also would like to thank John Coates for his interest and Ramdorai Sujatha for her valuable comments. The referee is heartily acknowledged for a very careful reading and many useful suggestions. Finally I am indebted to David Burns, Pedro Luis del Angel, Matthias Flach, Annette Huber, Adrian Iovita, Bruno Kahn and Guido Kings for helpful discussions. This survey was written during a stay at Centro de Investigacion en Matematicas (CIMAT), Mexico, as a Heisenberg fellow of the Deutsche Forschungsgemeinschaft (DFG) and I want to thank these institutions for their hospitality and financial support.

1 Noncommutative determinants The (absolute) TNC compares integral structures of Galois cohomology with values of complex L-functions. For this purpose the determinant is the adequate tool as is illustrated by the following basic

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Example 1.1 Let T be a Zp -lattice in a finite dimensional Qp -vector space V and f : T → T a Zp -linear map which induces an automorphism of V . Then the cokernel of f is finite with cardinality | det(f )|−1 p where | − |p denotes the p-adic valuation normalized as usual: |p|p = 1/p. This is an immediate consequence of the elementary divisor theorem. Since the equivariant TNC involves the action of a possibly noncommutative ring R one needs a determinant formalism over an arbitrary (associative) ring R (with unit). This can be achieved by either using virtual objects a la Deligne as Burns and Flach [9, §2] do or by Fukaya and Kato’s adhoc construction [23, 1.2], both approaches lead to an equivalent description. Let P(R) denote the category of finitely generated projective R-modules and (P(R), is) its subcategory of isomorphisms, i.e. with the same objects, but whose morphisms are precisely the isomorphisms. Then there exists a category CR and a functor dR : (P(R), is) → CR which satisfies the following properties: a) CR has an associative and commutative product structure (M, N ) → M · N or written just M N with unit object 1R = dR (0) and inverses. All objects are of the form dR (P )dR (Q)−1 for some P, Q ∈ P(R). b) all morphisms of CR are isomorphisms, dR (P ) and dR (Q) are isomorphic if and only if their classes in K0 (R) coincide. There is an identification of groups AutCR (1R ) = K1 (R) and MorCR (M, N ) is either empty or a K1 (R)-torsor where α : 1R → 1R ∈ K1 (R) acts on α·φ

φ : M → N as αφ : M = 1R · M → 1R · N = N. c) dR preserves the “product” structures: dR (P ⊕ Q) = dR (P ) · dR (Q). This functor can be naturally extended to complexes. Let Cp (R) be the category of bounded (cohomological) complexes in P(R) and (Cp (R), quasi) its subcategory of quasi-isomorphisms. For C ∈ Cp (R) we set C + = i even C i and C − = i odd C i and define dR (C) := dR (C + )dR (C − )−1 and thus we obtain a functor dR : (Cp (R), quasi) → CR with the following properties for all objects C, C , C ∈ Cp (R) : d) If 0 → C → C → C → 0 is a short exact sequence of complexes, then there is a canonical isomorphism dR (C) ∼ = dR (C )dR (C ) which we take to be an identification;

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Otmar Venjakob e) If C is acyclic, then the quasi-isomorphism 0 → C induces a canonical isomorphism in CR of the form 1R = dR (0) → dR (C); f) For any integer r there is a canonical morphism dR (C[r]) ∼ = r dR (C)(−1) in CR which we take to be an identification, here C[r] denotes the r-fold shift of C; g) the functor dR factorizes through the image of Cp (R) in the category Dp (R) of perfect complexes (as full triangulated subcategory of the derived category Db (R) of the homotopy category of bounded complexes of R-modules), and extends to (Dp (R), is) (uniquely up to unique isomorphisms) 1 ; h) If C ∈ Dp (R) has the property that all cohomology groups Hi (C) belong again to Dp (R), then there is a canonical isomorphism dR (C) ∼ =

i

dR (Hi (C))(−1) .

i

Moreover, if R is another ring, Y a finitely generated projective R -module endowed with a structure as right R-module such that the actions of R and R on Y commute, then the functor Y ⊗R − : P(R) → P (R ) extends to give a diagram

(Dp (R), is)

dR

Y ⊗LR −

(Dp (R ), is)

dR

/ CR

Y ⊗R −

/ CR

which commutes (up to canonical isomorphism). In particular, if R → R is a ring homomorphism and C ∈ Dp (R), we just write dR (C)R for R ⊗R dR (C). Now let R◦ be the opposite ring of R. Then the functor HomR (−, R) induces an anti-equivalence between CR and CR◦ with quasi-inverse induced by HomR◦ (−, R◦ ); both functors will be denoted by −∗ . This extends to give a diagram 1 But property d) does not in general extend to arbitrary distinguished triangles, thus from a technical point of view all constructions involving complexes will have to be made carefully avoiding this problem. We will neglect this problem but see [9] for details.

From the BSD to the MC via the ETNC / CR

dR

(Dp (R), is) RHomR (−,R)

(Dp (R◦ ), is)

337

dR◦

−∗

/ CR◦

which commutes (up to unique isomorphism); similarly we have such a commutative diagram for RHomR◦ (−, R◦ ). For the handling of the determinant functor in practice the following considerations are quite important: Remark 1.2 (i) For objects A, B ∈ CR we often identify a morphism f : A → B with the induced morphism 1R

A · A−1

f ·idA−1

/ B · A−1 .

Then for morphisms f : A → B and g : B → C in CR , the composition g ◦ f : A → C is identified with the product g · f : 1R → C · A−1 of g : 1R → C · B −1 and f : 1R → B · A−1 . Also, by this identification a map f : A → A corresponds uniquely to an element in K1 (R) = AutCR (1R ). Furthermore, for a map f : A → B in CR , we write f : B → A for its inverse with respect to composition and f −1 =: idB −1 · φ · idA−1 : A−1 → B −1 for its inverse with respect to the multiplication in CR , i.e. f · f −1 = id1R . Obviously, for a map f : A → A both inverses f and f −1 coincide if all maps are considered as elements of K1 (R) as above. Convention: If f : 1R → A is a morphism and B an object in CR , then we write B f

B

·f

/ B · A for the morphism idB · f. In particular, any morphism

/ A can be written as B

· (idB −1 · f )

/A. φ

(ii) The determinant of the complex C = [P0 → P1 ] (in degree 0 and 1) with def

1R and is defined even if φ is not P0 = P1 = P is by definition dR (C) an isomorphism (in contrast to dR (φ)). But if φ happens to be an isomorphism, acyc / dR (C) , i.e. if C is acyclic, then by e) there is also a canonical map 1R which is in fact nothing else than 1R

d(φ)−1 ·idd(P

dR (P1 )dR (P1 )−1

1)

−1

/ dR (P0 )dR (P1 )−1

dR (C)

(and which depends in contrast to the first identification on φ). Hence, the comacyc / dR (C) def 1R corresponds to dR (φ)−1 ∈ K1 (R) according posite 1R to the first remark. In order to distinguish the above identifications between

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1R and dR (C) we also say that C is trivialized by the identity when we refer def

to dR (C) 1R (or its inverse with respect to composition). For φ = idP both identifications agree obviously. We end this section by considering the example where R = K is a field and V a finite dimensional vector space over K. Then, according to [23, 1.2.4], ?top dK (V ) can be identified with the highest exterior product V of V and for an automorphism φ : V → V the determinant dK (φ) ∈ K × = K1 (K) can be identified with the usual determinant detK (φ). In particular, we identify 1K = K with canonical basis 1. Then a map 1K the value ψ(1) ∈ K × .

ψ

/ 1K corresponds uniquely to

Remark 1.3 Note that every finite Zp -module A possesses a free resolution C as in Remark 1.2 (ii), i.e. dZp (A) ∼ = dZp (C)−1 = 1Zp . Taking into account the above and Example 1.1 we see that modulo Z× p the composite 1Qp

/ dZp (C)Qp

acyc

def

1Qp

corresponds to the cardinality |A| ∈ Q× p.

2 K-Motives over Q In this survey we will be mainly interested in the Tamagawa Number Conjecture and Iwasawa theory for the motive M = h1 (E)(1) of an elliptic curve E or the slightly more general M = h1 (A)(1) of an abelian variety A defined over Q. But as it will be important to consider certain twists of M we also recall basic facts on the Tate motive Q(1) and Artin motives. We shall simply view (pure) motives in the naive sense, as being defined by a collection of realizations satisfying certain axioms, together with their motivic cohomology groups. The archetypical motive is hi (X) for a smooth projective variety X over Q with its obvious e´ tale cohomology Hie´t (X ×Q Q, Ql ), singular cohomology Hi (X(C), Q) and de Rham cohomology HidR (X/Q), their additional structures and comparison isomorphisms. More generally, let K be a finite extension of Q. A K-motive M over Q, i.e. a motive over Q with an action of K, will be given by the following data, which for M = hn (X)K arise by tensoring the above cohomology groups by K over Q :

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2.1 The l-adic realisation Ml of M (for every prime number l) For a place λ of K lying above l we denote by Kλ the completion of K with respect to λ. Then Mλ is a continuous finite dimensional Kλ -linear representa tion of the absolute Galois group GQ of Q. We put Kl := K ⊗Q Ql = λ|l Kλ and we denote by Ml the free Kl -module λ|l Mλ .

2.2 The Betti realisation MB of M Attached to M is a finite dimensional K-vector space MB which carries an action of complex conjugation ι and a Q-Hodge structure (of weight w(M )) MB ⊗Q C ∼ =

Hi,j

(over R) with ιHi,j = Hj,i and Hi,j = 0 if i + j = w(M ), where Hi,j are free KC := K ⊗Q C ∼ = CΣK -modules and where ΣK denotes the set of all embeddings K → C. E.g. the motive M = hn (X) has weight w(M ) = n.

2.3 The de Rham realisation MdR of M MdR is a finite dimensional K-vector space with a decreasing exhaustive filk 0 tration MdR , k ∈ Z. The quotient tM = MdR /MdR is called the tangent space of M.

2.4 Comparison between MB and Ml For each prime number l there is an isomorphism of Kl -modules Kl ⊗K MB

gl ∼ =

/ Ml

(2.1)

which respects the action of complex conjugation; in particular it induces canonical isomorphisms gλ+ : Kλ ⊗K MB+ ∼ = Mλ+ and gl+ : Kl ⊗K MB+ ∼ = Ml+ .

(2.2)

Here and in what follows, for any commutative ring R and R[G(C/R)]-module X we denote by X + and X − the R-submodule of X on which ι acts by +1 and −1, respectively.

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Otmar Venjakob 2.5 Comparison between MB and MdR

There is a G(C/R)-invariant isomorphism of KC -modules C ⊗Q MB

g∞ ∼ =

/ C ⊗Q MdR

(2.3)

(on the left hand side ι acts diagonally while on the right hand side only on C) such that for all k ∈ Z k g∞ ( Hi,j (M )) = C ⊗Q MdR . i≥k

This induces an isomorphism (C ⊗Q MB )+ ∼ = R ⊗Q MdR

(2.4)

/ R ⊗Q tM .

(2.5)

and the period map R ⊗Q MB+

αM

We say that M is critical if this happens to be an isomorphism2 .

2.6 Comparison between Mp and MdR Let BdR be the filtered field of de Rham periods with respect to Qp /Qp , which is endowed with a continuous action of the absolute Galois group GQp of Qp , and set as usual DdR (V ) = (BdR ⊗Qp V )GQp for a finite-dimensional Qp vector space V endowed with a continuous action of GQp . The (decreasing) i i i filtration BdR of BdR induces a filtration DdR (V ) = (BdR ⊗Qp V )GQp of DdR (V ). Then there is an GQp -invariant isomorphism of filtered Kp ⊗Qp BdR modules g BdR ⊗Qp Mp dR / BdR ⊗Q MdR (2.6) ∼ =

which induces an isomorphism of filtered Kp -modules by taking GQ -invariants DdR (Mp )

gdR / ∼ =

Kp ⊗K MdR ,

(2.7)

an isomorphism of Kp -modules 0 t(Mp ) := DdR (Mp )/DdR (Mp )

t gdR

∼ =

/ Kp ⊗K tM

(2.8)

2 By [13, Lem. 3] M is critical if and only if one of the following equivalent conditions holds: a) both infinite Euler factors L∞ (M, s) and L∞ (M ∗ (1), −s) (see section 5) are holomorphic at s = 0, b) if j < k and Hj,k = {0} then j < 0 and k ≥ 0, and, in addition, if Hk,k = {0}, then ι acts on this space as +1 if k < 0 and by −1 if k ≥ 0. See also [15, Lem. 2.3] for another criterion.

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and, for each place λ of K over p, an isomorphism of Kλ -vector spaces 0 t(Mλ ) := DdR (Mλ )/DdR (Mλ )

t gdR

∼ =

/ Kλ ⊗K tM .

(2.9)

The tensor product M ⊗K N of two K-motives is given by the data which arises from the tensor products of all realizations and their additional structures. Similarly the dual M ∗ of the K-motive M is given by the duals of the corresponding realizations. In particular, we denote by M (n), n ∈ Z, the twist of M by the |n|-fold tensor product Q(n) = Q(1)⊗n of the Tate motive if n ≥ 0 and of its dual Q(−1) = Q(1)∗ if n < 0. By det(M ) we denote the K-motive whose realisations arise as the highest exterior products of the realisations of the K-motive M. For the motive M = hi (X)(j) where the dimension of X is d, Poincar´e duality gives a perfect pairing hi (X)(j) × h2d−i (X)(d − j) → h2d (X)(d) ∼ =Q which identifies M ∗ with h2d−i (X)(d − j). Here Q = h0 (Spec(Q))(0) denotes the trivial Q-motive. Example 2.1 A) The Tate motive Q(1) = h2 (P1 )∗ should be thought of as h1 (Gm ) even though the multiplicative group Gm is not proper. Its l-adic realisation is the usual Tate module Ql (1) = Ql on which GQ acts via the cyclotomic character χl : GQ → Z× l . The action of complex conjugation on Q(1)B = Q is by −1, its Hodge structure is of weight w(M ) = −2 and given by H−1,−1 . The filtration Q(1)kdR of Q(1)dR = Q is either Q or 0, according as k ≤ −1 or k > −1, in particular we have tQ(1) = Q. Finally, g∞ sends 1 ⊗ 1 to 2πi ⊗ 1 while gdR sends 1 ⊗ 1 to t ⊗ 1, where t = “2πi” is the p-adic period analogous to 2πi. B) For the Q-motive M = h1 (A)(1) of an abelian variety A over Q we have Ml = He´1t (AQ¯ , Ql (1)) = HomQl (Vl A, Ql (1)) ∼ = Vl (A∨ ) via the Weil pairing. More generally, the Poincare bundle on A × A∨ induces isomorphisms M ∗ (1) ∼ = h1 (A∨ )(1) and M ∼ = h1 (A∨ )∗ , while by fixing a (very) ample symmetric line bundle on A, whose existence is granted by [33, cor. 7.2], it is sometimes convenient to identify M with h1 (A) := h1 (A)∗ using the hard Lefschetz theorem ([43, 1.15, Thm. 5.2 (iii)], see also [30]) (but in general better to work with the dual abelian variety A∨ ). Then Ml can be identified with Vl (A), while MB = H 1 (A(C), Q)(1) can be identified with H1 (A(C), Q), the Hodge-decomposition (pure of weight −1) is given by H0,−1 = H 0 (A(C), Ω1A )(∼ = HomC (H1 (A(C), Ω0A ), C)) and H−1,0 = −1 1 0 ∼ 1 H (A(C), ΩA )(= HomC (Ω (A), C)). Furthermore, we have MdR = MdR ,

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0 1 MdR is the image of Ω1A/Q (A)(∼ = 0. In particu= H1 (A, Ω0A/Q )∗ ) and MdR 1 0 ∨ lar, tM = H (A, ΩA/Q ) = Lie(A ) (e.g. [31, Thm. 5.11]) the Lie-algebra of A∨ , can be identified with th1 (A) = HomQ (Ω1A/Q (A), Q) = Lie(A). The map αM for the motive M = h1 (A), which is in fact ; an isomorphism, is induced by sending a 1-cycle γ ∈ H1 (A(C), Q)+ to γ ∈ HomQ (Ω1A/Q (A), R) = ; Lie(A)R which sends a 1-form ω to γ ω ∈ R. C) Artin motives [ρ] (with coefficients in a finite extension K of Q) are direct summands of the K-motive h0 (Spec(F )) ⊗Q K but can also be identified with the category of finite-dimensional K-vector spaces V with an action by GQ , i.e. representations ρ : GQ → AutK (V ) with finite image. We write [ρ] for the corresponding motive and have [ρ]l = V ⊗K Kl with GQ acting just on V, ¯ GQ , [ρ]B = V with Hodge-Structure pure of Type (0, 0) and [ρ]dR = (V ⊗Q Q) k where GQ acts diagonally. Since [ρ]dR is either [ρ]dR or 0 according as k ≤ 0 of k > 0, we have t[ρ] = 0. The inverse of g∞ is induced by the natural ¯ GQ ⊆ V ⊗Q Q. ¯ E.g. if ψ denotes a Dirichlet character of inclusion (V ⊗Q Q) conductor f considered via (Z/f Z)∗ ∼ = G(Q(ζf )/Q) as a character G → K × where K = Q(ζϕ(f )) ) and ϕ denotes the Euler ϕ-function, then we obtain a basis of [ψ]dR over K given by the Gauss sum ¯ GQ , ψ(n) ⊗ e−2πin/f ∈ (K(ψ) ⊗Q Q) 1≤n
where K(ψ) denotes the 1-dimensional K-vector space on which GQ acts via ψ. Of course, h0 (Spec(F )) ⊗Q K corresponds to the regular representation of G(F/Q) on K[G(F/Q)] considered as representation of GQ . Other examples arise by taking symmetric products or tensor products of the above examples. In particular, we will be concerned with the motives D) [ρ] ⊗ h1 (A)(1), where ρ runs through all Artin representations. E) Finally, the motive M (f ) of a modular form f is a prominent example, see [18, §7] and [42]. 2.7 Motivic cohomology The motivic cohomology K-vector spaces H0f (M ) := H0 (M ) and H1f (M ) may be defined by algebraic K-theory or motivic cohomology a la Voevodsky. They are conjectured to be finite dimensional. Instead of a general definition we just describe them in our standard examples. Example 2.2 A) For the Tate motive we have H0f (Q(1)) = H1f (Q(1)) = 0 and for its Kummer dual H0f (Q) = Q while H1f (Q) = 0.

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B) If M = h1 (A)(1) for an abelian variety A over Q one has H0f (M ) = 0 and H1f (M ) = A∨ (Q) ⊗Z Q. C) For M = h0 (Spec(F )) we have H0f (M ) = Q and H1f (M ) = 0 while for M ∗ (1) = h0 (Spec(F ))(1) one has H0f (M ∗ (1)) = 0 and H1f (M ∗ (1)) = OF× ⊗Z Q. More generally, for a K-Artin motive [ρ] one has H0f ([ρ]) = K n , where n is the multiplicity with which K occurs in [ρ]. Unfortunately the functor Hif does not behave well with tensor products, i.e. in general one cannot derive H∗f ([ρ] ⊗ h1 (A)(1)) from H∗f ([ρ]) and H∗f (h1 (A)(1)) (e.g. in the form of a K¨unneth formula).

3 The Tamagawa Number Conjecture - absolute version In [5] Bloch and Kato formulated a vast generalization of the analytic class number formula and the BSD-conjecture. While the conjecture of Deligne and Beilinson links the order of vanishing of the L-function attached to a motive M to its motivic cohomology and claims rationality of special L-values or more general leading coefficients (up to periods and regulators) the Tamagawa number conjecture of Bloch and Kato predicts the precise L-value in terms of Galois cohomology (assuming the conjecture of Deligne-Beilinson). Later, Fontaine and Perrin-Riou [22] found an equivalent formulation using (commutative) determinants instead of (Tamagawa) measures3 . In this section we follow closely their approach. Let us first recall the definition of the complex L-function attached to a Kmotive M. We fix a place λ of K lying over l and an embedding K → C. For every prime p take a prime l = p and set Pp (Mλ , X) = detKλ (1 − ϕp X|(Mλ )Ip ) ∈ Kλ [X], where ϕp denotes the geometric Frobenius automorphism of p in GQp /Ip and Ip is the inertia subgroup of p in GQp ⊆ GQ . It is conjectured that Pp (X) belongs to K[X] and is independent of the choices of l and λ. For example this is known by the work of Deligne proving the Weil conjectures for M = hi (X) for places p where X has good reduction; by the compatibility of the system of l-adic realisations for abelian varieties [21, Rem. 2.4.6(ii)] and Artin motives it is also clear for our examples A)-D). Then we have the L-function of M as Euler product LK (M, s) = Pp (Mλ , p−s )−1 , p

defined and analytic for ((s) large enough. 3 The name comes from an analogy with the theory of algebraic groups, see [5].

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Example 3.1 A) The L-function LQ (Q(1), s − 1) of the Tate motive is just the Riemann zeta function ζ(s). In general, one has LK (M (n), s) = LK (M, s + n) for any K-motive M and any integer n. B) If M = h1 (A)(1) for an abelian variety A over Q, then LQ (M, s − 1) is the classical Hasse-Weil L-function of A∨ , which coincides with that for A because A and A∨ are isogenous. C) LK ([ρ], s) coincides with the usual Artin L-function of ρ, in particular we retrieve the Dedekind zeta-function ζF (s) as LQ (h0 (Spec(F )), s). D) The L-functions LK ([ρ] ⊗ h1 (A)(1), s) will play a crucial role for the interpolation property of the p-adic L-function. Also, the meromorphic continuation to the whole plane C is part of the conjectural framework. The Taylor expansion LK (M, s) = L∗K (M )sr(M ) + . . . defines the leading coefficient L∗K (M ) ∈ C× and the order of vanishing r(M ) ∈ Z of LK (M, s) at s = 0. The aim of the conjectures to be formulated now is to express L∗K (M ) and r(M ) in terms of motivic and Galois cohomology. Conjecture 3.2 (Order of Vanishing; Deligne-Beilinson) r(M ) = dimK H1f (M ∗ (1)) − dimK H0f (M ∗ (1)) According to the remark in [19] the duals of Hif (M ∗ (1)) should be considered as “motivic cohomology with compact support H2−i c (M )” and thus r(M ) is just their Euler characteristic. This explains why the Kummer duals M ∗ (1) are involved here. The link between the complex world, where the values L∗K (M ) live, and the p-adic world, where the Galois cohomology lives, is formed by the fundamental line in CK following the formulation of Fontaine and Perrin-Riou [22]: ΔK (M ) := dK (H0f (M ))−1 dK (H1f (M ))dK (H0f (M ∗ (1))∗ ) dK (H1f (M ∗ (1))∗ )−1 dK (MB+ )dK (tM )−1 . The relation of ΔK (M ) with the Betti and de Rham realisation of M is given by the following

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Conjecture 3.3 (Fontaine/Perrin-Riou) There exist an exact sequence of KR := R ⊗Q K-modules 0 h

/ H0f (M )R

c

/ H1f (M )R

rB

/ ker(αM )

∗ rB

/ Coker(αM )

/ (H1f (M ∗ (1))R )∗ c∗

/ (H0f (M ∗ (1))R )∗

/0

where by −R we denote the base change from Q to R (respectively from K to KR ), c is the cycle class map into singular cohomology, rB is the Beilinson regulator map and (if both H1f (M ) and H1f (M ∗ (1)) are nonzero so that M is of weight −1, then) h is a height pairing. Example 3.4 A),C) For the motive M = h0 (Spec(F )) the above exact sequence is just the R-dual of the following 0

/ O × ⊗Z R

r

F

/ Rr1 × Rr2

Σ

/R

/0,

where r is the Dirichlet(=Borel) regulator map (see [6] for a comparison of the Beilinson and Borel regulator maps) and r1 and r2 denote the number of real and complex places of F, respectively. B) The N´eron height pairing (see [4] and the references therein) <, >: A∨ (Q) ⊗Z R × A(Q) ⊗Z R → R induces an isomorphism A∨ (Q) ⊗Z R → HomZ (A(Q), R), the inverse of which gives the exact sequence for the motive M = h1 (A)(1). We assume this conjecture. Using property e) and change of rings of the functor d, it induces a canonical isomorphism (period-regulator map) ϑ∞ : KR ⊗K ΔK (M ) ∼ = 1KR .

(3.1)

Conjecture 3.5 (Rationality; Deligne-Beilinson) There is a unique isomorphism ζK (M ) : 1K → ΔK (M ) such that for every embedding K → C we have L∗K (M ) : 1C

ζK (M )C

/ ΔK (M )C

(ϑ∞ )C

/ 1C

In other words, the preimage ϑ∞ (L∗K (M )) of L∗K (M ) with respect to ϑ∞ generates the K-vector space ΔK (M ) if the determinant functor is identified

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with the highest exterior product. Thus up to a period and a regulator (the determinant of ϑ∞ with respect to a K-rational basis) the value L∗K (M ) belongs to K. The rationality enables us to relate L∗K (M ) to the p-adic world which we will describe now. Let S be a finite set of places of Q containing p, ∞ and the places of bad reduction of M, i.e. U := Spec(Z[ S1 ]) is an open dense subset of Spec(Z). Then we have complexes RΓc (U, Mp ), RΓf (Q, Mp ) and RΓf (Qv , Mp ) calculating the (global) cohomology Hic (U, Mp ) with compact support, the finite part of global and local cohomology, Hif (Q, Mp ) and Hif (Qv , Mp ), respectively, see appendix, section 7. These complexes fit into a distinguished triangle (see (7.5)) / RΓf (Q, Mp ) / v∈S RΓf (Qv , Mp ) /. RΓc (U, Mp ) (3.2) On the other hand motivic cohomology specializes to the finite parts of global Galois cohomology: Conjecture 3.6 There are natural isomorphisms H0f (M )Ql ∼ = H0f (Q, Ml ) 1 1 (cycle class maps) and Hf (M )Ql ∼ = Hf (Q, Ml ) (Chern class maps). ∼ H3−i (Q, M ∗ (1))∗ for all i, this Hence, as there is a duality Hif (Q, Ml ) = l f conjecture determines all cohomology groups Hif (Q, Ml ). Using properties d), g) and change of rings of the determinant functor, Conjecture 3.6 for l = p, the canonical isomorphisms (see appendix (7.9)) ηp (Mp ) :

1Kp

→ dKp (RΓf (Qp , Mp )) · dKp (t(Mp )),

(3.3)

ηl (Mp ) :

1Kp

→ dKp (RΓf (Ql , Mp )), l = p,

(3.4)

and the comparison isomorphisms (2.2) and (2.8) as well as (3.2), we obtain a canonical isomorphism in CKp (p-adic period-regulator map) ϑp (M ) : ΔK (M )Kp ∼ = dKp (RΓc (U, Mp ))−1 ,

(3.5)

which induces for any place λ above p an isomorphism in CKλ ϑλ (M ) : ΔK (M )Kλ ∼ = dKλ (RΓc (U, Mλ ))−1 .

(3.6)

Now let Tλ be a Galois stable Oλ -lattice of Mλ and RΓc (U, Tλ ) its Galois cohomology with compact support, see section 7. Here, Oλ denotes the valuation ring of Kλ . Note that by Artin-Verdier/Poitou-Tate duality (see (7.6)) the “cohomology” RΓc (U, Tλ ) with compact support can also be replaced by the complex RΓ(U, Tλ∗ (1))∗ ⊕ (Tλ∗ (1))+ where RΓ(U, Tλ∗ (1)) calculates as usual the global Galois cohomology with restricted ramification.

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The following Conjecture, for every prime p, gives a precise description of × the special L-value L∗K (M ) ∈ C× up to OK , i.e. up to sign if K = Q, where OK denotes the ring of integers in K: Conjecture 3.7 (Integrality; Bloch/Kato, Fontaine/Perrin-Riou) Assume Conjecture 3.5. Then for every place λ above p there exist a (unique) isomorphism ζOλ (Tλ ) : 1Oλ → dOλ (RΓc (U, Tλ ))−1 which induces via Kλ ⊗Oλ − the following map ζOλ (Tλ )Kλ : 1Kλ

ζK (M )Kλ

/ ΔK (M )K ϑλ (M ) / dK (RΓc (U, Mλ ))−1 . λ λ (3.7)

If we identify again the determinant functor with the highest exterior product, this conjecture can be rephrased as follows: ϑλ ϑ∞ (L∗K (M )) generates the Oλ -lattice dOλ (RΓc (U, Tλ ))−1 of dKλ (RΓc (U, Mλ ))−1 . In other words, this generator is determined up to a unit in Oλ . It can be shown that this conjecture is independent of the choice of S and Tλ . Example 3.8 (Analytic class number formula) For the motive M = h0 (Spec(F )) we have that r(M ) = r1 + r2 − 1 if F ⊗Q R ∼ = Rr1 × Cr2 −|Cl(OF )|R ∗ and that LQ (M ) = for the unit regulator R. Thus Conjectures 3.2, |μ(F )| 3.5 and 3.7 are all theorems in this case! For other known cases of these conjectures we refer the reader to the excellent survey article [19], where in particular the results of Burns-Greither [10] and Huber-Kings [25] are discussed. Another example will be discussed in the following section.

3.1 Equivalence to classical formulation of BSD I am very grateful to Matthias Flach for some advice concerning this section, in which we assume p = 2 for simplicity. In order to see that the above conjectures for the motive M = h1 (A)(1) of an abelian variety A are equivalent to the classical formulation involving all the arithmetic invariants of A one has to consider also integral structures for the finite parts of global and local Galois cohomology. For Tp we take the Tate-module Tp (A∨ ) of A∨ . In particular one can define perfect complexes of Zp -modules RΓf (Q, Tp ) and RΓf (Qv , Tp )

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such that the analogue of (3.2) holds, see [8, §1.5]. We just state some results concerning their cohomology groups Hif in the following Proposition 3.9 ([8, (1.35)-(1.37)]) (a)(global) If the Tate-Shafarevich group X(A/Q) is finite, then one has H0f (Q, Tp ) = 0 H1 (Q, Tp ) ∼ = A∨ (Q) ⊗Z Zp

H3f (Q, Tp ) ∼ = HomZ (A(Q)tors , Qp /Zp ), Hif (Q, Tp ) = 0 for i = 0, 1, 2, 3

f

and an exact sequence of Zp -modules 0

/

X(A/Q)(p)

/ H2f (Q, Tp )

/ HomZ (A(Q), Zp )

/ 0.

(b)(local) For all primes l one has H0f (Ql , Tp ) = 0, H1f (Ql , Tp ) ∼ = A∨ (Ql )∧p , Hif (Ql , Tp ) = 0 for i = 0, 1, where A∨ (Ql )∧p denotes the p-adic completion of A∨ (Ql ). Note that one has Hif (−, Tp ) ⊗Zp Qp ∼ = Hif (−, Mp ) for both the local and global versions. Recall from Example 2.2 B) that we have ΔQ (M ) = dQ (A∨ (Q) ⊗Z Q))dQ (HomZ (A(Q), Q))−1 dQ (H1 (A∨ (C), Q)+ )dQ (Lie(A∨ ))−1 In order to define the period and regulator we have to choose bases: we first fix P1∨ , . . . , Pr∨ ∈ A∨ (Q) (respectively P1 , . . . , Pr ∈ A(Q)), where r = rkZ (A∨ (Q)) = rkZ (A(Q)), such that setting TA∨ := ZPi∨ (respec d d d tively TA := ZPi ⊆ HomZ (A(Q), Z), where Pi denotes the obvious dual “basis”) we obtain A∨ (Q) ∼ = A∨ (Q)tor ⊕ TA∨

HomZ (A(Q), Z) = TAd .

(3.8)

Similarly we fix a Z-basis γ + = (γ1+ , . . . , γd++ ) of TB+ := H1 (A∨ (C), Z)+ and a Z-basis δ = (δ1 , . . . , δd+ ) of the Z-lattice LieZ (A∨ ) := Lie(B) = HomZ (Ω1B/Z (B), Z) of Lie(A∨ ), respectively. Here B/Z denotes the (smooth, but not proper) N´eron model of A∨ over Z. Thus we obtain an integral structure of ΔQ (M ) : ΔZ (M ) := dZ (TA∨ )dZ (TAd )−1 dZ (TB+ )dZ (LieZ (A∨ ))−1

(3.9)

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together with a canonical isomorphism 1Z

canZ

/ ΔZ (M )

(3.10)

induced by the above choices of bases.4 Define the period Ω+ ∞ (A) and the regulator RA of A to be the determinant of the maps αM and h with respect to the bases chosen above, respectively. Then Conjecture 3.5 tells us that ζQ (M ) =

L∗Q (M ) Ω+ ∞ (A) · RA

· canQ ,

(3.11)

where canQ : 1Q → ΔQ (M ) is induced from canZ by base change. Indeed, we have by definition of the period and regulator Ω+ ∞ (A)RA = (ϑ∞ )C ◦ (canQ )C and by Conjecture 3.5 L∗Q (M ) = (ϑ∞ )C ◦ (ζQ (M ))C L∗ (M )

in AutCC (1C ) = C× and thus ζQ (M ) differs from canQ by Ω+ Q(A)·R . A ∞ On the other hand, using among others property h) of the determinant functor, Proposition 3.9 and the identification TB+ ⊗Z Zp ∼ = Tp+ (induced from (2.2)) one easily verifies that there is an isomorphism ΔZp (M ) := ΔZ (M )Zp ∼ = dZp (RΓf (Q, Tp ))−1 dZp (Tp+ )dZp (LieZp (A∨ ))−1 · dZp (X(A/Q)(p))dZp (A(Q)(p))−1 dZp (A∨ (Q)(p))−1 where LieZp (A∨ ) := LieZ (A∨ ) ⊗Z Zp is a Zp -sublattice of t(Mp ) ∼ = H1 (A, OA ) ⊗Q Qp . In order to compare this with the integral structure RΓc (U, Tp ) of RΓc (U, Mp ) we have to introduce the local Tamagawa numbers cl (Mp ) [22, I §4]. We first assume l = p and we write φl ∈ G(F¯l /Fl ) for the geometric Frobenius automorphism at l. Then there is an exact sequence (cf. [22, I §4.2]) 0

/ TpIl

1−φl

/ TpIl

/ H1f (Ql , Tp )

/ H1 (I , T )GQl l p tors

/0

4 The choice of the basis Pi∨ of TA∨ induces a map Zr → TA∨ and, taking determinants, the isomorphism canP ∨ : dZ (Zr ) → dZ (TA∨ ). Similarly, we obtain canonical isomorphisms d , T + and Lie (A∨ ), respectively. Set can := can ∨ · canP d , canγ + and canδ for TA Z Z P B −1 −1 canP d · canγ + · canδ .

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which induces an isomorphism ψl : 1Zp → dZp ([ TpIl

/ TpIl ]) ∼ =

1−φl

GQ dZp (RΓf (Ql , Tp ))dZp (H1 (Il , Tp )torsl )−1 ∼ = dZp (RΓf (Ql , Tp )).

Here the first map arises as trivialization by the identity, the second comes from the above exact sequence (interpreted as short exact sequence of complexes) while the last comes again from trivializing by the identity according to Remark 1.3. GQ We define cl (Mp ) := |H1 (Il , Tp )torsl | and remark that (ψl )Qp differs # $ acyc def Ql / dZ H1 (Il , Tp )G 1Qp , from ηl (Mp ) precisely by the map 1Qp tors Qp p which appealing to Remark 1.3 we also denote by cl (Mp ). In other words we have (ψl )Qp = cl (Mp ) · ηl (Mp ).

(3.12)

Note also, that one has |A∨ (Ql ) ⊗Z Zp | = |Pl (Mp , 1)|−1 p · cl (Mp ) and that cl (Mp ) = 1 whenever A has good reduction at l. It can be shown [44, Exp. IX,(11.3.8)] that cl (Mp ) is the order of the p-primary part of the group of Fl -rational components (E/E 0 )(Fl ) ∼ = 0 ∼ E(Fl )/E (Fl ) = A(Ql )/A0 (Ql ) of the special fibre E := AFl of the smooth (but not necessarily proper) N´eron model A of A over Z.5 Here we write G 0 for the identity component of an algebraic group G while A0 (Ql ) denotes the image of A0 (Zl ) under the canonical isomorphism A(Zl ) ∼ = A(Ql ). For an elliptic curve E over Q this group can be identified with the set of points W 0 (Zl ) of W(Zl ) ∼ = E(Ql ) with non-singular reduction, where W ⊆ P2Z is the closed subscheme defined by a minimal Weierstrass equation over Z for E/Q and W 0 /Z is the smooth part of W/Z (cf. [46, IV cor. 9.1,9.2]). Now let l = p. Similarly one defines maps (both depending on the choice of δ) ψp : 1Zp → dZp (RΓf (Qp , Tp ))dZp (LieZp (A∨ )), cp (Mp ) : 1Qp → 1Qp , 5 The first isomorphism is a consequence of the theorem of Lang [32] that the map x → φ(x)x−1 on the k-rational points of a connected algebraic group over a finite field k (with Frobenius φ) is surjective.

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such that (ψp )Qp = cp (Mp ) · ηp (Mp ) holds.6 Using the integral version of (3.2), the maps ψl (analogously as ηl for ϑp (3.5)) induce a canonical map κp : ΔZp (M ) → dZp (RΓc (Q, Tp ))−1 where all of the terms dZp (X(A/Q)(p)), dZp (A(Q)(p))−1 and dZp (A∨ (Q)(p))−1 are trivialized by the identity. Hence, using again Remark 1.3, we have |X(A/Q)| (κp )Qp = cl (Mp ) · ϑp (3.13) |A(Q)tors ||A∨ (Q)tors | × modulo Z× p . Since ζZp (Tp ) equals κp ◦ canZp up to an element in Zp , it follows immediately from (3.13),(3.11) and (3.7) that

L∗Q (M ) Ω+ ∞ (A)

· RA

|X(A/Q)| cl (Mp ) ∨ |A(Q)tors ||A (Q)tors |

∼

mod Z× p.

(3.14)

be the formal group of A∨ over Zp , i.e. the for6 Assume that A has dimension d and let B mal completion of B along the zero-section in the fibre over p. Note that LieZp (A∨ ) can be with values in Zp (a good reference for foridentified with the tangent space tB (Zp ) of B mal groups is [17]). Furthermore we write Ga for the formal additive group over Zp , B◦ ◦ for the connected component of the identity of B and its fibre B over p, respecand B Again by Lang’s theorem we have tively, and Φ for the group of connected components of B. p )/B ◦ (Fp ). Moreover there are exact sequences Φ(Fp ) = B(F

/

0

/

B ◦ (Zp )

/

B(Zp )

Φ(Fp )

/

0

and

/

0

/ B◦ (Zp )

p) B(Z

/

◦ (Fp ) B

/

0.

Now the logarithm map a (Zp )d LieZp (A∨ ) ⊇ (pZp )d = G

o

∼ = log

p ) ⊆ A∨ (Qp )∧p = H1 (Qp , Tp ) B(Z f

induces the map ψp by trivializing all finite subquotients of the above line by the identity. Note that the first subquotient on the left has order pd . Using [5, ex. 3.11], which says that the Bloch-Kato exponential map coincides, up to the identification induced by the Kummer map, with the usual exponential map of the corresponding formal group, it is easy to see that cp (Mp ) := ηp−1 · (ψp )Qp = η¯p ◦ (ψp )Qp equals modulo Z× p cp (Mp )

=

◦ p−d |P (Mp , 1)|−1 p #B (Fp )(p)#Φ(Fp )(p) = #Φ(Fp )(p),

◦ (Fp )(p). For ellipwhere we used the relation |P (Mp , 1)|p = |P (Ml , 1)|p = p−d #B tic curves this is well known [45, appendix §16], the general case is an exercise using the description of the reduction of abelian varieties in [44, Exp. IX].

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For all primes p this implies the classical statement of the BSD-Conjecture up to sign (and a power of 2 due to our restriction p = 2).

4 The TNC - equivariant version The first equivariant version of the TNC with commutative coefficients (other than number fields) was given by Kato [28, 27] observing that classical Iwasawa theory is, roughly speaking, nothing else than the ETNC for a “big” coefficient ring. Inspired by Kato’s work Burns and Flach formulated an ETNC where the coefficients of the motive are allowed to be (possibly non-commutative) finite-dimensional Q-algebras, using for the first time the general determinant functor described in section 1 and relative algebraic Kgroups. Their systematic approach recovers all previous versions of the TNC and more over all central conjectures of Galois module theory. Huber and Kings [24] were the first to realize that the formulation of the ETNC by relative K-groups is equivalent to the perhaps more suggestive use of “generators,” i.e. maps of the form 1R → dR (?) in the category CR for various rings R instead, see also Flach’s survey [19, §6]. They used this approach to give - for motives of the form M ∗ (1 − k) with k big enough, i.e. with very negative weight the first version of a ETNC over general p-adic Lie extensions, which they call Iwasawa Main Conjecture (while in this survey we reserve this name for versions involving p-adic L-functions). While Burns and Flach use “equivariant” motives and L-functions in their general formalism, Fukaya and Kato realized that, at least for the connection with Iwasawa theory which we have in mind, it is sufficient to use non-commutative coefficients only for the Galois cohomology, but to stick to number fields as coefficients for the involved motives. In this survey we closely follow their approach. To be more precise, consider for any motive M the motive h0 (Spec(F )) ⊗ M (both defined over Q) for some finite Galois extension F of Q with Galois group G = G(F/Q). This motive has a natural action by the group algebra Z[G] and thus will be of particular interest for Iwasawa theory where a whole tower of finite extensions Fn of Q is considered simultaneously. Since there is an isomorphism of K-motives (for K sufficiently big) h0 (Spec(F ))K ⊗ M ∼ [ρ∗ ]nρ ⊗ M = ρ∈G

where ρ runs through all absolutely irreducible representations of G and nρ denotes the multiplicity with which it occurs in the regular representation of G on K[G], it suffices - on the complex side - to consider the collection

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of K-motives [ρ∗ ] ⊗ M and their L-functions or more precisely the corresponding leading terms and vanishing orders. Indeed, the C-algebra C[G] can be identified with ρ∈G Mnρ (C) and thus its first K-group identifies with × ∼ × C = center(C[G]) . In contrast, on the p-adic side, even more when ρ∈G integrality is concerned, such a decomposition for Zp [G] is impossible in general. This motivated Fukaya and Kato to choose the following form of the ETNC. In fact, in order to keep the presentation concise, we will only describe a small extract of their complex and much more general treatment. Let F be a p-adic Lie extension of Q with Galois group G = G(F/Q). By Λ = Λ(G) we denote its Iwasawa algebra. For a Q-motive M over Q we fix a GQ -stable Zp -lattice Tp = Tp (M ) of Mp and define a left Λ-module T := Λ ⊗Zp Tp on which Λ acts via multiplication on the left factor from the left while GQ acts diagonally via g(x ⊗ y) = x¯ g −1 ⊗ g(y), where g¯ denotes the image of g ∈ GQ in G. This is a “big Galois representation” in the sense of Nekovar [34]. Choose S as in the previous section and such that T is unramified outside S and denote, for any number field F , by GS (F ) the Galois group of the maximal outside S unramified extension of F . Then by Shapiro’s Lemma the cohomology of RΓ(U, T) with U = Spec(Z[1/S]) for example is nothing else than the perhaps more familiar Λ(G)-module HiIw (F, Tp ) := limF Hi (GS (F ), Tp ) ←− where the limit is taken with respect to corestriction and F runs over all finite subextensions of F/Q. Let K be a finite extension of Q, λ a finite place of K above p, Oλ the ring of integers of the completion Kλ of K at λ and assume that ρ : G → GLn (Oλ ) is a continuous representation of G which, for some suitable choice of a basis, is the λ-adic realisation Nλ of some K-motive N. We also write ρ for the induced ring homomorphism Λ → Mn (Oλ ) and we consider Oλn as a right Λ-module via the action by the transpose ρt on the left, viewing Oλn as set of column vectors (contained in Kλn ). Note that, setting M (ρ∗ ) := N ∗ ⊗ M, we obtain an isomorphism of Galois representations Oλn ⊗Λ T ∼ = Tλ (M (ρ∗ )), where Tλ (M (ρ∗ )) is the Oλ -lattice ρ∗ ⊗ Tp := Oλn ⊗Zp Tp of M (ρ∗ )λ on which GQ acts diagonally, via the contragredient(=dual) representation ρ∗ of ρ on the left factor.

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Now the equivariant version of Conjecture 3.7 reads as follows Conjecture 4.1 (Equivariant Integrality; Fukaya/Kato) There exists a (unique)7 isomorphism ζΛ (M ) := ζΛ (T) : 1Λ → dΛ (RΓc (U, T))−1 with the following property: For all K, λ and ρ as above the (generalized) base change Oλn ⊗Λ − sends ζΛ (M ) to ζOλ (Tλ (M (ρ∗ ))).

Note that this conjecture assumes Conjecture 3.7 for all K-motives M (ρ∗ ) with varying K. Furthermore, it is independent of the choice of S and of the lattices Tp (M ) and Tλ (M (ρ∗ )). One obtains a slight modification - to which we will refer as the Artinversion - of the above conjecture by restricting the representations ρ in question to the class of all Artin representations of G, (i.e. having finite image). If F/Q is finite, both versions coincide. Moreover, it is easy to see8 that in this situation the conjecture is equivalent to (the p-part of) Burns and Flach’s equivariant integrality conjecture [9, Conj. 6] for the Q-algebra Q[G] with Z-order Z[G]. Also, T = Zp [G] ⊗ Tp (M ) identifies with the induced representation F IndG GQ Tp (M ). Assume now that F = n Fn is the union of finite extensions Fn of Q with Galois groups Gn . Putting ζQp [Gn ] (M ) = Qp [Gn ] ⊗Λ ζΛ (M ) one recovers the “generator” δp (Gn , M, k)9 (for k big enough) in [24] as ζQp [Gn ] (M ∗ (1 − k)). Hence, up to shifting and Kummer duality, the Artin-version of Conjecture 4.1 7 In fact, Fukaya and Kato assign such an isomorphism to each pair (R, T) where R belongs to a certain class of rings containing the Iwasawa algebras for arbitrary p-adic Lie extensions of Q as well as the valuation rings of finite extensions of Qp and where T is a projective R-module endowed with a continuous GQ -action. Then ζ? (?) is supposed to behave well under arbitrary change of rings for such pairs. Moreover they require that the assignment T → ζR (T) is multiplicative for short exact sequences. Only this full set of conditions leads to the uniqueness [23, §2.3.5], while e.g. for finite a group G the map K1 (Zp [G]) → K1 (Qp [G]) need not be injective and thus ζZp [G] (?) might not be unique if considered alone. 8 If we assume that K is big enough such that K[G] decomposes completely into matrix algebras with coefficients in K, then the equivariant integrality statement (inducing (absolute) integrality for M (ρ∗ ) for all Artin representations of G) amounts to an integrality statement for the generator ζKλ [G] (M ) := Kλ [G] ⊗Zp [G] ζZp [G] (M ) and thus Burns and Flach’s version for the Q-algebra K[G] with order OK [G]. Using the functorialities of their construction [9, Thm. 4.1], it is immediate that taking norms leads to the conjecture for the pair (Q[G], Z[G]). 9 To be precise, this is only morally true, since Huber and Kings take for the definition of their generators the leading coefficients of the modified L-function without the Euler factors in S. It is not clear to what extent this is compatible with our formulation above.

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for F is (morally) equivalent to [24, Conj. 3.2.1]. Hence, using [24, Lem. 6.0.2] we obtain the following Proposition 4.2 (Huber/Kings) Assume Conjectures 3.2, 3.5 and 3.7 for all M (ρ∗ ) where ρ varies over all absolutely irreducible Artin representations of G. Then the existence of ζΛ(G) (M ) satisfying the Artin-version of Conjecture 4.1 is equivalent to the existence of ζΛ(Gn ) (M ) for all n. In general, as remarked in footnote 7, the isomorphism ζZp [Gn ] (M ) might not be unique (if it is considered alone). But it is realistic to hope uniqueness for infinite G (cf. [29]) and then the previous zeta isomorphism would be unique by the requirement that ζZp [Gn ] (M ) is equal to Zp [Gn ] ⊗Λ(G) ζΛ(G) (M ). Indeed, this is true at least if G is big enough, see [23, Prop. 2.3.7]. Moreover, as Huber and Kings [24, §3.3] pointed out, by twist invariance (over trivializing extensions F/Q for a given motive) arbitrary zeta-isomorphisms ζΛ (M ) are reduced to those of the form ζZp [G(F/Q)] (Q) for the trivial motive Q and where F runs through all finite extensions of the field Q. Question: Does the Artin-version imply the full version of Conjecture 4.1 ?

5 The functional equation and -isomorphisms The L-function of a Q-motive satisfies conjecturally a functional equation, which we want to state in the following way LQ (M, s) = (M, s)

L∞ (M ∗ (1), −s) LQ (M ∗ (1), −s) L∞ (M, s)

where the factor L∞ at infinity is built up by certain Γ-factors and certain powers of 2 and π depending on the Hodge structure of MB . The -factor decomposes into local factors (M, s) = v (M, s), v∈S

whose definition for finite places v is recalled in footnotes 11 and 14; ∞ (M, s) is a constant equal to a power of i.10 We assume this conjecture. Then, taking leading coefficients induces L∗Q (M ) = (−1)η (M )

L∗∞ (M ∗ (1)) ∗ LQ (M ∗ (1)) L∗∞ (M )

where (M ) = v (M ) with v (M ) = (M, 0) and η denotes the order of vanishing at s = 0 of the completed L-function L∞ (M ∗ (1), s)LQ (M ∗ (1), s). 10 We fix once and for all the complex period 2πi, i.e. a square root of −1, and, for every l, the l-adic period t = “2πi”, i.e. a generator of Zl (1).

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Example 5.1 For the motive M = h1 (A)(1) of an abelian variety one has L∞ (M, s) = L∞ (M ∗ (1), s) = 2(2π)−(s+1) Γ(s + 1), L∗∞ (M ) = L∗∞ (M ∗ (1)) = π −1 , ∞ (M ) = −1 and η = 0. It is in no way obvious that the ETNC is compatible with the functional equation and Artin-Verdier/Poitou-Tate duality. The following discussion is a combination and reformulation of [9, section 5] and [37, Appendix C]. In order to formulate the precise condition under which the compatibility holds we first return to the absolute case and define “difference” terms L∗dif (M ) := L∗Q (M )L∗Q (M ∗ (1))−1 = (−1)η (M )

L∗∞ (M ∗ (1)) L∗∞ (M )

and Δdif (M ) := dQ (MB )dQ (MdR )−1 . We obtain an isomorphism ϑP D : ΔQ (M ) · ΔQ (M ∗ (1))∗ ∼ = Δdif (M ) which arises from the mutual cancellation of the terms arising from motivic cohomology, the following isomorphism MB+ ⊕ (MB∗ (1)+ )∗ ∼ = MB+ ⊕ MB (−1)+ ∼ = MB , where the last map is (x, y) → x + 2πiy, and from the Poincare duality exact sequence 0

/ (tM ∗ (1) )∗

/ MdR

/ tM

/ 0.

(5.1)

On the other side define an isomorphism ∼ ϑdif ∞ : Δdif (M )R = 1R applying the determinant to (2.4) and to the following isomorphism (C ⊗Q MB )+ = (R ⊗Q MB+ ) ⊕ (R(2πi)−1 ⊗Q MB− ) ∼ = (MB )R

(5.2)

where the last map is induced by R(2πi)−1 → R, x → 2πix. Due to the autoduality of the exact sequence of Conjecture 3.3 (see [9, Lem. 12]) we have a commutative diagram ΔQ (M )R · ΔQ (M ∗ (1))∗R

/ Δdif (M )R ϑdif ∞

ϑ∞ (M )·ϑ∞ (M ∗ (1))∗

1R

D ϑP R

id1R

/ 1R

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Thus we obtain the following Proposition 5.2 (Rationality) Assume that Conjecture 3.5 is valid for the Qmotive M. Then it is also valid for its Kummer dual M ∗ (1) if and only if there exists a (unique) isomorphism ζ dif (M ) : 1Q → Δdif (M ) such that we have L∗dif (M ) : 1C

ζ dif (M )C

(ϑdif ∞ )C

/ Δdif (M )C

/ 1C .

Putting tH (M ) := := dimK grr (MdR ) r∈Z rh(r) with h(r) r (= dimKλ gr (DdR (Mλ ))) for a K-motive M and noting that tH (M ) = tH (det(M )), we have in fact the following Theorem 5.3 (Deligne [18, Thm. 5.6], Burns-Flach [9, Thm. 5.2]) If the motive det(M ) is of the form Q(−tH (M )) twisted by a Dirichlet character, then ζ dif (M ) exists. Deligne [18] conjectured that the condition of the theorem is satisfied for all motives. It is known to hold in all examples A)-E). See (5.6) below for the rationality statement which is hidden in the formulation of this theorem. Now we have to check the compatibility with respect to the p-adic realizations. To this aim we define the isomorphism ϑdif : Δdif (M )Qp ∼ dQp (RΓ(Ql , Mp )) = dQp (Mp ) · p SS∞

as follows: Apply the determinant to (2.1) and multiply the resulting isomorphism by iddQ (MdR )−1 · Θl (Mp ) : dQ (MdR )−1 dQp (RΓ(Ql , Mp )) Qp → Qp

SS∞

l∈SS∞

where Θl (Mp ) = ηl (M ) · ηl (M ∗ (1)) is defined in the appendix, section 7. On the other hand Artin-Verdier/Poitou-Tate Duality induces the following isomorphism dQp (RΓc (U, Mp ))

−1

−1 ∼ = dQp (RΓ(U, Mp )) dQp (

RΓ(Qv , Mp ))

v∈S ∗ ∗ ∗ + ∗ ∼ = dQp (RΓc (U, Mp (1)) )dQp ((Mp (1) ) )

dQp (RΓ(Qv , Mp ))

v∈S ∗ ∗ + + ∼ = dQp (RΓc (U, Mp (1)) )dQp (Mp (−1) )dQp (Mp )

l∈SS∞

dQp (RΓ(Ql , Mp )).

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Using the identification Mp+ ⊕ Mp (−1)+ = Mp+ ⊕ Mp− (−1) ∼ = Mp ,

(5.3)

where the last map is induced by multiplication with the p-adic period t = “2πi” : Mp− (−1) → Mp− , we obtain ϑAV : dQp (RΓc (U, Mp ))−1 · dQp (RΓc (U, Mp∗ (1))∗ )−1 ∼ = p dQp (Mp ) · dQp (RΓ(Ql , Mp )). SS∞

Again one has to check the commutativity of the following diagram (cf. [9, Lem. 12]) ΔQ (M )Qp ·ΔQ (M ∗ (1))∗ Qp ϑp (M )·ϑp (M ∗ (1))∗

(ϑP D )Qp

dQp (RΓc (U,Mp ))−1 dQp (RΓc (U,Mp∗ (1))∗ )−1

ϑAV p

/

/

dQp (Mp )

Δdif (M )Qp

SS∞

ϑdif p dQp (RΓ(Ql ,Mp ))

Note that analogous maps exist and analogous properties hold also if we replace Mp by a Galois stable Zp -lattice Tp or even by the free Λ-module T. Thus we obtain the following Proposition 5.4 (Integrality) Assume that Conjecture 3.7 is valid for the Q-motive M. Then it is also valid for its Kummer dual M ∗ (1) if and only if there exists a (unique) isomorphism ζZdif (T ) : 1 → d (T ) · dZp (RΓ(Ql , Tp )). p Z Z p p p p SS∞

which induces via Qp ⊗Zp − the following map 1Qp

ζ dif (M )Q

p

/

Δdif (M )Qp

dif ϑp

/

dQp (Mp ) ·

SS∞

dQp (RΓ(Ql , Mp )).

If this holds we have, using the above identifications, the functional equation ζZp (Tp ) = (ζZp (Tp∗ (1))∗ )−1 · ζZdif (Tp ) p Note that (ζZp (Tp∗ (1))∗ )−1 is the same as ζZp (Tp∗ (1))∗ ·iddZp (RΓc (U,Tp∗ (1))∗ ) according to Remark 1.2(i). Needless to say, all of the above has an analogous version for K-motives whose formulation we leave to the reader. Then it is clear how the equivariant version of this proposition looks like:

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Proposition 5.5 (Equivariant Integrality) Assume that Conjecture 4.1 is valid for the Q-motive M. Then it is also valid for its Kummer dual M ∗ (1) if and only if there exists a (unique) isomorphism ζΛdif (M ) : 1Λ → dΛ (T) · dΛ (RΓ(Ql , T)) SS∞

with the following property (∗) : For all K, λ and ρ as before Conjecture 4.1 the (generalized) base change dif Oλn ⊗Λ − sends ζΛdif (M ) to ζO (Tλ (M (ρ∗ ))), the analogue of ζZdif (Tp ) for λ p ∗ the K-motive M (ρ ). If this holds we have the functional equation ζΛ (M ) = (ζΛ (M ∗ (1))∗ )−1 · ζΛdif (M ). Thus we formulate the Conjecture 5.6 (Local Equivariant Tamagawa Number Conjecture) The isomorphism ζΛdif (M ) in the previous Proposition exists (uniquely).

5.1 -isomorphisms One obtains a refinement of the above functional equation if one looks more closely at which part of the Galois cohomology (and comparison isomorphisms) the factors occurring in L∗dif (M ) belong precisely. We first recall from [37] the equality L∗∞ (M ∗ (1)) d− (M )−d+ (M ) −(d− (M )+tH (M )) = ±2 (2π) Γ∗ (−j)−hj (M ) L∗∞ (M ) j∈Z

(5.4) where Γ (−j) is defined to be Γ(j) = (j − 1)! if j > 0 and lims→j (s − j)Γ(s) = (−1)j ((−j)!)−1 otherwise. The factor (2π)−(d− (M )+tH (M )) arises as follows. Assume for simplicity that det(M ) = Q(−tH (M )). Then fixing a Q-basis γ = (γ + , γ − ) of MB and ω = (δM , δM ∗ (1) ) of MdR which induce the canonical basis (cf. Example A)) of det(M )B and det(M )dR , respectively, gives rise to a map ∗

1Q

canγ,ω

/ dQ (MB )dQ (MdR )−1 .

Base change and the comparison isomorphism (2.3) induce (canγ,ω )C

d(g∞ )

1C −−−−−−→ dQ (MB )C dQ (MdR )−1 −−−→ 1C C −

(5.5)

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× whose value Ωdif ∞ in C is nothing else than the inverse of the determinant over C of the comparison isomorphism

C ⊗Q det(M )B → C ⊗Q det(M )dR −tH (M ) and thus Ωdif . But note that due to the definition of (5.2) ∞ = (2πi) dif the above map differs from (ϑ∞ )C ◦ (canγ,ω )C by the factor (2πi)−d− (M ) . Thus we obtain an explanation of the factor (2πi)−(tH (M )+d− (M )) . Moreover, Theorem 5.3 and Proposition 5.2 tell us that

L∗dif (M ) (2πi)−d− (M ) Ωdif ∞

= ±2d− (M )−d+ (M )

∞ (M ) −(t (M )+d− (M )) H i ·

l∈SS∞

l (M )

Γ∗ (−j)−hj (M )

j∈Z

(5.6) is rational and that ζdif (M ) is the map canγ,ω multiplied by this rational number. The factor 2d− (M )−d+ (M ) arises as quotient of the Tamagawa factors of M and M ∗ (1) at infinity. One can either cover it by defining Θ∞ (see below) or changing the last map of the identification in (5.3) as follows: on the summand Mp+ multiply with 2 and on Mp− by 12 (as Fukaya and Kato do). The map (5.5) induces 1Qp → Δdif (M )Qp ∼ = dQp (Mp )dQp (DdR (Mp ))−1 and furthermore dQp (gdR )BdR

(canγ,ω )B

dR 1BdR −−−−−−−− → Δdif (M )BdR −−−−−−−−−→ 1BdR

× whose value Ωdif in BdR is nothing else than the inverse of the determinant p over BdR of the comparison isomorphism

BdR ⊗Qp DdR (det(M )p ) → BdR ⊗Qp det(M )p and thus Ωdif = (2πi)−tH (M ) , where we consider the p-adic period t = “2πi” p nr , the completion of the as an element of BdR . Note that (BdR )Ip = Q p nr maximal unramified extension Qp of Qp . We need the following Lemma 5.7 ([23, Prop. 3.3.5],[37, C.2.8]) The map p (M ) · Ωdif p · (can)BdR comes from a map −1 dR (Mp ) : 1Q . nr → dQ nr (Mp )dQ nr (DdR (Mp )) p

p

p

Moreover, let L be any finite extension of Qp . Then a similar statement holds for any finite dimensional L-vector space V with continuous GQp -action

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instead of Mp 11 . We write dR (V ) for the corresponding map, which is defined nr ⊗ := Q over L Qp L (see [23, Prop. 3.3.5] for details). p For any V as in the lemma we define an isomorphism # $ p,L (V ) : 1L → dL (RΓ(Qp , V ))dL (V ) L as product of ΓL (V ) := Z Γ∗ (j)−h(−j) , Θp (V ) (see (7.10) in the appendix) and dR (V ), where h(j) = dimL grj DdR (V ). := W (Fp )⊗Z Now let T be a Galois stable O := OL -lattice of V and set O p O, where W (Fp ) denotes the Wittring of Fp . The following conjecture is a local integrality statement Conjecture 5.8 (Absolute -isomorphism) There exists a (unique) isomorphism # $ p,O (T ) : 1O → dO (RΓ(Qp , T ))dO (T ) O which induces p,L (V ) by base change L ⊗O −. This conjecture, which is equivalent to conjecture CEP (V ) in [22, III 4.5.4], or more precisely its equivariant version below is closely related to the conjecture δZp (V ) [37] via the explicit reciprocity law R´ ec(V ), which was conjectured by Perrin-Riou and proven independently by Benois [1], Colmez [16], and Kurihara/Kato/Tsuji [26]. In particular, the above conjecture is known for ordinary crystalline p-adic representations [37, 1.28,C.2.10] and for certain semi-stable representations, see [3]. To formulate an equivariant version, define # $ nr [[G]] = lim W (F ) ⊗ := Z Λ p Zp Zp [G/Gn ] , p ←− n nr = W (F ) denotes the ring of integers of Q nr . We assume L = Q where Z p p p p and set as before T := Λ ⊗Zp T (but later T might differ from our global Tp ). We write T (ρ∗ ) for the O-lattice ρ∗ ⊗ T of ρ∗ ⊗ V, which we assume de Rham.

11 p (V ) = (Dpst (V )) where Dpst (V ) is endowed with the linearized action of the Weil-group and thereby considered as a representation of the Weil-Deligne group, see [20]. Furthermore, we suppress the dependence of the choice of a Haar measure and of t = “2πi” in × the notation. The choice of t = (tn ) ∈ Zp (1) determines a homomorphism ψp : Qp → Qp 1 with ker(ψp ) = Zp sending pn to tn ∈ μpn .

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Conjecture 5.9 (Equivariant -isomorphism) There exists a (unique)12 isomorphism # $ p,Λ (T) : 1Λ → dΛ (RΓ(Qp , T))dΛ (T) Λ such that for all ρ : G → GLn (O) ⊆ GLn (L), L a finite extension of Qp with valuation ring O, we have On ⊗Λ p,Λ (T) = p,O (T (ρ∗ )). If T = Tp ⊆ Mp is fixed we also write p,Λ (M ) for p,Λ (T). This equivariant version has recently been proved by Benois and Berger [2] in the cyclotomic situation for crystalline representations. Similarly we proceed in the case l = p, formulating the analogues of Conjectures 5.8 and 5.9 just in one Conjecture 5.10 There exists a (unique) 13 isomorphism l,Λ (T) : 1Λ → dΛ (RΓ(Ql , T))Λ such that for all ρ : G → GLn (L), L a finite extension of Qp , we have On ⊗Λ l,Λ (T) = l,O (T (ρ∗ )). Here l,O (T (ρ∗ )) is the analogue of the above with respect to O instead of Λ and required to induce L ⊗O l,O (T (ρ∗ ))) = Θl (V ) · l (V (ρ∗ )) and its existence is part of the conjecture14 . For commutative Λ this conjecture was proved by S. Yasuda [48] and it seems that he can extend his methods to cover the non-commutative case, too. 12 Again, Fukaya and Kato assign such an isomorphism to each triple (R, T, t), where R is as before, T is a projective R-module endowed with a continuous GQp -action and t is a generator of Zp (1). Then p,? (?) is supposed to behave well under arbitrary change of rings for such pairs. Moreover they require that the assignment T → p,R (T) is multiplicative for short exact sequences, that it satisfies a duality relation when replacing T by T∗ (1), that the group Gab Qp acts on a predetermined way (modifying t) compatible in a certain sense with the Frobenius induced from the absolute Frobenius of Fp . Only this full set of ring homomorphism on Λ conditions may lead to the uniqueness in general. 13 A similar comment to that in footnote 12 applies here. 14 l (V ) = (V ) where V is considered as representation of the Weil-Deligne group of Ql and where we suppress the dependence of the choice of a Haar measure and of t = 2πi in the × notation. The choice of t = (tn ) ∈ Zl (1) determines a homomorphism ψl : Ql → Q¯l with Ker(ψl ) = Zl sending l1n to tn ∈ μln . The formulation of this conjecture is equivalent to [23, conj. 3.5.2] where the constants 0 are used instead of and where θl does not occur. More precisely, our l,Λ (T) equals 0,Λ (Ql , T, ξ) · sl (T ) in [23, 3.5.2,3.5.4].

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If T = Tp ⊆ Mp we also write l,Λ (M ) for l,Λ (T). ∞ (M ) Finally we set ∞,Λ (M ) = ±2d− (M )−d+ (M ) −(tH(M )+d− (M )) where the i sign is that which makes (5.4) correct. The following result is now immediate. Theorem 5.11 (cf. [23, Conj. 3.5.5]) Assume Conjectures 5.8, 5.9 and 5.10. Then Conjecture 5.6 holds, ζΛdif (M ) = v∈S v,Λ (M ) and we have the functional equation ζΛ (M ) = (ζΛ (M ∗ (1))∗ )−1 · v,Λ (M ). v∈S

6 p-adic L-functions and the Iwasawa main conjecture A p-adic L-function attached to a Q-motive M should be considered as a map on a certain class of representations of G which interpolates the L-values of the twists M (ρ∗ ) at s = 0. The experience from those cases where such p-adic L-functions are known to exist, shows that one has to modify the complex L-values by certain factors before one can hope to obtain a p-adic interpolation (cf. [12, 13] or [37] ). The reason for this becomes clearer if one considers the Galois cohomology involved together with the functional equation; in fact, that was the main motivation of the previous section. In order to evaluate e.g. the ζ-isomorphism or a modification of it at a representation ρ over a finite extension L of Qp , one needs that the complex ρ⊗Λ C, where C is (a modification of) RΓc (U, T), becomes acyclic: then the induced map 1L → dL (ρ ⊗LΛ C) → 1L can be considered as value in K1 (L) = L× at ρ. In general, RΓc (U, T) does not behave well enough and will have to be replaced by some Selmer complex, which we well achieve in two steps. This modification corresponds to a shifting of certain Euler- and -factors from one side of the functional equation to the other such that both sides are ‘balanced’. Though the following part of the theory holds in much greater generality (e.g. in the ordinary good reduction case, but not in the supersingular good reduction case) we just discuss the case of abelian varieties in order to keep the situation as concise as possible. Thus let A be an abelian variety over Q with good ordinary reduction at a fixed prime p = 2 and set M = h1 (A)(1) as before. Let F∞ be an infinite p-adic Lie extension of Q with Galois group G. For simplicity we assume also that G has no element of order p, hence its Iwasawa algebra Λ = Λ(G) is a regular ring. Due to our assumption on the reduction type of A, we have the following fact: There is a unique Qp -subspace Vˆ of V = Mp which is stable under the

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action of GQp and such that 0 DdR (Vˆ ) ∼ (V ). = DdR (V )/DdR

(6.1)

More precisely, Vˆ = Vp (A∨ ) where A∨ denotes the formal group of the dual abelian variety A∨ , i.e. the formal completion of the N´eron model A/Zp of Then (6.1) arises from the A∨ along the zero section of the special fibre A. 0 0 ∼ unit root splitting DdR (Vp (A)) = DdR (V ) ⊆ DdR (V ) (see [35, 1.31]) which 0 is induced by applying DdR (−) to the exact sequence of GQp -modules / V (A∨ ) p

0

/ Vp (A∨ )

/ Vp (A)

/ 0.

Let T be the GQ -stable Zp -lattice Tp (A∨ ) of V and set Tˆ := T ∩ Vˆ , a GQp -stable Zp -lattice of Vˆ . As before let T denote the big Galois represenˆ := Λ ⊗Z Tˆ similarly. Then T ˆ is a GQ -stable tation Λ ⊗Zp T and put T p p sub-Λ-module of T. In fact, it is a direct summand of T and we have an isomorphism of Λ-modules15 ∼ = ˆ β : dΛ (T+ ) − → dΛ (T).

(6.5)

In this good ordinary case one can now first replace RΓc (U, T) by the ˆ T) (see (7.11)) which fits into the following Selmer complex SCU := SCU (T, 15 which arises as follows: Choose a basis γ + = (γ1+ , . . . , γr+ ) of H1 (A∨ (C), Q)+ and γ − = (γ1− , . . . , γr− ) of H1 (A∨ (C), Q)− , which gives rise to a Zp -basis of T + ∼ = (H1 (A∨ (C), Z) ⊗Z Zp )+ and T − ∼ = (H1 (A∨ (C), Z) ⊗Z Zp )− respectively, where r = d+ (M ) = d− (M ). Then we obtain isomorphisms Λr ∼ = T+ and φ : dΛ (Λr ) → dΛ (T + ) using the Λ-basis 1+ι ⊗ + 1−ι ⊗ γj− , 1 2 2 choose a Q-basis δ = (δ1 , . . . , δr ) of Lie(A∨ ) γj+

such that the isomorphism

(6.2)

≤ j ≤ r, of T+ . On the other hand one can (e.g. a Z-basis of LieZ (A∨ ) as in section 3.1)

∼ ∼ ˆ ∼ ˆ dQp (Qrp )Q nr = dQp (Qp ⊗Q tM )Q nr = dQp (DdR (V ))Q nr = dQp (V )Q nr , p

p

p

(6.3)

p

which is induced by δ, (2.8), (6.1) and dR (Vˆ ) (according to Lemma 5.7, but note that

p (Vˆ ) = 1 due to the good reduction), comes from an isomorphism ∼ ˆ dZp (Zrp )Z nr = dZp (T )Z nr , p

p

nr := W (F ). Then base change Λ ⊗ where Z − induces an isomorphism p p Znr p

ˆ . ∼ ψ : dΛ (Λ )Λ = dΛ (T)Λ r

Now β = βγ,δ is ψ ◦ φ.

(6.4)

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distinguished triangle / SCU

RΓc (U, T)

/ T+ ⊕ RΓ(Qp , T) ˆ

/

(6.6)

and thus induces an isomorphism ˆ dΛ (RΓc (U, T))−1 ∼ = dΛ (SCU )−1 dΛ (T+ )dΛ (RΓ(Qp , T)). The p-adic L-function LU = LU,β (M, F∞ /Q) arises from the zetaisomorphism ζΛ (M ) by a suitable cancellation of the two last terms, compatible with the functional equation. This is achieved by putting LU := LU,β := ˆ −1 · ζΛ (M )) : 1 → dΛ (SCU )−1 . (β · iddΛ (SCU )−1 dΛ (T) ˆ −1 ) ◦ (p,Λ (T) Λ Λ

(6.7) In order to arrive at a p-adic L-function which is independent of U one has ˆ T) (see (7.12)), to replace SCU by another Selmer complex SC := SC(T, which fits into a distinguished triangle / SC

SCU

/

l∈S{p,∞}

ˆ RΓf (Ql , T)

/,

(6.8)

where for l = p ˆ ∼ ˆ Il −−−→ T ˆ Il ] RΓf (Ql , T) = [T 1−ϕl

(6.9)

ˆ Il by a projective resolution if necessary in the derived category. Replacing T and using the identity isomorphism of it we obtain isomorphisms ˆ −1 ζl (M ) = ζl (M, F∞ /Q) : 1Λ → dΛ (RΓf (Ql , T)) and we define the p-adic L-function as L = Lβ (M ) = LU,β · ζl (M ) : 1Λ → dΛ (SC)−1 . Λ

(6.10)

(6.11)

l∈S{p,∞}

Let Υ be the set of all primes l = p such that the ramification index of l in F∞ /Q is infinite. Note that Υ is empty if G has a commutative open subgroup. ˆ Il = 0 and thus ζl (M ) = 1 in K1 (Λ) for Lemma 6.1 [23, Prop. 4.2.14(3)] T all l in Υ. Let us derive the interpolation property of LU and L. Whenever Ln ⊗LΛ SCU is acyclic for a continuous representation ρ : G → GLn (OL ),

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nr L a finite extension of Qp , we obtain an element LU (ρ) ∈ L isomorphism

1L˜

Ln ⊗Λ LU

/ d ˜ (Ln ⊗L SCU )−1 Λ L

acyclic

/ 1L˜

×

from the

(6.12) ×

nr ) can be considered as an element of L nr . ˜ → K1 (L which via K1 (L) Let K be a finite extension of Q, ρ : G → GLn (OK ) an Artin representation, [ρ∗ ] the Artin motive corresponding to ρ∗ . Fix a place λ of K above p, put L := Kλ and consider the L-linear representation of GQ or its restriction to GQl

W := M (ρ∗ )λ = [ρ∗ ]λ ⊗Qp Mp and the GQp -representation ˆ := [ρ∗ ]λ ⊗Q Vˆ . W p For a GQp -representation V define PL,l (V, u) := detL (1−ϕl u|V Il ) ∈ L[u] if l = p and PL,p (V, u) := detL (1 − ϕp u|Dcris (V )) ∈ L[u] otherwise. Some conditions for acyclicity are summarized in the next Proposition 6.2 ([23, 4.2.21, 4.1.6-8]) Assume the following conditions: (i) Hfj (Q, W ) = Hfj (Q, W ∗ (1)) = 0 for j = 0, 1, (ii) PL,l (W, 1) = 0 for any l ∈ Υ (respectively for any l ∈ S {p, ∞}). ˆ , u)−1 }u=1 = 0 and PL,p (W ˆ ∗ (1), 1) = 0. (iii) {PL,p (W, u)PL,p (W n L Then the following complexes are acyclic: L ⊗Λ,ρ SC (respectively Ln ⊗LΛ,ρ SCU ), RΓf (Ql , W ) = Ln ⊗LΛ,ρ RΓf (Ql , T), for any l ∈ Υ (respectively for any l ∈ S {p}). Furthermore, there are quasi-isomorphisms ˆ ) → RΓf (Qp , W ) and RΓf (Qp , W ˆ ) → RΓ(Qp , W ˆ ). RΓ(Qp , W Finally, assuming Conjectures 3.2 and 3.6, LK (M (ρ∗ ), s) has neither a zero nor a pole at s = 0. Henceforth we assume the conditions (i)-(iii). We define Ω∞ (M (ρ∗ )) ∈ C× to be the determinant of the period map C ⊗R αM (ρ∗ ) with respect to the K-basis which arise from γ (respectively δ) and the basis given by ρ. It is easy to see that we have d+ (ρ) − Ω∞ (M (ρ∗ )) = Ω+ Ω∞ (M )d− (ρ) , ∞ (M )

(6.13) ∼ =

± ± where d± (ρ) = dimK ([ρ]± → B ) and Ω∞ (M ) is the determinant of C ⊗Q MB − ± C ⊗Q tM with respect to the basis γ and δ. Assuming Conjecture 3.5 we have

LK (M (ρ∗ ), 0) ∈ K ×. Ω∞ (M (ρ∗ ))

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We claim that, using Proposition 7.2, the isomorphism LU (ρ) 1

ζΛ (M )(ρ)= ∗

ϑλ ◦ζ(M (ρ ))

ˆ

/ d(RΓc (U, W ))−1 · p,Λ (T)

−1

(ρ)

ˆ )−1 =· p,L (W

ˆ , W ))−1 d(W ˆ )−1 d(W + ) d(SCU (W

id·β(ρ)

/

/ d(SC (W ˆ , W ))−1 U

acyc

/1

(we suppress for ease of notation the subscripts and remind the reader of our convention in Remark 1.2) is the product of the following automorphisms of 1 : (1) LK (M (ρ∗ ), 0)Ω∞ (M (ρ∗ ))−1 , ˆ )−1 = ΓQ (Vˆ )−1 , (2) ΓL (W p (3) Ωp (M (ρ∗ )) which is, by definition, the composite ˆ) d(W

ˆ )−1 · dR (W

/ d(DdR (W ˆ ))

# $ d (M (ρ∗ )+ B )L

+ d(gλ )

t d(gdR )

· canγ,δ

/ d(tM (ρ∗ ) )

/

(6.14) / d(W ) +

β(ρ)

/ d(W ˆ)

0 ˆ ) = 0 for the second isomorphism and where we apply where we use DdR (W Remark 1.2 to obtain an automorphism of 1,16 ηl (W ) / d(RΓf (Ql , W )) acyc / 1 (4) l∈S{p,∞} PL,l (W, 1) : 1 where the first map comes from the trivialization by the identity and the second from the acyclicity, ˆ , u)−1 }u=1 : (5) {PL,p (W, u)PL,p (W ˆ )−1 ηp (W )·ηp (W

quasi / d(RΓf (Qp , W ))d(RΓ(Qp , W / 1, ˆ ))−1 ˆ ) = t(W ˆ ) and the quasi-isomorphisms where we use that t(W ) = DdR (W mentioned in Proposition 6.2, and

1

ˆ

∗

∗

ηp (W (1)) acyc ˆ ∗ (1), 1) : 1 − ˆ ∗ (1))) − (6) PL,p (W −−−−−−→ d(RΓf (Qp , W −−→ 1, ∗ 0 ˆ ˆ where we use again that t(W (1)) = DdR (W ) = 0.

16 Using Remark 1.2(i) it is easy to see that this amounts to taking the product of the following isomorphisms and identifying the target with 1 afterwards 1

# $ −1 , d (M (ρ∗ )+ B )L d(tM (ρ∗ ) )

canγ,δ

/

t id− ·d(gdR )

/

ˆ ))−1 d(tM (ρ∗ ) ), d(DdR (W

1 1

id− ·β(ρ)

/

1

+ id− ·d(gλ )

# $ −1 d(W + )d (M (ρ∗ )+ , B )L

/

ˆ )−1 dR (W

1

/

ˆ )−1 d(DdR (W ˆ )), d(W

ˆ )d(W + )−1 , d(W

# $ −1 ˆ ))−1 and d(W + )−1 , where the identity maps are those of d (M (ρ∗ )+ , d(DdR (W B )L respectively.

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In order to describe the interpolation property for L we need in addition to (6.8) and Lemma 6.1 another Lemma 6.3 ([23, Lem. 4.2.23]) Let l = p be a prime that does not belong to ˆ is acyclic if and only if PL,l (W, 1) = 0. If this Υ. Then Ln ⊗LΛ,ρ RΓf (Ql , T) holds then we have ζl (M )(ρ) = PL,l (W, 1)−1 . Thus we obtain the following Theorem 6.4 ([23, Thm. 4.2.26]) Under the conditions (i)-(iii) from Propoˆ the sition 6.2 and assuming Conjecture 4.1 for M and Conjecture 5.9 for T value L(ρ) (respectively LU (ρ)) is equal to LK (M (ρ∗ ), 0) · Ωp (M (ρ∗ )) · ΓQp (Vˆ )−1 · Ω∞ (M (ρ∗ )) ˆ , u)−1 }u=1 · PL,p (W ˆ ∗ (1), 1) · · {PL,p (W, u)PL,p (W PL,l (W, 1), l∈B

where B = Υ ⊆ S {p, ∞} (respectively B = S {p, ∞}). Remark 6.5 Note that conditions (ii) and (iii) are satisfied in the case of an abelian variety with good ordinary reduction at p. Furthermore, the quotient Ωp (M (ρ∗ ))/Ω∞ (M (ρ∗ )) is independent of the choice of basis γ and δ. Also, it is easy to see17 that for some suitable choice we have Ωp (M (ρ∗ )) = ˆ )−1 which, according to standard properties of -constants (cf. [23, p (W §3.2]) using that Vˆ is unramified as module under the Weil-group, is equal to p (ρ∗ )−k · ν −fp (ρ) where k = dimQ (tM ) = dimQp (Vˆ ), ν = detQp (ϕp |Dcris (Vˆ )) and where fp (ρ) is the p-adic order of the Artinconductor of ρ. Due to the compatibility conjecture CW D in [21, 2.4.3], which is known for abelian varieties (loc.cit., Rem. 2.4.6(ii)) and for Artin motives, one obtains the - and Euler-factors either from Dpst (W ) or from the corresponding l-adic realisations with l = p. Furthermore, we have PL,p (W, 1) = 0 ˆ , 1) = 0 for weight reasons. Thus, noting that for abelian varieties and PL,p (W ˆ Γ(V ) = 1, the above formula becomes ˆ ∗ (1), 1) LK,Υ (M (ρ∗ ), 0) PL,p (W · p (ρ∗ )−k · ν −fp (ρ) · , ∗ ˆ , 1) Ω∞ (M (ρ )) PL,p (W

(Int)

17 Note that in the definition of β and thus in β(ρ) the epsilon factor p (Vˆ ) in dR (Vˆ ) equals 1 ˆ )−1 · β(ρ)). and thus Ωp (M (ρ∗ )) = β(ρ) ◦ ( p (W

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where LK,Υ denotes the modified L-function without the Euler-factors in Υ := Υ ∪ {p}.18 Proof We consider the case LU . First observe that due to the vanishing of the motivic cohomology the map −1 ζK (M (ρ∗ )) : 1K → d(Δ(M (ρ∗ ))) = d(M (ρ∗ )+ B )d(tM (ρ∗ ) ) −1 is just the map canγ,δ : 1 ∼ , induced by the bases = d(M (ρ∗ )+ B )d(tM (ρ∗ ) ) ∗ arising from γ and δ, multiplied with LK (M (ρ ),0)Ω∞ (M (ρ∗ ))−1 . Secondly, since d(RΓf (Q, W )) = 1, the isomorphism ϑλ (M (ρ∗ )) : d(Δ(M (ρ∗ )))L ∼ = ∼ d(RΓc (U, W ))−1 corresponds up to the identification d(M (ρ∗ )+ ) B L = + d(W ) to the product of

d(tM (ρ∗ ) )−1 L

t ) d(gdR

−1

/ d(t(W ))−1

·ηλ (W )

/ d(RΓf (Qp , W ))

ˆ ) is equal to with B PL,l (W, 1). Thirdly, the contribution from p,L (W ˆ ) · ηp (W ˆ ∗ (1))∗ · ΓL (W ˆ ) · dR (W ˆ ) up to the canonical local duality isoηp (W morphism. Together with β(ρ) we thus obtain all the factors (1)-(6) above, i.e. after revealing all definitions and identifications, in particular all comparison isomorphisms, the same constituents show up in both expressions, the product of the factors (1)-(6) and LU (ρ); hence by Remark 1.2 which says that all compositions of maps in CL˜ can be rephrased in terms of products, which do not depend on any particular order, we have proven the interpolation property for LU . To finish the proof in the case L use Lemmata 6.1, 6.3 and equation (7.14).

6.1 Leading terms In the interpolation formula of Theorem 6.4 both sides are equal to zero if the (analytic and algebraic) rank of the motive M (ρ∗ ) is strictly positive (assuming here and henceforth the validity of the conjectures involved in that theorem). In this case one might still wonder whether one can describe the ‘leading term’ of the p-adic L-function L in terms of the leading term L∗K (M (ρ∗ )) of the complex L-function of M (ρ∗ ). To this end in [11] the notion of the ‘leading term at ρ’ for elements of a suitable localized K1 -groups and for representations ρ has 18 In order to compare this formula with (107) in [14] we remark that, with the notation of (loc. cit.), u = det(φl |V (−1)) = pν = pω −1 . Then by [23, Rem. 4.2.27] one has

(ρ∗ )−d ν −fp (ρ) = (ρ)d u−fp (ρ) (strictly speaking one has to replace the period t by −t in the second epsilon factor). But it seems that one has to interchange ρ and ρ on the right hand side of (107).

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been introduced in terms of the Bockstein homomorphisms that have already played significant roles (either implicitly or explicitly) in work of Perrin-Riou [36, 37], of Schneider [41, 40, 39, 38] and of Burns and Greither [10, 7] and have been systematically incorporated into Nekov´aˇr’s theory of Selmer complexes [34]. We briefly sketch the result in the case M = h1 (A)(1) of an abelian variety A over Q with good ordinary reduction at a fixed prime p = 2, for more general results and the details see [11]. We assume that the Galois group G of F∞ over Q has a quotient isomorphic to Zp . For an arbitrary Artin representation ρ : G → GLn (OL ), L a ˆ T) need not be acyclic. finite extension of Qp , the complex Ln ⊗LΛ SCU (T, But nevertheless the above assumption on G grants the existence of Bockstein homomorphisms Bi = Bi (SCU ) : Hi (G, ρ∗ , SCU ) → Hi−1 (G, ρ∗ , SCU ) ˆ T)) for the (co)homology groups Hi (G, ρ∗ , SCU ) := H−i (Ln ⊗LΛ SCU (T, ˆ T) giving rise to a complex (H• (G, ρ∗ , SCU ), B• ) and we say that SCU (T, is semisimple at ρ if this complex is acyclic. If this condition holds, using property h) of the determinant functor one easily constructs a trivialization ˆ T)) → 1L . t(ρ∗ , SCU ) : dL (Ln ⊗LΛ SCU (T, Analogously to the evaluation LU (ρ) of LU at ρ using (6.12) we thus can × nr given by define the leading term L∗U (ρ) of LU at ρ as the element in ∈ L the isomorphism L

∗

−1

L ⊗ LU t(ρ ,SCU ) ˆ T))−1 − 1L˜ −−−−Λ−−→ dL˜ (Ln ⊗LΛ SCU (T, −−−−−−−→ 1L˜ n

and similarly for L∗ (ρ). On the other hand one can show that there exists an isomorphism in Dp (L) of the form ˆ T) ∼ ˆ ,W) ∼ Ln ⊗LΛ SCU (T, = SCU (W = RΓf (Q, W ) under which the Bockstein map B−1 (SCU ) coincides with the map ad(hp (W )) : H1f (Q, W ) → H2f (Q, W ) ∼ = H1f (Q, Z)∗ which is induced from Nekov´aˇr’s p-adic height pairing hp (W ) : H1f (Q, W ) × H1f (Q, Z) → L and global duality with Z = W ∗ (1). Since the cohomology of RΓf (Q, W ) is ˆ T) is semisimple at ρ concentrated in degrees 1 and 2, it turns out that SCU (T, if and only if the pairing hp (W ) is non-degenerate, which we will assume

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henceforth. We also assume that the archimedean height pairing for N = M (ρ∗ ) is non-degenerate. Fixing K-bases of Hf1 (N ) and Hf1 (N ∗ (1))∗ we define the regulators R∞ (N ) and Rp (N ) as the determinant of the inverse of # $ h∞ (N ) : Hf1 (N ∗ (1))∗ C → Hf1 (N )C and as the determinant of ad(hp (W )) with respect to the induced bases, respectively; then the ratio Rp (N )R∞ (N )−1 is independent of the choice of the bases. Finally, we set r := r(N ) := dimL H1f (Q, W ). Theorem 6.6 ([11, Thm. 6.5]) Let M = h1 (A)(1) be the motive of an abelian variety A over Q with good ordinary reduction at a fixed prime p = 2. We assume that the archimedean and p-adic height pairing for the motive ˆ that M (ρ∗ ) are non-degenerate and that the morphisms ζΛ (M ) and p,Λ (T) are described in Conjecture 4.1 and Conjecture 5.9 exist. ˆ T) and SC(T, ˆ T) are semisimple at ρ and the leading Then both SCU (T, ∗ ∗ term L (ρ) (respectively LU (ρ)) is equal to the product (−1)r

ˆ ∗ (1), 1) L∗K,B (M (ρ∗ )) PL,p (W ∗ ∗ ·Ω (M (ρ ))·R (M (ρ ))· , p p ˆ , 1) Ω∞ (M (ρ∗ )) · R∞ (M (ρ∗ )) PL,p (W

where L∗K,B (M (ρ∗ )) denotes the leading term at s = 0 of the B-truncated complex L-function of M (ρ∗ ) with B := Υ ∪ Sp (respectively B := S S∞ ).

6.2 Interlude - Localised K1 The following construction of a localized K1 by Fukaya and Kato is one of the differences to the approach of Huber and Kings [24]19 . For a moment let Λ be an arbitrary ring with unit and let Σ be a full subcategory of C p (Λ) satisfying (i) if C is quasi-isomorphic to an object in Σ then it belongs to Σ, too, (ii) Σ contains the trivial complex, (iii) all translations of objects in Σ belong again to Σ and (iv) any extension C in C p (Λ) (by an exact sequence of complexes) of C , C ∈ Σ is again in Σ. Then Fukaya and Kato construct a group K1 (Λ, Σ) whose objects are all of the form [C, a] with C ∈ Σ and an i isomorphism a : 1Λ → dΛ (C) (in particular, [C] := i∈Z [C](−1) = 0 in K0 (Λ)) satisfying certain relations, see [23, 1.3]. This group fits into an exact sequence K1 (Λ)

/ K1 (Λ, Σ)

∂

/ K0 (Σ)

/ K0 (Λ),

(6.15)

19 Instead of the localised K1 they work with K1 of the ring lim Qp [G/Gn ], which occurs in ←− n the context of distributions, see [16].

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where K0 (Σ) is the abelian group generated by [[C]], C ∈ Σ and satisfying certain relations, see (loc. cit.). Here the first map is given by sending the class of an automorphism Λr → Λr to [[Λr → Λr ], can], where can denotes the trivialization of the complex [Λr → Λr ] by the identity according to Remark 1.2, ∂ maps [C, a] to −[[C]] while the last map is given by [[C]] → [C]. If S is an left denominator set of Λ, ΛS := S −1 Λ the corresponding localization and ΣS the full subcategory of C p (Λ) consisting of all complexes C such that ΛS ⊗Λ C is acyclic, then K1 (Λ, ΣS ) and K0 (ΣS ) can be identified with K1 (ΛS ) and K0 (S-torpd ), respectively. Here S-torpd denotes the category of finitely generated S-torsion Λ-modules with finite projective dimension.

6.3 Iwasawa main conjecture I Let O be the ring of integers of the completion at any place λ above p of the ab,p maximal abelian outside p unramified extension F∞ of Q inside F∞ . Note that the latter extension is finite because every non-finite abelian p-adic Lie extension of Q contains the cyclotomic Zp -extension, which is ramified at p. ˆ and thus L are already defined over Then by [23, Thm. 4.2.26(2)] p,Λ (T) ΛO := O ⊗Z Λ(G) instead of Λ. p

Now let Σ = ΣSC be the smallest full subcategory of C p (ΛO ) containing SC and satisfying the conditions (i)-(iv) above. Then the evaluation of L factorizes over its class in K1 (ΛO , Σ) which we still denote by L. By the construction of L we have Theorem 6.7 ([23, Thm. 4.2.22]) Assume Conjectures 4.1 for (M, Λ) and 5.9 ˆ Λ). Then the following holds: for (T, (i) ∂(L) = [[SC]] (ii) L satisfies the interpolation property (Int). Question: If one knows the existence of L ∈ K1 (ΛO , Σ) with the above properties, what is missing to obtain the zeta-isomorphism?

6.4 The canonical Ore set Now assume that the cyclotomic Zp -extension Qcyc is contained in F∞ and set H := G(F∞ /Qcyc ). In this situation there exists a canonical left and right denominator set of ΛO @ S∗ = pi S i≥0

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with S = {λ ∈ ΛO |ΛO /ΛO λ is a finitely generated ΛO (H)-module} as was shown in [14]. In this case we write MH (G) for the category of S ∗ -torsion modules and identify K0 (MH (G)) with K0 (ΣS ∗ ) recalling that ΛO is regular. We write X = Sel(A/F∞ )∨ for the Pontryagin dual of the classical Selmer group of A over F∞ , see [14]. Conjecture 6.8 ([14, Conj. 5.1]) X ∈ MH (G). It is shown in [23, Prop. 4.3.7] that Conjecture 6.8 is equivalent to SC belonging to ΣS ∗ . We assume this conjecture. Observe that then Σ ⊆ ΣS ∗ , which induces a commutative diagram

K1 (ΛO )

/ K1 (ΛO , Σ)

∂

/ K0 (Σ)

/ K0 (ΛO )

K1 (ΛO )

/ K1 ((ΛO )S ∗ )

∂

/ K0 (MH (G))

/ K0 (ΛO ).

Question: What is the (co)kernel of the middle vertical maps?

6.5 Iwasawa main conjecture II In [14, §3] it is explained how to evaluate elements of K1 ((ΛO )S ∗ ) at representations. By [23, Lem. 4.3.10] this is compatible with the evaluation of elements in K1 (ΛO , Σ) as explained above. The following version of a Main Conjecture was formulated in [14]. Conjecture 6.9 (Noncommutative Iwasawa Main Conjecture) There exists a (unique) element L in K1 ((ΛO )S ∗ ) such that (i) ∂L = [XO ] in K0 (MH (G)) and (ii) L satisfies the interpolation property (Int).

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Otmar Venjakob

The connection with the previous version is given by the following Proposition 6.10 Let F∞ be e.g. Q(A(p)) or Q(μ(p), Q× μ (false Tate curve)20 . Then

√

p∞

α) for some α ∈

[[X]] = [[SC]] in K0 (ΣS ∗ ). In particular, Conjecture 6.9 is a consequence of Conjecture 4.1 ˆ Λ). for (M, Λ) and Conjecture 5.9 for (T, The advantage of the localisation (ΛO )S ∗ relies on the fact that one has an explicit description of its first K-group since the natural map (ΛO )× S∗ → K1 ((ΛO )S ∗ ) induces quite often an isomorphism of the maximal abelian quotient of (ΛO )× S ∗ onto K1 ((ΛO )S ∗ ), see [14, Thm. 4.4]. On the other hand, the localized K1 (ΛO , Σ) exists without the assumption that G maps surjectively onto Zp , e.g. if G = SLn (Zp ). Also, if G has p-torsion elements, i.e. if ΛO (G) is not regular, one can still formulate the Main Conjecture using the complex SC instead of the classical Selmer group X (which could have infinite projective dimension). Question: To which extent does a p-adic L-function L together with Conjecture 6.9 determine the ζ-isomorphism in Conjecture 4.1? In other words, does the Main conjecture imply the ETNC?

7 Appendix: Galois cohomology The main reference for this appendix is [23, §1.6], but see also [9, 8]. For simplicity we assume p = 2 throughout this section. Let U = Spec(Z[ S1 ]) be a dense open subset of Spec(Z) where S contains Sp := {p} and S∞ := {∞} (by abuse of notation). We write GS for the Galois group of the maximal outside S unramified extension of Q. Let X be a topological abelian group with a continuous action of GS . Examples we have in mind are X = Tp , Mp , T, etc. Using continuous cochains one defines a complex RΓ(U, X)21 whose cohomology is Hn (GS , X). Then RΓc (U, X) is defined by the exact triangle / RΓ(U, X) / / RΓc (U, X) (7.1) v∈S RΓ(Qv , X) where the RΓ(Ql , X) and RΓ(R, X) denote the continuous cochain complexes calculating the local Galois groups Hn (Ql , X) and Hn (R, X). Its cohomology is concentrated in degrees 0, 1, 2, 3. 20 See [23, Prop. 4.3.15-17] for a more general statement. 21 For ease of notation we do not distinguish between complexes and their image in the derived category, though this is sometimes necessary in view of the correct use of the determinant functor and exact sequences of complexes.

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Let L be a finite extension of Qp with ring of integers O. Now we define the local and global “finite parts” for a finite dimensional L-vector space V with continuous GQv - and GQ -action, respectively. For Qv = R we set RΓf (R, V ) := RΓ(R, V ) while for a finite place RΓf (Ql , V ) is defined as a certain subcomplex of RΓ(Ql , V ), concentrated in degree 0 and 1, whose image in the derived category is isomorphic to RΓf (Ql , V ) ∼ = ⎧ ⎪ ⎨[ V Il 1−ϕl / V Il ] (1−ϕ ,1) ⎪ ⎩[ Dcris (V ) p / Dcris (V ) ⊕ DdR (V )/D0 (V ) ] dR

if l = p,

(7.2)

if l = p.

Here ϕl denotes the geometric Frobenius (inverse of the arithmetic) and the 0 induced map DdR (V )/DdR (V ) → Hf1 (Qp , V ) is called the exponential map expBK (V ) of Bloch-Kato, where we write Hnf (Ql , V ) for the cohomology of RΓf (Ql , V ). Defining RΓ/f (Ql , V ) as a mapping cone RΓf (Ql , V )

/ RΓ(Ql , V )

/ RΓ/f (Ql , V )

/

(7.3)

we finally define RΓf (Q, V ), whose cohomology is concentrated in degrees 0, 1, 2, 3, as mapping fibre RΓf (Q, V )

/ RΓ(U, V )

/

l∈SS∞

/.

RΓ/f (Ql , V )

(7.4) This is independent of the choice of U. The octahedral axiom induces an exact triangle RΓc (U, V )

/ RΓf (Q, V )

/

v∈S

/.

RΓf (Qv , V )

(7.5)

7.1 Duality Let G, Λ = Λ(G), T as in section 4. By abuse of notation we write −∗ for both (derived) functors RHomΛ (−, Λ) and RHomΛ◦ (−, Λ◦ ). Then Artin-Verdier/Poitou-Tate duality induces the existence of the following distinguished triangle in the derived category of Λ-modules RΓc (U, T)

/ RΓ(U, T∗ (1))∗ [−3]

/ T+

/

(7.6)

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and similarly for T (a Galois stable O-lattice of V ) and V as coefficients (with Λ replaced by O and L, respectively).22 For the finite parts one obtains from Artin-Verdier/Poitou-Tate and local Tate-duality the following isomorphisms RΓf (Ql , V ) ∼ = (RΓ(Ql , V ∗ (1))/RΓf (Ql , V ∗ (1)))∗ [−2],

(7.7)

RΓf (Q, V ) ∼ = RΓf (Q, V ∗ (1))∗ [−3].

(7.8)

0 Set t(V ) := DdR (V )/DdR (V ) if l = p and t(V ) = 0 otherwise. TrivializIl ing V and Dcris (V ), respectively, in (7.2) by the identity induces, for each l, an isomorphism

ηl (V ) :

→ dL (RΓf (Ql , V ))dL (t(V )).

1L

(7.9)

Then, setting Dl (V ) = DdR (V ) if l = p and Dl (V ) = 0 otherwise, the isomorphism Θl (V ) : 1L → dL (RΓ(Ql , V )) · dL (Dl (V ))

(7.10)

is by definition induced from ηl (V )·(ηl (V ∗ (1))∗ ) followed by an isomorphism 0 induced by local duality (7.7) and using the analogue DdR (V ) = t(V ∗ (1))∗ of (5.1) if l = p.23

7.2 Selmer complexes For l = p we define RΓf (Ql , T) as in (7.2) and RΓ/f (Ql , T) as in (7.3) with V replaced by T, see also (6.9). We do not define RΓf (Qp , T) since there is in general no integral version of Dcris (V ). 22 A more precise form to state the duality is the following. Let RΓ(c) (U, T) be defined like RΓc (U, T) but using Tate cohomology RΓ(R, T) instead of the usual group cohomology RΓ(R, T). Then one has isomorphisms RΓ(U, T∗ (1))∗ ∼ = RΓ(c) (U, T)[3] ∼ = RΓ(U, T∨ (1))∨ where −∨ = Homcont (−Qp /Zp ) denotes the Pontryagin dual. 23 More explicitly, θp (V ) is obtained from applying the determinant functor to the following exact sequence

/ H0 (Qp , V )

0

H1 (Qp , V )

/

/

expBK (V ∗ (1))∗

H2 (Qp , V )

Dcris (V )

/

Dcris (V ) ⊕ t(V )

/

Dcris (V ∗ (1))∗ ⊕ t(V ∗ (1))∗

/

0

/

expBK (V )

/

Dcris (V ∗ (1))∗

which arises from joining the defining sequences of expBK (V ) with the dual sequence for expBK (V ∗ (1)) by local duality (7.7).

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ˆ T) is by definition the mapping fibre The Selmer complex SCU (T, ˆ ⊕ / RΓ(U, T) / RΓ(Qp , T/T) ˆ T) SCU (T, l∈S(Sp ∪S∞ ) RΓ(Ql , T) (7.11) ˆ T) is the mapping fibre while SC(T, ˆ T) SC(T,

/ RΓ(U, T)

ˆ ⊕ / RΓ(Qp , T/T) l∈S(Sp ∪S∞ ) RΓ/f (Ql , T) .

(7.12) Thus by the octahedral axiom one obtains the distinguished triangles (6.6), (6.8) and, using Artin-Verdier/Poitou-Tate duality, ˆ ⊕ ˆ T) / RΓ(Qp , T) RΓ(U, T∨ (1))∨ / SC(T, l∈S(Sp ∪S∞ ) RΓf (Ql , T) . (7.13) ˆ , W ) and With the notation of section 6 the Selmer complexes SCU (W ˆ , W ) are defined analogously and satisfy analogous properties. SC(W The following properties [23, (4.2),propositions 1.6.5, 2.1.3 and 4.2.15] are necessary conditions for the existence of the zeta-isomorphism ζΛ (T) in Conjecture 4.1 and the p-adic L-functions LU (6.11) and L (6.7). ˆ T) are perfect24 and Proposition 7.1 The complexes RΓc (U, T) and SCU (T, in K0 (Λ) we have ˆ T)] = 0. [RΓc (U, T)] = [SCU (T, ˆ T) is also perfect and we have If G does not have p-torsion, then SC(T, ˆ [SC(T, T)] = 0. 7.3 Descent properties For the evaluation at representations one needs good descent properties of the complexes involved. Proposition 7.2 [23, Prop. 1.6.5] With the notation as in section 6 we have canonical isomorphisms (for all primes l) Ln ⊗LΛ,ρ RΓ(U, T) ∼ = RΓ(U, W ), n L L ⊗ RΓ(c) (U, T) ∼ = RΓ(c) (U, W ), Λ,ρ

Ln ⊗LΛ,ρ RΓc (U, T) ∼ = RΓc (U, W ), n L L ⊗ RΓ(Ql , T) ∼ = RΓ(Ql , W ), Λ,ρ

ˆ T) ∼ ˆ , W ). Ln ⊗LΛ,ρ SCU (T, = SCU (W For l ∈ Υ ∪ Sp we also have: Ln ⊗LΛ,ρ RΓf (Ql , T) ∼ = RΓf (Ql , W ). 24 RΓc (U, T) is even perfect for p = 2, this is the reason that it is better for the formulation of the ETNC than RΓ(U, T).

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ˆ for l ∈ Υ, and thus also SC(T, ˆ T), But note that the complex RΓf (Ql , T) does not descend like this in general. Instead, according to [23, Prop. 4.2.17] one has a distinguished triangle ˆ T) Ln ⊗LΛ,ρ SC(T,

/ SC(W ˆ ,W)

/

l∈Υ

RΓf (Ql , W ) . (7.14)

Bibliography [1] [2] [3] [4] [5]

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[16] P. Colmez, Th´eorie d’Iwasawa des repr´esentations de de Rham d’un corps local, Ann. of Math. (2) 148 (1998), no. 2, 485–571. [17] B. Conrad and M. Lieblich, Galois representations arising from p-divisible groups, http://www.math.lsa.umich.edu/˜ bdconrad/papers/pdivbook.pdf. [18] P. Deligne, Valeurs de fonctions L et p´eriodes d’int´egrales, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 313–346. [19] M. Flach, The equivariant Tamagawa number conjecture: a survey, Stark’s conjectures: recent work and new directions, Contemp. Math., vol. 358, Amer. Math. Soc., Providence, RI, 2004, With an appendix by C. Greither, pp. 79–125. [20] J.-M. Fontaine, Repr´esentations l-adiques potentiellement semi-stables, Ast´erisque (1994), no. 223, 321–347. [21] J.-M. Fontaine, Repr´esentations p-adiques semi-stables, Ast´erisque (1994), no. 223, 113–184. [22] J.-M. Fontaine and B. Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 599–706. [23] T. Fukaya and K. Kato, A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society, Vol. XII (Providence, RI), Amer. Math. Soc. Transl. Ser. 2, vol. 219, Amer. Math. Soc., 2006, pp. 1–86. [24] A. Huber and G. Kings, Equivariant Bloch-Kato conjecture and non-abelian Iwasawa main conjecture, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, pp. 149–162. [25] A. Huber and G. Kings, Bloch Kato Conjecture and Main Conjecture of Iwasawa Theory for Dirichlet Characters, Duke Math. Journal 119 (2003), no. 3, 393–464. [26] K. Kato, M. Kurihara and T. Tsuji, Local Iwasawa theory of Perrin-Riou and syntomic complexes, 1996. [27] K. Kato, Iwasawa theory and p-adic Hodge theory, Kodai Math. J. 16 (1993), no. 1, 1–31. [28] K. Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR . I, Arithmetic algebraic geometry (Trento, 1991), Lecture Notes in Math., vol. 1553, Springer, Berlin, 1993, pp. 50–163. [29] K. Kato, K1 of some non-commutative completed group rings, K-Theory 34 (2005), no. 2, 99–140. [30] S. L. Kleiman, Algebraic cycles and the Weil conjectures, Dix espos´es sur la cohomologie des sch´emas, North-Holland, Amsterdam, 1968, pp. 359–386. [31] S. L. Kleiman, The picard scheme, arXiv:math.AG/0504020v1 (2005). [32] S. Lang, Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563. [33] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 103–150. [34] J. Nekov´aˇr, Selmer complexes, to appear in Ast´erisque. Available at http:// www.math.jussieu.fr/∼nekovar/pu/

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[35] J. Nekov´aˇr, On p-adic height pairings, S´eminaire de Th´eorie des Nombres, Paris, 1990–91, Progr. Math., vol. 108, Birkh¨auser Boston, Boston, MA, 1993, pp. 127– 202. [36] B. Perrin-Riou, Th´eorie d’Iwasawa et hauteurs p-adiques (cas des vari´et´es ab´eliennes), S´eminaire de Th´eorie des Nombres, Paris, 1990–91, Progr. Math., vol. 108, Birkh¨auser Boston, Boston, MA, 1993, pp. 203–220. [37] B. Perrin-Riou, p-adic L-functions and p-adic representations, SMF/AMS Texts and Monographs, vol. 3, American Mathematical Society, Providence, RI, 2000. [38] P. Schneider, Height pairings in the Iwasawa theory of abelian varieties., Theorie des nombres, Semin. Delange-Pisot-Poitou, Paris 1980-81, Prog. Math. 22, 1982, pp. 309–316. [39] P. Schneider, p-adic height pairings. I., Invent. Math. 69 (1982), 401–409. [40] P. Schneider, Iwasawa L-functions of varieties over algebraic number fields. A first approach, Invent. Math. 71 (1983), no. 2, 251–293. [41] P. Schneider, p-adic height pairings. II., Invent. Math. 79 (1985), 329–374. [42] A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419–430. [43] A. J. Scholl, Classical motives, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 163–187. [44] SGA7, Groupes de monodromie en g´eom´etrie alg´ebrique. I, Springer-Verlag, Berlin, 1972, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1967–1969 (SGA 7 I), Dirig´e par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim, Lecture Notes in Mathematics, Vol. 288. [45] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. [46] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. [47] O. Venjakob, From classical to non-commutative Iwasawa theory - an introduction to the GL2 main conjecture, 4ECM Stockholm 2004, EMS, 2005. [48] S. Yasuda, Local constants in torsion rings, Ph.D. thesis, University of Tokyo, 2001.

The Andr´e-Oort conjecture - a survey Andrei Yafaev University College London, Dept. of Mathematics, 25 Gordon Street, WC1H 0AH, U.K. [email protected] a

The author was supported by an EPSRC grant and the European Research Training Network “Arithmetic Algebraic Geometry”.

1 Introduction The purpose of this paper is to review the strategies and methods developed over the past few years in relation to the Andr´e-Oort conjecture. We would like bring the attention of the reader to a survey by Rutger Noot (see [20]) based on his talk at the S´eminaire Bourbaki in November 2004. In this paper, we have tried to avoid overlapping too much with Noot’s survey. This paper is based on the author’s talk at the Durham Symposium in the summer of 2004. Laurent Clozel gave a talk on an approach to the Andr´e-Oort conjecture involving ergodic-theoretic methods. We will touch upon the contents of his lecture in the last section of this paper. Let us recall the statement of the Andr´e-Oort conjecture. Conjecture 1.1 (Andr´e-Oort) Let S be a Shimura variety and let Σ be a set of special points in S. The irreducible components of the Zariski closure of Σ are special subvarieties (or subvarieties of Hodge type). In the next section we will review the notions of Shimura varieties, special points and special subvarieties. In this introduction we review some of the results on this conjecture obtained so far. This conjecture was stated by Yves Andr´e in 1989 (Problem 9 in [1]) for one dimensional subvarieties of Shimura varieties and in 1995 by Frans Oort in [22] for subvarieties of the moduli space Ag of principally polarised abelian varieties of dimension g. The statement above is the obvious generalisation of these two conjectures and is now refered to as the Andr´e-Oort conjecture. One of the motivations for the Andr´e-Oort conjecture was the Manin-Mumford conjecture about the distribution of torsion points on subvarieties of abelian varieties. The Manin-Mumford conjecture has been first proved by Raynaud in 1983 and since then, the number of new proofs of this conjecture has been increasingly growing. One may 381

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consult [26], [28], [36], [13] or [25] for some of the proofs of this conjecture. The strategies to attack the Andr´e-Oort conjecture that we will review in this survey are inspired by some of the proofs of the Manin-Mumford conjecture. Although the conjecture remains open, a number of results on this conjecture have been obtained in the course of the past ten years or so. Let us review the most significant of them. (i) Moonen proved the following result (see [17] and [18]). Let Σ be a set of special points in Ag . Suppose that there exists a prime p such that for every s in Σ, the corresponding abelian variety As has good ordinary reduction at some place above p of which As is the canonical lift. Then the components of the Zariski closure of Σ are special. This result has been generalised by Yafaev in 2003 to the case of arbitrary Shimura varieties, see [34]. We will come back to this result and its proof in the section 5. (ii) Edixhoven (see [10]) and Andr´e (see [3]) proved the Andr´e-Oort conjecture for products of two modular curves. Edixhoven’s proof is conditional on the Generalised Riemann Hypothesis (GRH) for imaginary quadratic fields while Andr´e’s proof is unconditionnal (it relies on some diophantine approximation results on the j-function, see section 5.2 of [20] for a sketch of his proof). However Edixhoven’s proof offers a definite strategy suitable for further generalisations. This generalisation has been subsequently carried out by Edixhoven and the author of the present article. (iii) Edixhoven proved, assuming the GRH, the Andr´e-Oort conjecture for subvarieties of products of an arbitrary number of modular curves and for curves contained in the Hilbert modular surfaces. In 2005, Ullmo and Yafaev proved the Andr´e-Oort conjecture for products of modular curves, also assuming the GRH using a different method. The paper [31] will be released shortly. (iv) Edixhoven and Yafaev proved the Andr´e-Oort conjecture for curves in Shimura varieties containing an infinite set of special points lying in one Hecke orbit. See [12]. One of the motivations for doing this was an application to a problem in transcendence theory of hypergeometric functions. One can consult, for example, the section 4.2 of [20] for more on this application. (v) In 2002 Yafaev proved the Andr´e-Oort conjecture for curves in Shimura varieties assuming the generalised Riemann hypothesis (see [35]). This was the original question of Yves Andr´e.

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(vi) Clozel and Ullmo (see [5]) proved a result on equidistribution of strongly special subvarieties which has implications for the Andr´e-Oort conjecture. This result has very recently been generalised by Ullmo (see [33]), thus obtaining a nearly optimal result that can be reached using ergodic theoretic methods. This will be explained in more detail in the section 5. The strategy, orginated by Edixhoven and used in [10], [11], [12], [35] and [34] to obtain results on the Andr´e-Oort conjecture will be reviewed in detail in this survey. This strategy relies on two principles. The first, is that a subvariety Z of a Shimura variety (satisfying certain conditions, but we are not being specific here) is special if and only if it is contained in its image by some suitable Hecke correspondence. This characterisation will be reviewed in the third section. With this characterisation in hand, one tries to find a suitable Hecke correspondence by using the Galois action on special points. Key points here are that the Galois conjugates of a special point lie in its Hecke orbit and that Hecke correspondences commute with the Galois action. The important ingredient here is a theorem on lower bounds for Galois orbits of special points. In the fourth section we will review a more general theorem, on lower bounds for non-strongly special subvarities. The most recent developments on the conjecture exploit the dichotomy between the Galois-theoretic properties of special subvarieties and equidistribution discovered by Ullmo and Yafaev in [31]. This is the object of papers [31] and [14] in preparation. Finally a proof of the Andr´e-Oort conjecture assuming the Generalised Riemann Hypothesis and unconditionally under the assumption that the set Σ lies in one (generalised) Hecke orbit has been announced by Bruno Klingler and the author of this survey. We will review the strategy of the announced proof in the last section. We would like to point out that there is a generalisation of the Andr´e-Oort conjecture to mixed Shimura varieties which combines both the Andr´e-Oort conjecture and the Mordell-Lang conjecture. We will not touch upon this topic in this paper but we refer the interested reader to the papers [23] and [24] by Richard Pink available on his web-page. Acknowledgements The author would like to express his gratitude to the organisers of the Durham Symposium for having given him the opportunity to give a lecture and write this survey as well as for stimulating atmosphere during the conference. The author thanks the referee for pointing out many inaccuracies and suggesting improvements.

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Andrei Yafaev 2 Preliminaries.

The purpose of this section is to briefly review the notions of Shimura varieties, special subvarieties, special points and Hecke correspondences. The detailed exposition of these topics can be found in [17] but one can also consult [11] or [20]. The exposition of the general theory of Shimura varieties can be found in [7], [8] or [16]. Let S denote the real torus ResC/R GmC . A Shimura datum is a pair (G, X) where G is a connected reductive group over Q and X is a G(R)-orbit in Hom(S, GR ) satisfying the conditions 2.1.1(1-3) of [8]. These conditions imply that connected components of X are hermitian symmetric domains and faithful representations of G induce variations of polarisable Q-Hodge structures. Let K be a compact open subgroup of G(Af ). Let ShK (G, X) be the following double coset set ShK (G, X) := G(Q)\(X × G(Af )/K) Let X + be a connected component of X and let G(Q)+ be the stabiliser of X + in G(Q). It is easy to see that ShK (G, X) is a disjoint union of quotients of X + by the arithmetic groups Γg := G(Q)+ ∩gKg −1 where g ranges through a set of representatives for the finite double coset set G(Q)+ \G(Af )/K. A wellknown theorem of Baily and Borel implies that ShK (G, X) has a structure of a quasiprojective complex algebraic variety. The varieties ShK (G, X) form a projective system indexed by compact open subgroups K ⊂ G(Af ) : the inclusions K ⊂ K of compact open subgroups induce finite morphisms ShK (G, X) −→ ShK (G, X). Let Sh(G, X) denote the projective limit of the ShK (G, X), it is a scheme naturally endowed with the action of G(Af ). Let Z be the centre of G. The set of complex points of Sh(G, X) is Sh(G, X)(C) =

G(Q) \(X × G(Af )/Z(Q)− ) Z(Q)

where Z(Q)− denotes the closure of Z(Q) in Z(Af ). Let s be a point of Sh(G, X) and ShK (G, X). There exists an element (x, t) of X × G(Af ) such that s is the image (x, t) in the quotient. We will use this notation throughout this paper. An element g of G(Af ) acts on the point (x, t) by sending this point to (x, tg). It is important to note that the action of g on Sh(G, X) induces an algebraic correspondence Tg on ShK (G, X) for every compact open subgroup K of G(Af ). These correspondences are called Hecke correspondences. One can describe Tg more explicitly. Let π1 be the morphism ShK∩gKg−1 (G, X) −→

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ShK (G, X) induced by the inclusion K ∩ gKg −1 ⊂ K and let π2 be the morphism induced by the inclusion preceeded by right multiplication by g on the second factor. Then, for every subvariety Z of ShK (G, X), by definition Tg (Z) := π2 π1−1 (Z). Let S = Γ\X + be an irreducible component of ShK (G, X). Elements of G(Q)+ induce Hecke correspondences on S in an analogous way except for left multiplication by elements of G(Q)+ versus right multiplication by elements of G(Af ). We can now define special subvarieties and special points. Let f : (H, Y ) → (G, X) be a morphism of Shimura data. By this we mean a morphism of algebraic groups H −→ G which, by composition induces a map Y −→ X. Such a morphism induces a morphism of algebraic varieties Sh(f ) : Sh(H, Y ) −→ Sh(G, X). Definition An irreducible subvariety Z of ShK (G, X) is called special or of Hodge type if there exists a morphism of Shimura data f : (H, Y ) −→ (G, X) and an element g of G(Af ) such that Z is an irreducible component of the image of the following composition Sh(f )

·g

Sh(H, Y ) −→ Sh(G, X) −→ Sh(G, X) −→ ShK (G, X). The terminology ‘of Hodge type’ comes from the fact that these subvarieties are loci of Hodge classes in the variations of Hodge structure coming from rational representations of G. Details on this can be found in [17]. By definition, a special point is a special subvariety of dimension zero. We can give an equivalent (and more useful) definition involving Mumford-Tate groups. Definition Let x be a point of X. View x as a morphism x : S −→ GR . By definition the Mumford-Tate group of x, denoted MT(x), is the smallest algebraic subgroup H of G such that x(S) ⊂ HR . A point s = (x, g) of ShK (G, X) is called special if MT(x) is a torus. Note that this is independent of g. Given a special subvariety Z of ShK (G, X) the special points of ShK (G, X) contained in Z form a dense subset for the archimedian (and in particular Zariski) topology. This follows from two facts: the first is that a special subvariety contains a special point and the second is that for a reductive group H, H(Q) is dense in H(R). We now make a couple of important remarks regarding special subvarieties. The first property is that the open compact subgroup K does not play any

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role. More precisely, let Z be a subvariety of ShK (G, X) and K a compact open subgroup of K. Then Z is special if and only if one (equivalently any) component of the preimage of Z in ShK (G, X) is special. The second one is that irreducible components of intersections of special subvarieties are special. This is easily seen by using the definition of special subvarieties as ‘loci of Hodge classes’. An obvious consequence of this fact is that, given a subvariety Z of ShK (G, X), there exists a smallest special subvariety of ShK (G, X) containing Z. One can describe the Shimura datum defining this special subvariety. This involves the notion of generic Mumford-Tate group. Suppose that Z is contained in the component Γ\X + of ShK (G, X). Choose a faithul representation of G on a Q-vector space V , this be the preimage of Z induces a variation of Hodge structures over X + . Let Z let MT(Vx ) be the Mumford-Tate group of the in X + . For every point x of Z, Hodge structure on V defined by x. There exists a countable union of analytically closed subsets Σ ⊂ Z, such that MT(Vx ) is independent of x in Z − Σ. We denote this constant group by MT(Z) and call it the generic MumfordTate group. Points of Z lying outside the image of Σ are called Hodge generic. Let XZ be the MT(Z)(R)-orbit of a Hodge generic point x. This defines a morphism of Shimura data ShK∩MT(Z)(Af ) (MT(Z), XZ ) −→ ShK (G, X) and the image of this morphism is the smallest special subvariety containing Z. The subvariety Z of Γ\X + is called Hodge generic if it is not contained in a proper special subvariety. In other words, the smallest special subvariety containing Z is Γ\X + . The last important thing that we need to mention is the existence of canonical models of Shimura varieties over certain number fields. We will not go into detail here and refer the reader to the section 6.1 of [20] and references therein for a more detailed exposition. To a Shimura datum is associated a number field E(G, X) called the reflex field. The Shimura variety Sh(G, X) (projective limit) admits a canonical model over E := E(G, X). What is meant by this is that there exists an E-scheme Sh(G, X)E such that Sh(G, X)E × C = Sh(G, X) and the action of G(Af ) on Sh(G, X) is defined over E. It follows that for every compact open subgroup K of G(Af ), the variety ShK (G, X) admits a canonical model over E in such a way that Hecke correspondences commute with the Galois action. Furthermore, special points are Q-valued points and the Galois action on them is explicitly described via the reciprocity morphism. For details on the reciprocity morphism, we refer to the section 6.1 of [20]. Let us give a brief overview. Consider a special Shimura datum (T, x), where T is a torus and x is a morphism x : S −→ TR satisfying the axioms for Shimura datum. Suppose that T is the Mumford-Tate group of x.

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One constructs a surjective morphism of Q-tori rx : ResE/Q GmE −→ T. Consider now a special point (x, g) of ShK (G, X). Let Ex := E(MT(x), x). The action of Gal(Ex /Ex ) on (x, g) factors through the maximal abelian quotient Gal(Exab /Ex ). Class field theory gives a surjection (A ⊗ Ex )∗ to Gal(Exab /Ex ). Let σ be an element of Gal(Exab /Ex ) and let s = (sf , s∞ ) be an idele in the preimage of σ in (A ⊗ Ex )∗ where sf ∈ (Af ⊗ Ex )∗ and s∞ ∈ (R ⊗ Ex )∗ . Then, by definition σ((x, g)) = (x, rx (sf ) · g) In particular σ((x, g)) ∈ Tg−1 rx (sf )g ((x, g)) Let us note, that in the case of Shimura varieties parametrising families of abelian varieties, special points correspond to abelian varieties of CM type. The Galois action on them is then described by the main theorem of complex multiplication by ShimuraTaniyama. In particular, Galois conjugates of a CM abelian variety lie in one isogeny class. The reciprocity morphism is the one induced by “reflex type norm”. A reference for the Shimura-Taniyama theorem is [15].

3 Images under Hecke correspondences. The strategy developed by Edixhoven and the author of the present survey relies on the characterisation of special subvarieties in terms of their images by some Hecke correspondences. In this section we suppose that we are in the following situation. Let (G, X) be a Shimura datum with G semi-simple of adjoint type. Let K be a neat open compact subgroup of G(Af ) which is the product of compact open subgroups Kl of G(Ql ) and let X + be a connected component of X. Let S be the image of X + × {1} in ShK (G, X), then S = Γ\X + where Γ := G(Q)+ ∩ K (with G(Q)+ being the stabiliser of X + in G(Q)). Let Z be an irreducible Hodge generic subvariety of S containing a smooth special point. First one proves the following irreducibility result which combines the results of section 5 of [12] and section 3 of [34]. Theorem 3.1 There exists an integer M depending on G, X, Z and K such that the following holds. Let l be a prime. There is a compact open subgroup K of K (K is again a product of compact open subgroups Kp of G(Qp ) and Kp = Kp for p = l)

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and a component Z of the preimage of Z in S := Γ \X + (with Γ := K ∩ G(Q)+ ) with the property that for any q in (G(Ql ) × p=l Kp ) ∩ G(Q)+ (the intersection is being taken in G(Af )) the image Tq Z of Z by the Hecke correspondence Tq is irreducible. Furthermore, if l does not divide M , then K = K. Note that this theorem is far from being effective : one has no a priori control over M or over K . This causes significant difficulties while trying to deal with the general case of the Andr´e-Oort conjecture. In the case of a product of two modular curves (or in the case of arbitrary products of modular curves) however, one can gain control over these quantities. Indeed, Edixhoven in [10] proves the following theorem. Theorem 3.2 Let Z be a Hodge generic curve contained in C2 , viewed as the Shimura variety associated to the Shimura datum (GL2 × GL2 , H± × H± ). Suppose that both projections of Z are dominant. Let l be a prime larger than 13 and the degrees of the two projections of Z. Then Tl Z is irreducible. Let us now briefly sketch the proof of Theorem 3.1. Let ξ : G −→ GL(V ) be a faithful rational representation of G where V is a finite dimensional Qvector space. Choose a Γ-invariant lattice VZ in V . This induces a variation of Z-Hodge structures over S and over the smooth locus Z sm of Z (this is where the assumption that K is neat is used). Let π1 (Z sm ) be the topological fundamental group of Z sm with respect to some Hodge generic point s of Z sm . There is a monodromy representation ρ : π1 (Z sm ) −→ GL(VZ ) associated to the local system underlying the variation of Hodge structure. The image of ρ is contained in ξ(Γ). A theorem of Yves Andr´e (see [2, Thm. 1.4]) implies that ρ(π1 (Z sm )) is Zariski dense in ξ(G). Let l be a prime. By a theorem of Nori ([21]), the closure of ρ(π1 (Z sm )) in ξ(Kl ) is ξ(Kl ) for a compact open subgroup Kl of G(Ql ). Let K be a compact open subgroup of G(Af ) obtained as follows : K is a product K = p Kp where Kp are compact open subroups of G(Qp ) and Kp = Kp for p = l and Kl is as above. Let Γ and S be as in the statement of the theorem. There exists a component Z of the preimage of Z such that sm ρ(π1 (Z )) is equal to π1 (Z sm ). Let, for every integer k ≥ 0, Γ (lk ) be the kernel of Γ −→ GL(VZ /lk VZ ). Let Sk := Γ (lk )\X + . By construction, for sm every k > 0, the preimage of Z in Sk is irreducible. The conclusion of the theorem follows. As for the second statement, one uses again Nori’s theorem which implies that there exists an integer M such that for every l not dividing M , the images of Γ and π1 (Z sm ) in GLn (VZ /lk VZ ) are equal for all k. This finishes the proof.

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The second ingredient in the characterisation is the density of Hecke orbits. Strictly speaking, one considers the orbits for the correspondence Tq + Tq−1 where q is some well-chosen element of G(Q)+ . Theorem 3.3 Let p be a prime. Let q be an element of (G(Qp ) × l=p Kl ) ∩ G(Q)+ such that for every simple factor Gi of G, the image of q in Gi (Qp ) is not contained in a compact subgroup. Then for every s in S, the Tq + Tq−1 -orbit of s is dense for the archimedian topology. Proof We only sketch the proof. One reduces the proof to showing that the index of Γ in the group Γq generated by Γ and q is infinite. If this index is finite, then the index of the closure Γ of Γ in G(Qp ) in the group Γq generated by Γ and q is also finite. This implies that the group Γq is compact which contradicts our assumption. Regarding Hecke orbits, an equidistribution result of Tq -orbits is now available thanks to the work of Clozel, Oh and Ullmo (see [4]) and may be used in the charactersation instead of the density result above. Theorem 3.4 (Clozel, Oh, Ullmo) Let G be an almost simple simply connected algebraic group over Q such that G(R) is not compact. Let K∞ be a maximal compact subgroup of the neutral component G(R)+ and let X + be G(R)+ /K∞ . Let dμ be the natural normalised measure on Γ\X + induced by the Haar measure on G(R)+ and let qn be a sequence of elements of G(Q)+ such that the index [Γ : qn Γqn−1 ∩Γ] tends to infinity as n tends to infinity. Then the Tqn -orbits are equidistributed for the measure μ in the sense that for every x in Γ\X + and every continuous compactly supported function f on Γ\X + , 1 lim f (y) = f dμ −1 n−→∞ [Γ : qn Γqn ∩ Γ] y∈T (x) Γ\X + qn

In particular, for every x in Γ\X + , the set

n≥0

Tqn (x) is dense in Γ\X + .

From the irreducibility and density results, one derives the following characterisation of special subvarieties. Theorem 3.5 Let Z be a Hodge generic subvariety of S = Γ\X + . Suppose that there exists a prime l and an element m of G(Qp ) such that (i) For every simple factor Gi of G, m is not contained in a compact subgroup of G(Qp ).

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(ii) The variety Z is contained in Tm Z where Z is as in Theorem 3.1. Then Z is of Hodge type. Note that if l does not divide M with M as in Theorem 3.1 then K = K and the second condition becomes that Z is contained in Tm Z. To prove this theorem, one simply notes that the inclusion Z ⊂ Tm Z implies that Z is contained in its image by Tq and by Tq−1 where q is some element of G(Q)+ such that Tq Z and Tq−1 Z are both irreducible and the orbits of Tq + Tq−1 are dense in S. A Hodge generic subvariety Z of S is special if and only if Z = S.

4 Lower bounds for Galois orbits of special subvarieties. The purpose of this section is to present results on lower bounds for Galois orbits of non-strongly special subvarieties of Shimura varieties. In [12] lower bounds for Galois orbits of sets of special points lying in one Hecke orbits have been obtained. This result was subsequently generalised in [35] to arbitrary sets of special points and used to prove the Andr´e-Oort conjecture for curves in Shimura varieties. The result we present in this section is one of the main results of [31] which generalises these previous results. We begin by recalling some definitions regarding special subvarieties. As in the previous section, we let (G, X) be a Shimura datum where G is semisimple of adjoint type. Via a faithful representation, we view G as a closed subgroup of some GLnQ . Let K be a compact open subgroup of G(Af ) We also assume that K is a product of compact open contained in GLn (Z). subgroups Kp of G(Qp ). Let (H, XH ) be a sub-Shimura datum of (G, X). Let H = MT(XH ) be the generic Mumford-Tate group on XH . We have H der = H der . Let x be an element of XH and let XH be the H (R)-orbit of x. Then XH = XH and (H , XH ) is a sub-Shimura datum of (H, XH ). Definition A special subvariety Z ⊂ ShK (G, X) is called strongly special if the Shimura datum (H, XH ) defining Z is such that MT(XH ) is semisimple (in other words the connected centre of MT(XH ) is trivial). Examples of strongly special subvarieties abound : the modular curves Y0 (n) (and products thereof) embedded in products of modular curves, Hilbert modular varieties as subvarieties of Ag , etc. Typically these varieties admit canonical models over ‘small fields’. At the other end of the scale are the special points. Recall that the Mumford-Tate group of a special point is a torus.

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It has been shown in [12] and [35] that Galois orbits of the special points tend to get very large. Examples of non-strongly special subvarieties which are not zero-dimensional include, for example subvarieties of a product of three modular curves of the form {x} × Y0 (n) where x is a special point of a modular curve and Y0 (n) is embedded in a product of two modular curves. One can easily see that these subvarieties also have large Galois orbits (what is meant by the Galois orbit in this case is the number of Galois conjugates of geometrically irreducible components). Let us now explain a general result on Galois orbits of non-strongly special subvarieties obtained in the forthcoming paper [31]. So let (G, X), (H, XH ) and H be as above. We suppose that H is not semisimple. Let T be the connected centre of H , so that H is an almost direct product T H der . Let KH be the compact open subgroup H (Af ) ∩ K of H (Af ). Our aim is to give a lower bound for the number of Galois conjugates of geometrically irreducible components of the image of ShKH (H, XH ) in ShK (G, X). Let us briefly recall how Galois acts on the geometrically irreducible components of Shimura varieties. We refer to the sections 2.4-2.6 of [8] for details and proofs. Let π0 (H , KH ) (H , XH ). As be the set of geometrically irreducible components of ShKH a set, π0 (H , XH ) is H (Q)+ \H (Af )/KH where H (Q)+ is the stabiliser of a connected component of XH in H (Q). Let EH be the reflex field of (H , XH ). We refer to [8] for details on the definitions of reflex fields. We just mention here that EH is the number field of definition of the H(C)-conjugacy class of hC (z, 1) for h in XH . In particular, one notes that EH = EH . What follows is a generalisation of the Galois action on special points, overviewed in the second section. The action of Gal(Q/EH ) on π0 (H , KH ) is given by the reciprocity morphism r(H ,XH ) : Gal(Q/EH ) −→ π(H ) < (Af ). Here ρ : H < −→ H der is the univerwhere π(H ) is H (Af )/H (Q)ρH der ab sal covering of H . The morphism r(H ,XH ) factors through Gal(EH /EH ) which is identified with a quotient of π(ResEH /Q GmEH ) by class field theory. Let C be the torus H /H der . To (H , XH ) one associates two Shimura data (C, {x}) and (H ad , XH ad ) where x is induced by any element of XH . The field EH is the composite of E(C, {x}) and E(H ad , XH ad ). There are morphisms θ ab : (H , XH ) −→ (C, {x}) and θ ad : (H , XH ) −→ (H ad , XH ad ).

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Note that (C, {x}) is a special Shimura datum. Let r(C,{x}) be the reciprocity morphism associated to (C, {x}). The morphism θ ab induces a morphism π(H ) → π(C). This morphism, preceeded by r(H ,XH ) , is r(C,{x}) . We let F be the Galois closure of EH . Note that the degree of F over Q is bounded uniformly on (H, XH ). Note that there is an isogeny T −→ C with kernel T ∩ H der . One can easily see that the order of the group T ∩ H der is uniformly bounded as (H, XH ) ranges through the sub-Shimura data of (G, X). Let KH be the compact open subgroup H(Af ) ∩ K of H(Af ). The following lemma is proved in the forthcoming paper [31], lemma 4.2. The proof is essentially elementary. Lemma 4.1 Let f : ShKH (H, XH ) −→ ShK (G, X) be the morphism induced by the inclusion (H, XH ) into (G, X). The morphism f is finite onto its image of uniformly (on (H, XH )) bounded degree. Furthermore, if K is neat, then f is generically injective. In particular, the number of geometrically irreducible components of ShKH (H, XH ) is, up to a uniform (on (H, XH )) constant, equal to the number of components of its image in ShK (G, X). This lemma reduces the problem of giving a lower bound for the Galois orbit of a geometrically irreducible component of the image of ShKH (H, XH ) in ShK (G, X) to that of giving a lower bound for the Galois orbit of a component of the Shimura variety ShKH (H, XH ) itself. We can now forget about ShK (G, X) and work with ShKH (H, XH ). The main theorem on the Galois orbits is the following. Theorem 4.2 Assume the GRH for CM fields. Let N be a positive integer. There exists a real B > 0 such that the following holds. Let (H, XH ) be a subShimura datum of (G, X) and suppose that H is the generic Mumford-Tate group on XH . Let T be the connected centre of H and let KTm be the maximal compact open subgroup of T (Af ) and KT be the compact open subgroup m T (Af ) ∩ K. Let i(T ) be the number of primes p such that KT,p = KT,p . Let F be the composite of the reflex field (H, XH ) with the splitting field of T . The following inequality holds |{σ(Z), σ ∈ Gal(Q/F )}| B i(T ) |KTm /KT |(log(dT ))N . where dT denotes the absolute value of the discriminant of the splitting field of T and stands for ‘larger up to a uniform constant’.

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We now give some ingredients of the proof of this theorem. The lower bound naturally splits into two parts. The first corresponds to the factor |KTm /KT | and the second to the term in log(dT ). Let G be the image of Gal(Q/F ) (the field F being as in the statement) in π(H )/KH by r(H ,XH ) . The number of components of the Galois orbit of V is at least the size of G. The number of components of the Galois orbit of V is at least the size of G. m Let KT be the compact open subgroup T (Af ) ∩ KH of T (Af ) and let KC and KTm be the maximal compact open subgroups of C(Af ) and T (Af ) respectively. The following lemma, proved in [31], lemma 4.3 realises the splitting and explains the presence of the factor B i(T ) . Lemma 4.3 There exists a uniform constant B and an exact sequence 0 −→ W −→ KTm /KT −→ π(H )/KH where W is a finite group of order at most B i(T ) . Let x be a Hodge generic point of XH (recall that this means that MT(x) is H ). View, as usual, x as a morphism x : S −→ HR . We let μ be the character xC (z, 1) : GmC −→ HC and let μC be the character μ composed with the map HC −→ CC . The Lemma 4.4 of [31] proves a uniformity property of the characters defining this morphism

Lemma 4.4 There is a basis (χi ) of the character group X ∗ (C) such that, with respect to the natural pairing <, > : X ∗ (L) × X∗ (L) −→ Z the < χi , σ(μC ) > are bounded uniformly on (H, XH ). Previous lemmas allow to deal with the first part of the splitting. Lemma 4.5 There is a uniform integer B such that the size of the intersection i(T ) of G with the image of KTm /KT in π(H )/KH is at least B · |KTm /KT | To prove this lemma, one first notes that, using the lemma 4.3, it is enough to ˆ ⊗ OF )∗ ) in K m /KT . The give a lower bound for the preimage of r(C,{x}) ((Z T m m group KT /KT is the direct product of the |KT,p /KT,p | for the i(T ) primes m such that KT,p = KT,p . It follows that it is enough to give a lower bound m for the preimage of r(C,{x}) ((Zp ⊗ OF )∗ ) in KT,p /KT,p for one prime p. Using the fact that the isogeny T −→ C has uniformly bounded kernel and the lemma 4.4, one proves that the size of the preimage of r(C,{x}) ((Zp ⊗ OF )∗ ) m m in KT,p /KT,p is, up to a uniform constant, |KT,p /KT,p |.

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m Next, one deals with the image of rC ((Af ⊗ F )∗ ) in C(Q)\C(Af )/KC m where KC denotes the maximal compact open subgroup of C(Af ).

Proposition 4.6 Assume the GRH for CM fields. Let N be a positive integer. The size of the image of r(C,{x}) ((Af ⊗ LC )∗ ) in C(Q)\C(Af )/KCm is at least a constant depending on N only times log(dT )N . We will not sketch the proof of this proposition, but we refer to [35] for a detailed proof. The idea of the proof is to choose sufficiently many prime ideals P of the ring of inetegers OF of F lying over primes splitting completely in OF and small with respect to large power of log(dT ) (this is where the GRH is used). And then use the images of the corresponding ideles by r(C,{x}) to m generate a sufficiently large subgroup of C(Q)\C(Af )/KC .

5 Some cases of the Andr´e-Oort conjecture. In this section we sketch the results on the Andr´e-Oort conjecture that have been obtained by using the results from the previous sections and the strategy outlined in the end of the introduction. These theorems are proved in [12], [35] and [34] and in this section we will give an overview of the proofs. The first theorem deals with the one-dimensional subvarieties. Theorem 5.1 Let S be a Shimura variety defined by a Shimura datum (G, X) and a compact open subgroup K. Let Z be an irreducible algebraic curve in S. Make one of the following assumptions. (i) Assume the GRH for CM fields and that Z contains an infinite set of special points. (ii) Assume that Z contains an infinite set of special points Σ such that with respect to some faithful representation V of G, the Q-Hodge structures Vs with s ranging through Σ lie in one isomorphism class. Then Z is of Hodge type. Remark The second condition of this statement is satisfied if the points in Σ lie in one Hecke orbit. For example, if the Shimura variety is a moduli space for some family of abelian varieties, then the condition would be satisfied for sets of abelian varieties of CM type lying in one isogeny class. Even more explicitly, one can think of ellipic curves with complex multiplication by a fixed imaginary quadratic field K and varying conductor. The proof of this theorem under the first assumption is contained in [35] and under the second in [12].

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The second theorem deals with subvarieties of arbitrary dimension. The proof of this theorem is contained in the note [34]. Theorem 5.2 Let (G, X) be a Shimura datum and let K be a compact open subgroup of G(Af ). Let Σ be a set of special points of X. Let V be a faithful representation of G. Via this representation, we view G as a closed subgroup of GLnQ . For each x in Σ we let MT(x) be its Mumford-Tate group. We suppose that there is a prime p such that for any s in Σ, the torus MT(x)Qp has a subtorus Ms isomorphic to GmQp such that the weights in the representation VQp of Mx are bounded uniformly on x and for every nontrivial quotient T of MT(x), the image of Mx in TQp is nontrivial. Furthermore, we assume that the Zariski closure in GLnZp of each Mx is isomorphic to GmZp . Then for any g in G(Af ) the irreducible components of the Zariski closure of the image of Σ × {g} in ShK (G, X) are of Hodge type. This theorem is a generalisation of a theorem of Moonen (see [18]) which deals with the case where the Shimura variety is Ag . However the proof of this theorem is very different from Moonen’s. In fact the proof of this theorem uses the same ideas as the proof of the previous one. For the sake of completeness, we reproduce Moonen’s statement below. A survey of the ingredients of Moonen’s proof can be found in the section 5.1 of [20]. Theorem 5.3 Let g ≥ 1 be an integer and let Ag be the moduli space of principally polarised abelian varieties of dimension g. Let Σ be a set of special points in Ag (Q) such that there exists a prime p with the property that every s in Σ has good ordinary reduction at a place lying over p of which s is the canonical lift. Then all irreducible components of the Zariski closure of Σ are of Hodge type. In the second section of [34] it is proved that the theorem 5.2 above does indeed imply the result of Moonen. Although, there is no restriction on the dimension of the Zariski closure of Σ, the condition imposed on the set Σ is quite restrictive. As an exercise, an interested reader may want to try to construct an infinite set of elliptic curves with complex multiplication such that the condition of the statement is not satisfied for every prime p. The method used to prove the two theorems is basically the same. One uses the characterisation of Theorem 3.5 i.e. one tries to prove that Z is contained in its image by some suitable Hecke correspondence. To find such a Hecke correspondence, one uses the Galois action on special points.

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First one reduces oneself to the following case. (i) The group G is semisimple of adjoint type. (ii) The group K is neat and is a product of compact open subgroups Kp of G(Qp ). (iii) The variety Z is Hodge generic and is contained in S := Γ\X + . Let us briefly explain why these assumptions can be made. One replaces the ambient Shimura variety by the smallest special subvariety containing Z so that Z is Hodge generic. Given a Shimura datum (G, X), there exists an adjoint Shimura datum (Gad , X ad ) where X ad is the Gad (R)-conjugacy class of the morphism had obtained by composing any h ∈ X with GR −→ Gad R . There is a natural morphism of Shimura data (G, X) −→ (Gad , X ad ). By choosing a compact open subgroup K ad of Gad (Af ) containing the image of K, one obtains a finite morphism of Shimura varieties ShK (G, X) −→ ShK ad (Gad , X ad ). It is not hard to see that a subvariety Z of ShK (G, X) is of Hodge type if and only if its image in ShK ad (Gad , X ad ) is. This allows us to replace G with Gad . However, one has to be a bit careful when the set Σ consists of special points such that the Hodge structures associated to these points with respect to some representation of G are isomorphic. In this case, one needs to choose a faithful representation of Gad such that this property for the image of Σ in ShK ad (Gad , X ad ) is conserved. This is done in [12], construction 2.3. We do not reproduce it here. Lastly, as we have already noted, replacing K by another compact open subgroup changes nothing to the property of subvarieties being special. It follows that K can be chosen to be neat and to be a direct product of compact open subgroups of G(Qp ). By possibly replacing Z with its image by a suitable Hecke correspondence, which again changes nothing to the property of Z being special, one can assume that Z is contained in Γ\X + . It follows that every point s of Σ can be represented as s := (x, 1) with x some point in X + . We define the Mumford-Tate group of s as MT(s) := MT(x). Note that this is defined up to conjugation by Γ. As Z contains a Zariski dense set of special points which are Q-valued points, we can (and do) choose a number field F sufficiently large so that S admits a canonical model over F and Z is defined over F as an absolutely irreducible subscheme of SF . Suppose now that Z is a curve containing an infinite set of special points. Let us recall lower bounds for Galois orbits of special points (we specialise the theorem from the previous section to the case of special points).

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Theorem 5.4 Let Σ ⊂ S be an infinite set of special points. Assume either the GRH for CM fields or that the points in Σ are such that the Vs lie in one isomorphism class of Q-Hodge structures as s ranges through Σ. For every point s in Σ, let MT(s) be its Mumford-Tate group and let Ls be the splitting field of MT(s). We let ds be the absolute value of the discriminant of MT(s). We also let Ksm be the maximal compact open subgroup of MT(s)(Af ) and Ks be the compact open subgroup K ∩ MT(s)(Af ). These groups are products of compact open subgroups of MT(s)(Qp ) and we denote by i(MT(s)) the m number of primes p such that Ks,p = Ks,p . Let N be an integer. There exists a constant B such that the following inequality holds |Gal(Q/F ) · s| B i(MT(s)) |Ksm /Ks | log(ds )N . Recall that to every s in Σ is associated a reciprocity morphism rs : ResF/Q GmF −→ MT(s) which is a surjective morphism of algebraic tori. One has the following theorem which gives the ‘candidates’ for the Hecke correspondences such that Z ⊂ Tm Z. Theorem 5.5 There exists an integer k such that the following holds. Let s be m a point of Σ and p a prime splitting MT(s) and such that Ks,p = Ks,p . Then ∗ there exists an element m in rs ((Qp ⊗ F ) ) ⊂ MT(s)(Qp ) satisfying (i) For every simple factor Gi of G, the image of m in Gi (Qp ) is not contained in a compact subgroup. (ii) [Kp : mKp m−1 ∩ Kp ] ' pk . Another important theorem is the following. Again, we are not giving a proof here. This theorem can be thought of as a kind of Bezout’s theorem. Theorem 5.6 ([11], Thm. 7.2) Let Z be an irreducible algebraic curve in S and m an element of G(Af ). Suppose that the intersection Z ∩ Tm Z is proper, then |Z ∩ Tm Z| ' [K : mKm−1 ∩ K]. The important observation that we are making now is that if a point s of Σ is contained in the intersection Z ∩ Tm Z, then the Galois orbit Gal(Q/F ) · s is contained in the intersection Z ∩ Tm Z. Hence if the size of Gal(Q/F ) · s is large compared to [K : mKm−1 ∩ K], then the intersection Z ∩ Tm Z can not be proper and if m is as in Theorem 3.5, then we are finished.

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Suppose that Z is a curve. The three theorems stated above show that one needs to find a prime p and a point s satisfying the following inequalities. (i) The prime p splits MT(s) and p is not among the i(MT(s)) primes such that MT(s)Fp is not a torus. (ii) B i(MT(s)) |Ksm /Ks | log(ds )N pk . To find such a prime, one uses the Chebotarev density theorem. For any real x > 0, define π(x) = |{p ≤ x : p splits Ls }|. Assume the GRH. If x is larger than some absolute constant and larger than log(ds )3 , then x π(x) . log(x) If the points s in Σ are such that the Vs lie in one isomorphism class of QHodge structures, then the assumption of GRH can be dropped because in this case ds is constant and the usual Chebotarev theorem can be used. Is is shown in [31], that x := B i(MT(s)) |Ksm /Ks | log(ds )N is unbounded as s ranges through Σ. In particular, if x is large enough, then log(x) < x1/2k and consequently, π(x) x1/2k . It is also shown in [12], that if p is unramified in m Ls and Ks,p = Ks,p , then m |Ks,p /Ks,p | pnp

with np ≥ 1. If i(MT(s)) is bounded then for at least one p, the np ’s occuring in the inequalities, as s varies, are unbounded (this is proved in [31]). Using the obvious inequality, that the ith prime is larger than i, it is now easy to show that, provided that N is larger than 6k, a prime p satisfying the conditions above exists. This finishes the proof of the first theorem. Let us now turn to the second theorem. Suppose that Z is a subvariety (of arbitrary dimension) containing a Zariski dense set Σ of special points satisfying the condition from the second theorem (in the sense, Z is a component of the Zariski closure of the set Σ of special points satisfying the condition of the second theorem). We sketch the proof under a simplified condition, namely, that for some prime p, the groups MT(x)Fp are split tori. The proof is essentially the same. We start by replacing the group K with K as in Theorem 3.1 and Z with Z . We can now choose an element m of rx ((Qp ⊗ F )∗ ) for some x in Σ in such a way that for every s in a Zariski dense subset Σ ⊂ Σ, some Galois conjugate of s is in Tm (s). It follows that Z ∩ Tm Z contains Σ hence Z ⊂ Tm Z and hence Z is special.

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6 The equidistribution approach. In this section we explain another approach to the “Andr´e-Oort-type conjectures”. We will start by briefly recalling the principle (mainly following Ullmo’s survey [29]) and then we will review some of the results obtained so far in this direction. Ullmo has very recently written a more comprehensive set of notes [32] on the equidistribution approach to this kind of question which will appear in the proceedings of the summer school on equidistribution in number theory that took place at the Universit´e de Montreal in July 2005. We will only very briefly summarise it. Let S be a complex algebraic variety and E a finite subset of S. For every x in S, let δx be the Dirac measure at x. One defines a normalised measure ΔE associated to E by 1 ΔE := δx . |E| x∈E

Let μ be a probability measure on S. A sequence of finite subsets En is said to be equidistributed with respect to μ if for every continuous bounded function f in S, one has 1 f dΔEn = f (x) −→ f dμ. |En | S S x∈En

A sequence xn of points of S is said to be generic if for every proper subvariety Z of S, the set {n : xn ∈ Z} is a finite set. The reader may want to prove, as an exercise, that this is equivalent to saying that the sequence xn converges to the generic point of S for the Zariski topology. Let us now suppose that S is a Shimura variety. One says that a sequence of special points xn is strict if for every special subvariety Z, the set {n : xn ∈ Z} is finite. It is obvious that the Andr´e-Oort conjecture is equivalent to the statement that every strict sequence of special points is generic. In the introduction, we have mentioned an “abelian analog” of the Andr´eOort conjecture, the Manin-Mumford conjecture stating that the irreducible components of the Zariski closure of a set of torsion points in an abelian variety are translates of abelian subvarieties by torsion points. There is a number of proofs of this conjecture. One of the most striking is due to Ullmo [28] (subsequently generalised by Zhang [36]). They actually prove Bogomolov’s conjecture which is a stronger statement than Manin-Mumford. Their proof relies on equidistribution of generic sequences of points whose N´eron-Tate height tends to zero for two different measures. The proof crucially relies on Arakelov theory.

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Let us now go back to the case where S is a Shimura variety. As has already been mentioned in the third section, Clozel, Oh and Ullmo proved an equidistribution result regarding Hecke orbits. Consider a sequence xn of special points. Let O(xn ) denote the Galois orbit of xn . It is natural to ask the question whether or not the sequence of the O(xn ) is equidistributed with respect to the normalised measure on S induced by the Haar measure of G(R)+ . So far the only case where this question is known to have a positive answer is the case of a sequence of special points of a modular curve. This is due to Duke (see [9]). Let us briefly review Duke’s results. Let K be an imaginary quadratic field and let OK,f := Z + f OK be an order in K. We let ΣK,f be the set of isomorphism classes of pairs (E, α) where E is an elliptic curve and α : OK,f −→ End(E) is a ring isomorphism. The set ΣK,f is a Pic(OK,f )-torsor and the action of Gal(Q/K) on the set ΣK,f factors through the quotient Gal(HK,f /K) where HK,f is the ring class field of OK,f . The action of Gal(HK,f /K) is given by the class field theory isomorphism Gal(HK,f /K) ∼ = Pic(OK,f ), in particular Gal(Q/K) acts transitively on ΣK,f . The size of the Galois orbit of the point xK,f of SL2 (Z)\H representing the isomorphism class of (E, α) is exactly hK,f := |Pic(OK,f )|. The BrauerSiegel theorem implies that for every > 0, one has 1/2−

dK,f

1/2+

' |ΣK,f | ' dK,f

where dK,f denotes the absolute value of the discriminant of OK,f . Let μ be the probability measure on SL2 (Z)\H induced by the Poincar´e measure on H. Duke proved the following theorem. Theorem 6.1 As dK,f −→ ∞, the sets ΣK,f are equidistributed with respect to μ. A simpler question is whether the toric orbits are equidistributed. Note that in Duke’s case the Galois and toric orbits are equal. A very recent result regarding toric orbits is due to Zhang. Let s := (x, g) ∈ ShK (G, X) be a special point of ShK (G, X). The toric orbit of s is, by definition, the finite set (x, MT(x)(Af ) · g). It is obvious that the Galois orbit of s is contained in its toric orbit. However, the difference between the Galois orbit and the toric orbit is not understood well enough to deduce the equidistribution of Galois orbits from that of toric orbits. Regarding the toric orbits, Zhang recently announced equidistribution of toric orbits in Hilbert modular varieties or, more generally in Shimura varieties associated to quaternion algebras over totally

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real fields. Very recently Clozel and Ullmo proved that in the cases where the reciprocity morphisms attached to special points have connected kernels, toric equidistribution implies the equidistribution of Galois orbits. See [6]. The Haar measure on G(R)+ induces a measure μ on S := Γ\X + that we normalise by μ(S) = 1 (μ is a probability measure). To any special subvariety Z ⊂ S, is similarly associated a canonical probability measure μZ . A sequence μn of probability measures is said to be weakly convergent to μ if for any continuous function with compact support f on S, the sequence of real numbers μn (f ) converges to μ(f ). In general, one can formulate the following conjecture that one can call “equidistribution conjecture of Galois orbits of special subvarieties” which implies the Andr´e-Oort conjecture but actually is much stronger. The only known case of this conjecture is the case where S is a modular curve, thanks to Duke’s result. Conjecture 6.2 Let S be a Shimura variety and fix a number field F over which S admits a canonical model. To a special subvariety Z of S, one associates the canonical probability measure ΔZ defined by 1 ΔZ = μσ(Z) . |{σ(Z) : σ ∈ Gal(F /F )}| σ∈Gal(F /F )

Let Q(S) be the set of measures ΔZ with Z ranging through the set of special subvarieties of S. The set Q(S) is compact : every sequence of measures in Q(S) admits a weakly convergent subsequence. Furthermore, if ΔZn is a sequence in Q(S) weakly convergent to μZ , then for all n 0, the support of ΔZn is contained in the support of ΔZ . Note that it is essential to consider measures associated to Galois orbits of special subvarieties in the statement above. If one considers sequences of measures μZ with Z special then the conclusion of the conjecture above is wrong even in the case of a modular curve. A sequence of measures δxn where xn are special points on a modular curve can converge to δx where x is a nonspecial point. However, for some classes of special subvarities, one can prove the conclusion of the conjecture above. This is precisely the object of the work of Clozel and Ullmo. Clozel and Ullmo prove the following theorem (see [5]). Theorem 6.3 Let Zn be a sequence of strongly special subvarieties of S with associated measures μn (as explained above). There exists a strongly special subvariety Z of S and a subsequence μnk weakly convergent to μZ . Furthermore, Z contains Snk for every k large enough.

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The proof of the theorem of Clozel and Ullmo relies on some theorems from ergodic theory due to Ratner, Mozes-Shah and Dani-Margulis. An obvious consequence of this theorem is the following. Corollary 6.4 Let Y be a proper subvariety of the Shimura variety S. There exists a finite set {Z1 , . . . , Zn } of strongly special subvarieties of Y such that if Z is a strongly special subvariety of Y , then Z ⊂ Zi for some i. Let us prove that the theorem 6.3 implies the corollary above. So let Y be a proper subvariety of S. We will need to see that the set of maximal strongly special subvarieties of Y is finite. Consider the sequence of maximal strongly special subvarieties Zn contained in Y and let μn be the associated sequence of measures. After possibly replacing μn with a subsequence, we can assume that μn converges weakly to μZ for some strongly special subvariety Z of S. As the support of all of the Zn is contained in Y , Z is also contained in Y . From the facts that the Zn are maximal and that Zn is contained in Z for n large enough, we deduce that the sequence Zn is stationary. As an obvious consequence of the last corollary, we get the following. Corollary 6.5 Let Σ be a set of strongly special subvarieties of S. The components of the Zariski closure of Σ are special. The theorem 6.3 has been recently further generalised by Ullmo (see [33]). He defined a class of NF-special subvarieties (NF stands for Non-Factor). The definition is as follows. Definition A special subvariety Z of S is called a NF-special subvariety if there does not exist a special subvariety direct product Z1 × Z2 of S such that Z is a subvariety of the form {x} × Z2 of Z1 × Z2 where x is a special point of Z1 . It is clear that a strongly special subvariety is NF-special. As an easy exercise, the reader may prove that in the case where S is a product of modular or Shimura curves, the notions of strongly special and NF-special subvarieties are equivalent. In [33], Ullmo proved that the NF-special subvarieties are equidistributed in the above sense. We finish this section by explaining an equidistribution result of Ullmo and the author of this survey, which is one of the main results of [31]. Before we state the theorem, we need to introduce a definition. Definition Let (G, X) be a Shimura datum with G semisimple of adjoint type. Let T be a subtorus of G such that T (R) is compact.

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A T -sub-Shimura datum (H, XH ) of (G, X) is a sub-Shimura datum such that H der is non-trivial and T is the connected centre of the generic MumfordTate group on XH . Let K be compact open subgroup of G(Af ), a T -special subvariety of ShK (G, X) is a special subvariety defined by a T -Shimura datum. We show the following theorem, which is the equidistribution property for sequences of T -special subvarieties. Theorem 6.6 Let S be a connected component of ShK (G, X) with G semisimple of adjoint type. Fix T with T (R) compact. Let Zn be a sequence of T -special subvarieties of S and let μn be the associated sequence of probability measures. There exists a T -special subvariety Z of S and a subsequence Znk such that μnk converges weakly to μZ . Furthermore, Z contains Znk for all k large enough. We derive this theorem from the Clozel-Ullmo theorem. The idea is that T -special subvarieties are geometrically very similar to strongly special ones. The only thing which counts in their geometry is H der , the connected centre T does not count as it is constant. Theorem 6.7 Let S be a connected Shimura variety and F a number field over which S admits a canonical model. Assume the GRH for CM fields and let N be an integer. There exists a finite set {T1 , . . . , Tr } of Q-subtori of G with the following property. Let Z be a special subvariety of S defined and geometrically irreducible over a finite extension F of F of degree at most N . Then Zi is a Ti -special subvariety for some i ∈ {1, . . . , r}. To prove this theorem we use the lower bounds for Galois orbits. We express the fact that these lower bounds are bounded. This in particular implies that the discriminants log(dT ) are bounded and this implies that the connected centres T of H lie in finitely many GLn (Q) conjugacy classes with respect to some faithful representation G → GLn . Then we derive from the fact that the quantities B i(T ) |KTm /KT | are bounded, the fact that the tori T lie in finitely many G(Q) ∩ GLn (Z)-classes. This uses results of Gille and Moret-Bailly on algebraic actions of arithmetic groups (see the appendix to [31]). The two theorems above imply that a sequence of special subvarieties geometrically irreducible over a field extension of fixed degree, is equidistributed. We’ll explain in the next section how this fact yields a strategy for proving the Andr´e-Oort conjecture.

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Andrei Yafaev 7 The alternative and the strategy for proving the Andr´e-Oort conjecture.

The proof of the Andr´e-Oort conjecture has been very recently announced by Bruno Klingler and the author of the present paper. The idea is a combination of the Galois-theoretic approach and of the equidistribution approach. The results of [31] explained in the end of the previous section, show that the following alternative occurs. As usual, let S be a connected component of a special subvariety. Let Zn be a sequence of irreducible special subvarieties of S. Let F be a number field over which S admits a canonical model. After possibly replacing Zn by a subsequence and assuming the GRH for CM-fields, at least one of the following cases occurs. (i) The cardinality of the sets {σ(Zn ), σ ∈ Gal(Q/F )} is unbounded as n → ∞ (and therefore Galois-theoretic techniques can be used). (ii) The sequence of probability measures μn canonically associated to Zn weakly converges to some μZ , the probability measure canonically associated to a special subvariety Z of S. Moreover, for every n large enough, Zn is contained in Z. We now very briefly explain how this alternative gives a strategy for proving the Andr´e-Oort conjecture. Suppose now that we are given a Hodge generic subvariety X containing a Zariski dense sequence Zn of special subvarieties. Our aim to prove that X is special. We fix a number field F such that S admits a canonical model over F and such that X is defined over F . If the cardinality of sets {σ(Zn ), σ ∈ Gal(Q/F )} is bounded then the second case of the alternative occurs and X is special. If the sets {σ(Zn ), σ ∈ Gal(Q/F )} are unbounded, then the techniques of Edixhoven and the author apply. Using the induction on the dimension of subvarieties of X, we produce a subsequence Znk of Zn such that for all k, Znk is contained in a special subvariety Zn k with dim Zn k > dim Znk . We replace Zn with Zn k and re-iterate the process. We continue until we get either dim Zn = dim X or that the sets {σ(Zn ), σ ∈ Gal(Q/F )} are bounded. In each case, X is special. Technically, the induction is quite difficult, we refer to [14] for details. Bibliography Y. Andr´e, G-functions and geometry. Aspects of Mathematics, E13, Vieweg, 1989. [2] Y. Andr´e, Mumford-Tate groups of mixed Hodge structures and the theorems of fixed part. Compositio Math., 82, (1992), p. 1-24. [1]

The Andr´e-Oort conjecture - a survey [3] [4] [5] [6]

[7] [8]

[9] [10] [11]

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

[24]

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Y. Andr´e, Finitude de couples d’invariants modulaires singuliers sur une courbe algebrique plane non-modulaire. J. Reine Angew. Math., 505, p. 203-208, (1998). L. Clozel, H. Oh, E. Ullmo, Hecke operators and equidistribution of Hecke points. Invent. Math. 144, (2001), p. 327-351. L. Clozel, E. Ullmo, Equidistribution de sous-vari´et´es sp´eciales, Ann. of Math. (2) 161 (2005), no. 3, p. 1571–1588. L. Clozel, E. Ullmo, Equidistribution ad´elique des tores et equidistribution des points CM. Preprint 2005. Available at http://www.math.u-psud.fr/ ullmo/liste-prepub.html P. Deligne. Travaux de Shimura, S´eminaire Bourbaki, Expos´e 389, Fevrier 1971, Lecture Notes in Maths. 244, Springer-Verlag, Berlin 1971, p. 123-165. P. Deligne, Vari´et´es de Shimura: interpr´etation modulaire et techniques de construction de mod`eles canoniques, dans Automorphic Forms, Representations, and L-functions part. 2; Editeurs: A. Borel et W Casselman; Proc. of Symp. in Pure Math. 33, American Mathematical Society, 1979, p. 247–290. W. Duke, Hyperbolic distribution problems and half integral weights Maas-forms, Invent. Math. 92, no .1 (1988), p. 73-90 B. Edixhoven, Special points on products of modular curves Duke Math. J. 126 (2005), no. 2, 325–348. B. Edixhoven, On the Andr´e-Oort conjecture for Hilbert modular surfaces, Moduli of abelian varieties (Texel Island, 1999), 133–155, Progr. Math., 195, Birkhuser, Basel, 2001. B. Edixhoven, A. Yafaev, Subvarieties of Shimura varieties, Ann. Math. (2) 157, (2003), p. 621–645. M. Hindry, Autour d’une conjecture de Serge Lang. Invent. Math. 94 (1988) p. 575-603. B. Klingler, A. Yafaev, The Andre-Oort Conjecture. Preprint 2006, submitted. S. Lang, Complex Multiplication. Springer-Verlag, 1983. J. Milne, Shimura varieties. Lecture notes. Available at http://www.jmilne.org/math/ B. Moonen, Linearity properties of Shimura varieties I , Journal of Algebraic Geom. 7 (1998), p. 539–567. B. Moonen, Linearity properties of Shimura varieties II , Compositio Math. 114 (1998), no. 1, 3–35. S. Mozes, N. Shah, On the space of ergodic invariant measures of unipotent flows. Ergod. Th. and Dynam. Sys. 15 (1995), p. 149-159. R. Noot, Orbites de Galois, correspondances de Hecke et la conjecture de Andr´eOort (d’apr`es Edixhoven et Yafaev) Seminaire Bourbaki 2004. M. Nori, On subgroups of GLn (Fp ), Invent. Math. 88 (1987), no. 2, 257–275. F. Oort, Canonical liftings and dense sets of CM points. In Arithmetic Geometry, 1994, Vol. XXXVII of Sympos. Math. R. Pink, A Combination of the Conjectures of Mordell-Lang and Andr´e-Oort. In Geometric Methods in Algebra and Number Theory. Progress in Mathematics 253, Birkhauser (2005), 251-282. R. Pink, A Common Generalisation of the Conjectures of Andr´e-Oort, ManinMumford and Mordell-Lang. Preprint (April 2005), 13p. Available on author’s web-page.

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[25] R. Pink, D. Roessler, On ψ-invariant subvarieties of abelian varieties and the Manin-Mumford conjecture. J. Algebraic Geom. 13 (2004), n. 4, 771-798. [26] M. Raynaud, Sous-vari´et´es d’une variet´e ab´elienne et points de torsion. Arithmetic and Geometry, Vol. I, Progr. Math. 35, Birkhauser Boston, MA, (1988). [27] L. Szpiro, E. Ullmo, S.W. Zhang, Equir´epartition des petits points. Invent. Math, 127 (1997), p. 337-347. [28] E. Ullmo, Positivit´e et discretion des points alg´ebriques des courbes. Ann. Math., 147 (1998), p. 167-179. [29] E. Ullmo, Th´eorie ergodique et g´eometrie arithm´etique. ICM 2002. Vol III, 1-3. [30] E. Ullmo, Equidistribution des sous-vari´et´es sp´eciales II. Preprint. 2005 [31] E. Ullmo, A. Yafaev, Galois orbits and equidistribution of special subvarieties: towards the Andre-Oort conjecture With an appendix by P. Gille and L. MoretBailly. Preprint 2006. Submitted. [32] E. Ullmo, Manin-Mumford, Andr´e-Oort, the equidistribution viewpoint. preprint, 35 pages. http://www.math.u-psud.fr/ ullmo/liste-prepub.html [33] E. Ullmo, Equidistribution des sous-variet´es speciales II. Preprint, 2005. [34] A. Yafaev, On a result of Moonen on the moduli space of principally polarised abelian varieties. Compositio Math. 141 (2005) 1103-1108. [35] A. Yafaev, A conjecture of Yves Andr´e’s, Duke Math. J. 132 (2006), no. 3, 393– 407. [36] S.W. Zhang, Equidistribution of small points on abelian varieties, Ann. Math., 147, (1998). p.159-165. [37] S.W. Zhang, Equidistribution of CM-points on quaternion Shimura varieties, Int. Math. Res. Not. (2005), no. 59, 3657–3689.

Locally analytic representation theory of p-adic reductive groups: a summary of some recent developments Matthew Emerton Northwestern University Department of Mathematics 2033 Sheridan Rd. Evanston, IL 60208-2730 USA [email protected]

The purpose of this short note is to summarize some recent progress in the theory of locally analytic representations of reductive groups over padic ﬁelds. This theory has begun to ﬁnd applications to number theory, for example to the arithmetic theory of automorphic forms, as well as to the “p-adic Langlands programme” (see [3, 4, 5, 10, 11, 12]). I hope that this note can serve as an introduction to the theory for those interested in pursuing such applications. The theory of locally analytic representations relies for its foundations on notions and techniques of functional analysis. We recall some of these notions in Section 1. In Section 2 we describe some important categories of locally analytic representations (originally introduced in [20], [23] and [8]). In Section 3, we discuss the construction of locally analytic representations by applying the functor “pass to locally analytic vectors” to certain continuous Banach space representations. In Section 4 we brieﬂy describe the process of parabolic induction in the locally analytic situation, which allows one to pass from representations of a Levi subgroup of a reductive group to representations of the reductive group itself, and in Section 5 we describe the Jacquet module construction of [9], which provides functors mapping in the opposite direction. Parabolic induction and the Jacquet module functors are “almost” adjoint to one another. (See Theorem 5.19 for a precise statement.) Acknowledgments. I would like to thank David Ben-Zvi for his helpful remarks on an earlier draft of this note, as well as the anonymous referee, whose comments led to the clariﬁcation of some points of the text. The author would like to acknowledge the support of the National Science Foundation (award numbers DMS-0070711 and DMS-0401545)

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We begin by recalling some notions of non-archimedean functional analysis. A more detailed exposition of the basic concepts is available in [17], which provides an excellent introduction to the subject. Let K be a complete discretely valued ﬁeld of characteristic zero. A topological K-vector space V is said to be locally convex if its topology can be deﬁned by a basis of neighbourhoods of the origin that are OK submodules of V ; or equivalently, by a collection of non-archimedean semi-norms. (We will often refer to V simply as a convex space, or a convex K-space if we which to emphasize the coeﬃcient ﬁeld K.) The space V is called complete if it is complete as a topological group under addition. If V is any locally convex K-space, then we may complete V to obtain a complete Hausdorﬀ convex K-space Vˆ , equipped with a continuous Klinear map V → Vˆ , which is universal for continuous K-linear maps from V to complete Hausdorﬀ K-spaces. (See [17, Prop. 7.5] for a construction of Vˆ . Note that in this reference Vˆ is referred to as the Hausdorﬀ completion of V .) If V is a convex K-space, then we let V denote the space of K-valued continuous K-linear functionals on V , and let Vb denote V equipped with its strong topology (the “bounded-open” topology – see [17, Def., p. 58]; the subscript “b” stands for “bounded”). We refer to Vb as the strong dual of V . There is a natural K-linear “double duality” map V → (Vb ) ; we say that V is reﬂexive if this map induces a topological isomorphism V → (Vb )b . If V and W are two convex K-spaces, then we always equip V ⊗K W with the projective tensor product topology. This topology is characterized by the requirement that the map V × W → V ⊗K W deﬁned by (v, w) → v ⊗ w should be universal for continuous K-bilinear maps from V × W to convex K-spaces. (See [17, §17] for more details about the ˆ K W denote construction and properties of this topology.) We let V ⊗ the completion of V ⊗K W . A complete convex space V is called a Fr´echet space if it is metrizable, or equivalently, if its topology can be deﬁned by a countable set of seminorms. If the topology of the complete convex space V can be deﬁned by a single norm, then we say that V is a Banach space. Note that we don’t regard a Fr´echet space or a Banach space as being equipped with any particular choice of metric, or norm. If V and W are Banach spaces, then the space L(V, W ) of continuous linear maps from V to W again becomes a Banach space, when equipped with its strong topology. (Concretely, if we ﬁx norms deﬁning the topologies of V and W respectively, then we may deﬁne a norm on L(V, W )

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as follows (we denote all norms by || ||): for any T ∈ L(V, W ), set || T || = supv∈V s.t. ||v||=1 || T (v)||.) We say that an element T ∈ L(V, W ) is compact if it may be written as a limit (with respect to the strong topology) of a sequence of maps with ﬁnite dimensional range (see [17, Rem. 18.10]). If V is a Fr´echet space, then completing V with respect to each of the members of an increasing sequence of semi-norms that deﬁne its topology, we obtain a projective sequence of Banach spaces {Vn }n≥1 , and an isomorphism of topological K-vector spaces ∼

V −→ lim Vn , ←− n

where {Vn }n≥1 is a projective system of Banach spaces over K, and the right hand side is equipped with the projective limit topology. Conversely, any such projective limit is a Fr´echet space over K. Deﬁnition 1.1. A nuclear Fr´echet space over K is a K-space which admits an isomorphism of topological K-vector spaces ∼

V −→ lim Vn , ←− n

where {Vn }n≥1 is a projective system of Banach spaces over K with compact transition maps (and the right hand side is equipped with the projective limit topology). In fact there is a more intrinsic deﬁnition of nuclearity for any convex space [17, Def., p. 120], which is equivalent to the above deﬁnition when applied to a Fr´echet space (as follows from the discussion of [17, §16] together with [20, Thm. 1.3]). Proposition 1.2. Let V be a nuclear Fr´echet space. (i) V is reﬂexive. (ii) Any closed subspace or Hausdorﬀ quotient of V is again a nuclear Fr´echet space. Proof. See [17, Prop. 19.4].

We now introduce another very important class of locally convex spaces. Deﬁnition 1.3. We say that a convex K-space V is of compact type if there is an isomorphism of topological K-vector spaces ∼

V −→ lim Vn , −→ n

where {Vn }n≥1 is an inductive system of Banach spaces over K with compact and injective transition maps (and the right hand side is equipped with the locally convex inductive limit topology).

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Proposition 1.4. Let V be a space of compact type. (i) V is complete and Hausdorﬀ. (ii) V is reﬂexive. (iii) Any closed subspace or Hausdorﬀ quotient of V is again of compact type. Proof. See [20, Thm. 1.1, Prop. 1.2].

Proposition 1.5. Passing to strong duals yields an anti-equivalence of categories between the category of spaces of compact type and the category of nuclear Fr´echet spaces. Proof. This is [20, Thm. 1.3]. A proof can also be extracted from the discussion of [17, §16]. We now deﬁne an important class of topological algebras over K (originally introduced in [23]). Deﬁnition 1.6. Let A be a topological K-algebra. We say that A is ∼ a nuclear Fr´echet-Stein algebra if we may ﬁnd an isomorphism A −→ lim An , where {An }n≥1 is a sequence of Noetherian K-Banach algebras, ←− n

for which the transition maps An+1 → An are compact (as maps of KBanach spaces) and ﬂat (as maps of K-algebras), and such that each of the maps A → An has dense image (or equivalently, by [2, II §3.5 Thm. 1], such that each of the maps An+1 → An has dense image). If A is a nuclear Fr´echet-Stein algebra over K, then A is certainly a nuclear Fr´echet space. If A is a topological K-algebra, then any two representations of A as a projective limit as in Deﬁnition 1.6 are equivalent in an obvious sense. (See [8, Prop. 1.2.7].) Example 1.7. Let us explain the motivating example of a nuclear Fr´echet-Stein algebra. Suppose that X is a rigid analytic space over ∞ K that may be written as a union X = n=1 Xn , where {Xn }n≥1 is an increasing sequence of open aﬃnoid subdomains of X, for which the inclusions Xn → Xn+1 are admissible and relatively compact (in the sense of [1, 9.6.2]), and such that for each n the restriction map C an (Xn+1 , K) → C an (Xn , K) has dense image. (Here C an (Xn , K) denotes the Tate algebra of rigid analytic K-valued functions on Xn .) We will say that such a rigid analytic space X is strictly quasi-stein. (If one omits the requirement that the inclusions be relatively compact, one obtains the notion of a quasi-stein rigid analytic space, as deﬁned by Kiehl.) Since Xn → Xn+1 is an admissible open immersion for each n ≥ 1, the restriction map C an (Xn+1 , K) → C an (Xn , K) is ﬂat. The relative compactness assumption implies that it is furthermore compact, and hence that the space

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∼

C an (X, K) −→ lim C an (Xn , K) ←− n

of rigid analytic functions on X is naturally a nuclear Fr´echet-Stein algebra. Deﬁnition 1.8. Let A be a nuclear Fr´echet-Stein algebra over K, and ∼ write A −→ lim An as in Deﬁnition 1.6. We say that a Hausdorﬀ topo←− n

logical A-module M is coadmissible if the following two conditions are satisﬁed: (i) The tensor product Mn := An ⊗A M is a ﬁnitely generated An Banach module, for each n. (We regard the tensor product An ⊗A M as being a quotient of An ⊗K M , and endow it with the quotient topology induced by the projective tensor product topology on An ⊗K M .) (ii) The natural map M → lim Mn is an isomorphism of topological ←− n

A-modules. The preceding deﬁnition is a variation of [23, Def., p. 152], to which it is equivalent, as the results of [23, §3] show. Theorem 1.9. Let A be a nuclear Fr´echet-Stein algebra over K. (i) Any coadmissible topological A-module is a nuclear Fr´echet space. (ii) Any A-linear map between coadmissible topological A-modules is automatically continuous, with closed image. (iii) The category of coadmissible topological A-modules (with morphisms being A-linear maps, which by (ii) are automatically continuous) is closed under taking ﬁnite direct sums, passing to closed submodules, and passing to Hausdorﬀ quotients. Proof. This summarizes the results of [23, §3].

Remark 1.10. The category of all locally convex Hausdorﬀ topological A-modules is an additive category that admits kernels, cokernels, images and coimages. More precisely, if f : M → N is a continuous A-linear morphism between such modules, then its categorical kernel is the usual kernel of f , its categorical image is the closure of its settheoretic image (regarded as a submodule of N ), its categorical coimage is its set-theoretical image (regarded as a quotient module of M ), and its categorical cokernel is the quotient of N by its categorical image. Part (ii) of Theorem 1.9 implies that if M and N in the preceding paragraph are coadmissible, then the image and coimage of f coincide. Part (iii) of the Theorem then implies that the kernel, cokernel, and image of f are again coadmissible. Thus the category of coadmissible topological A-modules is an abelian subcategory of the additive category of locally convex Hausdorﬀ topological A-modules.

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Remark 1.11. If B is a Noetherian K-Banach algebra (for example, one of the algebras An appearing in Deﬁnitions 1.6 and 1.8), then the results of [1, 3.7.3] show that the natural functor from the category of ﬁnitely generated B-Banach modules (with morphisms being continuous B-linear maps) to the abelian category of ﬁnitely generated B-modules, given by forgetting topologies, is an equivalence of categories. Theorem 1.9 is an analogue of this result for the nuclear Fr´echet-Stein algebra A. It shows that forgetting topologies yields a fully faithful embedding of the category of coadmissible topological A-modules as an abelian subcategory of the abelian category of all A-modules. In light of this, one can suppress all mention of topologies in deﬁning this category (as is done in the deﬁnitions of [23, p. 152]). Deﬁnition 1.12. If A is a nuclear Fr´echet-Stein algebra over K, we say that a topological A-module M is strongly coadmissible if it is a Hausdorﬀ quotient of An , for some natural number n. Since A is obviously a coadmissible module over itself, Theorem 1.9 implies that any strongly coadmissible topological A-module is a coadmissible topological A-module. Example 1.13. Suppose that X is a strictly quasi-Stein rigid analytic space over K, as in Example 1.7. If M is any rigid analytic coherent sheaf on X, then the space M of global sections of M is naturally a coadmissible C an (X, K)-module, and passing to global sections in fact yields an equivalence of categories between the category of coherent sheaves on X and the category of coadmissible C an (X, K)-modules. The C an (X, K)module M of global sections of the coherent sheaf M is strongly coadmissible if and only if M is generated by a ﬁnite number of global sections. Fix a complete subﬁeld L of K. We close this section by recalling the deﬁnition of the space of locally analytic functions on a locally L-analytic manifold with values in a convex space. (More detailed discussions may be found in [14, §2.1.10], [20, p. 447], and [8, §2.1].) Deﬁnition 1.14. If X is an aﬃnoid rigid analytic space over L, and if ˆK W. W is a K-Banach space, then we write C an (X, W ) := C an (X, K) ⊗ (Here, as above, we let C an (X, K) denote the Tate algebra of K-valued rigid analytic functions on X, equipped with its natural K-Banach algebra structure.) If the set X := X(L) of L-valued points of X is Zariski dense in X, then C an (X, W ) may be identiﬁed with the space of W -valued functions on X that can be described by convergent power series with coeﬃcients in W .

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Now let X be a locally L-analytic manifold. A chart of X is a compact open subset X0 of X together with a locally analytic isomorphism between X0 and the set of L-valued points of a closed ball. We let X0 denote ∼ this ball (thought of as a rigid L-analytic space), so that X0 −→ X0 (L). By an analytic partition of X we mean a partition {Xi }i∈I of X into a disjoint union of charts Xi . We assume that X is paracompact; then any covering of X by charts may be reﬁned to an analytic partition of X. (Here we are using a result of Schneider [18, Satz 8.6], which shows that any paracompact locally L-analytic manifold is in fact strictly paracompact, in the sense of the discussion of [20, p. 446].) If V is a Hausdorﬀ convex space, then we say that a function f : X → V is locally analytic if for each point x ∈ X, there is a chart X0 containing x, a Banach space W equipped with a continuous K-linear map φ : W → V , and a rigid analytic function f0 ∈ C an (X0 , W ) such that f = φ ◦ f0 . (Replacing W by its quotient by the kernel of φ, we see that it is no loss of generality to require that φ be injective.) We let C la (X, V ) denote the K-vector space of locally analytic V -valued functions on X, and let Ccla (X, V ) denote the subspace consisting of compactly supported locally analytic functions. It follows from the deﬁnition that there are K-isomorphisms of vector spaces ∼ C la (X, V ) −→ lim C an (Xi , Wi ) −→ {Xi ,Wi ,φi }i∈I

and

∼

Ccla (X, V ) −→

lim −→

{Xi ,Wi ,φi }i∈I

i∈I

C an (Xi , Wi ),

i∈I

where in both cases the inductive limit is taken over the directed set of collections of triples {Xi , Wi , φi }i∈I , where {Xi }i∈I is an analytic partition of X, each Wi is a K-Banach space, and φi : Wi → V is a continuous injection. We regard C la (X, V ) and Ccla (X, V ) as Hausdorﬀ convex spaces by equipping them with the locally convex inductive limit topologies arising from the targets of these isomorphisms. Note that the inclusion Ccla (X, V ) → C la (X, V ) is continuous, but unless X is compact (in which case it is an equality) it is typically not a topological embedding. Given any collection {Xi , Wi , φi }i∈I as above, there is a natural map

C an (Xi , Wi ) =

i∈I

ˆ K Wi C an (Xi , K) ⊗

i∈I ˆ φi ⊕ id ⊗

−→

i∈I

ˆ K V −→ C an (Xi , K) ⊗

i∈I

ˆ K V. C an (Xi , K) ⊗

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(Note that if we were working with inductive, rather than projective, tensor product topologies, then the last map would be an isomorphism.) Passing to the inductive limit over all such collections yields a continuous map (1.15)

ˆ K V. Ccla (X, V ) → Ccla (X, K) ⊗

Proposition 1.16. If X is σ-compact (i.e. the union of a countable number of compact open subsets) and V is of compact type then the map (1.15) is a topological isomorphism and Ccla (X, V ) is again of compact type. Proof. If X is compact (so that Ccla (X, V ) = C la (X, V )) then this is [8, Prop. 2.1.28]. The proof in the general case is similar. 2. Categories of locally analytic representations Fix a ﬁnite extension L of Qp , for some prime p, as well a ﬁeld K that extends L and is complete with respect to a discrete valuation extending that on L. Let G be a locally L-analytic group (an analytic group over L, in the sense of [25, p. LG 4.1]). The identity element of G then has a neighbourhood basis consisting of compact open subgroups of G [25, Cor. 2, p. LG 4.23]. If H is any compact open subgroup of G, then Proposition 1.16 shows that the space C la (H, K) of locally L-analytic K-valued functions on H is a compact type convex K-space, and hence its strong dual is a nuclear Fr´echet space, which we will denote by D la (H, K). Any element h ∈ H gives rise to a “Dirac delta function” supported at h, which is an element δh ∈ Dla (H, K). In this way we obtain an embedding K[H] → Dla (H, K) (where K[H] denotes the group ring of H over K). The image of K[H] is dense in D la (H, K), and the K-algebra structure on K[H] extends (in a necessarily unique fashion) to a topological K-algebra structure on D la (H, K) [20, Prop. 2.3, Lem. 3.1]. Theorem 2.1. The topological K-algebra Dla (H, K) is a nuclear Fr´echet-Stein algebra. Proof. This is the main result of [23]. A diﬀerent proof is given in [8, §5.3]. We will now consider various convex K-spaces V equipped with actions of G by K-linear automorphisms. There are (at least) three kinds of continuity conditions on such an action that one can consider. Firstly, one may consider a situation in which G acts by continuous automorphisms of V . (Such an action is referred to as a topological action in [8];

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note that this condition does not make any reference to the topology of G.) Secondly, one may consider the case when the action map G × V → V is separately continuous. Thirdly, one may consider the case when the action map G × V → V is continuous. If V is barrelled (see [17, Def., p. 39]; for example a Banach space, a Fr´echet space, or a space of compact type) then any separately continuous action is automatically continuous, by the Banach-Steinhaus theorem. Proposition 2.2. If V is a compact type convex space, equipped with an action of G by continuous K-linear automorphisms, then the following are equivalent: (i) For some compact open subgroup H of G, the K[H]-module structure on V extends to a (necessarily unique) D la (H, K)-module structure on V , for which the map D la (H, K) × V → V describing this module structure is separately continuous. (i ) For every compact open subgroup H of G, the K[H]-module structure on V extends to a (necessarily unique) D la (H, K)-module structure on V , for which the map D la (H, K) × V → V describing this module structure is separately continuous. (ii) For some compact open subgroup H of G, the K[H]-module structure on Vb arising from the contragredient H-action on Vb extends to a (necessarily unique) topological D la (H, K)-module structure on Vb . (ii ) For every compact open subgroup H of G, the K[H]-module structure on Vb arising from the contragredient H-action on Vb extends to a (necessarily unique) topological D la (H, K)-module structure on Vb . (iii) There is a compact open subgroup H of G such that for any v ∈ V , the orbit map ov : H → V, deﬁned via h → hv, lies in C la (H, V ). (iii ) For any v ∈ V , the orbit map ov : G → V, deﬁned via g → gv, lies in C la (G, V ). Proof. The uniqueness statement in each of the ﬁrst four conditions is a consequence of the fact that K[H] is dense in D la (H, K), for any compact locally analytic L-analytic group. The equivalence of (i), (ii) and (iii) follows from [20, Cor. 3.3] and the accompanying discussion at the top of p. 453 of this reference. The equivalence of (iii) and (iii ) is straightforward. (See for example [8, Prop. 3.6.11].) Since (iii ) is independent of H, we see that (i ) and (ii ) are each equivalent to the other four conditions. Deﬁnition 2.3. If V is a compact type convex space equipped with an action of G by continuous K-linear automorphisms, then we say that V is a locally analytic representation of G if the equivalent conditions of Proposition 2.2 hold.

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We let Repla.c (G) denote the category of compact type convex spaces equipped with a locally analytic representation of G (the morphisms being continuous G-equivariant K-linear maps). Example 2.4. If G is compact, so that C la (G, K) is a compact type convex space (by Proposition 1.16), then the left regular action of G on C la (G, K) equips this space with a locally analytic G-representation. This is perhaps most easily seen by applying the criterion of Proposition 2.2 (ii). Indeed, the strong dual of C la (G, K) is equal to D la (G, K), and under the contragredient action to the left regular representation, an element g ∈ G acts as left multiplication by δg on Dla (G, K). Thus the required topological D la (G, K)-module structure on the strong dual of C la (G, K) is obtained by regarding the topological algebra Dla (G, K) as a left module over itself in the tautological manner. Similarly, the right regular action of G on C la (G, K) makes C la (G, K) a locally analytic G-representation. (Indeed, the topological automorphism f (g) → f (g −1 ) of C la (G, K) intertwines the left and right regular representations.) Remark 2.5. If V is an object of Repla.c (G), then since the orbit maps ov lie in C la (G, V ) for all v ∈ V they are in particular continuous on G. Thus the G-action on V is separately continuous, and hence (as was remarked above) continuous, by the Banach-Steinhaus theorem. Furthermore, we may diﬀerentiate the G-action on V and so make G a module over the Lie algebra g of G (or equivalently, over its universal enveloping algebra U(g)). The action g × V → V is again seen to be separately continuous (since the derivatives along the elements of g of a function in C la (G, V ) again lie in C la (G, V )), and hence (applying the Banach-Steinhaus theorem once more) is continuous. The U(g)-module structure on V admits an alternative description. Indeed, for any compact open subgroup H of G, there is a natural embedding U(g) → Dla (H, K), given by mapping X ∈ U(g) to the functional f → (Xf)(e). (Here X acts on f as a diﬀerential operator,1 and e denotes the identity of G.) Since V is an object of Repla.c (V ), it is a D la (H, K)-module (by part (i) of Theorem 2.1), and so in particular is a U(g)-module. This U(g)-module structure on V coincides with the one described in the preceding paragraph. 1 More precisely, the g action on C la (H, K) that we have in mind is the one obtained via diﬀerentiating the right regular action of H on C la (H, K). (By applying Example 2.4 to H, we ﬁnd that this H-action is locally analytic, and so may indeed be diﬀerentiated to yield a g-action.) It is given explicitly by the formula d (Xf )(h) = f (h exp(tX)), for any X ∈ g. dt |t=0

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Now suppose that Z is a topologically ﬁnitely generated abelian locally L-analytic group. If E is any ﬁnite extension of L, then we may ˆ consider the set Z(E) of E × -valued locally L-analytic characters on Z. Proposition 2.6. There is a strictly quasi-stein rigid analytic space Zˆ ˆ over L that represents the functor E → Z(E). Proof. This is [8, Prop. 6.4.5].

Example 2.7. Suppose that L = Qp , and that Z is the group Zp . Then Zˆ is isomorphic to the open unit disk centered at 1. (A character of Zˆ may be identiﬁed with its value on the topological generator 1 of Zp .) Example 2.8. Suppose that L = Qp , and that Z is the multiplicative group Q× p . There is an isomorphism ∼

∼

× Z Z Q× p −→ Zp × p −→ μ × Γ × p ,

where μ denotes the subgroup of roots of unity in Q× p , Γ denotes the subgroup of Z× consisting of elements congruent to 1 modulo p (respecp 2 Z tively p if p = 2), and p denotes the cyclic group generated by p ∈ Q× p. The group Γ is isomorphic to Zp , and so there is an isomorphism ∼ Zˆ −→ Hom(μ, Q× p ) × open unit disk around 1 × Gm .

Here Hom(μ, Q× p ) is the character group of the ﬁnite group μ, the open unit disk around 1 is the character group of Γ (see the preceding example), and Gm is the character group of pZ . (A character of the cyclic group pZ may be identiﬁed with its value on p). ˆ K) of The discussion of Example 1.7 shows that the K-algebra C an (Z, ˆ rigid analytic functions on Z is a nuclear Fr´echet-Stein algebra. Evaluation of characters at elements of Z induces an embedding of K-algebras ˆ K), with dense image (by [8, Prop. 6.4.6] and [20, K[Z] → C an (Z, Lem. 3.1]), and we have the following analogue of Proposition 2.2. Proposition 2.9. If V is a compact type convex space, equipped with an action of Z by continuous K-linear automorphisms, then the following are equivalent: (i) The K[Z]-module structure on V extends to a (necessarily unique) ˆ K)-module structure on V , for which the map C an (Z, ˆ K)×V → V C an (Z, describing this module structure is separately continuous. (ii) The K[Z]-module structure on Vb arising from the contragredient ˆ K)Z-action on Vb extends to a (necessarily unique) topological C an (Z, module structure on Vb .

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Proof. This follows from [8, Prop. 6.4.7]. If the Z-action on V satisﬁes the equivalent conditions of the preceding proposition, then it is separately continuous (as follows from condition (i)), and so is in fact continuous. If Z is a compact abelian locally L-analytic group (which is then necessarily topologically ﬁnitely generated [8, Prop. 6.4.1]), then we have the ˆ K), each containing two nuclear Fr´echet algebras Dla (Z, K) and C an (Z, the group ring K[Z] as a dense subalgebra. Proposition 2.10. If Z is a compact abelian locally L-analytic group, ∼ then there is an isomorphism of topological K-algebras D la (Z, K) −→ an ˆ C (Z, K), uniquely determined by the condition that it reduces to the identity on K[Z] (regarded as a subalgebra of the source and target in the natural manner). Proof. This is [8, Prop. 6.4.6]. It is proved using the p-adic Fourier theory of [22]. We now wish to tie together the two strands of the preceding discussion. We begin with the following strengthening of Theorem 2.1. Theorem 2.11. If H is a compact locally L-analytic group and Z is a topological ﬁnitely generated abelian locally L-analytic group, then the ˆ K) ⊗ ˆ K D la (H, K) (which by [17, p. 107] completed tensor product C an (Z, is a K-Fr´echet algebra) is a nuclear Fr´echet-Stein algebra. Proof. This follows from [8, Prop. 5.3.22], together with the remark following [8, Def. 5.3.21]. Suppose now that G is a locally L-analytic group, whose centre Z (an abelian locally L-analytic group) is topologically ﬁnitely generated. Deﬁnition 2.12. We let Repzla.c (G) denote the full subcategory of Repla.c (G) consisting of locally analytic representations V of G, the induced Z-action on which satisﬁes the equivalent conditions of Proposition 2.9. It follows from Propositions 2.2 and 2.9 that if V is a compact type convex space equipped with an action of G by continuous K-linear automorphisms, then the following are equivalent: (i) V is an object of Repzla.c (G). (ii) For some (equivalently, every) compact open subgroup H of G, the ˆ K)⊗ ˆ K D la (H, K)G-action on V induces a (uniquely determined) C an (Z, module structure on V for which the corresponding map ˆ K) ⊗ ˆ K D la (H, K) × V → V C an (Z, is separately continuous.

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(iii) For some (equivalently, every) compact open subgroup H of G, the contragredient G-action on Vb induces a (uniquely determined) ˆ K) ⊗ ˆ K D la (H, K)-module on Vb . structure of topological C an (Z, We can now deﬁne some important subcategories of the category Repzla.c (G). Deﬁnition 2.13. Let V be an object of Repzla.c (G). (i) We say that V is an essentially admissible locally analytic repreˆ K) ⊗ ˆ K D la (H, K)-module sentation of G if Vb is a coadmissible C an (Z, for some (equivalently, every) compact open subgroup H of G. (ii) We say that V is an admissible locally analytic representation of G if Vb is a coadmissible D la (H, K)-module for some (equivalently, every) compact open subgroup H of G. (iii) We say that V is a strongly admissible locally analytic representation of G if Vb is a strongly coadmissible Dla (H, K)-module for some (equivalently, every) compact open subgroup H of G. The equivalence of “some” and “every” in each of these deﬁnitions follows from the fact that if H ⊂ H is an inclusion of compact open subgroups of G then the algebra D la (H, K) is free of ﬁnite rank as a D la (H , K)-module (since H has ﬁnite index in H). Clearly, any strongly admissible locally analytic G-representation is admissible, and any admissible locally analytic G-representation is essentially admissible. The notion of strongly admissible (respectively admissible, respectively essentially admissible) locally analytic G-representation was ﬁrst introduced in [20] (respectively [23], respectively [8]). (Let us remark that any object V of Repla.c (G) for which Vb satisﬁes condition (ii) of Deﬁnition 2.13 automatically lies in Repzla.c (G), by [8, Prop. 6.4.10], and so the deﬁnitions of admissible and strongly admissible locally analytic representations of G given above do coincide with those of [23] and [20].) We let Repes (G) denote the full subcategory of Repzla.c (G) consisting of essentially admissible locally analytic representations, let Repad (G) denote the full subcategory of Repes (G) consisting of admissible locally analytic representations, and let Repsa (G) denote the full subcategory of Repad (G) consisting of strongly admissible locally analytic representations. These various categories lie in the following sequence of full embeddings: Repsa (G) ⊂ Repad (G) ⊂ Repes (G) ⊂ Repzla.c (G) ⊂ Repla.c (G). Both of the categories Repla.c (G) and Repzla.c (G) are closed under passing to countable direct sums (and more generally to Hausdorﬀ countable locally convex inductive limits), closed subrepresentations, Hausdorﬀ quotients, and completed tensor products [9, Lems. 3.1.2, 3.1.4].

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Theorem 2.14. Each of Repes (G) and Repad (G) is an abelian category, closed under the passage to closed G-subrepresentations, and to Hausdorﬀ quotient G-representations. Proof. This follows from Theorem 1.9.

The subcategory Repsa (G) of Repad (G) is closed under passing to ﬁnite direct sums and closed subrepresentations, but in general it is not closed under passing to Hausdorﬀ quotients. Remark 2.15. Let Z0 denote the maximal compact subgroup of Z, and let H be a compact open subgroup of G. Replacing H by Z0 H ifnecessary, we may assume that H contains Z0 (so that then Z0 = H Z). ∼ The K-algebra C an (Zˆ0 , K) −→ D la (Z0 , K) is a subalgebra of each of an ˆ la C (Z, K) and D (H, K). If V is an object of Repzla.c (G), then the two actions of C an (Zˆ0 , K) on each of V and Vb (obtained by regardˆ K) or Dla (H, K) respectively) coincide ing it as a subalgebra of C an (Z, (since both are obtained from the one action of Z0 on V ). Thus the ˆ K) ⊗ ˆ K D la (H, K)-action on each of V and Vb factors through C an (Z, ˆ K) ⊗ ˆ C an (Zˆ ,K) Dla (H, K). We take particuthe quotient algebra C an (Z, 0 lar note of two consequences of this remark. Example 2.16. If Z is compact (and so equals Z0 ), and if V lies in Repzla.c (G), then the preceding remark shows that the action of ˆ K) ⊗ ˆ K D la (H, K) on each of V and Vb factors through D la (H, K). C an (Z, Thus any essentially admissible locally analytic G-representation is in fact admissible. Also, in this situation, the categories Repla.c (G) and Repzla.c (G) are equal. Thus if the centre Z of G is compact, it can be neglected entirely throughout the preceding discussion. Example 2.17. If G is abelian, then G = Z. The preceding remark ˆ K) ⊗ ˆ K D la (H, K)shows that if V lies in Repzla.c (G), then the C an (Z, an ˆ action on each of V and Vb factors through C (Z, K). Example 1.13 then shows that passing to strong duals induces an antiequivalence of categories between the category Repes (Z) and the category of coherent ˆ Under this antiequivalence, the subcaterigid analytic sheaves on Z. gory Repad (Z) of Repes (Z) corresponds to the subcategory consisting of those coherent sheaves on Zˆ whose pushforward to Zˆ0 under the surjection Zˆ → Zˆ0 (induced by the inclusion Z0 ⊂ Z) is again coherent. (The point is that on the level of global sections, this pushforward corˆ K)-module as a C an (Zˆ0 , K)-module, via responds to regarding a C an (Z, an ˆ ˆ K).) the embedding C (Z0 , K) → C an (Z, Example 2.18. If G is compact, then C la (G, K) is an object of Repsa (G), and furthermore, any object of Repsa (G) is a closed subrepresentation of C la (G, K)n , for some n ≥ 0. (This follows directly from

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Deﬁnitions 2.13 (iii) and 1.12, and the fact that passing to strong duals takes closed subrepresentations of C la (G, K)n to Hausdorﬀ quotient modules of D la (G, K)n .) The following result connects the locally analytic representation theory discussed in this note with the more traditional theory of smooth representations of locally L-analytic groups. Theorem 2.19. If V is an admissible smooth representation of G on a K-vector space (in the usual sense), and if we equip V with its ﬁnest locally convex topology, then V becomes an element of Repad (G). Conversely, any object V of Repad (G) on which the G-action is smooth is an admissible smooth representation of G, equipped with its ﬁnest locally convex topology. Proof. See [8, Prop. 6.3.2] or [23, Thm. 6.5]. In the applications to the theory of automorphic forms, one typically assumes that G is the group of L-valued points of a connected reductive linear algebraic group G deﬁned over L. (Any such group certainly has topologically ﬁnitely generated centre.) In this case, we can make the following deﬁnition. Deﬁnition 2.20. If W is a ﬁnite dimensional algebraic representation of G deﬁned over K, then we say that a representation of G on a K-vector space V is locally W -algebraic if, for each vector v ∈ V , there exists an open subgroup H of G, a natural number n, and an H-equivariant homomorphism W n → V whose image contains the vector v. When W is the trivial representation of V , we recover the notion of a smooth representation of G. The following result generalizes Theorem 2.19. Theorem 2.21. Suppose that G = G(L), for some connected reductive linear algebraic group over L. If V is an object of Repad (G) that is also locally W -algebraic, for some ﬁnite dimensional algebraic representation W of G over K, then V is isomorphic to a representation of the form U ⊗B W, where B denotes the semi-simple K-algebra EndG (W ), and U is an admissible smooth representation of G deﬁned over B, equipped with its ﬁnest locally convex topology. Conversely, any such tensor product is a locally W -algebraic representation in Repad (G). Proof. This is [8, Prop. 6.3.10].

Remark 2.22. Taking the tensor product of ﬁnite dimensional representations and smooth representations is something that is quite unthinkable in the classical theory of smooth representations of G (in which the ﬁeld of coeﬃcients typically is taken to be C, or an -adic ﬁeld, with

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= p). In the arithmetic theory of automorphic forms, the role of smooth representations of p-adic reductive groups is to carry information about representations of the absolute Galois group of L on -adic vector spaces. (This is a very vague description of the local Langlands conjecture.) The consideration of locally algebraic representations of the type considered in Theorem 2.21 opens up the possibility of ﬁnding representations of p-adic reductive groups that can carry information about the representations of the absolute Galois group of L on p-adic vector spaces; in this optic, the role of the ﬁnite dimensional factor is to remember the “p-adic Hodge numbers” of such a representation. (See the introductory discussion of [3] for a lengthier account of this possibility.) 3. Locally analytic vectors in continuous admissible representations Let L, K and G be as in the preceding section. In this section we discuss an important method for constructing strongly admissible locally analytic representations of G, which involves applying the functor “pass to locally analytic vectors” to certain Banach space representations of G. We will begin by deﬁning that functor, but ﬁrst we must recall the notion of an analytic open subgroup of G. Suppose that H is a compact open subgroup of G that admits the structure of a “chart” of G; that is, a locally analytic isomorphism with the space of L-valued points of a closed ball. We let H denote the corresponding rigid analytic space (isomorphic to a closed ball) that has H as its space of L-valued points. If furthermore the group structure on H extends to a rigid analytic group structure on H, then, suppressing the choice of chart structure on H, we will refer to H as an analytic open subgroup of G. Since G is locally L-analytic, it has a basis of neighbourhoods consisting of analytic open subgroups. (See the introduction of [8, §3.5] for a more detailed discussion of the notion of analytic open subgroup.) Suppose now that U is a Banach space over K, equipped with a continuous G-action. If H is an analytic open subgroup of H, then we let UH−an denote the subspace of U consisting of vectors u for which the orbit map ou : H → U deﬁned by ou (h) = hu is (the restriction to H of) a rigid analytic U -valued function on H. Via the association of ou to a vector u ∈ UH−an , we may regard UH−an as a subspace of C an (H, U ), the Banach space of rigid analytic U -valued functions on H. Lemma 3.1. For any analytic open subgroup H of G, the space UH−an is a closed subspace of C an (H, U ). Proof. A rigid analytic function φ in C an (H, U ) belongs to UH−an if and only if its restriction to H is in fact of the form ou , for some u ∈ U

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(which will then certainly lie in UH−an ). This is the case if and only if φ satisﬁes the equation φ(h) = hφ(e) for all h ∈ H. (Here e denotes the identity element in H). These equations cut out a closed subspace of C an (H, U ), as claimed. We will always regard UH−an as being endowed with the Banach space topology it inherits by being considered as a closed subspace of C an (H, U ), as in the preceding lemma. The inclusion UH−an → U is thus continuous, but typically is not a topological embedding. Deﬁnition 3.2. We say that a vector u in U is locally analytic if the orbit map ou lies in C la (G, U ). (In fact, it suﬃces to require that ou be locally analytic in a neighbourhood of the identity, since the G-action on U is by continuous automorphisms). We let Ula denote the subspace of U consisting of locallyanalytic vectors; the preceding parenthetical remark shows that Ula = H UH−an , where H runs over all analytic open subgroups of G. We topologize Ula by endowing it with the locally convex ∼ inductive limit topology arising from the isomorphism Ula −→ lim UH−an −→ H

(the inductive limit being taken over the directed set of analytic open subgroups of G). This deﬁnition exhibits Ula as the locally convex inductive limit of a sequence of Banach spaces (and thus Ula is a so-called LB-space). The inclusion Ula → U is continuous, but typically is not a topological embedding. The map u → ou deﬁnes a continuous injection (3.3)

Ula → C la (G, U ).

Note that in [19] and [23], the topology on Ula is deﬁned to be that induced by regarding it as a subspace of C la (G, U ). In general, this is coarser than the inductive limit topology of Deﬁnition 3.2. We next introduce some terminology related to lattices in convex spaces. Deﬁnition 3.4. A separated, open lattice L in a convex K-space U is an open OK -submodule of U that is p-adically separated. We let L(U ) denote the set of all separated open lattices in U . Deﬁnition 3.5. If U is a convex space, then we say that two lattices L1 , L2 ∈ L(U ) are commensurable if aL1 ⊂ L2 ⊂ a−1 L1 for some a ∈ K ×. Clearly commensurability deﬁnes an equivalence relation on L(U ).

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Deﬁnition 3.6. If L ∈ L(U ) then we let {L} denote the commensurability class of L (i.e. the equivalence class of L under the relation of commensurability). We let L(U ) denote the set of commensurability classes of elements of L(U ). Example 3.7. If U is a Banach space over K, then L(U ) is non-empty, and in fact the elements of L(U ) form a neighbourhood basis of U . Furthermore, any two elements of L(U ) are commensurable, and so L(U ) consists of a single element. In general, if L ∈ L(U ), then L gives rise to a continuous norm sL on U , its gauge, uniquely determined by the requirement that L is the unit ball of sL . We let UL denote U equipped with the topology induced by ˆL the Banach space obtained by completing UL with respect sL , and let U to the norm sL . The identity map on the underlying vector space of U induces a continuous bijection U → UL , and hence a continuous injection ˆL . Given a pair of elements L1 , L2 ∈ L(U ), the topologies on UL U →U 1 and UL2 coincide if and only if L1 and L2 are commensurable. Suppose now that U is equipped with a continuous G-action. There is then an induced action of G on L(U ), deﬁned by (g, L) → gL for g ∈ G and L ∈ L(U ). This action evidently respects the relation of commensurability, and so descends to an action on L(U ). We write L(U )G (respectively L(U )G ) to denote the subset of L(U ) (respectively of L(U )) consisting of elements that are ﬁxed under the action of G. Passing to commensurability classes induces a map L(U )G → L(U )G . Lemma 3.8. If L is an element of L(U ), then the G-action on U inˆL ) if and only if the duces a continuous G-action on UL (and hence on U commensurability class {L} is G-invariant. Proof. It is immediate from the deﬁnitions that G acts on UL via continuous automorphisms if and only if {L} is G-invariant. Since the G-action on U is continuous by assumption, and since the natural bijection U → UL is continuous, the G-action on UL automatically satisﬁes conditions (i) and (iii) of [8, Lem. 3.1.1]. It thus follows from that lemma that if G acts on UL via continuous automorphisms, then the G-action on UL is in fact continuous. Lemma 3.9. Let H be an open subgroup of G. (i) If H is compact, then the map L(U )H → L(U )H is surjective. (ii) If L ∈ L(U ) is such that {L} ∈ L(U )H , then there is an open subgroup H of H such that L ∈ L(U )H . Proof. Suppose that L ∈ L(U ) is H-invariant. The H-action on U then induces a continuous H-action on UL , by Lemma 3.8. Part (i)

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of the present lemma is now seen to follow from [8, Lem. 6.5.3], while part (ii) follows immediately from the fact that the H-action on UL is continuous. In contrast to part (i) of the preceding lemma, if G is not compact then the map L(U )G → L(U)G is typically not surjective. For example, if U is a Banach space, then L(U )G = L(U ) (since the set on the right is a singleton). On the other hand, asking that L(U )G be non-empty is a rather stringent condition. Deﬁnition 3.10. A continuous representation of G on a Banach space is said to be unitary if L(U )G = ∅, that is, if U contains an open, separated lattice that is invariant under the entire group G (or equivalently, if its topology can be deﬁned by a G-invariant norm). Suppose now that L ∈ L(U )H for some open subgroup H of G. If π denotes a uniformizer of OK , then L/πL is a vector space over the residue ﬁeld OK /πOK , equipped with a smooth representation of H. Deﬁnition 3.11. If U is a convex space, equipped with a continuous Gaction of G, then we say that L ∈ L(U ) is admissible if it is H-invariant, for some compact open subgroup H of G, and if the resulting smooth H-representation on L/πL is admissible. Note that if L ∈ L(U) is admissible, and if H ⊂ G is a compact open subgroup that satisﬁes the conditions of the preceding deﬁnition with respect to L, then any open subgroup H ⊂ H also satisﬁes these conditions. Lemma 3.12. If L ∈ L(U ) is admissible, then every lattice in {L} is admissible. Proof. Let H be a compact open subgroup of G that satisﬁes the conditions of Deﬁnition 3.11 with respect to L. If L is an element of {L}, then by Lemma 3.9 (ii) (and replacing H by an open subgroup if necessary) we may assume that L is again H-invariant. Since L and L are commensurable, we may also assume (replacing L by a scalar multiple if necessary) that π n L ⊂ L ⊂ L for some n > 0. Thus L /πL is an H-invariant subquotient of L/π n+1 L. The latter H-representation is a successive extension of copies of L/πL, and so by assumption is an admissible smooth representation of H over OK /π n+1 OK . Any subquotient of an admissible smooth H-representation over OK /π n+1 OK is again admissible. (This uses the fact that the category of such representations is anti-equivalent – via passing to OK /π n+1 OK -duals – to the category of ﬁnitely generated modules over the completed group ring (OK /π n+1 OK )[[H]], together with a theorem of Lazard to the eﬀect that

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this completed group ring is Noetherian [16, V.2.2.4].2 ) In particular we conclude that L /πL is admissible. We say that a commensurability class {L} ∈ L(U ) is admissible if one (or equivalently every, by Lemma 3.12) member of the class is admissible in the sense of Deﬁnition 3.11. Proposition 3.13. If U is an object of Repes (G), then L(U ) contains an admissible lattice if and only if U is strongly admissible. Furthermore, if U is strongly admissible, then for any compact open subgroup H of G, we may ﬁnd an admissible H-invariant lattice in L(U ). Proof. See [8, Prop. 6.5.9]. Deﬁnition 3.14. Let U be a Banach space over K, equipped with a continuous action of G. We say that U is an admissible continuous representation of G, or an admissible Banach space representation of G, if one (or equivalently every, by Lemma 3.12) lattice in L(U ) is admissible, in the sense of the Deﬁnition 3.11. Theorem 3.15. The category of admissible continuous representations of G (with morphisms being continuous G-equivariant K-linear maps) is an abelian category, closed under passing to closed G-subrepresentations and Hausdorﬀ quotient G-representations. Proof. This is the main result of [21]. (See [8, Cor. 6.2.16] for the case when K is not local.) The key point is that if H is any compact open subgroup of G, then the completed group ring OK [[H]] is Noetherian [16, V.2.2.4].3 We let Repb.ad (G) denote the abelian category of admissible continuous representations of G. One important aspect of the preceding result is that maps in Repb.ad (G) are necessarily strict, with closed image. Example 3.16. If G is compact, then the space C(G, K) of continuous K-valued functions on G, made into a Banach space via the sup norm, and equipped with the left regular G-action, is an admissible continuous G-representation. Furthermore any object of Repb.ad (G) is a closed subrepresentation of C(G, K)n for some n ≥ 0. (See [8, Prop.-Def. 6.2.3].) If G is (the group of Qp -points of) a p-adic reductive group over Qp , then the admissible G-representations that are also unitary are perhaps the most important objects in the category Repb.ad (G). In [3, §1.3], 2 Strictly speaking, this reference only applies to the case when K = Q , so that p OK = Zp . However, the result is easily extended to the case of general K; see for example the proof of [8, Thm. 6.2.8]. 3 See the preceding note.

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Breuil explains the role that he expects these representations to play in a hoped-for “p-adic local Langlands” correspondence, in the case of the group GL2 (Qp ). For a discussion of how some of Breuil’s ideas might generalize to the case of a general reductive group, see [24, §5]. The following result provides a basic technique for producing strongly admissible locally analytic representations of G. Proposition 3.17. If U is an object of Repb.ad (G), then Ula is a strongly admissible locally analytic representation of G. Proof. This follows from the discussions of Examples 2.18 and 3.16, and the following two (easily veriﬁed) facts: (i) for any compact open sub∼ group H of G, there is a natural isomorphism C la (H, K) −→ C(H, K)la [8, Prop. 3.5.11]; (ii) if U and V are Banach spaces equipped with continuous G-representations, if U → V is a G-equivariant closed embedding, and if Vla is of compact type, then the diagram Ula

/ Vla

U

/V

is Cartesian in the category of convex spaces; in particular, the map Ula → Vla is again a closed embedding [8, Prop. 3.5.10]. See [8, Prop. 6.2.4] for the details of the argument. A version of the preceding theorem, working with the topology obtained on Ula by regarding it as a closed subspace of C la (G, U ), is given in [23, Thm. 7.1 (ii)]. We remark that if U is an object of Repb.ad (G), then the map (3.3) is in fact a topological embedding (see [5, Rem. A.1.1]). Thus, for such U, the topology on Ula induced by regarding it as a subspace of C la (G, U ) coincides with the inductive limit topology given by Deﬁnition 3.2. Lemma 3.18. If U is a convex space equipped with a continuous action of G, and if H is an open subgroup of G, then there exists a continuous H-equivariant injection U → W for some admissible continuous Hrepresentation W if and only if L(U ) contains an H-invariant admissible commensurability class. Proof. Given such a map U → W, the preimage of any lattice in W determines a commensurability class in L(U ) with the required properties. Conversely, given such a commensurability class {L}, it follows from Lemma 3.8 that the H-action on U extends to a continuous H-action ˆL , and so we may take W = U ˆL . on U

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Deﬁnition 3.19. An object V of Repad (G) is called very strongly admissible if V admits a G-equivariant continuous K-linear injection into an object of Repb.ad (G), or equivalently (by Lemma 3.18), if L(V ) contains a G-invariant admissible commensurability class. We let Repvsa (G) denote the full subcategory of Repad (G) consisting of very strongly admissible locally analytic G-representations. It is evidently closed under passing to subobjects and ﬁnite direct sums. Proposition 3.13 shows that it is a full subcategory of Repsa (G). It also follows from Proposition 3.13 that if G is compact, then every strongly admissible locally analytic G-representation is in fact very strongly admissible. The author knows no example of a strongly admissible, but not very strongly admissible, locally analytic G-representation (for any G). The following theorem of Schneider and Teitelbaum is fundamental to the theory of admissible continuous representations. Theorem 3.20. If L = Qp and if K is a ﬁnite extension of L then the map U → Ula yields an exact and faithful functor from the category Repb.ad (G) to the category Repvsa (G). Proof. See [23, Thm. 7.1]. (That the image of this functor lies in Repvsa (G) follows from Proposition 3.17 and the deﬁnition of Repvsa (G).) Given the exactness statement in the preceding result, the faithfulness statement is equivalent to the fact that Ula is dense as a subspace of U . In the context of Theorem 3.20, the functor U → Ula is not full, in general, as we now explain. If U is an object of Repb.ad (G), if L is an element of L(U ), and if we write Lla = L Ula , then {Lla } is a G-invariant and admissible commensurability class in Ula , which is evidently well-deﬁned independent of the choice of L (since all lattices in L(U ) are commensurable). Conversely, if V is an object of Repvsa (G), equipped with a Ginvariant and admissible commensurability class {M} ∈ L(V ), then the completion VˆM is an object of Repb.ad (G). In the case when (V, {M}) = (Ula , {Lla }) (in the notation of the previous paragraph), it follows from ∼ Theorem 3.20 (and the remark following that theorem) that VˆM −→ U. Thus, if we let C denote the category whose objects consist of pairs (V, {M}), where V is an object of Repvsa (G) and {M} ∈ L(V ) is a G-invariant admissible commensurability class (and whose morphisms are deﬁned in the obvious way), then the preceding discussion shows that U → (Ula , {Lla }) is a fully faithful functor Repb.ad (G) → C, to which the functor (V, {M}) → VˆM is left adjoint, and left quasi-inverse.

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On the other hand, the obvious forgetful functor C → Repvsa (G) (forget the commensurability class of lattices), while faithful, is not full. This amounts to the fact that a given very strongly admissible locally analytic representation of G can admit more than one G-invariant commensurability class of admissible lattices. Explicit examples are provided by the results of [3] (which show that the same irreducible admissible locally algebraic representation of GL2 (Qp ) can admit non-isomorphic admissible continuous completions, which are even unitary, in the sense of Deﬁnition 3.10). 4. Parabolic induction This section provides a brief account of parabolic induction in the locally analytic context. We let L and K be as in the preceding sections, and we suppose that G is (the group of L-valued points of) a connected reductive linear algebraic group over L. We let P be a parabolic subgroup of G, and let M be the Levi quotient of P . If V is an object of Repla.c (M ) (regarded as a P -representation through the projection of P onto M ), then we make the following deﬁnition: la IndG P V = {f ∈ C (G, V ) | f (pg) = pf (g) for all p ∈ P, g ∈ G},

equipped with its right regular G-action. (We topologize IndG P V by regarding it as a closed subspace of C la (G, V ).) Proposition 4.1. If V lies in Repla.c (M ) (respectively Repzla.c (M ), Repad (M ), Repsa (M ), Repvsa (M )), then IndG P V lies in Repla.c (G) (respectively Repzla.c (G), Repad (G), Repsa (G), Repvsa (G)). Proof. Although the proof of each of these statements is straightforward, altogether they are a little lengthy, and we omit them. Locally analytic parabolic induction satisﬁes Frobenius reciprocity. Proposition 4.2. If U and V are objects of Repla.c (G) and Repla.c (M ) respectively, then the P -equivariant map IndG P V → V induced by evalua∼ tion at the identity of G yields a natural isomorphism LG (U, IndG P V ) −→ LP (U, V ). (Here LG (– , – ) and LP (– , – ) denote respectively the space of continuous G-equivariant K-linear maps and the space of continuous P -equivariant K-linear maps between the indicated source and target.) Proof. This is a particular case of [14, Thm. 4.2.6], and also follows from [8, Prop. 5.1.1 (iii)]. Just as in other representation theoretic contexts, parabolic induction provides a way to obtain interesting new representations from old. The following result is due to H. Frommer [15]. (The case when G = GL2 (Qp ) was ﬁrst treated in [20].)

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Theorem 4.3. Suppose that L = Qp and that G is split, and let G0 be a hyperspecial maximal compact subgroup of G. If U is a ﬁnite dimensional irreducible object of Repla.c (M ) for which U(g) ⊗U(p) U is irreducible as U(g)-module, then IndG P U is topologically irreducible as a G0 -representation, and so in particular as a G-representation. (Here U denotes the contragredient to U , and U(p) is the universal enveloping algebra of the Lie algebra p of P .) One surprising aspect of this result is that it shows (in contrast to the cases of smooth representations of compact p-adic groups, and continuous representations of compact real Lie groups) that the compact group G0 can admit topologically irreducible inﬁnite dimensional locally analytic representations. 5. Jacquet modules Let L, K and G be as in the previous section, let P be a parabolic subgroup of G, and choose an opposite parabolic P to P . The intersection M := P P is then a Levi subgroup of each of P and P . Let N denote the unipotent radical of P . If U is an object of Repla.c (M ), then let Ccsm (N, U ) denote the closed subspace of Ccla (N, U ) consisting of compactly supported, locally constant (= smooth) U-valued functions on N . The projection map G → P \G restricts to an open immersion of locally analytic spaces N → P \G, and this immersion allows us to identify Ccla (N, U ) with the subspace of IndG U consisting of functions whose support is contained in P N . In P this way Ccla (N, U ) becomes a closed (U(g), P )-submodule of IndG U, P sm la and Cc (N, U ) is identiﬁed with the closed P -submodule of Cc (N, U ) consisting of elements annihilated by n (the Lie algebra of N ). Proposition 5.1. If U is an object of Repla.c (M ) then Ccsm (N, U ) is an object of Repla.c (P ). Proof. This follows from the identiﬁcation of Ccsm (N, U ) with a closed P -invariant subspace of IndG U, which Proposition 4.1 shows to be an P object of Repla.c (G). The formation of Ccsm (N, U ) is clearly functorial in U , and so we obtain a functor Ccsm (N, – ) from Repla.c (M ) to Repla.c (P ). Proposition 5.2. The restriction of Ccsm (N, – ) to Repzla.c (M ) (which is thus a functor from Repzla.c (M ) to Repla.c (P )) admits a right adjoint. Proof. See [9, Thm. 3.5.6].

As usual, let δ denote the smooth character of M that describes how right multiplication by elements of M aﬀects left-invariant Haar measure

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on P . Concretely, if m ∈ M , then δ(m) is equal to [N0 : mN0 m−1 ]−1 , for any compact open subgroup N0 of N . If U is an object of Repzla.c (M ), then let U (δ) denote the twist of U by δ. Deﬁnition 5.3. We let JP denote the functor from Repla.c (P ) to Repzla.c (M ) obtained by twisting by δ the right adjoint to the functor Ccsm (N, –). If V is an object of Repla.c (P ), we refer to JP (V ) as the Jacquet module of V . Thus for any objects U of Repzla.c (M ) and V of Repla.c (P ) there is a natural isomorphism (5.4)

∼

sm LP (Cc (N, U ), V ) −→ LM (U (δ), JP (V )).

Remark 5.5. If U is an object of Repzla.c (M ), then the natural map U (δ) → JP (Ccsm (N, U )) in Repzla.c (M ), corresponding via the adjointness isomorphism (5.4) to the identity automorphism of Ccsm (N, U ), is an isomorphism [9, Lem. 3.5.2]. Thus the isomorphism (5.4) is induced by passing to Jacquet modules (i.e. applying the functor JP ). Remark 5.6. Regarding a G-representation as a P -representation yields a forgetful functor from Repla.c (G) to Repla.c (P ). Composing this functor with the functor JP yields a functor from Repla.c (G) to Repzla.c (M ), which we again denote by JP . Theorem 5.7. The functor JP restricts to give a functor Repes (G) → Repes (M ). Proof. See [9, Thm. 0.5].

This theorem provides the primary motivation for introducing the notion of essentially admissible locally analytic representations. Indeed, even if V is an object of Repad (G), it need not be the case that JP (V ) lies in Repad (M ); however, see Corollary 5.24 below. Example 5.8. If G is quasi-split (that is, has a Borel subgroup deﬁned over L), and if we take P to be a Borel subgroup of G, then M is a torus, and so Repes (M ) is antiequivalent to the category of coherent sheaves ˆ . Thus if V is an object of on the rigid analytic space of characters M Repes (G), then we may regard JP (V ) as giving rise to a coherent sheaf on ˆ . This fact underlies the approach followed in [10] to the construction M of the eigencurve of [7], and of more general eigenvarieties. Example 5.9. If V is an admissible smooth representation of G, then there is a natural isomorphism between JP (V ) and VN , the space of N -coinvariants of V [9, Prop. 4.3.4]. This space of coinvariants is what is traditionally referred to as the Jacquet module of V in the theory of smooth representations.

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More generally, if V = U ⊗B W is an admissible locally W -algebraic representation of G, as in Theorem 2.21, then there is a natural iso∼ morphism JP (V ) −→ UN ⊗B W N (where W N denotes the space of N -invariants in W ) [9, Prop. 4.3.6]. Since U is an admissible smooth Grepresentation, the space UN is an admissible smooth M -representation [6, Thm. 3.3.1]. Thus JP takes admissible locally W -algebraic G-representations to admissible locally W N -algebraic M -representations. The remainder of this section is devoted to explaining the relation between the functor JP on Repla.c (G) and the process of locally analytic parabolic induction. We begin with the following remark. Remark 5.10. If V is an object of Repla.c (G), then the universal property of tensor products yields a natural isomorphism (5.11)

∼

sm sm LP (Cc (N, U ), V ) −→ L(g,P ) (U(g) ⊗U(p) Cc (N, U ), V ).

Thus for such V , the adjointness isomorphism (5.4) induces an isomorphism (5.12)

∼

sm L(g,P ) (U(g) ⊗U(p) Cc (N, U ), V ) −→ LM (U (δ), JP (V ).)

Deﬁnition 5.13. As above, we regard Ccsm (N, U ) as a closed subspace of IndG (U ). We let IPG (U ) (respectively Ipg (U )) denote the closed GP subrepresentation (respectively the U(g)-submodule) of IndG U that it P generates.

Note that Ipg (U ) admits the following alternative description: taking V to be IndG (U ), the isomorphism (5.11), applied to the inclusion P G sm Cc (N, U ) ⊂ IndP U , induces a (g, P )-equivariant map U(g) ⊗U(p) Ccsm (N, U ) → IndG (U ), P whose image coincides with Ipg (U ). In particular, there is a (g, P )equivariant surjection (5.14)

U(g) ⊗U(p) Ccsm (N, U ) → Ipg (U ).

Remark 5.15. The isomorphism of Remark 5.5 yields a closed embedding U (δ) → JP (IPG (U )), and hence for each object V of Repla.c (G), passage to Jacquet modules induces a morphism (5.16)

LG (IP (U ), V ) → LM (U (δ), JP (V )), G

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which is injective, by the construction of IPG (U ). Restricting elements in the source of this map to Ipg (U ) yields the left hand vertical arrow in the following commutative diagram LG (IPG (U ), V )

(5.16)

/ LM (U (δ), JP (V )) O ∼ (5.12)

g

L(g,P ) (Ip (U ), V )

/ L(g,P ) (U(g) ⊗U(p) Ccsm (N, U ), V )

whose bottom horizontal arrow is induced by composition with (5.14). Deﬁnition 5.17. Let U and V be objects of Repzla.c (M ) and Repla.c (G) respectively, and suppose given an element ψ ∈ LM (U (δ), JP (V )), corresponding via the adjointness map (5.12) to an element φ ∈ L(g,P ) (U(g) ⊗U(p) Ccsm (N, U ), V ). We say that ψ is balanced if φ factors through the surjection (5.14), and we let LM (U (δ), JP (V ))bal denote the subspace of LM (U (δ), JP (V )) consisting of balanced maps. (The property of a morphism being balanced depends not just on JP (V ) as an M -representation, but on its particular realization as the Jacquet module of the G-representation V .) Equivalently, LM (U (δ), JP (V ))bal is the image of the injection g L(g,P ) (Ip (U ), V ) → LM (U (δ), JP (V )) given by composing the right hand vertical arrow and bottom horizontal arrow in the commutative diagram of Remark 5.15. A consideration of this diagram thus shows that the image of (5.16) lies in LM (U (δ), JP (V ))bal . Deﬁnition 5.18. Let U be an object of Repzla.c (M ), and let H denote the space of linear M -equivariant endomorphisms of U . We say that U is allowable if for any pair of ﬁnite dimensional algebraic M-representations W1 and W2 , each element of LH[M ] (U ⊗K W1 , U ⊗K W2 ) is strict (i.e. has closed image). (Here each U ⊗K Wi is regarded as an H[M ]-module via the H action on the left hand factor along with the diagonal M -action.) It is easily checked that if U is an object of Repes (M ) and W is a ﬁnite dimensional algebraic M -representation, then M ⊗K W is again an object of Repes (M ). Thus objects of Repes (M ) are allowable in the sense of Deﬁnition 5.18. Theorem 5.19. If U is an allowable object of Repzla.c (M ) (in the sense of Deﬁnition 5.18) and if V is an object of Repvsa (G) (see Deﬁnition 3.19) then the morphism LG (IP (U ), V ) → LM (U (δ), JP (V )) induced by (5.16) is an isomorphism. G

bal

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The proof of Theorem 5.19 will appear in [13]. Remark 5.20. An equivalent phrasing of Theorem 5.19 is that (under the hypotheses of the theorem) the left hand vertical arrow in the commutative diagram of Remark 5.15 is an isomorphism. Remark 5.21. If U and V are admissible smooth representations of M and G respectively, then IPG (U) coincides with the smooth parabolic induction of U , while any M -equivariant morphism U (δ) → JP (V ) is balanced. The isomorphism of Theorem 5.19 in this case follows from Casselman’s Duality Theorem [6, §4]. Example 5.22. We consider the case when G = GL2 (Qp ) in some detail. We take P (respectively P ) to be the Borel subgroup of upper triangular matrices (respectively lower triangular matrices) of G, so that M is the maximal torus consisting of diagonal matrices in G. Let χ be a locally analytic K-valued character of Q× p , and let U denote the one dimensional representation of M over K on which M a 0 acts through the character → χ(a). Let k ∈ K denote the 0 d derivative of the character χ. Suppose ﬁrst that k is a non-negative integer. Let Wk denote the irreducible representation Symk K 2 of GL2 (Qp ) over K, and let χk denote the highest weight of Wk with respect to P (so χk is the character a 0 → ak of M ). If U (χ−1 k ) denotes the twist of U by the inverse 0 d of χk , then U (χ−1 k ) is a smooth representation of M . The G-representation IPG (U ) is a proper subrepresentation of IndG U; P it coincides with the subspace of functions that are locally polynomial of degree ≤ k when restricted to N = Qp under the open immersion N → P \G = P1 (Q), and may also be characterized more intrinsically as the subspace of locally algebraic vectors in IndG U. It decomposes as a P tensor product in the following manner: G IPG (U ) ∼ = (IndP U (χ−1 k ))sm ⊗K Wk ,

where the subscript “sm” indicates that we are forming the smooth parabolic induction of the smooth representation U (χ−1 k ). If V is any object of Repvsa (G), then we let VWk −lalg denote the closed subspace of Wk -locally algebraic vectors in V . (See PropositionDeﬁnition 4.2.2 and Proposition 4.2.10 of [8].) The closed embedding VWk −lalg → V induces a corresponding morphism on Jacquet modules (which is again a closed embedding; see [9, Lem. 3.4.7 (iii)]), which in turn induces an

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injection LM (U (δ), JP (VWk −lalg )) → LM (U (δ), JP (V )). It is not hard to check that LM (U (δ), JP (V ))bal is precisely the image of this injection. Now the space VWk −lalg admits a factorization VWk −lalg ∼ = X ⊗K Wk , where X is an admissible smooth locally analytic GL2 (Qp )-representation [8, Prop.4.2.4], and so by Example 5.9 there is an isomorphism JP (VWk −lalg ) ∼ = JP (X)(χk ). Thus Theorem 5.19 reduces to the claim that the natural map G −1 LG ((IndP U (χk ))sm ⊗K Wk , X ⊗K Wk ) → LM (U (δ), JP (X)(χk ))

induced by passing to Jacquet modules is an isomorphism. This map sits in the commutative diagram G −1 LG ((IndP U (χk ))sm ⊗K Wk , X ⊗K Wk ) ∼

G −1 LG ((IndP U (χk ))sm , X)

/ LM (U (δ), JP (X)(χk )) ∼

/ LM (U (χ−1 )(δ), JP (X)), k

where the bottom arrow is again induced by applying JP . Thus we are reduced to considering the case of Theorem 5.19 when U and V are both smooth. As noted in the preceding remark, this case of Theorem 5.19 follows from Casselman’s Duality Theorem. If k is not a non-negative integer, on the other hand, then IPG (U ) coincides with IndG U, and every element of LM (U (δ), JP (V )) is balanced. P In this case the proof of Theorem 5.19 is given in [5, Prop. 2.1.4]. (More precisely, the cited result shows that the left hand vertical arrow of the commutative diagram of Remark 5.15 is an isomorphism.) Corollary 5.23. Suppose that G is quasi-split, and subgroup of G. If V is an absolutely topologically strongly admissible locally analytic representation JP (V ) = 0, then V is a quotient of IPG (χ) for some K-valued character χ of the maximal torus M of G.

that P is a Borel irreducible 4 very of G for which locally L-analytic

Proof. We sketch the proof; full details will appear in [13]. Since JP (V ) ˆ (E) for is a non-zero object of Repes (M ), we may ﬁnd a character ψ ∈ M some ﬁnite extension E of K for which the ψ-eigenspace of JP (V ⊗K E) is non-zero. Taking U to be ψδ −1 in Deﬁnition 5.17, we let W denote the image of the map U(g) ⊗U(p) Ccsm (N, U ) → V ⊗K E corresponding via (5.12) to the inclusion of U (δ) into JP (V ⊗K E). If dψ denotes the 4 That is, E ⊗ V is topologically irreducible as a G-representation, for every ﬁnite K extension E of K.

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derivative of ψ (regarded as a weight of the Lie algebra m of M ) then Ccsm (N, U ) is isomorphic to a direct sum of copies of dψ as a U(p)-module, and so W is a direct sum of copies of a quotient of the Verma module U(g) ⊗U(p) dψ. Let W [n] denote the set of elements of W killed by n; this space decomposes as a direct sum of weights of m. Furthermore, for every weight α of m that appears, there is a corresponding character ψ˜ appearing in JP (V ⊗K E) for which dψ˜ = α. (Compare the proof of [9, Prop. 4.4.4].) The theory of Verma modules shows that we may ﬁnd a weight α of m appearing in W [n] such that α − β does not appear in W [n] for any element β in the positive cone of the root lattice of m. Let ψ˜ be a character ˜ and set U ˜ −1 . Our ˜ = ψδ of M appearing in JP (V ⊗K E) for which α = dψ, ˜ (δ) → JP (V ⊗K E) is choice of α ensures that the resulting inclusion U ˜ ) → V ⊗K E. balanced, and so Theorem 5.19 yields a non-zero map IPG (U Since V ⊗K E is irreducible by assumption, this map must be surjective. ˜ −1 must Since V is deﬁned over K, a simple argument shows that ψδ also be deﬁned over K. Corollary 5.24. Let G and P be as in Corollary 5.23. If V is an admissible locally analytic representation of G of ﬁnite length, whose composition factors are very strongly admissible, then JP (V ) is a ﬁnite dimensional M -representation. Proof. The functor JP is left exact (see [9, Thm. 4.2.32]), and so it sufﬁces to prove the result for topologically irreducible objects of Repvsa (G). One easily reduces to the case when V is furthermore an absolutely topologically irreducible object of Repvsa (G). If JP (V ) is non-zero then ˆ (K). AlCorollary 5.23 yields a surjection IPG (χ) → V for some χ ∈ M though JP is not right exact in general, one can show that the induced map JP (IPG (χ)) → JP (V ) is surjective. Thus it suﬃces to prove that the source of this map is ﬁnite dimensional. This is shown by a direct calculation. The details will appear in [13]. References 1. S. Bosch, U. G¨ untzer and R. Remmert, Non-archimedean analysis, SpringerVerlag, 1984. 2. N. Bourbaki, General Topology. Chapters 1-4, Springer-Verlag, 1989. 3. C. Breuil, Invariant L et s´ erie sp´ eciale p-adique, Ann. Scient. E.N.S. 37 (2004), 559–610. 4. C. Breuil, S´ erie sp´ eciale p-adique et cohomologie ´ etale compl´ et´ ee, preprint (2003). 5. C. Breuil, M. Emerton, Repr´ esentations p-adiques ordinaires de GL2 (Qp ) et compatibilit´ e local-global, preprint (2005). 6. W. Casselman, Introduction to the theory of admissible representations of p-adic reductive groups, unpublished notes distributed by P. Sally, draft May 7, 1993.

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7. R. Coleman, B. Mazur, The eigencurve, Galois representations in arithmetic algebraic geometry (Durham, 1996) (A. J. Scholl and R. L. Taylor, eds.), London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, 1998, pp. 1–113. 8. M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups, to appear in Memoirs of the AMS. 9. M. Emerton, Jacquet modules for locally analytic representations of p-adic reductive groups I. Construction and ﬁrst properties, Ann. Scient. E.N.S. 39 (2006), 775-839. 10. M. Emerton, On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math. 164 (2006), 1-84. 11. M. Emerton, p-adic L-functions and unitary completions of representations of p-adic reductive groups, Duke Math. J. 130 (2005), 353–392. 12. M. Emerton, A local-global compatibility conjecture in the p-adic Langlands programme for GL2/Q , Pure and Appl. Math. Quarterly 2 (2006), no. 2 (Special issue: In honor of John H. Coates, Part 2 of 2), 1-115. 13. M. Emerton, Jacquet modules for locally analytic representations of p-adic reductive groups II. The relation to parabolic induction, in preparation. 14. C. T. F´eaux de Lacroix, Einige Resultate u ¨ber die topologischen Darstellungen padischer Liegruppen auf unendlich dimensionalen Vektorr¨ aumen u ¨ber einem padischen K¨ orper, Thesis, K¨ oln 1997, Schriftenreihe Math. Inst. Univ. M¨ unster, 3. Serie, Heft 23 (1999), 1-111. 15. H. Frommer, The locally analytic principal series of split reductive groups, Preprintreihe des SFB 478 – Geometrische Strukturen in der Mathematik 265 (2003). 16. M. Lazard, Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965). 17. P. Schneider, Nonarchimedean functional analysis, Springer Monographs in Math., Springer-Verlag, 2002. 18. P. Schneider, p-adische analysis, Vorlesung in M¨ unster, 2000; available electronically at http://www.math.uni-muenster.de/math/u/schneider/publ/lectnotes. 19. P. Schneider, J. Teitelbaum, p-adic boundary values, Ast´erisque 278 (2002), 51– 123. 20. P. Schneider, J. Teitelbaum, Locally analytic distributions and p-adic representation theory, with applications to GL2 , J. Amer. Math. Soc. 15 (2001), 443–468. 21. P. Schneider, J. Teitelbaum, Banach space representations and Iwasawa theory, Israel J. Math. 127 (2002), 359–380. 22. P. Schneider, J. Teitelbaum, p-adic Fourier theory, Documenta Math. 6 (2001), 447–481. 23. P. Schneider, J. Teitelbaum, Algebras of p-adic distributions and admissible representations, Invent. Math 153 (2003), 145–196. 24. P. Schneider, J. Teitelbaum, Banach-Hecke algebras and p-adic Galois representations, Documenta Math. Extra Volume: John H. Coates’ Sixtieth Birthday (2006), 631-684. 25. J. -P. Serre, Lie algebras and Lie groups, W. A. Benjamin, 1965.

Modularity for some geometric Galois representations Mark Kisin Department of Mathematics University of Chicago USA kisin@ math.uchicago.edu with an appendix by Ofer Gabber Contents Introduction. 1. The Yoga of patching. 2. Barsotti-Tate representations. 3. Crystalline representations of intermediate weight. References. Appendix by Ofer Gabber.

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Introduction In our previous paper [Ki 1] we introduced a new technique for studying ﬂat deformation rings of local Galois representations, and we applied this to prove the following modularity lifting theorem for two dimensional potentially Barsotti-Tate representations: Theorem. Let p > 2, S a ﬁnite set of primes of Q, and GQ,S the Galois group of the maximal extension of Q unramiﬁed outside S. Let E/Qp be a ﬁnite extension, O its ring of integers, and denote by χ the p-adic cyclotomic character of GQ,S . Suppose ρ : GQ,S → GL2 (O) is a continuous representation such that (1) ρ is potentially Barsotti-Tate at each p, and det ρ · χ−1 has ﬁnite order. (2) The mod p representation ρ¯ obtained by reducing ρ modulo the radical of O is modular. (3) ρ¯|Q(ζp ) is absolutely irreducible. Then ρ is modular. The author was partially supported by NSF grant DMS-0400666 and a Sloan Research Fellowship Typeset by AMS-TEX

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The purpose of this note is to give an overview of the methods of [Ki 1], and to explain how, when combined with recent work of BergerBreuil [BB 1], they can also be used to prove a modularity lifting theorem for crystalline representations of “intermediate weight”. Speciﬁcally, we show the following Theorem. With the notation above, let ρ : GQ,S → GL2 (O) be a continuous representation. Suppose that for some 2 k 2p − 1 (1) ρ is crystalline with Hodge-Tate weights 0 and k − 1 at p. (2) ρ¯ arises from a modular form of weight k and prime to p level.1 (3) ρ¯|Q(ζp ) is absolutely irreducible, and ρ¯|GQp has only scalar endomorphisms. Then ρ is modular. In fact we prove a version of both the above theorems in the case where Q is replaced by a totally real ﬁeld in which p is totally split. This gives a mild extension of the main result of [Ki 1]. The paper is organized as follows. In the ﬁrst section, we explain the modiﬁcation of the Taylor-Wiles patching argument which was used in [Ki 1]. The main new idea is that one should patch global deformation rings as algebras over local deformation rings, and not just as O-algebras. This has the eﬀect of reducing diﬃculties in deformation theory to local questions. The most important consequence is that the method has a chance to work even if the local deformation ring at p is highly singular, whereas the original method of Taylor-Wiles could succeed only if this ring was a power series ring over O. Another upshot is that the minimal and non-minimal cases can be treated using the same argument. In particular the commutative algebra arguments of Wiles [W] comparing Fitting and congruence ideals completely disappear. The patching argument reduces modularity lifting theorems to a question about the components of a suitable local deformation ring R. In §2 we explain the method used in [Ki 1] to resolve this question for BarsottiTate representations (over a possibly ramiﬁed extension of Qp ). The basic idea is that rather than studying the deformation ring directly, one builds over it a kind of resolution G R → Spec R, which parameterizes ﬁnite ﬂat models of deformations. Although this resolution is not quite smooth, its singularities can be controlled, and the analysis of its connected components can eventually be reduced to a computation inside an aﬃne Grassmannian. Using this, we give a slight extension of the main result of [Ki 1] to the case of a totally real ﬁeld. 1 Using the weight part of Serre’s conjecture one can show that it suﬃces to assume simply that ρ¯ is modular of any weight and level.

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In §3, we study the question of components in the case of local deformation rings corresponding to crystalline deformations with weight 2p − 2. The basic idea is that, by Berger-Breuil, one knows the reductions of crystalline representations of medium weight. This result suggests that the generic ﬁbre of the corresponding deformation ring should be either a disc or an annulus, and so in particular connected. To really show this one ﬁrst constructs a map from Spec R[1/p] to the corresponding disc (resp. annulus). For the construction one uses an analogue of the space G R which appears in the Barsotti-Tate case. In that situation G R is a kind of moduli space for ﬁnite ﬂat models of deformations. In the higher weight case the role of the ﬁnite ﬂat model is played by a certain integral structure in the (ϕ, Γ)-module corresponding to a deformation. We call this structure a Wach lattice. The deﬁnition is motivated by the work of Berger [Be] which asserts that the (ϕ, Γ)-module associated to a Zp -lattice in a crystalline representation admits a unique Wach lattice. Having constructed the required map of complete local rings, one needs to show that it induces an isomorphism on generic ﬁbres over O. The results of Berger-Breuil guarantee that it induces a bijection on points with values in any ﬁnite extension of Qp . That this is enough follows from a general result about maps between complete local rings, whose proof is given in the appendix by Ofer Gabber. Acknowledgment: It is a pleasure to thank K. Buzzard, M. Emerton, T. Gee, and F. Herzig for useful remarks on various versions of this article. I would also like to thank the referee for a very thorough reading of the paper. §1 The Yoga of patching (1.1) Let p > 2 be a prime, F a totally real ﬁeld, and D be a quaternion algebra with center F which is ramiﬁed at all the inﬁnite places of F and at a set of ﬁnite places Σ, which does not contain any primes dividing p. We ﬁx a maximal order OD of D, and for each ﬁnite place ∼ v∈ / Σ, an isomorphism (OD )v −→ M2 (OFv ).

f × Let contained

U = × v Uv ⊂ (D⊗F AF ) be a compact open subgroup in v (OD )v . We assume that if v ∈ Σ, then Uv = (OD )× , v and that Uv = GL2 (OFv ) for v|p. ¯ p of Qp , and let E ⊂ Q ¯ p be a ﬁnite exFix an algebraic closure Q tension of Qp , with ring of integer O, and residue ﬁeld F. We assume ¯ p . Write Wk = that E contains the images of all embeddings F → Q k−2 2 ⊗F

Sym O , where k ≥ 2 is an integer. Then Wk is naturally →E a v|p GL2 (OFv )-module. We regard it as a representation of U by letting Uv act trivially for v p. We also ﬁx a continuous character ψ : (AfF )× /F × → O × such that the action of U ∩ (AfF )× on Wk is given

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by multiplication by ψ −1 . We think of (AfF )× as acting on Wk via ψ −1 , so that Wk becomes a U (AfF )× -module. Note that for a given U there may be no such ψ, however such a ψ will exist if U is suﬃciently small. Let Sk,ψ (U, O) denote the O-module of continuous functions f : D × \(D ⊗F AfF )× → Wk such that for g ∈ (D ⊗F AfF )× we have f (gu) = u−1 · f (g) for u ∈ U, and f (gz) = ψ(z)f (g) for z ∈ (AfF )× . If we write (D ⊗F AfF )× = f × f × × i∈I D ti U (AF ) for some ti ∈ (D ⊗F AF ) and some ﬁnite index set I, then we have ∼

× × (U (AfF )× ∩t−1 i D ti )/F

Sk,ψ (U, O) −→ ⊕i∈I Wk

.

For each ﬁnite prime v of F we ﬁx a uniformiser πv of Fv . Let S be a set of primes containing Σ, the primes dividing p, the inﬁnite primes, and the primes v of F such that Uv ⊂ Dv× is not a maximal compact subgroup. We denote by S p the complement in S of the primes dividing p. If v ∈ /S is a ﬁnite prime we consider the left action of (D ⊗F AfF )× on Wk -valued functions on (D ⊗F AfF )× given by the formula This

(gf )(z) = f (zg).

π

0

induces an action of the double cosets Uv 0v πv Uv and Uv π0v 01 Uv on Sk,ψ (U, O). We denote these operators by Sv and Tv respectively. They do not depend on the choice of πv . We denote by Tψ (U ) the Oalgebra generated by the endomorphisms Tv and Sv for v ∈ / S. It is commutative and ﬁnite over O. Similarly, for v | p there is a natural action of M2 (OFv ) on Wk -valued functions on (D ⊗F AfF )× , given by (usm · f )(g) = u · f (gu). This action is smooth on functions in Sk,ψ (U, O), and its restriction to Uv leaves Sk,ψ (U, O) invariant, and

hence induces

a well deﬁned action of the π

0

double cosets Uv 0v πv Uv and Uv π0v 01 Uv on Sk,ψ (U, O). We again denote these operators by Tv and Sv , and we write Tψ (U ) for the Tψ (U ) algebra generated by these endomorphisms. It is again commutative, and ﬁnite over O. (1.2) Fix an algebraic closure F¯ of F, and let GF,S be the Galois group of the maximal subﬁeld of F¯ which is unramiﬁed over F outside of S. A maximal ideal m ⊂ Tψ (U) is called Eisenstein if there exists a ﬁnite abelian extension F of F such that Tv − 2 ∈ m for each prime v which ¯ denotes an algebraic closure splits completely in F . Equivalently, if F ¯ and of F, then m is the kernel of an O-algebra map θm : Tψ (U ) → F, m is Eisenstein if there is a two-dimensional, reducible representation ¯ of GF,S on an F-vector space such for v ∈ / S, the trace of a Frobenius

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at v is given by θm (Tv ). We say that a maximal ideal m of Tψ (U ) is Eisenstein if its intersection with Tψ (U) is. Suppose that m ⊂ Tψ (U ) is a non-Eisenstein maximal ideal. Using the existence of Galois representations attached to Hilbert modular eigenforms [Ta 2], and the Jacquet-Langlands correspondence (see [Ta 1, 1.3]), one sees that there exists a continuous representation ρm : GF,S → GL2 (Tψ (U )m ) such that for v ∈ / S, the characteristic polynomial of ρm (Frobv ) is X 2 − Tv X + N(v)Sv . Here Frobv denotes an arithmetic Frobenius at v, and N(v) denotes the order of the residue ﬁeld at v. After replacing O by a ﬁnite extension, we may assume that the residue ﬁeld at m is F, and we denote by ρ¯m : GF,S → GL2 (F) the representation obtained by reducing ρm modulo m . Since m is non-Eisenstein, ρ¯m is an absolutely irreducible representation. In particular it admits a universal deformation O-algebra RF,S , and ψ we denote by RF,S the quotient of RF,S corresponding to deformations with determinant ψχ, where χ denotes the p-adic cyclotomic character. Here we regard ψ as a character of GF,S via the class ﬁeld theory isomorphism, normalized the take uniformizers to arithmetic Frobenii. The ψ representation ρm induces a map RF,S → Tψ (U )m . The formula for the characteristic polynomial of Frobenius at v ∈ / S shows that this map is surjective. We now ﬁx, once and for all, a basis for the underlying F-vector space VF of ρ¯m . Let Σp = Σ ∪ {v}v|p . For any v ∈ Σp we ﬁx a decomposition group GFv ⊂ Gal(F¯ /F ), and we denote by DVF ,v the functor which assigns to a local Artinian O-algebra A with residue ﬁeld F the set of isomorphism classes of pairs (VA , βv,A ) where VA is a lift of the GFv -representation VF to a continuous representation on a ﬁnite free A-module, and βv,A is an A-basis for VA lifting the chosen basis of VF . Then DVF ,v is pro-representable by a complete local O-algebra R v called ,ψ the universal framed deformation ring of VF . We denote by Rv the quotient of R v corresponding to deformations with determinant ψχ. We set ,ψ v∈Σp Rv,ψ RΣ,p =⊗

where all the completed tensor products are over O. Similarly, we denote by R,ψ F,S the complete local O-algebra, which represents the functor obtained by assigning to A as above the set of isomorphism classes of tuples (VA , {βv,A }v∈Σp ), where VA is a deformation of the GF,S -representation VF to A having determinant ψ, and for v ∈ Σp βv,A is a lifting of the

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,ψ chosen basis to VA . The O-algebra RF,S has an obvious structure of a ψ formally smooth RF,S -algebra. Choose a maximal ideal m of Tψ (U ) whose restriction to Tψ (U ) is m . We will slightly abuse notation, and write Tψ (U )m for the image of Tψ (U )m in Tψ (U )m . (The map Tψ (U )m → Tψ (U)m is not in general ,ψ injective.) We set T ψ (U )m = Tψ (U )m ⊗Rψ RF,S . The tensor product F,S

is formally smooth over Tψ (U )m , and hence O-ﬂat. We now make the following assumptions on ρ¯m (1) (2) (3) (4)

ρ¯m is unramiﬁed outside primes dividing p. The restriction of ρ¯m to GF (ζp ) is absolutely irreducible. If p = 5, then [F (ζp ) : F ] > 2. If v ∈ S \ Σp , then

(1 − N(v))((1 + N(v))2 det ρ¯(Frobv ) − (N(v))(tr¯ ρ(Frobv ))2 ) ∈ F× . The conditions (2) and (3) are needed to ensure that in the patching method of Taylor-Wiles, one can ﬁnd enough auxiliary primes to kill the dual Selmer group. The order of the Selmer group which controls ,ψ ,ψ the number of (topological) generators of RF,S as an RΣ,p -algebra is then given as a product of local terms, and (1) and (4) ensure that this order is suﬃciently small that the dimension of the patched deformation ring (denoted R∞ in (1.3) below) is equal to that of the patched Hecke algebra. We also make the following assumption on U :

(1.2.1)

For all t ∈ (D ⊗F AfF )× the group (U (AfF )× ∩ t−1 D× t)/F × has order prime to p.

This condition is always satisﬁed if U is suﬃciently

small [Ta 1, 1.1]. For example, it holds if U is a normal subgroup and U v∞ OF×v ∩Ddet=1 contains no non-trivial p-power roots of unity. Proposition (1.3). Let d = [F : Q]+3|Σp |. Then there exists an integer h ≥ d, a commutative diagram of complete local O-algebras O[[y1 , . . . , yh ]]

R,ψ Σ,p [[x1 , . . . , xh−d ]]

/ R∞

/ R,ψ F,S

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and an R∞ -module M∞ such that (1) The horizontal maps are surjective and the map on the right ∼ induces an isomorphism R∞ /(y1 , . . . , yh )R∞ −→ R,ψ F,S . (2) M∞ is a ﬁnite ﬂat O[[y1 , . . . , yh ]]-module, and the action of R∞ on the quotient M∞ /(y1 , . . . , yh )M∞ factors through T ψ (U )m , and makes it into a faithful T ψ (U )m -module. Proof. This follows from [Ki 1, 3.2.5, 3.1.13], and the arguments of [Ki 1, 3.3.1, 3.4.11]. Let us just indicate the ingredients. The O-algebra R∞ and the R∞ -module M∞ are constructed by patching deformation rings and Hecke modules at auxiliary levels as in the argument of TaylorWiles [TW], as modiﬁed by Diamond [Di]. The only diﬀerence is the ,ψ observation that all the rings involved are RΣ,p -algebras and we patch ,ψ RΣ,p -algebras rather than just O-algebras. More precisely, for suitably chosen collections of primes Qn = {v1 , . . . , vh } which are disjoint from Σp and at which U is maximal, and such that N(vi ) ≡ 1 (pn ) for all i, one deﬁnes a compact open subgroup UQn of (D ⊗F AfF )× by (UQn )v = Uv if v ∈ / Qn , and

(UQn )v = {g ∈ GL2 (OFv ) : g ≡

a b 0 d

(πv ), ad−1 → 1 ∈ Δv }

for v ∈ Qn , where πv is a uniformizer at v, and Δv denotes the maximal pro-p quotient of (OFv /πv )× . Set SQn = S ∪ Qn , and denote by mQn the preimage of m under the natural map Tψ (UQn ) → Tψ (U ). Then R∞ is deﬁned as an inverse limit ,ψ ,ψ of suitable ﬁnite length RΣ,p -algebra quotients of RF,S , and M∞ is Qn obtained by taking an inverse limit over certain ﬁnite length quotients ,ψ of Sk,ψ (UQn , O)mQn ⊗Rψ RF,S . Qn F,SQ

n

The freeness M∞ over O[[y1 , . . . , yh ]] may be deduced from the fact that under the assumption (1.2.1) Sk,ψ (UQn , O) is a free O[ΔQn ]-module, because if we write (D ⊗F AfF )× = i∈IQ D× ti UQn (AfF )× , then ΔQn n acts freely on IQn (see also [Ta 1, 2.3]). ¯ ,ψ Corollary (1.4). Suppose that for v ∈ Σp there exists a quotient R v ,ψ of Rv such that ¯ ,ψ . (1) The action of Rv,ψ on M∞ factors through R v ,ψ ,ψ ¯ ¯ (2) Rv is an O-ﬂat domain and Rv [1/p] is formally smooth over E and geometrically integral (that is it remains a domain after tensoring by any ﬁnite extension of E). ¯ ,ψ (3) If v ∈ Σ then R has relative dimension 3 over O, while if v|p v it has relative dimension 3 + [Fv : Qp ].

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¯ ,ψ = ⊗ ¯ ,ψ , and set R ¯ ,ψ = R,ψ ⊗ ,ψ R ¯ ,ψ . Then the v∈Σp R Let R v Σ,p F,S F,S Σ,p R Σ,p

¯ ,ψ → T (U ) is surjective with p-power torsion kernel. natural map R m ψ F,S ,ψ Proof. First note that (1) ensures that the map RF,S → T ψ (U )m factors ,ψ ¯ ¯ ,ψ → through R . We have to show that the resulting surjection R F,S

F,S

T ψ (U)m has p-power torsion kernel. ¯ ,ψ [[x1 , . . . , xh−d ]] in the endomorphisms of M∞ is The image of R Σ,p a faithful O[[y1 , . . . , yh ]]-module and hence of dimension at least h + 1. ¯ ,ψ is a domain.2 Since the dimension The condition (2) ensures that R Σ,p ¯ ,ψ [[x1 , . . . , xh−d ]] is h + 1, M∞ is a faithful R ¯ ,ψ [[x1 , . . . , xh−d ]]of R Σ,p

Σ,p

module, and this ring is a ﬁnite faithful O[[y1 , . . . , yh ]]-module. ¯ ,ψ [[x1 , . . . , xh−d ]][1/p], M∞ has depth ≥ At any maximal ideal of R Σ,p h, since it is non-zero and ﬁnite free over O[[y1 , . . . , yh ]]. It follows by ¯ ,ψ [[x1 , . . . , xh−d ]][1/p]the Auslander-Buchsbaum theorem that the R Σ,p module M∞ ⊗Zp Qp is projective. In particular M∞ /(y1 , . . . , yh )M∞ ⊗Zp Qp is a faithful module over ¯ ,ψ [[x1 , . . . , xh−d ]][1/p]/(y1 , . . . , yh )R ¯ ,ψ [[x1 , . . . , xh−d ]][1/p] R Σ,p Σ,p ∼

¯ ,ψ [1/p]. −→ R F,S Since the action of R,ψ F,S on M∞ /(y1 , . . . , yh )M∞ factors through Tψ (U )m , the corollary follows (cf. [Ki 1, §3.3]).

¯ v,ψ for v ∈ Σ. (1.5) We now explain how to construct the quotient R ,ψ ¯ For v|p, a quotient Rv with the required properties is known to exist only in certain situations. When k = 2, the construction of such a quotient is one of the main points of [Ki 1], and will be explained in ¯ ,ψ can the next section. When p is totally split in F the existence of R v sometimes be deduced from results of Berger and Breuil if k 2p − 1. Proposition (1.6). Suppose v ∈ Σ, and let γ : GFv → O× be a continuous character. We denote by γ¯ the composite of γ and the projection O× → F× . Suppose that ρ¯m |GFv is an extension of γ¯ by γ¯ χ. Then there exists a quotient Rvχγ,γ, of Rv with the following properties. (1) Rχγ,γ, is an O-ﬂat domain, and Rvχγ,γ, [1/p] is formally smooth v over E and of dimension 3. ¯ ,ψ [1/p] has connected spectrum. After see this it is enough to show that R Σ,p ¯ ,ψ enlarging E we may assume that each of the rings R [1/p] has an E-valued v 2 To

point. Then the result follows from the fact that if Y → X is a map of topological spaces, which has connected ﬁbres and admits a section, then Y is connected if X is connected.

446

Mark Kisin (2) If E /E is a ﬁnite extension, and ξ : Rv → E a map of Oalgebras, then ξ factors through Rvχγ,γ, if and only if the corresponding E -representation Vξ of GFv is an extension of γ by γχ.

Proof. This is proved in [Ki 1, §2.6]. The argument already illustrates ¯ ,ψ for v|p, which is that one one of the ideas behind the construction R v can study a deformation ring by studying a resolution of it. We give a sketch. Deﬁne a functor Lγv on O-algebras, as follows: If A is an O-algebra, and ξ : Rv → A a morphism of O-algebras, then ξ gives rise to a representation of GFv on a ﬁnite free A-module VA . We denote by Lγv (A) the set consisting of pairs (ξ, LA ), where ξ is a morphism as above, and LA ⊂ VA is a projective A-submodule of rank 1 such that VA /LA is projective over A. We have an obvious morphism of functors Lγv → DVF ,v given by sending (ξ, LA ) to ξ, and this morphism is easily seen to be represented by a projective map Θ : Lvγ → Spec Rv . This map becomes an closed immersion after inverting p, because if p is invertible in A then LA , if it exists, is unique. We deﬁne Rvχγ,γ, to be the quotient of Spec R v corresponding to the scheme theoretic image of Θ. From the deﬁnition one sees that Rvχγ,γ, satisﬁes (2). To see that Lvγ is formally smooth over O we remark that H 2 (GFv , Zp ) has no p-torsion, so that for any ﬁnite length Zp -module M, equipped with a continuous action of GFv , the map H 1 (GFv , Zp (χ)) ⊗Zp M → H 1 (GFv , M (χ)) is surjective. (Here M (χ) denotes M with GFv acting via χ.) In particular, if A is an Artin ring with residue ﬁeld a ﬁnite extension of F, the natural map Ext1O[GFv ] (O(γ), O(γχ)) ⊗O A → Ext1A[GFv ] (A(γ), A(γχ)) is a surjection and Lγv is formally smooth. Hence Rvχγ,γ, satisﬁes (1). The dimension of Lvγ [1/p] can be computed using Galois cohomology. It remains to show that Lvγ is connected. Since Lvγ is formally smooth over O it suﬃces to show that the ﬁbre of Θ over the closed point of Spec Rv is connected. However this ﬁbre is either a single point or a projective line. The second case occurs precisely, when the action of GFv on VF is scalar, in which case we must have γ¯ = γ¯ (1). Corollary (1.7). For each v ∈ Σ, there is a unique γ : GFv → O × as above such that the quotient Rvχγ,γ, satisﬁes the conditions of (1.4).

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Proof. By the local Langlands correspondence, ρ¯m is an extension of γ¯ by γ¯(1) for some unramiﬁed character GFv → O × . We necessarily have γ¯ 2 = ψ. Since p = 2, γ¯ lifts to a unique unramiﬁed character γ : GFv → O× such that γ 2 = ψ. We claim the corollary holds with this choice of γ. The conditions (2) and (3) of (1.4) follow from (1.6)(1), while (1.4)(1) follows from (1.6)(2) and the local Langlands correspondence, the point being that D is ramiﬁed at each v ∈ Σ, while M∞ is constructed using quaternionic forms on compact open subgroups of (D ⊗F AfF )× which are maximal compact at each such v. The last remark also shows that γ is the only character such that Rvχγ,γ, satisﬁes (1.4)(1). §2 Barsotti-Tate representations ¯ ,ψ (2.1) In this section we explain how to construct the quotient R v in (1.4) in the case that v|p and k = 2. Since this is essentially a local question, we work locally. Let k be a ﬁnite extension of Fp . Write W = W (k) for the ring of Witt vectors of k, K0 = W [1/p] and K for a ﬁnite totally ramiﬁed extension of K0 of degree e. Let OK be the ring of integers in K, π ∈ OK a ﬁxed uniformiser, and E(u) the Eisenstein polynomial of π. We ¯ of K, and write OK¯ for its ring of integers. ﬁx an algebraic closure K √ n ¯ ¯ be a root of π, such that We set GK = Gal(K/K). Let p π ∈ K √ √ √ n+1 n n p p p ( π) = π for all n ≥ 0. Set K∞ = ∪n≥1 K( p π), and write ¯ GK∞ = Gal(K/K ∞ ). Let S = W [[u]]. We equip S with a Frobenius endomorphism ϕ which acts as the usual Frobenius on W and takes u to up . Write OE for the p-adic completion of S[1/u]. The ring OE is a complete discrete valuation ring with uniformiser p, and residue ﬁeld k((u)). The endomorphism ϕ extends to OE by continuity. An ´etale ϕ-module over OE is a ﬁnite OE -module M together with ∼ an isomorphism of OE -modules ϕ∗ (M ) −→ M. There is an equivalence of categories between ﬁnite length, ´etale OE -modules and continuous representations of GK∞ on ﬁnite length Zp -modules. This is essentially a consequence of [Fo, §A]. The results of loc. cit are more often applied to the p-cyclotomic extension of K, when they give rise to the theory of (ϕ, Γ)-modules. Here we use the Kummer extension K∞ . Since this ﬁeld is far from being Galois over K, there is no immediate analogue of Γ, and this suggests that the Kummer extension may be less useful for studying representations of GK . However a result of Breuil [Br 3, 3.4.3], asserts that for representations arising from ﬁnite ﬂat OK -group schemes, the restriction functor from GK to GK∞ is fully faithful.

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To explain how to describe ﬁnite ﬂat models of Galois representations in terms ´etale ϕ-modules, we introduced the category (Mod FI/S). An object of this category consists of a ﬁnitely generated S-module M equipped with a S-linear map 1 ⊗ ϕ : ϕ∗ (M) → M such that M is a ﬁnite direct sum of modules of the form S/pn S for various n ≥ 1, and the cokernel of 1 ⊗ ϕ is killed by E(u). Then we have the following result [Ki 1, 1.1.13], a large part of which is due to Breuil [Br 4, 3.1.2, 3.1.3], [Br 3, 3.3.2]. Theorem (2.2). We have a commutative diagram of functors / {ﬁnite ﬂat OK -group schemes}

(Mod FI/S) M →M[1/u]

{ﬁnite length, ´etale OE -modules}

∼

¯ G →G(K)(−1)

/ {ﬁnite length GK -representations} ∞

where the lower horizontal functor has been discussed above. The top horizontal functor is exact and fully faithful, and an equivalence on objects killed by p. Using the theorem, and the full faithfulness of restriction from GK to GK∞ for ﬁnite ﬂat Galois representations, we see in particular, that two modules M and M give rise to ﬁnite ﬂat group schemes with iso∼ morphic generic ﬁbres if and only if M[1/u] −→ M [1/u]. Our method for understanding ﬂat deformation rings involves studying the modules in (Mod FI/S) which give rise to a given ﬁnite length ´etale OE -module. The above discussion shows that this is closely related to studying ﬁnite ﬂat models of a given Galois representations. We remark that the category (Mod FI/S) in the above theorem can be slightly enlarged so that the above functor extends to an equivalence with all ﬁnite ﬂat OK -group schemes [Ki 2, 2.3.5]. We denote this enlarged category by (Mod/S). (2.3) Now let F be a ﬁnite extension of Fp , as above, and VF a ﬁnite dimensional F-vector space of dimension d, equipped with a continuous action of GK . We denote by MF the ´etale ϕ-module attached to VF (−1)|GK∞ by the correspondence explained in (2.1). Then MF is naturally a free F ⊗Fp k((u))-module. We will assume that End GK VF = F. This assumptions is made only to simplify the exposition. The general case requires the language of groupoids. We denote by ARW (F) the category of Artinian, local W (F)-algebras with residue ﬁeld F. We denote by AugW (F) the category of W (F)algebras A equipped with a nilpotent ideal I such that pA ⊂ I. Note that if A is in ARW (F) then we may regard A as an object of AugW (F)

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by taking I to be the maximal ideal of A. This makes ARW (F) into a full subcategory of AugW (F) . Let A be a Zp -algebra. We denote by (Mod FI/S)A the category whose objects consist of a ﬁnite projective SA = S ⊗Zp A-module M together with a map of SA -modules ϕ∗ (M) → M whose cokernel is killed by E(u). Here we extend ϕ to SA by A-linearity. For (A, I) in AugW (F) , we denote by DS,MF (A, I) the set of isomorphism classes of pairs (MA , ψ), where MA is in (Mod FI/S)A , and ψ : OE ⊗S MA ⊗A ∼ A/I −→ MF ⊗F A/I is an isomorphism of A/I ⊗Fp k((u))-modules compatible with the action of ϕ. For A in ARW (F) we denote by DVF (A) the set of isomorphism classes of a ﬁnite free A-module VA equipped with a continuous action of GK ∼ and an isomorphism of F[GK ]-modules VA ⊗A F −→ VF . We denote by ﬂ DVF (A) ⊂ DVF (A) the subset consisting of those VA which are the generic ﬁbres of a ﬁnite ﬂat group scheme over OK . Both DVF and DVﬂF can be extended to functors on AugW (F) , by taking DVF (A, I) and DVﬂF (A, I) to be the direct limit of DVF (A ) and DVﬂF (A ) respectively, where A ⊂ A runs over Artinian local W (F)-algebras with residue ﬁeld F whose radical is contained in I. Our assumption on VF implies that the functors DVF and DVﬂF are pro-representable by complete local W (F)-algebras, RVF and RﬂVF respectively [Ma], [Ram]. Finally, for A in ARW (F) we denote by DMF (A) the set of isomorphism classes of ﬁnite free OE ⊗Zp A-modules MA equipped with an isomor∼ ∼ phism ϕ∗ (MA ) −→ MA and an F-linear isomorphism MA ⊗A F −→ MF respecting ϕ. As above, we may extend this to a functor on AugW (F) . The relationship among these functors is given by the following results [Ki 1, 2.1.4, 2.1.11, 2.4.8]. Proposition (2.4). We have a commutative diagram of functors on AugW (F) . / D MF DVﬂF O cFF FF FF ΘVF FF DS,MF where the vertical map is given by MA → OE ⊗S MA . The horizontal map is injective, so that ΘVF is uniquely determined by requiring the commutativity of the diagram. Proof. This is [Ki 1, 2.1.4]. The ﬁrst step in the proof is to show that the association by MA → OE ⊗S MA gives a well deﬁned object of DMF . This is not completely obvious, since for (A, I) in AugW (F) , DMF (A, I) is deﬁned by a limit process. One has to show that the module OE ⊗S MA

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can be descended to an object MA in DMF (A ) for some subring A ⊂ A with A in ARW (F) . Using (2.2) above, MA corresponds to a representation of GK∞ on a ﬁnite free A -module VA . One checks that MA actually arises from a module in (Mod/S). This implies that VA extends to ﬁnite ﬂat representation of GK . This extension is unique by Breuil’s result mentioned above. Proposition (2.5). The morphism ΘVF is represented by a projective morphism ΘVF : G R VF → Spec RﬂVF . After inverting p both sides of this map become formally smooth over W (F)[1/p], and ΘVF becomes an isomorphism. Proof. The statement is proved in several steps. First consider any (A, mA ) in ARW (F) , and ﬁx ξ = [VA ] ∈ DVﬂF (A). Let MA be the corresponding object in DMF (A). Regarding ξ as a functor on AugW (F) (represented by the ring A equipped with its radical), we can form the product ∼

DS,MF ,ξ = ξ ×DVfl DS,MF −→ ξ ×DMF DS,MF . F

(The fact that this product - as deﬁned - has reasonable properties depends crucially on the assumption that VF has only scalar endomorphisms. Without this assumption one needs to deﬁne it as a product of groupoids.) ∼ Fix an isomorphism of (A ⊗Zp W )((u))-modules MA −→ (A ⊗Zp W )((u))d . Recall that the aﬃne Grassmannian corresponding to MA is an Ind-projective A-scheme which to an A-algebra B associates the collection of (B ⊗ W )[[u]]-lattices in (B ⊗Zp W )((u))d . (A (B ⊗ W )[[u]]lattice is a ﬁnite projective (B ⊗W )[[u]]-submodule of rank d, which spans the (B ⊗Zp W )((u))-module (B ⊗Zp W )((u))d .) The ﬁrst step in proving the proposition is to show that DS,MF ,ξ is representable by a closed Ind-subscheme of the aﬃne Grassmannian. This is not surprising since if (A, mA ) → (B, I) is a morphism in AugW (F) and MB in DS,MF (B, I) maps to MA ⊗A B under the vertical functor in (2.3), then MB can be regarded as such a lattice. Next one shows that DS,MF ,ξ is actually representable by a closed subscheme, the key point being that multiplying a ϕ-stable lattice by u multiplies ϕ by up−1 . Since for any module MB in DS,MF (B, I) the cokernel of ϕ∗ (MB ) → MB is killed by E(u), if MB and MB are two lattices arising in this way then one ﬁnds that ek MB ⊂ u−i MB for some i p−1 , where k is the least integer such that k p · A = 0.

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Applying the above discussion to ﬁnite length quotients of RVﬂF and passing to the limit gives rise to a proper map of formal schemes G R VF → Spf RﬂVF . That it arises from a projective map of schemes follows from formal GAGA, and the fact that the aﬃne Grassmannian is equipped with a line bundle whose restriction to any closed subscheme of ﬁnite type is very ample. The claims regarding smoothness can be proved by giving a functor of points style description of the generic ﬁbres of G R VF and RVﬂF . (At least for points with values in Artin rings). In the case of RVﬂF this description is in terms of weakly admissible modules. Its availability is based on Breuil’s results [Br 2] which, in particular, imply that a weakly admissible module with Hodge-Tate weights equal to (0, 1) arises from a p-divisible group (see also [Ki 2]). Finally, the proof that ΘVF becomes an isomorphism after inverting p rests on two facts: that the category (Mod FI/S)Zp is equivalent to the category of p-divisible groups over OK (this is proved in [Ki 1, 2.2.22] using the results of [Br 2]; another proof is given in [Ki 2]) and on Tate’s theorem that a p-divisible group is determined by its generic ﬁbre. In fact, given that the morphism ΘVF is intended to classify ﬁnite ﬂat models of Galois representations, the fact that the generic ﬁbre of ΘVF is an isomorphism can be viewed as a geometric incarnation of Tate’s theorem. Actually, the proof given in [Ki 1] also uses the smoothness discussed above, but this can probably be avoided. (2.6) We now specialize to the case dimVF = 2. For (A, I) in AugW (F) , 0,1 we deﬁne DS,M (A, I) ⊂ DS,MF (A, I) as the subset of modules MA F such that the A ⊗Zp OK -submodule ϕ∗ (MA )/E(u)MA ⊂ MA /E(u)MA is Lagrangian. When MA /E(u)MA is free over A ⊗Zp OK , this means that under some (and hence any) non-degenerate symplectic pairing on MA /E(u)MA , the submodule ϕ∗ (MA )/E(u)MA is equal to its own annihilator. (Here we are identifying ϕ∗ (MA ) with a submodule of MA via 1 ⊗ ϕ.) In general we require that this condition hold locally on Spec A ⊗Zp OK . For A in ARW (F) we denote by DV0,1 (A) ⊂ DVﬂF (A) the subset consistF ing of deformations VA such that the action of inertia on det VA is given by the cyclotomic character. As usual we can extend this subfunctor to AugW (F) . We have the following result describing the relationship between these two subfunctors [Ki 1, 2.4.8]. 0,1 0,1 Proposition (2.7). ΘVF induces a morphism Θ0,1 VF : DS,MF → DVF , which is represented by a projective morphism

(2.7.1)

0,1 0,1 Θ0,1 VF : G R VF → Spec RVF .

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After inverting p, Θ0,1 VF becomes an isomorphism of formally smooth W (F)[1/p]-schemes of pure dimension [K : Qp ] + 1. The complete local rings at closed points on G R 0,1 VF are isomorphic to those on Hilbert modular schemes. In particular, G R 0,1 VF is a local complete intersection over W (F), and the reduction G R 0,1 VF ⊗Z Z/pZ is normal. Proof. The statements in the ﬁrst paragraph are not hard to deduce from (2.5). To see the claim on dimensions one describes the tangent space at a closed point x of Spec RV0,1 [1/p] in terms of crystalline self F extensions of the corresponding Galois representation Vx . The dimension of the corresponding Galois cohomology group can be computed using formulas found in [Ne, 1.24]. Consider some F-valued point y ∈ G R 0,1 VF (F). This corresponds to a lattice MF ⊂ MF . To describe the complete local ring at y one studies the problem of deforming MF to an object MA in (Mod FI/S)MA , for A in ARW (F) , subject to the condition that ϕ∗ (MA )/E(u)MA is Lagrangian. This problem can be broken into two parts: First one speciﬁes a Lagrangian submodule LA ⊂ MA /E(u)MA , and then a surjection ˜ A , where L ˜ A denotes the preimage of LA in MA . The ϕ∗ (MA ) → L problem of specifying LA is visibly related to the moduli problem describing integral models of Hilbert modular varieties [DP]. The extra ˜ A introduces only smooth parameters bedata of the map ϕ∗ (MA ) → L cause ϕ∗ (MA ) is a free SA -module. To see this one deduces the lifting criterion for smoothness from the lifting property for maps with source a free module. The ﬁnal two statements follow from the known local structure of Hilbert modular schemes [DP]. (2.8) If E/W (F)[1/p] is a ﬁnite extension, and x : RVF → E a map of W (F)-algebras, we denote by Vx the two dimensional E-representation of GK obtained by specializing the universal representation over RVF by x. Recall that a two dimensional, Barsotti-Tate representation is called ordinary if its restriction to inertia is an extension of E by E(1). Theorem (2.9). Let E/W (F)[1/p] be a ﬁnite extension and consider E-valued points x1 , x2 : RV0,1 → E. If the images of x1 and x2 in F Spec RV0,1 [1/p] lie in the same connected component then Vx1 and Vx2 F are either both ordinary or both non-ordinary. The converse statement holds under any of the following conditions (1) Vx1 and Vx2 are ordinary. (2) The residue ﬁeld of K is equal to Fp . (3) The ramiﬁcation index e(K/Qp ) is at most p − 1. Proof. For a space X, denote by H0 (X) its set of connected components.

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Using (2.7) we have ∼

∼

0,1 H0 (Spec RV0,1 [1/p]) −→ H0 (G R 0,1 VF ⊗Zp Qp ) −→ H0 (G R VF ⊗Zp Z/pZ) F

where the second isomorphism is easily deduced from the fact that 0,1 0,1 G R 0,1 VF ⊗Zp Z/pZ is reduced, and G R VF is Zp -ﬂat and proper over Spec RVF . By the theorem on formal functions, the ﬁnal term is also equal to 0,1 0,1 H0 (G R 0,1 VF ,0 ) where G R VF ,0 denotes the ﬁbre of ΘVF over the closed point of Spec RV0,1 . F The ﬁrst statement in the theorem can be proved by considering the quotient of RV0,1 corresponding to ordinary deformations, and showing F that after inverting p, the corresponding inclusion map on spectra becomes a local isomorphism. One can also show the corresponding statement for the scheme G R 0,1 VF , by deﬁning the notion of an ordinary module in DS,MF (A, I). The corresponding subscheme of G R 0,1 VF is a connected component, even without inverting p. To show the converse one has to compute H0 (G R 0,1 VF ,0 ), and show that it consists of at most two components. More precisely, one has to show that two points which are both either ordinary or non-ordinary lie on the same component. In the ordinary case one ﬁnds that, if End GK VF = F, 0,1 then the ordinary subscheme G R ord VF ,0 ⊂ G R VF ,0 consists of at most a single point. If one works without this assumption, then G R ord VF ,0 can also consist of two points (in certain cases where VF is a direct sum of two distinct characters) or it may be equal to a projective line (when the action on VF is scalar). In applications one can always reduce to the last situation by making a base change, so the case with two points does not cause a problem. In the case that the residue ﬁeld of K is Fp , one shows that the nonordinary subset of G R 0,1 VF ,0 is connected by constructing chains of rational curves between any two points. The connectedness result should be true without the condition on the residue ﬁeld, but we are unable to prove it. It is not hard to show that closed points on the non-ordinary part of G R 0,1 VF ,0 correspond to bi-connected (i.e connected with connected dual) ﬁnite ﬂat models of VF ⊗F F , for F a ﬁnite extension of F. If the ramiﬁcation degree e(K/Qp ) is at most p − 1 then VF has a unique bi-connected model by results of Raynaud [Ra, 3.3.5, 3.3.6], so in this case the nonordinary subset consists of a single point. (2.10) Now returning to the situation of §1, suppose that k = 2, and that v|p, and VF |GFv has only scalar endomorphisms. Taking K = Fv above, the ordinary (resp. non-ordinary) points on Spec RV0,1 [1/p] are a F union of connected components by (2.9). If Tv ∈ / m (resp. Tv ∈ m), we

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¯ v the quotient of R0,1 corresponding to the closure of the denote by R VF ordinary (resp. non-ordinary) components in Spec RV0,1 [1/p]. F Let RV0,1,ψ denote the quotient of RV0,1 = RV0,1 ⊗W (F) O corresponding F F ,O F to deformations with determinant ψχ. Note that since k = 2, ψ has ∼ ﬁnite order. We have RV0,1 −→ RV0,1,ψ [[X]], the isomorphism depending F ,O F on a choice of topological generator for the Galois group of the maximal ¯ψ = R ¯ v ⊗ 0,1 R0,1,ψ , one easily unramiﬁed extension of Fv . Setting R v VF ,O R VF

∼ ¯ψ ¯ v ⊗W (F) O −→ ¯ vψ [1/p] is sees that R Rv [[X]]. Hence by (2.9) and (2.7), R formally smooth over O[1/p] of dimension [Fv : Qp ], and it is a domain ¯ ,ψ either if Tv ∈ / m or Fv has residue ﬁeld Fp . Finally we denote by R v the ring representing the functor which to an Artinian O-algebra A with residue ﬁeld F, assigns the set of isomorphism classes of pairs (VA , βv ) where VA is a deformation of VF isomorphic to one obtained from a ¯ ψ → A, and βv is an A-basis for VA lifting some ﬁxed basis morphism R v of VF . ¯ ,ψ satisﬁes the conditions (2) and (3) of (1.4). It It is clear that R v remains to check the condition (1). The fact that the action of RVF on M∞ factors through RV0,1,ψ follows from the construction of the Galois F representations associated to Hilbert modular forms [Ca], [Ta 2], [Ta 3]; they are limits of Galois representations found in abelian varieties with good reduction at primes dividing p (because U is assumed maximal at ¯ v follows from the fact p). That this action actually factors through R that these representations satisfy a crystalline version of the EichlerShimura relation. Namely, if Vf is the representation associated to a Hilbert eigenform f, then the action of Tv on f is given by the trace of Frobenius on Dcris (Vf ) [Ki 1, 3.4.2]. Hence if f corresponds to a maximal ideal m as in §1, then Tv ∈ m if and only if Vf is non-ordinary at v. Finally we remark that if VF |GFv has non-scalar endomorphisms one can still carry out a variant of the theory of this section and deﬁne a ring ¯ ,ψ . It is a domain except in certain cases where Tv ∈ R / m and VF |GFv v is a sum of two distinct characters. In applications this does not cause any problems, since after a suitable base change we can assume that ¯ ,ψ is a domain; the scheme the action of GFv is trivial. In this case R v ord G R VF ,0 of (2.9) is a projective line.

Using (1.4), results on raising and lowering the level of quaternionic forms in the style of [DT] and [SW], as well as some base change arguments, one deduces the following modularity lifting theorem. To state ¯ = F¯ into Q ¯ p and C. If f is a Hilbert modular it, we ﬁx embeddings of Q eigenform, then we obtain a corresponding representation ρf : GF,S → GL2 (O) where O is the ring of integers of some suﬃciently large extension E of Qp . In the following we will always assume that O is large

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enough. Given any such representation ρ we will denote by ρ¯ the corresponding representation into GL2 (F), where F is the residue ﬁeld of O. Theorem (2.11). Let ρ : GF,S → GL2 (O) be a continuous representation. Suppose that (1) ρ is Barsotti-Tate at each v|p and det ρ · χ−1 has ﬁnite order. (2) There exists a Hilbert modular eigenform f over F of parallel weight 2 such that ρ¯ ∼ ρ¯f , and for each prime v|p of F, the representation of GL2 (AF ) generated by f is spherical at v, and the eigenvalue of Tv acting on f is a p-adic unit if and only if ρ|GFv is ordinary. (3) If v|p and ρ|GFv is non-ordinary then either the residue ﬁeld of v is Fp or the ramiﬁcation index e(Fv /Qp ) is at most p − 1. (4) ρ¯|F (ζp ) is absolutely irreducible, and [F (ζp ) : F ] > 2 if p = 5. Then ρ ∼ ρg for some Hilbert modular eigenform g over F. (2.12) By base change, the above also implies a modularity lifting theorem for potentially Barsotti-Tate representations. The condition that the eigenvalue of Tv is a unit if and only if ρ|GFv is ordinary is equivalent to asking that ρ is ordinary at v if and only if ρf is. This condition is sometimes inconvenient in applications. It is partially ameliorated by the following result. Corollary (2.13). Let F be a totally real ﬁeld in which p is totally split, and ρ : GF,S → GL2 (O) a continuous representation. Suppose that (1) ρ is potentially Barsotti-Tate at each v|p and det ρ·χ−1 has ﬁnite order. (2) There exists a Hilbert modular form over F of parallel weight 2 such that ρ¯ ∼ ρ¯f . (3) ρ¯|F (ζp ) is absolutely irreducible. Then ρ ∼ ρg for some Hilbert modular eigenform g over F. Proof. Let S denote the set of primes v|p of F such that ρ|GFv is not potentially ordinary (i.e. is absolutely irreducible), and S the set of v|p such that ρ|GFv is potentially ordinary. By (2.14) below we may choose f so that at each v ∈ S , ρf becomes ordinary and Barsotti-Tate over an extension Fv of Fv of degree p − 1. Thus after replacing F by a ﬁnite solvable extension we may assume that for each v ∈ S , ρf is ordinary and Barsotti-Tate. We, of course, no longer assume that p is split in F, and this hypothesis is needed only to apply (2.14). Now an argument using the mod p reductions of irreducible representations of GL2 (Zp ) as in [Ki 1, 3.5.7, 3.1.6] shows that there exists a Hilbert modular form f over F of parallel weight 2, such that ρ¯f ∼ ρ¯f , ρf is ordinary and Barsotti-Tate at each v ∈ S , and such that the corresponding GL2 (AF )-representations πf is cuspidal at each v ∈ S. After

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again replacing F by a solvable extension, we ﬁnd that ρf is BarsottiTate at each v|p, and ordinary if and only if ρf is ordinary. Thus the corollary follows from (2.11), because we can choose the ﬁnite extensions of F we have made so that the conditions (2.11)(3) and (2.11)(4) are satisﬁed. Lemma (2.14). Let F be a totally real ﬁeld in which p is totally split, and S a collection of places of F dividing p. Let f be a Hilbert modular cusp form over F of parallel weight 2, and suppose that ρ¯f is absolutely irreducible, and that for v ∈ S , ρ¯f |GFv is the reduction of a potentially ordinary, potentially Barsotti-Tate representation of GFv . Then there exists a Hilbert modular form f over F of parallel weight 2 and such that (1) ρ¯f ∼ ρ¯f . (2) ρf |GFv becomes ordinary and Barsotti-Tate over an extension of degree p − 1, for each v ∈ S . Proof. Choose f satisfying (1). Let πf denote the representation of GL2 (AF ) generated by f , and let S ⊂ S denote the subset of places at which πf is a twist by a character of a spherical representation. We may assume that f has been chosen so that S has largest possible order. An argument similar to that in [Ki 1, 3.5.7, 3.1.6], using [CDT, 3.1.1], shows that one may choose f such that for v ∈ S the local factor at v of πf has a corresponding Weil group representation which is tamely ramiﬁed and abelian (so that the inertia acts through a quotient of order p − 1), and is scalar for v ∈ S . The point is that if σ is an irreducible mod p representation of GL2 (Fp ) then σ appears as a subquotient of the mod p reduction of one of the following: × • The character θ ◦ det, where θ : F× p →O . GL (F )

• The subspace of Ind B 2 p θ consisting of functions with average value 0, where B denotes the subgroup of upper triangular matrices, and θ is as above. GL (F ) • The representations Ind B 2 p θ, where θ : B → E × is a character

given by θ a0 db = θ1 (a)θ2 (d) with θ1 = θ2 . We claim that πf is not special at any v ∈ S . To see this, suppose χ ∗ that πf is special at some such v. Then ρ¯f ∼ ρ¯f has the form on inertia at v, where χ is the mod p cyclotomic character. Our 0

assumption that ρ¯f |GFv is the reduction of a potentially ordinary, potentially Barsotti-Tate representation (which necessarily becomes BarsottiTate over an abelian extension of Fv = Qp ) implies that ∗ is peu ramiﬁ´e, because a tr`es ramiﬁ´e cocycle cannot become peu ramiﬁ´e over an abelian extension of Qp . Then Jarvis’ generalization of Mazur’s principle

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[Ja, Thm. 6.1] applied to a suitable twist of f , implies that there exists a Hilbert modular form f of parallel weight 2 such that ρ¯f ∼ ρ¯f and at each place of {v} ∪ S πf has a spherical twist. This contradicts the maximality of |S |. It follows that at each v ∈ S , ρf becomes Barsotti-Tate over Fv = Fv (ζp ). We claim that ρf |GF is ordinary. To see this we use the fact v that ρf |GF arises as the generic ﬁbre of a p-divisible group G. Then v either G is ordinary, or connected with connected dual. In the latter case G[p] would be the unique ﬁnite ﬂat model of ρ¯f |GF [Ra, 3.3.5, v 3.3.6]. However, since ρ¯f |GF is the reduction of a potentially ordinary, v potentially χ ∗ Barsotti-Tate representation, its restriction to inertia has the form 0 1 with ∗ peu ramiﬁ´e, as before. (Of course χ|GF is actually v trivial). Hence ρ¯f |GF has an ordinary ﬁnite ﬂat model. It follows that v G is ordinary, and so is ρf |GF . v

§3 Crystalline representations in intermediate weight (3.1) We continue to use the notation of the previous section, but we assume that K = K0 . We brieﬂy recall some facts from the theory of (ϕ, Γ)-modules [Fo]. As in the previous section we denote by OE the p-adic completion of W [[u]][1/u], however we now equip this ring with a continuous Frobenius endomorphism given by ϕ(u) = (1 + u)p − 1, and the usual Frobenius on W. The subring S is again stable by ϕ. We also equip OE with a continuous action of Γ = Gal(Qp (μp∞ )/Qp ) ∼ by setting γ(u) = (1 + u)χ(γ) − 1 where χ : Γ −→ Z× p denotes the cyclotomic character. The actions of ϕ and Γ on OE are easily seen to commute, and S is Γ-stable. Recall that an ´etale (ϕ, Γ)-module is an OE -module M of ﬁnite length, equipped with commuting semi-linear actions of ϕ and Γ, such that the 1⊗ϕ OE -linear map ϕ∗ (M ) → M is an isomorphism. The basic result of Fontaine asserts that the category of ´etale (ϕ, Γ)-modules is equivalent to the category of continuous representations of GK on ﬁnite length Zp -modules. Given an ´etale (ϕ, Γ)-module M, we denote by VOE (M ) the corresponding GK -representation. Conversely given a GK -representation V we denote by DOE (V ) the corresponding ´etale (ϕ, Γ)-module. There is an analogous result for (ϕ, Γ)-modules which are ﬁnite free over OE . They correspond to continuous representations of GK on ﬁnite free Zp -modules. We will again denote by VOE and DOE the functors which give rise to this equivalence of categories. (3.2) Our task in this section is to construct and analyze certain crystalline deformation rings. The existence of these rings follows immediately

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from the main result of [Be], however it will be useful to give another construction (again using the results of [Be]) which provides some extra crucial information about these rings. To this end we make the following deﬁnition: Fix a non-negative integer k. Suppose that A is a ﬁnite local Zp algebra, and that VA is a ﬁnite free A-module equipped with a continuous action of GK . An argument as in [Ki 1, 1.2.7(4)] shows that MA = DOE (VA ) is a free OE,A = OE ⊗Zp A-module. Now let B be any Aalgebra. We extend the action of ϕ and Γ to MB := MA ⊗A B by B-linearity. A Wach lattice with weights in [0, k] is a ﬁnite projective SB -submodule MB ⊂ MB such that (1) MB is stable under the action of ϕ and Γ. (2) MB spans MB as a OE ⊗Zp B-module. (3) The cokernel of the induced map ϕ∗ (MB ) → MB is killed by q k , p −1 where q(u) = (1+u) . u (4) The action of Γ on MB /uMB is trivial. The motivation for this deﬁnition is the following result of Berger [Be, III, 4.2, 4.5] (which depends on the hypothesis that K = K0 ), where we take A = B = Zp . Proposition (3.3). If V is a ﬁnite dimensional Qp -vector space with a continuous action of GK , and T ⊂ V is a GK -stable lattice, then V is crystalline with Hodge-Tate weights in [−k, 0] if and only if the (ϕ, Γ)module DOE (T ) contains a Wach lattice with weights in [0, k]. Moreover in this case, DOE (T ) contains a unique Wach lattice M, and if Dcris (V ) denotes the weakly admissible module associated to V, ∼ then there exists a natural isomorphism Dcris (V ) −→ M/uM ⊗Zp Qp . (3.4) Now let F and VF be as in §2, A in ARW (F) , and VA a deformation of VF to A. We set MA = DOE (VA∗ ), where VA∗ denotes the A-dual of VA . For an A-algebra B, we set MB = MA ⊗A B as above, and we denote by WVA ,k (B) the set of Wach lattices MB ⊂ MB with weights in [0, k]. Proposition (3.5). The functor WVA ,k is represented by a projective A-scheme WVA ,k . If R is any complete local Noetherian ring with residue ﬁeld F, and VR is a deformation of VF to R, then there exists a projective R scheme WVR ,k such that (1) If A is in ARW (F) and R → A is a map inducing the identity on residue ﬁelds, then there is a canonical isomorphism ∼

WVR ,k ⊗R A −→ WVA ,k where VA = VR ⊗R A.

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(2) The map ΘR : WVR ,k → Spec R becomes a closed embedding after inverting p. If A is a ﬁnite W (F)[1/p]-algebra, then a map R → A factors through the scheme theoretic image of ΘR if and only if VA = VR ⊗R A is a crystalline representation with HodgeTate weights in [0, k]. Proof. The proof of the ﬁrst claim is completely analogous to that of [Ki 1, 2.1.7]. The existence of WVR ,k satisfying (1) follows using formal GAGA as in [Ki 1, 2.1.10]. The properties in (2) can be deduced from Berger’s theorem (3.3): The fact that ΘR is a closed embedding follows from the uniqueness of the Wach lattice, while the description of the scheme theoretic image follows (after a little work) from the criterion for the existence of a Wach lattice. (cf. [Ki 3, §1]). (3.6) As in §2 we ﬁx a basis of VF . We denote by RVF the universal framed deformation ring of VF as in (1.2), and if End F[GK ] VF = F, we denote by RVF the universal deformation ring of VF . Suppose now that A is a ﬁnite local Qp -algebra, and that VA is a ﬁnite free A-module of rank d equipped with a continuous action of GK . We suppose that VA is crystalline with non-negative Hodge-Tate weights. Then Dcris (VA∗ ) is a free K0 ⊗Qp A-module [Ki 1, 1.3.2] on which ϕ[K0 :Qp ] acts linearly. We set PVA (T ) = det K0 ⊗Qp A (T − ϕ[K0 :Qp ] |Dcris (VA∗ )), and for i = 0, 1, . . . , d − 1, we denote by cVA ,i the coeﬃcient of T i in PVA (T ). Corollary (3.7). Suppose that R = RVF or RVF , and let Rk denote the scheme theoretic image of the map ΘR of (3.5). Then (1) Rk [1/p] is formally smooth over W (F). (2) For i = 0, 1, . . . d − 1 there is a unique ci ∈ Rk [1/p] such that for any ﬁnite W (F)[1/p]-algebra A, and any W (F)-algebra map h : Rk → A, the representation VA obtained from the universal representation over R by specialization by h satisﬁes cVA ,i = h(ci ). Moreover the ci are contained in the normalization of Im(Rk → Rk [1/p]). Proof. The ﬁrst part follows using an argument similar to that in [Ki 1, 2.3.9]. For the second part we consider (the algebraization of) the universal family of Wach modules MWR,k over WR,k (cf. [Ki 1, 2.4.17]), and we set DWR,k = MWR,k /uMWR,k . Then DWR,k is a vector bundle on

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WR,k , on which ϕ[K0 :Qp ] acts linearly, and we may form the characteristic polynomial of the endomorphism: P (T ) = det(T − ϕ[K0 :Qp ] : DWR,k ) ∈ Γ(WR,k , OWR,k ). ∼

Since ΘR induces an isomorphism WR,k ⊗Zp Qp −→ Spec Rk [1/p], we may regard the coeﬃcients ci of P (T ) as elements of R[1/p]. Since the map ΘR is proper the ci are contained in the normalization of Im(Rk → Rk [1/p]). That the ci have the property claimed in (2) follows from the fact that in the situation of (3.3), if V is crystalline, and T ⊂ V is a lattice corresponding to a Wach module M, then Dcris (V ) is canonically isomorphic to (M/uM)[1/p] [Be, Thm. III.4.4]. (3.8) We now specialize to the case K0 = Qp and d = dimF VF = 2, and we suppose that k ≥ 2. Let E/Qp be a ﬁnite extension, and let ap ∈ E satisfy val(ap ) > 0, where val denotes the p-adic valuation normalized so that val(p) = 1. We denote by Vk,ap the two dimensional E-representation of GQp which is crystalline with Hodge-Tate weights 0, k − 1 (that is, as a Qp -representation Vk,ap has [E : Qp ] Hodge-Tate weights equal to k − 1 and [E : Qp ] equal to 0), with determinant χk−1 , ∗ and such that the trace of Frobenius on Dcris (Vk,a ) is equal to ap . Here, p ∗ as usual, Vk,ap denotes the E-dual of Vk,ap . Let O denote the ring of integers in E, and choose a GQp -stable O-lattice in Vk,ap . Reducing this lattice modulo the radical of O, and taking its semi-simpliﬁcation, we obtain a two dimensional, semi-simple representation V¯k,ap of GQp , over the residue ﬁeld FE of E. To describe V¯k,ap , for i ≥ 1 denote by Qpi the unramiﬁed extension of Qp of degree i, and by ωi : GQpi → F× the character which is the pi fundamental character of level i on inertia, and sends the arithmetic ¯ E an algebraic Frobenius element corresponding to p ∈ Q× to 1. For F pi × × ¯ write μλ : GQ → F ¯ for the unramiﬁed closure of FE , and λ ∈ F p E E character which sends an arithmetic Frobenius to λ. For any x in O, we denote by x the image of x in FE . The following result was conjectured by Breuil [Br 1, Conj. 1.5], and established by Berger-Breuil [BB 1] using the unitary p-adic representations of GL2 (Qp ) associated to two dimensional crystalline representations of GQp [BB 2], as well as some explicit calculations with Wach modules, following those of [BLZ]. Theorem (3.9). (1) If 2 k p + 1, then V¯k,ap ∼ Ind ω2k−1 , where Ind denotes induction from GQp2 to GQp .

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(2) Suppose that k = p + 2. 2 • If val(ap ) < 1 then V¯k,ap ∼ Ind ω2 . μλ ω 0 a • If val(ap ) ≥ 1 then V¯k,ap ∼ with λ2 − pp λ + 0 μλ−1 ω 1 = 0. (3) Suppose p + 3 k 2p − 1 • If val(ap ) < 1 then V¯k,ap ∼ Indω2k−p . μλ ω k−2 0 ¯ • If val(ap ) = 1 then Vk,ap ∼ with λ = 0 μλ−1 ω (k−1)ap . p

• If val(ap ) > 1 then V¯k,ap ∼ Ind (ω2k−1 ). (3.10) Suppose now that VF is as above, and ﬁx k ≥ 2 as before. We will assume moreover, that End F[GK ] VF = F, since this is the only case in which we will be able to prove a result strong enough to have consequences for modularity. Lemma (3.11). There exist a unique quotient RV0,k−1 of RVF with the F following properties (1) R0,k−1 is p-torsion free, and R0,k−1 [1/p] is formally smooth over VF VF W (F)[1/p]. (2) If E/W (F)[1/p] is a ﬁnite extension, x : RVF → E a map of W (F)[1/p]-algebras, and Vx the representation of GK obtained by specializing the universal RVF -representation by x, then x factors through RV0,k−1 if and only if Vx is crystalline with Hodge-Tate F weights 0 and k − 1. Proof. Set R = RVF . Then the ring Rk−1 constructed in (3.7) satisﬁes (1) by (3.7), and its deﬁnition as the scheme theoretic image of ΘR implies that it satisﬁes a modiﬁed version of (2), where one requires only that the Hodge-Tate weights of Vx are in the interval [0, k − 1]. To see this one uses the description of the Hodge-Tate weights of a crystalline representation in terms of the associated Wach module [Be, III.2]. It is now easy to see, using either Wach lattices or the main result of [Se], that one can construct a quotient RV0,k−1 with the required properF ties by taking the closure in Spec Rk−1 of a suitable union of connected components of Spec Rk−1 [1/p]. (3.12) Next, we ﬁx E/Qp a ﬁnite extension as above, and a character ψ : GQp → O× whose restriction to inertia is χk−2 . We denote by RV0,k−1,ψ the quotient of RV0,k−1 ⊗W (F) O corresponding to deformaF F tions with determinant ψχ. As in (2.10), one sees using (3.11)(1) that RV0,k−1,ψ [1/p] is formally smooth over E. F

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Now suppose, moreover that 2 k 2p − 1, and that VF has semisimpliﬁcation equivalent to one of the representations listed in (3.9). In other words, we assume that VFss ∼ V¯k,ap with val(ap ) > 0. Proposition (3.13). Suppose that End F[GK ] VF = F. If RV0,k−1,ψ is not F the zero ring, then (1) The E-algebra RV0,k−1,ψ [1/p] is formally smooth of dimension 1. F (2) If 2 k 2p − 1, then RV0,k−1,ψ is a domain. F Proof. The ﬁrst claim follows as in [Ki 1, 2.3.9, 2.3.11]. For the second claim, we may suppose, after twisting that ψ = χk−2 . Moreover, in the case when VF is ordinary, the list in (3.9) shows that every closed point of RV0,k−1 [1/p] corresponds to an ordinary represenF tation. Since End F[GK ] VF = F, it is well known that the ordinary deformation ring with ﬁxed determinant is formally smooth of dimension 2. This can be seen by a calculation with Galois cohomology, although an argument analogous to that given below in the non-ordinary case would also work. Hence we may suppose that VF is non-ordinary, so that VF has semi-simpliﬁcation equivalent to one of the representations given in (3.9). Now suppose that V¯k,ap ∼ VFss . If E denotes the ﬁeld of deﬁnition of Vk,ap , then we claim that there is a unique map x : RV0,k−1.ψ [1/p] → E F such that the induced E -representation of GQp is isomorphic to Vk,ap . This is clear if VF is irreducible. Suppose that VF is reducible. The uniqueness of such a point follows from the condition End F[GK ] VF = F. For its existence, suppose that VF is a (necessarily non-trivial) extension of ψ1 by ψ2 for some characters ¯ is ψ1 , ψ2 . Let L ⊂ Vk,ap be a GQp -stable OE -lattice whose reduction L an extension of ψ1 by ψ2 . For diﬀerent choices of L the resulting extensions all lie on an F-line in Ext1GQp (ψ1 , ψ2 ). In particular, any two such extensions which are non-trivial are isomorphic as GQp -representations. ¯ is a non-trivial Since Vk,ap is irreducible, we can choose L such that L extension. In most cases considered in (3.9) the space of such extensions ¯ ∼ VF . The only exception is when k = p+3, is one dimensional, so that L val(ap ) = 1, and λ = ±1. In this case it follows from [BB 1, Thm. 2.2.5] ¯ is always a peu ramiﬁ´e extension. Hence, R0,k−1,ψ is non-zero that L VF ¯ and that Vk,a is only if VF is peu ramiﬁ´e, when we again have VF ∼ L, p a deformation of VF . Now let ap ∈ RV0,k−1,ψ [1/p] be the image of the function c1 ∈ F ˜ ˜ Rk−1 [1/p]. Let R denote the normalization of RV0,k−1,ψ . Then ap ∈ R, F 0,k−1,ψ because ap is a bounded function on the closed points of R [1/p] (this follows for example using Noether normalization). Alternatively,

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one could use the same argument as in the proof of (3.7). Moreover none of the representations in (3.9) are the reduction of an ordinary representation, so ap is in fact strictly bounded by 1, and we obtain a map X →ap ˜ O[[X]] → R. Suppose that VF ∼ Ind ω2k−p . By (3.9), we see that also

RV0,k−1,ψ [1/p] F

is a function which is strictly bounded by 1. Hence and we obtain a map (3.13.1)

p ap

∈ ˜ ∈ R,

p ap

˜ O[[X, Y ]]/(XY − p) → R.

By (3.9) and what we have seen above, the map obtained from (3.13.1) by inverting p induces a bijection on closed points, and an isomorphism on residue ﬁelds between corresponding closed points. In the appendix Gabber proves that any such map between two normal rings which are ﬁnite over a power series ring over O is an isomorphism, and (2) follows. When we are in one of the other cases given by (3.9) one ﬁnds, by a ˜ is isomorphic to a power series ring, so that similar argument, that R (2) again follows. For example, in the second case in (3), we have an ˜ given by sending X to (k−1)ap − [λ]. isomorphism O[[X]] → R p Theorem (3.14). Let F be a totally real ﬁeld in which p is totally split, and let ρ : GF,S → GL2 (O) be a continuous representation. Suppose that for some 2 k 2p − 1 (1) ρ is crystalline with Hodge-Tate weights 0 and k − 1 at each v|p. (2) There exists a Hilbert modular form over F of parallel weight k and prime to p level, such that ρ¯ ∼ ρ¯f . (3) ρ¯|F (ζp ) is absolutely irreducible, and ρ¯|GFv has only scalar endomorphisms for each v|p. Then ρ ∼ ρg for some Hilbert modular eigenform g over F. ¯ ψ = R0,k−1,ψ ⊗R R , where VF denotes the Proof. For v|p we set R v VF VF VF underlying F-vector space of ρ¯|GFv , and ψ = det ρ|GFv · χ−1 . Then (3.13) ¯ ψ satisﬁes (2) and (3) of (1.4). That it satisﬁes (1) follows implies that R v from an argument analogous to that made in the Barsotti-Tate case in (2.10). The theorem can now be deduced by combining (1.4) with some level lowering and raising arguments, exactly as for the proof of (2.11). We should remark that the level lowering arguments in [SW] are carried out only for weight two, although the argument in higher weight should be similar, as the authors remark. Thus, the cautious reader may want to admit the theorem as fully proved only in the case when F = Q, for which level lowering is available in greater generality.

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References [BB 1] L. Berger, C. Breuil, Sur la r´ eduction des repr´ esentations cristallines de dimension 2 en poids moyens, preprint (2005). [BB 2] L. Berger, C. Breuil, Repr´ esentations cristallines irr´ eductibles de GL2 (Qp ), preprint (2004). [Be] L. Berger, Limites de repr´ esentations cristallines, Compositio Math. 140 (2004), 1473-1498. [BLZ] L. Berger, H. Li, H.J. Zhu, Construction of some families of 2-dimensional crystalline representations, Math. Ann. 329 (2004), 365-377. [Br 1] C. Breuil, Sur quelques repr´ esemtations modulaires et p-adiques de GL2 (Qp ) II, J.Inst. Jussieu 2 (2003), 23-58. [Br 2] C. Breuil, Groupes p-divisibles, groupes ﬁnis et modules ﬁltr´ es, Ann. of Math. 152 (2000), 489-549. [Br 3] C. Breuil, Integral p-adic Hodge Theory, Algebraic Geometry 2000, Azumino, Adv. Studies in Pure Math. 36, 2002, pp. 51-80. [Br 4] C. Breuil, Schemas en groupes et corps des normes (unpublished) (1998), 13 pages. [Ca] H. Carayol, Sur les repr´ esentations p-adiques associ´ ees aux formes modulaires de Hilbert, Ann. Scient. de l’E.N.S 19 (1986), 409-468. [CDT] B. Conrad, F. Diamond, R. Taylor, Modularity of certain potentially BarsottiTate Galois representations, J. Amer. Math. Soc. 12(2) (1999), 521-567. [Di] F. Diamond, The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379-391. [DP] P. Deligne, G. Pappas, Singularit´ es des espaces de modules de Hilbert, en les caract´ eristiques divisant le discriminant, Compositio Math. 90 (1994), 59–79. [DT] F.Diamond, R. Taylor, Non-optimal levels of mod l modular representations, Invent. Math. 115 (1994), 253-269. [Fo] J-M. Fontaine, Repr´ esentations p-adiques des corps locaux, Grothendieck Festschrift II, Prog. Math. 87, Birkhauser, pp. 249-309, 1991. [Ja] F. Jarvis, Correspondences on Shimura curves and Mazur’s principle at p., Paciﬁc J. Math 213 (2004), 267-280. [Ki 1] M. Kisin, Moduli of ﬁnite ﬂat group schemes and modularity, preprint (2004), 74 pages. [Ki 2] M. Kisin, Crystalline representations and F -crystals, Algebraic geometry and number theory. In honour of Vladimir Drinfeld’s 50th birthday, Prog. Math. 253, Birkh¨ auser, pp. 459-496, 2006. [Ki 3] M. Kisin, Potentially semi-stable deformation rings, J.A.M.S, to appear. [Ma] B. Mazur, Deforming Galois representations, Galois groups over Q (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 16, Springer, New York-Berlin, pp. 395437, 1989. [Ne] J. Nekov´ aˇr, On p-adic height pairings, S´eminaire de Th´ eorie de Nombres, Paris 1990-91, Progr. Math. 108, Birkh¨ auser Boston, pp. 127-202, 1993. [Ram] R. Ramakrishna, On a variation of Mazur’s deformation functor, Compositio Math. 87 (1993), 269-286. [Ra] M. Raynaud, Sch´ emas en groupes de type (p, p, . . . , p), Bull. Soc. Math. France 102 (1074), 241-280. [Se] S. Sen, The analytic variation of p-adic Hodge Structure, Ann. of Math. 127 (2) (1988), 647-661. [SW] C. Skinner, A. Wiles, Base change and a problem of Serre, Duke Math. J 107 (2001), 15-25.

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[Ta 1] R. Taylor, On the meromorphic continuation of degree 2 L-functions, Documenta Math., Extra Volume: John H. Coates’ Sixtieth Birthday, 2006, 729779. [Ta 2] R. Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265-280. [Ta 3] R. Taylor, On Galois representations associated to Hilbert modular forms. II, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, Internat. Press, Cambridge, MA, 1995., pp. 185-191. [TW] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141(3) (1995), 553-572. [W] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141(3) (1995), 443-551.

Appendix by Ofer Gabber Consider a complete Noetherian local ring V with residue ﬁeld k and let C be the category of V -algebras which are ﬁnite over some power series ring V [[x1 , . . . , xn ]]. Every R ∈ Ob(C) is a ﬁnite product of complete Noetherian local rings and the maximal ideals of R are the prime ideals whose residue ﬁeld is a ﬁnite extension of k; in what follows mR denotes the Jacobson radical of R and Max(R) denotes the set of maximal ideals of R. It follows from ([Ma] Lemma 1 on p. 247) that for a Noetherian local domain R of dimension > 0 the set P1 R := {P ∈ Spec (R) | dim(R/P ) = 1} is dense in Spec (R). In particular for every R ∈ Ob(C), every open subscheme U of Spec (R) − Max(R) is a Jacobson scheme whose set of closed points is U ∩ P1 R. Lemma 0. Let f : R → R be a map in C, P ∈ Spec (R ) and P = f −1 (P ). If dim(R /P ) 1 then dim(R/P ) dim(R /P ). Proof. We may assume P = 0, P = 0. If dim(R ) = 0 then R is a ﬁnite ﬁeld extension of k and R is also. Suppose dim(R ) = 1 and R is not a ﬁeld. By the case dim(R /P ) = 0 of the lemma, f is a local homomorphism of local rings. If 0 = x ∈ mR then R /xR is of dimension 0. Hence R /mR R is ﬁnite dimensional over R/mR . If g1 , . . . , gn ∈ R lift an R/mR -basis of R /mR R it is easy to see that R = i Rgi . Thus dim(R) = dim(R ) by EGA 0IV (16.1.5). Theorem. Let f : R → R be a morphism in C with R reduced and R normal (i.e. a ﬁnite product of normal domains) and U an open subscheme of Spec (R) − Max(R) such that U is dense in Spec (R), U = Spec (f )−1 (U ) is dense in Spec (R ), and (∗) Spec (f ) induces a bijection {closed points of U } −→ {closed points of U } and isomorphisms on the residue ﬁelds. Then f is an isomorphism.

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Remark 1: Condition (∗) is equivalent to the assertion that for every closed point y of U , (U ×U y)red −→ y is an isomorphism. Remark 2: Let J ⊂ R be an ideal with JOU = OU . Consider the blow-ups X = BlJ (Spec R) and X = BlJR (Spec R ) and write X0 = m∈MaxR X ⊗R (R/m), and deﬁne X0 similarly. Then X is a closed subscheme of X ⊗R R , X0 is a closed subscheme of X0 ⊗R/mR (R /mR ), and X0 → X0 is ﬁnite. Note that for every P ∈ U ∩ P1 R the proper transform of Spec (R/P ) in X is Spec of some ring between R/P and its normalization. The morphism X,ξ ) −→ X ϕ: Spec (O ξ closed point of X0

becomes an isomorphism over any of those proper transforms. Now let U denote also the preimage of U in X. By Lemma 0, ϕ−1 (U ) → U sends closed points to closed points. Thus ϕ−1 (U ) → U induces a bijection between the sets of closed points and isomorphisms on their residue ﬁelds. Similarly for X , U . Let ξ ∈ X0 be a closed point and η1 , . . . , ηn its liftings to X0 , S =

n X ,η . Then S → S equipped with the inverse image OX,ξ , S = i=1 O i of U satisﬁes the conditions in the theorem except normality. Note that by properties of excellent rings S is reduced and its non-normal locus is contained in V (J). If S ν is the normalization of S then S ν → (S ⊗S S ν )/(J-power-torsion) satisﬁes the conditions of the theorem. Here (Jpower-torsion) denotes the ideal of elements annihilated by a power of J, and in our case it is equal to the (I-power-torsion) ideal for every ideal I ⊂ R such that V (J) ⊂ V (I) ⊂ Spec (R) − U , because the schematic denseness of U and the ﬂatness of Spec (S ) → X imply that the Ipower-torsion ideal of S is 0. ¯ be an The theorem will be proved by a ﬂattening technique. Let k algebraic closure of k. Lemma 1. If kα is a direct system of ﬁelds then − lim →α (kα [[x1 , . . . , xn ]]) is Noetherian for all n. Proof. Since the class of faithfully ﬂat homomorphisms is closed under ﬁltered direct limits, the map lim −→α (kα [[x1 , . . . , xn ]]) → (lim −→kα )[[x1 , . . . , xn ]] is faithfully ﬂat.

Modularity for geometric Galois representations Alternatively use EGA 0III (10.3.1.3).

467

In particular, the rings Pn = k[[x1 , . . . , xn ]] ⊗k k¯ are Noetherian. ¯ Let D be the class of k-algebras which are ﬁnite over some Pn . We say that the theorem holds for B ∈ D if the theorem holds whenever ¯ R is local and, for some V -embedding of the residue ﬁeld of R into k, ¯ ¯ the geometric ﬁber ring R ⊗R k is k-isomorphic to B. Note that for a morphism f : R → R in C with R local, the geometric ﬁber ring is in ¯ and it D, is independent up to k-isomorphism of the choice of R → k, is of dimension 0 if and only if f is ﬁnite. Of course, if B ∈ D is not ¯ “the theorem holds for B” means that B does not arise isomorphic to k, as a geometric ﬁber ring under the assumptions of the theorem. To prove the theorem, if suﬃces to show that if B ∈ D and the theorem holds for B/I for all non-zero ideals I, then the theorem holds for B. Lemma 2. The theorem holds when R → R is ﬁnite. Proof. We may assume that R is local. Let P1 , . . . , Pn be the minimal primes of R . Since U is Jacobson, the assumption gives that there exists i with R → R /Pi injective. Then as U ∩ Spec (R /Pi ) → U is surjective on closed points we ﬁnd that all closed points of U are in Spec (R /Pi ), so Pi = 0, n = 1. We have to prove that Frac(R) = Frac(R ). Let L/Frac(R) be the maximal separable subextension of Frac(R )/Frac(R) and R = R ∩ L. Then R → R satisﬁes the assumptions of the theorem and it is ﬁnite ´etale of degree d = [L : Frac(R)] over a dense open V ⊂ U. Taking a closed point of V gives d = 1, so it remains to consider the purely inseparable case, which holds by the next lemma. Lemma 3. Let R ⊂ R be a ﬁnite radicial extension of complete Noetherian local domains of characteristic p > 0 and dimension >0, with Frac(R ) purely inseparable of degree > 1 over Frac(R). Then for every open dense √ U ⊂ Spec (R), there is P ∈ U ∩ P1 R such that Frac(R/P ) → Frac(R / P R ) is of degree > 1. 1

Proof. We may assume R = R[f p ], with f in the maximal ideal of R, f ∈ / Frac(R)p , and that R has a system of parameters f, g1 , . . . , gn . Let F = Frac(R). Recall that if M/L/K is a tower of ﬁelds and M/L is ﬁnite then dimM Ω1M/K ≥ dimL Ω1L/K . Since R is a ﬁnite extension of K[[x1 , . . . , xn+1 ]] for some ﬁeld K and Frac(K[[x1 , . . . , xn+1 ]]) has n + 1 linearly independent derivations ∂/∂xi , we ﬁnd that dimF Ω1F ≥ n + 1. We can ﬁnd h1 , . . . , hn in the maximal ideal of R such that df, dh1 , . . . , dhn are linearly independent over F in Ω1F . Replacing gi by gi = gip (1 + hi ), we see that we can assume df, dg1 , . . . , dgn are linearly independent over F .

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1

1

Let R = R[f p , g1p , . . . , gnp ] := R[T0 , . . . , Tn ]/(T0p − f, T1p − g1 , . . . , Tnp − gn ) (which is a domain since f, g1 , . . . , gn are p-independent over Fp in F , see proof of [Ma] 26.5). By ([Ma], Cor. of 30.10) we may shrink U so that R and R are regular over U. Choose a coeﬃcient ﬁeld K ⊂ R, so R is a ﬁnite extension of a power series ring R0 = K[[f, g1 , . . . , gn ]]. There is a nonzero power series H ∈ K[[f, g1 , . . . , gn ]] with Spec(R[1/H]) ⊂ U. One can ﬁnd m1 , . . . , mn > 0 such that H(f, f pm1 , . . . , f pmn ) = 0. [Take the mi ’s such that (∀ i) pmi > a0 + pm1 a1 + · · · + pmi−1 ai−1 , where f a0 g1a1 . . . gnan is the smallest (for the reverse-lexicographic order) monomial which has a non-zero coeﬃcient in H.] Take P ∈ U which extends (g1 − f pm1 , . . . , gn − f pmn ) ⊂ R0 . The regularity of RP implies by EGA 0IV (22.5.4) (which is not correct as stated, but proved when p is in the square of the maximal ideal of A), or by other versions of the Jacobian criterion, that the diﬀerentials df, dg1 , . . . , dgn are linearly independent in Ω := Ω1RP ⊗RP κ(P ). Use the second fundamental exact sequence 0 → P RP /P 2 RP → Ω → Ω1κ(P ) → 0, and note that as RP is regular of dimension n, P RP /P 2 RP has κ(P )dimension n. Since the gi are pth powers in κ(P ) we ﬁnd that f is not. It remains to prove the theorem for B ∈ D of dimension > 0 (i.e. to get a contradiction), assuming it holds for proper quotients of B. Suppose we are in the situation of the theorem with R local and B is ¯ the geometric ﬁber ring of R → R for some k-embedding of R/mR in k. Let F be the set of ideals I ⊂ R such that R /IR is ﬂat over R/I. It follows from the lemma in [Fe] or [GR] 3.4.18 (iii) that F is closed under ﬁnite intersections, hence the set of I ∈ F with mnR ⊂ I has a smallest n element In . As In = In+1 + mnR , lim ←−(In /mR ) is an ideal J of R which induces In /mnR on R/mnR , and J ∈ F by [Ma] 22.3. It is easily seen to be the smallest element of F . This is an analogue of [RG, I. Thm. 4.1.2] and [HLT, Thm. 1 ]. Clearly J ⊂ I1 = mR . For every prime ideal J ⊂ P ⊂ R with dim(R/P ) = 1 we have dim(R /P R ) = 1 + dim(B), so R /P R ⊗R/P Frac(R/P ) has inﬁnitely many maximal ideals (e.g. by [Ma, Lem. 1, p. 247]). Hence P ∈ / U, so V (J) ⊂ Spec (R) − U . In particular, J = 0. Let K be the kernel of (R /mR R ) ⊗R/mR (J/mR J) JR /mR JR . Note that if J is an ideal of R with J 2 ⊂ J ⊂ J, we have by the local ﬂatness criterion ∼

((R /JR ) ⊗R/J (J/J ) −→ JR /J R ) ⇔ J ∈ F .

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469

We apply this to proper subideals of J containing mR J. The minimality of J implies that (∗∗) for every proper subspace W J/mR J, K (R /mR R ) ⊗R/mR W. In particular, K = 0. Note that K is a ﬁnitely generated R -module. Suppose the non-zero elements Φ1 , . . . , Φn generate K, and write each Φi in a minimal manner as Φi =

ni

[fij ] ⊗ [gij ] ,

fij ∈ R , gij ∈ J .

j=1

Thus for each i, the [fij ] ∈ R /mR R are linearly independent over R/mR and the [gij ] ∈ J/mR J are linearly independent over R/mR . By the deﬁnition of K, (∀ i) Σj fij gij ∈ mR JR , and by (∗∗) the elements gi,j generate J. Blow up and use notation as in Remark 2. For every ξ ∈ X0 , there exists (i, j) such that gi,j OX,ξ = JOX,ξ . Then a computation shows that the functions fi, (1 ni ) restricted to the ﬁber X ×X ξ are linearly dependent over the residue ﬁeld of ξ. So if ξ is a closed point of X, the geometric ﬁber ring of S → S is a proper quotient of B (for a suitable ¯ k-embedding κ(ξ) → k). ν Let X be the normalization of X, U the preimage of U in X ×X X ν ∼ (U −→ U ), and X the schematic closure of U in X ×X X ν . The assumption that the theorem holds for proper quotients of B gives that X → X ν is a bijection on closed points, and an isomorphism on the completions at these points. Set T = V (mR OX ν ) and T = V (mR OX ). All the closed points of ν X (resp. X ) are in T (resp. T ), T is a closed subscheme of T ⊗R/mR (R /mR ), and T, T are of ﬁnite type over k. The map T → T is ﬁnite, a bijection on closed points, and a closed immersion on the completions ∼ at such points. It follows that T → T is a closed immersion and Tred −→ Tred . Let I (resp. I ) be the ideal deﬁning Tred in X ν (resp. Tred in X ). It follows that IOX coincides with I in the completions of the local rings of X at closed points, hence in the local rings. So IOX = I . The set theoretic zero locus of mR on X is therefore the set theoretic zero locus of mR on X . But X → Spec (R ) is proper and an isomorphism over U , so it is surjective, and we get a √ contradiction, because mR R = mR as dim(B) > 0. This appendix beneﬁtted from many remarks of the referee.

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D. Ferrand, Descente de la platitude par un homomorphisme ﬁni, C.R. Acad. Sci. Paris 269 (1969), 946-949. [GR] O. Gabber, L. Ramero, Almost ring theory, LNM 1800, Springer, 2003. [HLT] H. Hironaka, M. Lejeune-Jalabert, B. Teissier, Platiﬁcateur local en g´ eom´ etrie analytique, Ast´erisque 7 & 8 (1973), 441-463. [Ma] H. Matsumura, Commutative ring theory, Cambridge Univ. Press, 1986. [RG] M. Raynaud, L. Gruson, Crit` eres de platitude et de projectivit´ e, Invent. Math 13 (1971), 1-89. [Fe]

The Euler system method for CM points on Shimura curves Jan Nekov´ aˇr Universit´e Pierre et Marie Curie (Paris 6) Institut de Math´ematiques de Jussieu Th´eorie des Nombres, Case 247 4, place Jussieu F-75252 Paris cedex 05 FRANCE [email protected] Let E be an elliptic curve over Q of conductor N ; let π : X0 (N ) −→ E be √ a modular parametrization sending the cusp ∞ to the origin. If K = Q( D) is an imaginary quadratic ﬁeld of discriminant D < 0 satisfying the “Heegner condition” of Birch [Bi] (“each prime dividing N splits in K/Q”), a choice of an ideal N ⊂ OK satisfying N N = N OK and (N , N ) = (1) determines a Heegner point x1 = [C/OK −→ C/N −1 ] ∈ X0 (N )(K[1]), deﬁned over the Hilbert class ﬁeld K[1] of K. Set y1 = π(x1 ) ∈ E(K[1]) and y = TrK[1]/K (y) ∈ E(K). A fundamental result of Kolyvagin [Ko, Thm. A] (explained in a simpliﬁed setting in [Gr 1]) states that (K) if y ∈ E(K)tors , then the abelian groups E(K)/Zy, X(E/K) are ﬁnite. More precisely, Kolyvagin showed that |X(E/K)| divides a certain multiple of [E(K) : Zy]2 . Kolyvagin’s result (K) has been generalized in several directions: Kolyvagin and Logaˇcev [Ko-Lo 1] considered higherdimensional simple quotients of J0 (N ) instead of elliptic curves, as well as [Ko-Lo 2] quotients of Jacobians of some Shimura curves over totally real number ﬁelds. Bertolini and Darmon [Be-Da] proved, under some additional assumptions, a “mod p” version of (K) over ring class ﬁelds K[c] ((c, N ) = 1). Tian [Ti] and Tian and Zhang [Ti-Zh] considered a common generalization of [Ko-Lo 2] and [Be-Da]. The more reﬁned “quantitative” results of Kolyvagin were generalized by Bertolini [Be] and Howard [Ho 1,2]. The main result of the present article is the following generalization of (K), which is one of the ingredients in the proof of the parity conjecture for Selmer groups of Hilbert modular forms ([Ne, Ch. 12]). 471

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Theorem. Let F be a totally real number ﬁeld, X a Shimura curve over F , and A an F -simple quotient of the Jacobian of X, which corresponds to (the Galois conjugacy class of) a Hilbert modular form f with trivial character (A is an abelian variety of GL2 -type and EndF (A) is an order in a totally real number ﬁeld of degree equal to dim(A)). Let x be a CM point on X by a totally imaginary quadratic extension K of F and α a character of the Galois group Gal(K(x)/K). Assume that A does not acquire CM over any totally imaginary quadratic extension K of F contained in K(x)Ker(α) (i.e., f is not a θ-series associated to a Hecke character of K ). If the α-component of the image y of x in A is not torsion, then the α-components of A(K(x))/EndF (A) · y and X(A/K(x)) are ﬁnite. See Theorem 3.2 for a precise formulation. The proof follows the main lines of the argument of [Be-Da]; the novelty lies in the fact that under our minimalist assumptions there is no “geometric” formula for the action of the complex conjugation on x. Unlike all previous works in this direction, our version of the Euler system argument (in particular, 7.6.4 below) does not require such a formula. Theorem 3.2 is expected to be related to arithmetic properties of the Rankin-Selberg L-function L(fK ⊗α−1 , s) at the central point s = 1 ([Zh 1,2], [Co-Va 1]). The assumption that f has trivial character ensures that fK ⊗ α−1 is self-dual in the sense that there is a “symmetric” functional equation relating L(fK ⊗ α−1 , s) (which turns out to be equal to L(fK ⊗ α, s) in our case) and L(fK ⊗ α−1 , 2 − s). As explained in [Gr 2] (see also [Co-Va 1]), it is possible for L(fK ⊗ α−1 , s) to be selfdual even if f has non-trivial character ω, provided that ω is equal to the restriction of α (considered as a character of A∗K /K ∗ ) to A∗F /F ∗ . It is an interesting question whether Theorem 3.2 can be generalized to such a more general self-dual setting. A substantial part of the present article was written during the author’s visit at the COE programme at Nagoya University in spring 2005. The author is grateful to K. Fujiwara for inviting him to Nagoya. He would also like to thank H. Carayol, C. Cornut and M. Dimitrov for useful discussions, and the referee for helpful comments. 1. Quaternionic Shimura curves and their Jacobians In this section we recall basic properties of Shimura curves associated to quaternion algebras over totally real ﬁelds. The standard reference is [Ca 1] (see also [Zh 1,2]), but we follow the notations and sign conventions of [Co-Va 1,2] (with the parameter = 1). In particular, the reciprocity map of class ﬁeld theory is normalized by making uniformizers correspond to geometric Frobenius elements.

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473

(1.1) Let F be a totally real number ﬁeld of degree d = [F : Q]. Fix an archimedean prime τ1 of F and a ﬁnite set SB of non-archimedean primes of F satisfying |SB | ≡ [F : Q] − 1 (mod 2). Let B be the (unique) quaternion algebra over F ramiﬁed at the set Ram(B) = {v|∞, v = τ1 } ∪ SB . For each real embedding τj : F → R (j = 1, . . . , d) put Bτj = B ⊗F,τj R. We ﬁx an isomorphism of R-algebras ∼

Bτ1 = B ⊗F,τ1 R −→ M2 (R)

(1.1.1)

(for j > 1, Bτj is isomorphic to the algebra of Hamilton quaternions, but there is no need to ﬁx a speciﬁc isomorphism). Fix an embedding F → C extending τ1 : F → R. (1.2) Let G be the algebraic group over Q satisfying G(A) = (B⊗Q A)∗ , for every commutative Q-algebra A. Denote by Z the centre of G and by nr : G(A) −→ (F ⊗Q A)∗ the reduced norm. For j = 1, . . . , d, denote by Gj the algebraic group over R given by Gj = G ⊗F,τj R; we have ∼ = A ⊗ Z. GR −→ G1 × · · · × Gd . We denote, for any abelian group A, A Quaternionic Shimura curves are associated to the Shimura datum (G, X), where G is as above and X is the G(R)-conjugacy class of the morphism h0 : S = ResC/R (Gm,C ) −→ G(R) = G1 (R) × · · · × Gd (R) x y

x + iy → , 1, . . . , 1 . −y x In concrete terms, X has a natural complex structure ([Mi 2, p. 320]) and the map Ad(g)h0 = gh0 g −1 → g(i)

(g ∈ G(R))

(where g acts on C − R as in 1.3 below) deﬁnes a holomorphic isomor∼ phism X −→ C − R. (1.3) The Shimura curves in question form a projective system {MH } ∗ . Each curve MH indexed by open compact subgroups H ⊂ G(Af ) = B is smooth over Spec(F ) and each transition map pr : MH −→ MH (for H ⊂ H ) is ﬁnite and ﬂat. The Riemann surface associated to MH an MH = (MH ⊗F,τ1 C) (C)

474

Jan Nekov´ aˇr ∼

∗ /H), where B ∗ ⊂ B ∗ acts on X −→ is identiﬁed with B ∗ \(X × B τ1 ∼ ∗ C − R via the isomorphism B −→ GL (R) induced by (1.1.1), and 2 τ1 a b the standard action ·z = az+b cz+d of GL2 (R) on C−R. We denote c d an ∗. by [z, b] = [z, b]H the point of MH represented by a pair (z, b) ∈ X × B For H ⊂ H , the transition map pr : MH −→ MH corresponds to the natural projection an an pr : MH −→ MH ,

[z, b]H → [z, b]H .

In the “classical case” F = Q, SB = ∅ ( ⇐⇒ B = M2 (Q)), the curves MH are the usual modular curves; we denote by ∗ MH = MH ∪ {cusps}H

their standard smooth compactiﬁcations; they form again a projective system with ﬁnite ﬂat transition maps. If B = M2 (Q), then the curves MH are proper over Spec(F ); we put ∗ MH = MH . ∗ acts on the right on the projective systems (1.4) The group G(Af ) = B ∗ ∗ {MH } and {M }. More precisely, the right multiplication by g ∈ B H

∼

induces an F -isomorphism [·g] : MH −→ Mg −1 Hg ; the corresponding holomorphic isomorphism ∼

an ∗ ∗ /H) −→ M an ∗ −1 Hg) [·g] : MH = B ∗ \(X × B g −1 Hg = B \(X × B /g

is given by the formula [x, b] → [x, bg]. ∗ acts on MH (and M ∗ ) via In particular, the centre F ∗ = Z(Af ) ⊂ B H ∗ ∗ ∗ ∗ ∩H); we denote by NH ⊂ N ∗ its ﬁnite quotient F /(F ∩H) = F /(O F H the corresponding quotient curves, which are again smooth over Spec(F ) [Ka-Ma, p. 508] (the curve MH (resp., NH ) was denoted by Y (resp., X) in [Zh 1]). The Riemann surface associated to NH an NH = (NH ⊗F,τ1 C) (C)

∗ /H F∗ ). In particular, NH (and N ∗ ) deis identiﬁed with B ∗ \(X × B H ∗ . pend only on the subgroup H F∗ ⊂ B In abstract terms, the projective system {NH } is associated to the Shimura datum (G/Z, X). ∗ , there exists an Note that, for each open compact subgroup H ⊂ B ∗ ∗ ∗ F∗ ⊂ H F ∗ , OF -order R ⊂ B such that R ⊂ H OF . In particular, R

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475

hence the curves NR ∗ (where R runs through all OF -orders of B) form a co-ﬁnal subsystem of {NH }. ∗ ∗ (1.5) Each curve MH (hence also NH ) is irreducible ([Ca 1, 1.3], [Co-Va 1, 3.2]), but not necessarily geometrically irreducible. Denote by F (MH ) ∗ (resp., F (NH )) the algebraic closure of F in the function ﬁeld of MH ∗ (resp., of NH ). The reciprocity law [De, 3.9], [Mi 2, II.5.1] (with the sign corrected, [Mi 3, 1.10]) implies that the ﬁelds F (NH ) ⊆ F (MH ) are abelian over F and that the reciprocity map recF : F ∗ −→ Gal(F ab /F ) induces isomorphisms (depending on a choice of embedding F (MH ) → F ab ) ∼

F+∗ \F∗ /nr(H) −→ Gal(F (MH )/F ), ∼ F+∗ \F∗ /F ∗2 nr(H) −→ Gal(F (NH )/F ), where F+∗ = Ker(F ∗ −→ π0 ((F ⊗Q R)∗ )) denotes the subgroup of totally positive elements of F ∗ . (1.6) Diﬀerentials and automorphic forms of “weight 2” ([CoVa 1, 3.6]) Denote by Ωan the sheaf of holomorphic 1-forms on any given Riemann surface. As in the classical case B = M2 (Q), the space of global holomorphic 1-forms ∗ an an lim −H → Γ((MH ) , Ω ),

∗ given by equipped with a canonical left action of B ∗ an [·g]∗ : Γ((Mg∗−1 Hg )an , Ωan ) −→ Γ((MH ) , Ωan ),

(1.6.1)

can be naturally identiﬁed with a certain space of automorphic forms on ∗ = B ∗ . G(A) = G(R) × B A ∗ we have (1) More precisely, for each compact open subgroup H ⊂ B G(A) =

G(Q) · (G(R)+ × αH),

α∈C

where G(R)+ = GL+ 2 (R) × G2 (R) × · · · × Gd (R), nr G(Q) = G(Q) ∩ G(R)+ = Ker B ∗ −→F ∗ −→ F ∗ /F+∗ +

(1)

As the case B = M2 (Q) is well-known, we discuss only the case ∗ B = M2 (Q), when MH = MH .

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∗ is a ﬁxed (ﬁnite) set of representatives of the double cosets and C ⊂ B + G(Q) \G(Af )/H. Put, for each α ∈ C, Γα = Im Γα −→ P GL+ 2 (R) .

Γα = αHα−1 ∩ G(Q)+ ⊂ G(R)+ ,

Writing H = {z ∈ C | Im(z) > 0} for the complex upper half plane, the map

∼

an ∗ an Γα \H −→ MH = (MH )

(g ∈ GL+ 2 (R))

α∈C

Γα · g(i) → [gh0 g

−1

, α]H

is a holomorphic isomorphism. Using the standard notation a b j(g, z) = det(g)−1/2 (cz + d), g= ∈ GL+ 2 (R), c d

z ∈ H,

the formula fω (γ(g + , αh)) = j(g1+ )−2 fα (g1+ (i)), (γ ∈ G(Q), g + = (g1+ , g2 , . . . , gd ) ∈ G(R)+ , h ∈ H) deﬁnes a bijection ∼

∗ an Γ((MH ) , Ωan ) −→

ω where S2H

∼

α∈C

Γ(Γα \H, Ωan ) −→

S2H

→ (fω : G(A) −→ C) , (1.6.2) denotes the space of functions f : G(A) −→ C satisfying

f γ g z∞

→

(fα (z) dz)α∈C

cos(θ) − sin(θ)

, g2 , . . . , gd h = e−2iθ f (g)

sin(θ) cos(θ) (γ ∈ G(Q), g ∈ G(A), z∞ ∈ Z(R), θ ∈ R, gj ∈ Gj (R) (j > 1), h ∈ H) and such that, for each g ∈ G(A), the function 1 x + iy → f g y

y

x

, 1, . . . , 1

0

1

is holomorphic on H (in the case B = M2 (Q) one has to impose an additional cuspidality condition on the function f ; this is automatic if B = M2 (Q)).

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Passing to the inductive limit with respect to all open compact subgroups ∗ , we obtain a B ∗ -equivariant bijection of B ∼ ∗ an an lim Γ((M ) , Ω ) −→ S := S2H , (1.6.3) 2 H −H → H

∗ acting (on the left) on S2 by right translations: (b·f )(g) = f (gb) with B (which means that S2H coincides, as the notation suggests, with the space of H-invariant elements of S2 ). Note that the centre Z(Af ) = F∗ acts on S2H via its ﬁnite quotient ∗ ∩ H), hence (1.6.2) induces a bijection F∗ /(F∗ ∩ H) = F∗ /(O F

∗ an Γ((NH ) , Ωan ) −→ S2H F , ∼

∗

(1.6.4)

where S2H F = {f ∈ S2H | (∀g ∈ G(A)) (∀z ∈ Z(A)) f (gz) = f (g)}. ∗

(1.7) Automorphic representations. The multiplicity one theorem ∗ for automorphic forms on BA ([Ge], VI.2) implies that the space S2 (as ∗ ) decomposes as a countable an admissible smooth representation of B direct sum ∼ S2 −→ πf , (1.7.1) π=σ2 ⊗πf

where π runs through all irreducible cuspidal (this is automatic if B = ∗ M2 (Q)) unitary automorphic representations of BA whose archimedean component π∞ is isomorphic to weight 2 holomorphic discrete trivial representation of σ2 = ⊗ . series representation of G1 (R) G2 (R) × · · · × Gd (R) In particular, (1.6.4) induces an isomorphism of admissible smooth rep ∗ /F ∗ resentations of B ∼ ∗ ∼ ∗ an an F lim −H → Γ((NH ) , Ω ) −→ S2 −→

πf ,

(1.7.2)

π=σ2 ⊗πf ωπ =1

where ωπ : Z(A) −→ C∗ denotes the central character of π. (1.8) The Hecke algebra. For each non-archimedean prime v of F , let dgv be the (bi-invariant) Haar measure on Bv∗ normalized so that Hv dgv = 1 for one (hence for every) maximal compact subgroup

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Hv ⊂ Bv∗ . The product dg = dgv then deﬁnes a bi-invariant Haar mea ∗ . We denote by vol(X) = sure on B dg the corresponding volume. X The Hecke algebra ∗ ) = C ∞ (B ∗ , C) = {α : B ∗ −→ C | loc. constant with cpt. support} H(B c with the convolution product (α ∗ β)(g) =

∗ B

α(gh−1 )β(h) dh

is isomorphic to the direct limit (over all open compact subgroups H ⊂ ∗ ) of the double coset algebras B ∼

∗ ∗ lim −H → C[H\B /H] −→ H(B ) HbH → vol(H)−1 1HbH ∗ /H] is as in [Sh 2, 3.1], [Miy, (2.7.3)] The product structure on C[H\B (hence is the opposite to that in [Co-Va 1, 3.4, 3.11]). ∗ gives rise to a left action of Any smooth representation (π, V ) of B ∗ ) on V , given by the formula H(B α·v =

∗ B

∗ ), v ∈ V ). (α ∈ H(B

α(h)π(h)v dh

∗ ) acts on S2 by In particular, α ∈ H(B (α · f )(g) = α(h)f (gh) dh. ∗ B ∗ and b ∈ B ∗, Note that, for any open compact subgroups H, H ⊂ B H H the action of 1HbH maps S2 to S2 . (1.9) Hecke correspondences. Let H, H be open compact subgroups ∗ and g ∈ B ∗ . The diagram of B ∗ MH∩gH g −1 pr

[·g]

/ Mg∗−1 Hg∩H pr

[HgH ] ∗ _ _ _ ∗ _ _ _ _/ MH MH

(1.9.1)

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deﬁnes a multivalued map (a “Hecke correspondence”) ∗ _ _ _ _/ M ∗ . [HgH ] : MH H

On non-cuspidal complex points, an ∗ /H) _ _ _ _ _/ M an = B ∗ \(X × B ∗ /H ) [HgH ] : MH = B ∗ \(X × B H

is given by the formula [x, b]H →

HgH F ∗ =

[x, bgi ]H ,

i

gi H F ∗ .

(1.9.2)

i

The degree of [HgH ] is the degree of pr, namely (HF ∗ : (H∩gH g −1 )F ∗ ) (which is equal to (H : H ∩ gH g −1 ), if OF∗ ⊂ H ∩ H ). If z ∈ Z(Af ) = F∗ , then we have ∗ ∗ [HzH] = [·z] : MH −→ MH ,

∗ ∗ [H zH ] = [·z] : MH −→ MH ,

[·z] ◦ [HgH ] = [HgH ] ◦ [·z]. For any “reasonable” cohomology theory H?∗ , the correspondences [HgH ] induce maps q ∗ ∗ [HgH ]∗ : H?q (MH ) −→ H (MH ), ?

∗ ∗ [HgH ]∗ : H?q (MH ) −→ H?q (MH ).

As the transpose of the diagram (1.9.1) is given by Mg∗−1 Hg∩H

[·g −1 ]

pr

∗ / MH∩gH g −1 pr

∗ _ _ [H ∗ _ g_ _H] _ _/ MH MH , −1

we have [HgH ]∗ = [H g −1 H]∗ ,

[HgH ]∗ = [H g −1 H]∗

(1.9.3)

If we replace H, H by H F∗ , H F∗ , then we obtain Hecke correspondences ∗ _ _ _ _/ ∗ NH NH with similar properties.

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(1.10) Action on holomorphic diﬀerentials. In the situation of ∗ -equivariance of (1.6.3) that (1.9.1), we deduce from (1.9.2) and the B the map ∗ an ∗ an [HgH ]∗ : Γ((MH , Ωan ) −→ Γ((MH ) , Ωan ) )

corresponds to the action of 1HgH , suitably normalized:

S2H −→ S2H f → vol(H )−1 1HgH · f . If z ∈ Z(Af ) = F ∗ and H = H , then ∗ an ∗ an [HzH ]∗ = [·z]∗ : Γ((MH ) , Ωan ) −→ Γ((MH ) , Ωan )

corresponds to the action of z on S2H , which we denote by [z]. (1.11) Adjoints. Up to a scalar multiple, the hermitian scalar product i ∗ an ω1 , ω2 = ω1 ∧ ω2 (ω1 , ω2 ∈ Γ((MH ) , Ωan )) 2 ∗ )an (MH

(1.11.1) corresponds, via the isomorphism (1.6.2), to the hermitian scalar product f1 , f2 = f1 (g)f2 (g) dg (f1 , f2 ∈ S2H ), + G(Q)\G(A)/Z∞ K∞

+ where Z∞ K∞ ⊂ G(R)+ consists of the elements z 0

cos(θ) − sin(θ) , g2 , . . . , gd (z ∈ R∗ , θ ∈ R, gj ∈ Gj (R)) 0 z sin(θ) cos(θ)

∗ ) and f1 , f2 ∈ S2 , we have For α ∈ H(B α · f1 , f2 = f1 , t α · f2 ,

t

α(g) = α(g −1 )

(1.11.2)

(i.e., t α is the adjoint of α). For example, t

1HgH = 1H g −1 H .

(1.11.3)

In particular, if H = H = Hv H v , where v ∈ SB , Hv ⊂ Bv∗ is a maximal ∗ , then compact subgroup and bv ∈ Bv∗ ⊂ B −1 Hv b−1 Hv bv Hv , v Hv = nr(bv )

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as Hv is isomorphic to GL2 (OF,v ) = GL2 (O) and GL2 (O)

t−1 1

0

0

t−1 2

GL2 (O) = (t1 t2 )

= (t1 t2 )

−1

−1

GL2 (O)

t1

0

0

t2

GL2 (O)

t2

0

0

t1

GL2 (O) =

GL2 (O),

hence t

1Hbv H = [nr(bv )−1 ] ◦ 1Hbv H = 1Hbv H ◦ [nr(bv )−1 ]

(1.11.4)

in this case. (1.12) Hecke operators and cup products. Fix an open compact ∗ . There exists a ﬁnite set S ⊃ SB of non-archimedean subgroup H ⊂ B

primes of F such that H = HS H S , where HS ⊂ v∈S Bv∗ and HS=

Hv = a maximal compact subgroup of G(Af )S =

v ∈S

v ∈S

Bv∗ .

(1.12.1) More precisely, for each non-archimedean prime v ∈ S there exists a unique maximal OF,v -order R(v) ⊂ Bv such that Hv = R(v)∗ (in other ∼ words, there exists an isomorphism of groups Bv∗ −→ GL2 (Fv ) inducing ∼ an isomorphism Hv −→ GL2 (OF,v )). For such v ∈ S, we deﬁne the Hecke correspondence T (v) as ∗ _ _ _ _/ ∗ MH T (v) = [Hbv H] : MH ,

(1.12.2)

∗ satisfying ordv (nr(bv )) = 1. for any element bv ∈ R(v) ∩ Bv∗ ⊂ Bv∗ ⊂ B For example, we can ﬁx a uniformizer v of OF,v and take the element ∼ bv which corresponds, under some isomorphism Hv −→ GL2 (OF,v ), to the matrix v 0 . 0 1 The Hecke operator T (v) is deﬁned as the induced map ∗ an ∗ an T (v) := [Hbv H]∗ : Γ((MH ) , Ωan ) −→ Γ((MH ) , Ωan ). ∼

∗ an According to 1.10, the action of T (v) on Γ((MH ) , Ωan ) −→ S2H is given by T (v) = [Hbv H]∗ corresponds to vol(H)−1 1Hbv H .

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Combining 1.10 with (1.11.4), we see that the adjoint t T (v) of T (v) with respect to the hermitian scalar product (1.11.1) corresponds to vol(H)−1 (t 1Hbv H ) = vol(H)−1 1Hb−1 = vol(H)−1 [v−1 ] ◦ 1Hbv H , v H hence t

∗ −1 −1 T (v) = [Hb−1 v H] = [v ] ◦ T (v) = T (v) ◦ [v ]

(1.12.3)

Fix a prime number and deﬁne ∗ an W := H 1 ((MH ) , Q),

W :=

1 ∗ Het (MH

∗ an WC := Γ((MH ) , Ωan ),

⊗F F , Q ),

1 ∗ WdR := HdR (MH /F ).

The comparison isomorphisms ∼

W ⊗Q C −→ WC ⊕ WC ,

∼

W ⊗Q Q −→ W

(1.12.4)

∗ -equivariant, hence Q[H\B ∗ /H]-equivariant (the second comparare B ison isomorphism is deﬁned by using the embedding F → C that was ﬁxed in 1.1). Up to a scalar multiple, the hermitian pairing (1.11.1) WC ⊗C WC −→ C is equal to the C-linear extension of the cup product ∪

∗ an ( , ) : W ⊗Q W −−→H 2 ((MH ) , Q)−−→Q(−1) Tr

(note that WC and WC are isotropic subspaces of W ⊗Q C with respect to ( , ) ⊗Q C). Similarly, ( , ) ⊗Q Q is identiﬁed with the ´etale cup product ∪

2 ∗ ( , ) : W ⊗Q W −−→Het (MH ⊗F F , Q )−−→Q (−1). Tr

Taking into account (1.12.3), this implies that the adjoint of the operator T (v) := [Hbv H]∗ acting on W (resp., on W ) with respect to the pairing ( , ) (resp., to ( , ) ) is equal to [v−1 ] ◦ T (v) = T (v) ◦ [v−1 ]. (1.13) Integral models. Integral models of the curves MH were studied, in an increasing level of generality, by Deligne-Rapoport [De-Ra], Katz-Mazur [Ka-Ma] and Carayol [Ca 1]. Fix a non-archimedean prime v ∈ SB of F ; denote by O(v) the local ∗ factorizes as H = Hv H v , ization of OF at v. Assume that H ⊂ B

The Euler system method for CM points on Shimura curves

483

where Hv (resp., H v ) is an open compact subgroup of Bv∗ (resp., of ∗ | gv = 1}). G(Af )v = {g ∈ B In the classical case B = M2 (Q), the curve MH is a coarse moduli space of elliptic curves with an H-level structure, for schemes over Spec(Q). Katz-Mazur [Ka-Ma] extended this moduli problem to schemes over Spec(O(v)), using the theory of Drinfeld bases. They obtained a regular model MH −→ Spec(O(v)) of MH , which they compactiﬁed - somewhat ∗ artiﬁcially - to a regular model M∗H of MH , proper over Spec(O(v)). Similar techniques apply in the case F = Q, B = M2 (Q), when MH is a coarse moduli space of abelian surfaces with quaternionic multiplication with a suitable level structure [Bu]; one obtains proper regular models ∗ MH = M∗H −→ Spec(O(v)) of MH = MH . If F = Q, then MH has no longer a moduli interpretation, but it can be related to a unitary Shimura curve parametrizing a certain class of abelian varieties of dimension 4[F : Q]. Carayol [Ca 1] extended this moduli problem to schemes over Spec(O(v)) and described, in the case when H v is suﬃciently small (this condition depends on Hv ), a regular ∗ model MH = M∗H of MH = MH , proper over Spec(O(v)). If Hv is a maximal open compact subgroup of Bv∗ , then M∗H is smooth over Spec(O(v)). As explained in [Co-Va 2, 3.1.3], Carayol’s results (1) can be used to ∗ construct regular models M∗H of MH , proper over Spec(O(v)), even if v H is not suﬃciently small: ﬁx a suﬃciently small open compact normal ∗ . The right action of the subgroup H v of H v and put H = Hv H v ⊂ B ∗ ﬁnite group H/H on MH extends naturally to an O(v)-linear action on M∗H ; one deﬁnes M∗H as the quotient of M∗H by this action of H/H . The model M∗H is regular, proper over Spec(O(v)), and independent on the choice of H v . Taking an additional quotient of M∗H by the action of the ﬁnite abelian group F∗ /(F∗ ∩ H), we obtain a regular model N∗H of ∗ NH , proper over Spec(O(v)). If Hv is a maximal open compact subgroup of Bv∗ , then M∗H and N∗H are again smooth over Spec(O(v)) (by [Ka-Ma, p. 508]). (1.14) The Eichler-Shimura congruence relation [Sh 1, Thm 2.2.3], [Oh, Thm. 3.4.3], [Ca 1, 10.3]. In the situation of (1.12.1), ﬁx v ∈ S and a prime number diﬀerent from the residue characteristic of v. As Hv is a maximal compact subgroup of Bv∗ , the model M∗H is proper and smooth over Spec(O(v)), hence the proper and smooth base change theorems yield a canonical isomorphism ∼

1 ◦ W −→ Het (M∗◦ H ⊗κ(v) κ(v), Q ) =: W , (1)

Note that Carayol [Ca 1,2] uses diﬀerent sign conventions. This is discussed in detail in [Co-Va 2, 3.3.1].

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∗ where κ(v) is the residue ﬁeld of v and M∗◦ H = MH ⊗O(v) κ(v) the special ∗ ﬁbre of MH . In particular, the natural action of GF = Gal(F /F ) on W is unramiﬁed at v. As the diagram (1.9.1) for g = bv from (1.12.2) naturally extends to the integral models, the graph of the Hecke correspondence T (v) (which is a ∗ ∗ divisor on MH ×F MH ) naturally extends to a divisor on M∗H ×O(v) M∗H ; ◦ ∗◦ denote by T (v) ⊂ M∗◦ H ×κ(v) MH its special ﬁbre. The generalized Eichler-Shimura congruence relation states that

T (v)◦ = Γϕ + Γ[v ·] ◦ t Γϕ ,

(1.14.1)

where Γϕ denotes the graph of the Frobenius morphism ϕ = ϕv : ∗◦ t M∗◦ H −→ MH and Γϕ its transpose. As the contravariant action of Γϕ (resp., of t Γϕ ) on W◦ coincides with the Galois action of F = Fr(v)geom (resp., with the action of t F = (N v)F −1 ), it follows that (1 − XF )(1 − X[v ](N v)F −1 ) = 1 − XT (v) + X 2 (N v)[v ] ∈ EndQ (W◦ )[X]. As [v ](N v)F −1 is the adjoint of F with respect to the pairing ( , ) , it follows from (1.12.4) and the compatibility of the various pairings discussed in 1.12 that det(1 − XF | W◦ ) = det(1 − XT (v) + X 2 (N v)[v ] | WC ).

(1.14.2)

∗ For the curve NH and its model N∗H , the congruence relation (1.14.1) simpliﬁes ([Zh 1, Prop. 1.4.10]) to

T (v)◦ = Γϕ + t Γϕ .

(1.14.3)

∗ (1.15) The L-function of MH . Combining the discussion from 1.14 (the notations and assumptions of which are still in force) with the decomposition (1.7.1), we obtain ∼ ∼ WC −→ S2H −→ πfH . π=σ2 ⊗πf

If π = σ2 ⊗ πf = ⊗v πv is an irreducible unitary cuspidal automor∗ phic representation of BA with πfH = 0, then T (v) acts on the onedimensional space πvHv of spherical vectors (recall that v ∈ S, by assumption) by a scalar λπ (v) ∈ C. The relation (1.14.2) then gives a formula for the local Euler factor ∗ Lv (h1 (MH ), s) := det(1 − (N v)−s Fr(v)geom | WIv )−1 ,

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485

namely ∗ Lv (h1 (MH ), s) =

(1 − λπ (v)(N v)−s + ωπ (v)(N v)1−2s )− dim(πf

H

)

π=σ2 ⊗πf

=

H

Lv (π, s − 1/2)dim(πf ) ,

π=σ2 ⊗πf

(1.15.1) where ωπ : A∗F /F ∗ −→ C∗ denotes the central character (of ﬁnite order) of π. Taking a product of (1.15.1) over all v ∈ S, we obtain an equality of partial L-functions

∗ LS (h1 (MH ), s) =

H

LS (π, s − 1/2)dim(πf ) ,

(1.15.2)

π=σ2 ⊗πf

where ∗ LS (h1 (MH ), s) =

∗ Lv (h1 (MH ), s),

LS (π, s) =

v ∈S

Lv (π, s).

v ∈S

Above, the automorphic L-functions are normalized as in [Ja-La] (the centre of symmetry of the functional equation lies at s = 1/2). It is a much deeper fact that the extreme left and right terms in (1.15.1) are equal even for v ∈ S. For v ∈ S − SB this is proved in [Ca 2]; for v ∈ SB one can reduce to the previous case by switching to a quaternion algebra unramiﬁed at v (possibly after a cyclic base change to a suitable totally real number ﬁeld in which v splits completely). The Jacquet-Langlands correspondence [Ja-La, §14] associates to each (irreducible, cuspidal) π = σ2 ⊗ πf with πfH = 0 an irreducible cuspidal representation JL(π) of GL2 (AF ), which corresponds to a Hilbert modular newform of weight (2, . . . , 2) and central character ωπ . This representation is characterized by the property ∼

∼

∗ (∀w ∈ Ram(B)) JL(π)w −→ πw as representations of GL2 (Fw )−→ Bw (=⇒ Lw (JL(π), s) = Lw (π, s))

(where w is a prime of F ). In particular, LS (JL(π), s) = LS (π, s). In fact, the equality Lw (JL(π), s) = Lw (π, s) holds also for w ∈ Ram(B) ([Ja-La], Rmk. before Thm. 14.4).

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∗ ∗ (1.16) The Jacobian of MH and NH . As in [Co-Va 1, 3.3], we deﬁne ∗ ∗ the Jacobian of MH (resp., NH ) to be the abelian variety over F ∗ ∗ ∗ J(MH ) := Pic◦ (MH /F ) = ResF (MH )/F (Pic◦ (MH /F (MH ))) ∗ ∗ ∗ J(NH ) := Pic◦ (NH /F ) = ResF (NH )/F (Pic◦ (NH /F (NH ))) ∗ (recall that F (MH ) (resp., F (NH )) denotes the ﬁeld of constants of MH ∗ (resp., of NH )). As Pic◦ (−/F ) is both covariant and contravariant for ﬁnite ﬂat morphisms of proper smooth curves over Spec(F ), each diagram (1.9.1) induces morphisms of abelian varieties ∗ ∗ [HgH ]∗ = [H g −1 H]∗ : J(MH ) −→ J(MH ), ∗ ∗ [HgH ]∗ = [H g −1 H]∗ : J(MH ) −→ J(MH )

(1.16.1)

∗ ∗ (and similarly for NH ). Each Jacobian J(MH ) has a canonical principal polarization. It follows from (1.16.1) that the corresponding Rosati involution ι acts by

ι ([HgH]∗ ) = [HgH]∗ = [Hg −1 H]∗ , ι ([HgH]∗ ) = [HgH]∗ = [Hg −1 H]∗ .

(1.16.2)

One can consider (1.16.2) as a “motivic” version of (1.11.3). We let the Hecke correspondences act on the Jacobians covariantly. As [·g] ◦ [·g ] = [·g g], the map ∗ ∗ /H]op −→ End(J(MH Z[H\B )) [HgH] → [HgH]∗

(1.16.3)

∗ /H] is a ring homomorphism (recall that the multiplication on Z[H\B used here is the opposite of the one considered in [Co-Va 1, 3.4, 3.11]). ∗ an The space Γ((MH ) , Ωan ) is canonically isomorphic to the cotangent ∗ an ∨ ∗ an space (T0 J(MH ) ) of the complex torus J(MH ) at the origin. For ∗ _ _ _ _/ M ∗ each Hecke correspondence [HgH ] : MH H , the induced map ∗ an ∗ an [HgH ]∗ : Γ((MH , Ωan ) −→ Γ((MH ) , Ωan ) ) corresponds to the dual of the diﬀerential of the morphism ∗ ∗ [HgH ]∗ : J(MH ) −→ J(MH )

at the origin.

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487

(1.17) The “physical” Hecke algebra. In the situation of (1.12.1), the spherical Hecke algebra ∗ /H] TSH := Z[H S \G(Af )S /H S ] ⊂ Z[H\B ∗ is commutative; we let it act on the Jacobian J(MH ) by the rule (1.16.3), which deﬁnes a ring homomorphism

op ∗ /H]op −→ End(J(M ∗ )). θ : TSH = TSH ⊂ Z[H\B H

(1.17.1)

As a ring, TSH is generated by the double cosets [Hbv H] and [Hv±1 H] (v ∈ S), where bv is as in 1.12. For each v ∈ S, the endomorphisms T (v) := θ([Hbv H]),

[v±1 ] := θ([Hv±1 H])

∗ ∗ an of J(MH ) induce the eponymous endomorphisms of Γ((MH ) , Ωan ) (which were deﬁned in 1.12)) via the contravariant action on the cotan∼ gent space at the origin. In other words, the action of TSH on S2H −→ ∗ an an Γ((MH ) , Ω ) factors through the morphism θ. Similarly, we let T (v) and [v±1 ] act on the spaces W , WdR and W deﬁned in 1.12 by contravariant functoriality, i.e. by [Hbv H]∗ and [Hv±1 H]∗ .

The commutative ring ∗ T := θ TSH ⊂ End(J(MH )) (the “physical” Hecke algebra) is a free Z-module of ﬁnite rank, generated (as a ring) by the endomorphisms T (v) and [v±1 ] (for v ∈ S). It follows from (1.16.2) that (∀v ∈ S)

ι (T (v)) = [v−1 ] T (v),

ι [v±1 ] = [v∓1 ],

(1.17.2)

∗ hence T is a commutative subring of End(J(MH )) stable by the Rosati involution ι. The positivity property of this involution implies ([Mu, §21]) that the Q-algebra T ⊗ Q is isomorphic to a ﬁnite product of number ﬁelds ∼ T ⊗ Q −→ Lj , J = J1 ∪ J2 , j∈J

where each factor Lj is ι-stable and (∀j ∈ J1 )

Lj is totally real and ι acts trivially on Lj

(∀j ∈ J2 )

Lj is a CM ﬁeld and ι acts on Lj by complex conjugation. (1.17.3)

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For each j ∈ J, we denote by θj the ring homomorphism θ

θj : TSH −−→T ⊂ T ⊗ Q Lj .

(1.17.4)

The image of θj is an order in Lj and T has ﬁnite index in j Im(θj ). ∗ If π = σ2 ⊗ πf is an irreducible cuspidal unitary representation of BA H S H satisfying πf = 0, then TH acts on πf through a ring homomorphism λπ : TSH −→ C, which factors as θj

σ

λπ : TSH −−→Lj → C, for a unique pair (j, σ) ∈ J × Hom(Lj , C). The strong multiplicity one ∗ theorem for automorphic representations of BA ([Ge], VI.2) implies that π = σ2 ⊗ πf = π = σ2 ⊗ πf , πfH = 0 = πfH =⇒ λπ = λπ . (1.17.5) ∗ ∗ The canonical map J(MH ) −→ J(NH ) (induced by the projection ∗ ∗ S S MH −→ NH ) is TH -equivariant and surjective; it follows that TH ⊗Q ∗ acts on J(NH ) ⊗ Q (in the category of abelian varieties up to isogeny) by a certain quotient of T ⊗ Q. Let be a prime number. For each prime L | of Lj , denote by VL (θj ) the L -adic Galois representation of GF,S associated to JL(π) (which is a Hilbert modular form of parallel weight (2, . . . , 2) over F ). This is a two-dimensional, absolutely irreducible ([Tay, Prop. 3.1]) representation of GF,S over (Lj )L , characterized by the property

(∀v ∈ S) det(1−X Fr(v)geom | VL (θj )) = 1−θj (T (v))X+ωπ (v)(N v)X 2 . ∗ ∗ (1.18) Proposition (Decomposition of J(MH ) and J(NH ) up to isogeny). (i) For each pair (j, σ) ∈ J × Hom(Lj , C) there exists a ∗ unique irreducible cuspidal unitary representation π = σ2 ⊗ πf of BA H satisfying πf = 0, λπ = σ ◦ θj . We denote π = π(σ ◦ θj ). 1 ∗ (ii) For each prime number , the T⊗Q [GF ]-module W = Het (MH ⊗F F , Q ) is isomorphic to ∼

W −→

j∈J

VL (θj )

aj

,

L |

where aj = dimC (π(σ ◦ θj )H f ) ≥ 1, L runs through all primes of Lj above , and T acts on VL (θj ) via θj and the embedding (Lj )L .

Lj → a ∗ (iii) There exists a T-linear isogeny u : J(MH ) −→ j∈J Aj j deﬁned over F , where each aj ≥ 1 is as in (ii), Aj is an abelian variety over F of dimension dim(Aj ) = [Lj : Q] satisfying OLj = EndF (Aj ) (hence

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1 Aj is F -simple) on which T acts via θj , Het (Aj ⊗F F , Q ) is isomorphic to L | VL (θj ) as an T ⊗ Q [GF ]-module and there is an equality of L-series (Euler factor by Euler factor) L(Aj /F, s) = L(π(σ ◦ θj ), s − 1/2). σ:Lj →C

The abelian variety Aj is unique up to an OLj -linear isogeny. (iv) If j, j ∈ J, j = j , then Aj is not isogeneous (over F ) to Aj . (v) For each j ∈ J1 (resp., j ∈ J2 ) and each polarization of Aj deﬁned over F , the corresponding Rosati involution acts on Lj = EndF (Aj ) ⊗ Q trivially (resp., by the complex conjugation).

a ∗ (vi) There exists a T-linear isogeny u1 : J(NH ) −→ j∈J1 Aj j (deﬁned over F ) ﬁtting into a commutative diagram (in which the right vertical arrow is the canonical projection)

u aj ∗ J(MH ) −−→ j∈J Aj ⏐ ⏐ ↓ ↓

u1 aj ∗ J(NH ) −−→ j∈J1 Aj . Proof. All statements are well-known, but as we are not aware of a good reference, we sketch the argument. In (i), the uniqueness is a consequence of (1.17.5). In order to prove the existence of π, note that ∗ W = H 1 (MH , Q) is a faithful T ⊗ Q-module, hence it is isomorphic to ⊕b ∼ W −→ Lj j (bj > 0). (1.18.1) j∈J

Both comparison isomorphisms ∼

W ⊗Q C −→ WC ⊕ WC ,

∼

WdR ⊗F,τ1 C −→ WC

∗ an 1 ∗ (where WC = Γ((MH ) , Ωan ) and WdR = HdR (MH /F )) are T ⊗ Clinear. As τ1 (F ) ⊂ R, the second one implies that the T ⊗ C-modules WC and WC are isomorphic, which yields an isomorphism of T ⊗ Cmodules ∼ ⊕a WC −→ (Lj ⊗Q C) j , aj = bj /2 > 0. j∈J

The isomorphism ∼

Lj ⊗Q C −→

σ:Lj →C

C,

a ⊗ z → (σ(a)z)σ

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implies that each morphism σ ◦ θj occurs as λπ in WC , with multiplicity aj > 0. In other words, π(σ ◦ θj ) exists and dimC π(σ ◦ θj )H f = aj . (ii) As W is a semi-simple Q [GF ]-module ([Fa, Satz 3]), it is suﬃcient to show that the traces of Fr(v)geom (v ∈ S) acting on both sides coincide, which follows from the T ⊗ Q -linear comparison isomorphism W ⊗Q ∼ Q −→ W and the formulas (1.14.2) and (1.18.1).

a ∗ (iii) The existence of a T-linear isogeny J(MH ) −→ j∈J Aj j (deﬁned over F ) such that Lj → EndF (A j ) ⊗ Q, T acts on Aj via θj 1 and Het (Aj ⊗F F , Q ) is isomorphic to L | VL (θj ) follows from (ii) and Faltings’ isogeny theorem [Fa, Satz 4] (formerly known as Tate’s conjecture). The endomorphism ring EndF (Aj ) contains Im(θj ), which is an order in Lj . Replacing Aj by the F -isogeneous abelian variety HomIm(θj ) (OLj , Aj ) (see [Co, Ch. 10] for a discussion of the formalism of O-transforms), we may assume that EndF (Aj ) contains OLj . Fix a prime number which splits in Lj /Q. If L and L are distinct primes above in Lj , then the Q [GF ]-modules VL (θj ) and VL (θj ) are not isomorphic, as Lj is generated (as a ﬁeld extension of Q) by the traces θj (T (v)) and determinants ωπ (v) of the Frobenius elements Fr(v)geom (v ∈ S) acting on VL (θj ) (as well as on VL (θj )), and the two embeddings Lj → (Lj )L , Lj → (Lj )L are distinct. Absolute irreducibility of VL (θj ) yields an inclusion Lj ⊗Q Q → EndF (Aj ) ⊗Q Q = EndQ [GF ]

L |

VL (θj ) = Q , L |

(1.18.2) which must be an equality, by comparing the dimensions of the two ﬂank terms; this proves that Lj = EndF (Aj ) ⊗ Q. As OLj ⊂ EndF (Aj ), we must have EndF (Aj ) = OLj ; as its endomorphism ring is an integral domain, the abelian variety Aj is F -simple. The statement about the local L-factor Lv (Aj /F, s) follows from (1.15.2) (resp., the discussion following (1.15.2)) if v ∈ S (resp., if v ∈ S), combined with (i) and ∗ ∗ (ii), since Lv (h1 (MH ), s) = Lv (J(MH ), s). The uniqueness of Aj (up to an OLj -linear isogeny) follows from Faltings’ isogeny theorem and the equality in (1.18.2). (iv) The same argument as in the proof of (iii) shows that, if is a prime number split in both Lj and Lj and L , L are primes above in Lj , Lj , respectively, (not necessarily distinct), then the Q [GF ]-modules VL (θj ) and VL (θj ) are not isomorphic, hence Aj is not isogeneous to Aj (over F ).

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(v) As EndF (Aj ) is commutative, the Rosati involution does not depend on the polarization. For the canonical polarization, the statement follows from (1.17.3). ∗ (vi) TSH ⊗ Q acts on J(NH ) ⊗ Q through

a certain quotient T ⊗ Q of T⊗Q, which is necessarily isomorphic to j∈J , for some subset J ⊂ J. ∗ ∼ As Γ((N ∗ )an , Ωan ) −→ S H F consists precisely of those automorphic H

2

forms from S2H which have trivial central character, the formula (1.17.2) implies that the Rosati involution acts trivially on T ⊗Q, hence J ⊂ J1 . ∗ The arguments used in the proof of (i), (ii) show that J(NH ) is T-linearly

aj isogeneous to j∈J Aj , where > 0. F aj = dimC π(σ ◦ θj )H f ∗

(∀j ∈ J ) (∀σ : Lj → C)

In particular, for each pair (j, σ) ∈ J × Hom(Lj , C), the representation ∗ π(σ ◦ θ ) satisﬁes π(σ ◦ θ )H F = 0, hence it has trivial central character, j

j f

which in turn implies that = dim π(σ ◦ θ )H = a . F aj = dimC π(σ ◦ θj )H C j f j f ∗

Conversely, if j ∈ J1 , then the existence of the Weil pairing implies that GF,S acts on Λ2 VL (θj ) by the cyclotomic character ([Cas, Cor. 5.5]; [Ri, Lemma 4.5.1]), hence ωπ = 1. It follows that J = J1 . ∗ ∗ ∗ ∗ (1.19) The maps MH −→ J(MH ) and NH −→ J(NH ). In the ∗ classical case B = M2 (Q), the class of the divisor (∞) in Pic(MH ) has the following properties: its pull-back to each irreducible component of ∗ MH ⊗Q Q has degree one and there exists an integer m ≥ 1 such that each Hecke operator [HbH]∗ acts on the class of m(∞) by multiplication by its degree. Indeed, we can take any m that annihilates the cuspidal group ∗ ∗ C(MH ) ⊂ Pic(MH ), which is deﬁned as the subgroup generated by the ∗ divisors whose pull-backs to each irreducible component of MH ⊗Q Q ∗ have degree zero and are supported at the cusps. The group C(MH ) is ﬁnite by the theorem of Manin-Drinfeld [Dr].

For B = M2 (Q), Zhang [Zh 1, Introduction] (see also [Co-Va 1, 3.5]) con∗ ∗ structed a certain class ξ(MH ) ∈ Pic(MH /F ) ⊗ Q (“the Hodge class”) ∗ such that the degree of the pull-back of ξ(MH ) to each irreducible com∗ ∗ ponent of MH ⊗F F is equal to one. More precisely, ξ(MH ) is con∗ ∗ structed, for a suitable integer m ≥ 1, as ξ(M ) = mξ(M ) ⊗ 1/m, H

H

∗ ) ∈ Pic(M ∗ /F ) has degree m on each irreducible comwhere mξ(M H H ∗ ponent of MH ⊗F F and each Hecke operator [HbH]∗ acts on it by multiplication by its degree ([Co-Va 1], Remark 3.7).

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In general, if ∗ MH ⊗F F =

Cα

α

is the decomposition into irreducible components, let mδα ∈ Pic(Cα ) ∗ )) if be the pull-back to Pic(Cα ) of the class of m(∞) (resp., of mξ(M H B = M2 (Q) (resp., if B = M2 (Q)). We denote by ∗ ∗ ι : MH −→ J(MH )

the morphism (deﬁned over F ), which is characterized by the formula (∀α) (∀P ∈ Cα (F ))

ι(P ) = m(P ) − mδα .

∗ The same discussion applies to the curves NH ; we obtain a morphism ∗ ∗ ι : NH −→ J(NH ).

2. CM points In this section we recall basic properties of CM points on the curves MH and NH . We follow the notation and conventions of Sect. 1. (2.1) Let K be a totally imaginary quadratic extension of F such that each prime v ∈ SB either ramiﬁes or is inert in K/F . This assumption implies that there exists an F -embedding (= an injective homomorphism of F -algebras) t : K → B; we ﬁx such an embedding and denote by → B) the induced embedding of the tv : K ⊗F Fv → Bv (resp., by t:K completions (resp., of the ﬁnite ad`eles). As in 1.1, we ﬁx an embedding K → C extending τ1 : F −→ R. (2.2) Lemma. (i) There exists a unique point z ∈ C with Im(z) > 0, ∼ which is ﬁxed by the action of t(K ∗ ) ⊂ B ∗ ⊂ Bτ∗1 −→ GL2 (R). ∗ ∗ (ii) We have {λ ∈ B | λ(z) = z} = t(K ). Proof. Easy exercise. (2.3) Deﬁnition. The set of CM-points by K on the curve MH (resp., on NH ) is the set ∗} CM (MH , K) = {x = [z, b] ∈ MH (C) | b ∈ B ∗} CM (NH , K) = {x = [z, b] ∈ NH (C) | b ∈ B = the image of CM (MH , K) in NH (C)

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(these sets do not depend on the choice of t, as two diﬀerent F embeddings of K into B are conjugate by an element of B ∗ , by the Skolem-Noether theorem). (2.4) The Galois action on CM-points. The reciprocity law [De, 3.9], [Mi 2, II.5.1] (with the sign corrected, [Mi 3, 1.10]) states that CM (MH , K) ⊂ MH (K ab ) and that the Galois action of Gal(K ab /K) ∗ −→ on CM (MH , K) is described, via the reciprocity map recK : K ab Gal(K /K), by the following formula: ∗) (∀a ∈ K

recK (a) [z, b] = [z, t(a)b].

(2.5) Proposition. Denote by K(x) ⊂ K ab the ﬁeld of deﬁnition over K of a CM point (by K) x = [z, b] on the curve MH (resp., on NH ). ∗/ Then the reciprocity map recK induces an isomorphism recK : K ∗ \K ∼ ∼ −1 −1 ∗ ∗ −1 ∗ −1 / t (bHb ) −→ Gal(K(x)/K) (resp., recK : K \K t (bH F b ) −→ Gal(K(x)/K)). ∗ , then we deduce from 2.4 Proof. If x = [z, b] ∈ CM (MH , K) and a ∈ K and Lemma 2.2(ii) that recK (a) [z, b] = [z, b] if and only if ⇐⇒ (∃λ ∈ B ∗ ) (∃h ∈ H) ⇐⇒ (∃μ ∈ K ∗ ) (∃h ∈ H) ⇐⇒ a ∈ K ∗ · t−1 (bHb−1 ).

∗ (z, t(a)b) = (λ(z), λbh) ∈ (C − R) × B t(a)b = t(μ)bh

The proof for x ∈ CM (NH , K) is similar. (2.6) Ring class ﬁelds of K (2.6.1) From now on, we shall consider only CM points on the curves NH . For such a point x = [z, b], the formula from Proposition 2.5 can be restated as an isomorphism ∼ ∗ /K ∗ F ∗ Z −→ recK : K Gal(K(x)/K),

∗ b−1 ) ⊂ O ∗ is an open (compact) subgroup of O ∗ where Z = t−1 (bH O F K K ∗ . containing O F ∗ , for an OF -order R ⊂ B, then (2.6.2) In the special case when H = R ∗ Z = O for some OF -order O ⊂ K. Such an order is necessarily of the form Oc = OF + cOK , where c ⊂ OF is a non-zero ideal of OF . The corresponding abelian extension K[c]/K satisfying ∼ ∗ /K ∗ F∗ O ∗ −→ recK : K Gal(K[c]/K) c

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is called the ring class ﬁeld of K of conductor c (note that Zhang [Zh 2] uses the same terminology, but Cornut and Vatsal [Co-Va 1,2] consider a slightly more general class of extensions). If c, c ⊂ OF are non-zero ideals, then we have ∗ · O ∗ = O ∗ O c c gcd(c,c ) ,

∗ ∩ K ∗ F∗ O ∗ ⊇ K ∗ F∗ O ∗ K ∗ F∗ O c c lcm(c,c ) , (2.6.2.1)

hence K[c] ∩ K[c ] = K[gcd(c, c )],

K[c] K[c ] ⊆ K[lcm(c, c )].

(2.6.3) Each prime of K not dividing cOK is unramiﬁed in K[c]/K. If is a prime of F which does not divide c and which is inert in K/F , then OK splits completely in K[c]/K and each prime above is totally tamely ramiﬁed in K[c]/K[c]. ∗ is an open subgroup containing O ∗ , we (2.6.4) In general, if Z ⊂ O K F ab denote by KZ ⊂ K the ﬁnite abelian extension of K satisfying ∼ ∗ /K ∗ F∗ Z −→ recK : K Gal(KZ /K).

It follows from the last paragraph in 1.4 and from Proposition 2.5 that we have KZ = K[c] =: K[∞], CM (NH , K) ⊂ NH (K[∞]). Z

c

(2.6.5) The reciprocity map recK is compatible with the action of the Galois group Gal(K/F ) = {1, ρ} (where ρ denotes the complex conjugation on K ⊂ C) on both sides. As ∗) (∀a ∈ K

a · ρ(a) ∈ F∗ ,

it follows that KZ /F is a Galois extension and the conjugation action of ρ on Gal(KZ /K) is given by ρgρ−1 = g −1 . In other words, ∼

Gal(KZ /F ) −→ Gal(KZ /K) {1, ρ} is a generalized dihedral group. (2.7) We shall now describe the Galois group ∼

Gal(KZ /KZ ) −→ K ∗ F∗ Z/K ∗ F ∗ Z ,

(2.7.1)

∗ are two open subgroups containing O ∗ . We use where Z ⊂ Z ⊂ O K F the elementary fact that, whenever A and B ⊃ C are subgroups of some

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abelian group, then the obvious inclusion maps give rise to an exact sequence A∩B B AB 0 −→ −→ −→ −→ 0, A∩C C AC from which we deduce ∼ ∗ ∗ = O ∗ =⇒ F∗ ∩ Z = F∗ ∩ Z = O ∗ =⇒ Z/Z −→ F∗ ∩ O F Z/F∗ Z K F F ∼

∗ ∗ ∗ ∗ F ∗ ∩ (OK ∩ Z) = F ∗ ∩ (OK ∩ Z ) = OF∗ =⇒ (OK ∩ Z)/(OK ∩ Z ) −→ ∼

∗ ∗ −→ F ∗ (OK ∩ Z)/F ∗ (OK ∩ Z )

and, by taking A = K ∗ , B = F∗ Z, C = F∗ Z , an exact sequence 0 −→

K ∗ ∩ F ∗ Z Z −→ −→ Gal(KZ /KZ ) −→ 0. Z K ∗ ∩ F ∗ Z

(2.7.2)

(2.7.3)

The next step is to determine the abelian group K ∗ ∩ F ∗ Z (and its counterpart for Z instead of Z). (2.8) Proposition. Put G = ) = {1, ρ}. Gal(K/F ∗ ∗ (i) Ker NK/F : OK −→ OF∗ = (OK )tors . ∼ ∗ )/F ∗ O∗ −→ (ii) There is an isomorphism (K ∗ ∩F ∗ O Ker(Pic(OF ) −→ K K Pic(OK )). (iii) ([Wa, Thm. 10.3]) The abelian group Ker(Pic(OF ) −→ Pic(OK )) ∗ naturally injects into H 1 (G, OK ). The order of the abelian group ∗ H 1 (G, OK ) is equal to 1 or 2. Proof. (i) As K is a CM ﬁeld and F its maximal real subﬁeld, the group ∗ ∗ Ker(NK/F : OK −→ OF∗ ) is ﬁnite, hence contained in (OK )tors . On the ∗ other hand, each root of unity ζ ∈ (OK )tors satisﬁes ζ · ρ(ζ) = 1. ∗ , then, for each non-archimedean prime v of F , (ii) If β ∈ K ∗ ∩ F∗ O K ∗ there exists αv ∈ Fv such that ordw (β) = ordw (αv ), for all primes w above v in K. Let I be the ideal of OF with divisor equal to ord (αv ) [v]; then IOK = (β). It follows that the map v v β → the class of IOK deﬁnes a homomorphism ∗ )/F ∗ O∗ −→ Ker(Pic(OF ) −→ Pic(OK )), (K ∗ ∩ F ∗ O K K the inverse of which is given by I → β, if IOK = (β). (iii) The exact sequence 0 −→ PK −→ IK −→ Pic(OK ) −→ 0

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(where PK (resp., IK ) denotes the group of principal (resp., of all) fractional ideals of K) and its counterpart over F give rise to a commutative diagram with exact rows 0 −→

PF ⏐ ⏐

G 0 −→ PK

−→

−→

IF ∩ ↓

G −→ IK

Pic(OF ) ⏐ ⏐

−→ Pic(OK )G

−→

0 ⏐ ⏐

−→ H 1 (G, PK ).

The Snake Lemma then implies that the group Ker(Pic(OF ) −→ G Pic(OK )) injects into Coker(PF −→ PK ). On the other hand, the cohomology sequence of ∗ 0 −→ OK −→ K ∗ −→ PK −→ 0

yields an exact sequence G ∗ 0 −→ OF∗ −→ F ∗ −→ PK −→ H 1 (G, OK ) −→ 0, ∼

G ∗ ∗ hence Coker(PF −→ PK ) −→ H 1 (G, OK ). The abelian group H 1 (G, OK ) ∗ ∗ is a quotient of the ﬁnite cyclic group Ker(NK/F : OK −→ OF ) = ∗ (OK )tors ; on the other hand, it is killed by |G| = 2, which implies that 1 ∗ |H (G, OK )| ≤ 2.

∗ containing (2.9) Proposition. If Z ⊂ Z are open subgroups of O K ∗ , then the natural inclusions give rise to an exact sequence O F 0 −→

∗ OK ∩Z K ∗ ∩ F∗ Z −→ −→ UZ,Z −→ 0, ∗ OK ∩ Z K ∗ ∩ F∗ Z

in which |UZ,Z | ≤ |Ker(Pic(OF ) −→ Pic(OK ))| ≤ 2. Proof. Injectivity of the ﬁrst arrow follows from the second isomorphism in (2.7.2) and the equality ∗ ∗ ∗ ∗ Z ) = F ∗ (OK ∩ Z) ∩ F ∗ Z = F ∗ (OK ∩ Z ∩ F∗ Z ) = F ∗ (OK ∩Z ∩O F ∗ = F ∗ (OK ∩ Z ).

Deﬁne UZ,Z to be the cokernel of this injective map. Assume ﬁrst that ∗ = F ∗ O∗ , by PropoKer(Pic(OF ) −→ Pic(OK )) = 0; then K ∗ ∩ F∗ O K K sition 2.8(ii), which implies that ∗ ∗ ∗ ∗ Z) K ∗ ∩ F∗ Z = F ∗ OK ∩ F∗ Z = F ∗ (OK ∩ F∗ Z) = F ∗ (OK ∩O F ∗ = F ∗ (OK ∩ Z)

(and similarly for Z ), hence UZ,Z = 0.

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If Ker(Pic(OF ) −→ Pic(OK )) = 0, then there exists β ∈ K ∗ such that ∗ = F ∗ O∗ ∪ βF ∗ O∗ , by Proposition 2.8(ii)-(iii); thus K ∗ ∩ F∗ O K K K ∗ ∗ K ∗ ∩ F∗ Z = F ∗ (OK ∩ Z) ∪ F ∗ (βOK ∩ F ∗ Z). ∗ ∗ ∗ If u, u ∈ OK satisfy βu, βu ∈ F ∗ Z, then u /u ∈ OK ∩ F ∗ Z = OK ∩ Z, which shows that |UZ,Z | ≤ 2.

(2.10) Proposition. Let I0 ⊂ OK be the non-zero ideal I0 := lcm {(u− ∗ 1) | u ∈ (OK )tors , u = 1}. If c ⊂ OF is a non-zero ideal such that cOK ∗ is a subgroup, then K ∗ ∩ F ∗ Z = F ∗ and O∗ ∩ Z = O∗ . I0 and Z ⊂ O c K F ∗ . Assume that β ∈ Proof. It is enough to consider the case Z = O c c∗ , β = ab, a ∈ F ∗ , b ∈ O c∗ . Then we have K ∗ ∩ F∗ O ∗ c∗ ⊂ OK u := β/ρ(β) = b/ρ(b) ∈ K ∗ ∩ O .

More precisely, u is contained in 1−ρ

∗ NK/F ∗ ∗ ∗ ∗ −→ (OK /cOK )∗ ; Ker (OK −−→OF ) ∩ O ⊆ (O ) ∩ Ker O tors c K K the condition cOK I0 then implies that u = 1, hence β ∈ F ∗ . It follows ∗ = F ∗ and O∗ ⊂ O ∗ ⊂ O∗ ∩ F∗ O ∗ ⊂ O∗ ∩ F ∗ = O∗ . that K ∗ ∩ F∗ O c c c F K K F ∗ are open subgroups containing O ∗ (2.11) Corollary. If Z ⊂ Z ⊂ O K F ∗ such that Z ⊂ Oc , for some non-zero ideal c ⊂ OF satisfying cOK I0 , then the Galois group Gal(KZ /KZ ) is isomorphic to Z/Z . Proof. Combine (2.7.3) with Propositions 2.9-10. 3. The main result We follow the notations deﬁned in Sect. 1-2. (3.1) Fix the following data: ∗. (3.1.1) An open compact subgroup H ⊂ B ∗ (3.1.2) An F -simple quotient Aj (j ∈ J1 ) of J(NH ) satisfying ∗ EndF (Aj ) = OLj and a non-trivial T-linear morphism J(NH )−→ Aj (deﬁned over F ). Denote by ιj the composite morphism ∗ ∗ ιj : NH −→J(NH ) −→ Aj . ι

(3.1.3) A totally imaginary quadratic extension K of F such that each prime v ∈ SB either ramiﬁes or is inert in K/F . (3.1.4) An F -embedding t : K → B.

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(3.1.5) A CM point x = [z, b] ∈ CM (NH , K); denote by K(x) its ﬁeld of deﬁnition over K. ∗ (3.1.6) A character α : Gal(K(x)/K) −→ OL , where L is a number ﬁeld containing the totally real ﬁeld Lj = EndF (Aj ) ⊗ Q. We denote eα =

α−1 (σ) σ ∈ OL [Gal(K(x)/K)]

σ∈Gal(K(x)/K)

and, for any OL [Gal(K(x)/K)]-module M , M (α) = {m ∈ M | ∀σ ∈ Gal(K(x)/K)

σ(m) = α(σ)m}.

The character α factors through an injective character ∗ β : Gal(K(α)/K) → OL ,

where K(α) = K(x)Ker(α) ⊂ K(x). (3.2) Theorem. Assume that we are given the data (3.1.1-6); put yj = ιj (x) ⊗ 1 ∈ Aj (K(x)) ⊗OLj OL . Assume, in addition, that the following condition is satisﬁed: () The abelian variety Aj does not acquire complex multiplication over any totally imaginary quadratic extension K of F contained in K(α).

If eα (yj ) ∈ Aj (K(x)) ⊗OLj OL

, then the following abelian groups tors

(more precisely, OL -modules) are ﬁnite:

(α)

(α) Aj (K(x)) ⊗OLj OL /OL · eα (yj ), X(Aj /K(x)) ⊗OLj OL

(α−1 )

(α−1 ) Aj (K(x)) ⊗OLj OL /OL · ρ(eα (yj )), X(Aj /K(x)) ⊗OLj OL (where the complex conjugation ρ acts on Aj (K(x)) ⊗OLj OL by ρ ⊗ id). (3.3) Previously known results in this direction ([Ko], [Ko-Lo 1,2], [BeDa], [Ho 1,2], [Be], [Ti], [Zh 3]) use several additional hypotheses which ensure, among other things, that there is a “geometric” formula relating ρ(yj ) to yj . Our main observation is that the Euler system argument does not require such a geometric relation. (3.4) As explained in 6.2.1 below, the condition () can be reformulated as follows: the Hilbert modular forms that occur in the factorization of L(Aj /F, s) do not have CM by K .

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(3.5) It will be convenient to rephrase Theorem 3.2 and its proof in terms of the abelian variety A = Aj ⊗OLj OL ,

OL → EndF (A)

(see [Con, §7] for the general formalism of such tensor products). We have yj ∈ Aj (K(x)) ⊗OLj OL = A(K(x)),

eα (yj ) = eβ (y) ∈ A(K(α)),

where y = TrK(x)/K(α) (yj ) ∈ A(K(α)) eβ = β −1 (σ) σ ∈ OL [Gal(K(α)/K)]. σ∈Gal(K(α)/K)

Fix a maximal ideal p ⊂ OL and a suﬃciently large integer M >> 0 such that the ideal pM is principal. Fixing its generator (which will be, by abuse of language, also denoted by pM ), the Galois cohomology of pM

0 −→ A[pM ] −→ A(F )−→A(F ) −→ 0 gives rise to the usual descent sequence δ

0 −→ A(K(α))/pM −→S −→ X(A/K(α))[pM ] −→ 0, where we have denoted by S = Sel(A/K(α), pM ) the classical Selmer group for the pM -descent on A. In order to prove Theorem 3.2, it will be enough to show, assuming that eβ (y) ∈ A(K(α))tors , that

(∃C(p) ≥ 0) (∀M >> 0) pC(p) · S (β) /OL · δ(eβ (y)) = 0. (3.5.1) For all but ﬁnitely many p, C(p) = 0.

(3.5.2)

(3.6) Let us brieﬂy outline the proof of (3.5.1), the main steps of which follow [Be-Da]. We denote Op = OL,p and use temporarily the notation . . “X = Y ” (resp., “x ∈ X”) as a shorthand for pC X = pC Y (resp., for pC x ∈ X), where C is a suitable constant, independent of M . (3.6.1) The Euler system. The ﬁrst step is to construct an Euler system, by letting various Hecke operators act on the CM point x. Applying Kolyvagin’s derivative and eβ , one obtains the derived cohomology classes κ1 = a suitable multiple of δ(eβ (y)) ∈ S (β) , (β)

κ1 ···r ∈ S{1 ,...,r } ⊂ H 1 (K(α), A[pM ])(β),

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where 1 , . . . , r are distinct primes of F , inert in K/F , which satisfy “Kolyvagin’s condition” modulo a suﬃciently high power of p. The subscript 1 , . . . , r indicates that the class κ1 ···r satisﬁes the “Selmer” local conditions only at primes distinct from 1 , . . . , r . j deﬁned (3.6.2) The Weil pairing. Fix a polarization ϕ : Aj −→ A over F ; it gives rise to the Weil pairing Tp (Aj ) × Tp (Aj ) −→ Zp (1) (where p is the residual characteristic of p). This extends naturally (see 5.19 below) to an Op -bilinear pairing Tp (A) × Tp (A) −→ Op (1), whose reduction modulo pM ( , )M : A[pM ] × A[pM ] −→ Op /pM (1) is skew-symmetric, and its kernel is killed by deg(ϕ). (3.6.3) The ﬁrst annihilation relation. For simplicity, assume that β = β := β −1 . Let s ∈ S (β) . For each Kolyvagin prime , we have a (β)

(β)

class κ ∈ S{} and its complex conjugate ρ κ ∈ S{} . Applying the reciprocity law of global class ﬁeld theory

∀a ∈ H 2 (K(α), Op /pM (1))

invv (av ) = 0 ∈ Op /pM

v

to the cup product a = s ∪ ρ κ , we obtain the relation

invv (sv ∪ (ρ κ )v ) = 0 ∈ Op /pM ,

(3.6.3.1)

v

to which only the primes of K(α) above contribute. Fixing one such a prime, say, λ, and using the fundamental relation 5.18 between the localizations (κ1 )λ , (κ )λ , one can rewrite (3.6.3.1) as [K(α) : K] (sλ , (κ1 )λ )M = 0 ∈ Op /pM (1), where one considers the localizations sλ , (κ1 )λ as elements of ∼

∼

1 Hur (K(α)λ , A[pM ]) −→ A[pM ] (−→ (Op /pM )2 ),

by using evaluation at the Frobenius element Fr(λ).

(3.6.3.2)

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The assumption eβ (y) ∈ Ators implies that . κ1 (= OL · κ1 ) = Op /pM . ˇ An application of the Cebotarev density theorem then shows that one can choose a Kolyvagin prime in such a way that . (κ1 )λ = Op /pM ∼

1 (inside Hur (K(α)λ , A[pM ]) −→ A[pM ]). As the pairing ( , )M on A[pM ] is almost symplectic, the annihilation relation (3.6.3.2) yields

. . sλ ∈ (κ1 )λ ⊥ = (κ1 )λ . If we put

(3.6.3.3)

S(β) = {s ∈ S (β) | sλ = 0},

then we deduce from (3.6.3.3) that . S (β) = κ1 ⊕ S(β) . (3.6.4) The second annihilation relation. Let s ∈ S(β) . In order . to complete the proof of (3.5.1), we must show that s = 0. If = is another Kolyvagin prime and λ a prime above in K(α), then the reciprocity law applied to the cup product s ∪ ρ κ

invv (sv ∪ (ρ κ )v ) = 0 ∈ Op /pM

v

simpliﬁes to [K(α) : K] (sλ , (κ )λ )M = 0 ∈ Op /pM (1).

(3.6.4.1)

As (κ1 )λ (resp., (κ )λ ) is unramiﬁed (resp., totally ramiﬁed) at λ, we have . (κ1 )λ , (κ )λ = (Op /pM )2 ⊂ H 1 (K(α)λ , A[pM ]), hence

. (β) κ1 , κ = (Op /pM )2 ⊂ S{} .

ˇ Another application of the Cebotarev density theorem shows that one can choose in such a way that . ∼ 1 (κ1 )λ , (κ )λ = (Op /pM )2 = Hur (K(α)λ , A[pM ]) −→ A[pM ] (3.6.4.2)

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and

. sλ = s.

(3.6.4.3)

The ﬁrst annihilation relation (3.6.3.2) applies with instead of , hence . sλ ∈ (κ1 )λ ⊥ . On the other hand, the second annihilation relation (3.6.4.1) yields . sλ ∈ (κ )λ ⊥ , thus

. . . sλ ∈ (κ1 )λ , (κ )λ ⊥ = A[pM ]⊥ = 0, . by (3.6.4.2), which in turns implies that s = 0, by (3.6.4.3). This ﬁnishes . . the proof that S(β) = 0, hence S (β) = κ1 = δ(eβ (y)). 4. The Euler system Assume that we are in the situation of 3.1. (4.1) According to Proposition 2.5, the ﬁeld of deﬁnition K(x) of x over K is characterized by the isomorphism ∼ ∗ /K ∗ F∗ Z −→ recK : K Gal(K(x)/K),

∗ b−1 ). Z= t−1 (bH O F

It follows from the ﬁrst formula in (2.6.2.1) that there exists a smallest non-zero ideal c(x) ⊂ OF (with respect to divisibility) such that Z ⊇ ∗ . Equivalently, K[c(x)] is the smallest ring class ﬁeld of K (with O c(x) respect to inclusion) containing K(x). (4.2) Fix S as in 1.12. The decomposition (1.12.1) implies that we have Z = ZS Z S ,

ZS ⊂

∗ OK,v ,

ZS =

Zv ,

v ∈S

v∈S

(∀v ∈ S)

∗ −1 ∗ Zv = t−1 v (bv Hv OF,v bv ) ⊂ OK,v ,

where we have denoted, slightly abusively, OK,v = OK ⊗OF OF,v =

OK,w .

w|v

For each v ∈ S, we have ∗ ordv (c(x)) = 0 ⇐⇒ Zv = OK,v .

The Euler system method for CM points on Shimura curves (4.3) Deﬁnition. Let S = ideals of OF : S0 = {(1)},

r≥0

503

Sr be the following set of square-free

S1 = { ∈ Spec(OF ) | is inert in K/F, ∈ S, (p)c(x), OK I0 }, (∀r > 1) Sr = {1 · · · r | j ∈ S1 distinct}, where I0 ⊂ OK is the ideal deﬁned in Proposition 2.10. ∗ be the following (4.4) Deﬁnition. For each n ∈ S , let h(n) ∈ B element: h((1)) = 1. For each ∈ S1 , we have H = R()∗ , for a maximal ∗ be any element OF, -order R() ⊂ B ; let h() ∈ R() ∩ B∗ ⊂ B∗ ⊂ B satisfying ord (nr(h())) = 1. Finally, for n = 1 · · · r (r > 1, j ∈ S1 ),

r ∗ . Having deﬁned h(n), we we put h(n) = h(1 ) · · · h(r ) ∈ j=1 B∗j ⊂ B deﬁne CM points (∀n ∈ S )

x(n) := [z, bh(n)] ∈ CM (NH , K).

(4.5) Proposition. For each n ∈ S , the ﬁeld of deﬁnition K(x(n)) is contained in K[c(x)n] and the reciprocity map induces an isomorphism ∼

∗ /K ∗ F ∗ Z(n) −→ Gal(K(x(n))/K), recK : K where F∗ h(n)−1 b−1 ) = Z ∩ O n∗ = ZS Z(n) = t−1 (bh(n)H O

|n

Z(n)

Z ,

∈S(n)

S(n) = S ∪ { ∈ S1 , | n}. The quotient Z/Z(n) is naturally isomorphic to ∗ ∼ ∼ (OK /OK ) Z/Z(n) −→ Z /Z(n) , Z /Z(n) −→ ; ∗ (OF /OF ) |n

in particular, each group Z /Z(n) (for ∈ S1 , | n) is cyclic of order N + 1. Proof. The isomorphism comes from Proposition 2.5. As h(n) = 1 for each v n, the abelian group Z(n) decomposes as required, with Z(n)v = Zv for each v S(n) and (∀ | n, ∈ S1 )

∗ −1 −1 Z(n) = t−1 b ). (b h()R() h() ∼

Applying Proposition 4.7(iv) below with E = F , R = R() ⊂ B −→ ∼ M2 (F ), i = Ad(b )−1 ◦t : K ⊗F F → B −→ M2 (F ) and r = 1 implies ∗ and that Z(n) = Z ∩ O n ∼

Z /Z(n) −→

(OK /OK )∗ . (OF /OF )∗

∗ that Z(n) ⊇ O ∗ ∩ O ∗ = O ∗ , Finally, it follows from Z ⊇ O n c(x) c(x) c(x)n hence K(x(n)) ⊆ K[c(x)n].

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(4.6) Proposition. Let v be a non-archimedean prime of F which does not divide c(x). (i) If n ∈ S and v n, then each prime of K above v is unramiﬁed in K(x(n))/K. (ii) If n ∈ S , v n and v is inert in K/F , then vOK splits completely in K(x(n))/K. (iii) If n ∈ Sr+1 ( ∈ S1 , n ∈ Sr , r ≥ 0), then each prime of K(x(n)) above is totally tamely ramiﬁed in K(x(n))/K(x(n)). Proof. (i) As K(x(n)) ⊆ K[c(x)n] and v c(x)n, the prime v is unramiﬁed in K(x(n))/K. (ii) The decomposition group of vOK in the extension K(x(n))/K is equal to the image of the composite map ∗ −→ K ∗ /K ∗ F∗ Z(n), Kv∗ := (K ⊗F Fv )∗ −→ K ∗ which factors through Kv∗ /OK,v Fv∗ = 0, as vOK is unramiﬁed in this extension. (iii) The inertia group of this extension at any prime λ of K(x(n)) above is equal to the image of the composite map

(OK /OK )∗ f3 = Z(n)/Z(n)−−→ (OF /OF )∗ f3 −−→Gal(K(x(n))/K(x(n))).

f1

f2

∗ ∗ f : OK(x(n)),λ −−→OK, −−→

The arrow f1 (given by the norm map) is surjective, since λ is unramiﬁed in K(x(n))/K, and the arrows f2 and f3 are given by the canonical projections (cf. (2.7.3)). It follows that f is surjective, hence λ is totally ramiﬁed; the ramiﬁcation is tame, as [K(x(n)) : K(x(n))] divides N +1, which is prime to N λ. (4.7) Proposition. Let E be a ﬁnite extension of Qp , E the unique unramiﬁed quadratic extension of E and π a common uniformizer of E and E . Let R ⊂ M2 (E) be a maximal OE -order and i : E → M2 (E) ∗ ∗ r ∗ an E-embedding. For r ≥ 1, put OE ,r := Ker (OE −→ (OE /π ) ). ∗ ∗ (i) R is a maximal compact subgroup of M2 (E) = GL2 (E). ∗ (ii) (∀γ ∈ GL2 (E)) i−1 (γR∗ γ −1 ) ⊂ OE . −1 ∗ ∗ (iii) If i (R ) = OE and if g ∈ R satisﬁes ordπ (det(g)) = 1, then, for each integer r ≥ 1, the map of sets ∗ ∗ ∗ ∗ ∗ r ∗ −r ir : OE ) /OE OE ,r −→ R /(R ∩ g R g

induced by i is bijective. (iv) Under the assumptions of (iii), we have, for each r ≥ 1, ∗ ∗ i−1 (g r R∗ g −r ) = i−1 (R∗ ∩ g r R∗ g −r ) = OE OE ,r ,

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hence the reduction modulo π r induces an isomorphism of abelian groups i−1 (R∗ )

∼

−→

i−1 (g r R∗ g −r )

∗ OE ∗ ∗ OE OE ,r

∼

−→

(OE /π r OE )∗ . (OE /π r OE )∗

In particular, reduction modulo π induces an isomorphism i−1 (R∗ ) i−1 (gR∗ g −1 )

∼

∗ ∗ −→ kE /kE ,

where kE and kE denote the residue ﬁelds of E and E , respectively. Proof. Without loss of generality, we can assume that R = M2 (OE ) and 1 0 g = . In this case (i) is well-known and (ii) follows from the 0 π properness of the map i (which implies that i−1 (γR∗ γ −1 ) is a compact ∗ subgroup of E ∗ , hence is contained in OE ). In order to prove (iii) and (iv), note ﬁrst that we have, for each r ≥ 0, (ii)

∗ −1 r ∗ −r i−1 (g r R∗ g −r ) = OE (g R g ) = i−1 (R∗ ) ∩ i−1 (g r R∗ g −r ) = ∩ i −1 = i (R∗ ∩ g r R∗ g −r ).

On the other hand, the intersection Rr := R ∩ g r Rg −r is equal to the Eichler order (of level π r ) Rr =

a

b

πr c

d

! " ! a, b, c, d ∈ OE ⊂ M2 (OE ) = R,

which implies that i−1 (Rr ) is an OE -order in OE , hence i−1 (Rr ) = OE + ∗ −1 π c(r) OE , for some c(r) ≥ 0. As i−1 (R∗ ) = OE (R) = , we must have i ∼ −1 −1 c(r) OE . For r ≥ 1, the OE -module i (R)/i (Rr ) −→ OE /π injects ∼ into R/Rr −→ OE /π r , hence c(r) ≤ r. If c(r) < r, then i(π r−1 OE ) ⊆ Rr , which implies that i(OE ) ⊆ R ∩ π 1−r Rr = R1 . In other words, i a b induces a kE -embedding of kE into the ring { | a, b, d ∈ kE }, 0 d which is impossible. This contradiction proves that c(r) = r, hence ∗ i−1 (R∗ ∩ g r R∗ g −r ) = i−1 (Rr∗ ) = i−1 (Rr ∩ R∗ ) = (OE + π r OE ) ∩ OE = ∗ ∗ = (OE + π r OE )∗ = OE OE ,r .

The remaining statements in (iii) and (iv) follow automatically.

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(4.8) Proposition (The norm relation). Let n ∈ Sr+1 ( ∈ S1 , n ∈ Sr , r ≥ 0). (i) K(x(n))/K(x(n)) is a cyclic extension of degree (N + 1)/u(r), ∗ where u(r) = 1 for r > 0 and u(0) = (K ∗ ∩ F∗ Z : F ∗ ) (= (OK ∩ Z : OF∗ ) ∗ ∗ or 2 (OK ∩ Z : OF )). ∗ (ii) We have an equality of divisors on NH ⊗F F T () x(n) = u(r) σ(x(n)), σ∈Gal(K(x(n)/K(x(n)))

where T () x(n) = [Hb H]∗ (x(n)) (which is also equal to [Hb H]∗ (x(n)), ∗ as we work with the curve NH ). ∗ (iii) The following equality holds in J(NH )(K(x(n))): T () (ι(x(n))) = u(r) TrK(x(n))/K(x(n)) (ι(x(n))). Proof. (i) We apply the exact sequence (2.7.3) in the case of the extension K(x(n))/K(x(n)) = KZ(n) /KZ(n) . As nOK I0 , Proposition 2.10 implies that K ∗ ∩ F∗ Z(n) = F ∗ . If r > 0, then K ∗ ∩ F∗ Z(n) = F ∗ as well, hence a combination of (2.7.3) with Proposition 4.5 yields an isomorphism (OK /OK )∗ ∼ −→ Gal(K(x(n))/K(x(n))). (OF /OF )∗ If r = 0 ( ⇐⇒ n = (1)), then the same argument shows that the kernel of the natural surjection (OK /OK )∗ −→ Gal(K(x(n))/K(x(n))) (OF /OF )∗ has order equal to u(0) = (K ∗ ∩ F∗ Z : F ∗ ), which is in turn equal to ∗ ∗ (OK ∩ Z : OF∗ ) or 2(OK ∩ Z : OF∗ ), thanks to Proposition 2.9. (ii) By deﬁnition, the divisor T () x(n) contains x(n). Proposition 4.7(iii) for r = 1 together with the proof of (i) imply that T () x(n) coincides with the orbit of x(n) under the action of Gal(K(x(n))/K(x(n))), with each point counted with multiplicity u(r). (iii) This is a consequence of (ii) and the fact that T () acts on the class ∗ )) by multiplication by deg T () = N + 1. of m(∞) (resp., on mξ(N H

(4.9) Proposition (The congruence relation). Let n ∈ Sr+1 ( ∈ S1 , n ∈ Sr , r ≥ 0). For each prime λ | above in K(n) the following congruence holds: x(n) ≡ Fr()arith x(n) ≡ Fr()geom x(n) (mod λ)

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(considered as an equality in N∗H (κ(λ))). Proof. Firstly, κ(λ)/κ() is a quadratic extension, by Proposition 4.6(ii)(iii), hence the actions of Fr()arith and Fr()geom on κ(λ) coincide. Combining this remark with the congruence relation (1.14.3), we deduce that the reduction modulo λ of each point in the support of T () x(n) (in particular, of x(n)) is equal to the reduction of Fr()arith x(n). (4.10) Proposition-Deﬁnition. For each n ∈ S , deﬁne the subﬁeld K(x(n)) ⊆ K(x(n)) by # K(x), n = (1) K(x(n)) = K(x(1 )) · · · K(x(r )), n = 1 · · · r (r ≥ 1, i ∈ S1 ) and put G(n) = Gal(K(x(n)) /K(x)). (i) For each ∈ S1 , the group G() is cyclic, of order (N + 1)/u(0). (ii) For each n = 1 · · · r ∈ Sr , the canonical map G(n) −→ G(1 ) × · · · × G(r ) is an isomorphism. (iii) For each n ∈ Sr (r ≥ 0), we have [K(x(n)) : K(x(n)) ] = u(r)u(0)r−1 . Proof. (i) This is a special case of Proposition 4.8(i). (ii) The case r = 0 is trivial. For r ≥ 1 the statement follows from Lemma 4.11 below, applied to G = Z/Z(n) = Z/Z(1 · · · r ), Gj = j = Z(j )/Z(n), A = (K ∗ ∩ F∗ Z)/F ∗ (we use (2.7.3) Z(n/j )/Z(n), G and Proposition 2.10 to view A ⊂ G, and then use Proposition 2.10 i ∩ A = 0). again for G (iii) The case r = 0 is again trivial. For r ≥ 1 we have [K(x(n)) : K(x(n)) ] = |A|r−1 = u(0)r−1 , by the proof of Lemma 4.11. (4.11) Lemma. Let G = G1 ⊕· · ·⊕Gr (r ≥ 1) be a ﬁnite abelian group, A ⊂ G a subgroup and π : G −→ G/A the canonical projection. Assume i ∩A = 0, where G i = Gi ⊂ G. that, for each i = 1, . . . , r, we have G j =i Then the canonical homomorphism π(G)/

r $ i=1

i ) −→ π(G

r

i ) π(G)/π(G

i=1

is an isomorphism. Proof. There is nothing to prove if r = 1, so we can assume that r > 1. As i )| = |G i | = |G|/|Gi | the homomorphism in question is injective and |π(G

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i ∩ Ker(π) = 0 by assumption), it is enough to show that (since G ! ! r r r !$ ! i )| |π(G |G| |A| ? ! i )!! = |π(G)| = = |A|r−1 . ! π(G ! ! |π(G)| |A| |Gi | i=1

i=1

i=1

Denote, for any subset I ⊂ {1, . . . , r}, GI = Gi ⊂ G, I 0 = {1, . . . , r} − I,

I = GI 0 . G

i∈I

Applying the Snake Lemma to the commutative diagrams with exact rows i G ⏐ ⏐

0

−→

−→

0

i ) −→ π(G) −→ π(G

G ⏐ ⏐π

π

i −→

−→ 0

Gi ⏐ ⏐

i ) −→ 0 −→ π(G)/π(G

(where πi : G −→ Gi , i = 1, . . . , r, denotes the natural projection with i ) and kernel G {1,...,k} {1,...,k−1} G −→ G ⏐ ⏐ ⏐α ⏐αk−1 k k k−1 0 −→ i=1 π(Gi ) −→ i=1 π(Gi ) 0 −→

π

k −→

Gk ⏐ ⏐π

−→ 0

k ) −→ π(G)/π(G ∼

(k = 2, . . . , r), we obtain isomorphisms A −→ πi (A) (i = 1, . . . , r), inclusions Ker(αr ) ⊆ Ker(αr−1 ) ⊆ · · · ⊆ Ker(α1 ) = 0 and exact sequences 0 −→ πk (A) −→ Coker(αk ) −→ Coker(αk−1 ) −→ 0

(k = 2, . . . , r),

which imply, by induction, the desired equality ! ! r r !$ ! ! ! |πi (A)| = |A|r−1 . ! π(Gi )! = |Coker(αr )| = ! ! i=1

i=2

(4.12) Deﬁnition of the Euler system. For each n ∈ Sr (r ≥ 0), deﬁne the point y(n) ∈ A(K(x(n)) ) = Aj (K(x(n)) ) ⊗OLj OL by y(n) =

u(0) TrK(x(n))/K(x(n)) ι(x(n)) ⊗ 1 u(r)

(in particular, y((1)) = ι(x) ⊗ 1 ∈ A(K(x))).

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(4.13) Proposition (The Euler system relations). Let n ∈ Sr+1 ( ∈ S1 , n ∈ Sr , r ≥ 0). Then: (i) TrK(x(n)) /K(x(n)) y(n) = a y(n), where a = θ(T ()) ∈ OLj is the eigenvalue of the Hecke operator T () acting on Aj . (ii) For each prime λ | above in K(x(n)) , we have y(n) ≡ u(0) Fr()arith y(n) ≡ u(0) Fr()geom y(n) (mod λ ).

Proof. (i) This follows from Proposition 4.8(iii). ∗ )) is deﬁned over F , it follows from (the (ii) As m(∞) (resp., mξ(N H proof of) Proposition 4.9 that we have, for each prime λ | λ above λ in K(x(n)), ι(x(n)) ≡ Fr()arith ι(x(n)) ≡ Fr()geom ι(x(n)) (mod λ),

hence (using Proposition 4.10(iii))

y(n) ≡

u(0) [K(x(n)) : K(x(n)) ] (ι(x(n)) ⊗ 1) ≡ u(r + 1) ≡ u(0)r+1 ι(x(n)) ⊗ 1 (mod λ)

and, similarly, y(n) ≡ u(0)r ι(x(n)) ⊗ 1 (mod λ), which proves the desired congruence.

5. The derivative classes We continue to use the notation of Sect. 1-4. (5.1) Fix a prime number p, a prime ideal p ⊂ OL above p and a prime element ∈ Op = OL,p . Set M0 = ordp (u(0)).

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(5.2) The set S1 (M ) of “Kolyvagin primes” (5.2.1) Deﬁnition. For each integer M ≥ 1, deﬁne S1 (M ) to be the set of maximal ideals ⊂ OF satisfying the following conditions: (p)c(x)dK/F ,

OK I0 ,

∈ S,

and the conjugacy class of the arithmetic Frobenius Fr()arith in the Galois group Gal(K(x)(A[pM +M0 ])/F ) coincides with the conjugacy class of the complex conjugation ρ. We also deﬁne S0 (M ) = {(1)},

Sr (M ) = {1 · · · r | i ∈ S1 (M ) distinct}

(r > 1).

(5.2.2) If (p)c(x)dK/F and ∈ S, then the ﬁeld K(x)(A[pM +M0 ]) is unramiﬁed over F at , so it makes sense to consider the conjugacy class of Fr()arith . ˇ (5.2.3) By the Cebotarev density theorem, the set S1 (M ) has positive density. (5.2.4) If ∈ S1 (M ), then is inert in K/F . In particular, S1 (M ) ⊆ S1 (hence Sr (M ) ⊆ Sr for each r ≥ 0). (5.2.5) If ∈ S1 (M ), then u(0) | (N + 1) (by Proposition 4.8(i)) and 1 − a X + (N )X 2 = det(1 − Fr()arith X | Tp A) ≡ ≡ det(1 − ρX | Tp A) = 1 − X 2 (mod pM +M0 ), which implies that the following congruences hold in OL : a ≡ 0 (mod pM +M0 ),

N + 1 ≡ 0 (mod u(0)pM ).

(5.3) From now on, we ﬁx a suﬃciently large integer M >> 0 divisible by the order of p in the ideal class group of OL , and a generator of the principal ideal pM (also to be denoted pM , by abuse of notation). (5.4) Deﬁnition. For each ∈ S1 (M ), ﬁx a generator σ of the cyclic group G() = Gal(K(x())/K(x)) and deﬁne |G()|−1

Tr =

i=0

|G()|−1

σi ,

D =

iσi ∈ Z[G()];

i=0

these elements satisfy (σ − 1)D = |G()| − Tr = (N + 1)/u(0) − Tr .

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511

For each n = 1 · · · r ∈ Sr (M ) (i ∈ S1 (M )), deﬁne Dn = D1 · · · Dr ∈ Z[G(1 )] ⊗ · · · ⊗ Z[G(r )] = Z[G(n)]. (5.5) Lemma. For each n ∈ Sr (M ), the image of the point Dn y(n) ∈ A(K(x(n)) ) in A(K(x(n)) )⊗Op /pM (which will be denoted by (Dn y(n) G(n) (mod pM ))) is contained in A(K(x(n)) ) ⊗ Op /pM . Proof. If n = 1 · · · r , then we have, for each i = 1, . . . , r, (σi − 1) Dn y(n) = = (N i + 1)/u(0) Dn/i y(n) − ai Dn/i y(n/i ) ∈ pM A(K(x(n)) ). (5.6) As pM is principal, the exact sequence of GF -modules pM

0 −→ A[pM ] −→ A(Q)−−→A(Q) −→ 0 induces, for each extension K of F , the cohomology sequence 0 −→ A(K ) ⊗ Op /pM −−→H 1 (K , A[pM ]) −→ H 1 (K , A)[pM ] −→ 0. δ

Kolyvagin’s derivative classes [c(n)] (which will be constructed in 5.7-9 below) are natural lifts of the classes δ(Dn y(n) (mod pM )) under the restriction maps (1) res : H 1 (K(x), A[pM ]) −→ H 1 (K(x(n)) , A[pM ])G(n)

(5.6.1)

(of course, there is nothing to do if n = (1), in which case one deﬁnes [c((1))] = δ(y((1))) = δ(ι(x) ⊗ 1)). (5.7) In order to construct the cohomology classes [c(n)] ∈ H 1 (K(x), A[pM ]) (where n = 1 · · · r ∈ Sr ), we ﬁx a point z(n) ∈ A(Q) satisfying pM z(n) = Dn y(n) ∈ A(K(x(n)) ). The cohomology class δ(Dn y(n) (mod pM )) ∈ H 1 (K(x(n)) , A[pM ]) is then represented by the 1-cocycle (g → (g − 1)z(n)) ∈ Z 1 (Gal(Q/K(x(n)) ), A[pM ]). (1)

As the kernel (resp., cokernel) of the map res is equal (resp., injects) to H i (G(n), A(K(x(n)) )[pM ]) for i =1 (resp., i = 2) and the torsion submodule A(K[∞])tors over K[∞] = K[c] is ﬁnite (cf. [Ne-Sch, 2.2]), one sees directly, without any calculation, that such natural lifts exist if we multiply the points y(n) by the square of any ﬁxed element of OL − {0} which annihilates A(K[∞])tors .

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Note that the formula g → (g−1)z(n) makes sense for g ∈ Gal(Q/K(x)); it deﬁnes a 1-cocycle in Z 1 (Gal(Q/K(x)), A(Q)). Denote by pr : Gal(Q/K(x)) −→ Gal(K(x(n)) /K(x)) = G(n) the natural surjection. We are going to deﬁne a 1-cocycle d (n) ∈ Z 1 (G(n), A(K(x(n)) )) such that the 1-cocycle c(n) : g → (inf(d (n)))(g) + (g − 1)z(n) =

= d (n)(pr(g)) + (g − 1)z(n) ∈ Z 1 (Gal(Q/K(x), A(Q)) will have values in A[pM ]. Its cohomology class [c(n)] ∈ H 1 (K(x), A[pM ]) will then satisfy res([c(n)]) = δ(Dn y(n) (mod pM )). In order to construct d (n), we apply the following Lemma to G = G(n), Gi = G(i ), σi = σi , X = A(K(x(n)) ). (5.8) Lemma. Let G = G1 ×· · ·×Gr be a ﬁnite product of ﬁnite cyclic groups Gi . For each G-module X, the evaluation of 1-cocycles at ﬁxed generators σi ∈ Gi induces an isomorphism of abelian groups ! % ! (∀i, j) (σi − 1)xj = (σj − 1)xi ! Z (G, X) −→ (x1 , . . . , xr ) ∈ X ! ! (∀i) (1 + σ + · · · + σ |Gi |−1 )x = 0 i i i #

∼

1

r

z → (z(σ1 ), . . . , z(σr )). Proof. Easy exercise. (5.9) Proposition-Deﬁnition. If n = 1 · · · r ∈ Sr (M ) (r ≥ 1, i ∈ S1 (M )), then: (i) The elements xi = Dn/i

ai N i + 1 y(n/i ) − y(n) M p u(0)pM

∈ A(K(x(n)) ) (i = 1, . . . , r)

satisfy (∀i, j = 1, . . . , r)

Tri xi = 0,

(σi − 1)xj = (σj − 1)xi .

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(ii) There exists a unique 1-cocycle d (n) ∈ Z 1 (G(n), A(K(x(n)) ) satisfying (∀i = 1, . . . , r) d (n)(σi ) = xi . (iii) The cohomology class of d (n) lies in H 1 (G(n), A(K(x(n)) ))[pM ]; more precisely, we have (∀i = 1, . . . , r)

pM d (n)(σi ) = −(σi − 1)Dn y(n).

(iv) Deﬁne d(n) = inf(d (n)) ∈ Z 1 (Gal(Q/K(x)), A(Q)), i.e. d(n)(g) = d (n)(pr(g)). (v) Deﬁne d((1)) = 0 ∈ Z 1 (Gal(Q/K(x)), A(Q)). Proof. (i) Firstly, the congruences 5.2.5 show that ai /pM and (N i + 1)/u(0)pM are contained in OL , so the deﬁnition of xi makes sense. Secondly, the relation 4.13(i) together with 4.10(i) imply that we have ai N i + 1 Tri xi = Dn/i |G(i )| − a y(n/i ) = 0 pM u(0)pM i and, if i = j, a j N i + 1 N j + 1 (σi − 1)xj = Dn/i j − Tri y(n/j ) − y(n) = u(0) pM u(0)pM (N i + 1)aj (N i + 1)(N j + 1) = Dn/i j − y(n) + y(n/j )+ u(0)2 pM u(0)pM ai aj (N j + 1)ai + y(n/i ) − M y(n/i j ) = (σj − 1)xi . u(0)pM p (ii) This follows from (i) and Lemma 5.8. (iii) We compute, using again 4.10(i) and 4.13(i), that pM d (n)(σi ) = Dn/i (ai y(n/i ) − (N i + 1)/u(0) y(n)) = = −Dn/i (σi − 1)Di y(n) = −(σi − 1)Dn y(n). As d (n) is a 1-cocycle, it follows that pM d (n) is a coboundary. (5.10) Proposition-Deﬁnition. Let n = 1 · · · r ∈ Sr (M ) (r ≥ 0, i ∈ S1 (M )). The 1-cocycle c(n) ∈ Z 1 (Gal(Q/K(x)), A(Q)), deﬁned by the formula c(n) : g → d(n)(g) + (g − 1)z(n), has the following properties: (i) c(n) ∈ Z 1 (Gal(Q/K(x)), A[pM ]).

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(ii) The cohomology class [c(n)] ∈ H 1 (K(x), A[pM ]) does not depend on the choice of z(n). (iii) The image of [c(n)] under the restriction map (5.6.1) is equal to res([c(n)]) = δ(Dn y(n) (mod pM )) ∈ H 1 (K(x(n)) , A[pM ]). (iv) The image of [c(n)] in H 1 (K(x), A(Q)) is equal to [d(n)]. Proof. (i) As c(n) is a 1-cocycle, it is suﬃcient to check that pM c(n)(g) = 0, for each element of Ker(pr) = Gal(Q/K(x(n)) ) and for each g ∈ pr−1 (σi ) (i = 1, . . . , r). If g ∈ Ker(pr), then pM c(n)(g) = (g − 1)Dn y(n) = 0. If pr(g) = σi , then we have, by Proposition 5.9(iii), pM c(n)(g) = −(σi − 1)Dn y(n) + (g − 1)Dn y(n) = 0. (ii) Two choices of z(n) diﬀer by an element of A[pM ]. (iii) For each element g ∈ Gal(Q/K(x(n)) ), we have (res(c(n))(g) = (g − 1)z(n) = δ(Dn y(n) (mod pM ))(g). (iv) The cohomology class of (g − 1)z(n) vanishes in H 1 (K(x), A(Q)). (5.11) We now investigate the localizations [c(n)v ] ∈ H 1 (K(x)v , A[pM ]), [d(n)v ] ∈ H 1 (K(x)v , A(Q))[pM ] = H 1 (K(x)v , A)[pM ] of the cohomology classes [c(n)] and [d(n)] at various primes v of K(x). (5.12) Proposition (Localization outside n). Let v be a nonarchimedean prime of K(x) lying above a prime vF of F . Denote by v the special ﬁbre at v of the N´eron model of A ⊗F K(x) over OK(x) , A v ) the OL -module of the connected components of A v , and deﬁne by π0 (A v )p = 0}. C1,v (p) = min{c ≥ 0 | pc · π0 (A If vF n, then

pC1,v (p) · [d(n)v ] = 0.

In particular, if vF n and v ∈ S, then [d(n)v ] = 0. Proof. If vF n, then v is unramiﬁed in K(x(n)) /K(x), hence the cohomology class [d(n)v ] is inﬂated from an element of the cohomology group

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1 v ))[pM ], where Hur (K(x)v , A)[pM ], which is isomorphic to H 1 (κ(v), π0 (A κ(v) denotes the residue ﬁeld of v ([Mi 1, I.3.8]). This implies that the anv )p kills [d(n)v ]. If, in addition, v ∈ S, then A has good nihilator of π0 (A v ) = 0. reduction at vF , A ⊗F K(x) has good reduction at v and π0 (A

(5.13) If ∈ S1 (M ), ﬁx primes w | v | λ | in the ﬁelds K(x()) ⊃ K(x) ⊃ K ⊃ F, respectively (of course, λ is unique and w is uniquely determined by v, by Proposition 4.6). We denote by Fr(v) = Fr(v)geom etc. the various geometric Frobenius elements. 2

The assumption ∈ S1 (M ) implies that Fr(v) = Fr(λ) = Fr() acts trivially on A[pM ], hence the evaluation map evFr(v)

1 Hur (K(x)v , A[pM ])=Hom(Fr(v), A(K(x)v )[pM ])−−−−→A(K(x)v )[pM ] f → f (Fr(v))

is an isomorphism. (5.14) Proposition. If ∈ S1 (M ) and if v | is a prime of K(x) above , then all maps in the following diagram are isomorphisms and the diagram is commutative. A(K(x)v ) ⊗OL OL /pM

δv

/ H 1 (K(x)v , A[pM ]) ur

redv

v (κ(v)) ⊗O OL /pM A L

evFr(v)

/ A(K(x)v )[pM ] redv

((N +1)−a Fr())/p

M

/A v (κ(v))[pM ]

Proof. The map δv is an isomorphism, as A has good reduction at and v p. The evaluation map evFr(v) is an isomorphism, by 5.13. Both vertical maps are isomorphisms, as the kernel of the surjective reducv (κ(v)) is a pro-char(κ(v))-group and tion map redv : A(K(x)v ) −→ A v p. It remains to show that the diagram is commutative (which will imply that the bottom horizontal map is also an isomorphism). As the composite map f : A(K(x)v )[p∞ ] → A(K(x)v ) A(K(x)v ) ⊗OL OL /pM is surjective, it is suﬃcient to compute the value of evFr(v) ◦ δv ◦ f (P ) for P ∈ A(K(x)v )[p∞ ]. If we ﬁx a point Q ∈ A(K(x)v ) such that pM Q = P ,

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then 2

evFr(v) ◦ δv ◦ f (P ) = (Fr(v) − 1)(Q) = (Fr() − 1)(Q) = =

a Fr() − (N + 1) a Fr() − (N + 1) (Q) = (P ) = N pM N (N + 1) − a Fr() = (P ), pM

as

Q ∈ A(K(x)v )[p∞ ]

N ≡ −1 (mod pM ), and

−2

Fr()

− a Fr()

−1

+ N = 0

on

Tp (A).

(5.15) In the situation of 5.13 we identify the OL /pM -modules H 1 (K(x)v , A[pM ]) H 1 (K(x)v , A[pM ]) ∼ = −→ H 1 (K(x)v , A)[pM ]. 1 M Hur (K(x)v , A[p ]) Im(δv ) The evaluation at the ﬁxed generator σ ∈ G() deﬁnes an isomorphism evσ :

H 1 (K(x)v , A[pM ]) ∼ M Fr(v)=1 −→ H 1 (K(x)ur = v , A[p ]) 1 (K(x) , A[pM ]) Hur v ∼

Hom(Ivt , A[pM ])Fr(v)=1 = Hom(Ivt , A[pM ]) ←− Hom(G(), A(K(x)v )[pM ]) evσ M −−→A(K(x) v )[p ] (where G() is identiﬁed with the quotient Gal(K(x())w K(x)ur v / t −1 K(x)ur ) of the tame inertia group I at v). We denote by Φ = ev v v v σ ◦ evFr(v) the isomorphism deﬁned by the following commutative diagram: H 1 (K(x)v ,A[pM ]) 1 (K(x) ,A[pM ]) Hur v

Φ

v 1 Hur (K(x)v , A[pM ]) −→ ⏐ ⏐evFr(v)

A(K(x)v )[pM ]

⏐ ⏐evσ

A(K(x)v )[pM ]

==

As N ≡ −1 (mod pM ), the conjugation action of Fr() on Ivt ⊗ OL /pM is given by multiplication by −1. In particular, Φv ◦ Fr() = −Fr() ◦ Φv .

(5.15.1)

(5.16) Proposition. If n ∈ Sr+1 (M ) (n ∈ Sr (M ), ∈ S1 (M ), r ≥ 0) and if v | is a prime of K(x) above , then the localization ∼

[d(n)v ] ∈ H 1 (K(x)v , A)[pM ] ←−

H 1 (K(x)v , A[pM ]) 1 (K(x) , A[pM ]) Hur v

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is equal to [d(n)v ] = −Φv (Fr() [c(n)v ]) . Proof. The commutative diagram red

w H 1 (K(x())w /K(x)v , A(K(x())w ))[pM ] −→ ⏐ ⏐

v (κ(w)))[pM ] H 1 (G(), A !

ur M H 1 (K(x())ur w /K(x)v , A(K(x)v ))[p ] ⏐ ⏐

v (κ(v))[pM ]) Hom(G(), A & ⏐red ⏐ v

M H 1 (K(x)ur v , A(K(x)v ))[p ] & ⏐ ⏐

Hom(G(), A(K(x)v )[pM ]) ⏐ ⏐

M H 1 (K(x)ur v , A[p ])

==

Hom(Ivt , A(K(x)v )[pM ])

ur (in which K(x())ur w = K(x())w K(x)v ) implies that

redv ◦ evσ ([d(n)v ]) = redv ([d (n)v ](σ )) = a N + 1 4.12(ii) = redv Dn y(n) − y(n) == pM u(0)pM a (N + 1)Fr() = Dn − redv y(n) = pM pM a Fr() − (N + 1) 5.14 = Fr() redv (Dn y(n)) == pM = −Fr() redv evFr(v) ◦ δv (Dn y(n) (mod pM )) 2

(where we have used the fact that Fr() = Fr(λ) = Fr(v) acts trivially on κ(v) and K(x)v ). As the reduction map redv : A(K(x)v )[pM ] −→ v (κ(v))[pM ] is an isomorphism, it follows that A [d(n)v ] = −ev−1 σ ◦ evFr(v) (Fr() [c(n)v ]) = −Φv (Fr() [c(n)v ]) . (5.17) If ∈ S1 (M ), ﬁx a prime vα of K(α) above . As OK splits completely in K(x)/K (by Proposition 4.6(ii)), we have, for each prime v | vα of K(x), natural identiﬁcations K(α)vα = K(x)v and isomorphisms ∼

1 evFr(vα ) : Hur (K(α)vα , A[pM ]) −→ A(K(α)vα )[pM ] ∼

1 evσ : H 1 (K(α)vα , A[pM ])/Hur (K(α)vα , A[pM ]) −→ A(K(α)vα )[pM ] ∼

1 M Φvα = ev−1 σ ◦ ev Fr(vα ) : Hur (K(α)vα , A[p ]) −→ ∼

1 −→ H 1 (K(α)vα , A[pM ])/Hur (K(α)vα , A[pM ]).

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(5.18) Proposition. If n ∈ Sr+1 (M ) (n ∈ Sr (M ), ∈ S1 (M ), r ≥ 0) and if vα | is a prime of K(α) above , then the localization '

corK(x)/K(α) d(n)

( vα

∼

∈ H 1 (K(α)vα , A)[pM ] ←−

H 1 (K(α)vα , A[pM ]) 1 (K(α) , A[pM ]) Hur vα

is equal to ' ' ( ( corK(x)/K(α) d(n) v = −Φv Fr() corK(x)/K(α) c(n) v . α

α

Proof. This follows from Proposition 5.16 and the fact that (∀X = A, A[pM ]) (∀c ∈ H 1 (K(x), X)) corK(x)/K(α) (c) v = α = corK(x)v /K(α)vα (cv ) = cv . v|vα

v|vα

(5.19) The Weil pairings. The Weil pairing )j ) −→ Zp (1) ( , )Aj : Tp (Aj ) × Tp (A is GF -equivariant and satisﬁes (∀f ∈ EndF (Aj ) = OLj )

(f (x), y)Aj = (x, f(y))Aj .

)j deﬁned over F . As the corresponding Fix a polarization ϕ : Aj −→ A Rosati involution acts trivially on OLj (by Proposition 1.18(v)), the induced skew-symmetric pairing Tp (Aj ) × Tp (Aj ) −→ Zp (1),

x, y → (x, ϕ(y))Aj

factors through Tp (Aj ) ⊗OLj Tp (Aj ) and preserves the decomposition Tp (Aj ) =

TP (Aj ),

P |p

where P runs through all primes of Lj above p. Denote by ( , )pj : Tpj (Aj ) ⊗Opj Tpj (Aj ) −→ Zp (1) its pj -component, where pj = p ∩ OLj and Opj = OLj ,pj . −1 Fix a generator d ∈ DO p

j

∼

/Zp

of the inverse diﬀerent; the map

λ : Opj −→ HomZp (Opj , Zp ),

x → (y → TrOpj /Zp (dxy))

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is a symmetric isomorphism of Opj -modules. The composition of the maps Tpj (Aj ) −→ HomOpj (Tpj (Aj ), HomZp (Opj , Zp (1))) x → (y → (a → (x, ay)pj )) and HomOpj (Tpj (Aj ), HomZp (Opj , Zp (1))) Hom(id,λ−1 )

−−−−−−→HomOpj (Tpj (Aj ), Opj (1)) deﬁnes, by adjunction, a skew-symmetric Opj -bilinear GF -equivariant pairing ( , )j : Tpj (Aj ) × Tpj (Aj ) −→ Opj (1) characterized by the property (∀x, y ∈ Tpj (Aj ))(∀a ∈ Opj ) TrOpj /Zp (da(x, y)j ) = (x, ay)pj = (ax, y)pj . On tensoring with Op = OL,p , we obtain a skew-symmetric Op -bilinear GF -equivariant pairing on Tp (A) = Tpj (Aj ) ⊗Opj Op : ( , ) : Tp (A) × Tp (A) −→ Op (1) x ⊗ a, y ⊗ b → ab(x, y)j

(x, y ∈ Tpj (Aj ); a, b ∈ Op ).

The adjoint map Tp (A) −→ HomOp (Tp (A), Op (1)) is injective and its cokernel is killed by deg(ϕ). Reducing ( , ) modulo pM yields a skewsymmetric Op /pM -bilinear GF -equivariant pairing ( , )M : A[pM ] × A[pM ] −→ Op /pM (1),

(5.19.1)

whose kernel is killed by deg(ϕ). This implies that, for each Op submodule Y ⊂ A[pM ] isomorphic to Op /pM −c , we have 2 deg(ϕ) pc Y ⊥ ⊆ Y.

(5.19.2)

(5.20) Proposition. In the situation of 5.16, let vα be the prime of K(α) induced by v and ζv ∈ K(α)vα = K(x)v be a root of unity whose image under the reciprocity map recv : K(x)∗v −→ Gab K(x)v induces σ ∈ 1 Gal(K(x())w /K(x)v ). Then we have, for each x ∈ Hur (K(α)vα , A[pM ]), ζv ⊗ invvα x ∪ corK(x)/K(α) c(n) v = α = − evFr(vα ) (x), Fr() evFr(vα ) corK(x)/K(α) c(n) v ∈ Op /pM (1), α

where ∪ denotes the cup product

M

∪

H 1 (K(α)vα , A[pM ]) × H 1 (K(α)vα , A[pM ])−−→H 2 (K(α)vα , Op /pM (1)) invvα −−−−→ Op /pM induced by the pairing ( , )M . Proof. Combine Proposition 5.18 with Lemma 5.21 below (using the identiﬁcation K(α)vα = Kλ , where λ = OK ).

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(5.21) Lemma. Let ∈ S1 (M ), λ = OK . If ζ ∈ Kλ∗ is a root of unity 1 of order prime to N , then we have, for each x ∈ Hur (Kλ , A[pM ]) and 1 M y ∈ H (Kλ , A[p ]), ζ ⊗ invλ (x ∪ y) = (x(Fr(λ)), y(recλ (ζ)))M ∈ Op /pM (1), where recλ : Kλ∗ −→ Gab Kλ denotes the reciprocity map (normalized in such a way that the geometric Frobenius element Fr(λ) corresponds to a uniformizer). Proof. We can assume that ζ is a primitive root of unity of order n = ∼ N λ − 1. Denoting by δ : Kλ∗ ⊗ Z/nZ −→ H 1 (Kλ , μn ) the standard Kummer map, then we have 1 1 x = δζ ⊗ a ⊗ ζ ⊗−1 ∈ Hur (Kλ , μn ) ⊗ A[pM ] ⊗ μ⊗−1 = Hur (Kλ , A[pM ]) n

y = δu ⊗ b ⊗ ζ ⊗−1 ∈ H 1 (Kλ , μn ) ⊗ A[pM ] ⊗ μ⊗−1 = H 1 (Kλ , A[pM ]) n for some a, b ∈ A[pM ] and u ∈ Kλ∗ (recall that GKλ acts trivially on A[pM ], since ∈ S1 (M )). As the reciprocity map is normalized by letting the uniformizers correspond to geometric Frobenius elements, x(Fr(λ)) = (δζ)(Fr(λ)) ⊗ a ⊗ ζ ⊗−1 = ζ −1 ⊗ a ⊗ ζ ⊗−1 = −a. For each character χ ∈ Homcont (GKλ , Q/Z) = H 1 (GKλ , Q/Z), we have ([Se 1, Prop. XI.2]) (∀z ∈ Kλ∗ )

χ(recλ (z)) = invλ (z ∪δχ) = −invλ (δz ∪χ) = invλ (χ∪δz), (5.21.1) where δχ ∈ H 2 (GKλ , Z) denotes the coboundary associated to the exact sequence 0 −→ Z −→ Q −→ Q/Z −→ 0. Using (5.21.1) with z = ζ and z = u, we obtain y(recλ (ζ)) = (δu)(recλ (ζ)) ⊗ b ⊗ ζ ⊗−1 = −(δζ)(recλ (u)) ⊗ b ⊗ ζ ⊗−1 (= ordλ (u)b) ζ ⊗invλ (x∪y) = invλ (δζ ∪δu) (a, b)M ⊗ζ ⊗−1 = (δζ)(recλ (u)) (a, b)M ⊗ζ ⊗−1 = (x(Fr(λ)), y(recλ (ζ)))M . 6. Linear algebra In order to simplify the notation, we denote H = K(α),

HM = H(A[pM +M0 ]),

T = Tp (A),

Δ = Gal(H/K),

V = Tp (A) ⊗Op Lp , ∼

UM = Gal(HM /H) ⊆ AutOp (A[pM +M0 ]) −→ GL2 (Op /pM +M0 ).

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(6.1) The constant C2 (p) (6.1.1) Proposition. (i) (Bogomolov [Bo]) The group Zp (A/H) := Z∗p ∩ Im GH −→ AutZp (Tp (A) is an open subgroup of Z∗p . (ii) (Serre [Se 3]) There exists a constant c(A/H) such that the inequality (Z∗p : Zp (A/H)) ≤ c(A/H) holds for all prime numbers p. (6.1.2) Proposition. Consider the restriction map M res : H 1 (H, A[pM ]) −→ H 1 (HM , A[pM ])UM = HomUM (Gab HM , A[p ]).

There exists an integer C2 (p) ≥ 0 which is equal to zero for all but ﬁnitely many p and such that pC2 (p) Ker(res) = 0 for all M . Proof. Proposition 6.1.1(i) implies that there exists u ∈ Z∗p − {1} such that (∀M ≥ 0) u (mod pM +M0 ) ∈ Z(UM ). It follows from “Sah’s Lemma” ([Sa], proof of Prop. 2.7(b)) that u − 1 kills H 1 (UM , A[pM ]) = Ker(res), hence pC2 (p) Ker(res) = 0 for C2 (p) = ordp (u−1). According to Proposition 6.1.1(ii), we can take u ≡ 1 (mod p) (hence C2 (p) = 0) if p − 1 > c(A/H). (6.2) The constant C3 (p) (6.2.1) Proposition. The following conditions are equivalent. (i) V is an absolutely irreducible representation of GH . (ii) EndLp [GH ] (V ) = Lp . (iii) For every non-trivial character η : Gal(H/F ) −→ {±1}, the Lp [GF ]-modules V and V ⊗ η are not isomorphic. (iv) For every totally imaginary quadratic extension K of F contained in H, the abelian variety Aj does not acquire complex multiplication over K . (v) For every K as in (iv), every totally imaginary quadratic extension Kj of Lj , every algebraic Hecke character ψ : A∗K −→ Kj∗ and every embedding τ : Kj → C, the L-series L(ψτ , s) (where ψτ : A∗K /K ∗ −→ C∗ is the id`ele class character asociated to ψ at τ ) is not equal to any of the L-series L(π(σ ◦ θj ), s − 1/2) from Proposition 1.18. Proof. If F /F is any quadratic extension, denote by ηF /F : GF −→ {±1} the associated quadratic character with kernel Ker(ηF /F ) = GF . (i) ⇐⇒ (ii): Irreducibility of V implies that its restriction to GH is semi-simple. The equivalence of (i) and (ii) is then well-known ([Cu-Re, Thm. 3.43]). (ii) =⇒ (iii): According to [Tay], Prop. 3.1 (and using the fact that each complex conjugation acts on V (f ) by a matrix with distinct eigenvalues

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±1 ∈ Lp ), V is an absolutely irreducible representation of GF , hence EndLp [GF ] (V ) = Lp , and both conditions (ii) and (iii) are invariant under ﬁnite extensions of the ﬁeld Lp . We can assume, therefore, that Lp contains all roots of unity of order [H : K]. The Frobenius reciprocity yields

GK F EndLp [GK ] (V ) = HomLp [GF ] V, IndG = GK ResGF (V ) = HomLp [GF ] V, V ⊕ (V ⊗ ηK/F ) = Lp ⊕ HomLp [GF ] V, V ⊗ ηK/F . (6.2.1.1) Irreducibility of V implies that any non-trivial element of HomLp [GF ] ∼ (V, V ⊗ ηK/F ) is an isomorphism of Lp [GF ]-modules V −→ V ⊗ ηK/F , which proves the equivalence (ii) ⇐⇒ (iii) in the case H = K. Similarly,

GH K EndLp [GH ] (V ) = HomLp [GK ] V, IndG = GH ResGK (V )

= HomLp [GK ] V, V ⊗χ = HomLp [GK ] (V, V ⊗ χ) , χ∈Δ χ∈Δ (6.2.1.2) ∗ where Δ = Hom(Δ, Lp ). Assume that there exists an isomorphism of ∼ Lp [GF ]-modules V −→ V ⊗ η, for a non-trivial character η : Gal(H/F ) −→ {±1}. If η = ηK/F , then (6.2.1.1) implies that EndLp [GK ] (V ) = Lp , hence EndLp [GH ] (V ) = Lp . If η = ηK/F , then is non-trivial, hence EndL [G ] (V ) = Lp , by the character χ := η|Δ ∈ Δ p H (6.2.1.2). (iii) =⇒ (ii): Assume that EndLp [GH ] (V ) = Lp . If EndLp [GK ] (V ) = Lp , ∼ then V −→ V ⊗ ηK/F , by (6.2.1.1). If EndLp [GK ] (V ) = Lp , then V is an absolutely irreducible representation of GK and, thanks to (6.2.1.2), and a non-trivial morphism there exists a non-trivial character χ ∈ Δ of Lp [GK ]-modules V −→ V ⊗ χ, which is necessarily an isomorphism. As det(V ) = det(V )χ2 , the values of χ are contained in {±1}. It follows that χ extends to a character χF : Gal(H/F ) −→ {±1} (χF = 1, ηK/F , since χ = 1). Applying again the Frobenius reciprocity 0 = HomLp [GK ] (V ⊗ χ, V )

GK F = HomLp [GF ] V ⊗ χF , IndG = GK ResGF (V ) = HomLp [GF ] V ⊗ χF , V ⊕ (V ⊗ ηK/F ) , we deduce that there exists a character η ∈ {χF , χF ηK/F } (hence η = 1) and a morphism of Lp [GF ]-modules V −→ V ⊗ η, which is necessarily an isomorphism.

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(ii) =⇒ (iv): If Aj acquires complex multiplication over K as in (iv), then Im Lp [GK ] −→ EndLp (V ) is a commutative subalgebra of ∼ EndLp (V ) −→ M2 (Lp ), hence EndLp [GH ] (V ) ⊇ EndLp [GK ] (V ) Lp . (v) =⇒ (iv): This follows from the Main Theorem of Complex Multiplication ([Sh-Ta], [Mi 2, I.5]). (iv) =⇒ (v): If the L-function L(π(σ ◦ θj ), s − 1/2) is given by a Hecke character of K , then Im(Lp [GK ] −→ EndLp (V )) is a commutative sub∼ algebra of EndLp (V ) −→ M2 (Lp ). Faltings’ isogeny theorem then implies that EndK (Aj ) ⊗ Q contains a commutative subalgebra bigger than Lj , hence Aj acquires CM over K . (v) =⇒ (iii): Assume that η : Gal(H/F ) −→ {±1} is a non-trivial character such that the Lp [GF ]-modules V and V ⊗ η are isomorphic. Then K = H Ker(η) is a quadratic extension of F and EndLp [GK ] (V ) = Lp ⊕ Lp , hence V is a semi-simple L-rational abelian representation of GK with inﬁnite image. This implies, according to [He, Thm. 2] (see also [Se 2, III.2.3, Thm. 2]) that K is totally imaginary and GK acts ∗ on V ⊗Lp Lp by ψp ⊕ ψp ◦ ρ, where ψp : GK −→ Lp is the Galois representation associated to an algebraic Hecke character ψ of K (cf. [Ne], proof of 12.6.5.2), hence Aj acquires complex multiplication over K .

(6.2.2) Proposition. If the equivalent conditions (i)-(v) of Proposition 6.2.1 hold, then there exists an integer C3 (p) ≥ 0, which is equal to zero for all but ﬁnitely many p and which satisﬁes Im Op [GH ] −→ EndOp (T ) ⊇ pC3 (p) EndOp (T ). Proof. Im Op [GH ] −→ EndOp (T ) is an Op -lattice in the Lp -algebra A = Im Lp [GH ] −→ EndLp (V ) ⊆ EndLp (V ). As V is a faithful simple A-module satisfying EndA (V ) = EndLp [GH ] (V ) = Lp , a theorem of Burnside ([Cu-Re, Thm 3.32]) implies that A = EndLp (V ), which proves the existence of C3 (p). It remains to show that, for all but ﬁnitely many p, the map Op [GH ] −→ EndOp (T ) is surjective. By the Nakayama Lemma, this amounts to the surjectivity of k[GH ] −→ Endk (T )

(k = Op /p, T = T /pT ),

H which would follow from absolute irreducibility of ResG GF (T ). We can assume that p [H : F ] (hence p = 2) and, after replacing L by a ﬁnite extension, that L contains all roots of unity of order [H : K] (hence k does, too). According to [Di, Prop. 3.1], for all but ﬁnitely many p, T

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is an absolutely irreducible k[GF ]-module, hence Endk[GF ] (T ) = k. For such p, T is a semi-simple k[GH ]-module, by Cliﬀord’s Theorem ([CuRe, Thm. 11.1(i)]; this applies to an arbitrary group G and its normal subgroup H of ﬁnite index), hence H ResG GF (T ) is absolutely irreducible

⇐⇒ Endk[GH ] (T ) = k.

The same argument as in the proof of 6.2.1 (ii) ⇐⇒ (iii) shows that # % ∃η : Gal(H/F ) −→ {±1}, η = 1 Endk[GH ] (T ) = k ⇐⇒ . ∼ T −→ T ⊗ η as k[GF ] − modules If this were true for inﬁnitely many p’s, we could ﬁnd a common nontrivial character η : Gal(H/F ) −→ {±1} for which the congruences (∀g ∈ GF )

Tr(g | T ) ≡ Tr(g | T ⊗ η) (mod p)

held for inﬁnitely many p, hence (∀g ∈ GF )

Tr(g | V ) = Tr(g | V ⊗ η), ∼

which would imply that V −→ V ⊗ η (by irreducibility of V ). This contradiction with 6.2.1(iii) shows that C3 (p) = 0 for all but ﬁnitely many p. (6.3) Galois groups (6.3.1) Let W ⊂ H 1 (H, A[pM ]) be an Op /pM -submodule of ﬁnite type. M Denote by W = res(W ) ⊂ HomUM (Gab HM , A[p ]) its image under the ⊥ restriction map from Proposition 6.1.2, by W ⊂ Gab HM the annihilator ab W ⊥ of W , and set HM (W ) := (HM ) . The natural injective map G := Gal(HM (W )/H) → HomOp (W, A[pM ]) g → (evg : w → w(g)) is a morphism of Z[UM ]-modules, where the group UM = Gal(HM /H) ⊂ AutOp (A[pM +M0 ]) acts trivially on W and by conjugation on G. Denote by X = Op · G = Op · Gal(HM (W )/H) ⊂ HomOp (W, A[pM ]) the Op /pM -submodule of HomOp (W, A[pM ]) generated by the image of G and by j : X → HomOp (W, A[pM ]) the inclusion map. By construction, j is Op [UM ]-linear and the natural map W −→ HomOp (X, A[pM ]) is injective.

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(6.3.2) Proposition. If the equivalent conditions of Proposition 6.2.1 are satisﬁed, then (∀M, W )

pC3 (p) Coker(j) = 0.

Proof. This is a special case of Proposition 6.4.3 below, for B = Im Lp [GH ] −→ EndLp (V ) = EndLp (V ), D = Lp , C3 (p) Λ = Im Op [GH ] −→ EndOp (T ) ⊇ p EndOp (T ), R = Op . (6.3.3) Corollary. If the equivalent conditions of Proposition 6.2.1 are satisﬁed, then the map Hom(res,id)

j

j : X −−→HomOp (W, A[pM ])−−−−−−−−→HomOp (W , A[pM ]) satisﬁes pC2 (p)+C3 (p) Coker(j ) = 0. Proof. It follows from Proposition 6.1.2 that Coker(Hom(res, id)) is killed by pC2 (p) . (6.3.4) Corollary. If the equivalent conditions of Proposition 6.2.1 are satisﬁed and if W is ρ-stable (where ρ ∈ GF = Gal(F /F ) denotes the complex conjugation with respect to the ﬁxed embedding F → C extending τ1 ), so are W , HM (W ) and X. The maps j, j are ρ-equivariant and, denoting (−)± = (−)ρ=±1 , we have 2X + ⊆ Op G+ ⊆ X + pC2 (p)+C3 (p) Coker j : X + → HomOp (W , A[pM ])+ = 0. Moreover, the cokernel of the map HomOp (W , A[pM ])+ → HomOp ((W )+ , A[pM ]+ )⊕ HomOp ((W )− , A[pM ]− ) is killed by 4, and the cokernel of the map j

Op ·2G+ → X + −−→HomOp ((W )+ , A[pM ]+ )⊕HomOp ((W )− , A[pM ]− ) is killed by 24 pC2 (p)+C3 (p) . (6.4) In this section we prove a general abstract version of Proposition 6.3.2.

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(6.4.1) Assume that V is a Qp -vector space of ﬁnite dimension, B ⊂ EndQp (V ) a Qp -subalgebra, Λ ⊂ B a Zp -order in B and T ⊂ V a Zp -lattice such that ΛT = T . The Qp -algebra D = EndB (V )op acts on V on the right; its action commutes with the left action of B. The ring R = {x ∈ D | T x ⊂ T } is a Zp -order in D. (6.4.2) In the situation of 6.4.1, assume that we are given the following data: • • • •

An integer N ≥ 1. A right (R/pN R)-module W = WR . A left (Λ/pN Λ)-module X = Λ X. A Zp -bilinear pairing , : X × W −→ T /pN T satisfying λx, w = λx, w x, wr = x, wr

(∀x ∈ X, ∀w ∈ W, ∀r ∈ R, ∀λ ∈ Λ)

such that the induced homomorphisms i = iW : WR −→ HomΛ

Λ X, Λ (T /p

N

T )R

j = jX : Λ X −→ HomR WR , Λ (T /pN T )R are both injective (i is a morphism of right R-modules, while j is a morphism of left Λ-modules). (6.4.3) Proposition. Assume that we are given the data 6.4.2 and that B is a semi-simple Qp -algebra. Then there exists a maximal Zp -order Rmax ⊃ R of D; ﬁxing Rmax , then Λmax := {b ∈ B | bT · Rmax ⊂ T · Rmax } is a maximal Zp -order in B containing Λ. Let c, d ∈ Z(Rmax ) be non-zero central elements in Rmax such that cΛmax ⊂ Λ and dRmax ⊂ R (they always exist). Then cd Coker(j) = 0.

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Proof. We begin by reducing to the case R = Rmax , Λ = Λmax . First of all, Rmax exists by [Cu-Re, Thm. 26.5] and Λmax is a maximal Zp -order in B, by [Cu-Re, Thm. 26.20, Thm. 26.23(iii)]. Consider the maps γ i WR −→HomΛ Λ X, Λ (T /pN T )R −→ γ −→HomΛ Λ X, Λmax (T Rmax /pN T Rmax )Rmax ; then

*R W := (γ ◦ i)(WR )Rmax max

is a right Rmax -module. Similarly, consider j Λ X −→HomR

δ WR , Λ (T /pN T )R −→

δ *R , Λ (T Rmax /pN T Rmax )R −→HomRmax W max max max and put

:= Λmax (δ ◦ j(Λ X));

Λmax X

this is a left Λmax -module and the canonical maps *R i : W −→ HomΛmax max

Λmax X, Λmax (T Rmax /p

N

T Rmax )Rmax

−→ HomR *R , Λ (T Rmax /pN T Rmax )R j : Λmax X W max max max max are injective. Assume that the statement has been proved for the maximal orders Rmax , Λmax and c = d = 1. Then j is surjective, hence

* , T Rmax /pN T Rmax , δ ◦ j(Λ X) ⊃ cΛmax (δ ◦ j)(Λ X) = c HomRmax W i.e. c kills Coker(δ ◦ j). As (T ∩ pN T Rmax )d ⊂ pN T , d kills Ker(δ) = HomR (W, (T ∩ pN T Rmax )/pN T ). The exact sequence Ker(δ) −→ Coker(j) −→ Coker(δ ◦ j) then implies that cd kills Coker(j). We can assume, therefore, that R = Rmax and Λ = Λmax . The next step is the reduction to the case of a simple Qp -algebra B. In general there is a ﬁnite decomposition B = ei B, where each ei B = Bi is a simple Qp -algebra and e are the corresponding orthogonal idempotents. i Then we have V = Vi , Vi = ei V . According to [Cu-Re, Thm. 26.20], Λ = Λi , where each Λi = ei Λ is a maximal Zp -order in Bi . As

ΛT ⊂ T , we have T = Ti with Ti = ei T , which implies that D = Di ,

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Di = EndBi (Vi )op and R = Ri , Ri = {x ∈ Di | Ti x ⊂ Ti }. Similarly, W = ei W and X = ei X. This implies that we can assume that B is a simple Qp -algebra, hence B = Mn (D) for a skew-ﬁeld D = EndB (V )op and V = Dn . In this case D has a unique maximal order R; every left or right ideal in R is m m bilateral and is of the form R R = R R (m ≥ 0), where R is a ﬁxed prime element of R ([Cu-Re, Thm. 26.23]). There is an isomorphism of right R-modules ∼

WR −→

k

ni R/R R,

i=1 ni ni −N where R | pN for each i. Using the fact that multiplication by R p ni N −ni N induces an isomorphism between T · p R /p T and T /T R , we deﬁne a left Λ-module Λ Y as the ﬁbre product of the maps

T k −→

k Λ

ni (T /T R )

i=1

and j Λ X −→HomR

k ∼ ∼ N −ni N WR , Λ (T /pN )R −→ Λ T · p R /p T −→ i=1 ∼

−→

k Λ

ni (T /T R ).

i=1

The injectivity of iW implies that Y has the following property: (∀r1 , . . . , rk ∈ R s.t. ∃i ri ∈ R∗ ) (∃y1 , . . . , yk ∈ Y )

k

yi ri ∈ T R

i=1

(k ) We shall prove, by induction on k, that any Λ-submodule Λ Y ⊂ Λ (T k )R satisfying (k ) is equal to T k . This will imply that jX is surjective, as claimed. Let k = 1. If Λ Y satisﬁes (1 ), then Y ⊂ T R . The lattice T is free over R and, in a suitable R-basis of T , Λ = Mr (R)op . As (Y + T R )/T R ⊂ T /T R is a non-zero Λ/R = Mr (R/R )-submodule of the simple Λ/R -module T /T R , we have Y + T R = T , hence Y = T by the Nakayama Lemma. Suppose that k > 1 and that any Λ Y satisfying (k−1 ) is equal to T k−1 . If Λ Y ⊂ Λ (T k )R satisﬁes (k ), then its image under the projection

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prk : T k −→ T on the k-th factor satisﬁes (1 ), hence prk (Y ) = T . As Λ is a maximal order in a semi-simple Qp -algebra, it is hereditary, i.e. T is a projective Λ-module [Cu-Re, Thm. 26.12(ii)]. It follows that the (surjective) projection prk : Y −→ T admits a Λ-linear section prk s s : T −→ Y . For each i = 1, . . . , k − 1, the map T −→Y → T k −→T is an element of EndΛ (T ) = Rop , hence is given by right multiplication by some ai ∈ R. It follows that Y = {(xa1 +y1 , . . . , xak−1 +yk−1 , x) | x ∈ T, y1 , . . . , yk−1 ∈ Y ∩ker(prk )}. (6.4.3.1) If r1 , . . . , rk−1 ∈ R and (∃i) ri ∈ R∗ , then there exist y1 , . . . , yk−1 ∈ Y ∩ ker(prk ) such that k−1 i=1

y i ri =

k−1 i=1

(xai + yi )ri − x

k−1

ai ri ∈ T R ,

i=1

since Y satisﬁes (k ). This implies that Y ∩ ker(prk ) satisﬁes (k−1 ), hence Y ∩ ker(prk ) = T k−1 by inductive hypothesis. It follows from (6.4.3.1) that Y = T k , which concludes the proof of the Proposition. (6.5) Galois groups and Frobenius elements (6.5.1) In the situation of 6.3.1, assume that W is ρ-stable. Given an arbitrary element g ∈ G+ = Gal(HM (W )/HM )+ , ˇ then h = g 2 = (ρ + 1)g ∈ 2G+ . The Cebotarev density theorem implies that there exist inﬁnitely many non-archimedean primes L (W ) of HM (W ) satisfying the following properties: (6.5.1.1) L (W ) is unramiﬁed in HM (W )/F . (6.5.1.2) L (W ) is prime to (pu(0))c(x)N (I0 ) and the prime of F induced by L (W ) does not lie in S. (6.5.1.3) FrHM (W )/F (L (W )) = ρg ∈ Gal(HM (W )/F ). Denote by , λ, λH and L , respectively, the primes of F , K, H and HM induced by L (W ). (6.5.2) Lemma. (i) ∈ S1 (M ). (ii) The prime λ = OK splits completely in HM /K. (iii) FrHM (W )/K (L (W )) = (ρg)2 = ρgρg = ρ g g = (ρ + 1)g = g 2 = h ∈ 2G+ . (iv) The decomposition group of L in HM /F is equal to Gal(HM /F )L = {1, ρ}. In particular, ρ(L ) = L . Proof. The statements (i) and (ii) follow from (6.5.1.2) and the equalities FrHM /F (L ) = ρg|HM = ρ, FrHM /K (L ) = FrHM /F (L )2 = ρ2 = 1. The

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statement (iii) is clear and (iv) follows from the fact that Gal(HM /F )L is generated by FrHM /F (L ) = ρ. (6.5.3) In the situation of 6.5.1, the element Fr(L ) := FrHM (W )/HM (L (W )) ∈ G depends only on L ; more precisely, Fr(L ) = g 2 = h ∈ 2G+ , by Lemma 6.5.2(ii). Its image via the map j

j : G+ → X + −→HomOp (W, A[pM ])+ −→ HomOp (W , A[pM ])+ is given by the evaluation map j(Fr(L ))(w ) = res(w )(Fr(L ))

(w ∈ W )

(6.5.3.1)

(6.5.4) The prime L determines identiﬁcations A(HλH )[pM ] = A((HM )L )[pM ] = A(HM )[pM ] = A(F )[pM ] (=: A[pM ]). The image of W ⊂ H 1 (H, A[pM ]) under the localization map resλH : H 1 (H, A[pM ]) −→ H 1 (HλH , A[pM ]) 1 is contained in Hur (HλH , A[pM ]) and the composite map resλ

evFr(λ

H H 1 M M W −−−−→H ur (HλH , A[p ])−−−−→A[p ] )

coincides with the map (6.5.3.1) j(Fr(L )) : w → res(w )(Fr(L )). We shall use repeatedly the fact that this map is ρ-equivariant (= is contained in HomOp (W , A[pM ])+ ). In order to stress the dependence on L , we shall write wL := resλH (w )

(w ∈ W ).

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(6.6) Linear and quadratic forms (6.6.1) Lemma. Let f (x) =

n

f i xi ,

g(x) =

i=1

gij xi xj

1≤i≤j≤n n

n N fi , gij ∈ Op /p , x = xi ei ∈ i=1

Op ei = Op⊕n

i=1

be a linear and a quadratic form, respectively, in n ≥ 1 variables, with n ⊕n coeﬃcients in Op /pN . Writing Z⊕n = Z p i=1 p ei ⊂ Op , we have, for each m ∈ {0, . . . , N }: m N m N (i) f (Z⊕n p ) ⊆ p (Op /p ) ⇐⇒ (∀i = 1, . . . , n) fi ∈ p (Op /p ) ⇐⇒ ⊕n m N f (Op ) ⊆ p (Op /p ). m N (ii) g(Z⊕n ⇐⇒ (∀1 ≤ i ≤ j ≤ n) gij ∈ p ) ⊆ p (Op /p ) ⊕n m N p (Op /p ) ⇐⇒ g(Op ) ⊆ pm (Op /pN ). m N (iii) If U Z⊕n is a proper Zp -submodule and g(Z⊕n p p ) ⊂ p (Op /p ) (1 ≤ m < N − ordp (2)), then (∃x ∈ Z⊕n p , x ∈ U )

g(x) ∈ 2 pm (Op /pN ) = pm+ordp (2) (Op /pN ).

Proof. The statements (i) and (ii) follow from the equalities fi = f (ei ), gii = g(ei ), gij = g(ei + ej ) − g(ei ) − g(ej ) (i < j). (iii) There exists an integer r ≥ 1 and a Zp -basis e1 , . . . , en of Z⊕n such p that r n U⊆ pZp ei ⊕ Zp ej . Writing x = g(x) =

n

i=1

i=1

xi e i =

n

j=r+1

i=1 xi ei ,

we have

gij xi xj , gii = g(ei ), gij = g(ei +ej )−g(ei )−g(ej ) (i < j).

1≤i≤j≤n

It follows from (ii) that (∃k ≤ l)

gkl ∈ pm (Op /pN ).

We distinguish three cases: (1) k ≤ l ≤ r: if k = l, then x := ek ∈ U and g(x) = gkk ∈ pm (Op /pN ). If k < l, m N then g(ek + el ) − g(ek ) − g(el ) ∈ p (Op /p ), hence there exists x ∈ {ek , el , ek + el } (=⇒ x ∈ U ) such that g(x) ∈ pm (Op /pN ).

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(2) k ≤ r < l and (∀i ≤ j ≤ r) the congruence

gij ∈ pm (Op /pN ):

(∀λ ∈ Zp ) g(ek +λel ) = gkk +λgkl +λ2 gll ≡ λgkl +λ2 gll (mod pm (Op /pN ))

implies that there exists λ ∈ {1, 2} such that x := ek + λel ∈ U satisﬁes g(x) ∈ 2 pm (Op /pN ). (3) (∀i ≤ r) (∀j ≥ i) gij ∈ pm (Op /pN ): in this case k > r and we have

∀y ∈

n

Op ej

g(e1 + y) ≡ g(y) (mod pm (Op /pN )).

j=r+1 If k = l, then x := e1 + ek ∈ U satisﬁes g(x) ≡ gkk ≡ 0 (mod pm (Op /pN )). If k < l, then

g(e1 + ek + el ) − g(e1 + ek ) − g(e1 + el ) ≡ g(ek + el ) − g(ek ) − g(el ) ≡ ≡ gkl ≡ 0 (mod pm (Op /pN )),

hence there exists x ∈ {e1 + ek + el , e1 + ek , e1 + el } (=⇒ x ∈ U ) such that g(x) ∈ pm (Op /pN ). (6.6.2) Proposition. Let a, b, N ≥ 0 be integers such that a, 2b + ordp (2) < N . Let Z ⊂ (Op /pN )⊕5 be a Zp -submodule satisfying Op · ⊕4 Z ⊇ pa (Op /pN ) ⊕ pb (Op /pN ) . Then there exists an element z = (z0 , . . . , z4 ) ∈ Z (zi ∈ Op /pN ) such that Op (Op /pN )/(Op /pN )z0 ≤ a Op (Op /pN )⊕2 /(Op /pN )(z1 , z2 ) + (Op /pN )(z3 , z4 ) ≤ 2b + ordp (2) (where Op (C) denotes the length of a ﬁnite Op -module C), hence (Op /pN )z0 ⊇ pa (Op /pN ), (Op /pN )(z1 , z2 ) + (Op /pN )(z3 , z4 ) ⊇ 2 p2b (Op /pN )⊕2 . (i)

(i)

Proof. Fix a set of Zp -generators z (i) = (z0 , . . . , z4 ) ∈ Z (1 ≤ i ≤ n) of Z and set ! n n (i) ! n ! i=1 xi z1(i) ! i=1 xi z2 ! ! (i) f (x) = xi z0 , g(x) = ! n ! (i) (i) n ! xi z xi z ! i=1

x=

n i=1

xi ei ∈

n i=1

i=1

3

Op ei = Op⊕n .

i=1

4

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533

The assumptions imply that f (Op⊕n ) ⊂ pa+1 (Op /pN ), hence a+1 U := {x ∈ Z⊕n (Op /pN )} p | f (x) ∈ p ⊕n is a proper (U Z⊕n p ) Zp -submodule of Zp , by Lemma 6.6.1(i). The N 2b+1 assumptions also imply that g(Op /p ) ⊂ p (Op /pN ), hence n

∃x = xi ei ∈ Z⊕n , x ∈ U p

g(x) ∈ p2b+1+ordp (2) (Op /pN ),

i=1

by Lemma 6.6.1(iii). The element z = desired properties.

n i=1

xi z (i) ∈ Z then has the

7. The main result We continue to use the notation from Sect. 3-6. (7.1) Selmer groups (7.1.1) Let F /F be a ﬁnite extension and Σ an arbitrary set of primes of F . We denote

SΣ (A/F , pM ) := Ker H 1 (F , A[pM ]) −→ H 1 (Fv , A[pM ])/Im(δv ) , v ∈Σ

where

δv : A(Fv ) ⊗ Op /pM → H 1 (Fv , A[pM ])

denotes the coboundary map arising from the cohomology exact sequence associated to pM

0 −→ A[pM ] −→ A(F )−−→A(F ) −→ 0. If v p∞ and if A has good reduction at the prime of F induced by 1 v, then Im(δv ) = Hur (Fv , A[pM ]). This implies that, if Σ is ﬁnite, so is SΣ (A/F , pM ). (7.1.2) The classical Selmer group S(A/F , pM ) := S∅ (A/F , pM ) sits in the standard exact sequence 0 −→ A(F ) ⊗ Op /pM −→S(A/F , pM ) −→ X(A/F )[pM ] −→ 0. δ

M The inductive limit − lim → S(A/F , p ) is an Op -module of co-ﬁnite type. M

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(7.1.3) If F /F is a ﬁnite Galois extension, then the restriction map induces a canonical homomorphism S(A/F , pM ) −→ S(A/F , pM )Gal(F

/F )

,

whose kernel and cokernel are killed by [F : F ]. (7.1.4) The Tate local duality [Mi 1, I.3.4-5] implies that, for each prime v of F , Im(δv ) is an isotropic subspace of H 1 (Fv , A[pM ]) with respect to the cup product ∪

v M H 1 (Fv , A[pM ]) × H 1 (Fv , A[pM ])−−→H 2 (Fv , Op /pM (1))−−→O p /p

inv

induced by the pairing ( , )M from (5.19.1). (7.1.5) If Σ is a ﬁnite set of primes of F , s ∈ S(A/F , pM ) and c ∈ SΣ (A/F , pM ), then the reciprocity law ∀a ∈ H 2 (GF , Op /pM (1)) invv (av ) = 0 ∈ Op /pM v

applied to a = s ∪ c yields (thanks to 7.1.4) invv (sv ∪ cv ) = 0 ∈ Op /pM . v∈Σ

(7.2) Annihilation relations (7.2.1) Kolyvagin’s classes. As in Sect. 6, we denote H = K(α). For each n = 1 · · · r ∈ Sr (M ) (r ≥ 0, i ∈ S1 (M )), we deﬁne κn = C1 (p) eβ corK(x)/H (c(n)) ∈ H 1 (H, A[pM ])(β) , where C1 (p) = maxv C1,v (p) (the constants C1,v (p) were deﬁned in 5.2). For r = 0, the cohomology class κ1 := κ(1) is equal to κ1 = C1 (p) δ(eβ (y)) ∈ S(A/H, pM )(β) , where y = corK(x)/H (yj ) ∈ A(H). For r ≥ 1, Proposition 5.12 implies that κn ∈ S{v|n} (A/H, pM )(β) . (7.2.2) Assume that n = 1 · · · r ∈ Sr (M ) (r ≥ 1, i ∈ S1 (M )). For each i ∈ {1, . . . , r}, ﬁx a prime Li | i of HM such that FrHM /F (Li ) = ρ (as an element of Gal(HM /F ), not only as a conjugacy class) and denote by vi the prime of H induced by Li . As in 6.5.4, Li determines identiﬁcations A(Hvi )[pM ] = A((HM )Li )[pM ] = A(F )[pM ]

(=: A[pM ]),

and the localization map resv

evFr(v

i i 1 M M Wi := S{v|(n/i )} (A/H, pM )−−−−→H ur (Hvi , A[p ])−−−−−−→A[p ] )

coincides with the evaluation map w → wL := (res(w ))(Fr(Li )). i

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(7.2.3) Proposition. In the situation of 7.2.2, assume that the generators σi ∈ G(i ) have been chosen in the following compatible manner: there exists a global root of unity ζ ∈ μp∞ (HM ) such that, for each i = 1, . . . , r, the image of σi in G(i ) ⊗ Op /pM coincides with the image of ζ under the composite map ∼

μp∞ (HM )⊗Op /pM −→ μp∞ ((HM )Li )⊗Op /pM ←− μp∞ (Hvi )⊗Op /pM recv ⊗id

i M −−−−→G( i ) ⊗ Op /p

(such a compatible choice always exists). Then we have, for each s ∈ S(A/H, pM )(β) , an equality [H : K]

r

sLi , (κn/i )Li

M

=0

i=1

in μp∞ (HM ) ⊗ Op /pM = Op /pM (1). Proof. Applying 7.1.5 to F = H, Σ = {v | n}, ρ(s) and c = κn , we obtain r

invσ(vi ) resσ(vi ) (ρ(s)) ∪ resσ(vi ) (κn ) = 0 ∈ Op /pM , (7.2.3.1)

i=1 σ∈Δ

where Δ = Gal(H/K). As σ(ρ(s)) = β −1 (σ)ρ(s),

(∀σ ∈ Δ)

σ(κn ) = β(σ)κn ,

we deduce from (7.2.3.1) the equality [H : K]

r

invvi (resvi (ρ(s)) ∪ resvi (κn )) = 0 ∈ Op /pM .

(7.2.3.2)

i=1

Combined with the formula from Proposition 5.20 (with ζv = ζ), (7.2.3.2) yields [H : K]

r ρ(s)Li , Fr(i )(κn/i )Li M = 0 ∈ Op /pM (1). i=1

Finally, it follows from ρ(s)Li = ρ(sLi ) = Fr(i )sLi ∈ A[pM ]

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that (ρ(s)Li , Fr(i )(κn/i )Li M = Fr(i )sLi , Fr(i )(κn/i )Li M = = Fr(i ) sLi , (κn/i )Li M = − sLi , (κn/i )Li M . (7.2.4) (i) If we do not choose the generators σi compatibly, then there exist ζ ∈ μp∞ (HM ) and u1 , . . . , ur ∈ Z∗p such that the image of σi in G(i ) ⊗ Op /pM is equal to the image of ζ ⊗ ui (i = 1, . . . , r). The statement of Proposition 7.2.3 then becomes [H : K]

r

sLi , (κn/i )Li

M

⊗ui = 0 ∈ μp∞ (HM )⊗Op /pM = Op /pM (1).

i=1

(ii) We shall apply Proposition 7.2.3 only in situations when it is known a priori that (∀i = 2, . . . , r)

sLi , (κn/i )Li

M

= 0.

In such a case one does not have to worry about the compatibility of the σi ’s. (7.3) Theorem. Assume that the condition () from Theorem 3.2 holds. If eβ (y) ∈ A(H)tors , then there exists an integer C(p) ≥ 0 which is equal to zero for all but ﬁnitely many p and such that

(∀M >> 0) pC(p) S(A/H, pM )(β) /Op · κ1 = 0. (7.4) Thanks to 7.1.2, Theorem 7.3 implies Theorem 3.2 (the statements about the α−1 -components follow by applying ρ to the α-components). The proof of Theorem 7.3 will occupy the rest of Sect. 7. Recall that the constants C1 (p), C2 (p) and C3 (p) were introduced in 7.2.1, 6.1 and 6.2, respectively. We set C5 (p) := ordp ([H : K]),

C6 (p) := ordp (deg(ϕ)),

)j is the isogeny from 5.19. If β 2 = 1, then another where ϕ : Aj −→ A constant C4 (p) will be deﬁned in 7.6.1 below. The assumption eβ (y) ∈ A(H)tors implies that the constant C0 (p) := max{c ∈ Z≥0 | eβ (y) ∈ A(H)tors + pc A(H)} is deﬁned (and C0 (p) = 0 for all but ﬁnitely many p).

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537

In order to simplify the notation, we write Ci = Ci (p) (i = 0, . . . , 6). We also denote, for each Op /pM -module Y and an element y ∈ Y , exp(y) := min{c ∈ Z≥0 | pc y = 0},

exp(Y ) := max{exp(y) | y ∈ Y }.

Using this notation, we have, for M >> 0, exp(κ1 ) ≥ M − C0 − C1 .

(7.4.1)

(7.5) Proof of Theorem 7.3 in the case β 2 = 1. In this case S := S(A/H, pM )(β) is ρ-stable; we denote S ± := S ρ=±1 . As 2κ1 = (1 + ρ)κ1 + (1 − ρ)κ1 ,

(1 ± ρ)κ1 ∈ S ± ,

it follows that (∃ε ∈ {±1})

exp((1 + ερ)κ1 ) ≥ M − C0 − C1 − ordp (2).

Fix such an ε; then xε := (1 + ερ)κ1 ∈ S ε . (7.5.1) Bounding the exponent of S −ε . Fix s ∈ S −ε with maximal exp(s) and set x−ε = s. Choosing the ﬁrst Kolyvagin prime. We apply the discussion from 6.3 and 6.5 to W = S. Set U±ε := {h ∈ 2G+ | exp(j (h))(x±ε ) < exp(x±ε ) − (C2 +C3 + 4 ordp (2))}. It follows from Corollary 6.3.4 that U±ε 2G+ are proper subgroups of 2G+ , hence there exists h = g 2 ∈ 2G+ − (Uε ∪ U−ε ). Applying the discussion in 6.5.1 to the element h = g 2 , choose a prime L (W ) satisfying (6.5.1.1-3). The induced primes L | λH | λ | of HM , H, K, F , respectively, satisfy ∈ S1 (M ) and ρ(L ) = L . The deﬁnition of U±ε implies that exp ((x±ε )L ) ≥ exp (x±ε ) − (C2 + C3 + 4 ordp (2)), hence exp (((1 + ερ)κ1 )L ) ≥ M −

3

Ci − 5 ordp (2)

i=0 −ε

exp (sL ) ≥ exp S

(7.5.1.1)

− (C2 + C3 + 4 ordp (2)). (7.5.1.2)

The ﬁrst annihilation relation. Applying Proposition 7.2.3 with n = to s and ρ(s), we obtain (using the ρ-equivariance of the map w → wL ) that [H : K] (sL , ((1 + ερ)κ1 )L )M = 0 ∈ Op /pM (1).

(7.5.1.3)

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Combining (7.5.1.3) with (7.5.1.1) and (5.19.2), we obtain 26 pC0 +C1 +C2 +C5 +C6 sL ∈ Op · ((1 + ερ)κ1 )L ⊂ A[pM ]ρ=ε . On the other hand, sL ∈ A[pM ]ρ=−ε (since s ∈ S −ε ) and 2(A[pM ]ρ=ε ∩ A[pM ]ρ=−ε ) = 0, hence 27 pC0 +C1 +C2 +C5 +C6 sL = 0. Combined with (7.5.1.2), we obtain pM1 S −ε = 0,

M1 = C0 + C1 + 2C2 + 2C3 + C5 + C6 + 11 ordp (2). (7.5.1.4) In particular, pM1 (1 − ερ)κ1 = 0, hence 2 pM1 κ1 = pM1 (1 + ερ)κ1 . (7.5.2) Bounding the exponent of S ε /Op · (1 + ερ)κ1 . Denote S = Ker resλH : S −→ H 1 (HλH , A[pM ]) and ﬁx s ∈ Sε = Sρ=ε with maximal exp( s). It follows from Proposition 5.18 combined with (5.15.1) that exp((1 − ερ)κ ) ≥ exp (f (((1 − ερ)κ )L )) = exp (((1 + ερ)κ1 )L ) ≥ ≥M−

3

Ci − 5 ordp (2),

i=0

where f denotes the map 1 f : H 1 (HλH , A[pM ]) −→ H 1 (HλH , A[pM ])/Hur (HλH , A[pM ])

(cf. 7.6.4 below). Choosing the second Kolyvagin prime. We apply the discussion from 6.3 and 6.5 to the submodule W = S{v|} (A/H, pM )(β) . Set xε = s ∈ (W )ε and x−ε = (1 − ερ)κ ∈ (W )−ε . The argument as in 7.5.1 shows that there exists a prime ∈ S1 (M ) ( = ) and a prime L | of HM such that ρ(L ) = L and exp (((1 − ερ)κ )L ) ≥ M −

3

Ci − 5 ordp (2) − (C2 + C3 + 4 ordp (2)),

i=0

exp ( sL ) ≥ exp Sε − (C2 + C3 + 4 ordp (2)).

(7.5.2.1) (7.5.2.2)

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539

The second annihilation relation. Applying Proposition 7.2.3 with n = to s and ρ( s), we obtain (using the ρ-equivariance of the map w → wL L = 0) that and the assumption s [H : K] ( sL , ((1 − ερ)κ )L )M = 0 ∈ Op /pM (1).

(7.5.2.3)

Combining (7.5.2.3) with (7.5.2.1) and (5.19.2), we deduce that 210 pC0 +C1 +2C2 +2C3 +C5 +C6 sL ∈ Op · ((1 − ερ)κ )L ⊂ A[pM ]ρ=−ε . On the other hand, s ∈ Sε implies that sL ⊂ A[pM ]ρ=ε , hence 211 pC0 +C1 +2C2 +2C3 +C5 +C6 sL = 0. Applying (7.5.2.2), we obtain 215 pC0 +C1 +3C2 +3C3 +C5 +C6 Sε = 0.

(7.5.2.4)

1 (7.5.3) Bounding the exponent of S/Op κ1 . As 2Hur (HλH , A[pM ])ε is a cyclic Op -module, it follows from (7.5.1.1) that

25 pC0 +C1 +C2 +C3 S ε / Sε + Op · (1 + ερ)κ1 = 0.

Putting this together with (7.5.1.4) and (7.5.2.4), we deduce 221 p2C0 +2C1 +4C2 +4C3 +C5 +C6 (S/Op κ1 ) = 0, which ﬁnishes the proof of Theorem 7.3 in the case β 2 = 1. (7.6) Proof of Theorem 7.3 in the case β 2 = 1. We write β := β −1 = β, S := S(A/H, pM ) (for M >> 0) and, as in 7.5, Ci := Ci (p) (i = 0, . . . , 6). (7.6.1) Lemma-Deﬁnition. Deﬁne # 0, C4 (p) = ordp (p)/(p − 1),

p order of β 2 p | order of β 2 .

(i) C4 (p) ∈ Z≥0 . (ii) C4 (p) ≤ C5 (p)/(p − 1). (iii) For any Op [Δ]-module Y (where Δ = Gal(H/K)), pC4 (p) (Y (β) ∩ Y (β) ) = 0. Proof. (i) If p divides the order of β 2 , then Lp contains the values of β 2 , hence Lp ⊃ Qp (μp ).

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(ii) This is clear. (iii) (∃σ ∈ Δ) β(σ) = β(σ); then Y (β) ∩Y (β) is killed by β 2 (σ)−1 = 0, but # ordp (β (σ) − 1) = 2

0,

β 2 (σ) ∈ μp∞

≤ ordp (p)/(p − 1),

β 2 (σ) ∈ μp∞ .

(7.6.2) Deﬁnition. (i) Fix t± ∈ T ± = T ρ=±1 such that T ± = Op t± and let t±,M be the image of t± in (T /pM T )± = A[pM ]± . (ii) exp(t±,M ) ≥ M − ordp (2); ﬁx u± ∈ (Op /pM )t±,M with exp(u± ) = M − ordp (2). (iii) Let W be as in 6.3.1; let f ∈ HomOp (W , A[pM ])+ . Then 2f ((W )± ) ⊂ (Op /pM −ordp (2) )u± ; we deﬁne the morphisms f± (W )± −→ Op /pM −ordp (2) ,

(∀w ∈ (W )± ) f± (w)u± = 2f (w).

(iv) For each w ∈ W , 2w = (1 + ρ)w + (1 − ρ)w, hence 4f (w) = f+ ((1 + ρ)w)u+ + f− ((1 − ρ)w)u− . (7.6.3) Choosing the ﬁrst Kolyvagin prime. We have κ1 ∈ S (β) ,

ρ(κ1 ) ∈ S (β) ,

exp(κ1 ) = exp(ρ(κ1 )) ≥ M − C0 − C1 .

As pC4 S (β) ∩ S (β) = 0, it follows that the elements (1 ± ρ)κ1 ∈ S ± = S ρ=±1 satisfy exp((1 ± ρ)κ1 ) ≥ M − C0 − C1 − C4 − ordp (2).

(7.6.3.1)

Applying Corollary 6.3.4 to W = S, we obtain, as in 7.5.1, primes ∈ S1 (M ) and L | in HM such that ρ(L ) = L and exp (((1 ± ρ)κ1 ))L ) ≥ exp((1 ± ρ)κ1 ) − (C2 + C3 + 4 ordp (2)) ≥ M − M2 , (7.6.3.2) where 4 M2 = Ci + 5 ordp (2). i=0

As 2κ1 = (1 + ρ)κ1 + (1 − ρ)κ1 and 2(A[pM ]+ ∩ A[pM ]− ) = 0, it follows that exp ((κ1 )L ) , exp ((ρ(κ1 ))L ) ≥ M − M2 − 2 ordp (2).

(7.6.3.3)

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541

The ﬁrst annihilation relation. Let s ∈ S (β) . Applying Proposition 7.2.3 with n = , we obtain [H : K] (sL , (κ1 )L )M = 0 ∈ Op /pM (1), hence, using (5.19.2) and (7.6.3.3), 23 pM2 +C5 +C6 sL = 28 p Set

6 i=0

Ci

sL ∈ Op (κ1 )L .

(7.6.3.4)

(res )

v S = Ker S −−→ H 1 (Hv , A[pM ]) ; v|

then

S(β) = Ker resλH : S (β) −→ H 1 (HλH , A[pM ]) ,

and the inclusion (7.6.3.4) can be reformulated as follows:

23 pM2 +C5 +C6 S (β) / S(β) + Op κ1 = 6

= 28 p i=0 Ci S (β) / S(β) + Op κ1 = 0.

(7.6.3.5)

(7.6.4) Choosing the second Kolyvagin prime. In the exact sequence of Op /pM -modules f

1 0 −→ Hur (HλH , A[pM ]) −→ H 1 (HλH , A[pM ])−→

H 1 (HλH , A[pM ]) −→ 0, 1 (H M Hur λH , A[p ])

∼

both ﬂank terms are isomorphic to A[pM ] −→ (Op /pM )⊕2 , hence the sequence splits and the middle term is isomorphic, as an Op /pM -module, to (Op /pM )⊕4 . We know that ∼

1 (ρ(κ1 ))L = ρ((κ1 )L ) ∈ Hur (HλH , A[pM ]) −→ A[pM ], ((1 ± ρ)κ1 )L ∈ A[pM ]±

and exp (((1 ± ρ)κ1 )L ) ≥ M − M2 . On the other hand, (5.15.1) and Proposition 5.18 imply that f (((1 ± ρ)κ )L ) = −ΦλH (((1 ∓ ρ)κ1 )L ) , hence exp (f (((1 ± ρ)κ )L )) ≥ M − M2 .

(7.6.4.1)

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Set (res )

v * = Ker W −−→ W H 1 (Hv , A[pM ]) ;

W = S{v|} (A/H, pM ),

v|

then * )(β) = S(β) , (W

* )(β) = S(β) , (W

κ1 , κ ∈ (W )(β) ,

ρ(κ1 ), ρ(κ ) ∈ (W )(β) . Denote by U ⊆ W the Op /pM -submodule generated by (1 + ρ)κ1 , (1 − ρ)κ1 , (1 + ρ)κ and (1 − ρ)κ . It follows from (7.6.3.2) and (7.6.4.1) that resλH (U ) ⊇ pM2 H 1 (HλH , A[pM ]), (7.6.4.2) hence

* ) = 0 pM2 (U ∩ W

(7.6.4.3)

and U contains an Op /pM -submodule isomorphic to (pM2 (Op /pM ))⊕4 . * )(β) = S(β) with maximal exp( Fix an element s ∈ (W s); then the argument used in the proof of (7.6.3.1) shows that exp((1 + ρ) s) ≥ exp( s) − C4 − ordp (2).

(7.6.4.4)

Deﬁne a homomorphism of Op /pM -modules ⊕5 z : HomOp (W , A[pM ])+ −→ Op /pN

(N = M − ordp (2))

by the formula z(f ) = (f+ ((1 + ρ) s), f+ ((1 + ρ)κ1 ), f− ((1 − ρ)κ1 ), f+ ((1 + ρ)κ ), f− ((1 − ρ)κ ))) and set Z = (z ◦ j )(2G+ ) = (z ◦ j )(2 Gal(HM (W )/HM )+ ), which is a Zp -submodule of (Op /pN )⊕5 . It follows from (7.6.3.2), (7.6.4.1), (7.6.4.3) and (7.6.4.4) that

Im(z) ⊇ 2 pa (Op /pN ) ⊕ (2 pM2 (Op /pN ))⊕4 , where

N − a ≥ exp( s) − (M2 + C4 + ordp (2)).

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543

Corollary 6.3.4 implies that Op · Z ⊇ pa (Op /pN ) ⊕ pb (Op /pN ))⊕4 , where N − a ≥ exp( s) − (M2 + C2 + C3 + C4 + 6 ordp (2)), b = M2 + C2 + C3 + 5 ordp (2). According to Proposition 6.6.2, there exists an element h = g 2 ∈ 2G+ such that the corresponding vector (z0 , . . . , z4 ) = (z ◦ j )(h) ∈ (Op /pN )⊕5 satisﬁes z0 ∈ pa+1 (Op /pN ),

! ! z1 ! ! ! z3

! z2 !! ! ∈ 2 p2b+1 (Op /pN ). z4 !

(7.6.4.5)

Applying the discussion from 6.5.1, we choose L (W ) satisfying (6.5.1.13) and not dividing . We denote by L , λH , ∈ S1 ( = ) the induced primes of HM , H and F , respectively. By construction, we have 2 ((1 + ρ) s)L = z0 u+ , 2 ((1 + ρ)κ1 )L = z1 u+ ,

2 ((1 − ρ)κ1 )L = z2 u− ,

2 ((1 + ρ)κ )L = z3 u+ ,

2 ((1 − ρ)κ )L = z4 u− ,

hence 4(κ1 )L = z1 u+ + z2 u− ,

4(κ )L = z3 u+ + z4 u− .

It follows from (7.6.4.5) that exp (((1 + ρ) s)L ) ≥ M − a ≥ exp( s) − (M2 + C2 + C3 + C4 + 5 ordp (2)) = exp( s) − (b + C4 ) (7.6.4.6) and ∼

1 (HλH , A[pM ]) −→ p2b A[pM ]. (7.6.4.7) resλH (Op κ1 + Op κ ) ⊇ p2b Hur

The second annihilation relation. Applying Proposition 7.2.3 with n = to the above element s = s ∈ S(β) , we obtain, as in (7.5.2.3), [H : K] ( sL , (κ )L )M = 0 ∈ Op /pM (1).

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Applying Proposition 7.2.3 with n = to s = s (i.e., the ﬁrst annihilation relation with instead of ), we obtain [H : K] ( sL , (κ1 )L )M = 0 ∈ Op /pM (1). Combining these two relations with (7.6.4.7), we deduce that the element p2b+C5 sL ∈ A[pM ] lies in the kernel of the pairing ( , )M , hence p2b+C5 +C6 sL = 0. It follows from (7.6.4.6) that exp S(β) = exp( s) ≤ exp (((1 + ρ) s)L ) + (b + C4 ) ≤ (2b + C5 + C6 ) + (b + C4 ) = 3b+C4 +C5 +C6 = 3C0 +3C1 +6C2 +6C3 +4C4 +C5 +C6 +30 ordp (2). Combined with (7.6.3.5), this relation yields

238 p4C0 +4C1 +7C2 +7C3 +5C4 +2C5 +2C6 S (β) /Op κ1 = 0, which concludes the proof of Theorem 7.3 (hence also the proof of Theorem 3.2) in the case β 2 = 1.

References [Be] M. Bertolini, Selmer groups and Heegner points in anticyclotomic Zp -extensions, Compositio Math. 99 (1995), 153–182. [Be-Da] M. Bertolini, H. Darmon, Kolyvagin’s descent and MordellWeil groups over ring class ﬁelds, J. Reine Angew. Math. 412 (1990), 63–74. [Bi] B. Birch, Heegner points of elliptic curves, In: Symposia Mathematica, Vol. XV (Convegno di Strutture in Corpi Algebrici, INDAM, Rome, 1973), pp. 441–445. Academic Press, London, 1975. [Bo] F.A. Bogomolov, Sur l’alg´ebricit´e des repr´esentations l-adiques, C. R. Acad. Sci. Paris S´er. A-B 290 (1980), no. 15, A701–A703. [Bu] K. Buzzard, Integral models of certain Shimura curves, Duke Math. J. 87 (1997), 591–612. [Ca 1] H. Carayol, Sur la mauvaise r´eduction des courbes de Shimura, Compositio Math. 59 (1986), 151–230.

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Repr´esentations irr´eductibles de GL(2,F) modulo p Marie-France Vign´eras Institut de Mathematiques de Jussieu 175 rue du Chevaleret Paris 75013 France [email protected] R´esum´e. This is a report on the classiﬁcation of irreducible representations of GL(2, F ) over Fp when F is a local ﬁeld of ﬁnite residual ﬁeld contained in the algebraically closed ﬁeld Fp of characteristic p. 1 Toutes les repr´esentations des groupes seront lisses, i.e. chaque vecteur est ﬁxe par un sous-groupe ouvert. Soit p un nombre premier, F un corps local complet pour une valuation discr`ete de corps r´esiduel ﬁni Fq de caract´eristique p ayant q oture s´eparable de F et Fp une clˆ oture alg´ebrique de ´el´ements, F un clˆ Fq . Soit n ≥ 1 un entier. Pour un nombre premier = p, la correspondance semisimple de Langlands modulo , est une bijection “compatible avec la r´eduction modulo ” entre les classes d’isomorphisme des repr´esentations irr´eductibles de dimension n du groupe de Gal(F /F ) sur F et les classes d’isomorphisme des repr´esentations irr´eductibles supercuspidales de GL(n, F ) sur F , ´etendue de fa¸con a` inclure les repr´esentations semi-simples de dimension n de Gal(F /F ), et toutes les repr´esentations irr´eductibles de GL(n, F ) sur F [V2]. On s’int´eresse au cas = p. On connait bien les repr´esentations irr´eductibles de dimension ﬁnie de Gal(F /F ) sur Fp , mais quelles sont les repr´esentations irr´eductibles de GL(n, F ) sur Fp ? La r´eponse est connue uniquement pour le groupe GL(2, Qp ); c’est un probl`eme ouvert pour n ≥ 3 ou pour F = Qp . 2 Les repr´esentations irr´eductibles de dimension n du groupe de Galois Gal(F /F ) sur Fp se classent facilement [V3] 1.14 page 423. 2.1 Lorsque n = 1, l’isomorphisme de r´eciprocit´e du corps de classes, qui envoie une uniformisante pF de F sur un Frobenius g´eom´etrique FrobF , identiﬁe les caract`eres (repr´esentations de dimension 1) de F ∗ et de Gal(F /F ). Nous utiliserons syst´ematiquement cette identiﬁcation. Pour une extension ﬁnie F de F contenue dans F , la restriction du cˆot´e galoisien correspond a` la norme F ∗ → F ∗ . 2.2 Les repr´esentations irr´eductibles de Gal(F /F ) sur Fp de dimension n ≥ 2 sont induites par les caract`eres r´eguliers sur F du 548

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549

groupe multiplicatif de l’unique extension Fn de F non ramiﬁ´ee de degr´e n sur F contenue dans F . Un caract`ere de Fn∗ est r´egulier sur F si ses n conjugu´es par le groupe de Galois Gal(Fn /F ) sont distincts. Les repr´esentations ρ(χ), ρ(χ ) de Gal(F /F ) induites de deux caract`eres χ, χ de Fn∗ r´eguliers sur F sont isomorphes si et seulement si χ, χ sont conjugu´es par le groupe de Galois Gal(Fn /F ). Le d´eterminant de ρ(χ) est la restriction de χ ` a F ∗. ∗ 2.3 On note OF l’anneau des entiers de F ; un caract`ere OF∗ → Fp s’identiﬁe a` un caract`ere de F∗q . L’uniformisante pF de F est aussi une uniformisante de l’extension non ramiﬁ´ee Fn . Le caract`ere ωn de Fn∗ tel que ωn (pF ) = 1 et dont la restriction `a OF∗ n s’identiﬁe au plongement na∗ turel ιn : F∗qn → Fp , est appel´e un caract`ere de Serre; il est r´egulier. Pour ∗ λ ∈ Fp , on note μn,λ le caract`ere non ramiﬁ´e de Fn∗ tel que μn,λ (pF ) = λ. On supprime l’indice n = 1 pour F ∗ . Le compos´e de la norme Fn∗ → F ∗ n−1 et du caract`ere ω, resp. μλ , est le caract`ere ωn1+q+...+q , resp. μn,λn . ∗ a Les caract`eres de Fn sont μn,λ ωn pour un unique couple (λ, a) ∈ ∗ Fp × {1, . . . , q n − 1}. 2.4 Les repr´esentations irr´eductibles de dimension n ≥ 1 du groupe de Galois Gal(F /F ) sur Fp sont Gal(F /F )

ρn (a, λ) = μλ ⊗ indGal(F /F

n)

ωna = ind

Gal(F /F ) (μn,λn Gal(F /Fn )

ωna )

pour les entiers a ∈ Z/(q n − 1)Z tels que a, qa, . . . , q n−1 a sont distincts. Les isomorphismes sont les suivants ρn (a, λ) # ρn (aq i , ζλ) ∗

pour les entiers 1 ≤ i ≤ n − 1 et ζ ∈ Fp avec ζ n = 1. Le d´eterminant de ρn (a, λ) est ω a μλn o` u a ∈ Z/(q−1)Z est l’image de a. Le nombre de repr´esentations irr´eductibles avec det FrobF ﬁx´e est ﬁni, ´egal au nombre de polynˆomes irr´eductibles unitaires de degr´e n dans Fq [X] [V1] 3.1 (10). Lorsque n = 2, ce nombre est q(q − 1)/2. 2.5 Lorsque F = Qp , les repr´esentations irr´eductibles de dimension 2 du groupe de Galois Gal(Qp /Qp ) sur Fp sont Gal(F /F )

σ(r, χ) = χ ⊗ indGal(F /F ) ω2r+1 2

∗

pour les entiers r ∈ {0, . . . , p − 1} et les caract`eres χ : Q∗p → Fp . On a σ(r, χ) = ρ2 (a, λ) avec χ = μλ ω b , a = (p + 1)b + r + 1. Le d´eterminant de σ(r, χ) est χ2 ω r+1 . Les isomorphismes sont les suivants σ(r, χ) # σ(p − 1 − r, χ) # σ(r, χμ−1 ) # σ(p − 1 − r, χμ−1 ).

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3 Les repr´esentations irr´eductibles de GL(2, Qp ) sur Fp avec un caract`ere central sont class´ees [BL2] [Br]. On dispose d’une liste pour GL(2, F ) lorsque F = Qp [V0], [Pa], probablement non compl`ete. 3.1 Les repr´esentations irr´eductibles de GL(2, Fq ) sur Fp sont (χ ◦ 2 det) ⊗ Symr Fp pour un unique couple (r, χ) avec 0 ≤ r ≤ q − 1 et ∗

un caract`ere χ : F∗q → Fp ; on peut aussi remplacer χ par l’unique entier 1 ≤ a ≤ q − 1 tel que χ(?) =?a . On a le d´eveloppement p-adique r = r1 + pr2 + . . . + pf −1 rf avec 0 ≤ ri ≤ p − 1, q = pf , et 2

2

Symr Fp = ⊗fi=1 Symri Fp ◦ Fri−1

o` u Fr

a b c d

=

ap cp

bp dp

2

et Symr Fp est la repr´esentation de GL(2, Fq ) sur les polynˆomes homog`enes de degr´e r ≥ 0 dans Fp [X, Y ] v´eriﬁant a b X i Y j = (aX + cY )i (bX + cY )j (i, j ≥ 0, i + j = r). c d La repr´esentation triviale et la repr´esentation sp´eciale (ou de Steinberg) 2 2 sont Sym0 Fp et Symq−1 Fp . 2

La repr´esentation irr´eductible Symr Fp s’identiﬁe a` une repr´esentation irr´eductible de GL(2, OF ), ou a` une repr´esentation de Ko = GL(2, OF )pZ F triviale sur pF . On lui associe par induction compacte une repr´esentation lisse de GL(2, F ) 2

E(r) = indG Ko Symr Fp . 3.2 Les repr´esentations irr´eductibles de G = GL(2, F ) sur Fp sont: ∗ (i) Les caract`eres χ ◦ det pour les caract`eres χ : F ∗ → Fp . ere (ii) Les s´eries principales indG B (χ1 ⊗ χ2 ) induites par le caract` a b (χ1 ⊗ χ2 ) = χ1 (a)χ2 (d) 0 d du sous-groupe triangulaire sup´erieur B, pour les caract`eres distincts ∗ χ1 , χ2 : F ∗ → Fp , χ1 = χ2 . (iii) Les s´eries sp´eciales (appel´ees aussi de Steinberg) Sp ⊗(χ ◦ det), ∗ u Sp est le quotient de la repr´esentation pour les caract`eres χ : F ∗ → Fp , o` G induite indB idFp du caract`ere trivial de B, par le caract`ere trivial de G.

Repr´esentations irr´eductibles de GL(2, F )

551

(iv) Les repr´esentations irr´eductibles supersinguli`eres. Il n’y a pas d’isomorphisme entre ces repr´esentations. Les repr´esentations supersinguli`eres sont les repr´esentations supercuspidales (non sous-quotient d’une repr´esentation induite de B). La repr´esentation sp´eciale ne se plonge pas dans une repr´esentation induite parabolique; elle est cuspidale (l’espace de ses coinvariants par le radical unipotent N de B est nul) sans ˆetre supercuspidale. Barthel et Livn´e ([BL2] prop.8) montrent que EndFp G (indG Ko id) # pF 0 ; les alg`ebres Fp [T ] o` u T correspond a` la double classe de h = 0 1 EndFp G E(r) sont canoniquement isomorphes. Ils ont appel´e supersinguli`eres les repr´esentations irr´eductibles supercuspidales ayant un caract`ere central, et d´emontr´e que ce sont les quotients irr´eductibles de V (r, χ) = (χ ◦ det) ⊗

E(r) T E(r)

∗

pour les caract`eres χ : F ∗ → Fp et les entiers 0 ≤ r ≤ q − 1; le caract`ere central de V (r, χ) est χ2 ω r . La repr´esentation de G V (r, λ, χ) = (χ ◦ det) ⊗

E(r) (T − λ)E(r)

∗

(λ ∈ Fp )

est isomorphe `a r i) la s´erie principale irr´eductible (χ ◦ det) ⊗ indG B (μλ−1 ⊗ μλ ω ) si r μλ−1 = μλ ω , i.e. si λ = ±1 ou r = (0, . . . , 0), (p − 1, . . . , p − 1), ii) la repr´esentation (χ ◦ det) ⊗ indG ee, B μλ de longueur 2 non scind´ contenant χμλ ◦ det et de quotient (χμλ ◦ det) ⊗ Sp lorsque λ = ±1 et r = (p − 1, . . . , p − 1), iii) une repr´esentation de longueur 2 non scind´ee contenant (χμλ ◦ det) ⊗ Sp et de quotient χμλ ◦ det lorsque λ = ±1 et r = (0, . . . , 0). 3.3 Dans le cas particulier mais important F = Qp , Breuil [Br] 4.1.1, 4.1.4, a montr´e que les repr´esentations V (r, χ) sont irr´eductibles. Les repr´esentations irr´eductibles supersinguli`eres de GL(2, Qp ) sont ∗ les V (r, χ) pour 0 ≤ r ≤ p − 1 et χ un caract`ere F ∗ → Fp ; les isomorphismes sont : V (r, χ) # V (p − 1 − r, χω r ) # V (r, χμ1 ) # V (p − 1 − r, χω r μ−1 ). 3.4 Breuil [Br] 4.2.4 en d´eduit une bijection unique “compatible avec la r´eduction modulo p” de la correspondance donn´ee par la “cohomologie ´etale des courbes modulaires” σ(r, χ) ↔ V (r, χ)

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Marie-France Vign´eras

entre les classes d’isomorphisme des repr´esentations irr´eductibles de dimension 2 du groupe de Galois Gal(Qp /Qp ) sur Fp (2.5) et les classes d’isomorphisme des repr´esentations irr´eductibles supersinguli`eres de GL(2, Qp ) sur Fp (3.3), qu’il ´etend de fa¸con `a inclure les repr´esentations semi-simples de dimension 2 de Gal(Qp /Qp ), χ⊗(ω r+1 μλ ⊕ μλ−1 ) G r r −1 ss ↔ (χ ◦ det) ⊗ [indG )] , B (μλ−1 ⊗ ω μλ ) ⊕ indB ((ω μλ )ω ⊗ μλ−1 ω

notant ?ss la semi-simpliﬁ´ee d’une repr´esentation ? de longueur ﬁnie. Soit r ∈ {0, . . . , p − 2} congru a` p − 3 − r modulo p − 1; le membre de droite est aussi (3.2): V (r, λ, χ)ss ⊕ V (r , λ−1 , ωr+1 χ)ss . Le d´eterminant χ2 ω r+1 de la repr´esentation galoisienne ne coincide pas par l’isomorphisme de la th´eorie du corps de classes avec le caract`ere central χ2 ω r de la repr´esentation de GL(2, Qp ). 3.5 Une repr´esentation irr´eductible avec un caract`ere central de GL(2, Qp ) est caract´eris´ee par sa restriction au sous-groupe triangulaire B, qui est irr´eductible sauf pour une s´erie principale o` u elle est de longueur 2; ceci est d´emontr´e par Berger [Be] lorsque F = Qp en utilisant les repr´esentations de B(Qp ) construites par Colmez avec les (φ, Γ)-modules de Fontaine; une preuve non galoisienne est donn´ee dans [Vc] (voir 3.6) pour tout F , mais uniquement pour les s´eries principales et la Steinberg. Toute repr´esentation irr´eductible W de Gal(Qp /Qp ) de dimension ﬁnie sur Fp , d´eﬁnit une repr´esentation irr´eductible de dimension inﬁnie ΩW de B(Qp ) sur Fp . Deux repr´esentations W non isomorphes donnent des repr´esentations ΩW non isomorphes. Les repr´esentations irr´eductibles contenues dans la s´erie principale ou la s´erie sp´eciale (resp. dans les supersinguli`eres) de GL(2, Qp ) sont les ΩW pour dim W = 1 (resp. dim W = 2). Remarque. Notons P le sous-groupe mirabolique form´e des matrices de seconde ligne (0, 1) dans B. Il est isomorphe au produit semi-direct du groupe additif Ga et du groupe multiplicatif Gm (agissant naturellement sur Ga ). On identiﬁe Ga au radical unipotent de P ou de B. Sur un corps alg´ebriquement clos de caract´eristique diﬀ´erente de p, le groupe mirabolique P (F ) a une unique repr´esentation irr´eductible de dimension inﬁnie τ ; les autres sont des caract`eres. La restriction `a P (F ) d’une repr´esentation irr´eductible π de dimension inﬁnie de GL(2, F ) contient τ et π/τ est de longueur 2, 1, 0 selon que π est de la s´erie principale, sp´eciale, ou est supercuspidale [V4].

Repr´esentations irr´eductibles de GL(2, F )

553

3.6 On peut d´ecomposer les s´eries principales avec les arguments suivants [Vc]. La repr´esentation naturelle ρ de P (F ) sur l’espace des fonctions localement constantes `a support compact sur F et `a valeurs dans Fp est irr´eductible; ceci utilise que l’alg`ebre de groupe compl´et´ee d’un pro-p-groupe sur corps ﬁni de caract´eristique p est locale. Joint au fait que les F -coinvariants de ρ sont nuls car un pro-p-groupe n’a pas de mesure de Haar `a valeurs dans Fp , on obtient que la restriction `a B(F ) de indG ere χ1 ⊗ χ2 , et B χ1 ⊗ χ2 est de longueur 2, de quotient le caract` l’image de indG χ ⊗χ par le foncteur de Jacquet (les U (F )-coinvariants) 1 2 B est χ = χ1 ⊗ χ2 . Cette d´emonstration n’utilisant pas l’arbre de P GL(2), est g´en´eralisable `a GL(n). 4 G´en´eralit´es. 4.1 On dit que V est admissible si l’espace V K des vecteurs de V ﬁxes par K est de dimension ﬁnie pour tout sous-groupe ouvert compact K de G. Lorsque C = Fp , il suﬃt que ce soit vrai pour un seul pro-psous-groupe ouvert P de G. Voici la preuve simple et astucieuse due a` u Paskunas. La restriction de V ` a P se plonge dans (dimFp V P ) Inj 1Fp , o` Inj 1Fp est l’enveloppe injective de la repr´esentation triviale de P; pour tout sous-groupe ouvert distingu´e K de P, on a (Inj 1Fp )K = Fp [P/K], donc la dimension de V K est ﬁnie. Toutes les repr´esentations irr´eductibles de GL(2, F ) sur Fp connues sont admissibles, ont un caract`ere central et sont d´eﬁnies sur un corps ﬁni. 4.2 Tout pro-p-groupe agissant sur un Fp -espace vectoriel non nul a un vecteur non nul invariant. Ceci implique qu’une repr´esentation admissible non nulle V de G sur Fp , contient une sous-repr´esentation irr´eductible W . 4.3 Soit W une repr´esentation lisse sur un corps commutatif C de dimension ﬁnie d’un sous-groupe ouvert K de G. On lui associe la ebre de repr´esentation lisse indG K W de G, par induction compacte. L’alg` Hecke de (K, W ) dans G, H(G, K, W ) = EndCG indG KW s’identiﬁe a` l’alg`ebre de convolution des fonctions f : G → EndC W de support une union ﬁnie de doubles classes de G modulo K, satisfaisant f (kgk ) = k ◦ f (g) ◦ k pour g ∈ G et k, k ∈ K; la condition sur F = f (g) ∈ EndC W est F ◦ k = gkg −1 ◦ F pour tout k ∈ K ∩ g −1 Kg. On associe `a V le H(G, K, W )-module a` droite HomCG (indG K W, V ) # HomCK (W, V ), par adjonction.

554

Marie-France Vign´eras

4.4 Les alg`ebres de Hecke fournissent un crit`ere bien utile d’irr´eductibilit´e [V0] Criterium 4.5 page 344. Si P est un pro-p-sous-groupe ouvert, une repr´esentation V de G sur Fp engendr´ee par π P est irr´eductible, si π P est un H(G, P, idFp )-module simple. 4.5 Nous donnons maintenant deux propri´et´es g´en´erales et techniques utilis´ees dans la d´emonstration par Barthel et Livn´e qu’une repr´esentation irr´eductible V de GL(2, F ) sur Fp ayant un caract`ere central est quotient d’un V (r, λ, χ) (3.2), que nous expliquerons en (6.3). (i) Soit V irr´eductible tel que HomCK (W, V ) contienne un sousH(G, K, W )-module de dimension ﬁnie sur C (vrai si V est admissible); alors il existe un H(G, K, W )-module simple `a droite M tel que V est quotient de M ⊗H(G,K,W ) indG K W. (ii) Soit un quotient j : W → W de la repr´esentation W de K; G G l’application indG K (j) : indK W → indK W est surjective et induit une application injective G G ? ◦ indG K (j) : HomCG (indK W , V ) → HomCG (indK W, V )

pour toute repr´esentation V de G. Soit M un sous-espace de HomCG G G (indG K W , V ) dont l’image par ? ◦ indK (j) dans HomCG (indK W, V ) est H(G, K, W )-stable. Si tout morphisme h ∈ H(G, K, W ) se rel`eve en G un morphisme h ∈ H(G, K, W ) tel que h ◦ indG K (j) = indK (j) ◦ h, alors M est H(G, K, W )-stable. La condition sur les alg`ebres de Hecke signiﬁe que pour chaque g dans un syst`eme de repr´esentants des doubles classes K\G/K, tout morphisme F ∈ EndC W v´eriﬁant F ◦ f = gkg −1 ◦ F pour tout eriﬁant k ∈ K ∩ g −1 Kg se rel`eve en un morphisme F ∈ EndCK indG K W v´ la mˆeme relation tel que F ◦ j = j ◦ F . 5 Alg`ebre de Hecke du “pro-p-Iwahori” I(1). 5.1 L’image inverse dans K = GL(2, OF ) de B(Fq ) par la r´eduction K → GL(2, Fq ) est le groupe d’Iwahori I; celle du groupe strictement triangulaire sup´erieur est le pro−p-sous-groupe I(1); c’est un pro-p-Sylow de I. La Z-alg`ebre de Hecke du pro-p-Iwahori I(1) [V1] H(2, q) = EndZG Z[I(1)\GL(2, F )] ne d´epend que de (2, q). La Z-alg`ebre H(2, q) a une grosse sous-alg`ebre commutative de type ﬁni avec une action naturelle de S2 , style Bernstein, dont les S2 -invariants forment le centre de H(2, q). L’alg`ebre H(2, q) est un module de type ﬁni sur son centre. Le centre de H(2, q) est une Z-alg`ebre de type ﬁni.

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5.2 Un H(2, q) ⊗Z Fp -module simple (`a droite ou a` gauche) a un caract`ere central, est de dimension ﬁnie, et est d´eﬁni sur un corps ﬁni. C’est l’analogue pour l’alg`ebre de Hecke du pro-p-Iwahori de “une sous-repr´esentation d’une repr´esentation de type ﬁni de G sur Fp est de type ﬁni” et de “une repr´esentation irr´eductible de G sur Fp est admissible d´eﬁnie sur un corps ﬁni” (questions ouvertes). Cela r´esulte de: 5.3 Soit C un corps commutatif parfait, Z une C-alg`ebre commutative de type ﬁni, H un anneau qui est un Z-module de type ﬁni. Alors un H-module (` a droite ou a` gauche) simple est de dimension ﬁnie sur C. Preuve (implicite dans [V4] I.7.11): Un H-module M simple (`a droite par exemple) ´etant un Z-module de type ﬁni, admet un quotient Z-simple. Si C est alg´ebriquement clos, un Z-module simple est de dimension 1 sur C; il existe donc un morphisme χ : Z → C tel que M (χ) = {mz − χ(z)m, (m ∈ M, z ∈ Z)} est distinct de M et M (χ) est stable par A. Comme M est simple, M (χ) = 0 i.e. Z agit sur M par χ; ceci implique que M est de dimension ﬁnie sur C. Si C est parfait, il existe une extension ﬁnie galoisienne C /C et un morphisme χ : Z → C tel que M ⊗C C est distinct de (M ⊗C C )(χ). Le H ⊗C C -module M ⊗C C est une somme directe ﬁnie de modules simples, conjugu´es par Gal(C /C). L’un d’entre eux N est distinct de N (χ), donc N (χ) = 0 et N est de dimension ﬁnie sur C . La dimension de M ⊗C C sur C est donc de dimension ﬁnie; c’est aussi celle de M sur C. 5.4 D´eﬁnition d’un H(2, q) ⊗Z Fp -module simple (`a droite ou `a gauche) supersingulier [V1]. Un caract`ere du centre de H(2, q) a` valeurs dans Fp a n´ecessairement beaucoup de “z´eros”. Lorsqu’il y a des z´eros suppl´ementaires, le caract`ere est dit singulier (non singulier est appel´e r´egulier). Le pire cas plein de z´eros est appel´e supersingulier. La terminologie s’´etend a` un H(2, q) ⊗Z Fp -module simple via son caract`ere central (5.3). Le nombre de H(2, q) ⊗Z Fp -modules simples supersinguliers de dimension 2 connus avec une action de pF ﬁx´ee, est exactement le nombre de repr´esentations continues irr´eductibles de Gal(F /F ) de dimension 2 avec le d´eterminant de FrobF ﬁx´e. 6 Nous expliquons le rˆ ole du foncteur π → π I(1) des I(1)-invariants dans les d´emonstrations de (3.2), (3.3) et nous donnons des propri´et´es de ce foncteur [V0], [O]. 6.1 Le cas du groupe ﬁni GL(2, Fq ) [BL2] [Pa]. Un p-groupe de Sylow est le groupe des matrices strictement triangulaires sup´erieures U (Fq ); le foncteur ρ → ρU (Fq ) d´eﬁnit une bijection des repr´esentations irr´eductibles de GL(2, Fq ) sur Fp sur les modules simples de la Fp alg`ebre de Hecke de U (Fq ).

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6.2 Soit ρ une repr´esentation irr´eductible de GL(2, Fq ); le poids de ρ est le caract`ere η de T (F ) sur les invariants ρU (Fq ) (qui est de dimension 1); le copoids de ρ est le caract`ere η de T (F ) sur les coinvariants ρU (Fq ) (qui est de dimension 1); par adjonction, ρ est un quotient GL(n,Fq )

de indB(Fq )

GL(2,Fq )

η et une sous-representation de indB(Fq ) GL(n,F ) indB(Fq ) q

η . Comme la

GL(n,F ) est indB(Fq ) q −1 −1

contragr´ediente de η η −1 , les poids et copoids de la contragr´ediente de ρ sont (η ) et η . Explicitement, la repr´esen2 tation irr´eductible deta ⊗ Symr Fp pour 1 ≤ a ≤ q − 1, 0 ≤ r ≤ q − 1 a 2

pour contragr´ediente det−r−a ⊗ Symr Fp , et pour poids diag(1, ?) →?a ,

diag(?, ?−1 ) →?r ;

le poids est le caract`ere diag(x, z) → xc z d , les deux entiers 1 ≤ c, d ≤ q − 1 ´etant li´es aux deux entiers a, r par les congruences d ≡ a (mod q − 1), c ≡ a + r (mod q − 1). Les repr´esentations irr´eductibles non isomorphes 2 2 det a ⊗ Symr Fp , det a+r ⊗ Symq−1−r Fp , ont des poids conjugu´es par S2 ; le poids est ﬁxe par S2 si r = 0 ou r = q − 1. Le nombre de caract`eres de T (Fq ) modulo l’action naturelle du groupe sym´etrique S2 , i.e. de repr´esentations semi-simples de dimension 2 de F∗q sur Fp , est q(q − 1)/2 comme en (2.4). 6.3 Nous expliquons comment l’isomorphisme [BL2] prop.8: H(G, KpZ F , idFp ) # Fp [T ], le r´esultat crucial [BL1] prop.15, [BL2] prop.18: si E = 0 est une sousrepr´esentation de E(r), alors E I(1) est de codimension ﬁnie dans E(r)I(1) , et (4.5) impliquent qu’une repr´esentation irr´eductible V de G = GL(2, F ) sur Fp ayant un caract`ere central est quotient d’un V (r, λ, χ) (3.2). Le premier groupe de congruence K(1) des matrices congrues modulo pF `a l’identit´e dans K = GL(2, OF ), ´etant un pro-p-groupe, V contient une repr´esentation irr´eductible de K triviale sur K(1). On en d´eduit qu’il existe r, χ comme en (3.2) tel que V est quotient de E(r, χ) = (χ ◦ det) ⊗ E(r). On se ram`ene ` a χ trivial par torsion. Il existe un G-morphisme non nul E(r) → indG Bχ ∗

pour tout caract`ere χ : B → Fp , trivial sur diag(pF , pF ) et de restriction a` T (OF ) le copoids η de Symr (6.1), par adjonction [V4] I.5.7 et

Repr´esentations irr´eductibles de GL(2, F ) GL(2,Fq )

K(1) # indB(Fq ) l’isomorphisme (indG B χ)

557

η . Donc E(r) est r´eductible;

le r´esultat crucial implique que l’image de E(r)I(1) dans V I(1) est un sous-H(G, I(1), idFp )-module de dimension ﬁnie. Les alg`ebres de Hecke o H(G, I(1), id) et H(G, Ko , indK I(1) id) sont isomorphes et l’on a une sur-

2

o jection canonique indK I(1) id → Symr Fp . Les isomorphismes Fp [T ] # Z H(G, KpZ F , idFp ) # H(G, KpF , Symr ) et (4.5) impliquent alors que V est quotient d’un V (r, λ). 6.4 Structure de H(2, q) ⊗Z Fp et modules simples a` droite [V0]. L’alg`ebre H(2, q) ⊗Z Fp est une somme directe ⊕η1 ⊕η2 H(G, I, η), param´etr´e par les repr´esentations semi-simples η1 ⊕η2 de dimension 2 de u η = η1 ⊗η2 si η1 = η2 et η = F∗q sur Fp (les ”poids modulo S2 ” (6.1)), o` (η1 ⊗η2 )⊕(η2 ⊗η1 ) si η1 = η2 . Le nombre de facteurs est q(q−1)/2 comme en (2.4). Les modules simples sont g´en´eriquement d´etermin´es par leur ∗ caract`ere central; pour chaque facteur H(G, I, η), chaque (a, z) ∈ Fp ×Fp d´etermine un ou deux modules simples “jumeaux”; l’uniformisante pF agit par multiplication par z. a) Pour η1 = η2 , le facteur H(G, I, η) est isomorphe `a la Fp -alg`ebre de Hecke du groupe d’Iwahori I; elle est engendr´ee par les fonctions 0 1 0 1 respectivement, de support une S, T , ´egales `a 1 sur , 1 0 pF 0 double classe modulo I, v´eriﬁant les relations T 2 S = ST 2 , S 2 = −S; le centre est engendr´e par ST + T S + T, T ±2 . Les modules `a droite simples sont g´en´eriquement les modules M2 (a, z) de dimension 2, d´etermin´es par leur caract`ere central (ST + T S + T, T 2 ) → (a, z), o` u T, S agissent respectivement par 0 z −1 a , . 1 0 0 0

Les modules a` droite simples sont (i) M2 (a, z) si z = a2 , ∗ (ii) les caract`eres M1 (a, −1), M1 (a, 0) tels que T → a ∈ Fp et S → ε ∈ {−1, 0} sont respectivement contenus et quotients du module M2 (a, a2 ); ce sont des “jumeaux”. b) Pour η1 = η2 , le facteur H(G, I, η) est isomorphe `a M (2, R) o` uR ±1 e bre commutative engendr´ e e par Z , X, Y v´ e riﬁant XY = est la Fp -alg` 0 1 0. En eﬀet, normalise I et permute η1 ⊗ η2 et η2 ⊗ η1 , donc pF 0 G induit un isomorphisme indG I (η1 ⊗ η2 ) # indI (η2 ⊗ η1 ). L’isomorphisme ±1 support´ R # H(G, I, η1 ⊗ η2 ) envoie Z , X, Y sur les fonctions ees sur 0 1 0 p F ±1 une doubles classe modulo I et ´egales `a 1 sur pF , , , 0 1 0 pF respectivement. L’automorphisme de R d´eduit de l’isomorphisme

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H(G, I, η1 ⊗ η2 ) # H(G, I, η2 ⊗ η1 ) induit par

0 pF

1 0

ﬁxe Z et per-

mute X, Y . Les modules a` droite simples M2 (xa, ya, z) de M (2, R) sont de dimension 2, d´etermin´es par leur caract`ere central X → xa, Y → ya, Z → ∗ z, pour tout a ∈ Fp , z ∈ Fp et (x, y) = (1, 0) ou (0, 1). Si a = 0, les modules simples M2 (a, 0, z), M2 (0, a, z) sont “jumeaux”; ils n’ont pas le mˆeme caract`ere central. Si a = 0, on note M2 (0, z) = M2 (0, 0, z). L’isomorphisme R # H(G, I, η2 ⊗ η1 ) aurait permut´e les jumeaux. Comme les caract`eres η1 , η2 jouent des rˆ oles sym´etriques, il faut regrouper les “jumeaux” si l’on veut rester canonique. On d´eduit de a), b): Les modules simples supersinguliers de H(2, q) ⊗Z Fp sont les modu ules M2 (0, z) pour chacun des q(q − 1)/2 facteurs de H(2, q) ⊗Z Fp , o` ∗ z ∈ Fp est l’action de pF . On notera Mη,z le module simple supersinu pF agit par z. gulier H(2, q) ⊗Z Fp de poids η modulo S2 o` 6.5 On donne les I(1)-invariants des repr´esentations indG B χ, Sp, V (r, χ) de G sur Fp introduites en (3.2), en utilisant la classiﬁcation des H(2, q) ⊗Z Fp -modules simples (6.4). ∗ a) Soit χ : B → Fp un caract`ere de B d´eﬁni par un caract`ere χ1 ⊗χ2 du groupe diagonal. On pose a−1 = χ1 (pF ), z −1 = χ1 (pF )χ2 (pF ), η? = χ? |OF∗ . I(1) Le H(2, q) ⊗Z Fp -module `a droite (indG est de dimension 2, B χ) s’identiﬁe a` un module du facteur associ´e `a η1 ⊕ η2 ´egal `a M2 (a, z) si η1 = η2 et ` a M2 (a, 0, z) si η1 = η2 pour le choix de l’ordre (η1 , η2 ). On a I(1) I(1) M2 (a, 0, z) ⊕ M2 (0, a, z) = (indG ⊕ (indG B χ1 ⊗ χ2 ) B χ 2 ⊗ χ1 )

lorsque η1 = η2 . b) On a 1I = M1 (1, 0) et SpI = M1 (1, −1) [BL1] lemma 26. c) L’irr´eductibilit´e des repr´esentations V (r) = E(r)/T E(r) de GL(2, Qp ) , avec 0 ≤ r ≤ p − 1, se montre ainsi. La repr´esentation V (r) ´etant engendr´ee par V (r)I(1) , il suﬃt de montrer que V (r)I(1) est un H(2, p)-module simple. Breuil montre [Br] 4.1.2, 4.1.3: V (0) # V (p − 1), puis pour 0 ≤ r ≤ p − 2, que l’image par l’application E(r)I(1) → V (r)I(1) des fonctions A, B ∈ E(r)I(1) concentr´ees sur une classe de

Repr´esentations irr´eductibles de GL(2, F )

559

GL(2, OF )pZ F , avec r

A(id) = X ,

B

1 0

0

p−1 F

= Y r,

forment une base de V (r)I(1) [Br1 3.2.4, 4.1.4]; il est facile d’en d´eduire que le H(2, Qp ) ⊗Z Fp -module V (r)I(1) est le module simple M2 (0, 1) du facteur correspondant a` ω r ⊕ id. ∗ On tord par le caract`ere χ = ω a μλ avec 0 ≤ a ≤ p−2, λ ∈ Fp , et l’on voit que le H(2, Qp ) ⊗Z Fp -module V (r, χ)I(1) est le module M2 (0, λ2 ) du facteur correspondant a` ω a+r ⊕ ω a . 6.6 La d´ecomposition des induites paraboliques indG B χ (3.2), l’irr´eductibilit´e des V (r) et les isomorphismes entre les V (r, χ) si F = Qp (3.3), se d´eduisent de (6.3), (6.5), en appliquant le crit`ere d’irr´eductibilit´e (4.5). On obtient aussi les propri´et´es suivantes du foncteur des I(1)invariants: (i) Le foncteur π → π I(1) d´eﬁnit une bijection des repr´esentations irr´eductibles non supercuspidales de GL(2, F ) sur Fp , sur les H(2, q) ⊗Z Fp -modules simples non supersinguliers. (ii) Le foncteur π → π I(1) d´eﬁnit une bijection des repr´esentations irr´eductibles de GL(2, Qp ) sur Fp ayant un caract`ere central sur les H(2, p) ⊗Z Fp -modules simples. u pF (iii) Toute repr´esentation irr´eductible de GL(2, Qp ) sur Fp o` op`ere trivialement est admissible. e 6.7 Pour G = GL(2, F )/pZ F et p = 2, Rachel Ollivier [O] a montr´ que la bijection (6.6 (iii)) provient d’une ´equivalence de cat´egories: (i) Le module universel Fp [I\G] d’un Iwahori I est projectif sur l’alg`ebre de ses Fp [G]-endomorphismes. Le module universel Fp [I(1)\G] d’un pro-p-Iwahori I(1) est plat sur l’alg`ebre H(2, q) ⊗Z Fp de ses Fp [G]-endomorphismes, si et seulement si q = p. (ii) Lorsque F = Qp et p = 2, la cat´egorie des repr´esentations π de G sur Fp engendr´ees par π I(1) est ab´elienne et ´equivalente a` celle des H(2, p) ⊗Z Fp -modules a` droite sur lesquels p agit trivialement, par le foncteur des I(1)-invariants ?I(1) d’inverse ? ⊗H(2,p)⊗Z Fp Fp [I(1)\G]. (ii) est faux lorsque q = p ou lorsque p = 2 et F = Fp ((t)) est le corps des s´eries de Laurent en la variable t ` a coeﬃcients dans Fp , car l’espace des I(1)-invariants de M ⊗H(2,p)⊗Z Fp Fp [I(1)\G] est de dimension inﬁnie lorsque M est supersingulier. 7 Nous expliquons la construction par Paskunas [Pa] d’une repr´esentation irr´eductible admissible ayant un caract`ere central de G = GL(2, F ) sur Fp , tel que les I(1)-invariants du socle de sa restriction

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`a K = GL(2, OF ) est un H(2, q) ⊗Z Fp -module simple supersingulier quelconque (6.4). Une telle repr´esentation est supersinguli`ere (3.2), par (6.6)(i). 7.1 La construction part du principe que se donner une action de G sur un groupe ab´elien V est ´equivalent a` se donner une action sur V de 0 1 Z Z qui coincident Ko = pZ F K, et une action de K1 = pF I ∪ pF I pF 0 sur Ko ∩ K1 = pZ u I le sous-groupe d’Iwahori sup´erieur. Ceci se F I, o` d´emontre en utilisant l’arbre X de P GL(2, F ). Les groupes Ko , K1 sont les stabilisateurs dans GL(2, F ) d’un sommet et d’une arˆete contenant ce sommet; les actions compatibles de Ko et de K1 sur V d´eﬁnissent un syst`eme de coeﬃcients V sur X qui est G-equivariant, d’homologie a V ; la repr´esentation de G sur Ho (X, V) proHo (X, V) est isomorphe ` longe les actions de Ko et de K1 [Pa] 5.3.5. 7.2 Soit Mη,z le H(2, q) ⊗Z Fp -module `a droite simple supersingulier, de poids η modulo S2 o` u pF agit par z (6.3). Soit ρη la somme directe des deux repr´esentations irr´eductibles de GL(2, Fq ) de poids η modulo S2 (6.2), vue comme une repr´esentation de K triviale sur K(1); on note ρη,z l’action ´etendue `a Ko en faisant agir pF I(1) par z; les H(2, q) ⊗Z Fp -modules Mη,z et ρη,z sont isomorphes. Soit Inj ρη l’enveloppe injective de la repr´esentation ρη de K; on note Inj ρη,z l’action ´etendue `a Ko en faisant agir pF par z. Le point crucial est [Pa] 6.4 page 76: (i) Inj ρη,z est munie d’une action de K1 compatible avec celle de Ko , donc d’une action de G (7.1), telle que I(1)

(ii) l’inclusion ρη,z ⊂ (Inj ρη,z )I(1) est H(2, q) ⊗Z Fp -´equivariante. La repr´esentation de G recherch´ee est la repr´esentation πη,z engendr´ee par ρη,z dans la repr´esentation Inj ρη,z de G (i). Elle est irr´eductible par l’argument suivant. Si π est une sous-repr´esentation non nulle de πη,z , le socle de π |K est non nul et contenu dans le socle ρη,z I(1) de Inj ρη,z |K ; la simplicit´e de ρη,z comme H(2, q) ⊗Z Fp -module d´eduite I(1) de (ii), implique ρη,z ⊂ (π )I(1) ; comme ρη,z et πη,z sont engendr´es par I(1) ρη,z on d´eduit π = πη,z donc πη,z est irr´eductible. Les I(1)-invariants du socle de πη,z |K est isomorphe au H(2, q) ⊗Z I(1) Fp -module simple supersingulier Mη,z ; on ne sait pas si πη,z est isomorphe `a Mη,z . La repr´esentation Inj ρη,z de G est admissible (4.2), donc πη,z est admissible. 7.3 Paskunas [Pa] 6.2 construit un autre syst`eme de coeﬃcients G-´equivariant Vη,z sur l’arbre tel que les I(1)-invariants de tout quotient irr´eductible de la repr´esentation de G sur Ho (X, Vη,z ) contiennent I(1) Mη,z ; il est associ´e `a une action de K1 sur ρη telle que l’inclusion

Repr´esentations irr´eductibles de GL(2, F )

561

I(1)

ρη → ρη soit Ko ∩ K1 -´equivariante; la repr´esentation de K1 est iso1 morphe `a indK ηz , o` u η est relev´e en un caract`ere ηz de IpZ F sur lequel IpZ F pF agit par z. Toute repr´esentation irr´eductible π de G sur Fp telle que πI(1) = Mη,z , est quotient de Ho (X, Vη,z ). Si p = q ou si le poids η est ﬁxe par S2 , l’inclusion Mη,z ⊂ π I(1) suﬃt pour que π soit quotient de Ho (X, Vη,z ) [Pa] Cor.6.8, 6.10. 8 Nous montrons que la partie lisse de la contragr´ediente d’une Fp -repr´esentation irr´eductible lisse de GL(2, F ) ayant un caract`ere central et de dimension inﬁnie, est nulle. 8.1 Une forme lin´eaire lisse sur indG B χ est nulle, pour tout car∗ act`ere χ : B → Fp . Preuve. Pour tout entier n ≥ 1, on note K(n) le n-i`eme sous-groupe de congruence de GL(2, OF ). Le caract`ere χ est trivial sur le pro-p-groupe gK(n)g−1 ∩ B, aussi pour tout g ∈ G, il existe une fonction fg,n : G → Fp de support BgK(n) ´egale `a χ(b) sur bgK(n) pour tout b ∈ B. Comme K(n) normalise K(n+ 1), on a fgh,n+1 = h−1 fg,n+1 pour tout h ∈ K(n) et fg,n =

fgh,n+1

pour h ∈ (g −1 Bg ∩ K(n))K(n + 1)\K(n).

h

Une forme lin´eaire lisse L sur indG ee par un groupe de conB χ est ﬁx´ gruence assez petit. Il existe un entier r ≥ 1 tel que L(h−1 fg,n ) = L(fg,n ) pour touth ∈ K(n) et pour tout n ≥ r et g ∈ G. On en d´eduit L(fg,n ) = h L(h−1 fg,n+1 ) = 0, car les pro-p-groupes K(n) et (g −1 Bg ∩ K(n))K(n + 1) sont toujours distincts. Donc L = 0. 8.2 [L1] [L2] Une forme lin´eaire lisse sur E(0)/T E(0) (3.2) est nulle. Preuve. Soit Ko = pZ F GL(2, OF ). L’ensemble Xo des sommets de l’arbre de P GL(2, F ) est en bijection avec GL(2, F )/Ko . Une forme lin´eaire L sur E(0)/T E(0) s’identiﬁe a` une fonction f : Xo → Fp de somme nulle sur les voisins de chaque sommet. On note xo le sommet ﬁxe par Ko et C(?) l’ensemble des sommets `a distance ? de xo , pour tout entier ? ≥ 1; le sous-groupe de congruence K(?) ﬁxe chaque sommet de C(?) et agit transitivement sur les sommets de C(? + 1) se projetant sur le mˆeme sommet de C(?); tout sommet x ∈ C(?) est voisin d’un unique sommet x− ∈ C(? − 1) (avec C(0) = xo ) et de q sommets x1 , . . . , xq ∈ C(? + 1). Si L est lisse, il existe un entier r ≥ 1 tel que L est ﬁxe par K(r); alors pour tout x ∈ C(n + 1), on a L(x− ) + qL(x1 ) = L(x− ) = 0; donc L est nulle sur C(n) pour n ≥ r. Donc le support de L est ﬁni. Le mˆeme

562

Marie-France Vign´eras

argument montre que si L est nulle sur C(? + 1), alors L est nulle sur C(?) pour tout ? ≥ 0, car pour y ∈ C(?) il existe x ∈ C(? + 1) avec x− = y. Donc L = 0. 8.3 Il n’y a pas de forme lin´eaire lisse non nulle sur une repr´esentation irr´eductible de dimension inﬁnie de GL(2, F ) sur Fp ayant un caract`ere central. Preuve. Pour une s´erie principale ou une repr´esentation sp´eciale Sp par (8.1) et (3.2). Pour une supersinguli`ere par (3.2), (8.2) et sa g´en´eralisation que nous admettons: pour 0 ≤ r ≤ q−1, une forme lin´eaire lisse sur E(r)/T E(r) (3.2) est nulle (je ne l’ai pas v´eriﬁ´e si r = 0, mais Ron Livn´e dit l’avoir fait). 8.4 Soit π une repr´esentation irr´eductible de GL(2, F ) sur Fp de caract`ere central ωπ . Notons π ∗ = π ⊗ (ωπ−1 ◦ det). Avec les notations de (3.2), on a Sp∗ = Sp,

G −1 −1 ∗ (indG B (χ1 ⊗ χ2 )) = indB (χ2 ⊗ χ1 ).

Lorsque F = Qp , on a V (r, χ)∗ = V (r, ω −r χ−1 ). Pour π une repr´esentation irr´eductible de GL(2, F ) sur C, la contragr´ediente de π est isomorphe ` a π∗ [Bu] 4.2.2. 8.5 Lorsque F = Qp , le dual de Cartier d’une repr´esentation σ de Gal(Qp /Qp ) est son dual usuel tordu par le caract`ere ω (2.4). La correspondance σ ↔ π de Breuil (3.4) envoie le dual de Cartier de σ sur π ∗ et le d´eterminant de σ sur le caract`ere ωπ ω produit du caract`ere central de π par ω. 9 Remarques ﬁnales. Pour F = Qp , on s’attend a` ce qu’il existe d’autres Fp -repr´esentations irr´eductibles supersinguli`eres de GL(2, F ) que celles construites par Paskunas. Nous avons essay´e de d´egager les principes g´en´eraux des preuves de [BL], [Br], [Pa], dans le but d’une g´en´eralisation ´eventuelle. Certains r´esultats pr´esent´es ici sont d´eja ´etendus `a GL(3) [O], ou a` GL(n) ou mˆeme `a un groupe r´eductif g´en´eral.

Bibliographie modulo p Barthel Laure, Livn´e Ron, [BL1] Modular representations of GL2 of a local ﬁeld: the ordinary, unramiﬁed case. J. Number Theory 55 (1995), 1-27. [BL2] Irreducible modular representations of GL2 of a local ﬁeld. Duke Math. J. 75 (1994), 261-292.

Repr´esentations irr´eductibles de GL(2, F )

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[Be] Berger Laurent, Repr´esentations modulaires de GL2 (Qp ) et repr´esentations galoisiennes de dimension 2. Preprint 2005. [Br] Breuil Christophe, Sur quelques repr´esentations modulaires et padiques de GL2 (Qp ). I, Compositio Math. 138 (2003), 165-168. [L1] Livn´e Ron, lettre a` Joseph Bernstein Nov 1996. [L2] Livn´e Ron, lettre 31 oct 2000. [O] Ollivier Rachel, Modules sur l’alg`ebre de Hecke du pro-p-Iwahori de GLn (F ) en caract´eristique p. Th`ese 2005. [Pa] Paskunas Vytautas, Coeﬃcient systems and supersingular representations of GL2 (F ). M´emoires de la S.M.F. 99 (2004). Vign´eras Marie-France, [V0] Representations modulo p of the p-adic group GL(2, F ). Compositio Math. 140 (2004) 333-358. [V1] On a numerical Langlands correspondence modulo p with the pro-p-Iwahori Hecke ring. Mathematische Annalen 331 (2005), 523-556. Erratum: 333 (2005), 699-701. [Vc] Repr´esentations lisses irr´eductibles de GL(2, F ). Notes de cours. http://www.math.jussieu.fr/ vigneras/cours2MP22.pdf Bibliographie modulo = p [V2] Correspondance de Langlands semi-simple pour GLn (F ) modulo = p, Invent. Math. 144 (2001), 177-223. [V3] A propos d’une conjecture de Langlands modulaire. Dans “Finite Reductive Groups: Related Structures and Representations. Marc Cabanes, Editor. Birkhauser PM 141, 1997, 415-452. [V4] Repr´esentations -modulaires d’un groupe r´eductif p-adique avec = p. Birkh¨ auser PM137 (1996). [V5] Induced representations of reductive p-adic groups in characteristic l = p. Selecta Mathematica New Series 4 (1998) 549-623.

Bibliographie sur C [Bu] Bump Daniel, Automorphic forms and representations. Cambridge Studies in Advances Mathematics 55 (1997).

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