Hilbert Modular Forms and Iwasawa Theory (Oxford Mathematical Monographs)

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Author: Haruzo Hida

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OX FO R D M AT H E M AT I C A L M O N O G R A P H S Series Editors

J.M. BALL W.T. GOWERS N.J. HITCHIN L. NIRENBERG R. PENROSE A. WILES

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Hilbert Modular Forms and Iwasawa Theory HARUZO HIDA Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA

CLARENDON PRESS · OXFORD 2006

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Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With oﬃces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c H. Hida, 2006 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 0–19–857102–X

978–0–19–857102–5

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PREFACE When I was a toddler, my parents brought me to an esoteric Buddhist temple (Kongobu-ji “temple” of the Shingon Buddhist sect) in the southern hilly part of Osaka in Japan, where I saw a prototypical example of the set of twin mandala depicting Buddha’s twin universe of the inside and the outside, following the Shingon philosophy. I was utterly impressed by, or even obsessed with, the picture; afterwards, I was often bothered by nightmarish dreams somehow ﬁnding myself in one of the ghostly mandalas. This is something like placing oneself in between two mirrors, and then ﬁnding inﬁnitely many copies of oneself, and then one losing one’s identity of one’s whereabouts. One’s present state of existence is in confusion, common to ordinary people. When I started learning mathematics in the junior year of undergraduate study at Kyoto, I read a couple of books, starting with a book on linear partial diﬀerential equations, which is the ﬁrst serious book in mathematics I ever read (because of the student movement at the time, the university was virtually closed for my freshman and sophomore years; so, I was given almost no general undergraduate education including mathematics). I found in the books, a sort of universe neatly arranged, something like the mandala, but somehow, I felt that the Buddha sitting at the center (who presides over his world) was missing from the book. I then read, as the third book of mathematics, Shimura’s introduction to modular and automorphic forms [IAT], where I clearly saw a focus; so, I decided to pursue number theory, in particular, the theory of modular forms and automorphic forms. From that time on, I have been determined to create my own twin mandalas depicting my own mathematical twin worlds. I have revealed my determination/obsession only to a very small number of people in my life up until now, because I did not like to appear eccentric. If I remember correctly, in a queue at a cafeteria at Universit´e de Paris-Sud (Orsay) in 1984, I started a conversation with my fellow young French mathematicians about what kind of mathematicians we would like to be, and succinctly, I explained to them about the mandala and my obsession, and to my surprise, some of them (including Perrin-Riou and Tilouine) seemed somehow to understood the point, at least to some extent. When I arrived at Princeton (Institute for Advanced Study) as a postdoctoral fellow in 1979, I was fairly desperate, because I had not been able to ﬁnd even a clue about how to create a new universe cut out of, say, all elliptic modular forms (which appeared to me like looking into a pitch-dark well too deep to see through). I was solving small problems and giving answers as had been predicted. Small-problem solving gives me some pleasure but not much. After having spent a couple of months in Princeton, I was really desperate; so, I decided to do

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Preface

one more problem solving, ﬁnishing up the project (I started with Koji Doi) of relating congruences among Hecke eigenforms to (now called) the adjoint square L-value at s = 1. Trying to prove that mod p congruence of a Hecke eigenform with another implies that p is a factor of the L-value, somehow I found a p-adic projector (acting on modular forms) I named e for some reasons (which cut the clear surface out of the dark-well water) as the holomorphic projection of the L2 -space of functions on the hyperbolic Poincar´e upper half-plane kills all nonholomorphic functions, though I had only a guess of the precise meaning of the projector at the time. I admit that the non-p-ordinary modular forms are as equally important as the p-ordinary modular forms (which is in the image of e), as nonholomorphic automorphic forms are as important as holomorphic ones. The point is that this p-ordinary projector creates a world where p-adic deformation theory can be built in the neatest way. In this book, I try to describe the world of p-ordinary Hilbert modular forms and their deformation for which many theorems can be established easily, leaving the hard work of extending them to more general nonordinary automorphic forms to mathematicians more eﬃcient and ambitious. In this book, several results on ordinary modular forms are presented. First of all, I describe, in Chapter 3, Fujiwara’s (highly nontrivial) generalization [Fu] (to the Hilbert modular forms) of the proof by Wiles and Taylor of the identiﬁcation of an appropriate Hecke algebra and the corresponding universal Galois deformation ring (of Mazur). As a preparation to this, I give a detailed exposition of the theory of automorphic forms on a deﬁnite quaternion algebra, including the level-raising argument of R. Taylor. I do not touch the level-lowering arguments which might still be premature in book form. Thus the identiﬁcation of the Hecke algebra and the Galois deformation ring treated in this book is limited to minimally ramiﬁed deformations. After ﬁnishing this, we discuss three major applications that I found: 1. A description of Greenberg’s L-invariant of the adjoint square L-function, and its generic nonvanishing; 2. A solution to the integral basis problem of Eichler; 3. A proof of the torsion property of the (modular) adjoint square Selmer groups, and related Iwasawa modules. I have been studying all these topics since 1996 after I learned of Fujiwara’s work. I have written some papers on the subjects (at least for elliptic modular forms), but the treatment in this book is new and also covers more general cases. Some early chapters are from my graduate courses in 2002–2005 at UCLA and also from my lectures in Peking University in February 2004 and at the morning center of Mathematics at Beijing in August, 2004. I have been encouraged by many people (especially those who supported me in my desperate period).

Preface

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I would like to thank all these people including the audience in my lectures and the people at the above institutions. Haruzo Hida, Los Angeles, October, 2005

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Suggestions to the reader In the text, articles are quoted by abbreviating the author’s name, for example, three articles by Hida–Tilouine are quoted as [HT], [HT1] and [HT2]. There is one exception: articles written by myself are quoted, for example, as [H04a] and [H98] indicating also the year published (or the year written in the case of preprints). For these examples, [H04a] and [H98] are published in 2004 and in 1998, respectively. Books are quoted by abbreviating their title. For example, one of my earlier books with the title: Geometric Modular Forms and Elliptic Curves is quoted as [GME]. Our style of reference is slightly unconventional but has been used in my earlier books [MFG], [GME] and [PAF], and the abbreviation is (basically) common to all of the above three books. As for the notation and the terminology, we describe here some standard ones used at many places in this book. The symbol Zp denotes the p-adic integer ring inside the ﬁeld Qp of p-adic numbers, and the symbol Z(p) is used to indicate the valuation ring Zp ∩ Q. We ﬁx throughout the book an algebraic closure Q of Q. A subﬁeld E of Q is called a number ﬁeld (often assuming [E : Q] := dimQ E < ∞ tacitly). For a number ﬁeld E, OE denotes the integer ring of E, OE,p = OE ⊗Z Zp ⊂ Ep = E ⊗Q Qp and OE,(p) = OE ⊗Z Z(p) ⊂ E. Often we ﬁx a base ﬁeld denoted by F which is usually a totally real ﬁeld. For the base ﬁeld F , we simply write O = OF . A central simple algebra over F of dimension 4 is called a quaternion algebra over F , which is often denoted by D/F . A quadratic extension M/F is called a CM ﬁeld if F is totally real and M is totally imaginary. For a CM ﬁeld M , we write R for OM . The symbol W is exclusively used to indicate a valuation ring inside Q with residual characteristic p. The ring W could be of inﬁnite rank over Z(p) but with ﬁnite ramiﬁcation index over Z(p) ; so it is still discrete. The p-adic completion limn W/pn W is denoted by W , and we write Wm = W/pm W = W/pm W. ←− The symbol A denotes the adele ring of Q. For a subset Σ of rational primes, we set A(Σ∞) = {x ∈ A|x∞ = xp =0 for p ∈ Σ}. If Σ is empty, A(∞) denotes the ring of ﬁnite adeles. We put ZΣ = p∈Σ Zp and deﬁne Z(Σ) = ZΣ ∩Q. If Σ = {p} for a prime p, we write A(p∞) for A(Σ∞) . For a vector space of a number ﬁeld E, we write VA = V (A) and VA(Σ∞) = V (A(Σ∞) ) for V ⊗Q A and V ⊗Q A(Σ∞) , × respectively. We identify A(Σ∞) with the group of ideles x ∈ A× with xv = 1 for v ∈ Σ {∞} in an obvious way. The maximal compact subring of A(∞) is which is identiﬁed with the proﬁnite ring Zp = lim Z/N Z. denoted by Z, p ←−N (Σ) (Σ∞) (p) (p∞) We put Z =Z∩A and Z = Z ∩ A . For a module L of ﬁnite type, = lim L/N L, L (Σ) and L (p) . (Σ) = L ⊗Z Z (p) = L ⊗Z Z = L ⊗Z Z we write L ←−N An algebraic group T (deﬁned over a subring A of Q) is called a torus if its scalar extension T/Q = T ⊗R Q is isomorphic to a product Grm of copies of the multiplicative group Gm . The character group X ∗ (T ) = Homalg-gp (T/Q , Gm/Q ) is simply denoted by X(T ), and elements of X(T ) are often called weights of T .

ACKNOWLEDGEMENTS The author acknowledges partial support from the National Science Foundation (through the research grants: DMS 0244401 and DMS 0456252) and from the Clay Mathematics Institute as a Clay research scholar while he was ﬁnishing preparing the manuscript of this book at the Centre de Recherches Math´ematiques in Montr´eal (Canada) in September 2005.

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CONTENTS

1

Introduction 1.1 Classical Iwasawa theory 1.1.1 Galois theoretic interpretation of the class group 1.1.2 The Iwasawa algebra as a deformation ring 1.1.3 Pseudo-representations 1.1.4 Two-dimensional universal deformations 1.2 Selmer groups 1.2.1 Deligne’s rationality conjecture 1.2.2 Ordinary Galois representations 1.2.3 Greenberg’s Selmer groups 1.2.4 Selmer groups with general coeﬃcients 1.3 Deformation and adjoint square Selmer groups 1.3.1 Nearly ordinary deformation rings 1.3.2 Adjoint square Selmer groups and diﬀerentials 1.3.3 Universal deformation rings are noetherian 1.3.4 Elliptic modularity at a glance 1.4 Iwasawa theory for deformation rings 1.4.1 Galois action on deformation rings 1.4.2 Control of adjoint square Selmer groups 1.4.3 Λ-adic forms 1.5 Adjoint square L-invariants 1.5.1 Balanced Selmer groups 1.5.2 Greenberg’s L-invariant 1.5.3 Proof of Theorem 1.80

1 2 9 12 13 17 19 19 26 28 29 31 32 35 41 43 47 47 49 56 59 62 64 67

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Automorphic forms on inner forms of GL(2) 2.1 Quaternion algebras over a number ﬁeld 2.1.1 Quaternion algebras 2.1.2 Orders of quaternion algebras 2.2 A short review of algebraic geometry 2.2.1 Aﬃne schemes 2.2.2 Aﬃne algebraic groups 2.2.3 Schemes 2.3 Automorphic forms on quaternion algebras 2.3.1 Arithmetic quotients 2.3.2 Archimedean Hilbert modular forms 2.3.3 Hilbert modular forms with integral coeﬃcients 2.3.4 Duality and Hecke algebras

70 76 76 80 86 87 91 93 95 96 99 104 109

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Contents

2.3.5 Quaternionic automorphic forms 2.3.6 The Jacquet–Langlands correspondence 2.3.7 Local representations of GL(2) 2.3.8 Modular Galois representations The integral Jacquet–Langlands correspondence 2.4.1 Classical Hecke operators 2.4.2 Hecke algebras 2.4.3 Cohomological correspondences 2.4.4 Eichler–Shimura isomorphisms Theta series 2.5.1 Quaternionic theta series 2.5.2 Siegel’s theta series 2.5.3 Transformation formulas 2.5.4 Theta series of imaginary quadratic ﬁelds The basis problem of Eichler 2.6.1 The elliptic Jacquet–Langlands correspondence 2.6.2 Eichler’s integral correspondence

110 114 117 125 129 129 132 134 138 139 139 141 147 150 153 156 158

Hecke algebras as Galois deformation rings 3.1 Hecke algebras 3.1.1 Automorphic forms on deﬁnite quaternions 3.1.2 Hecke operators 3.1.3 Inner products 3.1.4 Ordinary Hecke algebras 3.1.5 Automorphic forms of higher weight 3.2 Galois deformation 3.2.1 Minimal deformation problems 3.2.2 Tangent spaces of local deformation functors 3.2.3 Taylor–Wiles systems 3.2.4 Hecke algebras are universal 3.2.5 Flat deformations 3.2.6 Freeness over the Hecke algebra 3.2.7 Hilbert modular basis problems 3.2.8 Locally cyclotomic deformation 3.2.9 Locally cyclotomic Hecke algebras 3.2.10 Global deformation over a p-adic ﬁeld 3.3 Base change 3.3.1 p-Ordinary Jacquet–Langlands correspondence 3.3.2 Base ﬁelds of odd degree 3.3.3 Automorphic base change 3.3.4 Galois base change 3.4 L-invariants of Hilbert modular forms 3.4.1 Statement of the result 3.4.2 Deformation without monodromy conditions

162 163 163 167 168 174 180 183 183 187 189 200 210 213 217 230 233 243 245 245 246 247 248 251 251 256

2.4

2.5

2.6

3

3.4.3 3.4.4 3.4.5 3.4.6 3.4.7

Contents

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Selmer groups of induced representations L-invariant of induced representations Adjoint square Selmer groups and diﬀerentials Proof of Theorem 3.73 Logarithm of the universal norm

262 265 274 279 283

4

Geometric modular forms 4.1 Modular curves 4.1.1 Modular curves and elliptic curves 4.1.2 Arithmetic Weierstrass theory 4.1.3 Moduli of level N 4.1.4 Toric action 4.1.5 Compactiﬁcation 4.1.6 Action of an adele group 4.2 Hilbert AVRM moduli 4.2.1 Abelian variety with real multiplication 4.2.2 AVRM moduli with level structure 4.2.3 Classical Hilbert modular forms 4.2.4 Toroidal compactiﬁcation 4.2.5 Tate AVRM 4.2.6 Hasse invariant 4.2.7 Geometric Hilbert modular forms 4.2.8 p-Adic Hilbert modular forms 4.2.9 Hecke operators 4.3 Hilbert modular Shimura varieties 4.3.1 Abelian varieties up to isogenies 4.3.2 Finite level structure 4.3.3 Modular varieties of level Γ0 (N) 4.3.4 Isogeny action 4.3.5 Reciprocity law at CM points 4.3.6 Hilbert modular Igusa towers 4.3.7 Finite level Hecke algebra 4.3.8 q-Expansion 4.3.9 Universal Hecke algebras 4.4 Exceptional zeros and extension 4.4.1 Λ-adic automorphic representations 4.4.2 Extensions of automorphic representations 4.4.3 Extensions of Galois representations

286 286 286 287 289 291 292 294 296 296 300 303 307 311 313 315 317 319 323 324 330 332 332 334 334 336 337 338 341 343 347 351

5

Modular Iwasawa theory 5.1 The cyclotomic tower of deformation rings 5.1.1 Control of deformation rings 5.1.2 K¨ ahler diﬀerentials as Iwasawa modules 5.1.3 Dimension of R∞

353 353 354 355 363

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Contents

5.2

5.3

Adjoint square exceptional zeros 5.2.1 Order of exceptional zeros 5.2.2 Base change of Selmer groups Torsion of Iwasawa modules for CM ﬁelds 5.3.1 Ordinary CM ﬁelds and their Iwasawa modules 5.3.2 Anticyclotomic Iwasawa modules 5.3.3 The L-invariant of CM ﬁelds

366 367 375 377 377 379 383

References

387

Symbol Index

397

Statement Index

399

Subject Index

401

1 INTRODUCTION

In this book, we study classical, p-adic and archimedean Hilbert modular and quaternionic automorphic forms and their Hecke algebras in detail, relating them to the (corresponding) universal Galois deformation rings and automorphic and modular L-values. So in this introductory chapter, let us describe some of the focal points treated in this book. This book is, in some sense, a sequel of [MFG] and [PAF], where a general introduction to the p-adic deformation theory of Galois representations and automorphic forms on Shimura varieties is given. In [MFG], a detailed proof (following Wiles’ fundamental work [Wi2]) of the identiﬁcation of the (elliptic modular) Hecke algebra and the Galois deformation ring is given (and this identity of the two naturally given algebras is one of the principal ingredients of his proof of Fermat’s last theorem). In [PAF], the emphasis is put more on the automorphic and geometric side of the theory of the p-adic Hecke algebra (and Shimura varieties). Here, we describe Iwasawa-theoretic aspects of the theory along with the generalization of Wiles’ identiﬁcation to the Hilbert modular case ﬁrst worked out by Fujiwara in [Fu]. To be more precise, for a nearly p-ordinary two-dimensional Galois representation ρ associated to a Hilbert modular Hecke eigenform, we discuss the following four topics: 1. the identiﬁcation of the local ring of ρ of an appropriate p-adic Hecke algebra with the universal Galois deformation ring of ρ (the Taylor–Wiles method worked out by Fujiwara [Fu]; Chapter 3); 2. torsion property (over the Iwasawa algebra) of the adjoint square Selmer groups of Hilbert modular forms and the anticyclotomic Iwasawa modules associated with CM ﬁelds (Chapter 5); 3. the L-invariant of the adjoint square of ρ (Section 3.4), and its relation to the tower of Galois deformation rings and Hecke algebras over the cyclotomic Zp -extension (modular Iwasawa theory; Chapter 5) and to nontrivial extensions of p-adic automorphic representations (Section 4.4); 4. the basis problem of Eichler (Sections 2.6 and 3.2.7). The second, third, and fourth items are direct applications of the ﬁrst. The principal new results besides the ﬁrst topic in this book are Theorems 3.47, 3.59, 3.73, 4.29, 5.9, 5.27 and 5.33 and Corollaries 3.74, 4.32 and 5.39. Chapter 4 is a long summary of the results on geometric Hilbert modular forms

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Introduction

described in [PAF] (which are used throughout this book), though added in Section 4.4 are some new results on the close relation between nontrivial extensions of automorphic representations and the exceptional zeros of adjoint square p-adic L-functions. A natural path to follow after describing these topics is (a) the problem of nonvanishing modulo p of abelian critical L-values as an application of the theory of Hilbert modular Shimura varieties treated in [PAF] Chapter 3 (see [H04b] and [H05b]); (b) the proof of the anticyclotomic main conjecture in Iwasawa theory for CM ﬁelds under appropriate assumptions (see [H05d]). In the near future, we hope to treat these two topics (a) and (b) in book form. In this chapter, we shall give a brief outline of the ﬁrst three topics (1–3). The fourth topic is to relate quarternionic automorphic forms and elliptic and Hilbert modular forms through theta series, which will be discussed in Chapter 2. Since this is an introductory chapter putting forward special cases of the results described in detail in the later chapters, often we give only a sketch of a proof and possibly even omit proofs (which will be given later in a more general setting), or we shall just content ourselves with indicating the place where a proof can be found. Many open questions are also discussed here. We ﬁx a prime p > 2, algebraic closures Q of Q and Qp of Qp and the ﬁeld embeddings ip : Q → Qp and i∞ : Q → C.

1.1 Classical Iwasawa theory Before starting with a review of classical Iwasawa theory, we describe the group of roots of unity as a group scheme. A brief summary of schemes and group schemes will be given in Section 2.2; so, here we limit ourselves to the group, denoted by µN , of N -th roots of unity (which is the main subject of research in classical Iwasawa theory). We regard µN for a positive integer N as a covariant functor from the category ALG of commutative algebras with identity into the category AB of abelian groups. See [MFG] Section 4.1 or [GME] Section 1.4 for a brief description of functors. Thus µN associates with an algebra A the commutative group µN (A) = {ζ ∈ A× |ζ N = 1} for the identity 1 ∈ A. For two algebras A and B, the set of homomorphisms Homalg (A, B) is made up of ring homomorphisms taking the identity of A to the identity of B. Then for φ ∈ Homalg (A, B), the corresponding group homomorphism φA : µN (A) → µN (B) is given by µN (A) ζ → φ(ζ) ∈ µN (B), and this gives rise to the covariant functoriality. For any given algebra R, we can think of a covariant functor ALG → SET S, written as Spec(R), given by Spec(R)(A) = Homalg (R, A). For φ ∈ Homalg (A, B) and ϕ ∈ Spec(R)(A), the corresponding map φA : Spec(R)(A) → Spec(R)(B) is given by φA (ϕ) = φ ◦ ϕ. The functor Spec(R) is called the aﬃne scheme of the algebra R; we will discuss schemes more fully in Section 2.2.

Classical Iwasawa theory

3

Let W be a complete discrete p-adic valuation ring ﬂat over Zp with residue ﬁeld F. If R is a p-proﬁnite W -algebra, we can think of a slightly diﬀerent covariant functor Spf(R) from the category of p-proﬁnite algebras into SET S called an aﬃne formal scheme under the p-proﬁnite topology. For any p-proﬁnite W -algebra R, Spf(R) is called the formal spectrum of R. This is to impose the p-proﬁnite continuity on morphisms we consider. Thus for any p-proﬁnite W -algebra A, Spf(R)(A) = HomW -alg (R, A) made up of continuous W -algebra homomorphisms. The category of p-proﬁnite W -algebras is then sent into the category of formal aﬃne spectra faithfully by the contravariant functor R → Spf(R). In particular, we have Hom(Spf(R), Spf(B)) = HomW -alg (B, R), where W -algebra homomorphisms are assumed to be continuous with respect to the p-proﬁnite topology. For two covariant functors Φ, Ψ : ALG → C into a category C, φ : Φ → Ψ is a morphism (of covariant functors) if we have morphisms φA : Φ(A) → Ψ(A) of C indexed by A ∈ ALG such that the following diagram is commutative for each algebra homomorphism ρ : A → B: φA

Φ(A) −−−−→ Ψ(A)   Ψ(ρ)  Φ(ρ) φB

Φ(B) −−−−→ Ψ(B), where Φ(ρ) : Φ(A) → Φ(B) is the functorial map associated with ρ; in particular, these maps satisfy Φ(ρ ◦ η) = Φ(ρ) ◦ Φ(η) by covariant functoriality. If we have an inverse morphism ψ : Ψ → Φ such that φA ◦ ψA and ψA ◦ φA are identity morphisms in C for all A, we call Φ and Ψ isomorphic. The covariant functor µN has a special property that ιA : µN (A) ∼ = Homald (Z[X]/((1 + X)N − 1), A)) via µN (A) ζ ↔ ζ ∈ Homald (Z[X]/((1 + X)N − 1), A)) if ζ(1 + X) = ζ. We verify that ιB ◦ φA (ζ) = φA ◦ ζ for any φ ∈ Homalg (A, B) (so, ζ ↔ ζ gives rise to an isomorphism of covariant functors). In such a case, we say that the functor µN is represented by the aﬃne scheme Spec(R) with R = Z[X]/((1 + X)N − 1). We write µN = Spec(Z[X]/((1 + X)N − 1)) and identify the two functors Spec(R) and µN . Since µN is actually a functor into the subcategory AB of SET S and is identiﬁed with Spec(Z[X]/((1 + X)N − 1)), we call µN an aﬃne commutative group scheme, whose A-points µN (A) are made of N -th roots of unity in A (e.g. [GME] Example 1.6.3). This is a point of view regarding a scheme S/Z as an association to each commutative algebra A of its A-points S(A). Mathematically, A → S(A) is a covariant functor from the category ALG of (commutative) algebras (with identity) into the category of sets. If a scheme G/Z actually has values in the category GP of groups, G is called a group scheme. Since the ring Z[X]/((1 + X)N − 1) is free of ﬁnite rank, µN is called a locally free (actually free) group scheme.

4

Introduction

Fixing a base algebra B, we can think of the notion of the B-aﬃne scheme S = Spec(R)/B for a B-algebra R, which is by deﬁnition the covariant functor A → S/B (A) = HomB-alg (R, A) from the category of B-algebras into SET S. If S has values in GP , S is called a group B-scheme. Extending the notion of local freeness, if an aﬃne group B-scheme A → G(A) = HomB-alg (R, A) over a base ring B is given by a B-algebra R which is locally free of ﬁnite rank, we call G locally free of ﬁnite rank over B (or ﬁnite ﬂat over B). We call two group schemes G/B and H/B isomorphic if they are isomorphic as two covariant functors (from B-algebras) into GP . We put µp∞ = n µpn . Thus µp∞ (A) is the group of all p-power roots of unity in A, and if A is suﬃciently large, µp∞ (A) is p-divisible; so, it is an example of a special group scheme called a Barsotti–Tate group. An aﬃne group scheme G/B is called a Barsotti–Tate group G over B if G/B is p-divisible, is given by limn G[pn ]/B for its pn -torsion subgroups G[pn ], and G[pn ] is locally free of ﬁnite −→ rank over B (see [T], [CBT] and [ARG] Chapter III for such groups). If µp∞ (A) is p-divisible, it is not ﬁnitely generated, though µN (A) for ﬁnite N is a cyclic group if A is a domain of characteristic prime to N . Exercise 1.1 An abelian group G is called divisible if x → nx is surjective for any nonzero integer n. Show that a nonzero divisible group G is not ﬁnitely generated as a group. Hint: use the fundamental theorem of abelian groups. We can think of the constant functor A → ΦN (A) = Z/N Z for any algebra A, which is a functor from ALG to GP (with ΦN (ρ) given by the identity map idZ/N Z of Z/N Z for any ρ ∈ Homalg (A, B)). This functor for N > 1 is not isomorphic to the group scheme µN , because µp (Fp ) = {1} for a prime p but Φp (Fp ) = (Z/pZ) = {1}. Exercise 1.2 Show that ΦN is not of the form Spec(R) for a commutative ring R (thus, ΦN is not an aﬃne scheme). Propose a modiﬁcation of ΦN which is an aﬃne scheme Spec(R) and Spec(R)(A) = ΦN (A) for any indecomposable commutative algebras A. If A is a subring of an algebraically closed ﬁeld B of characteristic diﬀerent from p (for example, B = Qp or C), we often write A[µpn ] for the extension of A generated by µpn (B). We have the p-adic cyclotomic character σ N (σ) for all ζ ∈ µp∞ (Q), and for the N : Gal(Q[µp∞ ]/Q) ∼ = Z× p given by ζ = ζ × unique subgroup µp−1 (Zp ) ⊂ Zp , we write Q∞ for the subﬁeld of Q[µp∞ ] ﬁxed by Zp ∼ ∼ N −1 (µp−1 (Zp )). Then Gal(Q ∞∞/Q) s= Γn = 1 + pZp . Note that Γ = (1 + p) = Zp s via Zp s → (1 + p) = n=0 n p ∈ Γ, and Q∞ is called the cyclotomic Zp -extension over Q. For a given number ﬁeld M , the composite M∞ = Q∞ M is called the cyclotomic Zp -extension of M . In the late 1950s, Iwasawa started studying the arithmetic of Zp -extensions (particularly the cyclotomic ones) and created many important proﬁnite modules X with a continuous action of Γ, which are often called Iwasawa modules, and

Classical Iwasawa theory

5

it turned out to be a success, bringing us new knowledge of the arithmetic of Zp -extensions (see, for example, [Iw]). The book [ICF] by Washington is a good introduction to Iwasawa theory. For any complete valuation ring W ﬁnite ﬂat over Zp with residue ﬁeld F, the completed group algebra n n W [[Γ]] := lim W [Γ/Γp ] = lim W [x]/((1 + x)p − 1) ∼ = W [[x]] ←− ← − n n

acts on X (if X is a W -module in addition to being a Γ-module). Indeed, by the continuity of the action, we may assume that X = limn Xn for a ﬁnite ←− n module Xn over the ﬁnite group Γ/Γp , and the action is compatible with respect n to the projections Γ Γ/Γp and X Xn . Passing to the limit, W [[Γ]] = pn limn W [Γ/Γ ] acts on X = limn Xn . We write Λ = ΛW for W [[Γ]], which is a ←− ←− local ring with maximal ideal mΛ = mW + (x) and Λ/mΛ = F. n n ∼ W [[x]]. Exercise 1.3 Prove limn W [Γ/Γp ] = limn W [x]/((1 + x)p − 1) = ←− ←− n pn Hint: choose a generator γ ∈ Γ and associate with (γ mod Γ ) ∈ W [Γ/Γp ] n n the element (1 + x mod (1 + x)p − 1) ∈ W [x]/((1 + x)p − 1). n n We will later give a proof of limn W [Γ/Γp ] = limn W [x]/((1 + x)p − 1) ∼ = ←− ←− W [[x]] as Corollary 1.20 via Galois deformation theory; so, the point of the above exercise is to give a ring theoretic proof of this fact. Let us discuss a prototypical example of Iwasawa modules. In this discussion, we assume W = Zp . For simplicity, we assume only in this section that M/Q is at most tamely ramiﬁed at p. Here a prime is tamely ramiﬁed in M/Q if its ramiﬁcation index is prime to p. Since p fully ramiﬁes in the union of p-extensions Q∞ /Q, M and Q∞ are linearly disjoint; so, we have Gal(M∞ /M ) ∼ = Γ, canonically, by restricting σ ∈ Gal(M∞ /M ) to Q∞ . The Zp -extension M∞ /M has layers n of intermediate ﬁelds Mn ﬁxed by Γn = Γp . Let Cn be the p-Sylow subgroup of the class group of Mn . By Galois conjugation, Gal(Mn /M ) = Γ/Γn ∼ = Z/pn Z acts on Cn . The norm map Nm,n : Cm → Cn (m > n) is compatible with the Galois action, and we can form the projective limit C∞ = limn Cn with respect ←− to the norm map. Since Cn is a ﬁnite p-group, C∞ is a p-proﬁnite (compact) group with a continuous Γ-action. Thus we get an Iwasawa module C∞ , which is a module over Λ. By class ﬁeld theory, Cn is isomorphic to the Galois group Gal(Hn /Mn ) of p-Hilbert class ﬁeld Hn by the Artin reciprocity map. Since Hm ⊃ Hn (m > n), we have the restriction map Resm,n : Gal(Hm /Mm ) → Gal(Hn /Mn ) which is surjective. By class ﬁeld theory, we have the following commutative diagram:

∼

Gal(Hm /Mm ) −−−−→   Resm,n ∼

Cm  N m,n

Gal(Hn /Mn ) −−−−→ Cn .

6

Introduction

Thus C∞ is canonically isomorphic to X = Gal(H∞ /M∞ ), where H∞ /M∞ is the maximal unramiﬁed p-abelian extension. The action of σ ∈ Gal(Mm /M ) on of σ to Hm /M and conjugating Gal(Hm /Mm ) is given by taking an extension σ : τ → σ τ σ −1 . Since Gal(Hm /Mm ) is commutative, this τ ∈ Gal(Hm /Mm ) by σ action of σ is independent of the choice of the extension σ . Since Hn Mm is a maximal abelian extension of Mn inside Hm , the Galois group Gal(Hm /Hn Mm ) is the derived group of Gal(Hm /Mn ). Writing γ = 1 + p and identifying Γ n with Gal(M∞ /M ), Gal(Mm /Mn ) is generated by γ p ; so, the derived group Gal(Hm /Hn Mm ) is generated by the commutators p τ γ −p τ −1 = (γ p − 1)τ [ γp , τ ] = γ n

n

n

n

(writing additively on the right-hand side). This shows Gal(Hm /Hn Mm ) ∼ = (γ p − 1)Cm . n

Since Mm /Mn fully ramiﬁes at p, we have Gal(Hn Mm /Mm ) ∼ = Gal(Hn /Mn ). Thus we have the following control theorem (cf. [Iw] Sections 3–4 and [ICF] Section 13.3): Theorem 1.4 (Control)

We have C∞ ⊗Λ Λ/(γ p − 1)Λ ∼ = Cn for all n > 0. n

n Note that Λ ∼ = Zp [[x]] by γ → 1 + x and (1 + x)p − 1 is in the unique maximal ideal of Λ, by Nakayama’s lemma, and the ﬁniteness of Cn tells us the following.

Corollary 1.5

The Iwasawa module C∞ is a torsion Λ-module of ﬁnite type.

Nakayama’s lemma we referred to is the third assertion of the following lemma, which is actually due to Krull and Azumaya (according to Nakayama as Nagata wrote in his books on ring theory): Lemma 1.6 (Krull–Azumaya, Nakayama) imal ideal mR and X be an R-module.

Let R be a local ring with max-

1. If X is an R-module of ﬁnite type and X = mR X, then X = 0. 2. If mN R X = 0 for a suﬃciently large integer N , then X = mR X implies X = 0. 3. Suppose that R = limn R/mnR is an mR -adically complete local ring with ←− ﬁnite R/mnR for all n > 0 and that X is a continuous R-module under the mR -adic topology. Then (a) X = mR X implies X = 0, (b) if X ⊗R R/mR is ﬁnite dimensional over R/mR , X is an R-module of ﬁnite type, (c) if further R is a noetherian normal domain, X is an R-module of ﬁnite type and X/xX is a torsion R/xR-module for a prime element x ∈ R, X is a torsion R-module of ﬁnite type.

Classical Iwasawa theory

7

Proof For the proof of the ﬁrst two assertions, see [MFG] Lemma 1.3. The assertion (a) of the third assertion follows from (2) applied to quotients X/mnR X and passing to the limit. As for (b), take elements x1 , . . . , xr in X whose classes modulo mR X give rise to a basis of X/mR X. Consider the R-linear map π : Rr → X given by n n r π(a1 , . . . , ar ) = i ai xi . Applying (2) to Coker((π mod mR ) : (R/mR ) → n n X/mR ), we ﬁnd that (π mod mR ) is surjective for all n, and passing to the limit, we ﬁnd that Im(π) is mR -adically dense in X. Since Rr is compact (because of ﬁniteness of R/mnR ), Im(π) is closed, and hence π is surjective; so, X is of ﬁnite type. As for (c), we note that the localization R(x) is a discrete valuation ring (because a normal noetherian integral domain of dimension 1 is a discrete valuation ring; e.g., [CRT]). Then that X(x) /xX(x) is a torsion R(x) /(x)-module means X(x) /xX(x) = 0, which implies X(x) = 0 by (2). Then X has to be a torsion R-module. 2 Let F be a totally real ﬁeld of ﬁnite degree with integer ring O. We now assume that M = F [µp ] (note that p is tamely ramiﬁed in M/Q). Then we have Gal(M/F ) → µp−1 by the Teichm¨ uller character ω. Again by conjugation, Gal(M/F ) acts on C∞ . Let Cn [ω i ] = {x ∈ Cn |σ(x) = ω i (σ)x} (n = 0, 1, . . . , ∞). n The limit ω = limn→∞ N p exists, giving a character ω : Gal(Q/F ) → n n−1 µp−1 (Zp ), because (Z/pn Z) has order pn − pn−1 (so, N p −p ≡ 1 mod pn ). ∞ The limit ω is the Teichm¨ uller character of Gal(M [µp ]/M ) and we regard it as an ideal character by class ﬁeld theory so that ω(l) = ω(F robl ) for the Frobenius element F robl for prime ideal l outside p. We put ω(p) = 0 if p|p (but ω n (p) = 1 for all p including p|p if ω n is the trivial character). Write χ = ω a for a ﬁxed integer a, and regard it as a complex valued character by composing i∞ ◦ i−1 p . Then we consider the complex L-function

−1

χ(l) −s 1− L(s, χ) = χ(a)N (a) = N (l)s a l

absolutely and locally uniformly convergent on the right half-plane deﬁned by Re(s) > 1, where l runs over all prime ideals of O and a runs over all integral ideals of O. This L-function can be continued to the entire complex plane as a complex meromorphic function having pole at s = 1 only when χ is the trivial character (e.g., [LFE] Theorem 2.5.1). Since F is totally real, as was shown by Siegel, Klingen and Shintani (see [LFE] Corollary 2.5.1), L(1 − n, χ) ∈ Z(p) [µp−1 (C)] for Z(p) = Q ∩ Zp in Qp . Embed Z(p) [µp−1 (C)] into Zp by ip ◦ i−1 ∞ , and regard L(1 − n, χ) ∈ Zp . In the late 1970s, Deligne and Ribet [DR] (and Barsky and Cassou-Nogu´es; see [LFE] Chapter 3) constructed

8

Introduction

a power series ΦF (x; ω i ) ∈ Λ such that for even integer i with 0 ≤ i < p − 1 ΦF (γ 1−n − 1; ω i ) (γ 1−n − 1) p|p (1 − ω −n (p)N (p)n−1 )L(1 − n, ω −n ) = i−n (p)N (p)n−1 )L(1 − n, ω i−n ) p|p (1 − ω

if i = 0 if i > 0

for all n > 0. Since L(s, χ) satisﬁes a functional equation of the form s ↔ 1 − s with the Γ-factor having a pole at 1 − n if χ = ω i−n for odd i (e.g., [LFE] Section 8.6), the value L(1 − n, ω i−n ) vanishes for odd i, and therefore, we only look into even i. Here we insert a general fact valid for integral domains of dimension ≤ 2. Over a principal ideal domain A, any torsion module X of ﬁnite type is isomorphic to e(P ) for ﬁnitely many prime elements P . For a regular local ring A of P A/P dimension two (including Λ), if X is a torsion A-module of ﬁnite type, we can ﬁnd a homomorphism ι : X → P A/P e(P ) A for ﬁnitely many prime elements P with small kernel and cokernel (e.g. [ICF] Section 13.2 or [BCM] VII.4.4). When A = Λ, the word “small” means “ﬁnite.” A generator of the ideal P P e(P ) is called the characteristic element in A and the characteristic power series of X when A = Λ, which is uniquely determined modulo Λ× . Here is a theorem of A. Wiles [Wi1] which was originally called the main conjecture of the Iwasawa theory for F : Theorem 1.7 (A. Wiles) Let F be a totally real number ﬁeld of ﬁnite degree. Then we have char(C∞ [ω 1−i ])(x) = ΦF (x; ω i ) for even i up to a unit multiple in Λ for an odd prime p. This theorem was originally conjectured by Iwasawa and was proved by Mazur and Wiles for F = Q in [MzW]. Later Wiles gave a proof valid for general totally real F in [Wi1]. It appears to have a mismatch because the ω 1−i -part C∞ [ω 1−i ] is described by the L-function of ω i . However, we may regard L(1 − n, ω i−n ) as a function of characters N n−1 ω i−n ≡ ω i−1 mod p; so, the ω 1−i -part C∞ [ω 1−i ] is described by the L-function whose domain is made up of characters congruent to ω i−1 modulo p. In [H05d], a theorem similar to this one is given, taking a p-ordinary CM ﬁeld to be a base ﬁeld in place of the totally real ﬁeld F in the above theorem. For such a generalization, we need to deal with modules over a complete regular local ring with dimension ≥2. Indeed, we can generalize the construction of the characteristic power series to a more general ring R. We consider a formal power series ring W [[x1 , . . . , xg ]] and assume that R is a normal integral domain ﬁnite and torsion-free over W [[x1 , . . . , xg ]]. Then a prime ideal P of R is called a prime divisor if dim R/P = dim R − 1 for the Krull dimension dim R of R. In this situation, the localization RP of R at a prime divisor P is a discrete valuation ring. Then for a torsion R-module X of ﬁnite type, we have that XP = X ⊗R RP has ﬁnite length over RP . We write P (X) for the

Classical Iwasawa theory

9

length. Then char(X) = P P P (X) is an ideal of R, which is called the characteristic ideal of X. When R = W [[x1 , . . . , xg ]], the characteristic ideal of X is generated by the characteristic power series char(X)(x) ∈ W [[x1 , . . . , xg ]] (see [BCM] Chapter 7). Basically by deﬁnition, we have a morphism of R-modules i : X → P R/P e(P ) whose cokernel and kernel vanish after localization at every prime divisor (and this gives a precise deﬁnition of the kernel and the cokernel of i being “small”). An R-module M is said to be pseudo-null if MP = 0 for all prime divisors P . Exercise 1.8 Prove that for a Λ-module M of ﬁnite type, MP = 0 for all prime divisors P of Λ if and only if the order of M is ﬁnite. 1.1.1 Galois theoretic interpretation of the class group In order to give a Galois theoretic generalization of the class group, we introduce here brieﬂy Galois cohomology groups. Let Tp = Qp /Zp . For any abelian p-proﬁnite compact or p-torsion discrete module X, we deﬁne the Pontryagin dual module X ∗ by X ∗ = Homcont (X, Tp ) and give X ∗ the topology of uniform convergence on every compact subgroup of X. By Pontryagin’s duality theory (cf. [FAN] or [LFE] Section 8.3), we have (X ∗ )∗ ∼ = X canonically. Exercise 1.9 Show that X ∗ ∼ = X noncanonically if X is ﬁnite. Exercise 1.10 Prove that X ∗ is a discrete module if X is p-proﬁnite and X ∗ is compact if X is discrete (e.g., [LFE] Lemma 8.3.1). By this fact, if X ∗ is the dual of a proﬁnite module X = limn Xn for ﬁnite ←− modules Xn with surjections Xm Xn for m > n, X ∗ = n Xn∗ is a discrete module which is a union of ﬁnite modules Xn∗ . For the (proﬁnite) Galois group G over a ﬁeld inside Q or a p-adic ﬁeld inside Qp and a continuous G-module X, we denote by H q (G, X) the continuous group cohomology with coeﬃcients in X. If X is ﬁnite, H q (G, X) is as deﬁned in [MFG] 4.3.3. If X = limn Xn for ﬁnite G-modules Xn , we ←− have H q (G, X) = limn H q (G, Xn ), and if X = n Xn is discrete with ﬁnite ← − Xn , we have H q (G, X) = limn H q (G, Xn ). For a general K-vector space V −→ with a continuous action of G and a G-stable W -lattice L of V , we deﬁne H q (G, V ) = H q (G, L) ⊗W K. For a ﬁnite set S of rational primes, let F S /F be the maximal extension unramiﬁed outside S and the archimedean place of F , and write GSF for Gal(F S /F ). For a ﬁnite set S of rational primes containing all ramiﬁed primes in F/Q, we have QS = F S . By a result of Tate (e.g., [MFG] 4.4.2), GSF has (virtual) cohomological dimension 2; so, only H 0 , H 1 , and H 2 are important. We have H 0 (G, X) = X G = {x ∈ X|gx = x for all g ∈ G},

10

Introduction

and if X is ﬁnite, the ﬁrst cohomology is deﬁned by c

H 1 (G, X) =

→ X : continuous|c(στ ) = σc(τ ) + c(σ) for all σ, τ ∈ G} {G − b

{G − → X|b(σ) = (σ − 1)x for x ∈ X independent of σ}

.

As for the second cohomology, 2-cocycles c : G×G → X are continuous functions satisfying the following relation: c(α, β) + c(αβ, γ) = α · c(β, γ) + c(α, βγ) for all α, β, γ ∈ G. For any continuous function b : G → X, ∂b(α, β) = b(αβ) − αb(β) − b(α) is easily checked to be a 2-cocycle by computation. Such 2-cocycles obtained from b : G → X are called a 2-coboundary. Then H 2 (G, X) =

{2-cocycles with values in X} . {2-coboundaries with values in X}

Exercise 1.11 Check that ∂b as above is a 2-cocycle. If G = Gal(Qp /K) for a ﬁnite extension K/Qp , by Tate duality (see [MFG] Example 4.42 and Theorem 4.43), H 2−i (G, X) ∼ = Hom(H i (G, X ∗ (1)), Q/Z) for ﬁnite X. For each Galois character ψ : Gal(Q/F ) → W × , we write ψ for the F× valued character ψ mod mW for the maximal ideal mW of W . For any W -module X, we write X(ψ) for the Galois module whose underlying W -module is X with Galois action given by ψ. We simply write X(i) for X(N i ). In particular Zp (1) ∼ µ n (Q) as Galois modules. Note that GK = GpK = Gal(F p /K) = lim ←−n p for any intermediate ﬁeld K of F p /F , where F p /F is the maximal extension unramiﬁed outside p and ∞. Fix an integer i as above, and put ψ = ω 1−i . We consider the Galois cohomology group H 1 (GF , Tp (ψ)) and deﬁne the Selmer group of Tp (ψ) over K by Res H 1 (Il , Tp (ψ))), (1.1.1) SelK (Tp (ψ)) = Ker(H 1 (GK , Tp (ψ)) −−→ l

where l runs over all prime ideals of K and Il is a chosen inertia subgroup at l of GK . By the inﬂation-restriction sequence (e.g., [MFG] 4.3.4), 0 → H 1 (Gal(M/F ), H 0 (GM , Tp (ψ))) → H 1 (GF , Tp (ψ)) → H 0 (Gal(M/F ), H 1 (GM , Tp (ψ))) → H 2 (Gal(M/F ), H 0 (GM , Tp (ψ))) is exact. Since Gal(M/F ) (M = F [µp ]) has order prime to p, H 1 (Gal(M/F ), H 0 (GM , Tp (ψ))) = H 2 (Gal(M/F ), H 0 (GM , Tp (ψ))) = 0.

Classical Iwasawa theory

11

Thus we have ∼ H 1 (GF , Tp (ψ)) ∼ → HomGF (GM , Tp (ψ)). = H 0 (Gal(M/F ), H 1 (GM , Tp (ψ))) − ι

Since Hm /Mm is the maximal p-abelian extension unramiﬁed everywhere and elements in SelF (Tp (ψ)) are made up of unramiﬁed cocycle classes, we ﬁnd that SelF (Tp (ψ)) ∼ = Hom(C0 [ψ], Tp ). Exercise 1.12 Give a detailed proof of the above identity. This shows that C0 [ψ] ∼ = SelF (Tp (ψ))∗ , which we write as Sel∗F (Tp (ψ)) hereafter. As a consequence of Theorem 1.7, we get the following very precise p-class number formula: Corollary 1.13 We have C0 [ψ] ∼ = Sel∗F (Tp (ψ)) canonically, and if ψ = ω |L(0, ψ)|−1 p |C0 [ψ]| = |SelF (Tp (ψ))| = −1 if ψ = ω, |(γ − 1)L(0, ψ)|p where |x|p = |ip (x)|p for the p-adic absolute value | · |p of Qp with |p|p = p−1 . When F = Q, the direction p|C0 [ψ]| ⇒ p|L(0, ψ) is due to Herbrand (1932), and the converse p|L(0, ψ) ⇒ p|C0 [ψ]| was later proven by Ribet (1976; see [Ri1]). The above precise formula follows from Theorem 1.7 because (a) if X is a torsion Λ-module of ﬁnite type and has no nontrivial ﬁnite Λ-submodule, charΛ (X)(α) = charW (X/(x − α)X) by ring theory as long as X/(x − α)X is ﬁnite, (b) C∞ does not have a nontrivial ﬁnite Λ-submodule as proven by R. Greenberg (cf., [Gr1]). Exercise 1.14 Let the notation and the assumption be as in (a) above. Using (b), prove that multiplication by f = x − α on X (f : X t → f t ∈ X) is injective. Then using this fact, show (a). By a similar argument, taking the Pontryagin dual, we ﬁnd Corollary 1.15 Let ψ∞ be the restriction of ψ to GM∞ . Then we have a canonical isomorphism: C∞ [ψ∞ ] ∼ = Sel∗F∞ (Tp (ψ)). If G is a group with subgroup H, for a given W [H]-module X, we deﬁne the induced G-module IndG H X = W [G] ⊗W [H] X on which we let g ∈ G act by g(g ⊗ x) = (gg ⊗ x). Identifying X with 1 ⊗ X ⊂ IndG H X, the action of H on 1 ⊗ X and X are identical. Thus IndG H X is an extension of the H-module X to the bigger group a complete representative set Q of G/H in G, we G. Choosing ∼ ∼ ﬁnd IndG HX = q∈Q q ⊗ X = q∈Q X with q ⊗ X = X as W -modules. G We can embed IndH X into HomW [H] (W [G], X) by sending ξ = q q ⊗ aq to φξ (q) = aq . If we let g ∈ G act on φ ∈ HomW [H] (W [G], X) by (gφ)(x) = φ(g −1 x),

12

Introduction

this injection is W [G]-equivariant. If (G : H) < ∞, this is a surjective isomorphism. Often HomW [H] (W [G], X) is called a coinduced module and written G G G G ∼ as CoindG H X; so, CoindH X = IndH X if [G : H] < ∞ and IndH X → CoindH X in general. When G = Gal(Q/F ) and H = Gal(Q/M ), we write simply IndF M X for × IndG H X. Since Γ = Gal(F∞ /F ), we can deﬁne a character ψ : Gal(M∞ /F ) → Λ s s by ψ(σ) = ψ(σ|M )(1 + x) if σ|F∞ = γ for s ∈ Zp . In this way, we get the GF -module Λ∗ (ψ). Exercise 1.16 Prove Λ∗ (ψ) ∼ = IndF F∞ Tp (ψ). Here is the so-called Shapiro lemma (whose proof for H 1 will be given later as Lemma 3.83). Lemma 1.17 Let X be a discrete G-module for a proﬁnite group G with a ∼ closed subgroup H. Then we have a canonical isomorphism H q (G, IndG H X) = H q (H, X). If G = GF , H is the stabilizer in G of F∞ and X = Tp (ψ), the above isomorphism is an isomorphism of Λ-modules, where Λ acts on the left-hand side ∼ ∗ through the coeﬃcients IndF F∞ Tp (ψ) = Λ (ψ). Proofs of this lemma can be found in any book on group or Galois cohomology (e.g., [CGP] III.6.2 or [MFG] Lemma 4.20). In these books, the lemma is stated G using CoindG H , which is isomorphic to IndH X if G/H is ﬁnite. Since Galois cohomology is deﬁned by continuous cocycles, the lemma is actually valid for IndK F even if K/F is an inﬁnite extension if the coeﬃcient module has discrete topology. As an immediate corollary of this lemma, we have Corollary 1.18

We have SelF∞ (Tp (ψ)) ∼ = SelF (Λ∗ (ψ)).

1.1.2 The Iwasawa algebra as a deformation ring We can interpret the group algebra Λ as a universal Galois deformation ring when F = Q. In this subsection, we assume F = Q. We write CLW for the category of p-proﬁnite local W -algebras A with A/mA = F. A character ρ : GQ → A× for A ∈ CLW is called a W -deformation (or just simply a deformation) of ψ if (ρ mod mA ) = ψ. A couple (R, ρ) made of an object R of CLW and a character ρ : GF → R× is called a universal couple for ψ if for any deformation ρ : GF → A of ψ, we have a unique morphism φρ : R → A in CLW (so it is a local W -algebra homomorphism) such that φρ ◦ ρ = ρ. By the universality, the couple (R, ρ) (if it exists) is determined uniquely up to isomorphisms. The ring R is called the universal deformation ring and ρ is called the universal deformation of ψ. Proposition 1.19

The couple (ΛW , ψ) is the universal couple for ψ.

Proof Since Q[µp∞ ] is the maximal abelian extension of Q unramiﬁed outside p and ∞ by class ﬁeld theory (or else, by the theorem of Kronecker and Weber), we have GQ /[GQ , GQ ] = Gal(Q(p) /Q[µp∞ ]). Since each deformation ρ : GQ → A×

Classical Iwasawa theory

13

factors through Gal(Q[µp∞ ]/Q) = Γ × Gal(Q[µp ]/Q), the character ρ is determined by ρ(γ), because ρ|Q[µp ] is given by ψ. Then we have φρ : ΛW = W [[x]] → A 2 by sending x to ρ(γ) − 1, and we have φρ ◦ ψ = ρ. We consider the completed group algebra W [[Γ]] and a character ψ : GQ → W [[Γ]]× given by ψ (σ) = ψ(σ|Q[µp ] )[σ|Q∞ ], where [σ|Q∞ ] ∈ Γ and ψ(σ|Q[µp ] ) ∈ W × . Then by the universality of the group algebra, (W [[Γ]], ψ) is a universal deformation ring of ψ. Thus we get a canonical isomorphism ι : W [[Γ]] ∼ = W [[x]] with ι ◦ ψ = ψ. In particular, for the ﬁxed generator γ = 1 + p, we ﬁnd ι(γ) = 1 + x. We have thus proved Corollary 1.20 We have a W -algebra isomorphism ι : W [[Γ]] ∼ = W [[x]] such that ι(γ) = 1 + x. Let us return to a general number ﬁeld F . For a given n-dimensional representation ρ : GSF → GLn (F), a deformation ρ : GSF → GLn (R) is a continuous representation with ρ mod mR ∼ = ρ. Suppose that R is a normal domain of ﬁnite type over W [[x1 , . . . , xg ]]. Corollary 1.18 suggests that we could generalize the notion of the Selmer group to a nonabelian representation ρ as a suitable subgroup of H 1 (GSF , V (ρ)∗ ), where V (ρ) = Rn on which GSF acts by ρ and V (ρ)∗ = V (ρ) ⊗R R∗ . Suppose that we manage to make a good deﬁnition of the Selmer group SelF (V (ρ)∗ ). If further the Pontryagin dual Sel∗F (V (ρ)∗ ) of SelF (V (ρ)∗ ) is a torsion R-module of ﬁnite type, we have the nontrivial characteristic ideal char(Sel∗F (V (ρ)∗ )) and then try to compare it with the p-adic L-function Lp (ρ) of ρ (if it exists), which has to be an element of R. In such an favorable circumstance, the “main” conjecture in this context would be that char(Sel∗F (V (ρ)∗ )) is generated by Lp (ρ). We describe in this book abelian and nonabelian generalizations of Iwasawa’s theory (of this type) relating the Selmer groups with automorphic forms and L-functions. 1.1.3 Pseudo-representations In the following subsection, we shall generalize the notion of the universal deformation ring to representations into GLn (F) for n = 2. In order to show the existence of the universal deformation ring, pseudo-representations are very useful. We recall the deﬁnition of pseudo-representations (due to Wiles) when n = 2. See [MFG] Section 2.2 for pseudo-representations for n > 2. In this subsection, the coeﬃcient ring A is always a local ring with maximal ideal mA . We write F = A/mA . We would like to characterize the trace of a representation of a group G. We describe in detail traces of degree 2 representations ρ : G → GL2 (A) when 2 is invertible in A and G contains c such that c2 = 1 and det ρ(c) = −1. Let V (ρ) = A2 on which G acts by ρ.

14

Introduction

Exercise 1.21 Let ρ : G → GL2 (A) be another representation with its space V (ρ ). Show that V (ρ) ∼ = V (ρ ) as A[G]-modules if and only if there exists an invertible 2 × 2 matrix X ∈ GL2 (A) such that Xρ(g)X −1 = ρ (g) for all g ∈ G. Since 2 is invertible, we know that V = V (ρ) = V+ ⊕ V− for V± = 1±c 2 V . For ρ = ρ mod mA , we write V = V (ρ). Then similarly as above, V = V + ⊕ V − and V ± = V± /mA V± . Since dimF V = 2 and det ρ(c) = −1, dimF V ± = 1. This shows that V ± = Fv ± for v ± ∈ V ± . Take v± ∈ V± such that v± mod mA V± = v ± , and deﬁne φ± : A → V± by φ(a) = av± . Then φ± mod mA V is surjective by Nakayama’s lemma (Lemma 1.6). Exercise 1.22 Show that φ± : A ∼ = V± as A-modules. In other words, {v− , v+ } is an A–basis of V . We write

a(r) b(r) ρ(r) = c(r) d(r)

−1 0 with respect to this basis. Thus ρ(c) = . Deﬁne another function x : 0 1 G × G → A by x(r, s) = b(r)c(s). Then we have (W1) a(rs) = a(r)a(s) + x(r, s), d(rs) = d(r)d(s) + x(s, r) and x(rs, tu) = a(r)a(u)x(s, t) + a(u)d(s)x(r, t) + a(r)d(t)x(s, u) + d(s)d(t)x(r, u); (W2) a(1) = d(1) = d(c) = 1, a(c) = −1 and x(r, s) = x(s, t) = 0 if s = 1, c; (W3) x(r, s)x(t, u) = x(r, u)x(t, s). These are easy to check: we have

a(r) b(r) a(s) c(r) d(r) c(s)

b(s) d(s)

=

a(rs) b(rs) . c(rs) d(rs)

Then by computation, a(rs) = a(r)a(s) + b(r)c(s) = a(r)a(s) + x(r, s). Similarly, we have b(rs) = a(r)b(s) + b(r)d(s) and c(rs) = c(r)a(s) + d(r)c(s). Thus x(rs, tu) = b(rs)c(tu) = (a(r)b(s) + b(r)d(s))(c(t)a(u) + d(t)c(u)) = a(r)a(u)x(s, t) + a(r)d(t)x(s, u) + a(u)d(s)x(r, t) + d(s)d(t)x(r, u). A triple {a, d, x} satisfying the three conditions (W1–3) is called a pseudorepresentation of Wiles of (G, c). For each pseudo-representation τ = {a, d, x}, we deﬁne Tr(τ )(r) = a(r) + d(r)

and

det(τ )(r) = a(r)d(r) − x(r, r).

By a direct computation using (W1–3), we see a(r) =

1 (Tr(τ )(r) − Tr(τ )(rc)), 2

d(r) =

1 (Tr(τ )(r) + Tr(τ )(rc)) 2

Classical Iwasawa theory

15

and x(r, s) = a(rs) − a(r)a(s),

det(τ )(rs) = det(τ )(r) det(τ )(s).

Thus the pseudo-representation τ is determined by the trace of τ as long as 2 is invertible in A. Proposition 1.23 ((A. Wiles, 1988)) Let G be a group and R = A[G]. Let τ = {a, d, x} be a pseudo-representation (of Wiles) of (G, c). Suppose either that there exists at least one pair (r, s) ∈ G × G such that x(r, s) ∈ A× or that x(r, s) = 0 for all r, s ∈ G. Then there exists a representation ρ : R → M2 (A) such that Tr(ρ) = Tr(τ ) and det(ρ) = det(τ ) on G. If A is a topological ring, G is a topological group, and all maps in τ are continuous on G, then ρ is a continuous representation of G into GL2 (A) under the topology on GL2 (A) induced by the product topology on M2 (A). Proof When x(r, s) = 0 for all r, s ∈ G, we see from (W1) that a, d : G → A satisﬁes a(rs) = a(r)a(s) and d(rs) = d(r)d(s). Thus a, d are characters of G, and we deﬁne ρ : G → GL2 (A) by

a(g) 0 ρ(g) = , 0 d(g) which satisﬁes the required property. We extend ρ to R = A[G] by linearity. We now suppose x(r, s) ∈ A× for r, s ∈ G. Then we deﬁne b(g) = x(g, s)/x(r, s) and c(g) = x(r, g) for g ∈ G. Then by (W3), b(g)c(h) = x(r, h)x(g, s)/x(r, s) = x(g, h). Put

a(g) b(g) ρ(g) = . c(g) d(g)

−1 0 By (W2), we see that ρ(1) is the identity matrix and ρ(c) = . By 0 1 computation,

a(g) b(g) a(h) b(h) ρ(g)ρ(h) = c(g) d(g) c(h) d(h)

a(g)a(h) + b(g)c(h) a(g)b(h) + b(g)d(h) = . c(g)a(h) + d(g)c(h) d(g)d(h) + c(g)b(h) By (W1), a(gh) = a(g)a(h) + x(g, h) = a(g)a(h) + b(g)c(h) d(gh) = d(g)d(h) + x(h, g) = d(g)d(h) + b(h)c(g). Now let us look at the lower left corner: c(g)a(h) + d(g)c(h) = x(r, g)a(h) + d(g)x(r, h).

16

Introduction

Now apply (W1) to (1, r, g, h) in place of (r, s, t, u), and we get c(gh) = x(r, gh) = a(h)x(r, g) + d(g)x(r, h), because x(1, g) = x(1, h) = 0. As for the upper right corner, we apply (W1) to (g, h, 1, s) in place of (r, s, t, u). Then we get b(gh)x(r, s) = x(gh, s) = a(g)x(h, s) + d(h)x(g, s) = (a(g)b(h) + d(h)b(g))x(r, s), which shows that ρ(gh) = ρ(g)ρ(h). We now extend ρ linearly to R = A[G]. This shows the ﬁrst assertion. The continuity of ρ follows from the continuity of each entry, which follows from the continuity of τ . 2 Start with an absolutely irreducible representation ρ : G → GLn (F). Here a representation of a group into GLn (K) for a ﬁeld K is called absolutely irreducible if it is irreducible as a representation into GLn (K) for an algebraic closure K of K. Exercise 1.24 Give an example of irreducible representations of a group G into GL2 (Q) which is not absolutely irreducible. We ﬁx an absolutely irreducible odd representation ρ : G → GL2 (F). Here the word “odd” means that det(ρ)(c) = −1. If we have a representation ρ : G → GL2 (A) with ρ mod mA ∼ ρ, then det(ρ(c)) ≡ −1 mod mA . Since c2 = 1, if 2 is invertible in A (⇔ the characteristic of F is diﬀerent from 2), det(ρ(c)) = −1. Thus we have a well-deﬁned pseudo-representation τρ of Wiles associated to ρ. Since ρ is absolutely irreducible, we ﬁnd r, s ∈ G such that b(r) ≡ 0 mod mA and c(s) ≡ 0 mod mA . Thus τρ satisﬁes the condition of Proposition 1.23. Conversely if we have a pseudo-representation τ : G → A such that τ ≡ τ mod mA for τ = τρ , again we ﬁnd r, s ∈ G such that x(r, s) ∈ A× . The correspondence ρ → τρ induces a bijection: {ρ : G → GL2 (A) : representation|ρ

mod mA ∼ ρ} / ∼↔

{τ : G → A : pseudo-representation|τ

mod mA = τ } , (1.1.2)

where τ = τρ and “∼” is the conjugation under GL2 (A). The map is surjective by Proposition 1.23 combined with Proposition 1.25 and is one to one by the following Proposition 1.25 which we admit, because a pseudo-representation is determined by its trace. Proposition 1.25 (Carayol, Serre, 1994) Let A be an proartinian local ring with ﬁnite residue ﬁeld F. Let R = A[G] for a proﬁnite group G. Let ρ : R → Mn (A) and ρ : R → Mn (A) be two continuous representations. If ρ = ρ mod mA is absolutely irreducible and Tr(ρ(σ)) = Tr(ρ (σ)) for all σ ∈ G, then ρ ∼ ρ . See [MFG] Proposition 2.13 for a proof of this result. This proposition is also valid for any artinian K-algebras A with residue ﬁeld K for a ﬁnite extension K/Qp and p-adically continuous ρ and ρ as above (see [C2] for the proof of the general case).

Classical Iwasawa theory

17

1.1.4 Two-dimensional universal deformations Hereafter we always assume that p is odd. We ﬁx an absolutely irreducible representation ρ : G → GL2 (F) for a proﬁnite group G. Assume that we have c ∈ G with c2 = 1 and det(ρ(c)) = −1. First we consider a universal pseudorepresentation. Let τ = (a, d, x) be the pseudo-representation associated to ρ. A couple consisting of an object Rτ ∈ CLW and a pseudo-representation T = (A, D, X) : G → Rτ is called a universal couple if the following universality condition is satisﬁed: (uv) For each pseudo-representation τ : G → A (A ∈ CLW ) with τ ∼ = τ mod mA , there exists a unique W -algebra homomorphism ιτ : Rτ → A such that τ = ιτ ◦ T. We now show the existence of (Rτ , T ) for a proﬁnite group G. First suppose uller character, that G is a ﬁnite group. Let ω : W × → µq−1 (W ) be the Teichm¨ that is, ω(x) = lim xq n→∞

n

(q = |F| = |W/mW |).

We also consider the following isomorphism: µq−1 (W ) ζ → ζ mod mW ∈ µq−1 (F) = F× . We write ϕ : F× → µq−1 (W ) ⊂ W × for the inverse of the above map. We look at the power series ring with variables Ag , Dh , X(g,h) indexed by g, h ∈ G: Λ = ΛG := W [[Ag , Dh , X(g,h) ; g, h ∈ G]]. We put A(g) = Ag + ϕ(a(g)), D(g) = Dg + ϕ(d(g)) and X(g, h) = Xg,h + ϕ(x(g, h)). We construct the smallest closed ideal I so that T = (g → A(g)

mod I, g → D(g)

mod I, (g, h) → X(g, h)

mod I)

becomes a pseudo-representation. Thus the ideal I of Λ is topologically generated by all the elements of the following type: (w1) A(rs) − (A(r)A(s) + X(r, s)), D(rs) − (D(r)D(s) + X(s, r)) and X(rs, tu) − (A(r)A(u)X(s, t) + A(u)D(s)X(r, t) + A(r)D(t)X(s, u) + D(s)D(t)X(r, u)); (w2) A(1) − 1 = A1 , D(1) − 1 = D1 , D(c) − 1 = Dc , A(c) + 1 = Ac and X(r, s) − X(s, t) if s = 1, c; (w3) X(r, s)X(t, u) − X(r, u)X(t, s). Then we put Rτ = Λ/I and deﬁne T = (A(g), D(h), X(g, h)) mod I. By the above deﬁnition, T is a pseudo-representation with T mod mRτ = τ . For a

18

Introduction

pseudo-representation τ = (a, d, x) : G → A with τ ≡ τ mod mA , we deﬁne ιτ : Λ → A by f (Ag , Dh , X(g,h) ) → f (τ (g) − ϕ(τ (g))) = f (a(g) − ϕ(a(g)), d(h) − ϕ(d(h)), x(g, h) − ϕ(x(g, h))). Since f is a power series of Ag , Dh , Xg,h and τ (g) − ϕ(τ (g)) ∈ mA , the value f (τ (g) − ϕ(τ (g))) is well deﬁned. Let us see this. If A is artinian, a suﬃciently high power mN A vanishes. Thus if the monomial of the variables Ag , Dh , X(g,h) is of degree higher than N , it is sent to 0 via ιτ , and f (τ (g) − ϕ(τ (g))) is a ﬁnite sum of terms of degree ≤ N . If A is proartinian, the morphism ιτ is just the projective limit of the corresponding ones well deﬁned for artinian quotients. By the axioms of pseudo-representation (W1–3), ιτ (I) = 0, and hence ιτ factors through Rτ . The uniqueness of ιτ follows from the fact that {Ag , Dh , X(g,h) |g, h ∈ G} topologically generates Rτ . Now assume that G = limN G/N for open normal subgroups N (so, G/N ←− is ﬁnite). Since Ker(ρ) is an open subgroup of G, we may assume that N runs over subgroups of Ker(ρ). Since ρ factors through G/ Ker(ρ), Tr(τ ) = Tr(ρ) factors through G/N . Therefore we can think of the universal couple (RτN , TN ) for (G/N, τ ). If N ⊂ N , the algebra homomorphism ΛG/N → ΛG/N taking (AgN , DhN , X(gN,hN ) ) to (AgN , DhN , X(gN ,hN ) ) induces a surjective W -algebra homomorphism πN,N : RτN → RτN with πN,N ◦ TN = TN . We then N deﬁne T = limN TN and Rτ = limN Rτ . If τ : G → A is a pseudo-representation, ←− ←− by Proposition 1.23, we have the associated representation ρ : G → GL2 (A) such that Tr(τ ) = Tr(ρ). If A is artinian, then GL2 (A) is a ﬁnite group, and hence ρ and Tr(τ ) = Tr(ρ) factors through G/N for a suﬃciently small open normal π

ιN

τ subgroup N . Thus we have ιτ : Rτ −−N → RτN −− → A such that ιτ ◦ T = τ . Since (A(g), D(h), X(g, h)) generates Rτ topologically, ιτ is uniquely determined. Writing for the representation : G → GLn (Rτ ) associated to the universal pseudo-representation T and rewriting Rρ = Rτ , for n = 2, we have proven by (1.1.2) the following theorem, which was ﬁrst proven by Mazur in 1989 (see [Mz2] for his original proof and [MFG] Theorem 2.26 for a proof similar to the one given here, valid for any n without assuming the existence of an element c with c2 = 1 and det ρ(c) = −1).

Theorem 1.26 (Mazur) Suppose that ρ : G → GLn (F) is absolutely irreducible. Then there exists the universal deformation ring Rρ in CLW and a universal deformation : G → GLn (Rρ ). Suppose further n = 2 and that we have c ∈ G with c = 1 and c2 = 1. If ρ is odd (with respect to c) and we write τ for the pseudo-representation associated to ρ, then for the universal pseudorepresentation T : G → Rτ deforming τ , we have a canonical isomorphism of W -algebras ι : Rρ ∼ = Rτ such that ι ◦ Tr() = Tr(T ).

Selmer groups

19

Let (Rρ , ) be the universal couple for an absolutely irreducible representation ρ : GQ → GLn (F). We can also think of (Rdet(ρ) , ν), which is the universal couple for the character det(ρ) : GQ → GL1 (F) = F× . As we have studied already, Rdet(ρ) ∼ = W [[Γ]] = ΛW . Note that det() : GQ → GL1 (Rρ ) satisﬁes det() mod mRρ = det(ρ). Thus det() is a deformation of det(ρ), and hence by the universality of (ΛW ∼ = Rdet(ρ) , ν), there is a unique W -algebra homomorphism ι : ΛW → Rρ such that ι ◦ ν = det(). In this way, Rρ naturally becomes a ΛW -algebra via ι. Corollary 1.27 The universal ring Rρ is canonically an algebra over the Iwasawa algebra ΛW = W [[Γ]]. When G = GQ (or more generally, GF or GSF for a ﬁnite set S of rational primes), it is known that Rρ is noetherian (see 1.3.3). We come back to this point after relating a certain Selmer group with the universal deformation ring. 1.2 Selmer groups We ﬁrst recall very brieﬂy Deligne’s conjecture on rationality of L-values. This is to let us understand better the use of ﬁltration in the deﬁnition of Selmer groups, and in this part only, an outline of the conjecture is described without any proof. Then we recall the deﬁnition due to R. Greenberg of the Selmer groups in terms of Galois cohomology groups. There is another deﬁnition due to Bloch and Kato (which include the nonordinary cases; e.g., [FoP]) in terms of yet another cohomology theory. The characteristic ideal of the Selmer groups is conjectured (by many outstanding mathematicians, notably, Bloch, Kato, and Greenberg) to be related to L-values and p-adic L-functions. The groups are also useful to prove ﬁniteness properties of the universal Galois deformation rings, because the Selmer group of a p-adic Galois representation is, after localization, naturally isomorphic to the tangent space at the point of the spectrum of the deformation ring associated with the Galois representation, and the ﬁniteness of the dimension of the group tells us the neotherian property of the localization-completion of the universal deformation ring at the Galois representation. 1.2.1 Deligne’s rationality conjecture The source of the conjecture is Deligne’s paper [D4]. Although the theory works whenever we have good theory of ´etale, de Rham, and Betti cohomology (so for “motives”; e.g., [MTV]), we restrict ourselves to a smooth projective variety V deﬁned over Q; so, V ⊂ PN is deﬁned to be the the zero set of ﬁnitely many homogeneous equations with coeﬃcients in Q. For simplicity, we assume that the space V (C) of complex points is connected. Then V (C) is a smooth compact complex K¨ahler manifold. Thus we have the de Rham cohomology group H n (V (C), C) = H n (V (C), Ω•V /C ) computed by the diﬀeo-geometric resolution 0 → C → Ω•V /C of C by the ahler metric, by sheaves of C ∞ -diﬀerentials. Since PN has a canonical K¨

20

Introduction

n harmonic analysis, HDR (V (C), C) splits into a product H p,q (V (C), C) (p+q = n) generated by harmonic forms which are linear combinations of

f (z)dz1 ∧ · · · ∧ dzp ∧ dz p+1 ∧ · · · ∧ dz n (the Hodge decomposition). More intrinsically, we have p H p,q (V (C), C) ∼ = H q (V (C), ΩV /C ). n In particular, HDR has a decreasing Hodge ﬁltration whose p-th ﬁlter is given p n by F∞ HDR (V (C), C) = p ≥p H p ,q (V (C), C). The ﬁltration has meaning over Q but not the (direct sum) Hodge decomposition (a canonical splitting of the ﬁltration), since antiholomorphy cannot be deﬁned for the algebraic variety V over Q. Grothendieck [Gt] computed the algebraic de Rham hyper-cohomology n (V/Q , Q) = HnDR (V/Q , Ω•V /Q ) for the projective variety V over Q and the HDR algebraic diﬀerential sheaf Ω•V /Q (indeed his computation is valid over any ﬁeld n n of characteristic 0), and he found HDR (V/Q , Q) ⊗Q C ∼ (V (C), C) canonic= HDR p n p n n (V/Q , Q), we have ally, and deﬁning F∞ HDR (V/Q , Q) by F∞ H (V (C), C) ∩ HDR p n p n HDR (V/Q , Q) ⊗Q C = F∞ HDR (V (C), C). Thus we have the Hodge ﬁltration F∞ n 0 n 1 n (V/Q , Q) = F∞ HDR (V/Q , Q) ⊃ F∞ HDR (V/Q , Q) ⊃ · · · HDR n n n+1 n ⊃ F∞ HDR (V/Q , Q) ⊃ F∞ HDR (V/Q , Q) = 0.

We have given an ad hoc deﬁnition of the ﬁltration over Q by taking the restriction of the ﬁltration over C (given by the Hodge decomposition) to the cohomology group with coeﬃcients in Q. The well-deﬁnedness of the ﬁltration over Q (that is, the rationality of the ﬁltration over C) follows from the usual spectral sequence relating Hodge cohomology and de Rham cohomology: p+q E1p,q = H q (X/k , ΩpX/k ) ⇒ HDR (X, k). n Similarly, V (C) has the Betti cohomology HB (V (C), Q) by using simplicial decomposition of V (C), and by the de Rham theorem (e.g., [CAL] Theorem 7.5.1), we have a canonical isomorphism n n I∞ : HB (V (C), C) ∼ (V (C), C). = HDR

Grothendieck also constructed ´etale cohomology theory, which has coeﬃcients in -adic rings A, like Z and Q , such that Gal(Q/Q) canonically acts on n (V/Q , A), and there is a canonical isomorphism Het n n Het (V/Q , A) ∼ (V (C), Z) ⊗Z A. = HB

See [ECH] or [SGA4 12 ] for ´etale cohomology. Thus whichever cohomology we n (V/Q , Q ) use, the dimension of H?n (V, K) is the same, which we write g. So Het is a vector space of dimension g over Q . In particular, the Galois action on (n) n Het (V/Q , Q ) gives a representation ρ = ρ : Gal(Q/Q) → GLg (Q ). When

Selmer groups

21

dim V = 1, the ´etale cohomology group is easy to describe. Take a ﬁeld F ⊂ C, and consider the function ﬁeld F (V ) of V . Then the algebraic fundamental group π1alg (V/F ) = limX/V Gal(F (X)/F (V )) where X runs over all (every←− where) unramiﬁed Galois coverings of V . Then H 1 (VQ , Z ) = Hom(π1alg (VQ ), Z ). Over C, all unramiﬁed coverings of V are given by U/Γ for a subgroup Γ of ﬁnite index of the classical fundamental group π1 (V ) for the universal covering U , and we have π1alg (V/C ) = limΓ π1 (V )/Γ, where Γ runs over all normal ←− 1 (V (C), A) = Hom(π1 (X), A), subgroups of π1 (V ) of ﬁnite index. Since HB 1 1 ∼ we have HB (V, Z ) = Het (V, Z ). It is known that π1alg (V/Q ) = π1alg (V/C ) by Gal(Q(X)/Q(V )) ∼ = Gal(C(X)/C(V )) (see [SGA1] X.1.8). For any ﬁnite Galois extension F/Q, the function ﬁeld extension F (V )/Q(V ) is everywhere an unramiﬁed extension of Q(V ) whose Galois group Gal(F (V )/Q(V )) is naturally isomorphic to Gal(F/Q). Thus for any given unramiﬁed Galois covering, X/F V/F , we have the following exact sequence of groups: 1 → Gal(F (X)/F (V )) → Gal(F (X)/Q(V )) → Gal(F/Q) → 1. Passing to the projective limit with respect to F and X, we get the following exact sequence: 1 → π1alg (V/Q ) → π1alg (V/Q ) → Gal(Q/Q) → 1. Thus σ ∈ Gal(Q/Q) acts on the abelianization π1alg (V/Q ))ab = π1alg (V )/(π1alg (V/Q )), π1alg (V/Q ))) ∈ π1alg (V/Q ) projecting down to σ ∈ Gal(Q/Q). Thus by στ = σ τ σ −1 for a lift σ 1 Gal(Q/Q) acts on Het (V/Q , Z ) = Hom(π1alg (V/Q ))ab , Z ) naturally. Example 1.28 Although Gm = P1 − {0, ∞} is not projective, we can think of 1 H?1 (Gm , Q). Since π1 (Gm ) = Z, we ﬁnd HB (Gm , Z) = Z · S 1 for the unit circle 1 1 Het (Gm , Z ) = Hom(π1 (Z), Z ) ∼ S with the standard orientation = Z . We and dt dt 1 have HDR (Gm , Q) = Q t and S 1 t = 2πi. n

Exercise 1.29 Using the fact that t → t gives an unramiﬁed covering [n ] : n Gm → Gm with Galois group Gal(Q(t1/ )/Q(t)), show that the Galois action 1 (Gm , Z ) is given by N −1 for the -adic cyclotomic character N = N on Het given by ζ σ = ζ N (σ) for -power roots of unity ζ ∈ µ∞ (Q). We deﬁne a Galois module Z (1) = limn µn (Q), and put ←− m

Z (m) = Z (1) ⊗ · · · ⊗ Z (1), on which Gal(Q/Q) acts by N m . Similarly we deﬁne Z (−m) to be the Galois 1 module on which the Galois group acts via N −m . Thus Het (Gm , Z ) ∼ = Z (−1)

22

Introduction

by the above exercise. We just write H1? (Gm , K) = K? (1), which is the dual of H?1 (Gm , K). Thus we have H1et (Gm , Z ) = Z (1), H1B (Gm , Q) = QS 1 , H1DR (Gm , Q) = HomQ (Qω, Q) (1.2.1) B , C) ∼ H DR (Gm , C) for ω = dt t . The comparison isomorphism I∞ : H1 (G m dt = 1 dt 1 takes the generator S to the linear map: 1 : u t → S 1 u t = (2πi)u. Thus I∞ (H1B (Gm , Q)) = (2πi)H1DR (Gm , Q), and we may identify H1B (Gm , Q) with 1 (Gm , C) with C by u dt (2πi)Q in C by S 1 → (2πi) (identifying HDR t ↔ u). Thus QB (1) = (2πi)Q ⊂ C. If one want to include H1? (Gm ) in the framework of projective varieties, one can use the following isomorphism H?1 (Gm ) ∼ = H?2 (P1 ). 1 Thus the Hodge ﬁltration of HDR (Gm ) is given by 1 1 (Gm , Q) = H 1,1 (P1 , Q) = HDR (Gm , Q) F 1 HDR 1 and F 2 HDR (Gm , Q) = 0. Let K be a ﬁeld of characteristic 0 admissible to each cohomology theory (K is Qp for ´etale cohomology, K = Q for Betti cohomology and de Rham cohomology). Then we write K? (0) = H?0 (P1 , K) and for m > 0, we put m

K? (m) = H1? (Gm , K) ⊗ · · · ⊗ H1? (Gm , K)

(1.2.2)

and m

K? (−m) = H?1 (Gm , K) ⊗ · · · ⊗ H?1 (Gm , K) . Then K? (−m) = HomK (K? (m), K(0)) by the Poincar´e duality. Since we can tensor a power of the cyclotomic character with Galois representations, to expand our world slightly, we introduce the Tate twists H? (V, K)(m) = H? (V, K) ⊗K K? (m) for integers m. The representation ρ ⊗ Nm (ρ ⊗ N m (σ) = Nm (σ)ρ (σ)) gives the Galois action on Het (V, Q )(m), and we have n HDR (V, Q)(m) = HDR (V, Q)

and n m (V, Q)(m) = ((2πi)Q)⊗m ⊗ HB (V, Q) = (2πi)m HB (V, Q). HB n (V, C). We deﬁne the The last identity of the above equation holds inside HB Hodge ﬁltration on HDR (V, Q)(m) by

F j (HDR (V, Q)(m)) = (F j+m HDR (V, Q)) ⊗Q Q(m)

Selmer groups

23

and the (p, q)-component by H p,q (V, C)(m) = H p+m,q+m (V, C) ⊗C C(m). This is compatible with the fact that K? (−1) = H?2 (P1 ) and m

K? (−m) =

H?2 (P1 )⊗m

→ H

2m

(P1 × · · · × P1 )

by the K¨ unneth formula. The Hodge number h(p, q) for H n (V )(m) is given by p+m,q+m (V (C), C). Such a tuple dim H n n (V, Q)(m), HDR (V, Q)(m), Het (V/Q , Q )) H n (V )(m) = (HB

is an example of algebro-geometric objects called pure motives (see [MTV]). In particular, H n (V ) is called the weight n motive associated to the projective variety V . For all cohomology groups, we have the Poincar´e duality pairing (perfect over the ﬁeld K of characteristic 0): (·, ·) : H?n (V, K) × H?2d−n (V, K)(d) → K

(1.2.3)

for d = dim V . B Exercise 1.30 Take the generator [V ] ∈ H2d (V, Q) and take a diﬀerential ω 2d whose class is in HDR (V, Q). Show Z ω ∈ (2πi)d Q.

Here K? (d) ∼ = K, and if ? = et, K = Q , and Gal(Q/Q) acts by the negative d-th σ N (σ) power -adic cyclotomic character N : Gal(Q/Q) → Z× given by ζ = ζ for all -power root ζ of unity. The pairing satisﬁes σ(x, y) = (σ(x), σ(y)) for σ ∈ Gal(Q/Q). If ? = DR, KDR (d) = K, and the pairing is a classical Poincar´e duality. If ? = B, KB (d) = (2πi)d K. Use these facts and (1.2.3) to solve the above exercise. We also have 2d−n n (V, Q))⊥ = F −p (HDR (V, Q)(d)). (F p HDR

If V is actually deﬁned by equations with coeﬃcients in Z(p) = Zp ∩ Q and n (V/Q , Q )(m) if smooth over Z(p) , the inertia group Ip at p acts trivially on Het = p. Thus the characteristic polynomial of the Frobenius action (m)

Pp (X) = det(1g − ρ

(F rob−1 p )X)

is well deﬁned, where F robp (x) = xp and F rob−1 p is its inverse, often called the geometric Frobenius map. For a more general p for which V may not be well deﬁned over Z(p) or not smooth over Z(p) , we can still think of the characteristic (m) 0 n polynomial Pp (X) of ρ (F rob−1 p ) on H (Ip , Het (V/Q , Q )(m)). A remarkable fact conjectured by Weil and proved by Dwork and Grothendieck (e.g., [ECH] or [SGA4 12 ]) is that if V is smooth over Z(p) , the characteristic polynomial Pp (X)

24

Introduction

is in Z[X] and is independent of . This fact is conjectured to be true for all p without restriction. Then we deﬁne the L-function of H n (V )(m) by L(s, H n (V )(m)) = Pp (p−s )−1 , p

which converges absolutely if Re(s) > 1 + − m. We supplement this L-function with a Γ-factor and deﬁne n 2

Λ(s, H n (V )(m)) = Γ(H n (V )(m), s) × L(s, H n (V )(m)). Here Γ(H n (V )(m), s) = Γ(H n (V ), s + m) and Γ(H n (V ), s) = ΓC (s − i)h(i,j) × ΓR (s + ε − i)hε (i,i) i+j=n,i<j

i

for ΓR (s) = π −s/2 Γ(s/2) and ΓC (s) = 2(2π)−s Γ(s), where for ε = 0, 1, hε (i, i) is the dimension of the (−1)ε -eigenspace of complex conjugation on H i,i (V (C), C) (the complex conjugation c acts on V (C) as an analytic automorphism; so, on the diﬀerential ω by ω c (z) = ω(z)). Exercise 1.31 Let us identify Spec(Q) with the origin 0 in P1 (Q). Show that H?0 (Spec(Q), K) = K? (0) and L(s, H 0 (Spec(Q))) is equal to the Riemann zeta function. The Hasse–Weil conjecture (formulated by Serre) tells us that the L-function L(n) (s, H n (V )(m)) has meromorphic continuation to the whole of s ∈ C satisfying an appropriate functional equation of the form Λ(s, H n (V )(m)) = ε(s, Hn (V )(m))Λ(n − 2m + 1 − s, H n (V )(m)), where ε(s, Hn (V )) is a function of the form wC (n−2m+1)/2−s for a constant (m) −m 0 < C ∈ R and w ∈ C with |w| = 1. Since ρ (F rob−1 ρ (F rob−1 p ) = p p ) n (because of N (F robp ) = p), we have L(s, H (V )(m)) = L(s + m, H n (V )). The (m) number n − 2m is called the weight of the Galois representation ρ (and the n weight of the motive H (V )(m)). The motive H n (V )(m) is called critical if the gamma factors Γ(H n (V ), s) and Γ(H n (V ), n − 2m + 1 − s) are ﬁnite at s = 0. An integer m is called critical for H n (V ) if H n (V )(m) is critical. Exercise 1.32 Let h(p, q) be the Hodge number for H n (V )(m). Show the following facts. 1. Suppose hp,p = 0 for all p. Then H n (V )(m) is critical if and only if h(p, q) = 0 ⇒ q ≥ 0 > p or p ≥ 0 > q. 2. m is critical for H 0 (Spec(Q)) if and only if m is even or odd accordingly as m > 0 or m ≤ 0.

Selmer groups

25

On cycles Z ∈ HnB (V (C), K), complex conjugation c acts by [Z] → [c(Z)]. n (V (C), K) = HomK (HnB (V (C), K), K). We By duality, c acts on the dual HB n ± write HB ((V (C), K)(m) for the ± eigenspace of c. Let x ∈ Z for x ∈ R be the maximal integer not exceeding x. We then deﬁne n n n HDR (V, K)(m)± = HDR (V, K)(m)/F ∓ (HDR (V, K)(m)), n (V, C)(m), and where F + = F − = F (n/2)−m if there is no (p, p)-factor in HDR n if a nontrivial (p, p)-factor appears in HDR (V, C)(m), we put F (n/2)−m +1 if p is even, + F = if p is odd, F (n/2)−m

and F−

F (n/2)−m +1 = F (n/2)−m

if p is odd, if p is even.

Since H p,q (V (C), C) is spanned by (p, q)-forms, the complex conjugate of the component H p,q (V (C), C) is given by H q,p (V (C), C). Here the action of complex conjugation c on a diﬀerential ω is given by the pullback: ω c (z) = ω(z). Using this fact, prove Exercise 1.33 If H n (V )(m) is critical, I∞ induces the following isomorphisms I

∞ n n n n (V, C)(m)± → HB (V, C)(m) −− → HDR (V, C)(m) HDR (V, C)(m)± . I± : HB

n n (V, Q)(m)± and ω1 , . . . , ωk of HDR (V, Q)(m)± , write Take bases v1 , . . . , vk of HB ± n × × I± (vi ) = i aij wj , and deﬁne c (H (V )(m)) = det(aij ) ∈ C /Q . Here if n ± ± n HB (V, C)(m) = 0, we simply put c (H (V )(m)) = 1.

Exercise 1.34 Suppose m = 0 and either that n is odd or n = 2p such that n (V, C). If v1∗ , . . . , vk∗ is the basis of HnB (V, Q)± the (p, p)-factor vanishes in HDR n ± dual to v1 , . . . , vk ∈ HB (V, Q) , prove c± (H n (V )) = det

vj∗

ωi

.

Here is a specialized form of the conjecture of Deligne in our setting. Conjecture 1.35 (Rationality) ﬁnite at s = m, and we have

If m is critical for H n (V ), L(s, H n (V )) is

L(m, H n (V )) ∈ Q. c+ (H n (V )(m)) The special value L(m, H n (V )) is called a critical value if m is critical for H n (V ).

26

Introduction

Exercise 1.36 For a positive integer m, prove c+ (Spec(Q)(2m)) = (2πi)2m and c+ (Spec(Q)(1 − 2m)) = 1. This shows that the conjecture holds for the Riemann zeta function. All known rationality results proven by Shimura (for example, [Sh3], [Sh4] and [Sh5]) and Deligne and others (e.g., [H94]) appear to follow this conjecture. Since our philosophy is that the L-value L(m, H n (V )) gives the size of an arithmetically deﬁned (m) group out of the Galois representation ρ , we should have a way of deﬁning such a group out of the Hodge ﬁltration, in view of the importance of the ﬁltration in the formulation of the rationality conjecture. We explore this point in the following subsection. 1.2.2 Ordinary Galois representations We recall the deﬁnition of ordinary p-adic Galois representations having a natural ﬁltration characterized by the local Galois action. If an ordinary Galois representation is given by the ´etale cohomology of a projective smooth variety over a number ﬁeld F , the ﬁltration coincides with the (p-adic) Hodge–Tate ﬁltration (which in turn is the pullback of the Hodge ﬁltration of the de Rham cohomology by the p-adic comparison map Ip , an analogue of I∞ , which is deﬁned after tensoring Fontaine ﬁeld BDR , an analogue of C; see the articles in [MTV] on p-adic periods). Let K be a p-adic local ﬁeld (i.e., a ﬁnite extension of Qp ). Let S be a ﬁnite set of rational primes containing p. We consider the maximal extension F S /F unramiﬁed outside S and ∞, and write GSF = Gal(F S /F ). Let V be a ﬁnite-dimensional vector space with a continuous action of GSF . We write Dp for the decomposition group of a prime ideal p|p in GSF . By local class ﬁeld theory, we have the Artin reciprocity map Fp× u → [u, Fp ] ∈ Dpab such that N ([u, Fp ]) = NFp /Qp (u)−1 if u is a p-adic unit. For each prime factor p|p of F , we assume we have a ﬁltration of the following type for integers a = a(p) and b = b(p) with b ≤ 0 < a: V = Fpb V ⊃ · · · ⊃ Fp0 V ⊃ Fp1 V ⊃ · · · ⊃ Fpa+1 V = {0}

(1.2.4)

stable under the decomposition group Dp and an open subgroup of the inertia group Ip ⊂ Dp acts on each subquotient Fpi V /Fpi+1 V by N i . The Galois module V is called nearly p-ordinary if V satisﬁes the above condition for all the prime factors p|p in F . We call V p-ordinary if V satisﬁes (1.2.4) with unramiﬁed Fp0 V /Fp1 V for p|p. If V is p-ordinary for all prime factors p|p, we simply call V p-ordinary. Once V satisﬁes (1.2.4), its dual V ∗ (1) = HomK (V, K) ⊗ N again satisﬁes (1.2.4) for Fp−i V ∗ (1) = (Fpi V )⊥ (1). If V is associated with a pure motive M , the ﬁltration as in (1.2.4) is given by its p-adic Hodge–Tate ﬁltration (e.g., [D4]) and the condition b ≤ 0 < a is almost equivalent to the criticality of M at 0 and M ∗ (1) at 1, though one may allow a = b = 0 for criticality at 0 of M if complex conjugation acts by the scalar −1 on V (resp. a = b = 1 for criticality of M ∗ (1) at 1 if complex conjugation acts trivially on V ).

Selmer groups

27

Example 1.37 Let E be an elliptic curve deﬁned over F . Let O be the integer ring of F and Op be the p-adic completion of Op . See 4.1.1 for a brief review of the deﬁnition of elliptic curves. If E extends to an elliptic curve over Op , we say that E has a good reduction. Here the word “extends” means that there exists an elliptic curve over Op whose generic ﬁber is isomorphic to E/Fp . If E extends to a smooth group scheme over Op such that its p-ﬁber Ep = E ⊗Op O/p is a group scheme isomorphic to Gm over a ﬁnite ﬁeld extension of O/p, E is said to have multiplicative reduction at p. If further Ep ∼ = Gm over O/p, E is said to have split multiplicative reduction. We call E semistable at p if either E has multiplicative reduction or good reduction. If E has good reduction, the geometric kernel Ep [p](Fp ) = Ker(x → px) for an algebraic closure Fp of O/p is either isomorphic to Z/pZ or to the trivial group {0}. The elliptic curve E is said to have ordinary good reduction if Ep [p](Fp ) ∼ = Z/pZ, and in such a case, the connected component of the locally free group scheme Ep [p]◦ is isomorphic to µp over Fp . If E has multiplicative reduction, Ep [p] ∼ = Gm [p] = µp over Fp , which is connected. In the case where Ep [p]◦ ∼ = µp over Fp (which covers the multiplicative reduction case as well as the ordinary good reduction case), E is said to have ordinary semistable reduction. Take an elliptic curve E/F with ordinary semistable reduction at all prime factors p of p in F . Then its Galois representation V (E) = T (E) ⊗Z Q given by the Tate module T (E) = limr E[pr ](Q) satisﬁes the ←− above condition for b = 0 and a = 1 with dimQp Fpi V /Fpi+1 V = 1 for 0 ≤ i ≤ 1. Here again E[pn ]/Op is the group scheme given by the kernel of the multiplication by pn . Indeed, the group scheme E[pn ]/Op ﬁts into the following exact sequence 0 → E[pn ]◦/Op → E[pn ]/Op → E[pn ]et /Op → 0 of group schemes with E[pn ]et (Q) ∼ = Z/pZ, where E[pn ]◦ is the connected n component of E[p ]/Op , and we have Fp1 T (E) = lim E[pn ]◦ (Q) ∼ lim E[pn ]et (Q) ∼ = Zp and T (E)/F 1 T (E) = ← = Zp . ← − − n n Example 1.38 Let f be an elliptic Hecke eigenform with f |T (n) = an f on Γ0 (N ) with Neben character ψ. Here T (n) indicates the Hecke operator for an integer n > 0 (see 2.4.1 for Hecke operators and the space of classical modular forms). Let Q(f ) be the subﬁeld of C generated by an for all n (the Hecke ﬁeld of f ), which is a number ﬁeld of ﬁnite dimension over Q. Then ip ◦ i−1 ∞ gives an embedding of Q(f ) into Qp and hence gives rise to a prime factor P of p in the integer ring Z(f ) of Q(f ); that is, P = {x ∈ Z(f )| ordp (ip (i−1 ∞ (x))) > 0} for the standard p-adic valuation ordp of Qp . Then the ﬁeld Qp (f ) generated by Hecke eigenvalues ip (i−1 ∞ (an )) over Qp (the p-adic Hecke ﬁeld of f ) is isomorphic to the P-adic completion of Q(f ) via ip ◦ i−1 ∞ , and we have a representation V = V (f ) of GSQ for K = Qp (f ) (see Theorem 2.43). Here S is the set of prime factors of N p. If further f has weight k ≥ 2 and the Hecke eigenvalue ap for the Hecke operator T (p) or U (p) is a p-adic unit (we call such a form P-ordinary), V (f )

28

Introduction

has a ﬁltration as above for (a, b) = (k − 1, 0) for the weight k of f , as long as ψ is unramiﬁed at p. In this example, dimK Fp0 V /Fp1 V = dimK Fpk−1 V = 1 and dimK V = 2 The Galois representation associated to Hecke eigenforms will be described in detail in 2.3.8. 1.2.3 Greenberg’s Selmer groups Let V be an ordinary p-adic Galois representation of GSF . Let W be the p-adic integer ring of K, and take a W -lattice L in V stable under GSF . Here is how to ﬁnd L. Take a basis of V and identify EndK (V ) = M2 (V ). The action of GSF gives rise to a continuous homomorphism ρ : GF → GL2 (K). Since GSF is compact, Im(ρ) sits in a maximal compact subgroup. Since each maximal compact subgroup of GL2 (K) is a conjugate g · GL2 (W )g −1 of GL2 (W ) (Corollary 2.5), if Im(ρ) ⊂ g · GL2 (W )g −1 , replacing ρ by the isomorphic g −1 ρg, we may take L = W 2 ⊂ K 2 = V . The isomorphism class of L is unique when L/mW L is an absolutely irreducible GSF -module (a result of Carayol and Serre: see Proposition 1.25), where m = mW ⊂ W is the maximal ideal. Let us prepare some notation to deﬁne the Selmer group of V . We write H q (F, ?) for H q (GSF , ?) and H q (Fl , ?) for H q (Dl , ?). We put Fp+ V = Fp1 V , Fp+ L = Fp+ V ∩ L and Fp+ (V /L) = Fp+ V /Fp+ L. For prime ideals q p and p|p of F , we put, for X = V and V /L, Uq (X) = Ker(Res : H 1 (Fq , X) → H 1 (Iq , X))

if q p,

Up (X) = Ker(Res : H (Fp , X) → H

if p|p.

1

1

(Ip , X/Fp+ (X)))

(1.2.5)

Here the restriction map Res = ResG/H with respect to a group G and a subgroup H is obtained by restricting the cocycles of G to the subgroup H, and in the above deﬁnition of Uq , G = Dq and H = Iq . Then we deﬁne, for A = K and K/W , H 1 (Fq , L ⊗W A) q Resq 1 SelF (L ⊗W A) = Ker H (F, L ⊗W A) −−−−−→ , Uq (L ⊗ A) q (1.2.6) where Resq is the restriction map deﬁned with respect to GSF ⊃ Dq . The standard Selmer group of V is given by SelF (V /L), equipped with the discrete topology. The strict Selmer group Selstr F (L ⊗ A) is deﬁned replacing Iq by Dq for q|p in the above deﬁnition. The minus “−” Selmer group Sel− F (L ⊗ A) is deﬁned by the same formula replacing F + (V ) by F − (V ) := F 0 (V ) in the deﬁnition of Up (V ). Exercise 1.39 Show that the Selmer groups deﬁned above are independent of the choice of the ﬁnite set S of rational primes as long as S contains p and all ramiﬁed primes for the Galois representation V .

Selmer groups

29

Example 1.40 Take an elliptic curve E deﬁned over F as in Example 1.37, and consider SelF (E)p = SelF (V (E)/T (E)). Then by Kummer’s theory (e.g., [Gr4] Section 2), we have the following exact sequence 0 → E(F ) ⊗Z Qp /Zp → SelF (E)p → WE (F )p → 0, where E(F ) is the group of F -rational points of E, and WE (F )p is the p-part of the Shafarevich–Tate group (deﬁned by the above exact sequence). Thus, taking F = Q, the Selmer group is a key ingredient of the study of the p-adic version of the conjecture of Birch and Swinnerton-Dyer. The original conjecture roughly predicts the order of zero of L(s, E) = L(s, V (E)) at s = 1 is equal to dimQ (E(Q) ⊗Z Q) and the leading coeﬃcient of the Taylor expansion at s = 1 of L(s, E) is equal to the order of WE (Q) up to the product of the canonical period ΩE = E(R) ω of the N´eron diﬀerential and the regulator of E. Note that ΩE is the Deligne’s period c+ (H 1 (E)(1)) integrally normalized, and if L(1, E) = 0, the regulator is equal to 1; thus, the conjecture is consistent with Deligne’s rationality conjecture for the critical value of L(s, E) at s = 1. See [MzTT] Section 10 for the precise conjecture and its p-adic version. 1.2.4 Selmer groups with general coeﬃcients In the above deﬁnition of the Selmer group, an essential point is to have a twostep ﬁltration V ⊃ Fp+ V ⊃ {0} stable under the local Galois group Dp . Thus if we have such a ﬁltration, we can at least generalize the deﬁnition to any continuous We describe it when L is an R[GS ]-module, where p-proﬁnite Galois module L. F we assume that R is a proﬁnite local ring over the power series ring W [[x]]. Let κ : GSF → W [[x]]× be the universal character deforming the identity character of GSF . √ Exercise 1.41 Show that there is a unique square root κ : GpQ → Λ× W of κ deforming the identity character. Hint: One needs to use the assumption p > 2. is R-free of ﬁnite rank. For each prime factor p|p of F , we We suppose that L assume we have a ﬁltration: ⊃ · · · ⊃ Fp0 L ⊃ Fp1 L ⊃ · · · ⊃ Fpa+1 L = {0} = Fpb L L

(1.2.7)

stable under the decomposition group Dp such that an open subgroup of the pi+1 L by κi . We suppose inertia group Ip ⊂ Dp acts on each subquotient Fpi L/F pi+1 L is R-free of ﬁnite rank. that Fpi L/F = F 1 L. For prime ideals Let R∗ be the Pontryagin dual of R. We put F + L q p and p|p of F , we put ⊗R R∗ ) → H 1 (Iq , L ⊗R R∗ )) if q p, Ker(Res : H 1 (Fq , L ∗ ⊗R R ) = Uq (L R R∗ L⊗ 1 ∗ 1 Ker(Res : H (Fq , L ⊗R R ) → H (Ip , F + (L)⊗ )) if p|p. R∗ p

R

30

Introduction

Then we deﬁne, ⊗R R∗ ) = Ker H 1 (F, L ⊗R R∗ ) → SelF (L

H 1 (Fq , L ⊗R R∗ ) q

⊗R R∗ ) Uq (L

.

(1.2.8)

Example 1.42 The Selmer group SelQ (Λ∗ (ψ)) we already discussed is isomorphic to SelQ (Λ(ψ)⊗Λ Λ∗ ), because Fp+ Λ(ψ) = {0} in this case. Since Λ∗ (ψ) = IndQ Q∞ Tp (ψ) by Exercise 1.16, we have by Shapiro’s lemma (Lemma 1.17) SelQ (Λ(ψ) ⊗Λ Λ∗ ) ∼ = SelQ (Λ∗ (ψ)) ∼ = SelQ∞ (Tp (ψ)). Thus the Selmer group SelQ∞ (Tp (ψ)) which appears to depend on the Zp -extension Q∞ /Q can be interpreted as the Selmer group deﬁned over the base number ﬁeld Q of the universal character ψ in Lemma 1.19. In other words, we can deﬁne the classical Iwasawa module, without referring to the inﬁnite Zp -extension Q∞ /Q (and instead, taking the Selmer group over Q of the universal character ψ deforming ψ), and this viewpoint could be more natural, freeing Iwasawa’s theory from the peculiar choice of Zp -extensions. Example 1.43

Starting with an elliptic curve E/Q , we can think of Q T (E) ⊗Zp Λ∗ (ψ) ∼ = IndQ∞ (V (E)/T (E)).

Mazur [Mz] ﬁrst studied the Selmer group SelQ∞ (V (E)/T (E)) over the Zp -tower Q∞ /Q via Iwasawa theoretic techniques, and again by Lemma 1.17, Mazur’s Selmer group can be identiﬁed with SelQ (T (E)⊗Zp Λ∗ (ψ)) (Λ-adic Selmer group) deﬁned relative to Q. See [Gr4] for Iwasawa’s theory for elliptic curves. Suppose that R = Λ or its normal ﬁnite extension. To distinguish this case Naive questions are from the other, we write L for L. (q1) When does SelF (L ⊗R R∗ ) have an R-torsion Pontryagin dual module Sel∗F (L ⊗R R∗ ) of ﬁnite type? (See [Gr3]). It is well known that Sel∗F (L ⊗R R∗ ) is of ﬁnite type over R. Thus the main point of this question is the torsion property. We expect that this should k := L ⊗W [[Γ]],k W is algebro-geometric for inﬁnitely many be true if L positive integers k (that is, the Galois representation can be realized inside the ´etale cohomology group of a smooth projective variety over F ; in other words, it is associated to a motive). Here k : W [[Γ]] → W is the W -algebra homomorphism linearly extending the character Γ γ → γ k ∈ W ; (q2) Suppose further that R = Λ. If Sel∗F (L ⊗Λ Λ∗ ) is Λ-torsion, what meaning does its characteristic power series fL (x) of Sel∗F (L ⊗Λ Λ∗ ) have? More generally, if R is a normal ﬁnite extension of Λ and if Sel∗F (L ⊗R R∗ ) is R-torsion, is char(Sel∗F (L ⊗R R∗ )) a principal ideal generated by fL ∈ R? If that is the case, what is the function fL in the structure sheaf of Spf(R)?

Deformation and adjoint square selmer groups

31

Is it related to a p-adic L-function? (As for analytic p-adic L-functions, see [LFE] Chapters 7 and 10 and [SGL].) If L is algebro-geometric in the sense of (q1), we expect to have a padic analytic method of constructing p-adic L-functions Lp (L, s) such that [[x]] and φL (x) = fL (x) up Lp (L, s) = φL (γ s − 1) (γ = 1 + p) for φL (x) ∈ W to units in W [[x]], where W is a discrete valuation ring containing W . 1.3 Deformation and adjoint square selmer groups Adjoint square Selmer groups (which we will deﬁne in this section) have a very direct relation to Galois deformation rings. Indeed, we ﬁrst describe an explicit expression of the adjoint Selmer group as a module of diﬀerentials of the spectrum of a Galois deformation ring (which will be proven in detail in 3.4.5 later). Since we need to have a ﬁltration to deﬁne the Selmer group, we consider the universality among Galois deformations, imposing the existence of such a ﬁltration (the nearly ordinary universal deformation rings), and then we describe Mazur’s idea of relating the diﬀerentials on the spectrum of such deformation rings to Galois 1-cocycles in the Selmer group of the adjoint square of the deformations. Once the relation of the Selmer group with the diﬀerentials is established, we will state a watered-down version of the theorem of Wiles, Taylor and Fujiwara (Theorem 1.52) which gives a precise description of the deformation ring and at the same time the ﬁniteness of the corresponding Selmer group. A stronger version of the theorem will be proven in Chapter 3 identifying the deformation ring with the corresponding Hecke algebra. At the end of this section, we give a minimal outline of the proof of Theorem 1.52 and an overview of such modularity results known for elliptic modular forms and the base ﬁeld F = Q. Recall the Galois group GF = GpF of the maximal extension F p /F unramiﬁed outside p and ∞. In this section we ﬁx V as a two-dimensional irreducible representation of GF over K and suppose V satisﬁes (1.2.4): V Fp+ V {0} with dimK Fp+ V = 1

(ord)

stable under the decomposition group Dp such that the inertia group Ip ⊂ Dp acts on the quotient V /Fp+ V trivially. We write p (resp. δp ) for the character giving the action of Dp on Fp+ V (resp. V /Fp+ V ). We also write p = (p mod mW ) and δ p = (δp mod mW ). Let Fpur be the maximal unramiﬁed extension of Fp inside Qp . We suppose the following two conditions throughout this section: (cyc) The characters det(V )|Ip and δp |Ip factor through Gal(Fpur [µp∞ ]/Fpur ) for each p|p; (ds) δ p = p for all p|p. We ﬁx a W -lattice L ⊂ V stable under GF . Choose a basis of L over W , and write ρ : GF → GL2 (W ), the resulting matrix representation. We write ν = det(ρ). The representation ρ we start with is called the initial Galois representation.

32

Introduction

1.3.1 Nearly ordinary deformation rings In this subsection, A denotes an object in CLW . We consider universality among the deformations of L of the following type. They are rank two free A-modules with continuous GF -action satisfying the following four properties: L AL ∼ (D1) L/m = L as GF -modules for L = L ⊗W F. (D2) Writing ι : W → A for the structure homomorphism of W -algebras, we have the identity of the determinant characters: ι ◦ ν = det L. (D3) Fix a decomposition group Dp of each prime factor p|p in GF . Then we →L → L/F p+ L → 0 stable under Dp have an exact sequence 0 → Fp+ L + for each prime factor p|p, where L/Fp L is free of rank one over A and the ⊗A F is given by p . action of Dp on Fp+ L p+ L factors through the local cyclotomic Galois (D4) The action of Ip on L/F ur ur ∞ group Gal(Fp [µp ]/Fp ) (giving rise to the nearly ordinary character δρ,p : Gal(Fpur [µp∞ ]/Fpur ) → A× ), where Fpur is the maximal unramiﬁed extension of Fp . is unrami is ordinary if L/F p+ L The condition (D3) is the near ordinarity, and L ﬁed for all p|p. The condition (D2) on the automorphic side corresponds to ﬁxing the central character of automorphic representations. The local cyclotomy condition (D4) is not necessary (so, it is not made in [MFG] in 5.6.3) but makes our argument simpler as we will see later. (resp. L and L) When we consider the matrix form of the representation L ﬁxing a basis of L over A, we write it as ρ : GF → GL2 (A) (resp. ρ : GF → GL2 (F) and ρ : GF → GL2 (W )). If every centralizer x ∈ GL2 (F) of ρ (that is, xρ(σ) = ρ(σ)x for all σ ∈ GF ) is scalar, we have a universal couple (RF , ρF : GF → GL2 (RF )) such that for any deformation ρ as above, we have a unique W -algebra homomorphism ϕ : RF → A such that ϕ ◦ ρF ∼ = ρ in GL2 (A) (see Theorem 1.26 and [MFG] 2.3 and 3.2.4). Here RF is a noetherian p-proﬁnite local W -algebra with residue ﬁeld F. Starting with a Galois representation L of GFp := Gal(Qp /Fp ), we can think of L taking GF in place of GF . Again of the universality among deformations L p we have a universal couple (RFp , ρFp ) similar to (RF , ρF ) if the centralizer Z(ρp ) = {x ∈ GL2 (F)|xρp x−1 = ρp } for ρp = ρ|GFp is made up of scalars. We write Φ(A) for the set of isomorphism classes of all deformations ρ : GF → GL2 (A) of ρ. Similarly we write Φp (A) for the set of isomorphism classes of all deformations ρ : GFp → GL2 (A) of ρp . Then Φ and Φp are covariant functors from the category CLW of proﬁnite W -local algebras with residue ﬁeld F into the category SET S of sets. We also consider the subfunctor Φn.ord,ν (A) ⊂ Φ(A), made up of A-valued deformations satisfying (D1–4). If the universal couple (XF , ξF : GF → GL2 (XF )) exists for any one Ψ of Φ, Φn.ord,ν and Φp , we say that

Deformation and adjoint square selmer groups

33

∼ Ψ(A) the functor is representable by (XF , ξF ), and we have HomW -alg (XF , A) = by HomW -alg (XF , A) ϕ → ϕ ◦ ξF ∈ Ψ(A). As before, let Z(ρ) = {x ∈ GL2 (F)|xρx−1 = ρ} be the centralizer of ρ. The condition Z(ρ) = F× ⊂ GL2 (F) follows from one of the following two conditions: (aiM ) ρ restricted to Gal(Q/M ) is absolutely irreducible (Schur’s lemma; e.g., [MFG] Proposition 2.5); (DS) ρ is reducible, and Im(ρ) contains at least one nontrivial unipotent element and at least one semi simple element with two distinct eigenvalues. Exercise 1.44 Prove Z(ρ) = F× ⊂ GL2 (F) under (DS) by computation. Exercise 1.45 Let ρ : GQp → GL2 (Fp ) be a Galois representation of the form

ψ ∗ ρ(σ) = 0 1 for a nontrivial character ψ : GQp → F× p . Suppose

1 1 u= ∈ Im(ρ). 0 1 Show that, for each isomorphism class in Φp (A), we can ﬁnd its member ρ such ρ) contains that ρ (σ) is upper triangular for all σ ∈ GQp and Im(

1 1 u= ∈ GL2 (A). 0 1 Theorem 1.46 (Mazur) Under the existence of the universal couple (R, ) for Φ and under (ds), the universal couple (Rn.ord,ν , ρn.ord,ν ) exists for Φn.ord,ν . The universal ring Rn.ord,ν is noetherian if it exists. This fact was proven in Mazur’s paper in [Mz2]. Under appropriate assumptions, the existence of the universal couple (Rρ , ) has already been shown in Theorem 1.26. Here we prove the existence of the universal couple (Rn.ord , ρn.ord,ν ) assuming the existence of a universal couple (R = Rρ , ) for F = Q. We will later prove in Theorem 1.57 the noetherian property of the universal ring. The proof for a general ﬁeld F is the same and is left to the reader. Proof An ideal a ⊂ R is called of type (n.ord, ν) if mod a satisﬁes (D1–4). Let an.ord,ν be the intersection of all ordinary ideals, and put Rn.ord,ν = R/an.ord,ν and n.ord,ν = mod an.ord,ν . If ρ : GQ → GL2 (A) satisﬁes (D1–4), we have a unique morphism ϕρ : R → A such that ( mod Ker(ϕρ )) ∼ = ϕρ ◦ ∼ = ρ. Thus Ker(ϕρ ) is of type (n.ord, ν), and hence n.ord,ν Ker(ϕρ ) ⊃ a . Thus ϕρ factors through Rn.ord,ν . The only thing we need to show is that mod an.ord,ν satisﬁes (D1–4). Since an.ord,ν is an intersection of ideals of type (n.ord, ν) ideals, we need to show that if a and b are of type (n.ord, ν), then a ∩ b is of type (n.ord, ν).

34

Introduction

be an A-module with an action To show this, we prepare some notation. Let L has a submodule F + L on which Dp of GQ satisfying conditions (D1–4). Thus L p + acts by a character p with p mod mA = p . Then Fp L = σ∈Dp (σ − p (σ))L. Now suppose that ρa = mod a and ρb = mod b are both ordinary. Let ρa∩b = mod a ∩ b, and write their representation spaces as La = L(ρa ), Lb = L(ρb ) and La∩b = L(ρa∩b ). By deﬁnition, La∩b /aLa∩b = La and La∩b /bLa∩b = Lb . We write the character on Fp+ La (resp. Fp+ Lb ) as a (resp. b ). By the Chinese remainder theorem, we have an exact sequence π

0 → R/a ∩ b → R/a ⊕ R/b − → R/a + b → 0 with π(a, b) = (a − b) mod (a + b). Since a ≡ b mod (a + b) by (ds), the diagonal character := a × b : Dp → (R/a)× × (R/b)× is sent to 0 by π. a∩b = Thus has values in R/(a ∩ b). We deﬁne Fp+ L σ∈Dp (σ − (σ))La∩b . + +L Then by deﬁnition and by (ds), we have (L/F p a∩b ) ⊗R R/a = La /Fp La and + + a∩b ) ⊗R R/b = L b . Then by Nakayama’s lemma (Lemma 1.6), b /Fp L pL (L/F + (L/Fp La∩b ) is generated by one element, thus it is a surjective image of A = R/a ∩ b. Since in A, a ∩ b = 0, we can embed A into A/a ⊕ A/b as above. a ∼ ∼ b /F + L a /Fp+ L Since L = A/a and L p b = A/b, the kernel of the diagonal map a∩b ) → (L a ) ⊕ (L b ) ∼ a /Fp+ L b /Fp+ L p+ L (L/F = A/a ⊕ A/b p+ L a∩b ) ∼ a∩b is a∩b /Fp+ L has to be zero. Thus (L/F = A. Since the quotient L a∩b is a direct A-summand of L a∩b ∼ A-free, Fp+ L = A2 and hence is A-projective. + Since A is local, Fp La∩b is A-free of rank 1. This shows (D3). a and L b /F + L a /Fp+ L By the above construction, if the p-inertia action on L p b ur ur + a∩b has to factor a∩b /Fp L factors through Gal(Qp [µp∞ ]/Qp ), its action on L ur ∞ through Gal(Qur [µ ]/Q ). Thus (D4) follows. p p p a × det L b with values As for (D2), again consider the diagonal character det L in R/a ⊕ R/b which is killed by π. Thus it has values in A and gives the char a = ν mod a and det L b = ν mod b, we conclude a∩b . Since det L acter det L a∩b = ν mod (a ∩ b), which shows (D2) and ﬁnishes the proof. 2 det L Hereafter we write RF (resp. ρF ) for Rn.ord,ν (resp. ρn.ord,ν ). By (D3), the space L(ρF ) of ρF has a natural ﬁltration, stable under Dp , L(ρF ) ⊃ Fp+ L(ρF ) ⊃ {0} with L(ρF )/F + L(ρF ) ∼ = RF . We write δ p (resp. εp ) for the character giving the action of Dp on L(ρF )/F + L(ρF ) (resp. F + L(ρF )). ∼ Thus δ p : Gal(Fpur [µp∞ ]/Fpur ) → R× F is a continuous character, and ρp |Dp =

εp ∗ . Since ϕρ ◦ δ p is trivial for ϕρ : RF → W inducing ρ = ϕρ ◦ ρF , the 0 δp image of δ p in R× F is p-proﬁnite. Thus δ p factors through the p-proﬁnite part of Gal(Fp [µp∞ ]/Fp ). The local cyclotomic character Np : Gal(Fp [µp∞ ]/Fp ) → Z× p

Deformation and adjoint square selmer groups

35

identiﬁes the p-proﬁnite part of Gal(Fpur [µp∞ ]/Fpur ) with a p-proﬁnite subgroup m Γp of 1 + pZp . Thus Γp is generated by γp := γ p for a suitable m for the generator γ = 1 + p of 1 + pZp = γ Zp , and δ p induces a W [[Γp ]]-algebra structure on RF . Let ΓF = p|p Γp , and hereafter we regard RF as a W [[ΓF ]]-algebra by this algebra homomorphism. Writing S = {p|p}, we have an isomorphism W [[ΓF ]] ∼ = W [[xp ]]p∈S . 1.3.2 Adjoint square Selmer groups and diﬀerentials ∈ Φn.ord,ν (A). Deﬁne Let L = φ ∈ EndA (L) Tr(φ) = 0 . Ad(L)

(1.3.1)

by conjugation v → ρ (σ)v We let σ ∈ GF act on v ∈ Ad(L) ρ(σ)−1 . Then Ad(L) has the following three-step ﬁltration stable under Dp for each prime ideal p|p of F : ⊃ Fp− Ad(L) ⊃ Fp+ Ad(L) ⊃ {0}, Ad(L) (1.3.2) where + + = {φ ∈ Ad(L)|φ(F Fp− Ad(L) p L) ⊂ Fp L}, + = {φ ∈ Ad(L)|φ(F Fp+ Ad(L) p L) = 0}.

containing a generator of Fp+ L and we identify EndK (L) If we take a basis of L − + with M2 (A) by this basis, Fp Ad(L) (resp. Fp Ad(L)) is made up of upper triangular matrices with trace zero (resp. upper nilpotent matrices). We thus have SelF (Ad(V /L)) for Ad(V /L) := Ad(L)⊗Zp Tp = Ad(V )/Ad(L), and we also have ⊗A A∗ ) for L ∈ Φn.ord,ν (A). SelF (Ad(L) Note that Dp acts trivially on Fp− Ad(V )/Fp+ Ad(V ). We often indicate this fact by writing Fp− Ad(V )/Fp+ Ad(V ) ∼ = K as Dp -modules. We quote Proposition 1.47 Suppose that Φn.ord,ν has a universal couple (RF , ρF ). Then the Pontryagin dual Sel∗F (Ad(V /L)) is canonically isomorphic to the module of 1-diﬀerentials ΩRF /W [[ΓF ]] ⊗RF ,ϕ W , where ϕ : RF → W is the W -algebra ∈ Φn.ord,ν (A), homomorphism such that ρ ∼ = ϕ ◦ ρF . More generally, for any L we have ⊗A A∗ ) = Hom(SelF (Ad(L) ⊗A A∗ ), Tp ) ∼ Sel∗ (Ad(L) = ΩR /W [[Γ ]] ⊗R ,φ A, F

F

F

F

where φ : RF → A is the W -algebra homomorphism such that ρ ∼ = φ ◦ ρF . This proposition will be proven as Proposition 3.87. We recall here the deﬁnition of K¨ ahler 1-diﬀerentials and some of their properties. Let A be a B-algebra, and suppose that A and B are objects in CLW . We consider the module of continuous 1-diﬀerentials ΩA/B with the universal B-derivation d : A → ΩA/B for a B-algebra A (A, B ∈ CLW ). Here the continuity is with respect to the proﬁnite topology on A.

36

Introduction

For a module M with continuous A-action (in short, a continuous A-module), let us deﬁne the module of B-derivations by   δ: continuous   DerB (A, M ) = δ : A → M ∈ HomB (A, M )δ(ab) = aδ(b) + bδ(a) .   for all a, b ∈ A Here the B-linearity of a derivation δ is equivalent to δ(B) = 0, because δ(1) = δ(1 · 1) = 2δ(1) ⇒ δ(1) = 0. As we will prove in Proposition 1.49, the module ΩA/B has the following universal property. For any given δ ∈ DerB (A, M ), there exists a unique continuous A-linear map φ : ΩA/B → M such that δ = φ ◦ d; in other words, we have HomA (ΩA/B , M ) ∼ = DerB (A, M ) by φ → φ ◦ d. Thus ΩA/B represents the covariant functor M → DerB (A, M ) from the category of continuous A-modules into Z-M OD. Since DerB (A, M ) only depends on the image of B in A under the algebra homomorphism ι : B → A giving the B-algebra structure of A, we have ΩA/B ∼ = ΩA/ι(B) .

(1.3.3)

The construction of ΩA/B is easy. The multiplication a ⊗ b → ab induces B A → A taking a ⊗ b to ab. We put a B-algebra homomorphism m : A⊗ B A. Then we deﬁne ΩA/B = I/I 2 . I = Ker(m), which is an ideal of A⊗ The map d : A → ΩA/B given by d(a) = 1 ⊗ a − a ⊗ 1 mod I 2 is a continuous B-derivation (Exercise 1.48). Thus we have a morphism of functors: HomA (ΩA/B , ?) → DerB (A, ?) given by φ → φ ◦ d. Since ΩA/B is generated by d(A) as A-modules (Exercise 1.48), the above map is injective. Exercise 1.48 Show that the map d(a) = 1 ⊗ a − a ⊗ 1 mod I 2 is a continuous B-derivation and that ΩA/B is generated by d(A) as an A-module. To show that ΩA/B represents the functor, we need to show the surjectivity of the above map as follows. Proposition 1.49 The above morphism of functors M → HomA (ΩA/B , M ) and M → DerB (A, M ) is an isomorphism, where M runs over the category of continuous A-modules. Thus the functor: M → DerB (A, M ) is represented by ΩA/B in the category of continuous A-modules. Proof We now prove the surjectivity. Deﬁne φ : A × A → M by (a, b) → aδ(b) for δ ∈ DerB (A, M ). Then φ(ab, c) = abδ(c) = bφ(a, c) and φ(a, bc) = aδ(bc) = abδ(c) = bφ(a, c) for a, c ∈ A and b ∈ B. Thus φ gives a continuous B-bilinear map. By the universality of the tensor product,

Deformation and adjoint square selmer groups

37

B A → M . Now we see that φ : A × A → M extends to a B-linear map φ : A⊗ φ(a ⊗ 1 − 1 ⊗ a) = aδ(1) − δ(a) = −δ(a) and φ((a ⊗ 1 − 1 ⊗ a)(b ⊗ 1 − 1 ⊗ b)) = φ(ab ⊗ 1 − a ⊗ b − b ⊗ a + 1 ⊗ ab) = −aδ(b) − bδ(a) + δ(ab) = 0. This shows that φ|I –factors through I/I 2 = ΩA/B and δ = φ ◦ d, as desired. 2 If we add more conditions on deformations besides (D1–4), we get a smaller universal couple (RF , ρF ). Then we often get an identity of the following type (similar to Proposition 1.47): ⊗A A∗ ) = ΩR /W [[Γ ]] ⊗R ,φ A. Sel∗F (Ad(L) F F F

(1.3.4)

If RF is small enough, we may be able to conclude that the Selmer group Sel∗F (Ad(V /T )) is ﬁnite and expect to have the following conjectural formula L(1, Ad(V )) −[K:Qp ] p] |Sel∗F (Ad(V /T ))| = |Lp (1, Ad(V ))|−[K:Q = (1.3.5) p c+ (Ad(V )(1)) p (a generalized class number formula), where Lp (s, Ad(V )) is a (conjectural) p-adic L-function and c+ (Ad(V )(1) is an integrally normalized period of Deligne. This type of formula has been veriﬁed (as we describe in 3.4.5) adding the fol satisfying (D1–4). Let Σp be the set of lowing condition to the deformations L all prime factors of p in O. Fix a pair of integers (κ1,p , κ2,p ) for each p ∈ Σp , and write κ for the tuple (κ1,p , κ2,p )p . We assume that [κ] = κ1,p +κ2,p is independent of p ∈ Σp . As an extra condition, we now consider p+ L, Gal(Fpur [µp∞ ]/Fpur ) acts by the character N κ1,p for all p|p, and (D5) On L/F = N [κ] on an open subgroup of Ip . det(L) We write Φκ (A) for the set of isomorphism classes of deformations ρ : GF → GL2 (A) of ρ satisfying (D1–5). Under (aiF ) or (ds), we have the universal couple (Rκ,F , ρκ,F ) among the deformations satisfying (D1–5). Exercise 1.50 Show Rκ,F = RF ⊗W [[ΓF ]],κ W , where κ : ΓF → W × is given by (γp ) → p N (γp )κ1,p . We call c ∈ GF a complex conjugation, if c is in the conjugacy class of a complex conjugation in Gal(Q/Q). Conjecture 1.51 Suppose (ds) and (aiF ) for ρ and that F is totally real. If det(ρ)(c) = −1 for any complex conjugation c, the universal ring Rκ,F is free of ﬁnite rank over W , and Rκ,F is a reduced local complete intersection if κ2,p − κ1,p ≥ 1 for all p ∈ Σp . is

Here a reduced algebra A free of ﬁnite rank over W [[x1 , . . . , xt ]] a local complete intersection over R = W [[x1 , . . . , xt ]] if

38

Introduction

A∼ = R[[T1 , . . . , Tr ]]/(f1 (T ), . . . , fr (T )) for r power series fi (T ), where r is the number of variables in R[[T1 , . . . , Tr ]]. Though the assertion of Rκ,F being a local complete intersection is technical, as we will see later, this claim is a key to relating the size of the Selmer group with the corresponding L-value. In the classical setting of Galois representations associated to elliptic modular forms of weight k (in Sk (Γ1 (N ))), we have κ = (0, k − 1). Thus the condition κ2,p − κ1,p ≥ 1 is equivalent to requiring k ≥ 2. Theorem 1.52 (Wiles, Taylor, Fujiwara) Suppose that the initial representation ρ is associated to a Hilbert modular form of p-power level (in this case, we call ρ modular). If (aiM ) holds for M = F [µp ], Conjecture 1.51 holds. A more general version of this theorem will be proven as Theorem 3.67 and Corollary 3.42 later, and a brief outline of how we reach such a conclusion is given in 1.3.4 for F = Q. Proposition 1.53

Assume Conjecture 1.51. Then

ρ) ⊗W W ∗ ) is ﬁnite for any ρ ∈ Φk (W ) if κ2,p − κ1,p ≥ 1 for all 1. SelF (Ad( p ∈ S. 2. RF is a reduced local complete intersection free of ﬁnite rank over W [[ΓF ]]. 3. Sel∗F (Ad(ρF ) ⊗RF R∗F ) is a torsion RF -module. 4. For an irreducible component Spf(I) of Spf(RF ), write ρI = π ◦ ρF for the projection π : RF I. Then Sel∗F (Ad(ρI ) ⊗I I∗ ) is a torsion I-module. Proof If R is reduced and free of ﬁnite rank over W , ΩR/W is a ﬁnite module. Thus the ﬁrst assertion follows. Note that Pκ = Ker(κ : W [[ΓF ]] → W ) is generated by ((1 + xp ) − N (γp )κ1,p ) for p ∈ S. Thus ∩κ Pκ = {0}. Since RF /Pκ RF ∼ = Rκ,F which is free of ﬁnite rank s over W , by Nakayama’s lemma (Lemma 1.6), RF is generated by s elements r1 , . . . , rs over W [[ΓF ]] which gives a basis of Rκ,F over W . Thus we have a surjective W [[ΓF ]]-linear map ι : W [[ΓF ]]s → RF sending (a1 , . . . , as ) to j aj rj . Taking another κ , we ﬁnd that RF /Pκ RF ∼ = Rκ ,F which is free over W ; so, it has to be free of rank s over W . Thus Ker(ι) ⊂ Pκs for all κ ; so, ι has to be an isomorphism. This shows the freeness in the second assertion. Let C be " the set of all κ = (κp )p such that κ2,p − κ1,p ≥ 1 for all p. Then we still have κ∈C Pκ = {0}. Thus the natural W -algebra homomorphism RF → R κ∈C κ,F is an injection. The right-hand side is reduced (i.e., no nilpotent radical), and RF is reduced. We write Rκ = R/Pκ =

W [[T1 , . . . , Tr ]] . (f 1 (T ), . . . , f r (T ))

Write tj ∈ mRκ for the image of Tj in Rκ . Take a lift tj in mR of tj so that tj = (tj mod Pκ R). Deﬁne ϕ : W [[ΓF ]][[T1 , . . . , Tr ]] RF by

Deformation and adjoint square selmer groups

39

ϕ(f (T1 , . . . , Tr )) = f (t1 , . . . , tr ). Since RF is W [[ΓF ]]-free, Ker(ϕ)⊗W [[ΓF ]],κ W = (f 1 , . . . , f r ); so, taking a lift fj ∈ Ker(ϕ) of f j , we ﬁnd Ker(ϕ) = (f1 , . . . , fr ) by Nakayama’s lemma (Lemma 1.6), and hence RF is a local complete intersection over W [[ΓF ]]. Since RF is reduced and ﬁnite over W [[ΓF ]], ΩRF /W [[ΓF ]] is a torsion RF -module. From this, the last two assertions follow. This ﬁnishes the proof. 2 Since RF is reduced and free of ﬁnite rank over W [[ΓF ]], its total quotient ring F := RF ⊗W [[Γ ]] I is a product of ﬁelds of ﬁnite dimension over the ﬁeld Q of R F K of fractions of W [[ΓF ]]. In particular, writing K for the ﬁeld of fractions of I, we have Q = K ⊕ X for a complementary ring direct summand X. Let I be F to X. Then Spf(R F ) = Spf(I) ∪ Spf(I ) (and Spf(I ) is the projection of R the union of irreducible components other than Spf(I)). We take the intersection Spf(C0 ) = Spf(I) ∩ Spf(I ); so, C0 = I ⊗R F I , which is a torsion I-module called the congruence module of I (or of Spf(I)). # $ ∼ (cf. [H88a] 6.3). Exercise 1.54 Show that I ⊗R F I = I/ (K ⊕ 0) ∩ RF By the above expression, the W [[ΓF ]]-freeness tells us that charI (C0 ) = (K ⊕ 0)∩ RF is an intersection of a power of prime divisors (cf. [BCM] 7.4.2). Suppose that I is a unique factorization domain (for example, if I ∼ = W [[ΓF ]] = W [[xp ]]p which is often the case), and hence char(C0 ) is a principal ideal generated by f ∈ I. For this conclusion, we do not need the isomorphism I ∼ = W [[ΓF ]] = W [[xp ]]p but a milder condition that I is a Gorenstein ring over W [[ΓF ]] is enough (that is, HomW [[ΓF ]] (I, W [[ΓF ]]) ∼ = I as I-modules; see [H88a] Theorem 6.8). Note that a local complete intersection over W [[ΓF ]] is a Gorenstein ring (e.g., [CRT] Theorem 21.3). Now by a theorem of Tate (e.g., [MFG] 5.3.4), char(ΩRF /W [[ΓF ]] ⊗RF I) = char(C0 ). Thus we ﬁnd Corollary 1.55 Suppose either that I is a unique factorization domain or that I is a Gorenstein ring. Then char(Sel∗F (Ad(ρI ) ⊗I I∗ )) is a principal ideal in I generated by f ∈ I. The characteristic ideal of a well-behaved Selmer group has to be principal, since our philosophy predicts that it is spanned by the corresponding p-adic L-function. Thus at least, I has to be a Gorenstein ring. By (1.3.4) (and Lemma 5.7), we have for any prime ideal P ∈ Spf(I) with ι : I/P ∼ = W , writing ρP = ι ◦ ρI : GF → GL2 (W ) Sel∗F (Ad(ρP ) ⊗W W ∗ ) ∼ = ΩRF /W [[ΓF ]] ⊗RF ,P W ∼ = Sel∗F (Ad(ρI ) ⊗I I∗ ) ⊗I I/P. This shows that if char(Sel∗F (Ad(ρI ) ⊗I I∗ )) = (f ) for f ∈ I, we have char(Sel∗F (Ad(ρP ) ⊗W W ∗ )) = (f (P )),

40

Introduction

where f (P ) = (f mod P ) ∈ W . Thus we get Corollary 1.56 Let the notation and the assumption be as in Corollary 1.55. −[K:Qp ] for all P ∈ Spf(I)(W ). Then we have |Sel∗F (Ad(ρP ) ⊗W W ∗ )| = |f (P )|p In this corollary, we do not preclude the case where Sel∗F (Ad(ρP ) ⊗W W ∗ ) is inﬁnite. In such an extreme case, simply f (P ) = 0 and, hence, −[K:Qp ] = ∞. |f (P )|p If we can show that for densely populated points P ∈ Spf(I)(W ), the size of L(1, Ad(ρP )) the Selmer group Sel∗F (Ad(ρP ) ⊗W W ∗ ) is given by the L-value for c+ (Ad(ρP )) L(1, Ad(ρP )) a normalized period c+ (Ad(ρP )), f (P ) = up to units in W ; so, f c+ (Ad(ρP )) gives a p-adic L-function of the adjoint square representation Ad(ρI ). This holds if F = Q and the initial ρ is modular (associated to an elliptic modular form; see [MFG] Theorem 5.20 and [DiFG] for more general and global results). For a generalization of this type of formula, see the papers (e.g., [G], [G1] and [G2]) by E. Ghate when [F : Q] = 2 and the paper [Dim] by M. Dimitrov for general F under some assumptions. The analytic p-adic L-function P → Lp (1, Ad(ρP )) and the arithmetic counterpart P → f (P ) are a new type of L-function, because the variable is not the cyclotomic variable s but the geometric points P of the spectrum of the universal deformation ring (this is another indication of the departure from the cyclotomic Zp -extension in our nonabelian theory). In any case, we expect to have a p-adic L-function Lp (Ad(ρI )) such that the characteristic ideal char(Sel∗F (Ad(ρI )⊗I I∗ )) ⊂ I is generated by Lp (Ad(ρI )) := f . L(1, Ad(ρk )) for ρk = k ◦ ρI : This L-function should interpolate the L-values period GF → GL2 (W ). Note that Ad(ρ ⊗ χ) = Ad(ρ) for any Galois character χ of Gal(F [µp∞ ]/F ), and thus, this interpolation does not involve the character twists Ad(ρ) ⊗ χ. In other words, we should have the Selmer group of |Σp | + 1 variables (Σp = {p|p}): |Σp | variables coming from I (or RF ) and another coming from the (cyclotomic) character twists. Accordingly, for F = Q, we should have two W W [[Γ]] = I[[x]] such that Lp (Pk , Pχ ) = Lp variable p-adic L-functions Lp ∈ I⊗ mod (Pk ⊗ Ker(χ)) giving rise to Lp (1, Ad(ρk ) ⊗ χ). In order to include the character twists on the arithmetic side, we need to study RFn for each layer of the cyclotomic Zp -extension F∞ (see Section 1.4 and Chapter 5). In this section, we have related the adjoint square Selmer group with the diﬀerentials of the deformation rings. (q3) It is an interesting problem to ﬁnd such a link relating the Selmer group of Galois representations with its universal deformation ring. As a more precise example of this type of question, one could ask: Is there any good ring invariant of the deformation ring which describe, for example, the standard Selmer group of ρ or ρF ?

Deformation and adjoint square selmer groups

41

1.3.3 Universal deformation rings are noetherian We prove here the following ﬁniteness result, whose proof gives some ﬂavor of how to relate diﬀerentials to a Galois cohomology group. Later in the proof of Proposition 3.87 (which also proves Proposition 1.47), we fully exploit such a construction. Theorem 1.57 (B. Mazur) Let (Rρ , ) as in Theorem 1.26 be a universal couple for a Galois representation ρ : GSF → GL2 (F) for a ﬁnite set S of rational primes. Then Rρ is noetherian, and hence Rn.ord,ν in Theorem 1.46 is also noetherian. To prove this result, we prepare some preliminary results. Lemma 1.58 Let A be a proﬁnite local W -algebra with A/mA = W/mW = F. Then A is noetherian if and only if mA /m2A is ﬁnite dimensional over F. Proof If dimF mA /m2A < ∞, by Nakayama’s lemma (Lemma 1.6), mA is is generated over F by generated by x1 , . . . , xr over A and hence mnA /mn+1 A degree n monomials of x1 , . . . , xr . In other words, the ring homomorphism π of W [[X1 , . . . , Xr ]] into A sending Xj to xj has a dense image under the mA -adic topology. Since W [[X1 , . . . , Xr ]] is a compact ring, Im(π) is compact and hence is closed. Thus π is surjective, and hence A is noetherian. The proof of the converse is plain and is left to the reader. 2 Write t∗A = mA /m2A + mW , which is called the cotangent space of Spec(A) at mA . We deﬁne tA = HomF (t∗A , F) (the tangent space). Write R = Rρ simply. To prove R is noetherian, we compute tR via Galois cohomology and prove dimF tR < ∞ by class ﬁeld theory. We let GSF act on M2 (F) by gv = ρ(g)vρ(g)−1 . This GSF -module will be written as ad(ρ). Lemma 1.59

We have tR = HomF (t∗R , F) ∼ = H 1 (GSF , ad(ρ)),

1 where Hct (GSF , ad(ρ)) is the continuous ﬁrst cohomology group of GSF with coeﬃcients in the discrete GSF -module ad(ρ).

Proof Let A = F[X]/(X 2 ). We write for the class of X in A. Then 2 = 0. We consider φ ∈ HomW −alg (R, A). Write φ(r) = φ0 (r) + φ (r). Then we have from φ(ab) = φ(a)φ(b) that φ0 (ab) = φ0 (a)φ0 (b) and φ (ab) = φ0 (a)φ (b) + φ0 (b)φ (a). Thus Ker(φ0 ) = mR because R is local. Since φ is W -linear, φ0 (a) = a = a mod mR , and thus φ kills m2R and takes mR W -linearly into mA = F. Moreover for r ∈ W , r = rφ(1) = φ(r) = r + φ (r), and hence φ kills W . Since R shares its residue ﬁeld F with W , any element a ∈ R can be written as a = r + x with r ∈ W and x ∈ mR . Thus φ is completely determined by the restriction

42

Introduction

of φ to mR , which factors through t∗R . We write φ for φ when we regard it as an F-linear map from t∗R into F. Then we can write φ(r + x) = r + φ (x), and φ → φ gives rise to a linear map HomW −alg (R, A) → HomF (t∗R , F). Note that R/(m2R + mW ) = F ⊕ t∗R . For any ∈ HomF (t∗R , F), we extend to R/m2R declaring its value on F to be zero. Deﬁne φ : R → A by φ(r) = r + (r). Since 2 = 0, φ is a W -algebra homomorphism. In particular, φ = , and hence φ → φ is surjective. Since algebra homomorphisms killing m2R + mW are determined by its values on t∗R , φ → φ is injective. By the universality, we have HomW −alg (R, A) ∼ = {ρ : GSF → GL2 (A)|ρ

mod mA = ρ}/ ∼ .

Then we can write ρ(g) = ρ(g) + uρ (g). By the multiplicativity, we have ρ(gh) + uρ (gh) = ρ(gh) = ρ(g)ρ(h) = ρ(g)ρ(h) + (ρ(g)uρ (h) + uρ (g)ρ(h)). Thus as a function uρ : GSF → M2 (F), we have uρ (gh) = ρ(g)uρ (h) + uρ (g)ρ(h).

(1.3.6)

Deﬁne a map uρ : GSF → ad(ρ) by uρ (g) = uρ (g)ρ(g)−1 . Then, we have guρ (h) = ρ(g)uρ (h)ρ(g)−1 from the GSF -module structure of ad(ρ). From the above formula (1.3.6), by a simple computation, we conclude that uρ (gh) = guρ (h)+uρ (g). Thus uρ : GSF → ad(ρ) is a cocycle. Starting with a cocycle u, we can reconstruct the representation reversing the above process. Then again by computation, ρ ∼ ρ ⇐⇒ ρ(g) + uρ (g) = (1 + x)(ρ(g) + uρ (g))(1 − x)

(x ∈ ad(ρ))

⇐⇒ uρ (g) = xρ(g) − ρ(g)x + uρ (g) ⇐⇒ uρ (g) = (1 − g)x + uρ (g). Thus the cohomology classes of uρ and uρ are equal if and only if ρ ∼ ρ . This shows: HomF (t∗R , F) ∼ = HomW −alg (R, A) ∼ = {ρ : GSF → GL2 (A)|ρ

mod mA = ρ}/ ∼ ∼ = H 1 (GSF , ad(ρ)).

In this way, we get a bijection between HomF (t∗R , F) and H 1 (GSF , ad(ρ)). By tracking down (in the reverse way) our construction, one can check that the map is an F-linear isomorphism. This ﬁnishes the proof. 2 Proof of the theorem. Let K be the ﬁxed ﬁeld of Ker(ρ). Then K is a ﬁnite extension. Then by restricting cocycles to GSK , we have an exact sequence 0 → H 1 (Gal(K/Q), H 0 (GSF , ad(ρ))) → H 1 (GSF , ad(ρ)) → Hom(GSK , M2 (F)). By class ﬁeld theory, the abelianization (GSK )ab is the image of the ray class ∞ ﬁeld modulo (ps) for s = ∈S,=p , whose p-primary part is a Zp -module of

Deformation and adjoint square selmer groups

43

ﬁnite rank. Thus Hom(GSK , M2 (F)) is ﬁnite dimensional over F. Since Gal(K/Q) and ad(ρ) are ﬁnite groups, H 1 (Gal(K/Q), H 0 (GSF , ad(ρ))) is a ﬁnite module; so, tR ∼ = H 1 (GSF , ad(ρ)) is ﬁnite dimensional over F, which shows the desired assertion. Exercise 1.60 Prove the exactness of the sequence in the above proof. Here the map Res : H 1 (GSF , ad(ρ)) → Hom(GSK , M2 (F)) is obtained from restricting cocycles of GSF to the subgroup GSK , and the inﬂation map Inf : H 1 (Gal(K/Q), H 0 (GSF , ad(ρ))) → H 1 (GSF , ad(ρ)) comes from pulling back cocycles of the quotient Gal(K/Q) by the projection: GSF Gal(K/Q). 1.3.4 Elliptic modularity at a glance First we recall a minimum outline of how we prove Theorem 1.52, which we will carry out fully in Chapter 3 over a general totally real base ﬁeld F . For that, recall the congruence subgroup & %

a b Γ0 (N ) = ∈ SL2 (Z) c ∈ N Z c d deﬁned for each positive integer N . We consider the space of holomorphic cusp forms Sk (Γ0 (N )) of weight k (whose deﬁnition is in [IAT] Chapters 2 and 3 and is also recalled in 2.4.1 in the text), which is a ﬁnite-dimensional vector space over C of holomorphic functions on the upper half complex plane satisfying the conditions (s1) and (s2) in Section 2.4. As we recall in 2.4.1, we have linear operators T (n) indexed by positive integers n (called Hecke operators) acting on the vector space Sk (Γ0 (N )) for each positive integer n. These operators are mutually commutative, and T (1) is the identity map. By Proposition 2.50, if f = 0 is a common eigenvector of all T (n) with f |T (n)= λ(n)f , f is a con∞ stant multiple of an absolutely convergent Fourier series n=1 λ(n) exp(2πinz). ∞ Such an eigenvector with the expansion n=1 λ(n) exp(2πinz) is called a Hecke eigenform. For a given Hecke eigenform f ∈ Sk (Γ0 (N )), write f |T (p) = λ(p)f . Let hk (Z) = hk (N ; Z) ⊂ End(Sk (Γ0 (N ))) be the algebra generated over Z by the Hecke operators T (n). The algebra hk (Z) is commutative, reduced (no nontrivial nilpotent elements) and free of ﬁnite rank over Z by Theorem 2.48 and Proposition 2.50. Thus the ﬁeld Q(f ) generated by λ(p) for all primes p is a number ﬁeld. For each prime ideal p of Q(f ), the Hecke eigenform f as in Example 1.38 has an associated p-adic Galois representation ρp : Gal(Q/Q) → GL2 (K) for the p-adic completion K of Q(f ) (see Theorem 2.43). Let p be the ip

prime associated to the embedding Q(f ) ⊂ Q −→ Qp , and write ρ for ρp . Let W be the p-adic integer ring of K. We may assume that ρ has values in GL2 (W ) by Corollary 2.5. Then the Galois representation ρ is ordinary if λ(p) is a p-adic unit

44

Introduction

in W . We call such a prime p ordinary for f . Strictly speaking, W is a completion of the integer ring of Q(f ) at a prime p|p, and we should say that p is ordinary for f . The scalar extension hk (W ) = hk (Z) ⊗Z W has a W -algebra homomorphism λ : hk (W ) → W induced by T (n) → λ(n). Since hk (W ) is semi-local (because of rankW hk (W ) < ∞), there is a unique local direct summand Tρ of hk (W ) through which λ factors. We hereafter consider λ as a W -algebra homomorphism of Tρ into W . Assume now that N |p. Then ρ is unramiﬁed outside p and ∞ and hence factors through GQ . By Proposition 1.25 and the Chebotarev density theorem (that asserts the density of Frobenius elements for primes outside S in GSQ ; e.g., [CFN] Theorem 6.4), ρ is characterized by Tr(ρ(F rob )) = λ() for primes p. Similarly, we can construct a deformation ρT : GQ → GL2 (Tρ ) such that Tr(ρT (F rob )) is given by the image of T () in Tρ (see Corollary 2.45). The proof of the structure theorem (Theorem 1.52) of the deformation ring is done by identifying the deformation ring with the elliptic modular Hecke algebra Tρ . When F = Q, this goes as follows. Starting with a two-dimensional mod p Galois representation ρ : GSF → GL2 (F) given by ρ mod mW , we identify a suitable deformation ring of ρ with the Hecke algebra Tρ . In this process of identiﬁcation, we discover the properties claimed in Theorem 1.52 of the Galois deformation ring. In the setting of Theorem 1.52, the result is formulated in terms of a double digit weight κ, which is given by (0, k − 1) for the single weight k in the present setting, as we will see in Theorem 2.43. In Chapter 3, we will give a proof by Fujiwara (following the argument of Wiles and Taylor) of this fact in the Hilbert modular case, allowing ramiﬁcation at primes q outside p (under some minimality ramiﬁcation conditions (h1–4) in 3.2.1). By the eﬀort of number theorists after the work of Wiles [Wi2] and Taylor and Wiles [TaW], we now know such an identiﬁcation in may other cases beyond the scope of this book. Here, assuming F = Q, we shall give a glimpse of the local conditions necessary to deﬁne the deformation ring from which often we have an isomorphism onto the corresponding Hecke algebra, and we try to formulate a theorem covering known cases when ramiﬁcation is “minimal” (this minimality we specify as Cases (A–D) outside p and Cases (FL)–(BT) at p; see below). We say ρ : Gal(Q/Q) → GL2 (F) is ﬂat at p if there exists a ﬁnite ﬂat (or equivalently, locally free) group scheme with an action of F as endomorphisms whose Galois representation on the generic (geometric) ﬁber is isomorphic to ρ restricted to Dp (see 2.2.2 for ﬁnite ﬂat group schemes). For a deformation ρ with values in an artinian local p-proﬁnite algebra A over W (in CLW ), it is called ﬂat if there exists a ﬁnite ﬂat (equivalently, locally free) group scheme over Zp on which A acts and whose Galois representation on the generic ﬁber is isomorphic to ρ |Dp . When ρ has values in a p-proﬁnite local W -algebra A, ρ is ﬂat if it is a projective limit of ﬂat representations ρ mod mnA (as n → ∞). One of the following local conditions at p (the residual characteristic) is imposed to

Deformation and adjoint square selmer groups

45

deﬁne the deformation ring: (FL) ρ is ﬂat at p, and we require ﬂatness also for deformations (the associated Selmer group is not of Greenberg type if ρ is ﬂat non-ordinary and is that of Bloch-Kato); −1

(ST) ρ is p-ordinary, p δ p = Np on Dp , and ρ|Ip is not semisimple (purely multiplicative reduction case: this case is called the “strict” case in [Wi2]); (SL) ρ is p-ordinary (this case is called the minimal Selmer case, and we require p-ordinarity for deformations also). The cases (SL) and (FL) may overlap for ρ, but in the case (FL), we insist on the ﬂat condition on deformations and in the case (SL), we insist of (D1–3) with unramiﬁed δ p for deformations. In the case (ST), we insist on p-ordinarity and p δ p−1 = Np for deformations ρ . Two more important cases (containing the ﬂat case as a special case) treated in [DiFG] and [Ki] are: (CR) ρ is crystalline at p satisfying the condition of Example 1 of [Fo] with Hodge ﬁltration Fp0 = V (ρ) and Fpk V (ρ) = 0 for k ≤ p + 1. Therefore, we assume in this case that we have a crystalline characteristic 0 lift ρ of ρ with det(ρ)|Ip = Npk−1 and 2 ≤ k ≤ p + 1 and require that deformations of ρ be crystalline at p. This case is called the crystalline case; (BT) ρ has a characteristic 0 modular lift ρ : Gal(Q/Q) → GL2 (W ) with det ρ = Np up to ﬁnite order character, and there exists an open subgroup D of ﬁnite index of Dp such that ρ|D is associated with a Barsotti–Tate pdivisible group deﬁned over the p-adic integer ring of the ﬁxed ﬁeld of D in Qp . This case is called the potential Barsotti–Tate case. The structure theorems in Case (CR) are treated in [DiFG] Section 3, and the modularity of any characteristic 0 lift in Case (BT) has been ﬁnished recently in [Ki] after some early work of others (e.g., [BrCDT]). We require in the case (CR) the crystalline property for deformations also, and in the case (BT) we require the potentially ﬂat condition on deformations, that is, for an open subgroup D ⊂ Gal(Qp /Qp ), we require that ρ |D comes from a ﬁnite ﬂat group scheme deﬁned over the p-adic integer ring of the ﬁxed ﬁeld of D in Qp . Since we assumed N |p, the cases (FL) and (BT) actually do not show up (because of dim S2 (SL2 (Z)) = 0); so, we forget about them. As for ramiﬁcation conditions at primes outside p, we list the following examples. If a prime ramiﬁes in ρ, the restriction of ρ to D is supposed to be isomorphic to a representation satisfying one of the following conditions:

χ1 ∗ ∼ with unramiﬁed characters χj with (A) ρ ramiﬁes at and ρ|D = 0 χ2 χ1 χ−1 uller character modulo p; 2 = ω, where ω is the Teichm¨

46

Introduction

(B) ρ|I

∼ =

χ 0

∗ with χ = 1; 1

(C) H 1 (Gal(Q /Q ), Ad(ρ)) = 0; (D) absolutely irreducible ρ|I = IndQ Q2 ϕ for the unique unramiﬁed quadratic extension Q2 /Q with a ramiﬁed character ϕ (the potentially ﬂat case). The above four -types basically exhaust all possibilities of minimal ramiﬁcation (that is, the Artin conductor of ρ|D has at most square -factor and is minimal up to character twists; see [GME] page 327). Then in Cases (A), (B) and (D), we require that deformations ρ satisfy the same property with ρ (I ) ∼ = ρ(I ). More general settings are treated in [DiFG] Section 3. Since we assumed N |p, the initial modular Galois representation ρ is unramiﬁed outside p; so, the conditions (A–D) are actually irrelevant under this assumption. Thus our deformation functor Φ assigns each local p-proﬁnite W algebra with residue ﬁeld F = W/mW the set Φ(A) of isomorphism classes of deformations ρ : GQ → GL2 (A) of ρ satisfying (D1–2) and either (FL), (ST) or (SL) accordingly as ρ satisﬁes (FL), (ST) or (SL). Then if (aiQ ) is satisﬁed, we have a universal deformation couple (Rρ , ρ) : GQ → GL2 (Rρ )). Here Rρ is a local p-proﬁnite W -algebra with residue ﬁeld F = W/mW , and ρ is the universal deformation. In other words, by the correspondence HomW -alg (Rρ , A) φ → φ ◦ ρ ∈ Φ(A), we have a functorial isomorphism Φ(A) ∼ = HomW -alg (Rρ , A). Then as a special case of the result of the papers already quoted, we get √ Theorem 1.61 Let M = Q[ p∗ ] for p∗ = (−1)(p−1)/2 p. Assume (aiM ), p ≥ 5, k ≥ 2 (⇒ κ2 − κ1 ≥ 1) and that N = p if k = 2 and otherwise N = 1. Further suppose that k ≤ p − 1 if we are in Case (CR). Then we have Rρ ∼ = Tρ and ρ∼ = ρT . It is known that if ρ = ρ mod mW for ρ associated to f ∈ Sk (SL2 (Z)), ρ ⊗ χ for a suitable mod p character is associated to a Hecke eigenform in Sk0 (SL2 (Z)) for 2 ≤ k0 ≤ p + 1 (cf. [Ed] or [Kh]). Since N = 1, k is even if Sk (SL2 (Z)) = 0; so, k = p does not occur. If k = p + 1, ρ is p-ordinary in Case (SL) (e.g., [LFE] Theorem 7.6.1). Thus the condition k ≤ p − 1 in Case (CR) is equivalent to k ≤ p+1 in Case (CR), and the above theorem basically exhausts all possibilities of ρ when N = 1. A short proof of this result in the case (SL) can be found in [MFG] as Theorem 3.31 there. We will generalize this to Hilbert modular forms in Cases (FL), (ST) and (SL) at p and Cases (A) and (B) outside p later in Chapter 3, for example, Theorem 3.28 and Theorem 3.50. Up to here, we only discussed the identity Rρ ∼ = Tρ , and to have well-deﬁned Rρ , we need the absolute irreducibility. If we ask a slightly weaker question: “does a given ρ ∈ Φ(W ) come from a modular form in Sk (Γ0 (N ))?” we do not need to assume (aiM ), and indeed, this is known to be true if k ≥ 2 in almost all p-ordinary cases under the condition (ds) (see [SW] and [SW1]).

Iwasawa theory for deformation rings

47

1.4 Iwasawa theory for deformation rings In this section, we study the relation among Rn = RQn (n = 0, 1, 2 . . . ) for the cyclotomic Zp -extension Q∞ /Q. We start with a two-dimensional modular representation ρ of GQ acting on a vector space V over K. Suppose that ρ satisﬁes (aiQ ), (ord), and (ds). Throughout this section, we assume Conjecture 1.51 for F = Q for the initial modular Galois representation ρ. Indeed, by Theorem 1.52, if ρ satisﬁes (aiQ[µp ] ), the conjecture holds even for ρ restricted to Gal(Q/Qn ) for all n ≥ 0, because (aiF [µp ] ) and (ds) over F = Q are equivalent to these conditions over F = Qn (Lemma 1.62), and if the modularity for ρ is valid over Q, it is valid over Qn by the existence of base-change of a given elliptic modular form to a Hilbert modular form over Qn (see [BCG] and 3.3.3 in the text). This means that ρ restricted to GQn is again associated to a Hilbert modular form of p-power level if ρ is associated to an elliptic Hecke eigenform of p-power level. 1.4.1 Galois action on deformation rings We ﬁrst prove Lemma 1.62 Let G be a group with a normal subgroup H of ﬁnite index and K be a ﬁeld. If G/H is a cyclic group of odd order, then the absolute irreducibility of a representation ρ : G → GL2 (K) is equivalent to the absolute irreducibility of ρ|H . Proof We assume that ∆ = G/H is cyclic of odd order. We prove that ρ cannot contain a character of H as a representation of H, which shows the equivalence, since ρ is two-dimensional. Suppose by absurdity that ρ restricted to H contains a character χ. Let H = {g ∈ G|χ(ghg −1 ) = χ}. Then χ can be extended to a : H → B × for a ﬁnite extension B/K. character of H . We pick one extension χ Let ρ = ρ|H . By Shapiro’s lemma (Lemma 1.17), we have

H 0 HomZ[H ] (ρ , IndH H χ) = Hct (H , HomZ (ρ , IndH χ)) 0 ∼ (H, HomZ (ρ |H , χ)) = HomZ[H] (ρ |H , χ), = Hct

(1.4.1)

where, by deﬁnition, IndH G M = HomZ[G] (Z[H], M ) and we let g ∈ H act on φ ∈ HomZ (M, N ) by (gφ)(x) = g(φ(g −1 x)) for two H -modules M and N . If ρ = ρ|H remains irreducible, this shows that ρ ⊂ IndH H χ. It is easy to check from the deﬁnition that

∼ IndH ξ, H χ = ⊕ξ χ ξ running over all characters of the cyclic group H /H. Thus ρ cannot be irreducible, and we may assume that H = H . Then conjugates of χ under ∆ are all G σ ∼ σ distinct. Since, by Shapiro’s lemma again, ρ ⊂ IndG H χ and ρ = ρ ⊂ IndH χ . Therefore ρ|H contains all conjugates of χ with equal multiplicity. Thus (G : 2 H )|2, which is absurd because (G : H) is odd.

48

Introduction

Exercise 1.63 Let K/F be a p-proﬁnite extension (for a prime p > 2) inside the maximal extension QS unramiﬁed outside S and ∞. Prove that the condition (ds) for ρ : GSF → GL2 (F) is equivalent to the condition (ds) for ρ|GSK . We write ρn and ρn for the restriction of ρ and ρ to GQn . Then we consider the universal couple (Rn = RQn , ρn = ρQn ) starting with ρn . By Lemma 1.62, (aiQ ) implies (aiQn ), and by Exercise 1.63, once (ds) holds for ρ, it is still valid for ρn . Since we have assumed Conjecture 1.51, Rn is a reduced algebra free of ﬁnite n n rank over Λn := W [[Γp ]] ⊂ W [[Γ]] = Λ, because ΓQn = Γp ⊂ ΓQ = Γ. We write the functor Φn.ord,ν for Qn as Φn . Let m > n. Starting with ρ ∈ Φn (A), the restriction ρ m of ρ to Gm = GQm is a deformation of ρm ; so, we have a natural transformation Φn → Φm . In particular, we have a unique W -algebra homomorphism πm,n : Rm → Rn such that ρQn |Gm = πm,n ◦ ρQm . Thus we get a tower of rings: πm,m−1

Rm −−−−−→ Rm−1 → · · · → Rn → · · · → R0 . Put R∞ = limn Rn and ρ∞ = limn ρn : G∞ → GL2 (R∞ ). ←− ←− Proposition 1.64 for G∞ .

The couple (R∞ , ρ∞ ) represents the functor Φ∞ deﬁned

We write πn : R∞ → Rn for the projection. Proof Let A be an artinian W -algebra in CLW . Then we may assume that A is a ﬁnite ring. Let ρ ∈ Φ∞ (A). We need to ﬁnd a W -algebra homomorphism φ∞ : R∞ → A such that φ∞ ◦ ρ∞ ∼ = ρ (because the uniqueness of φ∞ is ∼ ρ)/Q∞ with Gal(Q∞ ( ρ)/Q∞ ) − → plain if it exists). Then the splitting ﬁeld Q∞ ( ρ

Im( ρ) is a ﬁnite extension. Thus Q∞ ( ρ) = Q∞ [θ] for an element θ ∈ Q∞ ( ρ), and the coeﬃcients of the minimal polynomial of θ are contained in Qn for suﬃciently large n. Replacing n by a larger integer if necessary, we may assume that Qn (θ)/Qn is a Galois extension linearly disjoint from Q∞ over Qn . Thus ρ)/Qn ) → GL2 (A), and the action of Γ on ρ factors we may regard ρ : Gal(Qn ( n through Γp . In particular, ρ extends to a unique deformation ρ n of ρn over Gn . Then by the universality of Rn , we ﬁnd a unique algebra homomorphism φ : Rn → A such that φ ◦ ρn ∼ = ρ n . Then φ∞ := φ ◦ πn gives the desired W -algebra homomorphism. This ﬁnishes the proof. 2 For σ ∈ Gal(Qn /Q), write σ for the lifting σ to G0 ; so, σ |Qn = σ. Then the σ g σ −1 ) is uniquely determined by σ for ρ ∈ isomorphism class of ρ σ (g) = ρ ( Φn (A). Thus Gal(Qn /Q) acts on Φn . In particular, we have a unique W -algebra homomorphism σ : Rn → Rn such that ρσQn ∼ = σ ◦ ρn . Since σ ◦ σ −1 is the identity map, σ is an automorphism of the W -algebra Rn . Hereafter, if confusion is unlikely, we simply write σ for σ, and in this way, Gal(Qn /Q) and hence G0 acts on Rn . The following theorem follows from Theorem 3.69.

Iwasawa theory for deformation rings

49

Theorem 1.65 Let Rn (σ − 1)Rn be the ideal of Rn generated by σ(r) − r for all r ∈ Rn . Write γ = N −1 (1 + p) for the generator of Gal(Q∞ /Q). Then πm,n n induces an isomorphism Rn ∼ = Rm /Rm (γ p − 1)Rm for all m ≥ n including m = ∞. Since Gal(Qm /Q) acts on the ring Rm , it acts on the formal spectrum Spf(Rm ). Then the theorem is equivalent to the fact that Spf(Rn ) is identiﬁed with the n maximal closed formal subscheme of Spf(Rm ) ﬁxed by γ p . Out of this, we expect that the tangent space at a given point P of Spf(Rn ) is the subspace of n the tangent space at P of Spf(Rm ) ﬁxed by γ p (then, by duality, the cotangent space at P ∈ Spf(Rn ) is the coinvariant under Gal(Qm /Qn ) of the cotangent space at P of Spf(Rm ); Proposition 1.66). We prove this intuitive expectation as Proposition 5.6 later. 1.4.2 Control of adjoint square Selmer groups Out of Theorem 1.65, we can create well-controlled Iwasawa modules. We suppose that we have an algebra A in CLW so that (i) R∞ is an A-algebra ; (ii) Γ acts trivially on A. We can obviously take A to be W or A to be the subalgebra of R∞ topologically generated over W by δ ∞ ([pp , Q]) = δ ∞ ([p, Qp ]) because the Artin symbol [pp , Q] acts trivially on Q∞ . There seems no other canonical choice of A, but anyway we state our result in this generality. Take an A-algebra B in CLW , and suppose we have an A-algebra homomorphism π : R0 → B. Then we have a unique Galois representation ϕ = π ◦ ρQ . We then consider the module of continuous 1-diﬀerentials ΩRj /A and its B-part: ΩRj /A ⊗Rj B. Then from the above control theorem, as an intuitive consequence, we have Proposition 1.66 Let A be a closed W -subalgebra of R∞ (in CLW ) on which Γ acts trivially. Let B be an A-algebra in CLW and π : R0 → B be an A-algebra j homomorphism. Then we have for 0 ≤ j ≤ k ≤ ∞ and γj = γ p , Rk B ΩRk /A ⊗ ∼ Rj B. = ΩRj /A ⊗ Rk B (γj − 1)ΩRk /A ⊗

(Ct)

A proof of this proposition will be given as Proposition 5.6. Taking A = W and B = F, we see that ΩR∞ /W ⊗R∞ F is an F[[Γ]]-module of ﬁnite type (by Lemma 1.6). It is of torsion if and only if R∞ is noetherian (as we have seen just after Theorem 1.57). We might ask (q4) In addition to the assumptions (aiQ ) and (ds), suppose Conjecture 1.51 for ρ over Qn for all n. Then, is R∞ noetherian? Conjecturing R∞ to be noetherian would be too wild, because ΩR∞ /W ⊗R∞ F having a quotient isomorphic to F[[Γ]] is just the positivity of the µ-invariant of

50

Introduction

the characteristic power series of Sel∗Q∞ (Ad(ρ∞ ) ⊗R∞ R∗∞ ) which could possibly occur (although for good p-adic L-functions, such as, p-adic Dirichlet L-functions constructed by Kubota and Leopoldt, and Iwasawa, their µ-invariants do vanish; see [ICF] Chapter 7). We study this question in more detail in Chapter 5 (see particularly, Corollary 5.11). We write Λn for the subalgebra n topologically generated by the image of R εn ∗ over the decomposition group at the of δ ∞ over W (writing ρn = 0 δn unique prime above p). By (D4), δ n restricted to the p-wild inertia subgroup n factors through Γn = Γp and the tame part has values in W . Thus Λn = W [[δ n (F robp ) − δn (F robp )]] ⊂ Rn for any lift F robp of the Frobenius element in the decomposition group at p in G∞ . We can speciﬁcally take F robp = [p, Qp ] for the local Artin symbol [p, Qp ] = [pp , Q], where pp is the idele having p at the place p and 1 at all other places. We write δ 0 (F robp ) = a(p) ∈ R0 . As before we identify W [[Γ]] with W [[x]] so that (1 + x) corresponds to the ﬁxed generator γ ∈ Γ. Proposition 1.67 Suppose that R0 ∼ = Λ = W [[x]] and da(p)/dx ∈ Λ× . Then ∼ we have Rn ∼ R Λ for all n. = 0= Proof For a B-algebra A with A, B ∈ CLW , we have ΩA/B = 0 ⇔ A is a surjective image of B. Note that ΩW [[x]]/W = W [[x]]dx and by our assumption, d(a(p)) =

da(p) dx dx

generates ΩW [[x]]/W . Thus by Lemma 5.7 (i), we ﬁnd ΩΛ0 /Λ∞ =

W [[x]]dx = 0, W [[x]]d(a(p))

σ F robp σ −1 ) = δ n (F robp ), we ﬁnd and Λ∞ = Λ0 . Since σ(δ n (F robp )) = δ n ( that Γ acts trivially on Λn . Thus applying the above proposition to A = Λ∞ and B = R0 = Λ0 , we ﬁnd that Rn R0 ΩRn /Λ∞ ⊗ ∼ = ΩR0 /Λ∞ = 0. Rn R0 (γ − 1)ΩRn /Λ∞ ⊗ Rn R0 , we ﬁnd By Nakayama’s lemma applied to the W [[Γ]]-module ΩRn /Λ∞ ⊗ Rn R0 = 0. Again by Nakayama’s lemma applied to the Rn that ΩRn /Λ∞ ⊗ module ΩRn /Λ∞ , we ﬁnd that ΩRn /Λ∞ = 0; so, Rn = Λn = Λ∞ ∼ = W [[x]]. 2 The condition R0 = Λ follows if and only if the following two conditions are met: 1. ρ is associated to a Hecke eigenform f in Sk (Γ0 (p)); 2. f ≡ g mod mW for any other Hecke eigenform g ∈ Sk (Γ0 (p)).

Iwasawa theory for deformation rings

51

Note here that the weight κ in Conjecture 1.51 for the classical weight k modular form in Sk (Γ0 (p)) is given by (0, k − 1). Example 1.68 Here is an example discussed in [MFG] page 322. If we start with ρ associated to a Hecke eigenform f in S2 (Γ0 (p)), we can verify by computation the assumption of the corollary holds for primes p = 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61 and 67. Note that S2 (Γ0 (13)) = 0. As for p = 53, there are two Hecke eigenforms modulo Galois action, and one of them with f |T (p) = −f has nonunit da(p)/dx. In this case, for the cusp form g ∈ S54 (SL2 (Z)) of weight 54 giving a deformation of ρ, λ + 1 for g|T (p) = λg is divisible by a square of a prime factor of 53 in the Hecke ﬁeld of g; so, a(p)(γ p−1 − 1) = λ and a(p)(0) = −1, which imply that da(p)/dx ∈ mΛ . & %

a b ∈ Γ0 (N ) a − 1 ∈ N Z for a positive integer N . Serre Let Γ1 (N ) = c d has predicted Conjecture 1.69 Rational primes p below ordinary prime ideals for a Hecke eigenform f ∈ Sk (Γ1 (N )) without complex multiplication have Kronecker density 1. Here a Hecke eigenform with f |T (n) = λ(n)f has complex multiplication if there exists a nontrivial Dirichlet character φ such that λ(p) = φ(p)λ(p) for almost all primes. In particular, this implies λ(p) = 0 if φ(p) = 1; so, primes p with λ(p) = 0 has positive density. It is known that any Hecke eigenform in Sk (SL2 (Z)) does not have complex multiplication. Indeed, by Exercise 1.71 below, the Galois representation of f with complex multiplication is an induced representation of a Galois character of Gal(Q/M ) for a quadratic extension M/Q. Then f is a theta series of the norm form of an imaginary quadratic ﬁeld if k ≥ 2 (note S1 (SL2 (Z)) = 0; see Theorem 2.71), and as we will see later in 2.5.4, such theta series cannot be in Sk (SL2 (Z). Remark 1.70 By the Ramanujan bound (see [D] and [IAT] 7.4), we have √ |λ(p)σ | ≤ 2 pk−1 for all Galois conjugate λ(p)σ of λ(p). If k = 2 and p|λ(p), taking the norm N (λ(p)) for Q(f )/Q and writing d = [Q(f ) : Q], we have pd ≤ |N (λ(p))| ≤ 2d pd/2 as long as λ(p) = 0. Thus for suﬃciently large p, if all prime factors p|p (in the integer ring of Q(f )) are not ordinary for f , we have λ(p) = 0. By a result of Serre [Se3], the primes p with λ(p) = 0 has density 0 if f does not have complex multiplication. Thus the conjecture is valid for Hecke eigenforms f of weight 2 without complex multiplication.

52

Introduction

Now we may ask (q5) For a given Hecke eigenform f ∈ Sk (Γ1 (N )) with Galois representation ρ, assuming Conjecture 1.69, does da(p)/dx ∈ Λ× hold for density-one ordinary primes? Exercise 1.71 If a Hecke eigenform f has complex multiplication (with respect to a nontrivial Dirichlet character φ modulo N ), prove the following facts: 1. The p-adic Galois representation ρ of f satisﬁes ρ ⊗ φ ∼ = ρ regarding φ as × a Galois character φ : Gal(Q(µN )/Q) ∼ = (Z/N Z)× → Qp . 2. Using the above fact, φ is a quadratic character if ρ is irreducible (cf. [DoHI] Lemma 3.2). Assume N = 1; so, hk (Z) ⊂ EndC (Sk (SL2 (Z))). We put hk (W ) = hk (Z)⊗Z W . We ﬁx a weight k0 and pick a Hecke eigenform f of weight k0 in Sk0 (SL2 (Z)). Deﬁning λ : hk0 → W (for suﬃciently large W ) by f |T = λ(T )f for T ∈ hk0 , we have a W -algebra homomorphism λ : hk0 (W ) → W . We pick a unique local ring Tk0 ⊂ hk0 (W ) through which λ factors. Suppose the prime p = mW ∩ Q(f ) is ordinary for f . Then the p-adic Galois representation ρ associated to f is ordinary (see [GME] Theorem 4.2.7). Further suppose that ρ satisﬁes (aiQ[µp ] ) and (ds) (which hold for almost all primes p by a result of Serre and Ribet; e.g., [Ri3] and [Ri4]). In this classical setting, the Eisenstein series E of even weight k ≥ 4:

Ek (z) = (mz + n)−k (1.4.2) (m,n)∈Z2 −(0,0)

has the following q-expansion (e.g., [IAT] (2.2.1)) up to a constant:   ∞ (k − 1)! Bk  k−1  n + q (q = exp(2πiz)). Ek (z) = − d Gk (z) = 2(2πi)k 2k n=1 0
By the von Staudt theorem, Bp−1 ≡ E=−

− p1

mod Z, and hence

2(p − 1) Gp−1 Bp−1

satisﬁes E ≡ 1 mod p (if p ≥ 5). Multiplying by E m preserves the q-expansion modulo p and sends f to g = f · E m ∈ Sk (SL2 (Z)) for k = k0 + m(p − 1). Sincethe eﬀect of the Hecke operator on the q-expansion of modular forms ∞ f = n=1 a(n, f )q n of weight k is (see (2.4.6)) # mn $

a(n, f |T (m)) = dk−1 a ,f , d2 0
deﬁning a(α, f ) = 0 for α ∈ N, we conclude g|T ≡ λ(T )g mod mW from the fact that dp−1 ≡ 1 mod p for d prime to p. Thus we have a W -algebra homomorphism

Iwasawa theory for deformation rings

53

λ : hk (W ) → F by sending T to λ(T ) = (λ(T ) mod mW ) ∈ F. Then there exists a unique local ring Tk ⊂ hk (W ) through which λ factors. The ring Tk is free : GQ → GL2 (Tk ) of ﬁnite rank over W and carries a Galois representation ρord k with 2 k−1 det(X − ρord k (F rob )) = X − t()X +

for all primes p (e.g. [GME] Theorem 4.2.7), where t() is the image of k−1 . Since we have the Hecke operator T () in Tk . In particular det(ρord k ) = N ord ord Tr(ρk (F rob )) ≡ λ(T ()) ≡ Tr(ρk (F rob )) mod mW , by the Chebotarev density theorem (e.g., [CFN] Theorem 6.4), ρord ≡ ρ mod mW ; so, ρord satk k ord ∗ ∼ k isﬁes (D1–2) and (D4). Thus we may write ρord as in (D3). k |Dp = 0 δkord ord Then by Theorem 2.43 (see also [GME] Theorem 4.2.7), δk is unramiﬁed, and δkord (F robp ) = ak (p) for the unit root ak (p) of X 2 − t(p)X + pk−1 = 0. ord −1 ψ = det(ρord det(ρord Since det(ρord k ) ≡ det(ρk0 ) mod mW , we see k ) k0 ) ≡ 1 √ × mod mW ; so, we have a unique square root ψ : GQ → W congruent to the + √ ord )(σ) = ψ(σ)ρ identity character modulo mW . Then ρk (σ) := ( ψ ⊗ ρord k k (σ) n.ord,ν k0 −1 (W ) with ν = N . Since F robp = satisﬁes (D1–4), and hence ρk ∈ Φ and N ([pp , Q]) = 1, we [pp , Q] for the Artin symbol [z, Q] of an idele z ∈ A× ∗ k × ∼ as in (D3). have δk ([pp , Q]) = ak (p) ∈ Tk writing ρk |Dp = 0 δk Exercise 1.72 Prove N ([pp , Q]) = 1, where pp is the image of p ∈ Q× p in the idele group A× . Thus we have a unique algebra homomorphism τk : R0 → Tk with τk ◦ ρ0 ∼ = ρk . ∼ for all k as above Indeed, by Theorem 1.61, τk induces an isomorphism Rk,Q T = k (under the assumptions of the theorem). Thus R0 → k Tk , and writing T for its image, we have R0 ∼ = T (cf. Theorem 3.50). We write u(p) ∈ T for the unit element corresponding to a(p). This symbol comes from the fact that τk (u(p)) gives the unique unit root of X 2 − t(p)X + pk−1 which is actually the eigenvalue of the operator U (p) of level p acting on old forms. Here U (p) is the Hecke operator T (p)

acting on Sk (Γ0 (p)) (because the action U (p) of the double coset 1 0 Γ0 (p) Γ0 (p) on Sk (Γ0 (p)) does not restrict to the action T (p) of the 0 p

1 0 SL2 (Z) on the subspace Sk (SL2 (Z)) ⊂ Sk (Γ0 (p)). double coset SL2 (Z) 0 p See Remark 2.27 in 2.4.1 for more detailed comments on this fact. Without usingthe existence of R0 , we may deﬁne T to be the p-proﬁnite W -subalgebra of k Tk topologically generated by t(n) for all n prime to p (then, T is ΛW -free of ﬁnite rank; see [LFE] Chapter 7). If (aiQ ) holds for ρ, the trace is t() ∈ T; so, of F rob (for primes p) of the Galois representation k ρord k by Chebotarev density, it has trace in T over all GQ . Then by Proposition 1.23, assuming (ds), we get a p-ordinary representation ρord : G → GL2 (T) with T

54

Introduction

ord Tr(ρord T (F rob )) = t() for all p. In particular, we have , ρT = ψ ⊗ ρT −1 det ρ in Φ0 (T) for the unique ψ with ψ ≡ 1 mod mT and ψ = det(ρord T ) (see Proposition 3.49 for more details of this fact). Thus we have a surjective W -algebra homomorphism R0 T by the universality of R0 . Anyway, τk (u(p)) = ak (p) (k ≥ 3) are all distinct (because ak (p) is the unit root of X 2 − t(p)X + pk−1 ). As before we identify W [[Γ]] with W [[x]] so that the ideal ((1 + x) − γ k−k0 ) ⊂ Ker(τk ). This shows

Proposition 1.73 Let a(p) = δ 0 ([pp , Q]) ∈ R0 . Then da(p)/dx is neither zero nor a zero-divisor of T. Thus our question (q5) is about when da(p)/dx is a unit in T. Theorem 1.74 In addition to (ord), (aiQ ) and (ds), suppose that ρ is modular as above associated to a Hecke eigenform f ∈ Sk0 (SL2 (Z)) and Conjecture 1.51 for ρ. Let I be a residue ring of R0 modulo a minimal prime ideal; so, Spf(I) is an irreducible component of Spf(R0 ). Then we have R∞ J is a torsion J[[Γ]]-module. 1. For J = R0 or I, ΩR∞ /Λ∞ ⊗ 2. We have a pseudo-isomorphism of I[[Γ]]-modules: R∞ I ; Sel∗Q∞ (Ad(ρI ) ⊗I I∗ ) ∼ I × ΩR∞ /Λ∞ ⊗ in particular, Sel∗Q∞ (Ad(ρI ) ⊗I I∗ ) is a torsion I[[Γ]]-module. 3. If λ(τk0 (da(p)/dx)) = 0 in W , we have a W [[Γ]]-homomorphism R∞ ,λ◦τk W Sel∗Q∞ (Ad(ρ) ⊗W W ∗ ) ∼ W × ΩR∞ /Λ∞ ⊗ 0 with ﬁnite kernel and cokernel. Moreover Sel∗Q∞ (Ad(ρ) ⊗W W ∗ ) is a torsion W [[Γ]]-module whose characteristic power series is of the form xϕ(x) with ϕ(0) = 0. Here is a sketch of a proof following the lines of the proof given in [MFG] Theorem 5.51 and [H00] Theorem 6.3. We will give a diﬀerent proof in a more general setting as Theorem 5.27 in Chapter 5. Proof Conjecture 1.51 for ρ implies R0 ∼ = T (see the proof of Theorem 3.50 for this fact). Since da(p)/dx = 0, ΩR0 /Λ0 is a torsion R0 -module. Then by Proposition 1.66, ΩR∞ /Λ∞ ⊗R∞ R0 is a torsion R0 [[Γ]]-module. This shows (1). As long as we have λ(τk0 (da(p)/dx)) = 0, the same argument works, and we get R∞ ,λ◦τk W over W [[Γ]] and its characteristic the torsion property of ΩR∞ /Λ∞ ⊗ 0 power series ϕ does not vanish at 0, because ϕ(0) is the size of the ﬁnite module R0 ,λ◦τk W = ΩTk /W ⊗ Tk W . ΩR0 /Λ0 ⊗ 0 0 0 As for the product expression (3), writing J∞ = R∞ (γ − 1)R∞ , we look into the following commutative diagram with exact rows and columns for J = I or W

Iwasawa theory for deformation rings

55

(see Lemma 5.7 for the construction of maps and the exactness of the horizontal rows and vertical columns): 0  

−−−−→ ΩΛ∞ /W ⊗Λ∞ J  e

ΩΛ∞ /W ⊗Λ∞ J −→ 0  f b

2 (J∞ /J∞ ) ⊗I J −−−−→ ΩR∞ /W ⊗R∞ J −−−−→   g  d

ΩI/Λ∞ ⊗I J −→ 0  

−−−−→

0.

2 (J∞ /J∞ ) ⊗I J −−−−→ ΩR∞ /Λ∞ ⊗R∞ J −−−−→    

0

−−−−→

0

ΩI/W ⊗I J −→ 0   h

(1.4.3)

Since da(p)/dx = 0 implies injectivity of f and e, we have the following exact sequence for M = ΩR∞ /Λ∞ ⊗R∞ J: β

α

→ M × (ΩI/W ⊗I J) − → ΩI/Λ0 ⊗I J → 0, 0 → ΩR∞ /O[[I∞ ]] ⊗ J − where α(m, a) = d(m)−h(a) and β(a) = (g(a), b(a)). Since ΩI/Λ0 ⊗I J is J-torsion by da(p)/dx = 0, it is pseudo-null as a J[[x]]-module. Again by da(p)/dx = 0, Jd(a(p))\(ΩI/W ⊗I J) is J-torsion and hence is pseudo-null as a J[[x]]-module. 2 There is a two-variable analytic p-adic L-function Lp (Ad(ρI )) ∈ I[[Γ]] constructed in [H90]. Assuming an analytic formula of the L-invariant of Lp (Ad(ρI )) claimed by Greenberg and Tilouine (cf. [HTU]) for the initial cusp form having split multiplicative reduction at p, recently E. Urban has shown (cf., [U], [U1] and [HTU]): Theorem 1.75 (E. Urban) Let the assumption be as in Theorem 1.74 (but allowing ramiﬁcation outside p, that is, f ∈ S2 (Γ0 (N )) for N ≥ 1). Suppose that ρ is associated to an elliptic Hecke eigenform f of weight 2 of “Haupt” type (that is, with the identity Neben character) and that the residual Galois representation ρ is indecomposable under the action of the inertia subgroup Iq of a prime q = p. Then we have 1. (Lp (Ad(ρI ))) ⊃ char(Sel∗Q∞ (Ad(ρI ) ⊗I I∗ )) in I[[Γ]]. 2. If ρ is further associated to an elliptic curve E/Q with split multiplicative reduction modulo p, we have (Lp (Ad(ρI ))) = char(Sel∗Q∞ (Ad(ρI ) ⊗I I∗ )). The identity in the assertion (2) specialized to a modular elliptic curve has been conjectured in [CS] (though some modiﬁcation of their conjecture is necessary; see Section 4.4 for more details). The proof is based on the I-adic Eisenstein congruence between the p-adic analytic families of cusp forms and Klingen-Eisenstein series on GSp(4)/Q . To establish the congruence, some mod p

56

Introduction

nonvanishing of certain L-values studied by Vatsal in [V] is needed (and at this point, the “Haupt” type weight 2 assumption comes in). The proof of the stronger assertion (2) relies further on Theorem 1.74, the L-invariant formula of Greenberg and Tilouine, and the fact that da(p)/dxx=0 = 0 for ρ associated to the elliptic curve with multiplicative reduction (the theorem of St. Etienne; see Theorem 1.83). If this nonvanishing holds for any ρ associated to an abelian variety with multiplicative reduction at p, the assertion (2) remains true for such ρ. 1.4.3 Λ-adic forms We constructed a big Hecke algebra T isomorphic R0 = RQ (under Conjecture 1.51). Since ur ∼ ∼ × Gal(Qur p [µp∞ ]/Qp ) = Gal(Q[µp∞ ]/Q) = Zp , × ur we may regard the local character δ 0 : Gal(Qur p [µp∞ ]/Qp ) → R0 as a global character

δ cyc : Gal(Q[µp∞ ]/Q) → R× 0.

ord ε ∗ −1 ord ord ∼ Deﬁne ρ = δ cyc ⊗ ρ0 . Then we have ρ |Dp = , and δ ord is 0 δ ord unramiﬁed (thus ρord is p-ordinary). By our construction, for the projection πk : T Tk , we have πk ◦ ρord ∼ = ρord k . By this fact, we ﬁnd from the properties ord satisﬁed by ρk (see Theorem 2.43 in the text and [GME] Theorem 4.2.7): Theorem 1.76 Assume (aiQ ), (ord), and (ds) for the initial modular Galois representation ρ. The Galois representation ρord : GQ → GL2 (T) is p-ordinary and satisﬁes: • Tr(ρord (F robq )) = t(q) for all primes q p; ord • δ (F robp )) = u(p), where the image of u(p) in Tk is the unit root of the

equation X 2 − t(p)X + pk−1 .

The isomorphism class of the representation ρord is uniquely determined by the above two conditions. Let us recall here the deﬁnition of aﬃne formal schemes given at the beginning of this chapter. For any p-proﬁnite W -algebra A, we write Spf(A) for its formal spectrum. Thus for any p-proﬁnite W -algebra C, Spf(A)(C) = HomW -alg (A, C) made up of continuous W -algebra homomorphisms. The category of p-proﬁnite W -algebras is then sent into the category of formal aﬃne spectrums by the contravariant functor A → Spf(A). In particular, we have Hom(Spf(A), Spf(B)) = HomW -alg (B, A). For any closed ideal a of A, we identify Spf(A/a) with the closed formal subscheme of Spf(A). If A is an integral local noetherian domain, of A in the quotient ﬁeld Q of A is again a p-proﬁnite noeththe integral closure A erian domain (cf. [EGA] 0.23.1.5 or [BCM] IX.4.2). The formal spectrum Spf(A)

Iwasawa theory for deformation rings

57

is called the normalization of Spf(A). If A is reduced noetherian, it has ﬁnitely many minimal prime ideals. For a minimal prime ideal P of A, Spf(A/P ) as a closed formal subscheme of Spf(A) is called an irreducible component of Spf(A). See [ALG] for more of formal schemes. Let Spf(I) be the normalization of an irreducible component of Spf(T) for T∼ = RQ . Then I is a normal integral domain ﬁnite and torsion-free over Λ. In this case of dim I = 2, it is known that I is free of ﬁnite rank over Λ (e.g., [EGA] 0.23.1.5). For each positive integer n, split n = n(p) pe(n) for n(p) prime to p, and write A(n) ∈ I for the image of the Hecke operator t(n(p) )u(p)e(n) ∞ n in I. Consider a formal series FI (q) = ∈ I[[q]], which is called n=1 A(n)q an I-adic form associated to the irreducible component Spf(I). By the duality between the Hecke algebra and the space of cusp forms (see Proposition 2.50), for any given algebra homomorphism λ : Tk → W , the value λ(A(n)) is algebraic over Q (because rank hk (Z) < ∞), and i∞ (i−1 p (λ(A(n))) ∈ Q ⊂ C gives (SL (Z)). In particular, we have the eigenvalue of a modular form f ∈ S k 2 ∞ −1 n i (i (λ(A(n)))q for q = exp(2πiz). If λ ∈ Spf(I)(W ) factors f = ∞ p n=1 λ

through Rk,Q = Tk , λ induces → W, ∞ an algebra homomorphism λk : hk → Tk − which gives rise to fλ = n=1 λ(a(n))q n ∈ Sk (Γ0 (p)). Here fλ actually belong to Sk (Γ0 (p)) whose Hecke eigenvalue for the U (p)-operator is given by λ(u(p)) and is not primitive in the sense of [MFM] Section 4.6. The primitive form associated to fλ belongs to Sk (SL2 (Z)). In the Hecke algebra hk (p; Z) ⊂ EndC (Sk (Γ0 (p))) generated by U (p) and T (n) for n prime to p, on the space of old forms, U (p) gives a root of X 2 − T (p)X + pk−1 = 0 (e.g., Lemma 3.13 in the text and [MFG] Lemma 3.25), and hence Tk is covered by a local component of hk (p; W ) = hk (p; Z) ⊗Z W and the image of U (p) in Tk is equal to u(p). Thus the formal expansion FI gives rise to a p-adic analytic family of modular forms {fλ }k≡k0 mod p−1 of level p. Now we renormalize until the end of this section the W [[x]]-algebra structure of T so that T/((1 + x) − γ k ))T ∼ = Tk . In other words, the new W [[x]]-algebra structure is the composite of ιk0 with the original inclusion W [[x]] → T, because it was originally normalized so that T/xT = Tk0 . We might be tempted to ask what arithmetic meaning the specialization T/((1 + x) − ζγ k ))T for ζ ∈ µpn (W ) has. A remarkable point about the structure of the algebra T is that T/((1 + x) − ζγ k ))T gives rise to a local ring of the Hecke algebra over W of Sk (Γ0 (pn+1 ), ε) for the “Neben” character ε with ε(γ) = ζ. Indeed, as we will prove in 4.3.9, as long as Ker(λ) contains ((1 + x) − ε(γ)γ k ) r Z)× → W × , we can ﬁnd a Hecke eigenforms for k ≥ 2 and a character ε : (Z/p ∞ r fλ ∈ Sk (Γ1 (p )) such that fλ = n=1 λ(a(n))q n . The point λ ∈ Spf(T)(W [ε]) satisfying the condition: Ker(λ) ((1 + x) − ε(γ)γ k ) for k ≥ 2 and a character

58

Introduction

ε of (Z/pr Z)× is called arithmetic. Here W [ε] is a p-proﬁnite W -algebra generated over W by the values of ε. We write Spf arith (I) for the set of all arithmetic ) for the p-proﬁnite completion W of the integral closure of points in Spf(I)(W W in Qp . Thus the I-adic form FI gives rise to a p-adic family of Hecke eigenforms {fλ }λ∈Spf arith (I) . If I ∼ = Λ, an arithmetic point λ is given by the evaluation W [[x]] φ(x) → φ(ε(γ)γ k − 1) for an integer k ≥ 2; so, the Fourier coeﬃcients of fλ depend p-adic analytically on λ ∈ Spf(I). The family {fλ }λ∈Spf arith (I) is hence called a p-adic analytic family of p-ordinary Hecke eigenforms. If ε is non-trivial, fλ has a nontrivial p-power level pr . We write U (p) for the Hecke operator acting 1 0 on Sk (Γ1 (pr )) corresponding to the double coset Γ1 (pr ) Γ1 (pr ). Here is 0 p a result one can ﬁnd in [LFE] Chapter 7 (whose version in the Hilbert modular case will be discussed in 4.3.9): Theorem 1.77 If k ≥ 1, for any given Hecke eigenform f ∈ Sk (Γ1 (pr )) with f |U (p) = λ(U (p))f for λ(U (p)) ∈ W × , we have an irreducible component of Spf(T) giving rise to a ﬁnite ﬂat Λ-algebra I such that f = fλ for a unique arithmetic point λ in Spf(I). If k ≥ 2, I is uniquely determined by f , and the localization-completion of I at P = Ker(λ) for λ ∈ Spf arith (I) is isomorphic to the localization-completion of Λ at P (which is in turn isomorphic to K[[t]]). The I-adic form FI is called the I-adic lift of f , and the family {fλ }λ∈Spf arith (I) is called the p-adic analytic family of f . A naive question is (q6) If (1 + x) − γ ∈ Ker(λ), what is fλ , a true classical modular form of weight 1? The formal q-expansion fλ for each λ ∈ Spf(I) gives (the q-expansion of) a p-adic modular form in the sense of Serre. If its weight k is 1, it could be a classical modular form of weight 1 but rarely. A suﬃcient condition for this to happen can be found in [BuT]; see also [MzW1] Section 11. Since Question (q5) is related to the question about how the values λ(u(p)) for λ ∈ Spf arith (I) are distributed at random in Q, we might ask, for a given positive integer n, (q7) Is the ﬁeld Q[λ(t(n))|λ ∈ Spf arith (I)] an inﬁnite extension of Q[µp∞ ]? If one restricts λ to λk with trivial ε, this question would probably be equivalent to asking if the ﬁeld Q[λk (t(n))|k ≥ k0 , k ≡ k0 mod (p − 1)] is an inﬁnite extension of Q. A naive guess is that the answer is positive as long as FI (q) does not have complex multiplication (here F = FI has complex multiplication if there exists a non-trivial Dirichlet character χ such that a(, F)χ() = a(, F) for almost all primes ; see [LFE] 7.6). Related to this question, Maeda conjectured that the Hecke algebra over Q (of weight k and of level 1): Q[T (n)|n = 1, 2, . . . ] ⊂ End(Sk (SL2 (Z)))

Adjoint square L-invariants

59

is a ﬁeld whose Galois closure over Q has Galois group isomorphic to the symmetric group Sd of d letters for d = dim Sk (SL2 (Z)) (see [HM]). Corollary 1.78 For a given p-adic analytic family {fλ }λ∈Spf arith (I) associated to an I-adic form FI , writing FI |U (p) = a(p)FI , we have λ(da(p)/dx) = 0 except for ﬁnitely many λ. This tells us that char(Sel∗Q∞ (Ad(ρλ ) ⊗W W ∗ )) is divisible by x only once for almost all λ, where ρλ is the Galois representation of fλ . Proof We already know that da(p)/dx (in the fraction ﬁeld of I) is not a con ) is a ﬁnite set by the Weierstrass stant. Then the zero set of da(p)/dx in Spf(I)(W preparation theorem (see [ICF] Theorem 7.3 or [BCM] VII.3.8). 2 1.5 Adjoint square L-invariants After the Mazur–Tate–Teitelbaum conjecture [MzTT], many number theorists have proposed diverse deﬁnitions of the L-invariant which are expected to give the error term (or the diﬀerence) of the conjectural arithmetic part of the leading term of the Taylor expansion of a given p-adic motivic L-function at an exceptional zero from its archimedean counterpart. For an elliptic curve E/Q with multiplicative or ordinary good reduction modulo p, its p-adic L-function Lp (s, E) has the following evaluation formula at s = 1 (see [LFE] 6.5): Lp (1, E) = (1 − a−1 p )

L∞ (1, E) , period

where L∞ (s, E) is the archimedean L-function of E, and ap is the eigenvalue of the arithmetic Frobenius element at p on the unramiﬁed quotient of the p-adic Tate module T (E) of E. Thus if E has split multiplicative reduction, ap = 1, and Lp (s, E) has a zero at s = 1. This type of zero of a p-adic L-function resulted from the modiﬁcation Euler p-factor is called an exceptional zero, and it is generally believed that if the archimedean L-values do not vanish, the order of the zero is the number e of such Euler p-factors; so, in this case, e = 1. Then Lp (1, E) = dLp (s, E)/ds|s=1 is conjectured to be equal to the archimedean value L∞ (1,E) period times an error factor L(E), the so-called L-invariant: Lp (1, E) = L(E)

L∞ (1, E) . period

The problem of L-invariants is to ﬁnd an explicit formula (without recourse to p-adic L-functions) for general motivic p-adic Galois representations V . In the × case of E/Q split multiplicative at p, writing E(Qp ) = Qp /q Z for the Tate period q ∈ pZp , the solution conjectured by Mazur and Tate and Teitelbaum [MzTT] and proved by Greenberg and Stevens [GS] is L(E) =

logp (q) , ordp (q)

(L)

60

Introduction

which will be proven as Proposition 1.85 later. Here logp is Iwasawa’s logarithm, that is, for any q ∈ Qp , writing q = pα u (α ∈ Q) with a p-adic unit u, for a suitable positive integer N , the power series logp converges at uN , and we deﬁne logp (q) := N1 logp (uN ), which does not depend on the choice of N . Since E is modular (by the solution of the Shimura-Taniyama conjecture; see [Wi2], [BrCDT] and [Ki]), it is associated to an elliptic Hecke eigenform fE of ∞ weight 2 with q-expansion n=1 a(n, fE )q n . In particular, a(p, fE ) = ap = 1 and ) = 1. By Theorem 1.77, we can lift fE to a unique I-adic Hecke eigenform a(1, fE ∞ FI = n=1 a(n)q n ∈ I[[q]] for a ﬁnite ﬂat extension I of Zp [[x]] (´etale over an open neighborhood of x = 0) so that fE is a specialization of FI at x = 0. Then one of the key ingredients of the proof of (L) in [GS] is the following formula: L(E) = −2 logp (γ)

da(p) , dx x=0

where γ is the generator of Γ = 1 + pZp corresponding to 1 + x under the identiﬁcation: Zp [[Γ]] = Zp [[x]]. Greenberg has generalized in [Gr2] the conjectural formula of his L-invariant to general V when V is p-ordinary and has a subquotient on which Dp acts trivially. We write L(V ) for the L-invariant of Greenberg and recall his deﬁnition in the following sections. Suppose that V = V (ρ) is a modular ordinary two-dimensional Galois representation associated to a p-ordinary elliptic Hecke eigenform f of weight k ≥ 2 with “Neben” character having conductor prime to p. Thus we have a(p, f ) ∈ W × . By deﬁnition, Ad(V ) has the middle subquotient on which Dp acts trivially; so, Lp (s, Ad(V )) (see [H90]) has to have an exceptional zero at s = 1. Under this circumstance, a version of a conjecture of Greenberg is Conjecture 1.79

L(Ad(V )) = 0.

For a suitable ﬁnite ﬂat extension I of W [[x]], we have a unique I-adic Hecke eigenform FI lifting f so that its specialization at a point Pf of Spf(I) over (x) gives f (see Theorem 1.77). Then we have an I-adic ordinary Galois : GS → GL2 (I) associated to FI acting on L = I2 representation ρord I (Theorem 1.76). Assume p > 2 and S = {p, ∞} for simplicity. The Galois −1 det(ρ) has values in the p-proﬁnite group 1 + mI for the character det(ρord I ) maximal ideal mI of I, and hence we have its unique square root ψ ≡ 1 mod mI . Recall the representation ρI : GQ → GL2 (I) with det(ρI ) = det(ρ) satisfying ord ⊗ ψ)(σ) = ψ(σ)ρord mod Pf . (ρord I I (σ). Note that ρI ≡ ρI What we prove in this section is the following result originally proven in [H04a]: Theorem 1.80 Let p be an odd prime, and assume that f is p-ordinary, has weightk ≥ 2 and that the “Neben” character of f has conductor prime to p. Let ∞ FI = n=1 a(n)q n ∈ I[[q]] be the I-adic lift of f . Assume (aiQ ) and (ds) for the Galois representation ρ of f . Suppose that Rk,Q ∼ = T). Then we = Tk (so, R ∼

Adjoint square L-invariants

have

−1 da(p)

L(Ad(V )) = −2 logp (γ) a(p)

dx

61

. x=0

Here we normalize the variable x so that f is the specialization at x = 0 of the I-adic form. Corollary 1.78 tells us Corollary 1.81 Let the notation and the assumption be as in the theorem. Then L(Ad(ρλ )) = 0 for almost all arithmetic points λ ∈ Spf(I), where ρλ is the Galois representation of the member fλ in the p-adic analytic family associated to FI . There is one more corollary. Suppose that f is of weight 2 and is associated to an elliptic curve E/Q with multiplicative reduction modulo p. Then twisting E by with split a quadratic character χ, we can bring E to another elliptic curve E/Q multiplicative reduction modulo p. Write E (Qp ) = Gm (Qp )/q Z for q ∈ Q× p and V for V ⊗ χ (which is a factor of the p-adic Tate module Tp A ⊗Z Q). Greenberg logp (q) (in [Gr2] (24)) proved that L(Ad(V )) = , and Greenberg and Stevens ordp (q) [GS] identiﬁed the arithmetic L-invariant with the analytic one (see (L)). Thus we have Corollary 1.82 Let the notation be as above. Then

logp (q) da(p) . L(Ad(V )) = L(V ) = −2 logp (γ) a(p)−1 = dx ordp (q) x=0 This follows from Proposition 1.85 which we prove later. Indeed, one can prove this identity even for a modular abelian variety A with multiplicative reduction at p (associated with a weight 2 Hecke eigenform f ). The argument for a general abelian variety A is similar (see [GS] (3.11)), but the deﬁnition of q is slightly more involved (see [GS] Section 3). Replacing A by its character twist, we may assume that A is split multiplicative. Then either geometrically as in [GS] (3.5) or by an Iwasawa theoretic method in [Gr2] (24), we can deﬁne the number q ∈ Q× p logp (q) such that L(Ad(V )) = . In this case, A has endomorphisms induced by ordp (q) Hecke operators, and by changing A in its isogeny class, we may assume that the integer ring R of the Hecke ﬁeld Q(f ) generated by Hecke eigenvalues of f acts on A. Then the Galois representation V is realized on the P-adic Tate module of A for the prime factor P in R corresponding to our ﬁxed embedding, and q depends on P. We will prove later a Hilbert modular generalization of this fact as Corollary 3.74 and Theorem 3.93. We then ask (q8)

logp (q) = 0?

62

Introduction

As for this question, the only known result so far is the one in [BDGP]: Theorem 1.83 If A/Q is an elliptic curve with split multiplicative reduction at the place ip , then the Tate period q of A is transcendental over Q; in particular, logp (q) = 0. Since this theorem was proved by the mathematicians at St. Etienne in France, this theorem is sometimes called the theorem of St. Etienne. Since there is a categorical equivalence between deformations of L with determinant det(ρ) and ordinary deformations of L (if p > 2), the ring R0 = RQ can also be identiﬁed with the universal ordinary deformation ring of L, and we have ρ = ρord ⊗ ψ for the universal ordinary deformation ρord , where ψ is the unique square root of det(ρord )−1 det(ρ) with ψ ≡ 1 mod mR0 . See [MFG] Proposition 5.43 for more details of this fact. In the following section, we start with a brief review of the deﬁnition by Greenberg of the balanced Selmer group and the L-invariant of the adjoint square of a two-dimensional modular ordinary p-adic Galois representation. After the review, we shall give a proof of the theorem. Recall the assumption: p > 2. 1.5.1 Balanced Selmer groups As already remarked, if we take a basis of V containing a generator of F + V and we identify EndK (V ) with M2 (K) by this basis, F − Ad(V ) (resp. F + Ad(V )) is made up of upper triangular matrices with trace zero (resp. upper nilpotent matrices). Note that Dp acts trivially on F − Ad(V )/F + Ad(V ); so, F − Ad(V )/F + Ad(V ) ∼ = K as Dp -modules. In particular, the p-adic L-function of Ad(V ) has an exceptional zero at s = 1. Taking the dual Ad(V )∗ (1) = HomK (Ad(V ), K) ⊗ N ∼ = Ad(V )(1), we deﬁne subspaces F − Ad(V )∗ (1) = F + Ad(V )⊥ (1) = F − Ad(V )(1) F + Ad(V )∗ (1) = F − Ad(V )⊥ (1) = F + Ad(V )(1). Thus we have a three-step ﬁltration Ad(V )∗ (1) ⊃ F − Ad(V )∗ (1) ⊃ F + Ad(V )∗ (1) ⊃ {0}. On F − Ad(V )∗ (1)/F + Ad(V )∗ (1), F + Ad(V )∗ (1) ∼ = K(1). We deﬁne

Dp

acts

by

χ;

so,

(1.5.1)

F − Ad(V )∗ (1)/

Ad(L) = EndW (L) ∩ Ad(V ), Ad(L)∗ (1) = Hom(Ad(L), W ) ⊗ N ⊂ Ad(V )∗ (1). Taking the intersection of each ﬁlter with Ad(L) (resp. Ad(L)∗ (1)), we have a three-step ﬁltration of Ad(L) (resp. Ad(L)∗ (1)) induced from the above ﬁltration. We need balanced Selmer groups SelQ (Ad(V )) and SelQ (Ad(V )∗ (1)) introduced in [Gr2] (16) under the notation of SA (Q). We call this Selmer group

Adjoint square L-invariants

63

balanced since SelQ (Ad(V )) and the dual SelQ (Ad(V )∗ (1)) have equal dimension by a result of Greenberg (see Proposition 3.82). We deﬁne SelQ (Ad(V )) by slightly shrinking Up (Ad(V )) (in (1.2.5)) to U p (Ad(V )) ⊂ Up (Ad(V )) and keeping Uq (Ad(V )) = U q (Ad(V )) for q = p intact. Then we deﬁne U q (Ad(V )∗ (1)) by the orthogonal complement Uq (Ad(V ))⊥ under the Tate pairing for all q including p. The new Selmer group is deﬁned by the same formula as in (1.2.6): for X = Ad(V ) and Ad(V )∗ (1), H 1 (Qq , X) . (1.5.2) SelQ (X) = Ker H 1 (Q, X) → U q (X) q The new Selmer group SelQ (Ad(V )) often coincides with our Selmer group deﬁned by (1.2.6). Indeed, unless f is multiplicative at p, we set U q (Ad(V )) = Uq (Ad(V )) for all q; thus, SelQ (A) = SelQ (A) for A = Ad(V ) and Ad(V /L) in this case. Here we call f multiplicative at p if f is of weight 2 and is associated to an abelian variety with multiplicative reduction at p (not necessarily split). Thus we only need to deﬁne U p (Ad(V )) if f is multiplicative. In this case, we simply put U p (Ad(V )) = F + H 1 (Qp , Ad(V )) ⊂ Up (Ad(V )), which is the image of H 1 (Qp , F + Ad(V )) in H 1 (Qp , Ad(V )). The above deﬁnition coincides with the one given in [Gr2], because as we already remarked, F + Ad(V )∗ (1) (resp. F − Ad(V )/F + Ad(V )) is the smallest (resp. the largest) subspace of F − Ad(V )∗ (1) (resp. Ad(V )/(F + Ad(V ))) stable under Dp so that Dp acts on F − Ad(V )∗ (1)/F + Ad(V )∗ (1) (resp. on F − Ad(V )/(F + Ad(V )) by χ (resp. by the trivial character); so, F − Ad(V )∗ (1)/F + Ad(V )∗ (1) ∼ = K(1) and F − Ad(V )/F + Ad(V ) ∼ = K. There+ ∗ fore the space F Ad(V ) (1) (resp. F − Ad(V )) is the subspace written as F 11 Ad(V )∗ (1) (resp. F 00 Ad(V )) in [Gr2]. We now verify the following condition in [Gr2] necessary to deﬁne L(U ): Lemma 1.84 Suppose that f satisﬁes the assumptions of Theorem 1.80. Then we have SelQ (Ad(V )) = 0 and SelQ (Ad(V )) = SelQ (Ad(V )∗ (1)) = 0.

(V)

Proof Using the local and global Tate duality (e.g., [MFG] Theorem 4.50), Greenberg has shown in [Gr2] Proposition 2 that dimK SelQ (Ad(V )) = dimK SelQ (Ad(V )∗ (1)), which will be proven as Proposition 3.82 later. Since the standard Selmer group SelQ (Ad(V /L)) contains SelQ (V /L), we have |SelQ (Ad(V /L))| < ∞ (⇒ SelQ (Ad(V )) = 0). Then the lemma follows from Proposition 1.53 (1) or Corollary 1.56.

(V1) 2

64

Introduction

We have the Poitou–Tate exact sequence (e.g., [MFG] Theorem 4.50 (5)): 0 → SelQ (Ad(V )) → H 1 (G, Ad(V )) →

H 1 (Qp , Ad(V )) → SelQ (Ad(V )∗ (1))∗ . U p (Ad(V ))

Thus by (V), we have H 1 (GQ , Ad(V )) ∼ =

H 1 (Qp , Ad(V )) . U p (Ad(V ))

(1.5.3)

1.5.2 Greenberg’s L-invariant Greenberg deﬁned in [Gr2] his invariant L(Ad(V )) in the following way. By the deﬁnition of U p (Ad(V )), the subspace F − H 1 (Qp , Ad(V ))/U p (Ad(V )) inside the right-hand side of (1.5.3) is isomorphic to F − Ad(V )/F + Ad(V ) ∼ = K (see just below (21) in Section 2 of [Gr2]). By (1.5.3), we have a unique subspace H of H 1 (GQ , Ad(V )) projecting down isomorphically onto F − H 1 (Qp , Ad(V )) H 1 (Qp , Ad(V )) → . U p (Ad(V )) U p (Ad(V )) Then by the restriction, H gives rise to a subspace L of

− ab F Ad(V ) Hom Dp , + F Ad(V ) isomorphic to F − Ad(V )/F + Ad(V ) ∼ = K, because

−

2 − F Ad(V ) ab F Ad(V ) ∼ ∼ Hom Dp , + = = K2 F Ad(V ) F + Ad(V )

φ([u, Qp ]) , φ([p, Qp ]) for any u ∈ Z× canonically by φ → p of inﬁnite order (see logp (u) (3.4.4) and (3.4.27) for the full details of this point). Here [x, Qp ] is the local Artin symbol (suitably normalized). If a cocycle c representing an element in H is unramiﬁed, it gives rise to an element in SelQ (Ad(V )). By the vanishing (V) of SelQ (Ad(V )), this implies c = 0; so, the projection of L to the ﬁrst factor F − Ad(V )/F + Ad(V ) (via φ → φ([γ, Qp ]) is surjective. Thus this subspace L is a graph of a K-linear map logp (γ) L : F − Ad(V )/F + Ad(V ) → F − Ad(V )/F + Ad(V ), which is given by multiplication by an element L(Ad(V )) ∈ K. This number is by deﬁnition the L-invariant of Ad(V ) (of Greenberg). The cocycle c as above becomes unramiﬁed at p after restriction to GQ∞ (because the p-ramiﬁcation of c is consumed by Q∞ /Q), and it gives rise to an element c∞ in SelQ∞ (V ). The map c → c∞ is injective (under the triviality of H 0 (GQ∞ , V )) by the inﬂation-restriction sequence. Thus the image H∞ ⊂ SelQ∞ (V ) gives rise to a zero of the characteristic power series of

Adjoint square L-invariants

65

SelQ∞ (Ad(V /L)) at the augmentation ideal of W [[Γ]] (see [Gr2] Proposition 1). This fact is compatible with the fact that an Euler factor at p kills the value of Lp (s, Ad(V )) at s = 0, and therefore this gives a motivation that we should be able to deﬁne the L-invariant L(Ad(V )) somehow using this phenomenon. Suppose now that V = T (E) ⊗Q Qp with the p-adic Tate module T (E) for an elliptic curve E over Q with split multiplicative reduction modulo p. Then we × have 0 = q ∈ pZp such that E(Qp ) ∼ = Qp /q Z as modules over Gal(Qp /Qp ) by a result of Tate (see [GME] Theorem 2.5.1). This p-adic number q = qE is called the Tate period of E. The subspace L ⊂ Hom(Dpab , Fp− Ad(V )/Fp+ Ad(V )) ∼ = Hom(Dpab , Qp ) has a generator φ0 : Dpab → Qp . The ﬁxed ﬁeld of the kernel of φ0 is a Zp φ([γ, Qp ]) extension M∞ /Qp . Since L φ → ∈ Qp is surjective, M∞ ramiﬁes log γ "∞p fully. Then by local class ﬁeld theory, n=1 NMn /Qp (Mn× ) has a rank 1 torsionfree part, which contains q0 = pa u with a = 0 and u ∈ Z× p . The quantity logp (q0 ) ∈ Qp is determined uniquely independent of the choice of q0 . The ordp (q0 ) following result is [GS] (3.11) combined with [Gr2] (24): Proposition 1.85

Let E/Q be an elliptic curve with split multiplicative reduclogp (q0 ) logp (qE ) tion at p. Then we have L(E) = L(Ad(T (E))) = = ∈ Qp . ordp (q0 ) ordp (qE ) Since we have L(Ad(T (E))) ∼ = L(Ad(T (E ⊗ χ))) as Galois modules for any quadratic twist E ⊗ χ of an elliptic curve E/Q , choosing a suitable quadratic Galois character χ : Gal(Q/Q) → {±1}, E ⊗ χ has split multiplicative reduction at p if E has multiplicative reduction at p, because E is split multiplicative if and only if F robp acts trivially on T (E)Ip ∼ = Zp . In this sense, the above result can be applied to elliptic curves with nonsplit multiplicative reduction at p. Here Ker(χ) of χ, the quadratic twist E ⊗ χ is given writing M for the splitting ﬁeld Q by the quotient scheme Gal(M/Q)\(E ×Spec(Q) Spec(M )), where σ ∈ Gal(M/Q) acts diagonally on x × y by χ(σ)x × σ(y), regarding χ(σ) = ±1 ∈ Aut(E/Q ). Then we have T (E) ⊗ χ ∼ = T (E ⊗ χ) as Galois modules. Proof Let M∞ /Qp be the composite of all Zp -extensions of Qp ; so, by local class ﬁeld theory, Gal(M∞ /Qp ) ∼ = Z2p . Then [q0 , Qp ] ∈ Gal(M∞ /M∞ ) again by local class ﬁeld theory, and by deﬁnition, φ0 ([q0 , Qp ]) = 0. Since [q0 , Qp ] = [u, Qp ][p, Qp ]a (a = ordp (q0 )), we have 0 = φ0 ([q0 , Qp ]) = φ0 ([u, Qp ])+ ur aφ0 ([p, Qp ]). Writing M∞ /Qp for the unique unramiﬁed Zp -extension and + M∞ /Qp for the cyclotomic Zp -extension, the restriction of φ0 to Γ+ = + /Qp ) is a constant multiple of logp ◦Np for the cyclotomic character Gal(M∞

66

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−1 ), Np ; i.e., φ0 |Γ+ = x(logp ◦Np ) for x ∈ Q× p . Since logp (Np ([u, Qp ])) = logp (u log (q ) p 0 . we have x logp (u−1 ) + aφ0 ([p, Qp ]) = 0. Thus L(Ad(T (E)) = a/x = ordp (q0 ) To prove the second identity, we look into the following exact sequence of Dp -modules: j

×

0→Z− → Qp → E(Qp ) → 0, where j(1) = qE . This exact sequence comes from the theory of Tate curves (e.g., [GME] Theorem 2.5.1). We get from the above sequence another exact sequence of Dp -modules: 0 → µpn → E[pn ] → Z/pn Z → 0. The Weil pairing E[pn ]×E[pn ] → µpn gives a perfect Cartier self-duality of E[pn ] (e.g., [GME] Theorem 4.1.17) and µpn ⊂ E[pn ] is totally isotropic (because the Cartier dual of µpn is Z/pn Z). Passing to the projective limit over n and tensoring with Qp , we have a self-dual exact sequence of Dp -modules 0 → Qp (1) → V → Qp → 0. By Tate duality in Galois cohomology (e.g., [MFG] Section 4.4), we have a perfect cup product pairing (·, ·) : H i (Qp , Qp ) × H 2−i (Qp , Qp (1)) → Qp = H 2 (Qp (1)). We can make explicit this pairing using Kummer’s theory: for x ∈ Q× p , the Galois n cocycle σ → (q 1/p )σ−1 gives rise to a class γx ∈ H 1 (Qp , µpn ). Passing to the projective limit over n, we get a cocycle γx ∈ H 1 (Qp , Zp (1)) ⊂ H 1 (Qp , Qp (1)). Then + ur we have (ξ, γx ) = ξ([x, Qp ]). Write Γ+ = Gal(M∞ /Qp ) and Γur = Gal(M∞ /Qp ). Splitting H 1 (Qp , Qp ) = Hom(Dpab , Qp ) = Hom(Γ+ , Qp ) × Hom(Γur , Qp ), ∼ Hom(Γ+ , Qp ) (resp. ιur : Qp ∼ write ι+ : Qp = = Hom(Γur , Qp )) for the inverses of φ([u, Qp ]) and φ → φ([p, Qp ]). Then the adjoints ι∗+ , ι∗ur : H 1 (Qp , Qp (1)) → φ → logp (u) Qp are given by ι∗+ (γx ) = logp (x) and ι∗ur (γx ) = ordp (x). The connection map δ : H 1 (Qp , Qp ) → H 2 (Qp , Qp (1)) = Qp of the long exact sequence attached to the above exact sequence is the adjoint of the connection map δ ∗ : Qp = H 0 (Qp , Qp ) → H 1 (Qp , Qp (1)). Then by deﬁnition, we have δ ∗ (y) = γy . Thus δ induces an isomorphism: Hom(Γur , Qp ) ∼ = H 2 (Qp , Qp (1)) = −1 Qp , and deﬁning δur = δ ◦ ιur and δ+ = δ ◦ ι+ , we have L(V ) = −δur ◦ δ+ . By logp (qE ) taking the adjoint of these maps, we ﬁnd L(V ) = because ι∗+ (γx ) = ordp (qE ) logp (x) and ι∗ur (γx ) = ordp (x).

Adjoint square L-invariants

67

∂δ p : Dpab → Qp in the following section ∂x x=0 (see (1.5.4)), by a general result relating the derivative to the L-invariant in [GS] logp (q0 ) φ0 ([p, Qp ]) (3.14), we can express L(V ) as logp (γ) . Thus L(Ad(V )) = = φ0 ([γ, Qp ]) ordp (q0 ) logp (qE ) L(V ) = as desired. 2 ordp (qE ) Since we will prove that φ0 = δ −1 p

1.5.3 Proof of Theorem 1.80 We take a p-ordinary Hecke eigenform f of weight k ≥ 2 as in the theorem and its two-dimensional Galois representation V . We take a matrix form ρ : GQ → of the Galois representation L so that its restriction to Dp is given by M2 (W ) (σ) β(σ) . We now identify Ad(L) with the following subspace of ρ(σ) = 0 δ(σ) M2 (W ): . ξ ∈ M2 (W ) = EndW (L)Tr(ξ) = 0 . Then F − Ad(L) is the subspace of Ad(L) made up of upper triangular matrices, and F + Ad(L) is made up of upper nilpotent matrices on which Dp acts by the character δ −1 . Recall the versal nearly ordinary deformation ρ : GQ → GL2 (R) with det(ρ) = det(ρ) and the point P ∈ Spf(R)(W ) carrying ρ. Recall the subspace H of H 1 (GQ , Ad(V )) studied in the previous section. We know 1. dimK H = 1; 2. H is made up of the classes of cocycles c which is upper triangular on Dp and unramiﬁed at the -inertia group for ∈ S diﬀerent from p; 3. SelQ (Ad(V )) = {0} (Lemma 1.84). We recall the deﬁnition of the “−” Selmer group given just below (1.2.6): replacing Up (Ad(V )) in (1.2.6) by the bigger Up− (Ad(V )) = Ker(Res : H 1 (Qp , Ad(V )) → H 1 (Ip ,

Ad(V ) )) F − (Ad(V ))

(and keeping Uq (Ad(V )) intact for q = p), we deﬁned the bigger “−” Selmer group Sel− Q (Ad(V )) ⊃ SelQ (Ad(V )). By (2), we have H ⊂ Sel− Q (Ad(V )). By (1.5.3), Sel− Q (Ad(V )) = H ⊕ SelQ (Ad(V )) = H, Up− (Ad(V )) H 1 (Qp , Ad(V )) is given by . This fact Up (Ad(V )) Up (Ad(V )) can be also seen from [H00] Corollary 5.4.

because the image of H in

68

Introduction

By our assumption, the deformation functor Φ = Φn.ord,ν of L deﬁned by (D1–3) is representable by R. Then by an argument similar to the proof of ∗ Proposition 1.47 (see Corollary 3.79 for details), the K-dual Sel− Q (Ad(V )) of 1 Sel− Q (Ad(V )) is canonically isomorphic to ΩR/W ⊗R RP /P RP where P is the point of Spf(R) corresponding to ρ and RP is the localization completion of R at P (so RP /P RP = K). Here Ω1R/W is the module of continuous diﬀerentials of R over W under the proﬁnite topology of R. Thus H = Sel− Q (Ad(V )) is isomorphic to the tangent space at P of RP . To describe this point more fully, we ﬁrst remark that (RP , ρ : GQ → GL2 (RP )) is the universal couple pro-representing the following localized functor ΦP as we prove in 3.2.10. The functor ΦP associates to each local artinian K-algebra A with residue ﬁeld K the set of isomorphism classes of p-adically continuous deformations ρ : GQ → GL2 (A) satisfying the following three conditions: (P1) ρ mod mA ∼ = ρ for the maximal ideal mA of A; (P2) writing ι : K → A for the structure homomorphism of K-algebras, we have the identity of the determinant characters: ι ◦ det(ρ) = det ρ ; (P3) ρ |Dp ∼ =

∗ ∗ mod mA = δ, writing ρ|D ∼ with δ . = p 0 δ 0 δ

Indeed, if we have such ρ : GQ → GL2 (A), moving ρ by conjugation in GL2 (A) if necessary, we ﬁnd by continuity a compact W -subalgebra B ⊂ A such that ρ has an image in GL2 (B) and ρ mod (mA ∩ B) ∼ = ρ. This point can also be veriﬁed by using the technique of pseudo-representation (see 1.1.3). Then by the universality of R, we have a W -algebra homomorphism ϕ : R → B such that ϕ◦ρ ∼ = ρ in GL2 (B). This morphism induces ϕP : RP → A after localization. Since RP is topologically generated by the traces of ρ, the localized version ϕP is uniquely determined by ρ , and hence (RP , ρ) pro-represents ΦP . Here is a brief sketch of how the isomorphism of H with the tangent space will be computed in Proposition 3.87: each inhomogeneous cocycle c (representing an element of H) gives rise to an inﬁnitesimal nearly ordinary deformation ρ c with det( ρ) = det ρ: ρ c : GQ → GL2 (K[x]/(x2 )) given by ρ c (σ) = ρ(σ) + c(σ)ρ(σ)x. The tangent space at P of Spf(R)/W is isomorphic to ρ mod (x) = ρ} { ρ ∈ Φ(K[x]/(x2 ))| = ΦP (K[x]/(x)2 ) ∼ by a standard argument (see the proof of Lemma 1.59). Here “∼” is the conjugation under 1 + xM2 (K) ⊂ GL2 (K[x]/(x2 )) and (as one can easily check)

Adjoint square L-invariants

69

corresponds to the relation in cocycles giving rise to H modulo coboundaries. Thus c → ρ c induces the isomorphism from H to the tangent space of Spf(RP ) at P . Taking an inhomogeneous cocycle c : GQ → Ad(V ) representing a generator −a(σ) b(σ) of H, we write c(σ) = for σ ∈ Dp . If c restricted to Dp modulo 0 a(σ) upper nilpotent cocycles is unramiﬁed, it gives rise to a nontrivial element of SelQ (Ad(V )). By the vanishing of SelQ (Ad(V )) (Lemma 1.84), we ﬁnd a = 0 on logp (γ) for a generator γ of 1 + pZp . We Ip . Then L(Ad(V )) = a([p, Qp ]) · a([γ, Qp ]) therefore need to compute this value. Since P RP /P 2 RP ∼ = H as already remarked, dρ/dx gives rise to a generator of H by the universality of (RP , ρ = ρI ); so, we have dρ (σ) dx

(σ) β(σ) with a constant C ∈ K × . Writing ρ(σ) = for σ ∈ Dp , we have 0 δ(σ) therefore dδ([p, Qp ]) a([p, Qp ])δ([p, Qp ]) = C · , dx x=0 (1.5.4) dδ([γ, Qp ]) a([γ, Qp ])δ([γ, Qp ]) = C · . dx x=0 c(σ)ρ(σ) = C ·

ord For the modular p-ordinary deformation ρord I , its determinant det(ρI ) is the universal deformation on GQ of det(ρ) mod m (at least locally around Pf ). Recall the universal deformation of the trivial character κ of GQ : κ(γ s ) = (1+x)s . √ −1 Thus the character κ sends σ ∈ G inducing γ s on Gal(Q∞ /Q) to (1+x)−s/2 . √ −1 ⊗ κ after localization at P . Then the character δ is This shows ρ = ρI = ρord I given by δ([γ s , Qp ]) = κ−1 because the character δ ord at the lower right corner of ρord I |Dp is unramiﬁed. Then we have dδ([γ s , Qp ]) s 1 dδ([γ, Qp ]) . = − and logp (γ)−1 = − dx 2 dx 2 logp (γ) x=0 x=0

As for the value at [p, Qp ], we have δ ord ([p, Qp ]) = a(p) by Theorem 1.76. Since χ([p, Qp ]) = 1 and the character (1 + x)s/2 interpolates χm(p−1) for all integer m, (1 + x)s/2 has value 1 at [p, Qp ]. Thus we get the desired result from the deﬁnition of L(Ad(V )).

2 AUTOMORPHIC FORMS ON INNER FORMS OF GL(2)

The aim of this chapter is to give an introduction to the theory of automorphic forms on the multiplicative group of a quaternion algebra over Q and over totally real ﬁelds F . We know traditionally from the time of Gauss and Eisenstein that modular forms on a congruence subgroup Γ of SL2 (Z) contain a striking amount of arithmetic information. Some of them have already been discussed in the previous chapter. An easier way of constructing a modular form is to make an averaging sum of its factors of automorphy: an Eisenstein series (1.4.2). There is another explicit way of constructing modular forms. As an application of Poisson’s summation formula, an inﬁnite series attached to each quadratic form Q(x) on a Q-vector space (of dimension m) has been used to construct elliptic modular forms explicitly: a theta series. Since the theta series of Q is deﬁned by

θ(z) = exp(2πiQ(x)z) x∈Zm

if Q is positive deﬁnite, one is able to count the number of integer solutions of Q(x) = n for a given positive integer n by studying the theta series, which is a modular form of weight m/2 (see Theorem 2.65). For small m, one can prove an exact formula of the number of solutions as an mexplicitly given function of n. This is the case for the sum of squares Q(x) = j=1 x2j for 2 ≤ m ≤ 8, because one can explicitely write θ down as a constant multiple of Eisenstein series and Fourier coeﬃcients of an Eisenstein series can be computed explicitly. The idea of relating theta series and Eisenstein series to ﬁnd such a formula is classical going back to the days of Gauss and Jacobi and has been developed much by Siegel [SL], Weil [We], [We1], and Shimura [Sh7] more recently. Following the recent work of Shimura [Sh8], we shall give examples of the formula for the sums of squares (2 ≤ m ≤ 8). Write Sm (n) for the number of representations of an integer n > 0 as sums of m squares. Assuming for simplicity n to be odd square free (see [Sh8] 3.9 for the general cases), we have, for the q quadratic residue symbol (primitive with respect to q), p −1 (Lagrange, Gauss, Jacobi); • S2 (n) = 2 1 + −1 0
Automorphic forms on inner forms of GL(2)

71

√ = 27 (2π)−2 ( n)3 b5 (n)L(2, n ) (Eisenstein, Smith, Minkowski, Shimura). Here b5 (n) = 5 if n ≡ 3 mod 4, and b5 (n) = 2−3 · 3 · 5, 2−3 · 5 · 7 according as n ≡ 1, 5 mod 4 8; −1 2 • S6 (n) = ( −1 0
• S5 (n)

One can ﬁnd a slightly more involved formula valid for all n in [Sh8] 3.9. When m > 8, there is a small but nontrivial contribution of cusp forms; so, we cannot have such a precise formula. The contribution of cusp forms is quite subtle even when m = 4, which we study in this chapter for norm forms (of four variables) of quaternion algebras. If we start with the quaternion algebra H = Q + Qi + Qj + Qk with H = H ⊗Q R such that i2 = j 2 = k 2 = −1, ij = −ji = k, jk = −kj = i and ki = −ik = j, the norm form is exactly the sum of four squares: N (x) = xx = x21 + x22 + x23 + x24 for x = x1 + x2 i + x3 j + x4 k ∈ H and x = x1 − x2 i − x3 j − x4 k. To get the formula of S2 (n), a key point is that the ring of Gaussian √ integers √ Z[ −1] is a Euclidean domain (so, a PID) and has four units {±1, ± −1}. For an odd prime , as Fermat observed, = x21 + x22 with x1 , x2 ∈ Z √ ⇐⇒ = αα for α ∈ Z[ −1]

−1 = 1 (⇔ ≡ 1 mod 4). ⇐⇒ Thus S2 () = 4, 0 according as ≡ 1 mod 4 or not. As for S4 (), we need to look into the order R = Z + Zi + Zj + Zk ⊂ H and study right ideals of R. This order is not maximal; that is, there is a maximal subring OH containing R which is a lattice of the Q-vector space H and maximal among such subrings. The ring OH is called the Hurwitz order. We have the index [OH : R] = 2 with (1 + i + j + k)/2 ∈ OH (see [Hz]). Since OH is a noncommutative Euclidean domain, all right ideals of OH are principal, and hence a right R-ideal a is principal if N (a) = [R : a] is odd: a = αR for α ∈ R. Since the quaternion conjugation x → x turns right ideals into left ideals, we ﬁnd aa = RααR, which is a two-sided ideal generated by N (α) = αα ∈ Z. Thus S4 ()/8 = 1 + is the number of such factorizations = αα, because R has eight units: {±1, ±i, ±j, ±k}. We can think of another quaternion algebra D = M2 (Q). Then a maximal order is given by M2 (Z). The unit group of M2 (Z) is inﬁnite and given by 1 0 GL2 (Z) = SL2 (Z) SL2 (Z) for = . Again all right ideals of M2 (Z) 0 −1

72

Automorphic forms on inner forms of GL(2)

are principal. We deﬁne the norm form of M2 (Q) We to be N (x) = det(x).

a b d −b also have an M2 (Q)-conjugation given by ι :

→ . Then c d −c a ι N (x) = x x. We see easily that up to units of M2 (Z), we have 1 + elements α in M2 (Z) with N (α) = (see Lemma 2.46): %

1 0

0

and

0

u 1

& for u = 1, . . . , .

(2.0.1)

Thus we conclude that S4 ()/8 gives the number of solutions det(α) = in M2 (Z) up to units. This is the simplest example of intricate relations between diﬀerent quaternion algebras, which we study in detail in this chapter (as an introduction to the theory of quaternionic automorphic forms). Elliptic modular forms and Hilbert modular forms are particular cases of such quaternionic automorphic forms coming from M2 (Q) and M2 (F ) for a totally real ﬁeld F . To motivate our study of quaternionic automorphic forms, let us continue to give examples of classical theorems whose proofs rely essentially on elliptic modular forms and quaternion algebras. If one wants to solve a degree ﬁve nonsoluble rational equation, what we need is a few elliptic functions in addition to classical operation of taking radicals (a result of F. Klein), and the solution is given in terms of the coordinate of a ﬁve-torsion point on a rational elliptic curve (without complex multiplication; see [Du]). If one wants to ﬁnd explicit generators (behaving nicely under Galois action) of an abelian extension of the rational number ﬁeld Q, we only need the exponential function z → e(z) = exp(2πiz), which uniformizes the multiplicative group Gm (e : C → Gm (C) = C× is the universal covering; see [PAF] 2.1.3). The generators are roots of unity {e( N1 )}0=N ∈Z (a theorem of Kronecker and Weber, and of Hilbert; see [ICF] Chapter 14 or [PAF] 2.1.4). If one wants to generalize this to abelian extensions of an imaginary quadratic ﬁeld K, one needs to consider (all) torsion points of an elliptic curve E with complex multiplication by K. Thus the desired generator is given again by an elliptic function. This is the famous “Kronecker’s dream of his youth” and the origin of Hilbert’s twelfth problem (see [Hl] and [PAF] 2.1.4). Modular functions f : H → C (that is, modular forms of weight 0) on a congruence subgroup Γ of SL2 (Z) can be considered as classifying functions of “all” elliptic curves with some extra structures (for example, a point on the curve of a given order N ), because over C, any elliptic curve E can be uniformized as E(C) = C/Zz + Z for a point z ∈ H = {z ∈ C|i(z − z) > 0}. Thus all information we get as above can be formulated more naturally using elliptic modular forms and functions. Among elliptic modular forms, those forms f which are eigenforms of all Hecke operators T (n) are particularly important. As was shown by Hecke and Shimura, the eigenvalues an of T (n): f |T (n) = an f generate a number ﬁeld Q(f ) (that is a ﬁnite extension of Q called a Hecke ﬁeld). When Q(f ) = Q, we call f a rational Hecke eigenform.

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73

One of the spectacular achievements in the recent history of number theory is the proof of the Shimura–Taniyama conjecture by Wiles, Taylor et al. (see [BrCDT], [Ki], [GME] 5.2.4, and 1.3.4 in the text). This could be (in a rather oversimpliﬁed way) formulated as follows. Starting from a rational Hecke eigenform f of weight 2 on the congruence subgroup Γ0 (N ) of SL2 (Z), Eichler (for N = 11) and Shimura (in general) in the 1950s created a rational elliptic curve Ef /Q whose L-function L(s, Ef ) is identical to L(s, f ) (so L(s, Ef ) has analytic continuation to the whole s-plane, proving the Hasse-Weil conjecture for this particular Ef ; see [IAT] Chapter 7 and [GME] Section 4.2). If we use the classical deﬁnition of L-functions of elliptic curve, this could be formulated as 1 + − a = |E(F )| as long as (U1) the equation of the curve modulo gives an elliptic curve over the ﬁnite ﬁeld F (that is, E has good reduction modulo ). If we use a slightly more modern formulation, Gal(Q/Q) acts naturally and continuously on the ´etale cohomology group H 1 (Ef /Q , Zp ) ∼ = Z2p (for any prime p), and the Galois action is characterized so that Tr(F rob ) = a for almost all primes p = (independently of p diﬀerent from ), where F rob is the (geometric) Frobenius element of in the Galois group Gal(Q/Q). Thus the Galois action on H 1 (Ef /Q , Zp ) gives a family of Galois representations {ρp }p indexed by primes p with independent trace Tr(ρ (F rob )) = a ∈ Z as long as (U2) the image of the inertia group at under ρp is trivial (ρp is called unramiﬁed at in this case). The condition (U2) is actually a consequence of (U1) (a result of Hasse and Deuring, and of Shimura, e.g., [ACM] Chapter III) and (U2) implies (U1) (a later result of Serre and Tate, [SeT]). The conjecture then states that any rational elliptic curve E is isogenous over Q to Ef for a suitably chosen rational Hecke eigenform f . An isogeny is a morphism of group schemes: E → Ef which is surjective (so, having ﬁnite kernel because of dim E = dim Ef = 1). The L-function is an isogeny invariant. In the spirit of Shimura and Langlands, we may generalize this modularity problem to a general compatible family {ρp }p of Galois representations. Here p runs over all prime ideals of a number ﬁeld E, and ρp : Gal(Q/Q) → GL2 (Ep ) with Tr(ρp (F rob )) ∈ E is independent of p as long as ρp is unramiﬁed at . Such a family can be created for any given Hecke eigenform f so that Tr(ρ(F rob )) = a (so E = Q(f )). This is a result of Shimura when the weight k is equal to 2, of Deligne if k > 2 (although Shimura also obtained a slightly weaker form of Deligne’s result (of 1969) for more general automorphic forms: see [68c] in [CPS]) and of Deligne and Serre for k = 1 (see Theorem 2.43). Thus if det(ρp )/N k−1 is of ﬁnite order for the p-adic cyclotomic character N , one expects to have a Hecke eigenform f of weight k giving rise to the compatible family {ρp }p . This generalized form of the conjecture is also known in many

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cases of weight k ≥ 2 as was summarized in 1.3.4, and also, some cases of k = 1 have been successfully attacked by Langlands and R. Taylor et al. (see [BCG], [BuDST], [Ta3] and [Ta4]). We can extend such a principle even to mod p representations. As Serre did in [Se4], one would then conjecture any mod p two-dimensional odd Galois representation ρ is a reduction modulo p of a modular Galois representation associated to an elliptic Hecke eigenform of speciﬁc weight and level (see also [Di]). Taylor [Ta5] found a Hilbert modular Hecke eigenform associated to ρ restricted to Gal(Q/F ) for an unspeciﬁed totally real ﬁeld F . Very recently, a proof of the conjecture for ρ unramiﬁed outside p was announced by Khare [Kh], which uses essentially the results in [Ta5]. In the reverse direction, we can study the deformation of a mod p Galois representation ρ, creating a “big” Galois representation ρ into GL2 (T) for a big p-proﬁnite algebra T so that, for speciﬁc prime ideals P densely populated in Spf(T), ρP := ρ mod P gives rise to the modular p-adic Galois representation as above whose reduction modulo the maximal ideal containing P is isomorphic to ρ. Examples of the “big” Galois representations were ﬁrst constructed in [H86b] for elliptic modular forms and were later generalized to Hilbert modular forms in [Wi] and [H89b] after the earlier work on modular Galois representations described above. This construction essentially depends on the study of quaternionic automorphic forms. The abstract framework of Galois deformation theory was given by Mazur [Mz2], and the principle proposed by Mazur is that the “big” modular Galois representation is universal among all speciﬁc deformations. So it appears to be suﬃcient only to study elliptic modular forms and automorphic forms on the split GL(2). This is not the case for a general base ﬁeld F . We can consider an arbitrary base ﬁeld F and a compatible family ρ = {ρp }p of representations of Gal(Q/F ). We can formulate the conjecture that there should exist a Hecke eigenform f : GL2 (F )\GL2 (FA ) → C giving rise to the family, because we can naturally associate with each elliptic Hecke eigenform an adelic Hecke eigenform on GL2 (A) (see 2.3.2). This direction of the conjecture has been also proven when F is totally real, by K. Fujiwara [Fu], Skinner and Wiles [SW] and [SW1] and Kisin [Ki] under diﬀerent sets of assumptions (see Chapter 3). The “direction” is to ﬁnd a modular form on GL2 (FA ) out of a given (arithmetic family of) Galois representation. However, there are cases where we have no known way to create a Galois representation directly out of a Hecke eigenform on the split GL(2), without relying on some tricks moving to automorphic forms on some other algebraic groups (e.g., [Ta1] and [Ta6]). If F is not totally real, the modular variety GL2 (F )\GL2 (FA ) is just a Riemannian manifold (not an algebraic variety); so, there is no way to have subtle arithmetic on the manifold to create Galois representations. As was noticed in the 1960s by Shimura, even if F is totally real, the Hilbert modular variety does not yield the desired two-dimensional Galois representations (as can be checked in the real quadratic cases; see [BryL] for general totally real ﬁelds). Creating Galois representations (or even creating an elliptic curve from a given

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Hilbert modular rational Hecke eigenform of weight 2) could be more diﬃcult than ﬁnding modular forms out of arithmetic Galois representations or elliptic curves. A known systematic way of creating an arithmetic object (see, for example, [67b] in [CPS] and [H81a]) out of an automorphic form is to study Shimura curves and varieties obtained from quaternion algebras over a totally real ﬁeld F whose automorphic manifold is an algebraic variety deﬁned canonically over F . The cases where we get algebraic curves in this way have been proven to be most useful. There is another possibility of using quaternion algebras over a totally real ﬁeld producing Shimura surfaces (e.g., [Bl]), although the above question is still open in general. The utility of such quaternion algebras was ﬁrst noticed and studied by Shimura. They are not only useful in creating out of quaternionic Hecke eigenforms elliptic curves deﬁned over F (in the rational weight 2 case: [H81a]) and families of Galois representations (cf. [68c] in [CPS] I, [LFE] Chapter 7 and [GME] Chapter 4) but also in solving (cyclotomic and anticyclotomic) Hilbert’s twelfth problem for CM ﬁelds ([67b] in [CPS] I), using quaternionic automorphic functions. There are some other signiﬁcant applications of quaternionic automorphic forms (for example, the geometric proof of the local Langlands’ conjecture for GL(2) and the study of p-adic properties of Hilbert modular forms, cf., [C] and [C1]). If we start with a quaternionic Hecke eigen-automorphic form fD on a quaternion algebra D/F , we have the associated family ρ of Galois representations by the results of Shimura [CPS] [68c] and Carayol [C1]. Then in the cases where the modularity problem is solved, we ﬁnd a Hilbert modular form f having the same eigenvalue as fD . This suggests a natural question if the Hecke eigenvalues of each quaternionic automorphic form would be realized by a Hilbert modular form. In other words, as Langlands pointed out, the non-abelian reciprocity law in a rough form (if it exists) depends only on the Q-points of the starting algebraic group deﬁned over F (not on its F -form; see [MFG] 1.2.1). A genesis of this question can be found in a problem Eichler studied in the 1950s (Eichler’s basis problem [Ei], which came out in his thought, presumably, without deﬁnite knowledge of the non-abelian reciprocity law). As Gauss and Jacobi knew, positive deﬁnite quadratic forms Q(x) of fourvariables with coeﬃcients in Q give rise to modular forms of weight 2: θ(z) = n∈Z4 e(Q(n)z). Eichler studied the norm form of an ideal a of a deﬁnite quaternion algebra D over Q and asked which subspace of elliptic modular forms can be spanned by such theta series θ(a), and more generally, he tried to ﬁnd a natural basis of the space. His result in a special case is as follows. Suppose that D = D ⊗Q Q ∼ = M2 (Q ) for all but one prime, say p. Take a maximal order OD of D with OD ⊗Z Z = M2 (Z ) for = p. In this case, × × D∞ for D∞ = D ⊗Q R and DA = D ⊗Q A the automorphic variety D× \DA× /O D is zero-dimensional; so, it is a set in bijection to the OD -right ideal classes: {right OD -ideals a} modulo left multiplication by D× . For a right OD -ideal a, the conjugate aOD a−1 is another maximal order of D. Deﬁne ea by the order × × × of the unit group (aOD a−1 )× . Take a Hecke eigenform A /OD → C −1 f : D \D −1 ). Then we with eigenvalue a for T (), and form θb (f ) = a ea f (a)θ(ab

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can ﬁnd a basis of S2 (Γ0 (p)) in the set {θb (f )}f,b and θb (f )|T () = a θb (f ), as expected. Here b runs over right OD -ideals up to left equivalence. An integral and a more precise solution to his basis problem will be studied in Section 2.6. This fact has been proven in a far more general setting by Jacquet and Langlands for all quaternion algebras D over any number ﬁeld F [AFG] Chapter 3, and by their results, for any Hecke eigenform on f : D× \DA× → C, we can ﬁnd a Hecke eigenform θf : GL2 (F )\GL2 (FA ) → C with the same Hecke eigenvalues as f , though θf may not be uniquely determined by f under their formulation. This association is now called the Jacquet–Langlands correspondence (or the Jacquet–Langlands–Shimizu correspondence including Shimizu who studied the basis problem over totally real ﬁelds F ; cf., [Si]). We reformulate the correspondence in an arithmetic way in this chapter along with standard deﬁnitions of quaternionic automorphic/forms and 0 try to give a sketch of a proof 1 of the correspondence integral over Z in the above special case (in 3(p/ −01) 1 is proven by Emerton [Em], but this special case, a stronger result over Z 2 our method can be generalized to Hilbert modular forms, as we will look into it later in 3.2.7). Exercise 2.1 Prove that every right ideal of M2 (Z) with an element of non-zero determinant is principal. 2.1 Quaternion algebras over a number ﬁeld We recall basic structure theorems of quaternion algebras over a number ﬁeld. Since our description is limited to a minimum (necessary for understanding of the later discussions), we refer to [BNT] and [AAQ] for thorough descriptions of the arithmetic of simple algebras and quaternion algebras. 2.1.1 Quaternion algebras Let F be a ﬁeld of characteristic 0. A quaternion algebra D over F is a central simple algebra of dimension 4 over F . Here the word “central” means that F is the center of D and “simple” means that there are no two-sided ideals of D except for {0} and D itself. First suppose that D is not a division algebra. Thus D has a proper left ideal a D. Since a is also a vector F -subspace of D, its dimension over F is either 1, 2 or 3. If dim(a) = 1, then for a generator x of the subspace a, bx for b ∈ D is a constant multiple of x itself. Write this constant as ρ(b) ∈ F . Then ρ : D → F is an F -algebra homomorphism and hence surjective. Thus Ker(ρ) is a three-dimensional two-sided ideal, which contradicts the simplicity of D. Thus dim(a) > 1. Similarly, if dim(a) = 3, D/a is a one-dimensional vector space over F on which the algebra D acts. By the same argument as above, dim(a) = 3 is impossible. Thus dim(a) = 2. Choose a basis x1 , x2 of a over

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F . Deﬁne ρ : D → M2 (F ) (the 2 × 2 matrix algebra with entries in F ) by (x1 , x2 )ρ(b) = b(x1 , x2 ). Then ρ : D → M2 (F ) is an F -algebra homomorphism taking the identity to the identity. Thus Ker(ρ) is a two-sided ideal of D. Since ρ = 0, the simplicity (non-existence of non-trivial two-sided ideals) tells us that Ker(ρ) = {0}. Thus ρ is injective. Comparing the dimension, we conclude that ρ:D∼ = M2 (F ) is an isomorphism. Now assume that D is a division algebra (so every non-zero element has a left and right inverse). Pick x ∈ D which is not in the center F . Then the subalgebra K = F [x] ⊂ D has to be a ﬁeld. Thus D becomes a vector space over K via left multiplication by elements of K. Then we have 4 = dimF D = (dimK D) × [K : F ]. Thus [K : F ] is either 4 or 2. If [K : F ] = 4, K = D, and D becomes commutative. This contradicts the centrality of D over F , and [K : F ] = 2. Taking y ∈ D − K, we can consider the subspace K + Ky in D. These two spaces D and K + Ky have dimension 2 over K, and D = K + Ky. Then we deﬁne a representation ρ : D → M2 (K) by 1 1 b ρ(b) = b= . y y yb

α β This means α + βy = b and γ + δy = yb when ρ(b) = . Since D is simple, γ δ ρ is injective. Thus D is realized as an F -subalgebra of M2 (K). We now determine the image ρ(D) of D in M2 (K) explicitly. Since x is quadratic over F , we may assume that x satisﬁes x2 − ax + b = 0 for a, b ∈ F , which is the minimal equation of x over F . Thus X 2 − aX + b has two distinct roots: x, xτ which are conjugate each other under the generator τ of Gal(K/F ). Then ρ(x) satisﬁes the same equation, and the eigenvalues of ρ(x) are roots of X 2 − aX + b = 0. Since ρ(x) is not in the center of ρ(D), it is not a scalar matrix, and the eigenvalues of ρ(x) are two distinct roots of X 2 − aX + b = 0. Changing the basis (1, y) suitably (we write the new basis

as (1,

v)), we may assume x 0 1 1 that ρ(x) = . By the deﬁnition, ρ(b) = b, we see xτ v = vx 0 xτ v v in D. Here we do not need to change 1 because 1 is already an eigenvector for = ρ(v)ρ(x). This implies that for any a ∈ K, ρ(x)1 = 1x = x1. Thus ρ(xτ )ρ(v)

0 α τ ρ(a )ρ(v) = ρ(v)ρ(a) and ρ(v) = for α, β ∈ K. Replacing v by vα−τ , β 0

0 1 we may assume that ρ(v) = and v 2 = ξ. If ξ ∈ F , then F [v] in D is of ξ 0 degree 4 over F , which is impossible. Thus ξ ∈ F . Namely &

% a b a, b ∈ K . ρ(D) = ρ(a + bv) = ξbτ aτ

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Thus we can always realize D as a subalgebra of M2 (K) for any quadratic extension K/F embeddable into D and a suitable ξ ∈ F × in the above form. We deﬁne for β = a + bv ∈ D, N (β) = det(ρ(β)) = aaτ − ξbbτ ∈ F and Tr(β) = Tr(ρ(β)) = a + aτ ∈ F. Moreover, β = a + bv satisﬁes the equation X 2 − Tr(β)X + N (β) = 0 in D, and hence, Tr and N are independent of the choice of v and K. The map N is called the reduced norm and Tr is called the reduced trace. On the image ρ(D) ⊂ M2 (K), the reduced norm N coincides with the determinant map of M2 (K); so, N is multiplicative: N (ab) = N (a)N (b) for a, b ∈ D. Now we start with a subalgebra Dξ of M2 (K) for a quadratic extension K/F given by & %

a b a, b ∈ K . Dξ = ξbτ aτ Since K is two dimensional over F , Dξ is four dimensional over F . Obviously Dξ is stable under multiplication and addition. We also see easily that the center of Dξ is F . Moreover Dξ + Dξ δ = M2 (K) for any generator δ of K over F . Thus Dξ ⊗F K ∼ = M2 (K), which shows that Dξ is a central simple algebra over F . 0 1 ∈ Dξ , N (a + bv) = aaτ − αb(αb)τ . If ξ = αατ for α ∈ K, for v = ξ 0 Choosing a = αb, c = αb + bv has determinant 0 but is a non-zero matrix. Thus Dξ is not a division algebra and hence Dξ ∼ = M2 (F ). If ξ ∈ NK/F (K × ), then τ τ N (a + bv) = aa − ξbb = 0 implies a = b = 0. Therefore a + bv always has an inverse if a + bv = 0; so, Dξ is a division algebra. We have proven ξ ∈ NK/F (K × ) if and only if Dξ ∼ = M2 (F ). If ξ = αατ η for ξ, η ∈ F ×

1 0 a 0 α ξbτ

and α ∈ K × , we see

b 1 0 a = a 0 α−1 ξ(α−1 b)τ

α−1 b aτ

(2.1.1)

∈ Dξ .

Thus Dξ ∼ = Dη if ξ = αατ η for ξ, η ∈ F × and α ∈ K × . Therefore the map: ξNK/F (K × ) → Dξ induces a surjection: . F× onto −−−→ the isomorphism classes of Dξ in M2 (K) for ξ ∈ F × . × NK/F (F ) We ﬁnd that by (2.1.1), this map is actually a bijection. When F = R, then the only possibility of K is C. Since NC/R (C× ) = R× + , we have R× /NC/R (C× ) ∼ = {±1}, and there are only two isomorphism classes of quaternion algebras: one is M2 (R) = D1 and the other is the Hamilton quaternion algebra H = D−1 . When F = C, there is no quadratic extension of C, thus there is only one isomorphism class M2 (C) = D1 .

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Now we suppose that F is a p-adic ﬁeld, that is, a ﬁnite extension of Qp , and we study the quaternion algebras over F . Let be the prime element of the p-adic integer ring O of F . Then F× ∼ = O× × {n |n ∈ Z}. We deﬁne formally the logarithm map on O× by log(x) =

∞

(−1)n+1 (x − 1)n n n=1

as long as the above series converges p-adically. Since for suﬃciently large r, log [F :Q ] converges p-adically on 1 + r O in O× , O× ∼ = µ × Zp p for the subgroup µ of [F :Q ] roots of unity in O. This shows F × ∼ = µ × Zp p × Z and (µ/µ2 ) × Z/2Z if p > 2 × × 2 ∼ C2 = F /(F ) = (∗) 2 [F :Qp ]+1 if p = 2. (µ/µ ) × (Z/2Z) For any quadratic extension K/F , we can choose a generator δ of K over F such that δ 2 ∈ F . Since F [δ] = F [αδ] for α ∈ F × , we have a map: F [δ] → δ 2 mod (F × )2 that induces a bijection {isomorphism classes of quadratic extension K/F } ∼ = C2 . √ × 2 If p > 2, for any p-adic unit u ∈ (O ) , K = F [ u] is the unique unramiﬁed quadratic extension. Thus if we write UK for the group of p-adic units of K, UF = O× = NK/F (UK ) if K/F is unramiﬁed and p > 2. Thus F × /NK/F (K × ) ∼ = Z/2Z if K/F is unramiﬁed and p > 2. Suppose that K/F is ramiﬁed. Let OK be the p-adic integer ring of K and P × . be the maximal ideal of OK . Then NK/F (x) mod P = (x mod P)2 for x ∈ OK × × This shows that F /NK/F (K ) has a quotient group isomorphic to Z/2Z. Since a prime element of F is a norm of the prime element of K, F × /NK/F (K × ) is a proper quotient group of (µ/µ2 ) × Z/2Z ∼ = (Z/2Z)2 . Thus we know from (∗) × × ∼ that F /NK/F (K ) = Z/2Z even if K/F is ramiﬁed (as long as p > 2). Even if p = 2, we can prove by local class ﬁeld theory F × /NK/F (K × ) ∼ = Gal(K/F ) ∼ = Z/2Z.

(2.1.2)

Thus for a given quadratic extension K/F , there are only two isomorphism classes of quaternion algebras Dξ /F inside M2 (K). We return to unramiﬁed K/F . The unique division quaternion algebra of the

0 1 form Dξ in M2 (K) is isomorphic to D . Since v = and v 2 = , we 0 √ know that the ramiﬁed extension F [ ] is isomorphic to K = F [v] ⊂ D . Thus any ramiﬁed quadratic extension K /F is embeddable in D . A division quaternion algebra embeddable in M2 (K ) corresponds to the generator of F × /NK/F (K × ) ∼ = Z/2Z, which is unique. There is only one isomorphism

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class of division quaternion algebras over F . Thus we know that any quaternion algebra over F is either isomorphic to the unique division quaternion algebra or M2 (F ). Now we assume that F is a number ﬁeld. For each prime ideal p of the integer ring O of F , we write Fp for the p-adic completion of F , and we put Dp = D ⊗F Fp , which is a quaternion algebra over Fp . The prime p is called ramiﬁed in Dp if Dp is a division quaternion algebra. We take a quadratic extension K/F inside D and take ξ ∈ F × so that D ∼ = Dξ . Then ξ ∈ NKp /Fp (Kp× ) for all but ﬁnitely many p, where Kp = K ⊗F Fp . Indeed, if Kp is a ﬁeld extension of Fp , it is unramiﬁed for all but ﬁnitely many p (hence NKp /Fp (UKp ) = UFp for almost all p). Since ξ is a unit for all but ﬁnitely many p, we know ξ ∈ NKp /Fp (Kp× ) for all but ﬁnitely many p. If Kp = Fp ⊕ Fp , Dp ∼ = M2 (Fp ) and also ξ ∈ NKp /Fp (Kp× ). This shows that D ramiﬁes at only ﬁnitely many places. For each embedding σ : F → R, we say that D is ramiﬁed at σ, if Dσ = D ⊗F,σ R ∼ = H. We write S for the set of all ramiﬁed places of F in D. Then global class ﬁeld theory tells us the following facts found by Hasse: (H1) The cardinality |S| is even. (H2) For any given set S of places with even cardinality, there exists a unique quaternion algebra D ramifying exactly at S. One can ﬁnd a proof of this in [BNT] XIII.3, Theorem 2 and XIII.6, Theorem 4. Exercise 2.2 1. Prove that the center of Dξ is equal to F . ∼ M2 (K) for an F -algebra D, prove that D is a central simple 2. If D ⊗F K = algebra of dimension 4 over F . 3. Determine the radius of convergence of the p-adic logarithm. 4. For the p-adic integer ring O of a ﬁnite extension F/Qp , give a detailed :Qp ] ∼ µ × Z[F proof of O× = for the subgroup µ of roots of unity in O. p 5. Give a detailed proof of (∗). 6. Let p be an odd prime, and K/F be a quadratic extension of p-adic ﬁelds. Without using class ﬁeld theory, give a detailed proof of (2.1.2). 7. Let F be a number ﬁeld, and D be a division quaternion algebra containing a quadratic extension K/F . Prove that if Kp = Fp ⊕ Fp , Dp ∼ = M2 (Fp ) 2.1.2 Orders of quaternion algebras Let F be a ﬁeld and A be a subring of F . We assume that the ﬁeld of fractions of A coincides with F . Let D be a quaternion algebra over F . Let V be a ﬁnitedimensional vector space over F . An A-lattice L in V is an A-submodule of D which satisﬁes (L1) L is an A-module of ﬁnite type (i.e., L = i Aξi for ﬁnitely many ξi );

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(L2) L ⊗A F = V (i.e. L contains a basis of V over F ). If V is an F -algebra, an A-order R of V is an A-lattice in V which is a subring of V sharing the identity. Taking a quadratic extension K/F , we realize D as & %

a b a, b ∈ K , Dξ = α(a, b) = ξbσ aσ where σ is the generator of Gal(K/F ). Note that Dξ ∼ = DNK/F (η)ξ for η ∈ K × . If A is an A-order of K, we may therefore assume that ξ ∈ A by replacing ξ by NK/F (η)ξ for a suitable η if necessary. Then . Rξ = α(a, b) ∈ Dξ a, b ∈ A is an A-order in D. We deﬁne, for α = α(a, b) ∈ Dξ , αι = α(aσ , −b) = Tr(α) − α = N (α)α−1 . Then α → αι is an F -linear involution: (αβ)ι = β ι αι . Recall that N (α) = det(α). Then in D, N (α) = ααι ∈ F and Tr(α) = α + αι ∈ F . In particular, Pα (X) = X 2 − Tr(α)X + N (α) is the minimal polynomial of α in D if α ∈ F , i.e. Pα (α) = 0 and Pα is monic and has minimal degree among all monic polynomials Q(X) with Q(α) = 0. Now we assume F to be a p-adic ﬁeld and A to be the p-adic integer ring O of F . If R is an order of D, then R is free of rank 4 over O because O is a valuation ring. Then taking a base x = (x1 , x2 , x3 , x4 ) of R, we deﬁne the regular representation ρ : D → M4 (F ) by ρ(α)t x = t xα = t (x1 α, x2 α, x3 α, x4 α). Then ρ(R) is contained in M4 (O). Thus Q(α) = 0 for Q(X) = det(X14 − ρ(α)). Since Pα (X) is the minimal polynomial of α, Pα (X) is a factor of Q(X). Since Q(X) is monic and has coeﬃcients in O, by Gauss’ lemma, Pα (X) has coeﬃcients in O. Namely N and Tr induce N : R → O and Tr : R → O. Suppose now that D is a division algebra. We put . R0 = α ∈ DN (α) ∈ O . Since D is a division algebra, ξ ∈ NK/F (K × ). Note that NK/F (K × ) ⊂ N (D× ) because aaσ = N (α(a, 0)). Since F × /NK/F (K × ) ∼ = Z/2Z and D contains any quadratic extension of F , we know that N (D× ) = F × . We may in fact assume that ξ is a prime element of O. Then it is easy to see that . R0 ⊃ α(a, b)a, b ∈ OK = R for the integer ring OK of K. For ω = α(0, 1), we have ωRω −1 = R, ω 2 = ξ and N (ω) = −ξ. Thus for each α ∈ D× , we can ﬁnd the minimal exponent w(α) ∈ Z such that αω −w(α) ∈ R. Then w is a sort of an additive valuation: it satisﬁes w(αβ) = w(α) + w(β) and w(α + β) ≥ min(w(α), w(α)). We put w(0) = ∞. Then R = {α ∈ D|w(α) ≥ 0} ⊃ R0 , which is the (noncommutative) “valuation” ring of w. This shows that R0 = R is an order. Since

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on any order R , N has values in O, we know R0 ⊃ R . Thus R0 is the unique maximal order OD . Proposition 2.3 If F is local and D is a division algebra, then all O-orders are contained in one and only one maximal order OD . In general, we call an order R in D maximal if there is no order containing R properly. Thus there may be several maximal orders. Suppose now that D = M2 (F ). Let L be an O-lattice of F 2 . We put R = RL = {α ∈ M2 (F )|αL ⊂ L}. Then R is an O-order. In fact, we can ﬁnd a basis (x, y) of L. Since x and y are column vectors, we consider X = (x, y) as an invertible matrix in M2 (F ). Then deﬁning the regular representation ρ : D ∼ = M2 (F ) by Xρ(α)t X = t Xα, we have ρ(α) ∈ M2 (O) ⇐⇒ α ∈ RL . Therefore, we have RL = X −1 M2 (O)X. Conversely, for any given O-order R in D, we put L = π(R) for the projection:

a b b π : D = M2 (F )

→ ∈ F 2. c d d Then L is an O-lattice of F 2 and R · L ⊂ L. Namely R ⊂ RL . Thus for any order R of M2 (F ), there exists a maximal order RL of the form X −1 M2 (O)X. Proposition 2.4 If F is a p-adic local ﬁeld and D = M2 (F ), for any given order R, there exists a maximal order OD containing R which is a conjugate of the standard maximal order M2 (O). Thus all the maximal orders in M2 (F ) (for local F ) are conjugate to each other. Corollary 2.5 The group GL2 (O) is a maximal compact subgroup of GL2 (F ). If K is a maximal compact subgroup of GL2 (F ), then K is a conjugate of GL2 (O) in GL2 (F ). Let aK0 = GL2 (O). A double coset K0 xK0 (x ∈ GL2 (F )) is of the 0 form K0 K0 for a suitable integer a ≥ b by the theory of elementary 0 b divisors. Thus, the subgroup generated by K0 and any x outside K0 contains

a 0 with one of a and b nonzero. Then plainly the subgroup is not 0 b 2 compact; so, K0 is a maximal compact subgroup. Let L = O . Since K\K · K0 is discrete and compact, it is ﬁnite. Then L = u∈K\K·K0 u(L) is isomorphic to O2 as O-modules, and R = {x ∈ M2 (F )|xL ⊂ L } is a maximal order of M2 (F ), which is compact. Thus R× is a compact subgroup. Since R× ⊃ K, they are equal by the maximality of K. Since R = g · M2 (O)g −1 , we have K = R× = g · K0 g −1 for g ∈ GL2 (F ). 2 Proof

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We now suppose that F is a number ﬁeld. We take A to be the integer ring O of F . For any (ﬁnite dimensional) vector space V over F , we ﬁx a base x1 , . . . , xr and identify V with F r . We put L0 = O · x1 + · · · + O · xr , which is an O-lattice. Consider any O-lattice L. By deﬁnition, L = O · y1 + · · · + O · ym with m ≥ r. Since L0 ⊗O F = V , we can ﬁnd α, β ∈ F such that α · L0 ⊂ L ⊂ β · L0 . For each prime ideal p of O, we write Op for the p-adic completion of O and Fp = F ⊗O Op . Then for almost all p, α and β are both p-adic units, and thus L0,p = α · L0,p = αOp ⊗O L0 ⊂ Lp ⊂ β · L0,p = L0,p . This shows that for almost all p, Lp = L0,p . Conversely, let {Lp }p be a family of Op -lattices indexed by all prime ideals of O. We suppose that Lp = L0,p for almost all p. Such a collection {Lp }p is called admissible. This deﬁnition of admissibility does not depend on the choice of starting lattice L0 , because for any O-lattice, its p-adic completion is the same for almost all p. We now show that for any given admissible family {Lp }p of local lattices, there is a unique O-lattice L in V which gives rise to the given collection. We ﬁrst take α in F × so that Lp ⊃ α · L0,p for all p. We can always ﬁnd such α, because L0,p and Lp are diﬀerent for only ﬁnitely many p. Since V /α · L0 ∼ = (F/αO)r , we have a unique ﬁnite subgroup X in V /α · L0 corresponding to Lp /α · L0,p . Put L = {v ∈ V |v mod α · L0 ∈ X}. By deﬁnition, L satisﬁes the required property, and we have 1 V Lp L= p

in V (A(∞) ) = V ⊗F A(∞) for the ﬁnite part A(∞) of the adele ring A of Q. We apply the above argument to V = D for a quaternion algebra D. Let R be an O-order of D. Then Rp is an Op -order of Dp . First suppose that D = M2 (F ). Then M2 (Op ) is maximal at every p, and M2 (O) is maximal. Thus for any order R of D, Rp = M2 (Op ) for almost all p and for ﬁnitely many p with Rp = M2 (Op ), we can ﬁnd xp ∈ Dp× such that xp M2 (Op )x−1 = Rp . For other primes p, we p × simply put xp = 1. Thus x = (xp )p ∈ DA (the adelization of D). The family {xp M2 (Op )x−1 p } is admissible, and therefore there exists an O-lattice OD in D " −1 such that OD,p = xp M2 (Op )x−1 p for all p. Since OD = D p xp M2 (Op )xp in (∞)

DA , OD is a subring; namely, OD is an O-order. Since RD,p is maximal for all p, OD has to be maximal and OD ⊃ R. Now suppose that D is a division algebra over a number ﬁeld F . We embed D into M2 (K) for a quadratic extension K/F . Let A be the integer ring of K. Then R = M2 (A)∩D is an order of D. We shall show that Rp is a maximal order for almost all p. We may assume that D = Dξ . If Kp = K ⊗F Fp ∼ = Fp ⊕ F p , then obviously Rp ∼ = M2 (Ap ) if ξ is a p-adic unit (which is true for almost

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∼ M2 (Fp ). Then ξ is an all p). Suppose that Kp /Fp is a ﬁeld extension and Dp = × × integral norm: ξ = xxσ for x ∈ A = (A ⊗ O ) . This is true for almost all p. O p

p xp 0 , we may assume that ξ = 1. Then conjugating Conjugating Dξ by α = 0 1

δ −δ by β = for δ ∈ K with δ 2 ∈ F × , we know Rp r → βαrα−1 β −1 1 1 induces an isomorphism of Rp with M2 (Op ) if βα ∈ GL2 (Ap ). Since βα falls in GL2 (Ap ) for almost all p, Rp is maximal for almost all p. Thus if one gives oneself a collection of maximal orders {OD,p }p such that OD,p = Rp for almost all p, there exists a unique maximal order OD in D whose completion is equal to the given data OD,p . Conversely, if R is any A-order of D, then Rp is maximal for almost all p. We put OD,p = Rp if Rp is maximal and choose a maximal order OD,p in Dp containing Rp if Rp is not maximal. Then the family {OD,p }p gives rise to a unique maximal order OD in D containing R. We have Proposition 2.6 Let D be a quaternion algebra over a number ﬁeld or a p-adic local ﬁeld. Then for any given order R of D, there exists a maximal order containing R. From now on, D is any quaternion algebra over a number ﬁeld F (including M2 (F )). Fix a maximal order OD of D. Then for any other maximal order OD × −1 in D and for each prime p, we can ﬁnd xp ∈ Dp such that xp OD,p xp = OD,p . × Since OD,p = OD,p for almost all p, xp ∈ OD,p for almost all p. Therefore × x = (xp )p ∈ DA . Thus we have a bijection: × {maximal orders in D} ↔ DA× /U · D∞ × ∼ D⊗Q R for σ running over all archimedean for U = p OD,p and D∞ = σ Dσ = −1 × places of F , since xp OD,p xp = OD,p for all p if x ∈ U · D∞ . If x ∈ D× , then the −1 . Thus conjugation OD r → xrx ∈ OD is well deﬁned and hence OD ∼ = OD we have a surjection: × D× \DA× /U · D∞ {isomorphism classes of maximal orders in D} .

Let I = Homﬁeld (F, C). Now we need to quote some deeper results: Theorem 2.7 (Norm theorem) Let . × FD = x ∈ F × xσ > 0 if D ⊗F,σ Fσ ∼ = H for σ ∈ I . × . In particular, F × /N (D× ) ∼ Then N (D× ) = FD = {±1}r , where r is the number of inﬁnite places at which D is ramiﬁed.

A proof of this theorem is in Weil’s book: [BNT] Proposition 3 on page 206. The proof given there is four pages long but quite elementary and can be read without reading much of the material in the earlier sections of [BNT] (basic algebraic number theory suﬃces for that).

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× Theorem 2.8 (Approximation theorem) The set D× \DA× /U · D∞ is a ﬁnite set. In particular, isomorphism classes of maximal orders of D are ﬁnitely many.

A proof in the case where D = M2 (F ) is in [LFE] Section 9.1. An outline of the proof for division algebras D is as follows. We consider . DA1 = x ∈ DA |N (x)|A = 1 . By the product formula: |ξ|A = 1 for ξ ∈ F × (e.g., [LFE] (8.1.5)), D× is a subgroup of DA1 . Then D× can be shown to be a discrete subgroup of DA1 and D× \DA1 is compact (see [MFM] Lemma 5.2.4). A similar assertion for number ﬁelds is also true, i.e., . F × is discrete in FA1 = x ∈ FA |x|A = 1 , and FA1 /F × is compact. A proof of this fact for ﬁelds F can be found in [LFE] Theorem 8.1.1. All the arguments in the proof there work well replacing |x|A for division algebras D × × 1 U · D by |N (x)|A . Then DA1 / U · D∞ ∩ DA1 is discrete because ∞ ∩ DA is an × ∩ DA1 is discrete and compact open subgroup of DA1 . Thus D× \DA1 / U · D∞ and hence is ﬁnite. Note that × ∩ DA1 D× \DA1 / U · D∞ is the kernel of the norm map × × N : D× \DA× /U · D∞ → FA× /FD UF (UF =

Op× ),

p

whose right-hand side is a ray class group (e.g., [LFE] Corollary 8.1.1]) which is ﬁnite. This shows the above theorem. When D = M2 (F ) (or more generally, if D ⊗Q R is isomorphic to a product of copies of M2 (R)), by the following theorem, × | is equal to the class number of F (see (CL) below). |D× \DA× /U · D∞ Theorem 2.9 (Strong approximation theorem) Let OD be a maximal order of D. Let v beone place (either ﬁnite or archimedean) such that Dv ∼ = × , where we put OD,σ = M2 (Fσ ) if v is the inﬁnite M2 (Fv ). Let U (v) = p=v OD,p place σ ∈ I. Then × Γ(v) = γ ∈ D× N (γ) = 1 and γ ∈ OD,p for all p = v is dense in {x ∈ U (v) |N (x) = 1}. In other words, for any given x ∈ U (v) , we can ﬁnd γ ∈ Γ(v) such that γ ≡ x (v) for any ideal N prime to v and γ (v) is arbitrarily close to xσ for mod N · O D (v) = all inﬁnite places σ = v, where O D p=v OD,p . When v is an inﬁnite place, an elementary proof can be found in Miyake’s book [MFM] Theorem 5.2.10. Although Miyake gives a proof assuming that F = Q, his argument works well

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for general number ﬁelds without much modiﬁcation. See [Kn] for a more general statement for strong approximation. We see easily from the strong approximation theorem that the reduced norm N : D → F induces, for any open subgroup S of U with SF = N (S) ⊂ FA×(∞) , × × = D× \DA×(∞) /S ∼ (CL) D× \DA× /S · D∞ = FA×(∞) /SF · FD if D has at least one inﬁnite place v such that Dv ∼ = M2 (Fv ). The cardinality of the set on the left-hand side of (CL) is called the class number of D and is equal to the number of elements in the set {fractional right OD -ideals in D} modulo the following equivalence relation a ∼ b ⇐⇒ a = αb for α ∈ D× . Thus often the class number of D is given bythat of a ray class group of F . However when D ⊗Q R is isomorphic to HI = σ∈I H (such a quaternion algebra is called a deﬁnite quaternion algebra), the class number of D cannot be given by the class number of F . If D is a deﬁnite quaternion algebra over Q only ramifying at a prime p, one of our goals in this chapter is to prove that the class number of D is equal to the dimension of the space of holomorphic modular forms on Γ0 (p) of weight 2.

Exercise 2.10 1. Let F be a number ﬁeld. For an O-lattice L of F n , prove that there exists an = Lp , O-ideal a such that L ∼ = On−1 ⊕ a as O-modules. Hint: putting L p =O n−1 ⊕ ﬁnd x ∈ GLn (FA ) such that x · L a ⊂ (F n ⊗Q A(∞) ). 2. For a given α ∈ GL2 (F ), prove that α ∈ GL2 (Op ) for almost all p. 3. Using the outcome of Exercise 1, show that the class number of F gives the number |GLn (F )\GLn (FA )/GLn (O)GL n (F∞ )|. 4. Prove (CL) using Theorem 2.9. 5. Prove the statement just after (CL) about the class number of D. 6. Suppose that F is totally real. For each embedding σ : F → R, deﬁne Dσ = D ⊗F,σ R. Prove that D ⊗Q R ∼ = σ Dσ . Describe what happens if F is not totally real. 2.2 A short review of algebraic geometry Since modular forms and automorphic forms are generally deﬁned on the adele points of a given algebraic group G, a minimal amount of knowledge of schemes and algebraic groups is necessary to describe automorphic forms on quaternion algebras which we treat in the following section. We recall here very brieﬂy deﬁnitions and result in algebraic geometry as succinctly as possible. Our treatment is not conventional. In a standard text, one ﬁrst introduces the category of aﬃne schemes as the category of local ringed spaces constructed out of the spectra of commutative rings with identity. Then gluing aﬃne schemes, again as a local ringed space, one constructs the category of schemes. After all this endeavor, usually, one identiﬁes (by tautology) the schemes with the associated covariant functor from the category of commutative rings with identity into the category

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of sets, because a closed algebraic subvariety X in a projective space over a ring A associates with each A-algebra R the set X(R) of points with coordinates in R (which is the covariant functor). Obviously this way is a logically correct path. However, in order to make our exposition short and to rapidly reach the core of the theory necessary to our later treatment, we deﬁne schemes as covariant functors outright, and in this sense our exposition is unconventional. Since the spectrum Spec(A) over the base ring B (or equivalently, over the base scheme Spec(B)) is traditionally ﬁrst deﬁned as a local ringed space made up of prime ideals of A with Zariski topology, we write SA for the associated covariant functor deﬁned on the category of B-algebras R → SA (B) = HomB-alg (A, R) only in this section (later we write this functor as R → Spec(A)(R), identifying the functor and the scheme). This section is added to facilitate a quick (but certainly not intended to be deep) understanding of the necessary geometric tools. For more thorough treatments of the subject, the following books are suggested: [EGA], [ALG] and [GME] Chapter 1. All rings A we consider are commutative, have the identity element 1A , and we denote by 0A the zero element of A. 2.2.1 Aﬃne schemes Let B be a base ring, which is always assumed to be noetherian. Let A be a noetherian B-algebra. The aﬃne scheme S = SA = Spec(A) associated to A is a functorial rule of assigning to each B-algebra R the set given by S(R) = HomB-alg (A, R). The set SA (R) is called the set of R-rational points (or R-integral points) of SA . A B-morphism φ : SA → SC (or a morphism deﬁned over B) is given by φ(P ) = P ◦ φ for an underlying B-algebra homomorphism φ : C → A; in other words, we have the following commutative diagram: P

A −−−−→ 2  φ C

R 2 P ◦φ  C.

By deﬁnition, we have the following properties of the functor SA : f

g

(F1) If R − → R − → R are B-algebra homomorphisms, then we have maps f∗ : SA (R) → SA (R ) and g∗ : SA (R ) → SA (R ) given by f∗ (P ) = f ◦ P and g∗ (Q) = g ◦ Q, and we have (g ◦ f )∗ = g∗ ◦ f∗ . (F2) If R = R and g as above is the identity map iR : R → R , we have iR ,∗ ◦ f∗ = f∗ . If R = R and f as above is the identity map iR : R → R, we have g∗ ◦ iR,∗ = g∗ . (F3) For the identity map iR : R → R, iR,∗ : SA (R) → SA (R) is the identity map of the set SA (R). Thus R → SA (R) is a covariant functor of B-algebras into sets (see [GME] 1.4 for more about functors and categories). For two aﬃne schemes S and T over B, a morphism φ : S → T is a family of maps φR : S(R) → T (R) indexed

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by B-algebras R such that the following diagram commutes for any B-algebra homomorphism α : R → R : φR

S(R) −−−−→ T (R)   α  α∗ ∗ S(R ) −−−−→ T (R ). φR

We write HomB (S, T ) for the set of all morphisms from S into T . By deﬁnition, we also have the following properties of aﬃne schemes: φ

ψ

→C − → D are B-algebra homomorphisms, then we have morphisms (cf1) If A − ψ

φ

of schemes SD − → SC − → SA such that φ ◦ ψ is associated to ψ ◦ φ. (cf2) If A = C and φ in (cf1) is the identity map iA of A, we have iA ◦ ψ = ψ. If C = D and ψ in (cf1) is the identity map iC of C, we have φ ◦ iC = φ. (cf3) For the identity map iA : A → A, iA : SA (R) → SA (R) is the identity map for all B-algebras R. Thus the functor A → SA is a contravariant functor from B-algebras into aﬃne schemes. One of the most basic facts in functorial algebraic geometry is (e.g. [GME] 1.4.3): HomB-alg (A , A) ∼ = HomB (SA , SA ) via α ↔ α.

(2.2.1)

The main point of the proof of this fact is to construct from a given morphism φ ∈ HomB (SA , SA ) a B-algebra homomorphism ϕ : A → A such that ϕ = φ. Exercise 2.11 Let ϕ = φA (iA ) ∈ SA (A) = HomB-alg (A , A), where iA ∈ SA (A) = HomB-alg (A, A) is the identity map. Then prove that ϕ = φ. A morphism φ : SA → SA is called ﬂat if M → M ⊗A ,φ A is a left exact functor from the category of A -modules to the category of A-modules. Here the functor M → M ⊗A ,φ A is left exact if it preserves injective morphisms of A-modules (that is, Ker(ι ⊗ 1 : M ⊗A ,φ A → M ⊗A ,φ A) = 0 if Ker(ι : M → M ) = 0). The morphism φ is faithfully ﬂat if it is ﬂat and 0 → M → M → M → 0 is exact if 0 → M ⊗A ,φ A → M ⊗A ,φ A → M ⊗A ,φ A → 0 is exact for any sequence M → M → M of A -modules. A morphism φ is called ﬁnite if the A -module A (via φ) is of ﬁnite type. If SA is ﬁnite ﬂat, localizations Am at each prime ideal m of A are free of ﬁnite rank over Am , and therefore, SA is also called locally free of ﬁnite rank over SA if φ is ﬁnite ﬂat. For a maximal ideal m of A, φ is smooth at m if SAm is ﬂat over SAm and for any A -algebra C with a square-zero ideal N , SA/mn (C) → SA/mn (C/N ) is surjective for all positive integers n. Here Am is the localization of A at m and Am is the localization of A at φ−1 (m). We call φ : SA → SA smooth if φ is

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smooth at all maximal ideals m of A. For a maximal ideal m of A, φ is ´etale at m if SAm is ﬂat over SAm and for any A -algebra C with a square-zero ideal N , SA/mn (C) → SA/mn (C/N ) is bijective for all positive integers n. We call φ : SA → SA ´etale if φ is ´etale at all maximal ideals m of A. Here are some examples of aﬃne schemes: Example 2.12 Take f (X, Y, Z) = X p + Y p − Z p for a prime p, and let B = Z. Then consider A = Z[X, Y, Z]/(f (X, Y, Z)). For each algebra R, we claim SA (R) ∼ = {(x, y, z) ∈ R3 |xp + y p = z p }. Indeed, for each solution P = (x, y, z) of Fermat’s equation in R, we deﬁne an algebra homomorphism φ : B[X, Y, Z] → R by sending polynomials Φ(X, Y, Z) to their value Φ(x, y, z) = φ(Φ) ∈ R. Since Φ ∈ (f (X, Y, Z)) ⇔ Φ = Ψf , we ﬁnd that φ(Ψf ) = Ψ(x, y, z)f (x, y, z) = 0; so, φ factors through the quotient A getting φ ∈ SA (R). In this way, we get an injection from the right-hand side to SA (R). If we start from φ : A → R in SA (R), we ﬁnd 0 = φ(0) = φ(X p + Y p − Z p ) = φ(X)p + φ(Y )p − φ(Z)p . Thus (x, y, z) = (φ(X), φ(Y ), φ(Z)) is an element in the right-hand side, getting the isomorphism. By Fermat’s last theorem, we have SA (Z) ∼ = {(a, 0, a), (0, b, b), (c, −c, 0)|a, b, c ∈ Z} if p is a prime ≥ 3. There is a simpler example: We have SZ[X1 ,...,Xn ] (R) = Rn via φ → (φ(X1 ), . . . , φ(Xn )). Thus often SZ[X1 ,...,Xn ] is written as Gna and is called the aﬃne space of dimension n. We write Ga for G1a . We have an algebra homomorphism B[X, Y, Z] → A for A in Example 2.12 sending Φ to (Φ mod f (X, Y, Z)). This in turn induces a morphism i : SA → G3a , which is visibly injective. When we have a morphism of aﬃne schemes φ : SA → SC , and if φ : C → A is a surjective ring homomorphism, we call φ a closed immersion. Then φR is injective (for any B-algebra R), and we can identify SA ⊂ SC all the time. In this case, SA regarded as a subfunctor of SC is called B-closed in SC . As we will see in Exercise 2.13(2), if " Si ⊂ SC is closed for a ﬁnite number of aﬃne schemes Si , the intersection R → i Si (R) is again an aﬃne closed subscheme. Thus we can give a topology on SC (R) for each R so that the closed set is given by the empty set ∅ and those of the form SA (R) for closed immersion SA → SC . This topology is called the Zariski topology of SC . Pick 0 = f ∈ A. Then A = A/(f ) is a surjective image of A. Thus SA ⊂ SA is a closed subscheme. For each point φ ∈ SA (R), f : φ → φ(f ) gives rise to a map fR : SA (R) → Ga (R) = R. This collection of maps {fR }R can be easily checked to be a morphism SA → Ga of functors, which we again call f : SA → Ga . In this way, we regard f ∈ A as a function well deﬁned on SA with values in

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Ga . Any B-algebra homomorphism (P : A → R) ∈ SA (R) factoring through A satisﬁes f (P ) = P (f ) = 0. Thus SA (R) = {P ∈ SA (R)|f / 0 (P ) = 0} (which is the 1 zero-set of f ). We consider the quotient ring Af = A . Then the natural map f b b → is a B-algebra homomorphism of A into Af ; so, SAf ⊂ SA . If P ∈ SA (R) 1 factors through Af , f ∈ Af is invertible in Af ; so, f (P ) = P (f ) ∈ R× . Thus SA = SA SAf , and hence SAf is an open subscheme of SA , giving outside the zero set of the function f . Any nonempty open aﬃne subscheme SA ⊂ SA can be given by a ring A inverting elements in A (there exist nonaﬃne open subschemes; see Exercise 2.17). We have a unique sheaf OSA of functions on SA such that OSA (SA ) = H 0 (SA , OS ) = A . Indeed, for any open subscheme U ⊂ SA , we have H 0 (U, OA ) = HomB (U, Ga ). This sheaf is called the structure sheaf of SA . Then the stalk at P ∈ SA is given by OSA ,P = limU H 0 (U, OS ), −→ where U runs over aﬃne open neighborhoods around P ∈ SA . If B is a B-algebra, we may regard A = B ⊗B A as a B -algebra by a → a⊗1. Then we get a new scheme SA over the ring B , which is sometimes written as SB ×B SA and is called the ﬁbered product of SA and SB over B. If we have a point φ ∈ SA (R) for a B -algebra R, we can extend φ : A → R to φ : A = B ⊗B A → R by φ (a ⊗ b) = aφ(b). Thus φ → φ gives the natural map SA (R) → SA (R) for all B -algebras R. This map is an isomorphism, because for any given φ ∈ SA (R), φ(b) = φ(1 ⊗ b) gives a point φ ∈ SA (R) as long as R is a B -algebra (Exercise 2.13(3)). However a B -closed subset of SA may not be B-closed; so, the Zariski topology depends on the base ring B.

Exercise 2.13 1. Prove that a closed immersion i : SA (R) → SC (R) gives rise to an injection for any B-algebra R. 2. Prove that if i : SA ⊂ SC and j : SD ⊂ SC are closed, then R → SA (R) ∩ SD (R) is closed in SC and is isomorphic to SE for E = A ⊗C D, where the tensor product is taken with respect to the associated algebra homomorphisms i : C → A and j : C → D. 3. Prove SA (R) ∼ = SA (R) if A = B ⊗B A and R is a B -algebra, where B is another B-algebra. Here the left-hand side is regarded as an aﬃne B-scheme and the right-hand side is regarded as an aﬃne B -scheme. 4. For two B-algebras A and C, show that SA⊗B C (R) = SA (R) × SC (R) for any B-algebra R. Hint: for φ ∈ SA (R) and ψ ∈ SC (R), we associate φ⊗ψ ∈ SA⊗B C (R) given by (φ ⊗ ψ)(b ⊗ c) = φ(b)ψ(c). Thus a product of aﬃne schemes is again an aﬃne B-scheme. The scheme SA⊗B C is called the ﬁbered product of SA and SC over B.

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2.2.2 Aﬃne algebraic groups Let G be an aﬃne scheme over a ring B. Thus G is a covariant functor from B-algebras to sets. If the values G(R) for all B-algebras R are groups (having values in the category GP of groups) and φ∗ : G(R) → G(R ) for any B-algebra homomorphism φ : R → R is a group homomorphism (i.e., G : B-ALG → GP is a covariant functor), G is called an aﬃne group scheme or an aﬃne algebraic group deﬁned over B. The scheme SA = µN for A = B[X]/(1+X)N −1) discussed at the beginning of Section 1.1 is an example of ﬁnite ﬂat (equivalently, locally free of ﬁnite rank) aﬃne group schemes, which send a B-algebra R to its group of N -th roots of unity. Similarly if an aﬃne scheme R/B is a covariant functor from the category of B-algebras into the category of rings, R is called an aﬃne ring scheme. Example 2.14 1. Let A = Z[X1 , . . . , Xn ]. Then SA (R) = Rn (as already remarked), which is an additive group. Since φ∗ (r1 , . . . , rn ) = (φ(r1 ), . . . , φ(rn )) for each algebra homomorphism φ : R → R , φ∗ is a homomorphism of additive groups. Thus Gna is an additive group scheme. 2. More generally, we can think of C = Z[Xij ] for n2 variables. Then SC (R) = Mn (R), and SC is not just a group scheme but is a ring scheme. This scheme is often written as Mn . As additive group schemes (ignoring ring structure), 2 we have Mn ∼ = Gna . 1 3. Consider the ring D = Z[Xij , det(X) ] for n2 variables Xij and the variable matrix X = (Xij ). Then SD (R) = GLn (R) and SD is a group scheme under matrix multiplication, which is a subscheme of SC because GLn (R) ⊂ Mn (R) for all R. This scheme SD is written as GL(n). In particular, SZ[t,t−1 ] = GL(1) is called the multiplicative group and written as Gm . 4. For a given B-module X free of rank n, we deﬁne XR = X ⊗B R (which is R-free of the same rank n) and . GLX (R) = α ∈ EndR (XR )∃ α−1 ∈ EndR (XR ) . Then GLX is isomorphic to GL(n)/B by choosing a coordinate system of X; so, GLX is an aﬃne group scheme deﬁned over a ring B. We can generalize this to a locally free B-module X, but in such a case, it is a bit more diﬃcult to prove that GLX is an aﬃne scheme. 5. We can then think of E = Z[Xij ]/(det(X) − 1). Then SE (R) = {x ∈ GLn (R)| det(x) = 1}. This closed subscheme of Mn and also of GL(n) is written as SL(n) and is a group scheme (under matrix multiplication) deﬁned over Z.

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6. Let X a free B-module of ﬁnite rank. We ﬁx a non degenerate bilinear form S : X × X → B. Then we consider G(R) = {α ∈ GLX (R)|SR (xα, yα) = SR (x, y) for all x, y ∈ XR } , where SR (r ⊗ x, s ⊗ y) = rsS(x, y) for r, s ∈ R and x, y ∈ X. To see that this G is an aﬃne algebraic group deﬁned over B, we ﬁx a base x1 , . . . , xn of X over B and deﬁne a matrix S by S = (S(xi , xj )) ∈ Mn (B). Then every (ij) entry sij (X) of the matrix XS · t X − S (X = (Xij )) is a quadratic polynomial with coeﬃcients in B. Then we consider L = B[Xij , det(X)−1 ]/(sij (X)). By deﬁnition, . SL (R) = α ∈ GLX (R)αS t α = S ∼ = G(R).

7.

8.

9.

10.

We ﬁnd αS t α = S ⇒ S = α−1 S · t α−1 ; so, the inverse exists, and G is an aﬃne algebraic group. If X = B n and S(x, y) = xS t y for a nondegenerate symmetric matrix S, G as above is written as OS/B and is called the orthogonal group of S. If X = Y × Y and S is nondegenerate skew symmetric of the form S((y, y ), (z, z )) = T (y, z ) − T (z, y ) for a symmetric bilinear form T : Y × Y → B, we write G = SpT /B . In particular, if 0 −1n t y, the group G often written as Spn/B is called S(x, y) = x 1n 0 the symplectic group of genus n. We consider a quadratic polynomial f (T ) = T 2 + aT + b ∈ Z[T ]. Then deﬁne Sf (R) = Ga (R)[T ]/(f (T )). As a scheme Sf ∼ = G2a but its value is a ring all the time. If φ : R → R is an algebra homomorphism, φ∗ (r + sT ) = φ(r) + φ(s)T ; so, it is a ring homomorphism of Sf (R) = R[T ]/(f (T )) into Sf (R ) = R [T ]/(f (T )). Thus Sf√is a ring scheme, and writing O for the order of the quadratic ﬁeld Q[ a2 − 4b] generated by the root of f (T ), we have Sf (R) ∼ = R ⊗Z O. Since any given number ﬁeld F is generated by one element, we know F = Q[T ]/(f (T )) for an irreducible monic polynomial f (T ). For any Q-algebra R, deﬁne Sf (R) = R[T ]/(f (T )). Then in the same way as above, Sf is a ring scheme deﬁned over Q such that Sf (R) = F ⊗Q R. Let G be an aﬃne algebraic group deﬁned over a number ﬁeld F . Then we deﬁne a new functor G deﬁned over Q-algebras R by G (R) = G(Sf (R)) = G(F ⊗Q R). We can prove that G is an aﬃne group scheme deﬁned over Q, which we write G = ResF/Q G (see Exercise 2.15 (3)). Assume that f is a quadratic polynomial in Q[T ]. Then we have that Sf (Q) = F is a quadratic extension with Gal(F/Q) = {1, σ}. Let X be a ﬁnite-dimensional vector space over Q and let Gal(F/Q) act on XF = F ⊗Q X through F . We suppose we have a hermitian form H : XF ×XF → F such that H(x, y) = σ(H(y, x)). Then for the Q-algebra R . UH (R) = α ∈ GLX (Sf (R))HSf (R) (xα, yα) = HSf (R) (x, y)

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is an aﬃne algebraic group, which is called the unitary group of H. Note that UH is deﬁned over Q (not over F ). For two aﬃne algebraic groups G, G deﬁned over B, we write HomB-alg gp (G, G ) . = φ ∈ HomB (G, G )φR is a group homomorphism for all R . (2.2.2) Exercise 2.15 1. Let F be a number ﬁeld with the integer ring O. Is there any aﬃne ring scheme S deﬁned over Z such that S(R) = O ⊗Z R? 2. Let S : X × X → B is a bilinear form for a B-free module X, and suppose that X ∼ = HomB (X, B) by S. Then the matrix of S is in GLn (B) for any choice of basis of X over B. 3. For an aﬃne algebraic group G over a number ﬁeld F (that is, a ﬁnite extension of Q), prove that ResF/Q G is an aﬃne algebraic group deﬁned over Q. 4. Show that the unitary group UH as above is an aﬃne algebraic group. More generally than the above Exercise 2.15(3), we start with an aﬃne group scheme H over a ring R . For a subalgebra R of R , if the covariant functor C → H(C ⊗R R ) deﬁned on the category of R-algebras is isomorphic to a , we write H/R = ResR /R H and call it the Weil restriction of H scheme H/R with respect to R /R (this is not changing the base ring of H/R to the subalgebra R). Since C → H(C ⊗R R ) is a group functor, ResR /R H is a group scheme if it exists. Theorem 2.16 Let the notation be as above. If R /R is locally R-free of ﬁnite rank, an aﬃne group scheme ResR /R H always exists. For a proof, see [NMD] 7.6, Theorem 4. 2.2.3 Schemes Let f : S1 → S and g : S2 → S be morphisms of covariant functors S and Sj (j = 1, 2) of B-ALG into SET S. We then deﬁne S1 ×S S2 (R) = {(x, y) ∈ S1 (R) × S2 (R)|fR (x) = gR (y)}, which is a covariant functor from B-ALG into SET S. If T is the fourth functor with morphisms f : T → S1 and g : T → S2 such that g ◦ g = f ◦ f , we have a unique morphism h : T → S1 ×S S2 given by h(x) = (f (x), g (x)) such that f ◦ h = g ◦ h. In this sense, S1 ×S S2 satisﬁes the universality of the ﬁbered product over S. Here fR : S1 (R) → S(R) is the map induced by the functor morphism f (see the beginning of Section 1.1 for the deﬁnition of morphisms

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of covariant functors). A (noetherian) scheme S/B deﬁned over B is a covariant functor from the category B-ALG of B-algebras into SET S such that 1. there exist ﬁnitely many aﬃne schemes Si = Spec(B i ) for noetherian B-algebras Bi with inclusion Si → S such that S = i Si ; 2. Si ∩ Sj = Si ×S Sj is Zariski open in Si and Sj for all pairs (i, j) of indices. In practice the scheme S is often constructed by gluing the aﬃne schemes Si , and each φ ∈ S(R) is determined by its “restriction” φi to Si in the following way: For each B-algebra R, φ ∈ S(R) is determined by the tuple {φi ∈ Hom(SR ×S Si , Si )} such that φi and φj induces an identical morphism of SR ×S Si ×S Sj → Si ×S Sj (ignoring the indices i with SR ×S Si = ∅); then, we may deﬁne & % 3 φi ∈ Hom(SR ×S Si , Si ), and S(R) = . Si := (φi )i φi and φj coincide on SR ×S Si ×S Sj for all i, j i

A B-morphism φ : S → T of schemes is a morphism of the covariant functor S : B-ALG → SET S into the covariant functor T : B-ALG → SET S. Thus the totality of B-schemes forms a category (denoted by B-SCH) which contains the category of aﬃne B-schemes as a full subcategory. Let φ : S → T be a morphism of B-schemes. Covering T by aﬃne schemes {SAi }i for B-algebras Ai , that is, T = i SAi , then φ−1 (SAi ) can be covered again by suitable aﬃne schemes SAij for Ai -algebras Aij . If SAij is smooth (resp. ´etale) over SAi for all i and j, we call S is smooth (resp. ´etale) over T . In order to give a typical example of non-aﬃne schemes, we introduce graded algebras over a commutative algebra B. A graded algebra R over an algebra B is a direct sum d Rd for a B-subalgebra R0 and R0 -modules Rd for integers d such that Rn Rm ⊂ Rm+n . A B-morphism φ : R → R of graded B-algebras R and R is a B-algebra homomorphism with φ(Rd ) ⊂ Rd for all d. An element x of a graded algebra R is called homogeneous of degree d if x ∈ Rd . The polynomial ring B[T ] is a graded algebra with B[T ]d = T d B for d ≥ 0 and Bd = 0 for negative d. More generally, the polynomial ring B[T1 , . . . , Tn ] of n variables is a graded algebra such that each monomial of degree d/is 0homogeneous of degree d. 1 is a graded ring by putIf x ∈ R is homogeneous of degree d = 0, then R x / 0 1 = xj Rm (in other words, for a homogeneous element ting R x n# $ jd+m=n a a ∈ R, deg j = deg(a) − jd). Suppose that the base ring B is noetherian. x If a graded algebra R is noetherian, then there are ﬁnitely many homogeneous elements x0 , . . . , xn (of degree d0 , . . . , dn , respectively) in R which generate the B-algebra R. Thus the algebra homomorphism B[T0 , . . . , Tn ] → R sending a polynomial P (T0 , . . . , Tn ) to P (x1 , . . . , xn ) ∈ R is a surjective algebra homomorphism.

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Write Di = Spec(B[T0 , . . . , Tn ][ T1i ]0 ). Since we have B[T0 , . . . , Tn ][Ti−1 ]0 ∩ B[T0 , . . . , Tn ][Tj−1 ]0 = B[T0 , . . . , Tn ][(Ti Tj )−1 ]0 for i = j (in B[T0 , . . . , Tn ][ T10 , T11 , . . . , T1n ]0 ), we may identify Di ∩ Dj with Spec(B[T0 , . . . , Tn ][(Ti Tj )−1 ]0 ) canonically. n In this way, we can deﬁne the projective space of dimension n by Pn = j=0 Dj , which is not aﬃne. If A is a B-algebra which is either a ﬁeld or a valuation ring, by deﬁnition, we have Pn (A) ∼ = {(x0 , . . . , xn )|xj ∈ A× for at least one j}/A× . If R is a general noetherian graded B-algebra, taking a ﬁnite set of homogeneous generators x0 , . . . , xn of degree 1, taking the surjective B-algebra homomorphism B[T0 , . . . , Tn ] sending Tj to xj , the algebra homomorphism gives rise induces a surjection B[T0 , . . . , Tn ][Ti−1 ]0 → R[x−1 i ]0 , which in turn ] ) of D . Then we deﬁne Proj(R) = to a subscheme Vi = Spec(R[x−1 0 i i i Vi . We can generalize this deﬁnition to any graded B-algebra R generated by ﬁnitely many homogeneous elements (not necessarily of degree 1; see [GME] Section 1.3), because Proj(R) = Proj(R(n) ) for R(n) = d Rnd . If R× contains a homogeneous element of nonzero degree, we have Proj(R) ∼ = Spec(R0 ) by deﬁnition; however, if R has no negative degree elements, Proj(R) is not aﬃne. If a scheme G/B as a functor induces a covariant functor from B-ALG into the category GP of groups, G is called a group scheme. Then the group structure gives rise to morphisms of schemes, for example, the group multiplication induces the multiplication morphism m : G × G → G and the existence of an identity can be formulated to be the existence of a closed immersion Spec(B) → G, which satisﬁes the group law. For example, associativity is equivalent to the commutativity of the following diagram (x,y,z)→(xy,z)

G × G × G −−−−−−−−−−→ G × G   m  (x,y,z)→(x,yz) G×G

−−−−→ m

G.

Exercise 2.17 Let B = C and A = C[X, Y ] (the polynomial ring of the indeterminates X and Y ). Let C = A/(X, Y ). Then SC is a closed subscheme of SA . Prove that SA − SC is a scheme but is not aﬃne. 2.3 Automorphic forms on quaternion algebras In this section, we recall the deﬁnition of holomorphic automorphic forms on the multiplicative idele group of a quaternion algebra. Following the tradition of Gauss, Eisenstein, Kronecker, and Hilbert, if D = M2 (F ) (for a totally real ﬁeld F ), such functions are called modular forms. On the other hand, general quaternionic cases are more recent, for which we use a more general term: automorphic forms (see [Ca] for the distinction of modular and automorphic forms).

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2.3.1 Arithmetic quotients Let F be a totally real ﬁeld and I be the total set of embeddings of F into Q (the algebraic closure of Q in C). Recall the embeddings i∞ : Q → C and ip : Q → Qp ﬁxed in the introduction. Since F is totally real, i∞ ◦ σ has an image in R for all σ ∈ I. As before, O denotes the integer ring of F . We ﬁx a quaternion algebra D/F . Let ID be a subset of I consisting of σ : F → R such that at the inﬁnite place i∞ ◦ σ (which we write again as σ for simplicity), Dσ = D ⊗F,i∞ ◦σ R ∼ = M2 (R). We write ΣD for the set of all places at which D ramiﬁes and I D for the subset of ΣD consisting of inﬁnite places (thus I = ID I D ). We ﬁx a maximal order OD of D. Take a quadratic extension K/F inside D such that Kσ = K ⊗F Fσ ∼ = (R ⊕ R) if σ ∈ ID , and ﬁx ρ : (D ⊗F K) ∼ = M2 (K). D ∼ Then, Kσ = C if σ ∈ I . We may assume that ρ(OD ) is contained in M2 (OK ) for the integer ring OK , because every O-order in M2 (K) is contained in an adelic conjugate of M2 (OK ) (Proposition 2.6). Since OD is a Z-lattice of D∞ = D⊗Q R, × × is a discrete subset of D∞ . Let OD is a discrete subset of D∞ . Then we see OD 1 D∞ = {x ∈ D∞ |N (x) = 1}. We know that the natural map: 1 × × D∞ × F∞ → D∞ : (x, y) → xy (1) (1) × 1 has ﬁnite kernel (∼ ∩D∞ . Therefore OD is = {±1}I ) and cokernel. Put OD = OD × × 1 1 × discrete in D∞ . The natural map from D∞ to P D∞+ = D∞+ /F∞ is surjective × × is the subgroup of D∞ consisting and has ﬁnite kernel (again {±1}I ), where D∞+ (1) × of elements with totally positive norm. The image of OD in P D∞+ is discrete. We have an exact sequence: × 1 → OD → OD −→ O× → Coker(N ) → 1. (1)

N

The image of the norm map N contains (O× )2 . Since O× is a ﬁnitely generated abelian group by Dirichlet’s theorem (e.g., [LFE] Theorem 1.2.3), (O× : (O× )2 ) × is ﬁnite (a power of 2). Thus Coker(N ) is ﬁnite. This shows that the image OD+ (1) × × × × × of OD+ = OD ∩ D∞+ in P D∞+ (isomorphic to OD /O× ) has the image of OD ×

× as a subgroup of ﬁnite index. Thus OD+ is discrete in P D∞+ . Let C∞ be the 1 maximal compact subgroup of D∞ . Since D 1 ∼ D∞ = SL2 (R)ID × H1 I D for H1 = {x ∈ H|N (x) = 1}, we have C∞ ∼ = SO2 (R)ID × H1 I . Thus

× × 1 D∞+ /F∞ C∞ = D∞ /C∞ ∼ = H ID √ √ via g → g(i) for i = ( −1, . . . , −1) ∈ HID (H = {z ∈ C| Im(z) > 0}). For (1) (1) simplicity, we write ZD for HID . Since OD is discrete, OD ∩ C∞ is discrete and compact and hence ﬁnite. In particular, if ID = ∅, the reduced norm map N : D∞ → R is a positive deﬁnite quadratic form (on H/R , it is the sum of four

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squares), and C∞ is compact. Thus we have × /O× is a ﬁnite group. If ID = ∅, OD

(2.3.1)

× We say that a subgroup Γ of P D∞+ acts properly discontinuously on ZD , if for any point z ∈ ZD , we can ﬁnd an open neighborhood U of z such that {γ ∈ Γ|γ(U ) ∩ U = ∅} is ﬁnite. If this set is a singleton made up of the identity element for all z, the action is called free. The quotient Γ\ZD for Γ acting freely is a complex manifold. When the action is only properly discontinuous, the quotient is a complex analytic space (locally isomorphic to a zero set of ﬁnitely many (1) complex analytic functions in Cn ). The group OD acts properly discontinuously × (1) on ZD (cf. [MFM] Section 1.5) because OD ∩ C∞ is ﬁnite, and hence OD+ also acts properly discontinuously on ZD . We now claim: (1)

×

Proposition 2.18 For any subgroup Γ of ﬁnite index of OD or OD+ , the quotient analytic space Γ\ZD is compact if D is a division algebra. (1)

Proof We only need to prove the proposition for OD \ZD because Γ\ZD is (1) a covering of OD \ZD with ﬁnite ﬁber (the number of elements in the ﬁber is × × (1) less than or equal to the index of Γ in OD or OD+ ) and OD+ \ZD is covered by . (1) OD \ZD . We know that D× \DA1 is compact for DA1 = x ∈ DA |N (x)|A = 1 (see (1) (1) 2.1.2). Thus D(1) \DA /U (1) C∞ is compact, where DA = {x ∈ DA |N (x) = 1}, × (1) (1) U = p OD,p , U (1) = DA ∩ U and D(1) = DA ∩ D. We consider the map ι : (1)

ZD → DA /U (1) C∞ given by ι(z) = g∞ mod C∞ for g∞ with g∞ (i) = z. Then (1) it is easy to see that ι induces an inclusion OD \ZD into D× \DA1 /U C∞ . By the strong approximation theorem, ι is surjective and hence an isomorphism. 2 Exercise 2.19 Prove that the map ι in the above proof is an embedding. By the above proposition, if D is a division algebra, Γ\ZD has no cusps. Let . × × γ − 1 ∈ NOD (N) = γ ∈ OD OD × (N) ∩ OD . We put Γ(N) to be the for each ideal N of O and OD (N) = OD × (1) × × (N) in P D∞+ . Since OD acts intersection with OD = Γ(1) with the image of OD (1) properly discontinuously on ZD , for each point z ∈ ZD , the stabilizer OD,z = (1)

(1)

(1)

{γ ∈ OD |γ(z) = z} of z is a ﬁnite group. In particular, when D is deﬁnite, Γ(N) (1) (1) and OD are ﬁnite. Take a non-central element ζ ∈ OD,z . Then ζ m = 1 for some m > 2. Thus F [ζ] is a totally imaginary quadratic extension of F in D. There are only ﬁnitely many such quadratic extensions over F generated by roots of unity. Thus the order m of ζ is bounded independently of z. Since m-th roots of unity for a given m > 2 can be separated modulo N for suﬃciently small ideal N

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(1)

(1)

in O, OD (N) acts ﬁxed-point free on ZD . Any subgroup of Γ containing OD (N) for some ideal N is called a congruence subgroup of D× . Thus (FR) We can ﬁnd a congruence subgroup acting on ZD without ﬁxed points. Actually we can give an exact lower bound for N when Γ(N) acts freely on ZD (1) (e.g., [H88b] Lemma 7.1]). In particular, if N ⊂ (3), OD (N) acts freely on ZD . A natural question is (1)

Are all subgroups of ﬁnite index of OD congruence subgroups.

(CS)

This is the congruence subgroup problem for D× (see [Rk] for a survey of the problem for general semisimple groups). It is conjectured by Serre that if the number r = |ID | of inﬁnite places of F unramiﬁed in D is bigger than or equal to 2, the answer should be aﬃrmative. Serre proved this to be aﬃrmative for M2 (F ) in [Se] if F = Q (that is, r = [F : Q] ≥ 2). When r = 1, for small enough N, X = Γ(N)\ZD is a compact Riemann surface of genus g ≥ 1, and H 1 (X, Z) = Hom(π1 (X), Z) ∼ = Z2g . Thus the maximal abelian quotient π1ab (X) of the fundamental group π1 (X) is inﬁnite. On the other hand, it is easy to show that the maximal abelian quotient of Γ(N) is ﬁnite if (CS) is aﬃrmative in this case. Thus (CS) has a negative answer when r = 1. Exercise 2.20 Prove that the maximal abelian quotient of Γ(N) is ﬁnite, assuming (CS) is aﬃrmative. We associate with the algebra D an algebraic group D× deﬁned over F . As a group functor, D× (R) = (D ⊗F R)× for all F -algebra R. Then we consider an algebraic group GD deﬁned over Q given by GD = ResF/Q D× . If we ﬁx a maximal order OD , we can extend D× to a group scheme deﬁned over the integer ring O of F by D× (R) = (OD ⊗O R)× for all O-algebras R. Thus GD extends × to a group scheme over Z by taking GD /Z = ResO/Z D /O (see Theorem 2.16). The D center Z of G is an algebraic group satisfying Z(A) = (O ⊗Z A)× ; so, it is independent of D and Z = ResO/Z Gm/O . Exercise 2.21 Prove that the functor D× /F is an aﬃne algebraic group over F . Let GD (R)+ be the identity connected component of the real Lie group GD (R); then, GD (R)+ = {x ∈ GD (R)|N (x) ! 0}. We let g ∈ GD (R) act on ZD = HID by the linear fractional transformation of gσ = σ(g) ∈ GL2 (K ⊗K,σ R) = GL2 (R) √ component-wise. Write Cσ+ for the stabilizer of −1 in (D⊗F,σ R)× and deﬁne a D D closed subgroup Ci ⊂ G (R) by Z(R)· σ∈ID Cσ+ × σ∈I D H× , which is the √ √ stabilizer of i = ( −1, . . . , −1) ∈ HID in the connected component GD (R)+ . Thus we have ZD = HID ∼ = GD (R)+ /CiD by g(i) ↔ g for the identity-connected D + D component G (R) of G (R). Write simply G = GM2 (F ) = ResO/Z GL(2). Since

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G and GD have the common center Z canonically isomorphic to ResF/Q Gm , we use the same symbol Z to indicate the center of GD independently of D. For any open compact subgroup S ⊂ GD (A(∞) ) under the adele topology, we think of the automorphic manifold associated with the subgroup S: Y (S) = Y D (S) := GD (Q)