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)\GD (A)/Z(A)S · CiD . Exercise 2.22 Let Γg (S) = g −1 S · GD (R)+ g ∩ D× . 1. Show that Γg (S) is a discrete subgroup of GD (R). 4 2. Prove that Y (S) ∼ Γg (S)\ZD via the isomorphism = g

G (Q)\G (Q)gZ(A)S · GD (R)+ /Z(A)S · CiD ∼ = Γg (U )\ZD 4 × given by gx∞ → (gx)∞ (i) ∈ ZD if DA× = g GD (Q)g · Z(A)S · D∞+ by Theorem 2.8. 3. Prove that Y (S) is a complex analytic space of dimension r = |ID | and is a complex manifold if S is suﬃciently small. D

D

2.3.2 Archimedean Hilbert modular forms Let us recall the deﬁnition of the adelic Hilbert modular forms and their Hecke ring of level N for an integral ideal N of F (cf. [PAF] Chapter this

ι 4). Thus in a b d −b subsection, D = M2 (F ) for a totally real ﬁeld F , and = . c d −c a Let G = ResO/Z GL(2) as an algebraic group over Z; G(A) = GL2 (A ⊗Z O) for each commutative ring A. Let T0 = G2m/O be the diagonal torus of GL(2)/O , and put T = ResO/Z Gm and TG = ResO/Z T0 . Then TG contains the center Z of G. Write I = Homﬁeld (F, Q). Then the set of algebraic characters X(TG ) = Homalg gp (TG/Q , Gm/Q ) can be identiﬁed with Z[I]2 so that κ = (κ1 , κ2 ) ∈ Z[I]2 induces the following character on TG (Q) = F × × F × : ×

TG (Q) (ξ1 , ξ2 ) → κ(ξ1 , ξ2 ) = ξ1κ1 ξ2κ2 ∈ Q , × where ξ κj = σ∈I σ(ξj )κj,σ ∈ Q . We consider the following set of continuous “Neben” characters → C× , ε+ : Z(A)/Z(Q) → C× ). ε = (ε1 , ε2 : T (Z) → C× is continuous, it is of ﬁnite order, and we have If a character ψ : T (Z) an ideal c(ψ) maximal among integral ideals c satisfying ψ(x) = 1 for all x ∈ =O We call c(ψ) the conductor of ψ. × with x − 1 ∈ cO. T (Z) → C× is of ﬁnite Exercise 2.23 Prove that a continuous character ψ : T (Z) order (see [MFG] Proposition 2.2). The character ε+ : Z(A)/Z(Q) → C× is an arithmetic Hecke character such that and ε+ (x∞ ) = x−(κ1 +κ2 )+I . We can deﬁne the ε+ (z) = ε1 (z)ε2 (z) for z ∈ Z(Z)

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conductor c(ε+ ) in the same manner as above taking the restriction of ε+ to ∼ We deﬁne c(ε) = c(ε1 )c(ε2 ) ⊂ c(ε+ ). Z(Z) = T (Z). The standard level group of Γ0 (N)-type for an integral ideal N is given by & %

a b (2.3.2) ∈ G(Z) c ∈ NO . Γ0 (N) = c d We also deﬁne the principal congruence subgroup & %

a b a − 1, b, c, d − 1 ∈ NO . Γ(N) = ∈ G(Z) c d Exercise 2.24 Show that for any integral ideal 0 = N and any a ∈ G(A(∞) ), −1 ) ⊂ aΓ(N)a . there exists an integral ideal 0 = N ⊂ N such that Γ(N with dO = δ O. Let d be the absolute diﬀerent of F , and choosing an idele δ ∈ O We then deﬁne a variant of Γ0 (N):

−1

δ 0 δ 0 Γ0 (N) . (2.3.3) S0 (N) = 0 1 0 1 This type of level group has often been used by Shimura (e.g., [Sh5]). Hilbert modular forms on the level group S0 (N) have a simpler form of Fourier expansion 0 (N) in [SGL]. than those of level Γ

a b a b If c(ε− ) ⊃ N for ε− = ε−1 ε , →

ε = ε2 (ad − bc)ε− (aN ) 1 2 c d c d is a continuous character of the compact group S0 (N) (this type of “Neben” character was ﬁrst considered in [H89b] and [H91]). Here aN is the projection of a to the product FN of Fl over all prime factors l of N. Exercise 2.25 Show that ε(uu ) = ε(u)ε(u ) for u, u ∈ S0 (N) if c(ε− ) ⊃ N. We deﬁne the automorphy factor Jκ (g, z) of weight κ for z ∈ Z = HI by Jκ (g, z) = det(g)κ1 −I j(g, z)κ2 −κ1 +I for g ∈ G(R) and z ∈ Z.

(2.3.4)

Here j(g, z) = (cσ zσ + dσ )σ∈I ∈ CI = F ⊗R C, writing g = (gσ ) ∈ GL2 (R)I = GL2 (F∞ ) and z = (zσ ) ∈ Z. The power j(g, z)κ2 −κ1 +I is an abbreviation of κ2,σ −κ1,σ +1 , and similarly det(g)κ1 −I = σ det(gσ )κ1,σ −1 . Then σ (cσ zσ + dσ ) we deﬁne Sκ (N, ε; C) to be the space of functions f : G(A) → C satisfying the following three conditions (e.g., [SGL] Section 2.2 and [PAF] 4.3.1). A function in Sκ (N, ε; C) is called a Hilbert cusp form of level N and with character ε. (SA1) We have the following automorphy f (αxuz) = ε+ (z)ε(u)f (x)Jκ (u∞ , i)−1 for all α ∈ G(Q), z√∈ Z(A),√and u ∈ S0 (N)Ci for the stabilizer Ci in G(R)+ of i = ( −1, . . . , −1) ∈ Z = HI , where G(R)+ is the identity-connected component of G(R);

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(SA2) Choosing u ∈ G(R) with u(i) = z for each z ∈ HI , deﬁne a function fg : Z → C by fg (z) = f (gu∞ )Jκ (u∞ , i) for each g ∈ G(A(∞) ). Then fg is a holomorphic function on Z for all g;

1 u (∞) x du = 0 for all x ∈ GL2 (FA ). This is equivalent (SA3) FA /F f 0 1 to the following statement: The function: z → fg (z)| σ∈I Im(zσ )kσ /2 | is bounded independently of z ∈ Z, where kσ = κ2,σ − κ1,σ + 1. Here “du” is an additive Haar measure on FA /F . Replacing the word “bounded” in (SA3) by “slowly increasing” with polynomial growth as Im(z) → ∞, we deﬁne a larger space of modular forms Gκ (N, ε; C). The function fg in (SA2) satisﬁes the classical automorphy condition fg (γ(z)) = ε−1 (g −1 γg)fg (z)Jκ (γ, z) for all γ ∈ Γ0,g (N),

(2.3.5)

where Γ0,g (N) = g ·S0 (N)g −1 G(R)+ ∩G(Q), and G(R)+ is the subgroup of G(R) made up of matrices with totally positive determinant. Indeed, we have under the notation in (SA2), fg (γ(z)) = f (g(γu)∞ )Jκ ((γu)∞ , i) = f (γγ (∞) =ε

−1

(g

−1

−1

g · u∞ )Jκ ((γu)∞ , i)

γg)f (gu∞ )Jκ (γ, u∞ (i))Jκ (u∞ , i) = ε−1 (g −1 γg)fg (z)Jκ (γ, z).

The same computation applied to α ∈ G(Q)+ yields fg (α(z))Jκ (α, z)−1 = fα(∞) −1 g (z).

(2.3.6)

By (SA3) combined with (2.3.6), we conclude fg is decreasing rapidly towards 0 (N)g −1 ⊃ Γ(N ) for a suitable integral all cusps of Γ0,g (N). Since we have g Γ ideal N , the discrete congruence subgroup Γ0,g (N) contains & %

1 a a∈a Γ∞ (a) = 0 1

1 a for a suitable ideal a ⊃ N . The action of on Z is given by z → z + a; 0 1 so, fg in (2.3.5) satisﬁes fg (z + a) = fg (z) for all a ∈ a. Since F∞ /a is a compact abelian group isomorphic to (R/Z)d , we can apply the standard Fourier analysis for the group F∞ /a (e.g., [LFE] Section 8.4), and we get the following Fourier expansion of fg locally uniformly (and absolutely) convergent over Z:

a(ξ, fg )q ξ , fg (z) = a∈a∗

where q ξ is an abbreviation of exp(2πi

σ

ξ σ zσ ) and

a∗ = a−1 d−1 = {ξ ∈ F |TrF/Q (ξO) ⊂ Z}. Since fg decreases as Im(zσ ) → ∞ uniformly for σ ∈ I, we have a(ξ, f ) = 0 if σ(ξ) ≤ 0 for one embedding σ : F → R. This shows that fg decreases

102

Automorphic forms on inner forms of GL(2)

exponentially as Im(zσ ) → ∞ uniformly for σ ∈ I. Writing a∗+ for the subset of a∗ made up of totally positive elements (that is, elements ξ with σ(ξ) > 0 for all σ ∈ I), we thus have

fg (z) = a(ξ, fg )q ξ (2.3.7) a∈a∗ +

for f ∈ Sκ (N, ε; C). If f ∈ Gκ (N, ε; C), the expansion can have a nontrivial ), for a subgroup E of 0 (N)g −1 ⊃ Γ(N constant term a(0, fg ). Again by g Γ × ﬁnite index

in O (on which ε is trivial), Γ0,g (N) contains diagonal matrices 0 for all ∈ E. The eﬀect of these matrices on fg is given by 0 −1 fg (2 z) = κ1 −κ2 −I fg (z) by (2.3.5). Thus we have a(−2 ξ, fg ) = κ1 −κ2 −I a(ξ, fg )

(2.3.8)

for a suﬃciently small subgroup E ⊂ O× of ﬁnite index, and we conclude Gκ (N, ε; C) = Sκ (N, ε; C) unless κ1 − κ2 ∈ Z · I.

(2.3.9)

× → R× ) of the Indeed, E contains a basis over R of the kernel Ker(N : F∞+ × is the norm map N by Dirichlet’s theorem. Here F∞ = F ⊗Q R, and F∞+ × identity connected component of the multiplicative group F∞ . Similarly we have Gκ = 0 unless κ1 + κ2 = [κ1 + κ2 ]I

for [κ1 + κ2 ] ∈ Z. α 0 To see this, we note by (SA1), for scalar matrices with α ∈ F × , f ∈ 0 α Gκ (N, ε; C) satisﬁes f (x) = f (αx) = ε+ (α(∞) )ακ1 +κ2 −I f (x) by (2.3.5). Thus ε+ (α(∞) )ακ1 +κ2 −I has to be equal to 1 for all α ∈ F × , which implies the inﬁnity × , ε+ is of ﬁnite order m, type of the Hecke character ε+ is κ1 + κ2 − I. On O × (∞) m and hence for ∈ O , we have ε+ ( ) = 1 and hence m(κ1 +κ2 −I) = 1 for all ∈ O× . Again by Dirichlet’s theorem, we get

Gκ = 0 ⇒ κ1 + κ2 ∈ Z · I and ε+ (α(∞) )ακ1 +κ2 −I = 1 for all α ∈ F × . (2.3.10) We hereafter simply write [κ] for [κ1 + κ2 ] ∈ Z if κ1 + κ2 ∈ Z · I. Also, by (2.3.9) and (2.3.10) combined, Gκ = Sκ implies that κ1 ∈ Z · I; so, in this case, we write κ1 = [κ1 ]I for [κ1 ] ∈ Z. ∩ G(A(∞) ) by We deﬁne the level N semigroup ∆0 (N) ⊂ M2 (O) %

& a b × (∞) ∈ M2 (O) ∩ G(A ) aN ∈ ON , c ∈ NO . (2.3.11) ∆0 (N) = c d Here ON is the product of Ol over all prime factors l of N. The opposite semigroup ∆∗0 (N) is deﬁned to be the image of ∆0 (N) for the involution ι of M2 (F ) with x + xι = Tr(x). Thus %

& a b ∩ G(A(∞) )dN ∈ O× , c ∈ NO . ∆∗0 (N) = ∈ M2 (O) (2.3.12) N c d

Automorphic forms on quaternion algebras

103

We ﬁx a prime p, and write v for one of the rational places p or ∞. Let W be a p-adic valuation ring of a number ﬁeld inside C containing the values of ε on and Z(Z) and all the conjugates of O in Q. We ﬁx once and for all a prime T (Z) element q for each prime ideal q. Here we choose q inside W ∩ F . We extend our Neben character ε = (ε1 , ε2 , ε+ ) to T (A(∞) ) and ∆0 (N) as in [PAF] 4.2.6 as follows. (ex0) For each w ∈ Z[I], we extend the character xp → xw p of T (Zp ) to T (Qp ) w Z trivially on and write this extension as x

→ xp p , where xw p p = p|p p wp wσ × σ(x ) for x ∈ O = T (Z ). Thus p is a p-adic unit. This only p p p σ∈I applies to the place v = p, because x ∈ T (Z) has trivial component x∞ = 1 w ∞ at inﬁnity. In particular, xw ∞ := x∞ = 1 for any x ∈ ∆0 (N) and any w ∈ Z[I] in the following conditions. (ex1) We extend ε2 to the idele group T (A(∞) ) trivially on q qZ and then extend ε1 to T (A(∞) ) by ε1 ε2 (x) = ε+ (x(∞) ). We put ε− (a) = ε−1 2 (a)ε1 (a) for a 0 , we get an a ∈ T (A(∞) ). Thus identifying T 2 ∼ = TG by (a, d) → 0 d extension of ε to TG (A). ∩ S0 ((v) ∩ N) to a character of the (ex2) We extend the character ε of TG (Z) semigroup ∆0 ((v) ∩ N) by

v εv∆ (δ) = det(δ)−I ε2 (det(δ))ε− v (a(v)∩N ) v

a b ∈ ∆0 ((v) ∩ N) ∪ ∆0 ((v) ∩ N)ι . Here (v) ∩ N = (p) ∩ N if c d v = p and (v) ∩ N = N if v = ∞. −2Iv (ex3) Since the character Z(A(∞) )/Z(Q) z → |z (∞) |−2 ε+ (z) and εv∆ on A zv S0 ((v) ∩ N) coincide on S0 ((v) ∩ N) ∩ Z(A(∞) ), we may extend εv∆ |S0 ((v)∩N) to εS = εvS : Z(A)S0 ((v) ∩ N) = Z(A)S1 ((v) ∩ N) → B × by εvS (zs) = −2Iv ε+ (z)εv∆ (s) for z ∈ Z(A) and s ∈ S1 ((v) ∩ N). |z (∞) |−2 A zv for δ =

(N) Since ε∞ )) on ∆ is deﬁned over ∆0 (N) and coincides with z → ε+ (zN )ε2 (det(z (∞) (∞) Z(A )∩∆0 (N), we may extend it to the subgroup generated by Z(A )∆0 (N) (which contains ∆∗0 (N)) so that ε(zδ) = ε+ (zN )ε2 (det(z (N) ))ε∞ ∆ (δ) for δ ∈ ∆0 (N) and z ∈ Z(A(∞) ). For each y ∈ ∆0 (N), we can decompose 5 S0 (N)y ι S0 (N) = utS0 (N) (2.3.13) u,t

and t ∈ TG (A(∞) ) with det(t) = det(y) (see (2.3.33)). for ﬁnitely many u ∈ U (Z) Deﬁne

(∗) ι f |[S0 (N)y ι S0 (N)](g) = ε∞ ε(ut)−1 f (gut). ∆ ((ut) )f (gut) = ε(det(y)) u,t

h

(2.3.14)

Automorphic forms on inner forms of GL(2)

104

The second identity (∗) follows from det(t) = det(y), (ut)ι = det(t)(ut)−1 and multiplicativity of ε, and by this, the sum is independent of the choice of u and t as long as det(t) = det(y). It is easy to verify that the Hecke operator deﬁned by (2.3.14) preserves the space Gκ (N, ε; C) and Sκ (N, ε; C) by conﬁrming (SA1–3) for f |[S0 (N)y ι S0 (N)]. Fix an isomorphism Qp ∼ = C (compatible with the two ip

i∞

embeddings Qp ← Q → C). Then assuming N ⊂ (p), we can also deﬁne the action of [S0 (N)y ι S0 (N)]p taking v = p in place of ∞ and replacing ε∞ ∆ (y) by det(yp )−κ1 εp∆ (y). Since det(yp )−κ1 εp∆ (y) = det(yp )−Ip −κ1 ε∞ ∆ (y) (under the convention of (ex0)), the only diﬀerence of the two normalization is det(yp )−Ip −κ1 , and in particular, we have det(yp )−κ1 [S0 (N)y ι S0 (N)] = [S0 (N)y ι S0 (N)]p

1 0 I for y = 1 for our chosen uniformizer q , because qp = 1 (but p−κ1 = 0 q if κ1 = 0). We therefore simply write T (q ) (resp. Tp ( q )) for the operator 1 0 ι ι [S0 (N)y S0 (N)] (resp. [S0 (N)y S0 (N)]p ) with y = . When q is a factor 0 q of N, writing N(q) for the prime-to-q part of N and assuming that the conductor of ε− is a factor of N(q) , we ﬁnd that the operator T (q ) on Sκ (N, ε; C) does not preserve Sκ (N(q) , ε; C) ⊂ Sκ (N, ε; C) and does not induce the operator T (q ) on Sκ (N(q) , ε; C). Thus if it is necessary to distinguish two operators T (q ) of level N and of level N(q) , we write U (q ) (resp. Up (q )) for T (q ) (resp. Tp (q )) of level N if q ⊃ N. Note here Tp (q ) = T (q ) and Up (q ) = U (q ) if κ1 = 0 or q p. 2.3.3 Hilbert modular forms with integral coeﬃcients Let eF : FA /F → C× be thestandard additive character determined by the condition: eF (x∞ ) = exp(2πi σ xσ ) for x∞ ∈ F∞ (e.g., [LFE] Theorem 8.3.1). We start with Proposition 2.26 Each member f of Sκ (N, ε; C) has a Fourier expansion of the following form,

y x f = |y|A a∞ (ξy, f )(ξy∞ )−κ1 eF (iξy∞ )eF (ξx). (2.3.15) 0 1 0ξ∈F

More generally, each modular form f of Gκ (N, ε; C) with κ1 = [κ1 ]I for [κ1 ] ∈ Z can be expanded into  

 

y x −[κ ] a∞ (ξy, f )(ξy∞ )−κ1 eF (iξy∞ )eF (ξx) . f = |y|A a0 (y, f )|y|A 1 + 0 1   0ξ∈F

Here y → a∞ (y, f ) is a function deﬁned on y ∈ FA× only depending on its ﬁnite The function part y (∞) and satisﬁes a∞ (uy, f ) = ε1 (u)a∞ (y, f ) for u ∈ T (Z).

Automorphic forms on quaternion algebras

105

× F∞ ) ∩ F × of integral ideles. The function a∞ (y, f ) is supported by the set (O A × F × . y → a0 (y, f ) factors through the class group ClF = FA×(∞) /O

y∞ 0 −κ1 We note that the function eF (iy∞ ) is the restriction of the

→ y∞ 0 1

∗ 0 + canonical Whittaker function of G(R) to matrices of the form (whose 0 1 Mellin transform gives the optimal Γ-factor of the standard L-function of f ). Here is a sketch of a proof (which is diﬀerent from the one given in [MFG] Theorem 3.10 and [SGL] Sections 2.3–4). Proof Since the proof is basically the same for cusp forms and modular forms, we give an argument for cusp forms (see [H88b] Section 4 for modular forms). We consider the unipotent subgroup %

& 1 u u ∈ (R ⊗Q F ) U (R) = 0 1

1 α of G. Then for ∈ U (Q), 0 1

1 α y x y x+α y x f =f =f . 0 1 0 1 0 1 0 1

y x Thus the function x → f for a ﬁxed y is a function on FA /F , which 0 1 is a compact abelian group. Applying the standard Fourier analysis to this group FA /F (e.g., [LFE] 8.3–4), we can expand

y x f = c(ξ, y, f )eF (ξx). 0 1 ξ∈F

Taking α ∈ F × , by (SA1), we have

c(ξ, αy, f )eF (ξαx) = f

ξ∈F

=f =

αy 0 α 0

αx 1

0 y x y =f 1 0 1 0

x 1

c(ξ, y, f )eF (ξx).

ξ∈F

Thus c(ξ, y, f ) = c(ξα−1 , αy, f ) for all α ∈ F × . In other words, c(ξ, y, f ) only depends on ξy; so, writing (∞)

c(y, f ) = c(1, y, f ), we

have c(ξ, y, f ) = c(ξy, f ). y∞ x∞ y x(∞) Taking g = and u∞ = in (SA2), as in (2.3.7) we 0 1 0 1

Automorphic forms on inner forms of GL(2)

106

have fg (x∞ + iy∞ ) =

κ1 −I y∞ f

y 0

x = a(ξ, fg ) exp(2πi ξ σ (xσ + iyσ )). 1 ∗ σ

ξ∈a

This shows I−κ1 c(ξy, f ) = y∞ a(ξ, fg ) exp(−2π

ξ σ yσ ).

σ

∩ F ; so, Since Γ0,g (N) = g · S0 (N)g −1 G(R)+ ∩ G(Q), a is given by y (∞) δ −1 O ∗ (∞) −1 (∞) ) O ∩ F . This shows that c(ξy, f ) = 0 unless (ξy) ∈ O, and hence a = (y Thus we may deﬁne c(y, f ) = 0 unless y (∞) ∈ O.

κ1 (∞) −1 yσ )|y|−1 |A a(1, fg ) a∞ (y, f ) = c(y, f ) exp(2π A y∞ = |y

(∞)

(∞)

σ

y x for g = , which satisﬁes the desired properties. In particular, from 0

1

y x u 0 y x f = ε1 (u)f , we have a∞ (uy, f ) = ε1 (u)a∞ (y, f ) 0 1 0 1 0 1 for u ∈ T (Z). 2 In view of the decomposition (see Lemma 2.46)

q 0 S0 (N) S0 (N) 0 1   4 1 0 q     0 S0 (N) u∈δq−1 Oq /δq−1 q 0 q =  4 u  q    u∈δq−1 Oq /δq−1 q 0 1 S0 (N)

u S0 (N) 1

we can directly verify by computation the following fact: a∞ (yq , f ) + N (q)ε+ (q )a∞ ( yq , f ) a∞ (y, f |T (q )) = a∞ (yq , f )

if q N, if q|N,

if q N, if q|N.

(2.3.16)

Remark 2.27 The Hecke operator T (q ) for q|N acts slightly diﬀerently from other T (l ) for l prime to N. If N(q) is the prime-to-q part of N and if the conductor of ε is prime to q, the action of T (q ) on Sκ (N(q) , ε; C) is not the restriction of the action of U (q ) on Sκ (N, ε; C) to Sκ (N(q) , ε; C) though Sκ (N(q) , ε; C) is canonically a subspace of Sκ (N, ε; C). Since the Let us prove (2.3.16). We call an idele y ∈ FA× integral if y (∞) ∈ O. computation in the cases where q|N and q N is the same, we treat the case

q 0 of q N. By the above decomposition of S0 (N) S0 (N) (into a disjoint 0 1

Automorphic forms on quaternion algebras

union of right cosets) combined with

1 0

0 q

integral y |y|A a∞ (y, f |T (q ))eA (x) = ε+ (q )|q−1 y|A a∞

= q

y ,f q

q−1 0

107

0 , we have for 1

eA (x) 

+ |q y|A a∞ (q y, f )eA (x) 

 eA (yu)

−1 −1 u∈δq Oq /δq qOq

Note that u∈δq−1 Oq /δq−1 qOq eA (yu) = |q |−1 A = N (q), because y ∈ O. From this we get the desired formula. For each Q-algebra R ⊂ C containing the values of characters ε = (ε+ , ε1 , ε2 ) and κ on TG (Q), we deﬁne . Sκ (N, ε; R) = f ∈ Sκ (N, ε; C)a∞ (y, f ) ∈ R for all y , (2.3.17) . Gκ (N, ε; R) = f ∈ Gκ (N, ε; C)a0 (y, f ), a∞ (y, f ) ∈ R for all y . Recall the p-adic valuation ring W of a number ﬁeld inside C containing the and Z(Z) and all the Galois conjugates of O in Q. As we values of ε on T (Z) will see in (4.3.7) in Chapter 4 (see also [PAF] (4.43) and 4.3.1), for a W-algebra R ⊂ Q with i∞ : Q → C: Gκ (N, ε; R) ⊗R,i∞ C = Gκ (N, ε; C) and Sκ (N, ε; R) ⊗R,i∞ C = Sκ (N, ε; C). (2.3.18) We recall the embedding ip : Q → Qp . Then for any Qp -algebra R, we deﬁne, consistently with (2.3.18), Gκ (N, ε; R) = Gκ (N, ε; Q) ⊗Q,ip R and Sκ (N, ε; R) = Sκ (N, ε; Q) ⊗Q,ip R. There is a more intrinsic deﬁnition of these spaces as global sections of the automorphic line bundle ω κ,ε over Hilbert modular Shimura varieties (see [PAF] 4.2.6) as we revisit the theory later in Chapter 4. By linearity, (y, f ) → a∞ (y, f ) extends to functions on FA× × Gκ (N, ε; R) with values in R. Then the p-adic q-expansion coeﬃcients ap (y, f ) deﬁned in [PAF] (4.63) of f ∈ Gκ (N, ε; R) has the following relation to the archimedean Fourier coeﬃcients, yp−κ1

ap (y, f ) = yp−κ1 a∞ (y, f ).

(2.3.19)

−κ1,σ

= σ σ(yp ) does not follow the convention in (ex0). Even if we Here have divided by possibly a nonunit ypκ1 , the coeﬃcients ap (y, f ) reﬂects faithfully the p-integrality coming from the q-expansion. Indeed, writing y = ξc for ξ ∈ and an idele c with cp = c∞ = 1, we have ap (y, f ) = |c(∞) |−1 F+× A a(ξ, fg ) for c 0 g= , which is the q-expansion coeﬃcients of the classical modular form 0 1 fg up to p-adic unit |c(∞) |−1 A . By Proposition 2.26, we have 1 ap (uy, f ) = ε1 (u)u−κ ap (y, f ) for u ∈ T (Z). p

(2.3.20)

108

Automorphic forms on inner forms of GL(2)

The coeﬃcient ap behaves better then a∞ under the action of Hecke operators, because we have the following p-integral formula without any conditions on κ which follows from (2.3.16) and (2.3.19) combined (see also Proposition 4.27 in the text and [PAF] (4.65–66) and (4.79)): ap (y, f |Tp (q )) = ap (yq , f ) + N (q)ε+ (q )ap (

y , f ) if q pN, q

(2.3.21)

ap (y, f |Up (q )) = ap (yq , f ) if q|pN. The formal q-expansion of f has values in the space of functions on T (A(∞) ) with values in the formal monoid algebra R[[q ξ ]]ξ∈F × of the multiplicative semigroup F+× , which is given by f (y) =

+

ap (ξy, f )q ξ .

(2.3.22)

ξ0

Here ξ ! 0 implies σ(ξ) > 0 for all σ ∈ I. This is the p-adic analog of the archimedean Fourier expansion (2.3.15). In particular, if ap (y, f ) = 0 for all integral ideles y, the modular form f vanishes. In other words, the q-expansion: y → f (y) determines f uniquely (for any algebra R for which the space of R-integral modular forms is well-deﬁned). Let W = limn W/pn W be the p-adic completion of W. From the q-expansion ←− principle (see Corollary 4.16 and (q-exp) in 4.2.8), for any p-adically complete p (the p-adic completion of Qp ), we conclude that the space W -algebra R in Q p )ap (y, f ) ∈ R for all integral y (2.3.23) Sκ (N, ε; R) = f ∈ Sκ (N, ε; Q coincides with the space of R-integral cusp forms algebro-geometrically (later) deﬁned in 4.3.3. For the moment, we take the above description (2.3.23) as the deﬁnition of the space Sκ (N, ε; R) for W -algebras R. As a direct consequence of (2.3.21) (see also [PAF] Theorem 4.28), under the following condition either p|N or [κ] ≥ 0,

(2.3.24)

the space of R-integral modular forms Sκ (N, ε; R) is stable under Hecke operators. We then deﬁne the ﬁnite level Hecke algebra hκ (N, ε; R) by the R-subalgebra of EndR (Sκ (N, ε; R)) generated by Tp (l ) for all prime ideals l. Actually we have well-deﬁned linear operators Tp (y) for general integral ideles y as we will see in the following section, and Tp (y) is an integral polynomial of Tp (l ) and the operators given by the action of S0 (N) (via ε) (see Lemma 2.39 in the text and [MFG] Lemma 3.9). Thus hκ (N, ε; R) is the R-subalgebra of EndR (Sκ (N, ε; R)) generated by Tp (y) for all integral ideles y. = A ⊗W W for any W-algebra Since we have chosen q inside W, writing A A ⊂ Qp , we have a well-deﬁned A-integral subspace Sκ (N, ε; A) of Sκ (N, ε; A)

Automorphic forms on quaternion algebras

109

given by p (y, f ) ∈ A if y is a product of q for primes q}. {f ∈ Sκ (N, ε; A)|a Then we have = Sκ (N, ε; A) ⊗A A. Sκ (N, ε; A) 2.3.4 Duality and Hecke algebras The elementary duality theorem between the Hecke algebra and the space of modular forms westate now is quite useful in many diﬀerent 7 applications.

−1

δ 0 δ 0 det g = y . Decompose the Let T (y) = g∈ ∆0 (N) 0 1 0 1 double coset S04 (N)T (y)S0 (N) for an integral ﬁnite idele y into a disjoint union of right cosets α:det(α)=y αS0 (N). Then we extend the deﬁnition of the Hecke operator T (l ) to general integral ideles y by f |Tp (y)(g) = α yp−κ1 εp∆ (αι )f (gα). Here we have taken α so that det(α) = y (which tells us that the operator depends on y). Then by (2.3.21), we ﬁnd (e.g., (2.4.6)), for each integral ﬁnite idele y, ap (1, f |Tp (y)) = ap (y, f ).

(2.3.25)

e(q)

Since we have chosen q inside W, the operator Tp (y) for y = q q with e(q) ≥ 0 preserves Sκ (N, ε; A) for any W-algebra A. Thus we may extend the deﬁnition of the Hecke algebra hκ (N, ε; A) to W-algebras A so that it is the A-subalgebra of EndA (Sκ (N, ε; A)) generated by Tp (y) for all y of the form y = e(q) . Then we deﬁne an A-bilinear pairing q q ( , ) : hκ (N, ε; A) × Sκ (N, ε; A) → A by (h, f ) = ap (1, f |h). Theorem 2.28 (Duality) isomorphisms

Let A be a W-algebra. Then ( , ) induces

HomA (Sκ (N, ε; A), A) ∼ = hκ (N, ε; A),

HomA (hκ (N, ε; A), A) ∼ = Sκ (N, ε; A),

and the latter isomorphism is given by sending φ to f with ap (y, f ) = φ(Tp (y)). Proof Since Sκ (N, ε; A) = Sκ (N, ε; W) ⊗W A, we may assume that A = W. Since W is the p-adic completion of W, it is faithfully ﬂat over W; so, we may assume that A = W . Actually we prove the duality ﬁrst for the quotient ﬁeld K of W . The space Sκ (N, ε; K) is ﬁnite dimensional over Qp ; so, we need to prove nondegeneracy of the pairing. By (2.3.25), ap (1, f |Tp (y)) = ap (y, f ); so, if (h, f ) = 0 for all h, ap (y, f ) = (Tp (y), f ) = 0 for all integral y, and hence f = 0. If (h, f ) = 0 for all f , then 0 = (h, f |Tp (y)) = ap (1, f |Tp (y)h) = (Tp (y), f |h) = ap (y, f |h); so, f |h = 0 for all f , which implies h = 0. If φ ∈ HomW (hκ (N, ε; W ), W ), then we ﬁnd f ∈ Sκ (N, ε; K) with (h, f ) = φ(h), and ap (y, f ) = (Tp (y), f ) = φ(Tp (y)) ∈ W ; so, f ∈ Sκ (N, ε; W ). This shows

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Sκ (N, ε; W ) = HomW (hκ (N, ε; W ), W ). Since W is a discrete valuation ring, we 2 also have HomW (Sκ (N, ε; W ), W ) ∼ = hκ (N, ε; W ). This tells us Corollary 2.29 Let H = hκ (N, ε; A). Let V and V be H-modules free of ﬁnite rank over A with an A-bilinear pairing , : V × V → A. Deﬁne a formal q-expansion of Θ(v ⊗ v ) by ap (y, Θ(v ⊗ v )) = v|T (y), v . Then Θ gives an H-linear map of V ⊗A V into Sκ (N, ε; A) regarding V ⊗A V as an H-module through V . If V is H-free of rank 1 and HomA (V, A) ∼ = V by , and hv, v = v, hv for h ∈ H, Θ induces an isomorphism V ⊗H V ∼ = Sκ (N, ε; A). Proof Just apply the theorem to Θ(v ⊗ v ) ∈ HomA (H, A) = Sκ (N, ε; A) given 2 by Θ(v ⊗ v )(h) = hv, v . We will later give another (partial) proof of this fact when V is the space of quaternionic automorphic ﬁrms, following the argument given in [Sh6] II, Theorem 3.1 (see the discussions after Proposition 2.51). 2.3.5 Quaternionic automorphic forms We now generalize the deﬁnition (SA1–3) of Hilbert modular forms to automorphic forms on a quaternion algebra D/F . We ﬁrst deﬁne the standard level (open compact) subgroup in GD (A(∞) ). We assume that Dp = D ⊗Z Zp ∼ = M2 (Fp ) for the ﬁxed prime p. For each prime ideal l outside ΣD , we ﬁx an isomorphism OD,l ∼ = M2 (Ol ) so that for the p-adic place p|p coming from ip ◦ σ, this isomorphism is given by the one already ﬁxed: ip ◦σ

OD → M2 (OK ) −−−→ M2 (Op ). By means of these isomorphisms, we identify Dl with M2 (Fl ). Let d(D) be the product of prime ideals in ΣD . For an integral ideal N0 of F prime to d(D), putting N = N0 d(D), we deﬁne %

& a b × D x (N) = x ∈ O = with c ∈ N O , (2.3.26) Γ N 0 N 0 0 0 D c d D −1 D and ON = D = OD ⊗Z Z, where O Γ0 (N)s for 0 l|N0 Ol . We set S0 (N) = s

(d(D)) 0 δ ∩ F = d. Similarly we deﬁne with a ﬁnite idele δ such that δ O s= 0 1 × ∆D 0 (N) ⊂ DA(∞) so that it is the product of local components ∆l which coincide with the local components of ∆0 (N) as long as l d(D) and ∆l = OD,l if l|d(D). We can think of the double coset ring R(S0D (N), ∆D 0 (N)) (which is a collection of formal linear combinations of double cosets S0D (N)xS0D (N) for x ∈ ∆D 0 (N) with multiplication given by convolution product; see 2.3.7 for a precise deﬁnition D D∗ D∗ ∼ of the ring). We have R(S0D (N), ∆D 0 (N)) = R(S0 (N), ∆0 (N)). Here ∆0 (N) D ι −1 is the image of ∆0 (N) under the involution ι (x = N (x)x for the reduced norm map N : D → F ). We call modules over these isomorphic double coset rings Hecke modules.

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l 0 We have T (l) = S0D (N) and T (l, l) = S0D (N)l S0D (N) in 0 1 D R(S0D (N), ∆D∗ 0 (N)) for l d(D), because the local component ∆0 (N)l at l of D ∆0 (N) is identical to ∆0 (N)l . For l|d(D), we take αl ∈ OD,l so that its reduced norm generates lOl . Then we deﬁne T (l) = S0D (N)αl S0D (N) for l|d(D), and we have S0D (N)

∼ R(S0D (N), ∆D 0 (N)) = R(S0 (N), ∆0 (N)).

(2.3.27)

These elements T (l) and T (l, l) (indexed by primes l) are generators of the commutative ring R(S0D (N), ∆D∗ 0 (N)) over Z (see Lemma 2.39). The above isomorphism brings T (l) and T (l, l) to the corresponding elements in the right-hand side. As an operator on the space of automorphic forms, T (l, l) induces the central action l of l . A particular feature of quaternionic automorphic forms is that they are often vector valued, though Hilbert modular forms deﬁned in (SA1–3) are scalar valued. Here we deﬁne the space in which quaternionic automorphic forms have values. For a given ring R, we consider the following module L(κ∗ ; R) of the multiplicative semigroup M2 (R). Let κ∗ = (κ1 + I, κ2 ) and put n = κ2 − κ1 − I ∈ Z[I], which is the restriction of κ∗ ∈ X(TG ) to T ⊂ G1 , and we conﬁrm (κ|T )∗ = k − 2I = n. We suppose that n ≥ 0 (i.e., nσ ≥ 0 for all σ ∈ I), and we consider polynomials with coeﬃcients in R of (Xσ , Yσ )σ∈I homogeneous of degree The collection of all such polynomials forms an R-free nσ for each pair (Xσ , Yσ ). module L(κ∗ ; R) of rank σ (nσ + 1). As before, we write v for the ﬁxed place p or ∞; so, the base ring B is W if v = p and C if v = ∞. Suppose that R is a B-algebra. Then iv (σ(δv )) (which we write simply σ(δv )) for δ ∈ GD (A) can be regarded as an element in M2 (R). Take a Neben character ε as in (ex1–4) of Section 2.3.2 with ε− |T (Z) factoring × through (O/N0 ) . ∆ 2) as in (ex1) and extend ε to ∆D We deﬁne εj (j = 1, 0 (N) by εD (δ) = a b ε2 (N (δ))ε− (a) if δN0 = . Since s → ε∆ D (s) and z → ε+ (z) coincide on c d Z(A(∞) ) ∩ S0D (N), we may extend ε to a character εSD : S0D (N)Z(A(∞) ) → R× (∞) by εSD (zu) = ε∆ ) and u ∈ S0D (N). We let ∆D 0 (N) and D (u)ε+ (z) for z ∈ Z(A D D + ∗ Z(A)S0 (N)G (R) act on L(κ ; R) as follows.

Xσ Xσ ∆ −1 κ1 s ι δ·Φ = εD (δ) N (δv ) Φ σ(( δv ) ) , Yσ Yσ

Xσ Xσ (zu) · Φ = εSD (z (∞) u(∞) )−1 N (zv uv )κ1 Φ σ(s (zv uv )ι ) . Yσ Yσ (2.3.28) Here z ∈ Z(A), u ∈ S0D (N)GD (R), and δ ι = N (δ)δ −1 and s δ = sδs−1 for s given just below (2.3.26). We write L(κ∗ ε; R) for the module L(κ∗ ; R) with this D D (∆D 0 (N), Z(A)S0 (N)G (R))-action.

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By computation, z ∈ Z(A) acts on L(κ∗ ε; R) through scalar multiplication × ⊂ Z(A) acts trivially on L(κ∗ ε; R). by ε+ (z)−1 zvκ1 +κ2 −I ; in particular, ∈ O+ × in If S is a suﬃciently small open compact subgroup so that S ∩ Z(Q) ⊂ O+ (∞) −1 D ∗ G(A ), central elements in Γx = xSx ∩ G (Q) act trivially on L(κ ε; R). Assume that v = ∞. We state a deﬁnition of automorphic forms on GD (A) D now for a division algebra D. For each κ, we deﬁne κD ∈ Z[I]2 by κD = (κD 1 , κ2 ) D D 2 for κD = κ σ. Thus κ is the projection of κ to Z[I ] . Similarly, we j σ∈I D j,σ deﬁne κD by the projection of κ to Z[ID ]2 ⊂ Z[I]2 . With each κ ∈ Z[I]2 = X(TG ), we associate an automorphy factor, JκD (g, z) = det(g)κD,1 −ID j(g, z)κD,2 −κD,1 +ID ,

(2.3.29)

for g ∈ GD (R) and z ∈ ZD . We write κ∗,D for the projection of κ∗ to Z[I D ]. We take a subset Θ ⊂ ID and split ID as ID = Θ Θ. Deﬁne for z ∈ ZD z σ if σ ∈ Θ, zσΘ = zσ if σ ∈ Θ. D (N, ε; C) to be the space of functions f on GD (A) Then, if ID = ∅, we deﬁne Sκ,Θ D with values in the left Ci -module L(κ∗,D , C) satisfying the following conditions. (SB1) We have the following automorphy (∞) −1 f (αxuz) = ε+ (z)ε∆ )u∞ · f (x)JκD (u∞ , iΘ )−1 D (u

for all α ∈ GD (Q), z ∈ Z(A), and u ∈ S0D (N)CiD , where GD (R)+ is the identity-connected component of GD (R). Here f (x) → u∞ · f (x) is the ∗,D ; C); action of the I D -component uD ∞ of u∞ on L(κ D D (SB2) Choosing u ∈ G (R) with u = 1 and u(i) = z for each z ∈ ZD , deﬁne a function fg : ZD → C by fg (z) = f (gu∞ )Jκ (u∞ , i) for each g ∈ G(A(∞) ). Then, for all g, fg is a function on ZD holomorphic in zσ for σ ∈ Θ and antiholomorphic in zσ for σ ∈ Θ. D (N, ε; C), which is the space of If Θ = ID , we simply write SκD (N, ε; C) for Sκ,I D D holomorphic automorphic forms on G (A) of level S0 (N) and of weight (κ, ε). When I D = I (⇔ ID = ∅), the variety Y D (S) is a ﬁnite set of points; the condition (SB2) is empty, and we may replace C in (SB1) by any ring A ⊂ C with values of ε. Writing MD κ (N, ε; A) for the space of functions satisfying (SB1) in this deﬁnite case, we need to take SκD (N, ε; A) to be the following quotient: SκD (N, ε; A) = MD κ (N, ε; A)/Iv(N, ε; A), where Iv(N, ε; A) is the subspace made up of functions in SκD (N, ε; A) factoring through the reduced norm map N : GD (A) → T (A). If κ2 − κ1 = I or ε is nontrivial for some x ∈ GD (A) with N (x) = 1, Iv(N, ε; A) = 0; so, no is necessary. Decomposing 4 modiﬁcation D (N) into hS (N), we shall deﬁne the action of S0D (N)y ι S0D (N) for y ∈ ∆D 0 0 h D (N)) on S (N, ε; A) by R(S0D (N), ∆D κ 0

D ι D f |[S0 (N)y S0 (N)](g) = h · f (gh) = h−ι · f (gh−ι ). (2.3.30) h

h

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When D = M2 (F ), we simply write Sκ,Θ (N, ε; C) for the space of functions on GL2 (FA ) satisfying (SB1–2) and (SA3). As we have stated for holomorphic Hilbert modular forms, the Hecke algebra R(S0 (N), ∆0 (N)) naturally acts on Sκ (N, ε; C). We now extend the deﬁnition of automorphic forms on GD (A) and on ZD D Let DH = D Dσ ∼ to more general level groups U ⊂ GD (Z). = HI . For σ∈I × be an open subgroup. Thus simplicity, we write V for L(κ∗,D ; C). Let U ⊂ O D D D + the image Γ(U ) of U ∩ G (Q) in P G (R) is a congruence subgroup of Γ(1). An automorphic form of weight κ and of analytic type (Θ, Θ) (of level U ) is a function f : GD (A) → V satisfying the following two conditions (sb1–2). D Θ −1 (sb1) f (γxs) = s−1 for γ ∈ GD (Q) and s ∈ U · CiD . The ∞ · f (x)Jκ (s∞ , i ) × on V . action of s∞ : f (x) → s∞ · f (x) is through the action of DH

For each g ∈ GD (A) with g∞ = 1, as before, we deﬁne a function fg : ZD → V by fg (z) = s∞ f (gs∞ )JκD (s∞ , iΘ ) for s∞ ∈ GD (R)+ with s∞ (i) = z. If z = s∞ (i) = s∞ (i), we can write s∞ = s∞ c × C∞ . Thus with c ∈ F∞ s∞ f (gs∞ )JκD (s∞ , iΘ ) = s∞ cf (gs∞ c)JκD (s∞ c, iΘ ) = s∞ cc−1 f (gs∞ c)JκD (c, iΘ )−1 JκD (c, iΘ )JκD (s∞ , iΘ ) = fg (z). This shows that fg is well deﬁned independently of the choice of s∞ . For γ ∈ Γg (U ) = g −1 U GD (R)+ g ∩ GD (Q) which is a congruence subgroup of the unit group of another maximal order g OD = g −1 OD g ∩ D, we see fg (γ(z)) = γ∞ s∞ · f (gγ∞ s∞ )JκD (γ∞ s∞ , iΘ ) = γ∞ s∞ · f (γgγ (∞)

−1 −1

g

gs∞ )JκD (γ∞ s∞ , iΘ )

= γ∞ s∞ · f (γgs∞ )JκD (γ∞ s∞ , iΘ ) (gγ (∞)

−1 −1

g

∈ U)

= γ∞ s∞ · f (gs∞ )JκD (s∞ , iΘ )JκD (γ, z Θ ) = γ∞ · fg (z)JκD (γ, z Θ ). Thus fg is a modular form on Γg (U ). Now, for σ ∈ IB , we impose the holomorphy or the antiholomorphy condition: (sb2)

∂fg ∂zσ

= 0 for σ ∈ Θ and

∂fg ∂z σ

= 0 for σ ∈ Θ.

Except for the cases where D is deﬁnite and dim V = 1 and D = M2 (F ), the space of functions satisfying the above conditions (sb1–2) will be denoted as

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D Sκ,Θ (U ; C). If Θ = ID , we drop the subscript Θ. When D = M2 (F ), we write Sκ,Θ (U ; C) (dropping the superscript D = M2 (F )) for the space of functions satisfying the above conditions (sb1–2) and (SA3). We write Gκ,Θ (U ; C) for the space of functions satisfying (sb1–2) with polynomial growth towards the cusps of Y D (U ) when D = M2 (F ). In order to deﬁne the space SκD (U ; C) as before, when D is deﬁnite and dim V = 1, we write M(U ; C) for the space of functions: GD (A) → V satisfying (sb1– 2). Then writing Iv(U ; C) for the subspace of functions in M(U ; C) factoring through the reduced norm map N : GD (A) → T (A), we deﬁne SκD (U ; C) = M(U ; C)/Iv(U ; C). We let each double coset [U xU ] of x ∈ GD (A(∞) 4 ) act on Sκ (U ; C) as a Hecke operator. In other words, decomposing U xU = y yU , the action is given by

f |[U xU ](g) = f (gy). g

2.3.6 The Jacquet–Langlands correspondence (d(B)∞)

(d(B)∞)

with M2 (FA ). We state the theorem of As before, we identify DA Jacquet and Langlands, and of Shimizu in the following way. (d(D)∞)

Theorem 2.30 For an open compact subgroup U (d(D)) of GL2 (FA (∞) deﬁne open subgroups of GD (A(∞) ) and GL2 (FA ) by × UD = U (d(D)) × OD,q ⊂ GD (A(∞) ),

),

q|d(D) (∞)

U0 (d(D)) = U (d(D)) × S0 (d(D))d(D) ⊂ GL2 (FA

).

Then we have a C-linear embedding i : SκD (UD ; C) → Sκ (U0 (d(B)); C) for all weights κ with kσ ≥ 2 and for all σ ∈ I D , where kσ = κ2,σ − κ1,σ + 1. For (d(D)∞) all x ∈ GL2 (FA ), we have i ◦ [UD xUD ] = [U0 (d(B))xU0 (d(B))] ◦ i. The image of this embedding only depends on d(D) and is made up of cusp forms in Sκ (U0 (d(B)); C) new at all primes q|d(D). In particular, if d(D) = 1, the above morphism is a surjective isomorphism. The word “new” means that a cusp form in the image of i cannot be in the space generated by S (U (d(D)/q); C) under the action f (h) → x · f (h) = q|d(D) κ 0 q 0 ∈ G(Fd(D) ); in other words, any f ∈ Im(i) is orthogonal f (hx) of x = 0 1 to x·g(h) = g(hx) for any g ∈ q|d(D) Sκ (U0 (d(D)/q); C) and x ∈ G(Fd(D) ), that is, G(Q)\G(A)/Z(A) f (h)x · g(h)dh = 0 for a right invariant Haar measure dh on G(Q)\G(A)/Z(A). We write Sκnew (U0 (d(D)); C) for the subspace of new forms. Some more comments on the new forms will be given after the classiﬁcation lemma (Lemma 2.41) of local representations. Applying the above theorem to U = S0D (N), we have

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Corollary 2.31 Suppose N = N0 d(D) for an integral ideal N0 prime to d(D). Identify R(S0 (N), ∆0 (N)) and R(S0D (N), ∆D 0 (N)) as in (2.3.27). Then we have an R(S0 (N), ∆0 (N))-linear embedding ι : SκD (N, ε; C) → Sκ (N, ε; C) for all weights κ with kσ ≥ 2 for all σ ∈ I D . The image of this embedding only depends on d(D) and is made up of cusp forms in Sκnew (N, ε; C) new at all primes q|d(D). In particular, if d(D) = 1, the above morphism is a surjective isomorphism. Since known proofs of this theorem require harmonic analysis and a good knowledge of representation theory of adele groups (especially the theta correspondence), we do not give a full proof of this fact, and we quote here some references where one can ﬁnd a proof. The above formulation of the theorem was given in [PAF] Section 4.3, [H81a] Section 2 and [H88b] Theorem 2.1, where one can ﬁnd an exposition of how to deduce this result from the original result of Jacquet and Langlands (whose exposition can be found in [AFG] Chapter 3 and [AAG] Section 10). The result of Jacquet and Langlands (e.g., [AFG] Chapter 3) is more general than the above theorem in the following points: (JL1) It covers any real-analytic automorphic forms including Maass forms and × any level group U (not necessarily under the condition Uq = OD,q for q|d(D)). (JL2) It also gives a description of the relation of the local representations of Dq× and GL2 (Fq ) for q|d(D). (JL3) Their approach is more representation theoretic: it gives a precise correspondence of automorphic representations πD realized on the L2 -space of functions on GD (Q)\GD (A) (with a given central character χ) and π realized on the L2 -space of functions on GL2 (F )\GL2 (FA ) with the same central character χ. If we factorize πD = ⊗v πv (D) and π = ⊗v πv , πv (D) ∼ = πv as long as Dv ∼ = M2 (Fv ). For v with division Dv , the correspondence πv (D) → πv is given by the Weil representation πv associated to πv (D) with respect to the norm form of D/F . We will give a slightly more detailed description of the correspondence πD → π in (JL3) at the end of the following subsection. Shimizu originally proved a result close to this theorem when d(D) = 1 (and a weaker assertion when d(D) = 1). Later he explictly realized the correspondence using theta series (see [Si]). The method of proof in the original work of Shimizu and Jacquet and Langlands is to compute the trace of the operator [U xU ] on both sides by means of the Eichler–Selberg trace formula. Eichler initiated the comparison of such traces (and he proved results slightly weaker than the above corollary for deﬁnite quaternion algebras over Q; [Ei]). We will give an exposition later of this for F = Q along a line closer to the treatment of Eichler, Shimizu, and Shimura. D We can actually embed Sκ,Θ (U ; C) into Sκ (U0 (d(D)); C) in a way compatible with the Hecke operator action. This fact follows from Theorem 2.30 and the

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following result: Proposition 2.32 For each Θ ⊂ ID , there is an isomorphism: SκD (U ; C) ∼ = D Sκ;Θ (U ; C) which is linear under the action of [U xU ] for all x ∈ GD (A(∞) ).

−1 0 × by Proof Let j = ∈ GL2 (R). We deﬁne t(Θ) ∈ D∞ 0 1 j if s ∈ Θ, t(Θ)σ = 1 otherwise. Then f (x) → f (xt(Θ)) gives the isomorphism iΘ . Left multiplication by t(Θ) only takes eﬀect at inﬁnite places and hence commutes with [U xU ]. 2 In the previous section, we have only deﬁned the space SκD (N, ε; C) of quaternionic automorphic forms with coeﬃcients in C. However, we have a good integral structure on SκD (N, ε; C) over W (as we will see for some D later in Chapter 3). Now we assume we have such an integral structure, and for each W-algebra A, we write SκD (N, ε; A) for the space of A-integral automorphic forms. We thus suppose (R1) SκD (N, ε; A) = SκD (N, ε; W) ⊗W A for every W-algebra R; (R2) SκD (N, ε; W) ⊂ SκD (N, ε; C) is stable under the Hecke operator action of R(S0D (N), ∆D 0 (N)). D We then extend the Hecke action of R(S0D (N), ∆D 0 (N)) to Sκ (N, ε; A) by the D identity (R1). Thus Sκ (N, ε; W ) becomes a module over the Hecke algebra hκ (N, ε; W ). Let SκD (N, ε; A)∗ be the A-linear dual of SκD (N, ε; A). We regard SκD (N, ε; A)∗ as a hκ (N, ε; A)-module by the adjoint action. Then by Proposition 2.51, we have a morphism of hκ (N, ε; W )-modules

fA : SκD (N, ε; A) ⊗A SκD (N, ε; A)∗ → Sκ (N, ε; A). For a suitable choice of φ ∈ SκD (N, ε; C)∗ , g → fC (g ⊗ φ) induces an embedding ι : SκD (N, ε; C) → Sκ (N, ε; C) in Corollary 2.31. By the faithful ﬂatness of C over the ﬁeld of fractions K of W, ι induces an embedding ι : SκD (N, ε; K) → Sκ (N, ε; K) which is hκ (N, ε; K)-linear. Then extending scalar to K-algebra A, we ﬁnd Corollary 2.33 Suppose N = N0 d(D) for an integral ideal N0 prime to d(D). Let A be a K-algebra. Then we have an hκ (N, ε; A)-linear embedding ιA : SκD (N, ε; A) → Sκ (N, ε; A) for all weights κ with kσ ≥ 2 for all σ ∈ I D . In particular, if d(D) = 1, the above morphism is a surjective isomorphism. Exercise 2.34 1. If χ : H× → C× is a continuous character, prove that χ factors through the reduced norm N : H× → R× .

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2. Prove that v ∈ Z[I] is an integer multiple of σ σ if v = 1 for all in a subgroup of ﬁnite index in O× . 3. Prove that if U is an open compact subgroup of GD (A(∞) ), the double coset U yU for y ∈ GD (A(∞) ) contains only ﬁnitely many left (and right) cosets 4. Give a detailed proof that (f |j)x (z) = fx (−z) for any x ∈ GD (A(∞) ) if D = M2 (Q) and f ∈ Sκ (U ; C), where f |j(g) = f (gj) and j is as in the proof of Proposition 2.32. 5. Prove that for a given κ and U , there are only ﬁnitely many Hecke characκ1 +κ2 −I ters χ : F × UF \FA× → C× with χ(z∞ ) = z∞ , where UF = U ∩Z(A(∞) ) (∞) × regarded as an open subgroup of (FA ) . 2.3.7 Local representations of GL(2) We summarize here well-known results from representation theory of the local group G = GL2 (Fq ) for a prime ideal q (see [AFG] Chapter 1) in order to supplement the representation theoretic description in (JL1-3) of the Jacquet– Langlands correspondence. A representation V of G with coeﬃcients in a ﬁeld K of characteristic 0 is called smooth if for each v ∈ V , we ﬁnd an open subgroup S ⊂ G that leaves v invariant. If furthermore, H 0 (S, V ) is ﬁnite dimensional for all open subgroups S, V is called admissible. Let π be an admissible semisimple representation of G on a vector space V = V (π) over a ﬁeld K of characteristic 0. Let U(Fq ) be the maximal upper unipotent subgroup of G, and write %

& 1 u u ∈ Oq ⊂ U(Fq ). U(Oq ) = 0 1 Let B(Fq ) (resp. B(Oq )) be the normalizer of U(Fq ) in G (resp. GL2 (Oq )). Then B(A) = T (A)U(A) for the subgroup T (A) of diagonal matrices (for A = Fq and Oq ). They are made up of upper triangular matrices. We have a Haar measure du of U(Fq ) with U (Oq ) du = 1. We deﬁne . V (B) = V (B, π) = v − π(n)v ∈ V (π)v ∈ V (π), n ∈ U(Fq ) as a T (Fq )-module, and put VB = VB (π) = V /V (B), which is called the Jacquet module of V . The notion of Jacquet module is useful in the classiﬁcation of admissible representations of a local group like G (see [BeZ]). By this deﬁnition, V → VB is a right exact functor. For each w = v − π(n)v ∈ V (B), we take a suﬃciently large open compact subgroup Uw ⊂ U(Fq ) containing n. Then we see that π(u)vdu = 0 for every open subgroup U with Uw ⊂ U ⊂ U(Fq ). If U π(u)vdu = 0 for every suﬃciently large open subgroup U of U(Fq ), for the U stabilizer U of v in U, we ﬁnd

du v = du v − π(u)vdu = du (v − π(n)v) U

U

U

U

n∈U /U

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which is in V (B). Thus V (B) is the collection of v with U π(u)vdu = 0 for every suﬃciently large open subgroup U of U(Fq ). By this fact, the functor V → VB is left exact, and we conclude that the association is an exact functor. Exercise 2.35 Explain why V → VB is exact (sending exact sequences of representations to exact sequences of K-vector spaces). For a smooth character λ of T (Fq ) (regarding it as a character of B via the projection B T ∼ = B/U), the smooth induction from B of λ is deﬁned by IndGB (λ) = {f : G → V (λ)|f : smooth, f (bg) = λ(b)g(g) ∀b ∈ B(Fq )}, (2.3.31) on which we let G act by g · f (x) = f (xg). Here the word “smooth” means that for each f ∈ IndGB V (λ), we ﬁnd an open compact subgroup S such that f (xk) = f (x) for all k ∈ S. Thus IndGB λ is smooth as a representation of G by deﬁnition. Since the smooth induction preserves admissibility (see [BeZ] 2.3), V stable under the V = IndGB λ has composition series {0} ⊂ V1 ⊂ V2 ⊂ · · · ⊂ action of G, and hence its semisimpliﬁcation (IndGB λ)ss = j Vj+1 /Vj is well deﬁned as an admissible G-module. = δ 1/2 λ for the module character δB of B: Put λ B φ(u)du = δB (b) φ(bxb−1 )du (for all φ). U (Fq )

U (Fq )

a 0 = |a−1 d|q for the standard absolute value 0 d (x ∈ Oq − {0}).

Exercise 2.36 Prove δB |x|q = |Oq /xOq |−1

is irreducible, we call the induced representation principal (or in the If IndGB λ is reducible, only one factor of the composition series principal series). If IndGB λ is inﬁnite dimensional, which is called special

(or a Steinberg representation). a 0 Since T is diagonal, we can write λ = λ1 (a)λ2 (d). Then a principal 0 d is denoted by π(λ1 , λ2 ) (resp. σ(λ1 , λ2 )). (resp. special) representation IndGB λ The following results are well known (e.g., [BeZ]). (π1) HomB (VB , V (λ)) ∼ = HomG (V, IndGB V (λ)) [BeZ] 1.9 (reciprocity); (π2) if π is absolutely irreducible, then dimK VB ≤ |W| = 2, where W is the Weyl group of T in G (see [BeZ] 2.9); (π3) if π is absolutely irreducible and VB = 0, then we have a surjective linear map IndGB λ V of G-modules for a character λ : T → K × (see [BeZ] 2.4–5); # $ss # $ss w ∼ for w ∈ W (see [BeZ] 2.9); (π4) IndGB λ = IndGB λ8 where we have written λw (t) = λ(wtw−1 ).

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On V U , we have the Hecke operator T () for

a prime element of Oq given by 0 T ()v = αU ⊂U ξU /U π(α)v for ξ = , where U − U(Oq ). We generalize 0 1 this action of Hecke operators as follows. Write B = B(Oq ) and U = U(Oq ) for simplicity. Let . D = x ∈ T (Fq )xUx−1 ⊃ U (2.3.32) . D∗ = x ∈ T (Fq )xUx−1 ⊂ U . The semigroup D is called the expanding semigroup in T (Fq ), and D∗ is called the shrinking semigroup. The set is ∆U = U · D · U and ∆∗U = U · D∗ · U are also multiplicative semigroups. Exercise 2.37 Show that D (resp. D∗ ) is generated by T (Oq ) and

e 0 1 for integers e1 ≤ e2 (resp. e1 ≥ e2 ). 0 e2 Deﬁne the so-called Iwahori subgroups by . I0 (r) = u ∈ GL2 (Oq )u mod qr ∈ B(O/qr ) , . I1 (r) = u ∈ I0 (r)u mod qr ∈ U(O/qr ) . These subgroups S have the Iwahori decomposition S = U T U ∼ = U × T × U for open compact subgroups T ⊂ T (Oq ) and U in the opposite unipotent t U. Each x ∈ D shrinks t U: xt Ux−1 ⊂ t U. From this, ∆S = S · D · S and ∆∗S = S · D∗ · S are again multiplicative sub-semigroups of G (this statement includes ∆B = B·D·B = ∆I0 (∞) and ∆∗B = B·D∗ ·B). We call ∆S (resp. ∆∗S ) the expanding (resp. shrinking) semigroup with respect to (S, U). When G = GL(2), ∆S for S = I0 (r) (r > 0) is almost equal to the q-part of the image of the semigroup ∆0 (qr ) introduced in (2.3.11) under the involution ι (strictly speaking, we have ∆0 (qr )p = ∆S modulo center). Exercise 2.38 Prove I0 (r) = t U (qOq )T (Oq )U (Oq ) = t U (qOq ) × T (Oq ) × U (Oq ), & %

1 u u ∈ a for an ideal a of Oq . where U (a) = 0 1 By the Iwahori decomposition, we have, for X = U, B, and an Iwahori subgroup S, 5 5 XξX = Xξu = Xuξ for ξ ∈ D u∈ξ −1 U ξ\U

XξX =

5

u∈ξU ξ −1 \U

u∈U \ξU ξ −1

uξX =

5

u∈U \ξ −1 U ξ

ξuX for ξ ∈ D∗ .

(2.3.33)

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By this fact, the double coset ring (or the Hecke ring; e.g., [IAT] 3.1) generated additively over Z by double cosets of X in ∆X have the following homomorphic relations as algebras: ∼ R(S, ∆S ) = RB R = R(U, ∆U ) R(B, ∆B ) = #↓ #↓ #↓ R∗ = R(U, ∆∗U ) R(B, ∆∗B ) ∼ = R(S, ∆∗S ) = R∗B via UξU → BξB → SξS for ξ ∈ ∆U (resp. ξ ∈ ∆∗U ) and R UξU → Uξ ι U ∈ R∗ for xι x = det(x). These algebras are commutative: T (ξ)T (η) = T (ξη) for T (ξ) = UξU and ξ, η ∈ D or D∗ . Here is a proof of these facts. Since we only need to deal with D or D∗ , in the following lemma, we prove the result for D∗ (and the result for D follows by applying the involution “ι”). Lemma 2.39 Let the notation and the assumption be as above. The algebras R(S, ∆∗S ) for Iwahori subgroups S are commutative, and if S ⊃ B(Zp ), they −1 are all isomorphic ring

to the polynomial

Z[t1 , t2 , t2 ] (with t2 inverted) for 0 0 S and t2 = . If S ⊃ U, we have R(S, ∆∗S ) ∼ t1 = S = 0 1 0 −1 Z[TS ][t1 , t2 , t2 ] for the quotient group TS = T (Oq )S/S, where Z[TS ] is the group algebra of TS . Proof

For ξ ∈ D∗ , we consider the double coset BξB. Decompose 5 B= η(ξBξ −1 ∩ B). η∈Ξ(ξ)

4 Multiplying by ξBξ −1 from the right, we get BξBξ −1 = η∈Ξ(ξ) ηξBξ −1 ⇔ a

4 1 0 BξB = η∈Ξ(ξ) ηξB. If ξ = , we have 0 a2 %

& a b ∈ Bb ∈ a1 −a2 Oq . ξBξ −1 ∩ B = 0 d We may choose the subset Ξ(ξ) inside U so that

1 b Ξ(ξ)

→ b mod a1 −a2 Oq ∈ O/a1 −a2 O 0 1 4 is a bijection. Then we have BξB = η∈Ξ(ξ) ηξB and a formula for the index: with [ξ] = (a1 − a2 ). Writing deg(BξB) for the (B : ξBξ −1 ∩ B) = |[ξ] |−1 q number of right cosets of B in BξB, we ﬁnd deg(BξζB) = deg(BξB) deg(BζB), because [ξζ] = [ξ] + [ζ] for ξ, ζ ∈ D∗ . Since BξBζB ⊃ BξζB, if we can show that deg(BξBζB) = deg(BξζB), we get BξBζB = BξζB and (BξB) · (BζB) = BξζB in the double coset ring R(B, ∆∗B ), which in particular shows the commut∗ ativity of R(B, ∆B ). To see deg(BξBζB) = deg(BξζB), we note BξBζB = η∈Ξ(ξ) η ∈Ξ(ζ) ηξη ζB. This implies deg(BξBζB) ≤ deg(BξB) deg(BζB) = deg(BξζB),

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and hence we get the identity deg(BξBζB) = deg(BξζB). Since the αj s give independent generators of D∗ /T (Oq ), the monoid algebra Z[D∗ /T (Oq )] is isomorphic

0 −1 to a polynomial ring with two variables Z[α, 12 , (12 ) ] with α = 0 1 (because 12 ∈ Z(Oq ) is invertible in D∗ ). The association α → T () = BαB and 12 → BB therefore induces a surjective algebra homomorphism Z[D∗ /T (Oq )] → R(B, ∆∗B ), which can be easily seen to be an isomorphism. Replacing D∗ /T (Oq ) by D∗ /(T (Oq ) ∩ S) = TS × (D∗ /T (Oq )) in the above argument, the same proof works well for any S with I0 (r) ⊃ S ⊃ I1 (r) and yields R(S, ∆∗S ) ∼ = Z[TS ][D∗ /T (Oq )] ∼ = Z[TS ][t1 , t2 , t−1 2 ], where Z[TS ] is embedded into ∗ R(S, ∆S ) by sending t ∈ TS to StS. 2 We let R∗ act on v ∈ V U = H 0 (U, V ) by

T (ξ)v = [UξU]v = π(uξ)v = u∈ξU ξ −1 \U

π(ξ)π(u)vdu,

(2.3.34)

ξ −1 U ξ

and similarly for v ∈ VB in place of v ∈ V U ; then the projection: V U → VB is R∗ -linear. To see the last identity of (2.3.34), it is suﬃcient to recall that we have normalized the measure du so that U du = 1. We may regard the above action as an action of R via the isomorphism R ∼ = R∗ :

v|T (ξ) = v|[UξU] = π((ξu)ι )v = π(ξ ι )π(uι )vdu. (2.3.35)

u∈ξ −1 U ξ\U

ξU ξ −1

0 −j j , we have U(Fq ) = ∪∞ j=0 α Uα . Thus writing T () = [UαU] 0 1 and T (ξ) = [UξU] for ξ ∈ D∗ as an operator on V U = H 0 (U, V ), we see easily from (2.3.34) that T (αj ) = T ()j and for each ﬁnite-dimensional subspace X ⊂ V (B), T ()|X is nilpotent on X U by (2.3.34). For any R-eigenvector v ∈ V U with tv = λ(t)v (t ∈ T (Fq ), v = v mod V (B)), we get /

0 9

−1 :

a 0 a 0 a 0 a 0 a 0 −1 v| U U = U: U λ v = |a d|q λ v, 0 d 0 d 0 d 0 d 0 d For α =

(2.3.36) where | |q is the standard absolute value of Oq such that ||−1 q = N (q). Lemma 2.40 If V = V (π) is admissible, we have a canonical splitting V U ∼ = VB ⊕ V (B)U as Hecke modules, where V U = H 0 (U(Oq ), V ). An absolutely irreducible admissible representation π is called supercuspidal if VB = 0. In other words, by (π1), an absolutely irreducible supercuspidal representation can never appear in a subquotient of an induced representation IndGB λ.

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I1 (r) . The subspace Proof We have by deﬁnition, V U = V U (Zp ) = rV Vr = V S1 (r) is ﬁnite dimensional and stable under R. By the Jordan decomposition applied to T (), we can decompose uniquely such that Vr = Vr◦ ⊕ V nil so that T () is an automorphism on Vr◦ and is nilpotent on V nil . We may replace T () by T (αa ) = T ()a for any positive a in the deﬁnition of the above splitting. Since T () is nilpotent over any ﬁnite-dimensional subspace of V (B), Vr◦ injects into VB ; so, dim Vr◦ is bounded by dim VB ≤ |W| = 2. For any T -eigenvector v ∈ VB , lift it to v ∈ V . Then for suﬃciently large j, π(α−j )v is in V U . Since π(α−j )v is a constant multiple of v, we may replace v and v by π(α−j )v and π(α−j )v, respectively. Then for suﬃciently large k, w = T (αk )v ∈ Vr◦ , and T ()−k w is equal to v for the image w in VB . This ﬁnishes the proof when the action of T on VB is semisimple. In general, take a suﬃciently large r so that Vr surjects down to VB . We apply the above argument to thesemisimpliﬁcation of Vr under the action of the Hecke algebra. Thus V ◦ = r Vr◦ ∼ = VB , and this ﬁnishes the proof of V U = VB ⊕ V (B)U as R-modules. 2 Lemma 2.41

We have

1. If an absolutely irreducible admissible representation π of G is ﬁnite dimensional, it is one dimensional and is a character. 2. If IndGB λ is absolutely reducible, λj has to satisfy λ1 /λ2 (x) = |x|±1 q . G 3. If λ1 /λ2 (x) = |x|±1 q , IndB λ is reducible, the length of the composition series is 2. The inﬁnite dimensional irreducible subquotient in IndG λ is of IndGB λ B denoted by σ(λ1 , λ2 ) and is called the special (or Steinberg) representation of G. G 4. If λ1 /λ2 (x) = |x|−1 q , IndB λ contains the subspace on which G acts 1/2

by the character x → λ1 (det(x))| det(x)|q , and if λ1 /λ2 (x) = |x|q , w contains the quotient on which G acts by the character x → IndGB λ 1/2 λ2 (det(x))| det(x)|q . If the conductor of λ1 is given by qr , the subspace of the inﬁnite-dimensional irreducible subquotient on which I0 (r) acts by a b

→ λ1 (a)λ2 (d) is one dimensional. c d Each vector in the subspace of the inﬁnite-dimensional subquotient (the Steinberg representation) described by the fourth assertion is called a minimal vector, which is uniquely determined up to scalar multiples. Indeed, for any induced repG resentation Ind B λ, we

can have two types of minimal

vectors v1 and v2 on which a b a b v = λ1 (a)λ2 (d)v1 and v = λ2 (a)λ1 (d)v2 if qr is I0 (r) acts by c d 1 c d 2 the smallest ideal among the conductors of λ1 and λ2 . When λ1 is unramiﬁed for a Steinberg representation σ(λ1 , λ2 ), the minimal vector v1 ∈ σ(λ1 , λ2 ) coincides

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with the new vector. A cusp form in the image of the Jacquet–Langlands correspondence in Theorem 2.30 regarded as an element of the representation space of GL2 (Fq ) for q ∈ ΣD is a linear combination of such new vectors. This is another aspect of the word “new” in Theorem 2.30. In particular, a nonzero new vector is not a linear combination of translations by elements of G of vectors ﬁxed by a maximal compact subgroup of G (because in this Steinberg representation, there is no vector ﬁxed by any maximal compact subgroup). Proof Suppose

that the representation space V (π) of π is ﬁnite dimensional. 0 Let α = for a prime element of Oq , and deﬁne 0 1 %

& 1 u n −n n u ∈ Oq . Un = α Uα = 0 1 Since {Un }n gives a fundamental system of neighborhoods of U(Fq ), we ﬁnd a nonzero vector v ﬁxed by Un for some n ∈ Z. Then π(α−m )v is ﬁxed by α−m Un αm = Un−m . This shows that H 0 (Un , V (π)) = 0 for all n. Since Un ⊂ Un−1 , we have an inﬁnite sequence of nontrivial subspace H 0 (U0 , V (π)) ⊃ H 0 (U−1 , V (π)) ⊃ · · · ⊃ H 0 (U−n , V (π)) ⊃ · · · . " From dim V < ∞, we conclude H 0 (U(Fq ), V (π)) = n H 0 (Un , V (π)) = 0, which is stable under B(Fq ) because B normalizes U. Since π is admissible, the stabilizer of 0 = v ∈ H 0 (U(Fq ), V (π)) contains an open subgroup S of G. In particular, the orbit S(∞) of the inﬁnity under S in the one-dimensional projective space P1 (Fq ) = Fq {∞} is open. Since U(Fq ) acts transitively on P1 (Fq )−{∞} = Fq , the subgroup H of G generated by U(Fq ) and S acts P1 (Fq ) transitively. In particular, H contains all conjugates of U(Fq ) and hence all unipotent elements. As is well known, SL2 (Fq ) is generated by unipotent elements (e.g., [PAF] Lemma 4.46). Thus π factors through G/SL2 (Fq ) ∼ = Fq× . Then by Schur’s lemma, π is one dimensional. This proves (1). is reducible. Take a proper subspace V ⊂ W stable Suppose that W = IndGB λ and under G. Then VB is a proper subspace of WB , because V → VB is exact

0 −1 ±1 w ⇔ λ1 /λ2 = | · | for w = (π1). Since dim WB = 2 by (π4), if λ8 = λ q 1 0 (cf. Exercise 2.36), we may assume that VB = V (λ). Since the action of w preserves V , VB also has a nonzero B-eigenvector belonging to λw ; so, VB = WB , a contradiction. Thus if W is reducible, λ1 /λ2 = | · |±1 q . G contains x → λ1 (det(x))| det(x)|1/2 If λ1 /λ2 = | · |−1 , IndG (λ) q ; so, Ind (λ) q

B

B

w ) contains x → λ (det(x))| det(x)|1/2 ; so, is reducible. If λ1 /λ2 = | · |q , IndGB (λ8 q 2 is reducible by (π4). IndGB (λ) As for the last assertion, by an explicit computation, the subspace of minimal (on which I0 (r) acts as speciﬁed) is two dimensional. Indeed, vectors in IndGB λ by Lemma 2.40, the space is isomorphic to VB under the projection V → VB , and

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hence at most two dimensional. One can easily create two linearly independent vectors in the space; so, it is two dimensional. Then by the third assertion, the one-dimensional subquotient takes the one-dimensional subquotient of this twodimensional subspace. The rest gives the desired one-dimensional subspace of the inﬁnite-dimensional irreducible subquotient. 2 All admissible absolutely irreducible representations of G are classiﬁed into four disjoint classes of representations: characters, principal representations, Steinberg representations, and supercuspidal representations. To give a more precise description of the correspondence πD → π in (JL3), we ﬁrst note a consequence of Proposition 2.3: Lemma 2.42 If Dq is a division algebra, each admissible irreducible representation of Dq× over C is ﬁnite dimensional. If it has a nonzero vector ﬁxed by the unit group of maximal order, it is isomorphic to a character λ ◦ N for an admissible character λ : Fq× → C× , where N : Dq → Fq× is the reduced norm map. Proof By Proposition 2.3, the unique maximal order R = ODq of Dq has a unique maximal two-sided ideal m with R/m = O/q. Then mn are all two-sided ideals, and hence 1 + mn is a normal subgroup of Dq× . Let V = V (π) be an admissible irreducible representation of Dq× . Since 1 + mn give a fundamental system of open neighborhoods of the identity of Dq× , for suﬃciently large n, H 0 (1 + mn , V ) = 0. Since 1 + mn is normal in Dq× , H 0 (1 + mn , V ) is stable under Dq× . Then by the irreducibility of V , H 0 (1 + mn , V ) = V . Since V is admissible, H 0 (1 + mn , V ) is ﬁnite dimensional. If H 0 (R× , V ) = 0, by the above argument, we have H 0 (R× , V ) = V , and hence the action of Dq× on V factors through the abelian group Dq× /R× ∼ = Z. Thus by the irreducibility, V is one dimensional. Since the reduced norm map N has kernel inside R× , the character π factors through Fq× ; so, π = χ ◦ N . This ﬁnishes the proof. 2 Secondly, we note that by the strong multiplicity-one theorem (e.g., [AAG] Theorems 5.14 and 10.10), for a given quaternion algebra D over F , the representation ΠD of GD (A(∞) ) on the space generated by ΠD (g)h(x) = h(xg) for all g ∈ GD (A(∞) ) and all h ∈ SκD (N, ε; C) is a direct sum of ﬁnitely many irreducible representations πD with multiplicity one. Here is a slightly more precise description of the correspondence πD → π in (JL3). Start with an eigenform 0 = f ∈ SκD (N, ε; C) of all Hecke operators in hκ (N, ε; C). By the multiplicity-one theorem, the representation πD of GD (A(∞) ) on the space generated by πD (g)f (x) = f (xg) for all g ∈ GD (A(∞) ) is irreducible. We can factor πD = ⊗q πq (D) for local irreducible representations πq (D) of Dq× (for the localization-completion Dq = D ⊗F Fq ). Thus the space V (N, ε; πD ) is the tensor product of the corresponding subspace V (N, εq ; πq (D)), and if qr

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exactly divides N and q is prime to d(D), . V (N, εq ; πq (D)) = v ∈ V (πq (D))u · v = ε(u)v for all u ∈ I0 (r) . We decompose V (N, ε; π) = ⊗q V (N, εq ; πq ) for subspaces V (N, εq ; πq ) ⊂ V (πq ) similarly deﬁned for π = πM2 (F ) . If Dq is a division algebra, our deﬁnition of × S0D (N) implies S0 (N)q = OD,q ; so, πq (D) = λ ◦ N as above. Then the associated (∞)

automorphic representation π = πM2 (F ) of GL2 (FA ) is given as follows: πq (D) if Dq ∼ = M2 (Fq ) ∼ πq = −1 σ(λ, λ| · |q ) if Dq is division and πq (D) is a character λ ◦ N , (2.3.37) where N : Dq× → Fq× is the reduced norm map. Then writing N = d(D)N0 , d(D) is prime to N0 . Then by Lemma 2.42, πq (D) = λ ◦ N for q|d(D), and U (q ) acts on the one-dimensional space V (πq (D)) via multiplication by λ(q ). Since 0 πq = σ(λ, λ| · |−1 q ), H (I0 (1), V (πq )) is one-dimensional, and the eigenvalue of U (q ) on this space is again given by λ(q ) as easily computable by the expression of the representation as an induced representation in Lemma 2.41. If q is prime to d(D), the representations πq and πq (D) are isomorphic, identifying Dq with M2 (Fq ); thus, V (N, εq ; πq (D)) ∼ = V (N, εq ; πq ) as Hecke modules. Thus the action of T (q ) of the corresponding representations match, and hence we have an hκ (N, ε; C)-linear isomorphism V (N, ε; πD ) ∼ = V (N, ε; πM2 (F ) ). The representation π associated to πD appears as an automorphic representation spanned by π(g)f0 for a Hecke eigenform f0 ∈ Sκ (N, ε; C) unique up to scalar multiples. In other words, ﬁxing the isomorphism V (N, ε;πD ) ∼ = V (N, ε; πM2 (F ) ), we D get an inclusion Sκ (N, ε; C) = πD V (N, ε; πD ) ∼ = πM (F ) V (N, ε; πM2 (F ) ) → 2 Sκ (N, ε; C) which is the linear map in Corollary 2.31, and the correspondence f → f0 gives rise to the embedding SκD (N, ε; C) → Sκ (N, ε; C). A Hecke eigenform in SκD (N, ε; C) or Sκ (N, ε; C) is called q-new (resp. q-minimal) for a prime q of F , if it gives a new vector (resp. a minimal vector) of the local q-component of the automorphic representation generated by the automorphic form. 2.3.8 Modular Galois representations For almost all ideals a of the Hecke algebra h = hκ (N, ε; W ), we can associate a modular two-dimensional Galois representation ρa . Thus h may be considered to be a deformation ring parameterizing all “modular” deformations of given level and given “Neben” characters. By the techniques invented by Wiles (and Taylor), under suitable assumptions, we can prove that a local ring of h is the universal deformation ring we studied in Theorem 1.52, and in this way, we will actually prove Theorem 1.52 as Theorem 3.28, Corollary 3.42 and Theorem 3.50. First we describe the representation ρP for prime ideals P of h. In the following description, we normalize the local Artin symbol [u, Fq ] so that [q , Fq ] modulo the inertia subgroup is the arithmetic Frobenius element in Gal(Fq /FN (q) ) for q > 0 with (q) = Z ∩ q.

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Theorem 2.43 Suppose k = κ2 − κ1 + I ≥ I. Let P be a prime ideal of h = hκ (N, ε; W ) and write k(P ) for the quotient ﬁeld of h/P . We assume that k(P ) has characteristic diﬀerent from 2. Then we have a semisimple Galois representation ρP : Gal(F /F ) → GL2 (k(P )) unramiﬁed outside pN such that 1. ρP is continuous with respect to the proﬁnite topology on k(P ) induced from the proﬁnite subring h/P . 2. We have Tr(ρP (F robl )) = T (l ) mod P for all prime ideals l prime to pN and det(ρP ) = ε+ N [κ] for the p-adic cyclotomic character N , where we regard ε+ as a Galois character by global class ﬁeld theory. 3. Let m be the unique maximal ideal containing P . Either if k ≥ 2I and Tp (p) ∈ m or if k = 2I and T p (p ) ∈ m for a prime factor p of p p ∗ in F , we have ρP |Dp ∼ for the decomposition subgroup Dp at = 0 δp p, and δp ([y, Fp ]) = Up (y) mod P for the local Artin symbol [y, Fp ]; in ε1,p (u)u−κ1 for u ∈ Op× . Here we have written particular, δp ([u, F p ]) = y 0 S0 (p∞ N)] for y ∈ Fp× . Up (y) = [S0 (p∞ N) 0 1 4. Write N = N0 c(ε− ). If the prime to p-part N0 of N0 is square free and is − ∼ prime to

c(ε ), for each prime factor l of N0 prime to p, we have ρP |Dl = l ∗ for the decomposition subgroup Dl at l, and δl ([y, Fl ]) = U (y) 0 δl mod P for the local Artin symbol [y, Fl ] (y ∈ Fl× ); in particular, δl ([u, Fl ]) = ε1,l (u) for u ∈ Ol× . (p)

The representation ρP can be realized to have values in GL2 (h/P ) which is unique up to isomorphisms over h/P if ρm for the maximal ideal m containing P is absolutely irreducible. Now we describe the local structure of automorphic representations via local Galois properties. Proposition 2.44 Let the notation and the assumption be as in the theorem. Suppose P = Ker(λ) for the algebra homomorphism λ : hκ (N, ε; W ) → Qp satisfying f |T (q ) = λ(T (q ))f for all prime ideals q for a Hecke eigenform f ∈ Sκ (N, ε; Q). Write π = ⊗q πq for the irreducible representation generated by f . Let l p be a prime. × 1. If πl ∼ , then the = π(η1 , η2 ) for characters ηj : Fl× → Q

restriction of ρP η2 0 to the decomposition group Dl is isomorphic to , where we abused 0 η1 ×

notation so that ηj is identiﬁed with the Galois character Dl → Q inducing the local characters ηj via local class ﬁeld theory.

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2. If πl ∼ = σ(η, η| · |−1 l ), the restriction of ρP to

the decomposition group Dl ηNl ∗ is isomorphic to a non-semisimple for the cyclotomic character 0 η −1 Nl : Dl → Z× p with Nl ([u, Fl ]) = |u|l . 3. If πl is supercuspidal and N0 = N/c(ε− ) is prime to c(ε− ), then l2 |N0 and ρP restricted to Dl is absolutely irreducible. Here is a brief outline of the proof of the above results. When F = Q, all these follow from the theorems of [GME] in 4.2.3–7 (see also [IAT] 7.6). First suppose that k(P ) has characteristic 0. By Theorem 2.28, the projection π : h → h/P (regarded as a linear form π : h → h/P ) gives rise to a Hecke eigenform f ∈ Sκ (N, ε; k(P )) with a∞ (l , f ) = π(T (l )) for all primes l; so, we are in the setting of Proposition 2.44. Let N0 = N/c(ε− ). By looking at the Fourier expansion, if a prime factor l of N satisﬁes l2 N0 and l c(ε− ), we can show that the Hecke operator T (l ) on Sκ (N, ε; C) is invertible (e.g., [MFM] Theorem 4.6.17 or [H88b] Lemma 12.2). Then by Lemma 2.40, πl cannot be supercuspidal, which implies that l2 |N0 when πl is supercuspidal (in (3) of Proposition 2.44). If there exists a quaternion algebra D/F with dim ZD = 1 such that f is in the image of the Jacquet–Langlands correspondence ιh/P , the existence of ρP satisfying the conditions in Theorem 2.43 and Proposition 2.44 follows from the work of Carayol [C1] (see also [68c] in [CPS] and [H81a] Theorem 4.12). In particular, if [F : Q] is odd, we have ρP by the work of Carayol. Even if [F : Q] is even, Blasius and Rogawski [BlR] realized the representation ρP in the p-adic ´etale cohomology group on the Shimura variety of a unitary group of dimension 3, and at the same time, R. Taylor [Ta] (and [Ta2]) generalized the method of Wiles in [Wi] to obtain the representation ρP by gluing together Galois representations coming from quaternion algebras D as above, using Wiles’ observation (see [Ta]) that the image of the Jacquet–Langlands correspondence is p-adically dense in the space of p-adic modular forms (if we vary quaternion algebras D). This density follows from Theorem 3.16 after a nontrivial argument. Once ρP is constructed, for the reduced part hred (i.e., hred = h modulo the nilradical), we can ﬁnd a pseudo-representation π : Gal(Q/F ) → hred unramiﬁed outside pN in the sense of Wiles (see 1.1.3) with Tr(π(F robl )) = T (l ) for all l outside pN (by the gluing technique described in [GME] 4.2.5). Out of the pseudo-representation π mod P , we can always create a representation as in (1) and (2) by Proposition 1.23. Since Taylor’s construction comes from gluing mod pn representations of Carayol which satisﬁes (4), it supplies us the assertion (4) of the theorem for all ρP . As for the assertion (3) of the theorem, it is veriﬁed in [Wi] and [H89b]. It remains to prove the existence of ρm and the assertions (3) and (4) of the theorem for each maximal ideal m of h. Take a prime ideal P ⊂ m (so, k(P ) has characteristic 0). Then by conjugating ρP , we may assume that ρP has values in the maximal compact subgroup GL2 (WP ) in GL2 (k(P )) (Corollary 2.5), where WP is the p-adic integer ring of k(Q). Let ρm = ρP mod mP for the maximal

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ideal mP of WP . The representation ρm satisﬁes the four conditions for the coeﬃcient ring k(mP ) = WP /mP in place of k(m). Since Tr(ρP ) = Tr(π) mod P , the trace of ρm has values in h/m. Since the characteristic of k(m) is odd, by the theory of pseudo-representations, if ρm is irreducible, the representation ρ : Gal(Q/F ) → GL2 (h/m) constructed out of the pseudo-representation π mod m has the same trace as ρm . Since ρP is also constructed by the same pseudorepresentation π, if we follow the same procedure as described in Proposition 1.23 to construct ρP and ρ , we ﬁnd that ρ = ρP mod mP = ρm . Thus in this case, ρm satisﬁes the assertions (3) and (4); so, we put ρm = ρm . If ρm is reducible, we deﬁne ρm by the semisimpliﬁcation of ρm . Thus ρm ∼ = ⊕δ for two global characters and δ. Suppose that T (p ) ∈ m for all p|p. Then we may deﬁne a character δ p : Dp → k(m)× by δ p ([y, Fp ]) = Up (y) mod m for y ∈ Fp× . Then ρm for the maximal ideal mP of WP satisﬁes the assertions (3) and (4) for k(mP ) in place of k(P ). Put p = det(ρP )δp−1 mod mP . Since det ρP has values in h/P , p has values in k(m)× . In particular, we have ρm |Dp ∼ = p ⊕δ p for the decomposition subgroup Dp at p. By the Brauer–Nesbitt theorem (e.g., [MFG] Corollary 2.8; see also Proposition 1.25 in the text), we have {p , δ p } coincides with {|Dp , δ|Dp } as sets; so, the assertion (3) follows for our choice of ρm . The assertion (4) of the theorem can be proven similarly. The uniqueness under absolute irreducibility follows again from the Brauer– Nesbitt theorem. We consider the following condition, N/c(ε− ) is square-free and is prime to c(ε− ).

(sf)

We can generalize the result in the above theorem from a prime ideal to any ideal of a local ring of h under mild assumptions including (sf): Corollary 2.45 Suppose (sf). Let T be the localization of hκ (N, ε; W ) at a maximal ideal m. If ρm is absolutely irreducible, for any ideal a of T containing the nilradical of T, we have a unique Galois representation ρa : Gal(Q/F ) → GL2 (T/a) up to isomorphisms over T/a such that 1. ρa is continuous with respect to the proﬁnite topology on T/a. 2. We have Tr(ρa (F robl )) = T (l ) mod a for all prime ideals l prime to pN and det(ρa ) = ε+ N [κ] for the p-adic cyclotomic character N , where we regard ε+ as a Galois character by global class ﬁeld theory. Moreover, if we assume further that T (p ) ∈ m for all prime factors p of p in p ∗ F and that p = δ p for all p|p, then ρa |Dp ∼ for the decomposition = 0 δp subgroup Dp at p, and δp ([y, Fp ]) = Up (y) mod a for the local Artin symbol [y, Fp ]; in particular, p ≡ p mod m and δ p ≡ δ p mod m. As for the restriction ∗ l to Dl for a prime l outside p, if ρm |Dl ∼ with l = δ l for a prime factor = 0 δl

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l ∗ l of N0 prime to p, we have ρa |Dl for the decomposition subgroup 0 δl Dl at l, and δl ([y, Fl ]) = Up (y) mod a for the local Artin symbol [y, Fl ]; in particular, δl ([u, Fl ]) = ε1,l (u) for u ∈ Ol× . ∼ =

Proof Let π : Gal(Q/F ) → hred be the pseudo-representation as in the above sketch of the proof of Theorem 2.43. Let πT be the projection of π to Tred for the reduced part Tred of T. Let t(l ) and u(y) for the projection to Tred of the Hecke operator T (l ) and Up (y). By the irreducibility of ρm , by the method of Wiles, we have a representation ρ : Gal(Q/F ) → GL2 (Tred ) unramiﬁed outside pN and Tr(ρ(F robl ) = T (l ) for all prime ideals l outside pN (Proposition 1.23). The uniqueness under absolute irreducibility follows from Chebotarev’s density theorem and the theorem of Carayol and Serre (see Proposition 1.25). By this uniqueness, we have ρm = ρ mod m. Since Tred ⊗Z Q = P k(P ), where P runs through prime ideals of T of residual characteristic P , we have ρ ∼ = P ρP over Tred ⊗Z Q. Let δp : Dp → Tred be the character given by δ([y, Fp ]) = u(y), and deﬁne p = det(ρ)δp−1 . Writing the representation space of ρ as V (ρ), we deﬁne V (δp ) by the quotient of V (ρ) by {x ∈ V (ρ)|σv = (σ)v ∀σ ∈ Dp }. By (3) of Theorem 2.43, V (δp ) ⊗Z Q is free of rank 1 over Tred ⊗Z Q. Since the semisimpliﬁcation of ρm |Dp is isomorphic to m ⊕ δ p , we ﬁnd from m = δ p that V (δp ) is free of rank 1 over Tred . This shows the second assertion. The third assertion follows from a similar argument. 2 2.4 The integral Jacquet–Langlands correspondence In this section, assuming that F = Q and ﬁxing a division quaternion algebra D/Q , we describe, in a more elementary manner, the Jacquet–Langlands correspondence, its integral version, and its relation to cohomology groups over Shimura varieties and automorphic manifolds. More general cases of quaternion algebras over totally real ﬁelds will be described in 3.2.7 (see also [PAF] Section 4.3.2). By assuming F = Q, things become a lot simpler at least for the notation concerned, but still the idea of making the correspondence integral would be clear. We normalize the Jacquet–Langlands correspondence using cohomology groups. 2.4.1 Classical Hecke operators In this section, we assume that D = M2 (Q). For a positive integer k, we choose κ = k = (0, k − 1), let N = (N ) for a positive integer N , and choose ε = 1 made up of the identity characters εj and ε+ (x) = |x|2−k A . The space S k (N, 1; C) is isomorphic to the classical space of elliptic modular forms Sk (Γ0 (N )). In this section, we keep this choice of κ, ε, and N. We shall ﬁrst establish a ﬁrm link between the classical space of modular forms with adelic ones.

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130

By our deﬁnition for N = (N ), we have & %

a b ∈ GL2 (Z) c ∈ N Z . Γ0 (N ) = S0 (N ) = c d

(2.4.1)

By the strong approximation theorem, we ﬁnd 0 (N )GL2 (R)+ , GL2 (A) = GL2 (Q)Γ where GL2 (R)+ = {g ∈ GL2 (R)| det(g) > 0}. Recall that we have√deﬁned f1 : H → C (for √the ﬁnite idele g = 1) by f1 (z) = f (s∞ )Jκ (s∞ , −1) = det(s∞ )−1 f (s∞ )(c −1 + d)k for f ∈ Sk (N, 1; C), where s∞ ∈ GL2 (R)

with √ y x s∞ ( −1) = z. Writing z = x + iy, we may take s∞ = , then 0 1

y x f1 (z) = y −1 f . Then in this case, we have a canonical association: 0 1 Sk (N, 1; C) f → f1 (z) ∈ Sk (Γ0 (N )), where f1 (z) is as in (SA2) for g = 1 and Sk (Γ0 (N )) is the space of holomorphic functions deﬁned on H satisfying the following two conditions:

az + b a b 0 (N ) ∩ SL2 (Q); (s1) f ∈ Γ0 (N ) = Γ = (cz + d)k f (z) for c d cz + d

az + b a b (s2) f |α(z) = f (cz + d)−k for all α = ∈ SL2 (Q) has its c d cz + d Fourier expansion without terms e(nz) for nonpositive n ≤ 0. This condition is equivalent to the boundedness of (f |α(z)) Im(z)k/2 over all H. We now compute explicitly the eﬀect of Hecke operators T (p) and U (p). Lemma 2.46 have

or Γ0 (N ) ⊂ GL2 (Z). Then we 0 (N ) ⊂ GL2 (Z) Let Γ be either Γ   p    Γ 0

4p−1 1 0 j=0 Γ

0 1 1 0 Γ Γ= 0 p  4p−1 1 j     j=0 Γ 0 p

j p

if p N , (2.4.2) if p|N .

Proof Since the argument in each case is essentially the same,

we only deal a b ∈ M2 (Z) and with the case where p N and Γ = Γ0 (N ). Take any γ = c d ad − bc = p. If c is divisible by p, then ad is divisible by p; so, one of a and d has a factor p. We then have

a b a/p b p 0 p 0 γ= = ∈ Γ0 (N ) c d c/p d 0 1 0 1

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131

if a is divisible by p. If d is divisible by p and a is prime to p, choosing an integer

−1 1 u ∈ GL2 (Z). If c is not u with 1 ≤ u ≤ p with ua ≡ b mod p, we have γ 0 p divisible by p but a is divisible by p, we can interchange a and c via multiplication 0 −1 by from the left side. If a and c are not divisible by p, choosing an 1 0

1 0 integer u so that ua ≡ −c mod p, we ﬁnd that the lower left corner of γ u 1 is equal to ua + c and is divisible by p. This ﬁnishes the proof. 2 From this we deﬁne    (∞)    p−1 1 0   + g p f g  j=0 f   0 p 0  f |T (p)(g) = (∞)    p−1 p j    f g    j=0 0 1

(∞)  

j 1

if p N ,

if p|N . (2.4.3)

Note that

f

x 1 1 0

y 0

0 p

1 This shows that g →

f g 0 −1

f1 (pz) = (py)

f

p Similarly g →

f g 0

j 1

py 0

(∞)

0 p

y x =f 0 1

1 0 y x =f 0 p−1 ∞ 0 1

y x −1 p 0 = f p∞ 0 1 ∞ 0 1

py px = pk−2 f . 0 1

1 0 0 p

(∞)

(∞)

gives rise to pk−1 f1 (pz), because

px 1

=p

1−k −1

y

f

y 0

x 1 1 0

0 p

(∞) .

(∞) gives rise to f1 ( z+j p ). Thus we have

 # $ pk−1 f1 (pz) + p−1 f1 z+j j=0 p (f |T (p))1 (z) = p−1 # z+j $  j=0 f1 p

if p N , if p|N .

(2.4.4)

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132

This is exactly the classical Hecke operator acting on Sk (Γ0 (N )) (as deﬁned in [IAT] Chapter 3 and [LFE] Chapter 6). ∞ Writing f1 = n=1 a(n, f )q n for q = e(z), we ﬁnd (as already remarked earlier in (2.3.16))

m k−1 a(m, f |T (p)) = a(mp, f ) + p a ,f . p More generally, using the decomposition (see [IAT] Proposition 3.36): %

=

a b c d

5 d−1 5 a b=0

& and (ad − bc)Z = nZ c ∈ N Z ∈ M2 (Z)

(∞) a b Γ0 (N ) (a > 0, ad = n, (a, N ) = 1, a, b, d ∈ Z), (2.4.5) 0 d

we get an identity of Hecke operators a(m, f |T (n)) =

0
dk−1 a

# mn d2

$ ,f .

(2.4.6)

Exercise 2.47 1. Prove (2.4.6) assuming (2.4.5). 2. Prove that f is a Hecke eigenform if and only if a(mn, f ) = a(m, f )a(n, f ) for all mutually prime positive integers m and n (hint: use (2.4.6)). 2.4.2 Hecke algebras We keep the notation introduced in the previous subsection; in particular, κ = (0, k − 1) = k, N = (N ), and 1 = (id, id, | · |2−k A ). For any subalgebra A ⊂ C, by our deﬁnition of the space of A-integral elliptic modular forms, we have . Sk (Γ0 (N ); A) = Sk (N, 1; A) = f ∈ Sk (N, 1; C)a(n, f ) ∈ A . (2.4.7) We admit the following important result which is the integral version of (2.3.18): Theorem 2.48 (G. Shimura) over C. In particular, we have

The space Sk (Γ0 (N ); A) spans Sk (Γ0 (N ); C)

Sk (Γ0 (N ); C) ∼ = Sk (Γ0 (N ); Z) ⊗Z C and Sk (Γ0 (N ); A) ∼ = Sk (Γ0 (N ); Z) ⊗Z A canonically. See [IAT] Theorem 3.52 (and Chapter 4 in the text) for the proof of the above theorem. By (2.4.6), the Hecke operator T (n) preserves A-integrality of modular forms. We deﬁne the Hecke algebra Hk (Γ0 (N ); A) with coeﬃcients in A by the A-subalgebra of EndA (Sk (Γ0 (N ); A)) generated by all Hecke operators T (n).

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Then Hk (Γ0 (N ); A) is a torsion-free commutative A-algebra of ﬁnite type (the commutativity follows from (2.4.6)). Exercise 2.49 Show T (m)T (n) = T (n)T (m) using (2.4.6). We deﬁne an A-bilinear pairing , : Hk (Γ0 (N ); A) × Sk (Γ0 (N ); A) → A by h, f = a(1, f |h). Proposition 2.50

We have the following canonical isomorphism: HomA (Sk (Γ0 (N ); A), A) ∼ = Hk (Γ0 (N ); A)

and HomA (Hk (Γ0 (N ); A), A) ∼ = Sk (Γ0 (N ); A), and the latter is given by sending an A-linear form φ : Hk (Γ0 (N ); A) → A to ∞ the q-expansion n=1 φ(T (n))q n . Proof By Shimura’s theorem, we may assume that A = Z. Actually, we start with proving the result for A = Q. Since Hk (Γ0 (N ); Q) and Sk (Γ0 (N ); Q) are both ﬁnite dimensional, we only need to show the nondegeneracy of the pairing. By (2.4.6), we ﬁnd T (n), f = a(n, f ); so, if h, f = 0 for all n, we ﬁnd f = 0. If h, f = 0 for all f , we ﬁnd 0 = h, f |T (n) = a(1, f |T (n)h) = a(1, f |hT (n)) = T (n), f |h = a(n, f |h). Thus f |h = 0 for all f , implying h = 0 as an operator. ∞ As for A = Z, we only need to show that φ → n=1 φ(T (n))q n is well deﬁned and is surjective onto Sk (Γ0 (N ); Z) from Hk (Γ0 (N ); Z). The cusp form f ∈ Sk (Γ0 (N ); Q) corresponding to φ satisﬁes h, f = φ(h); so, a(n, f ) = T (n), f = ∞ φ(T (n)). Thus f = n=1 φ(T (n))q n ∈ Sk (Γ0 (N ); Q). However f ∈ Sk (Γ0 (N ); Z) ⇐⇒ φ ∈ Hk (Γ0 (N ); Z), because Hk (Γ0 (N ); Z) is generated by T (n) over Z. This is enough to conclude surjectivity. 2 We now restate Corollary 2.29 for elliptic modular forms: Proposition 2.51 Let A be a subalgebra of C, and write simply H for the algebra Hk (Γ0 (N ); A) and S for the space Sk (Γ0 (N ); A). Let V be an H-module and V be an A-module of ﬁnite type with an A-bilinear product ( , ) : V × V → A. Then we have: (1) The formal q-expansion for v ∈ V and w ∈ V : f (v ⊗ w) =

∞

n=1

(v|T (n), w)q n

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Automorphic forms on inner forms of GL(2)

gives a unique element of S. (2) The map v ⊗ w → f (v ⊗ w) gives an A-linear map of V ⊗A V into S with f ((v|T (n)) ⊗ w) = f (v ⊗ w)|T (n). If further V is an H-module and (v|h, w) = (v, w|h) for all v ∈ V , w ∈ V and h ∈ H, then the map f induces an H-linear map: V ⊗H V → Sk (Γ0 (N ); A). (3) Suppose that R is an A-algebra direct summand of H, and put V (R) = RV and S(R) = RS. If V (R) is R-free of ﬁnite rank and HomA (V (R), A) is embedded into V by the pairing ( , ), then the map f : V (R) ⊗A V → S(R) is surjective. Taking V and V to be appropriate cohomology groups over Shimura varieties (made out of a quaternion algebra D), we will prove later that the theta series of the norm form of D actually give the correspondence v ⊗ w → f (v ⊗ w) when we apply the technique of Theorem 3.1 of [81c] in [CPS]. Proof We have an isomorphism ι : HomA (H, A) ∼ = S given by a(n, ι(φ)) = φ(T (n)) (Proposition 2.50), which is an H-linear map (that is, ι(φ ◦ h) = ι(φ)|h). Since V is an H-module, h → (v|h, w) gives an element of HomA (H, A) and hence an element in S. The element has the expression as in (1) by the above explicit form of ι. The assertion (2) is then clear from (1). As for (3), by the isomorphism HomA (V (R), A) → V , each element of Hom(R, A) ∼ = S(R) is a ﬁnite A-linear combination of h → (v|h, w) for v ∈ V (R) and w ∈ V ; so, the surjectivity follows. 2 ∞ Exercise 2.52 Prove that Hk (Γ0 (N ); Z) = n=1 ZT (n). ∞ Here is a hint to this exercise. Show ﬁrst n=1 ZT (n) ∼ = HomZ (Sk (Γ0 (N ); Z), Z) and use Proposition 2.50. 2.4.3 Cohomological correspondences Since the isomorphism of Theorem 2.30 is noncanonical, we try to rigidify it using some ideas of Eichler, Shimizu, and Shimura. Recall the automorphic manifold, for an open subgroup U ⊂ GD (Z): × Y (U ) = Y D (U ) = GD (Q)\GD (A)/R× U · C∞ ∼ \GD (A)+ /Z(R)U · C∞ , = D+

where GD (R)+ = {g ∈ D∞ |N (g) > 0}, GD (A)+ = GD (A(∞) ) × GD (R)+ and × × × = DA+ ∩ GD (Q). If D is deﬁnite, D∞ = R× · C∞ , so, Y (U ) ∼ D+ = {tj }j=1,...,h 4 h D D × if G (A) = j=1 G (Q)tj U · D∞ by the approximation theorem. When D is × we have indeﬁnite, then by the strong approximation theorem, if N (U ) = Z × DA+ = GD (Q)+ U · GD (R)+ .

Since D is indeﬁnite, D∞ ∼ = M2 (R); so, GD (R)+ ∼ = GL2 (R)+ = {g ∈ GL2 (R)| det(g) > 0}.

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Then we ﬁnd × , Y (U ) ∼ Proposition If N (U ) = Z = Γ(U )\H √ 2.53 Assume D to be indeﬁnite. D × by g → g∞ ( −1), where Γ(U ) = Γ1 (U ) = G (Q)+ ∩ U · D∞ . More generally if 4h 4h A× GD (Q)tj U · GD (R)+ with tj,∞ = 1, then Y (U ) ∼ = j=1 Γtj (U )\H by D = j=1 √ tj s → s∞ ( −1) for s ∈ U · GD (R)+ . × When F = Q, the D∞ -module L( k ∗ ; C) is simply the space of homogeneous polynomials in (X, Y ) of degree n = k − 2 with coeﬃcients in C. Thus we simply write it as L(n; C), and for any commutative ring A with identity, we write L(n; A) for the space of homogeneous polynomials in (X, Y ) of degree n = k − 2 with coeﬃcients in A. For a subalgebra A of C, suppose that the image of OD in M2 (C) falls in M2 (A). Then we#let OD$act on L(n; A) through the projection ι X to D∞ ⊂ M2 (C) by g · Φ X . If U is suﬃciently small so that the Y = Φ g Y (∞) ×

(this image Γ(U )x of Γ(U )x in GD (Q)/Q× acts freely on ZD for all x ∈ AD means that Γ(U )x = {1} if ID = ∅), and the action of Γ(U )x on L(n; A) factors through Γ(U )x . Then we can deﬁne an ´etale space over Y (U ): L(n; A) = GD (Q)\ GD (A) × L(n; A) /Z(R)U · C∞ , D D etale space where γ(x, P )uz = (γxu, u−1 ∞ P ) for u ∈ U · Ci and γ ∈ G (Q). This ´ gives rise to the sheaf L(n; A)/Y (U ) of locally constant sections. We consider the sheaf cohomology group H q (Y (U ), L(n; A)). If f ∈ H 0 (Y (U ), L(n; A)) is a D function f : GD (A) → L(n; A) with f (γxu) = u−1 ∞ f (x) for all u ∈ U · Ci and D 0 D γ ∈ G (Q). Thus if D is deﬁnite, H (Y (U ), L(n; A)) = Mk (U ; A). Since Y (U ) ∼ = x Γx \ZD for ﬁnitely many x with xp = 1, we have a canonical isomorphism (cf. [IAT] Chapter 8 and [LFE] Appendix): ; H q (Y (U ), L(n; A)) ∼ H q (Γ(U )x , L(n; A)) for q = |ID |, (2.4.8) = x

where the right-hand side is the direct sum of the group cohomology of the Γx – module L(n; A). Here q = 0, 1 and H 0 (Γ(U )x , L(n; A)) is the subspace made up of P ∈ L(n; A) with γP = P for all γ ∈ Γ(U )x . The cohomology group H 1 (Γ(U )x , L(n; A)) is given as follows: a map c : Γ(U )x → L(n; A) is called a 1-cocycle if c(αβ) = αc(β) + c(α). If c(α) = (α − 1)P for a ﬁxed P ∈ L(n; A), such a cocycle is called 1-coboundary. Then H 1 (Γ(U )x , L(n; A)) =

1-cocycles . 1-coboundaries

See [IAT] Chapter 8, [LFE] Appendix and [MFG] Chapter 4 for expositions on diverse cohomology groups. The kernel E = Ker(Γ(U )x → Γ(U )x ) is a subgroup of units Z× = {±1}. The action of ∈ E on L(n; A) is the multiplication by n = 1; so, if n is odd, E has to be trivial. Even if Γ(U )x does not act freely on the module L(n; A), we

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4 still have Y (U ) ∼ = x Γx \ZD for ﬁnitely many x with xp = 1, we can deﬁne the left-hand side of (2.4.8) by the right-hand side of (2.4.8). If U ⊂ U , we have the trace map TrU /U (that is, the transfer map in group cohomology; see [MFG] 4.3.1) and the restriction map ResU /U : TrU /U : H q (Y D (U ), L(n; A)) → H q (Y D (U ), L(n; A)) ResU /U : H q (Y D (U ), L(n; A)) → H q (Y D (U ), L(n; A)). The restriction map is obtained by restricting the cocycle of Γx (U ) to the 4d subgroup Γx (U ). To get the trace map, we split Γx (U ) = j=1 Γx (U )αj . Then αj γ = γ (j) αj for γ (j) ∈ Γx (U ) and x (U ). Then for a cocycle γ ι∈ Γ(j) c : Γx (U ) → L(n; A), we deﬁne Tr(c)(γ) = j αj c(γ ). Exercise 2.54 Check by computation that Tr(c) : Γx (U ) → L(n; A) is a 1-cocycle of Γx (U ). The map c → Tr(c) induces the trace map on cohomology groups. Write ∆ for d(D). Let N be an integer prime to ∆, and put N = N ∆. Recall D (∆) ). We put 0 (N )(∆) × O× , identifying O (∆) with M2 (Z (N ) = Γ Γ 0 D D,∆ %

& a b ∈ M2 (ZN ) with c ∈ N ZN , N (x)Z = nZ . TD (n) = x ∈ OD xN = c d We have a natural action of the operator TD (n) on H q (Y0D (N ), L(n; A)), because the Hecke operator on the cohomology group corresponding to the double coset D (N ) and g ∈ O D ∩ GD (A(∞) ) is given by U gU ⊂ TD (n) for U = Γ 0 [U gU ] = (TrU/(gU g−1 ∩U ) ◦ [g] ◦ ResU/(U ∩g−1 U g) ).

(2.4.9)

Here [g] : L(n; A) → L(n; A) is induced from the action of g on L(n; A); in other words, for a cocycle c, c|[g](γ) = g ι c(gγg −1 ) when q = 1. We recall the deﬁnition of the action when q = 0. In this case, we may regard each cohomology class as a global section f : GD (A) → L(n; A) with f (αxu) = D (N )D× . Decomposing Γ D (N )y Γ D (N ) = f (x) for α ∈ GD (Q) and u ∈ Γ u−1 ∞ 0 0 0 4∞ D g g Γ0 (N ), we have

D D f (xg). (2.4.10) f |[Γ 0 (N )y Γ0 (N )] =

Obviously the above operator preserves Iv(N, ε; A) and hence induces a linear operator on SkD (N, ε; A). Write q = |ID |. Suppose that D is a division algebra. Then Y (U ) is a q-dimensional compact complex analytic space (and is a manifold if Γ(U )x D (N )) as Y D (N ). Let V be the image of acts freely on ZD ). We write Y (Γ 0 0 q D q D H (Y0 (N ), L(n; A)) in H (Y0 (N ), L(n; A ⊗Z Q)). By the Eichler–Shimura isomorphism (as we describe in the following subsection) combined with

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137

Theorem 2.30, the above cohomology groups have the natural action of the Hecke algebra Hk (Γ0 (N ); A). We consider the duality pairing [ , ] on L(n; A):   −1

n−j j n−j j  j n  = bj X Y , aj X Y (−1) bn−j aj . (2.4.11) j j 0≤j≤n

0≤j≤n

× D D Set ΓD 0 (N ) = Γ0 (N ) ∩ D+ . Then as Γ0 (N )-modules, this pairing satisﬁes:

[γP, γP ] = [P, P ].

(2.4.12)

Then the pairing [ , ] induces a pairing [ , ] : L(n; A) × L(n; A) → A of the sheaves if it has values in A. Then by the compactness of Y0D (N ), the cup product pairing induces a nondegenerate pairing: ( , ) : H q (Y0D (N ), L(n; AQ )) × H q (Y0D (N ), L(n; AQ )) → AQ , where AQ = A ⊗Z Q. Thus we obtain from Proposition 2.51 the following result: Proposition 2.55 Let V = H q (Y0D (N ), L(n; W)). Suppose q = |ID | and that D is a division algebra, and write K for the quotient ﬁeld of W. Let V ∗ be the dual W-lattice of V in H q (Y0D (N ), L(n; K)) under the Poincar´e duality: ( , ) : H q (Y0D (N ), L(n; W)) × H q (Y0D (N ), L(n; W)) → K. Then, if n > 0 or q > 0, we have an Hk (Γ0 (N ); W)-linear map f : V ⊗W V ∗ → Sκ (Γ0 (N ); W) deﬁned by the q-expansion: f (v ⊗ w) =

∞

(v|T (n), w)q n ,

n=1 ∗

where we regard V ⊗W V as an Hk (Γ0 (N ); W)-module through the left factor V . If q = n = 0, replacing V by S0D (N, 1; W) and V ∗ by the dual of S0D (N, 1; W) induced by (·, ·), we have the same conclusion. We will give later in 2.5.3 (see Remark 2.66) a direct argument (when q = 0) which (partially) proves the above theorem without recourse to the Jacquet– Langlands–Shimizu correspondence.

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Exercise 2.56 1. Suppose that ID = ∅. Prove that γ n = 1 for a positive integer

n if z z D = γ ∈ G (Q) with N (γ) = 1 ﬁx a point z ∈ H. Hint: γ 1 1

z z ∗ ∗ cz + d 0 if γ = . c d 1 1 0 cz + d 2. Suppose that ID = ∅. Prove that if U is suﬃciently small, Γ(U )x acts freely on H without ﬁxed points for all x ∈ GD (A(∞) ), and the action of Γ(U )x on L(n; A) factors through Γ(U )x . Hint: Use Exercise 1. 3. Prove that (i) Tr(c) is a 1-cocycle (which might depend on the choice of the 4d decomposition: Γx (U ) = j=1 Γx (U )αj ) and (ii) the trace map on cohomology groups 4dis well deﬁned, independent of the choice of the decomposition: Γx (U ) = j=1 Γx (U )αj . 4. Prove (2.4.12). 2.4.4 Eichler–Shimura isomorphisms 0 (N ); C) and the Supposing that D is a division algebra, we shall relate SkD (Γ q D cohomology group H (Y0 (N ), L(n; C)). The isomorphism is called the Eichler– Shimura isomorphism. When |ID | = 0, we have trivially 0 (N ); A) = H 0 (Y0D (N ), L(n; A))/Iv(N, 1; A). SkD (Γ So, there is nothing left to do. Thus we assume |ID | = 1 and D is a division D algebra. Then Y0D (N ) is a Riemann surface given by ΓD 0 (N )\H for Γ0 (N ) = D (N )×GD (R)+ ). Thus Shimura’s theory described in [IAT] Chapter 8 GD (Q)∩(Γ 0 applies, which yields the following canonical isomorphism D ∼ 1 D Sk (ΓD 0 (N ); C) ⊕ S k (Γ0 (N ); C) = H (Γ0 (N ), L(n; C))

∼ = H 1 (ΓD 0 (N )\H, L(n; C)),

(2.4.13)

∼ D (Γ D (N ); C) is made up of complex conjugates where S k (ΓD 0 (N ); C) = S 0 k,∅ D of functions in Sk (ΓD 0 (N ); C), and Sk (Γ0 (N ); C) is made up of holomorphic functions f : H → C satisfying the following automorphy condition:

f (γ(z)) = f (z)JkD (γ, z) for all γ ∈ ΓD 0 (N ).

(2.4.14)

D D The fact that Sk,∅ (Γ0 (N ); C) ∼ = S k (ΓD 0 (N ); C) follows by a computation similar to the one done just below (SB1–2). The last isomorphism in (2.4.13) is the comparison isomorphism of sheaf cohomology and group cohomology (see [LFE] Appendix). The ﬁrst isomorphism in (2.4.13) is by sending f ∈ Sk (ΓD 0 (N ); C) γ(z) n f (z)(X + zY ) dz ∈ L(n; C) and sending f ∈ to a 1-cocycle cf (γ) = z γ(z) D S k (Γ0 (N ); C) to a 1-cocycle cf (γ) = z f (z)(X + zY )n dz ∈ L(n; C). A

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139

detailed proof of this fact is given in [IAT] Chapter 8 (see also [LFE] Chapter 6); so, we omit details of the proof of the following theorem. Theorem 2.57 (G. Shimura) Suppose |ID | = 1 and D is a division algebra. Then we have a canonical isomorphism of Hecke modules: D (N ); C) ⊕ S D (Γ D (N ); C) ∼ SkD (Γ = H 1 (Y0D (N ), L(n; C)). 0 0 k,∅ Exercise 2.58 By a direct computation, show that cf (αβ) = αcf (β) + cf (α) for α, β ∈ ΓD 0 (N ). 2.5 Theta series We ﬁrst interpret our theta series f (v ⊗ w) in terms of classical theta series (supposing that D is deﬁnite), and then we prove that f (v ⊗ w) is a modular form directly without using the Jacquet–Langlands theorem (see Remark 2.66). 2.5.1 Quaternionic theta series We start with a deﬁnite quaternion algebra D/Q . We ﬁx a maximal order OD and identify OD,p with M2 (Zp ) once and for all for all prime p unramiﬁed in D/Q. Let ∆ be the product of primes ramifying in D/Q; so, Dp for p|D is a division D (∆). For simplicity, we assume that N = ∆. × = Γ algebra. Note that O 0 D We ﬁx complete representative sets {a1 , . . . , ah } for D (R) GD (Q)\GD (A)/GD (Z)G × R× = (A(∞) )× R× , × )N (D× ) = Q× Z with ai,∆ = ai,∞ = 1. Since N (D× )N (O ∞ + + + D we may also assume that N (ai ) = 1, multiplying ai by suitable γ ∈ GD (Q) and D (R) from the left and the right side. Recall that the right ideal u ∈ GD (Z)G classes right fractional ideals of OD ClD = ∼ for a ∼ b ⇔ a = αb for α ∈ GD (Q) is in bijection with {a1 , . . . , ah } by associating D ∩ D, which is a lattice of D. We consider to aj the right ideal aj = aj O D × −1 × −1 ai a−1 j = ai · OD aj ∩ D and Γi = G (Q) ∩ ai OD ai D∞ .

(2.5.1)

If γ ∈ Γi , then N (γ) is a positive invertible integer, because N (γ)−1 = N (γ −1 ) ∈ Z and N (γ) ∈ N (D∞ ). Thus Γi is the intersection of the discrete set GD (Q) and 1 1 × a−1 D∞ for D∞ = {g ∈ D∞ |N (g) = 1}, which is therefore a compact set ai O D i ﬁnite. Thus Γi is a ﬁnite group and is the unit group of another maximal order D a−1 ∩ D. OD,i = ai O i D (∆); C); so, we may regard Let φ, φ∗ ∈ SkD (Γ 0 φ : GD (Q)\GD (A) → L(k − 2; C)

Automorphic forms on inner forms of GL(2)

140

D × × with φ(γxu) = u−1 ∞ φ(x) for u ∈ OD D∞ and γ ∈ G (Q). Then

(φ, φ∗ ) =

h

∗ e−1 i [φ(ai ), φ (ai )] for ei = |Γi |.

i=1

Pick a positive integer n. We consider = nZ}. D |N (g)Z T(n) = {g ∈ O We choose a decomposition T(n) =

5

× . αO D

α

× , and we can choose α so that ααι = N (α) = × ) = Z By the norm theorem, N (O D n. Then

φ|T (n)(x) = φ(xα). α

× D× , we can write ai α = αi aj ui for αi ∈ ai a−1 Since ai α ∈ j GD (Q)aj · O j D ∞ × × D∞ ) = 1, we have and ui ∈ O . Since N (a i D n = N (ai α) = N (αi )N (ui ) in GD (A). × R× ), we ﬁnd N (ui ) = 1, and hence N (αi ) = n. Since all Since N (ui ) ∈ (Q ∩ Z + −1 −1 D a ∩ D with N (β) = n can be written as ai αa−1 for some β ∈ ai aj = ai O j j D with N (α) = n, β is of the form γi αi γj for γi ∈ Γi and γj ∈ Γj . We α∈O have α∞ = 1; so, taking the inﬁnity part of ai α = αi aj ui , we ﬁnd α1,∞ ui = 1 ⇔ −1 . This shows that ui,∞ = αi,∞ φ(ai α) = φ(αi aj ui ) = φ(aj ui ) = ui,∞ φ(aj ) = αi φ(aj ). If we replace αi by β = γi αi γj for γi ∈ Γi and γj ∈ Γj , we ﬁnd ai α = γi−1 βaj ui −1 with ui = a−1 j γj aj ui , and by the same computation using this new formula ai α = γi−1 βaj ui , we ﬁnd φ(ai α) = βφ(aj ) = αi φ(aj ); so, the value obtained is independent of the choice of β ∈ Γi αi Γj . Then we see, up to totally positive units,

1 βφ(aj ), φ|T (n)(ai ) = ei ej −1 β∈ai aj ;N (β)=n

where ei = |Γi |. Thus we have Θ(φ ⊗ φ∗ ) =

1 ei ej i,j

γ∈ai a−1 j

[γφ(aj ), φ∗ (ai )]q N (γ) .

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Deﬁne, for v ∈ L(k − 2; C) and w ∈ L(k − 2; C), a theta series by

1 Θij (v ⊗ w) = [γv, w]q N (γ) , ei ej −1 γ∈ai aj

which is a theta series of the quadratic form N : D → Q with respect to a lattice ai a−1 j ⊂ D. We thus ﬁnd

Θij (φ(aj ), φ∗ (ai )). (2.5.2) Θ(φ ⊗ φ∗ )(z) = i,j

If D is indeﬁnite, we need to use the Poincar´e duality pairing (·, ·) on the cohomology group H 1 (Y0D (∆), L(n; C)) to deﬁne the theta series, and the outcome is basically the same. 2.5.2 Siegel’s theta series We study general theta series of a quadratic form q(x) on a (row) Q-vector space V with bilinear from S(x, y) = q(x + y) − q(x) − q(y) in this subsection. We will have formulas involving exp(πix) in many places; so, to make things simple, we write e(x) = exp(πix) (only in this and the following subsection). We ﬁx a base {vi }i of a lattice L ⊂ V and write S = (S(vi , vj )) which is an n × n symmetric matrix. A positive deﬁnite symmetric matrix P ∈ Mn (R) (or the symmetric bilinear form on VR associated to P ) is called a positive majorant of S if P S −1 = SP −1 (⇔ S −1 P = P −1 S).

(2.5.3)

We put ∆ = det(S). Here are some examples: Example 2.59 1. Suppose that S is diagonal with diagonal entries aj from the top; so, write S = diag[a1 , . . . , an ]. Then the diagonal matrix P = diag[|a1 |, . . . , |an |] gives a positive majorant, although as we will see, there are lots others. More generally, if S = X diag[a1 , . . . , an ]t X for X ∈ GLn (R), then P = X diag[|a1 |, . . . , |an |]t X is a positive majorant. 2. Suppose that S has signature (λ, µ) (so n = λ + µ). Then for any decomposition VR = W ⊕ W ⊥ for a subspace W with dimR W = λ on which S is positive deﬁnite, then PW (x, y) = S(xW , yW ) − S(x , y ) is a positive majorant of S, where x = xW + x and y = yW + y for xW , yW ∈ W and x , y ∈ W ⊥ . Here W ⊥ is the orthogonal complement of W in VR . By ⊥ Witt’s theorem (cf. [EPE] 1.2), if VR = W ⊕ W is another decomposition as above, then we ﬁnd g ∈ OS (R) such that W = W g and hence ⊥ W = W ⊥ g. 3. By (2.5.3), we ﬁnd (P − S)(P −1 + S −1 ) = 0. Deﬁning W = Ker(P − S), we ﬁnd that P is given by PW .

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142

Deﬁne

. . - Y = W ∈ GλV (R)S|W > 0 ∼ = P P :positive majorants of S ,

where GλV is the Grassmannian variety of index λ and the last isomorphism is given by W → PW . Then OS (R) acts on Y by W → W g. By Example 2.59 (2–3), OS (R) acts transitively on Y. It is easy to check that the actions of g ∈ OS (R) on positive majorants are P → gP · t g. If we ﬁx one positive majorant P0 , we ﬁnd that for the maximal compact subgroup C = OS (R) ∩ OP0 (R), Y∼ = C\OS (R). So Y is the realization as a real manifold of the symmetric space of OS (R). For z = x + iy ∈ H = {z ∈ C|y = Im(z) > 0}, we consider ΦP (v; z) = e(S[v]x + P [v]iy), where S[v] = S(v, v) and P [v] = P (v, v). The quadratic form in question is v → q(x) = 12 S[v]. If S is positive deﬁnite, we have S = P and ΦP (v; z) = e(S[v]x + P [v]iy) = exp(2πiq(v)z), which is a term of the classical theta series of the quadratic form q. When V = D for a quaternion algebra, we have S(x, y) = Tr(xι y) and q(x) = N (x) for the reduced trace Tr : D → Q and the reduced norm N : D → Q. As a function of v ∈ VR , ΦP (v; z) is a Schwartz function on VR . Thus for any homogeneous polynomial Q(v) of degree k, the following theta series is absolutely convergent:

θ(z, P ; Q) = Q()ΦP (; z). ∈L

We would like to show that θ(z, P ; Q) satisﬁes an automorphic property like an elliptic modular form and to determine its level. Actually, it is better to do it (half) adelically. A Schwartz–Bruhat function Φ for a big = L ⊗ Z on VA(∞) = V ⊗Q A(∞) is a function vanishing outside L ∼ lattice L and factoring through L /L = L /L for a smaller lattice L. The adelic theta series is given by

θ(z, P ; Φ) = Φ(∞) (v)Q(v)e(S[v]x + P [v]iy) v∈V

=

Φ(∞) (v)θ(z, gP ; v, Q ◦ g, L).

v∈L /L

It is well known that any homogeneous polynomial can be written uniquely [k/2] as H(v) = j=0 S[v]j ηk−2j (v) for a spherical polynomial ηj of degree j (see ∂2 , [Hk1] Section 5). A function η is spherical if ∇η = 0 for ∇ = i,j sij ∂xi ∂xj where S −1 = (sij ) and we write v = x1 v1 + · · · + xn vn for the base vi of L. Applying ∇j to θ(z, P ; η) for a spherical polynomial η as above, we get S[v]j in front of each term η(v)e(S[v]x + P [v]iy); so, we can deduce the result for

Theta series

143

θ(z, P ; ηS[·]j ) from the result concerning θ(z, P ; η) for a spherical polynomial η. Thus we only need to study θ(z, P ; H) for a spherical polynomial H. Any spherical polynomial is a linear combination of the following type of functions H : v → S(v, w+ ) S(v, w− )m for nonnegative integers and m. Here the (nonzero) vectors w± ∈ VC are such that S(v, w± ) = ±P (v, w± ) for all v ∈ VC and S[w± ] = 0. When nonzero w+ does not exist, we just suppose = 0 and ignore the factor S(v, w+ ). We take this convention also for w− . We thus ﬁx H(v) = S(v, w+ ) S(v, w− )m for non-negative integers and m. For simplicity, we write θ(z; v, L) for θ(z, P ; v, H, L). Let L∗ = {v ∈ V |S(v, L) ⊂ Z} (the dual lattice). We assume that S(L, L) ⊂ Z (replacing L by a smaller lattice if necessary); so, L∗ ⊃ L. Here is an easy lemma whose proof is left to the reader: Exercise 2.60 Let 0 = c ∈ Z. Prove the following facts: 1. If v ∈ L∗ and a ∈ c−1 Z, θ(z + a; v, cL) = e(aS[v])θ(z; v, cL). 2. If v ∈ L∗ , θ(z; v, L) = w∈(v+L)/cL θ(z; w, cL). 3. θ(c2 z; v, L) = c−−m θ(z; cv, c2 L). The Poisson summation formula in [LFE] Section 8.4 yields: Proposition 2.61

For 0 = c ∈ Z, we have

√ 1 θ − ; w, cL = (−1)+m ( −1)(µ−λ)/2 |c|−n |∆|−1/2 z

× z +(λ/2) z m+(µ/2) e(S(w, v))θ(z; v, cL), (2.5.4) v∈c−1 L∗ /cL

where z s = |z|s exp(iσs) writing z = |z| exp(iσ) with −π < σ < π and ∆ = det(S). Writing ψH (z; v) = H(v)e(S[v]x + P [v]iy), the idea of the proof is classical; indeed, we compute the Fourier transform of ψ and apply the Poisson summation formula to the sum over the lattice L which is the θ-function. The computation follows Hecke’s technique in [Hk] (see also [MFM] Section 4.9). Proof We start computing the Fourier transform of ψH . Here ψ1 indicates that we take H = 1. Here is a well-known formula. For z ∈ H = {z ∈ C| Re(z) > 0} and a ∈ R× :

∞ + −1 π|a|w2 . (2.5.5) exp(−π|a|zv 2 )e(awv)dv = |a| z −1/2 exp − z −∞

Automorphic forms on inner forms of GL(2)

144

We can always ﬁnd B ∈ GLn (R) such that BS · t B = diag[a1 , . . . , an ] and BP · t B = diag[|a1 |, . . . , |an |]. We write wi = j bij vj (so, for the real base, wi , S and P are diagonal). Then writing v = α1 w1 + · · · + αn wn (so, α(v) = (α1 , . . . , αn ) is the coordinate of v with respect to the base {wi }). Then

√ (αi2 |ai |y − αi2 ai −1x)). ψ1 (z; v) = exp(−π i

We write β for the coordinate with respect to {vi }; so, v = i βi vi . Then we write+ dv = dβ1 · · · dβn (a Haar measure on VR ). Then dv= Cdα1 dα2 · · · dαn for C = (|a1 · · · an ||∆|−1 ) for ∆ = det(S). Writing w = i γi wi (α(w) = γ), we ﬁnd ψ ∗ (z; w)

√ =C exp −π (y|ai |αi2 − −1xai αi2 ) e ai αi γi dα1 dα2 · · · dαn . VR

i

i

By applying (2.5.5) ψ1∗ (z; w)

−1/2

= |∆|

√ √ 1 λ/2 µ/2 (− −1z) ( −1z) ψ1 − ; w . z

(2.5.6)

∗ In order to compute ψH , we write β(w+ ) = r = (r1 , . . . , rn ) and β(w− ) = s and deﬁne

∂

∂ ∂+ = ri and ∂− = si . ∂βi ∂βi i i

Then by a simple computation, we get ∂± S[v] = 2S(v, w± ), ∂± P [v] = 2P (v, w± ) = ±2S(v, w± ), and ∂± S(u, v) = S(u, w± ).

(2.5.7)

From this, we get ∂± H(v) = S(v, w+ )−1 S(w+ , w± )S(v, w− )m + mS(v, w− )m−1 S(w− , w± )S(v, w+ ) . Since S[w± ] = 0 and S(w± , w∓ ) = 0, we have ∂± H(v) = 0. From this, we have m ∂− ψ1 (z; w) = (2πi)+m ψH (z; w)z z m . ∂+

(2.5.8)

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m Then, using the fact that ∂+ ∂− e(S(v, w)) = (2πi)+m H(v)e(S(v, w)), applying m the diﬀerential operator ∂+ ∂− to the formula (2.5.6), we get

1 ∗ −−λ/2 −m−µ/2 ψH (z; w) = C,m z z ψH − ; w (2.5.9) z √ for C,m = ( −1)(λ−µ)/2 (−1)m+ |∆|−1/2 . Then by the Poisson summation formula (e.g., [LFE] Section 8.4):

f (v + l) = f ∗ (l∗ )e(S(−v, l∗ )), l∗ ∈L∗

l∈L

we get θ(z, w; L) =

ψH (z; w + v) =

∗ ψH (z; v)e(S(−w, v))

v∈L∗

v∈L

= C,m z −−λ/2 z −m−µ/2

e(S(−w, v))ψH

v∈L∗

1 − ;v , z

because e(S(−w, v)) only depends on v mod L if w ∈ L∗ . Now we make a variable change z → − z1 and use Exercise 2.60(3) to get the desired formula for c = 1. For general cL, we just replace L by cL and do the same argument. 2 Applying Proposition 2.61 and Exercise 2.60 to:

for γ =

a b c d

a 1 az + b = − cz + d c c(cz + d)

∈ SL2 (Z) with c = 0, we ﬁnd

θ(γ(z); v, L) = C,m c−(n/2) (c/|c|)λ (cz + d)+(λ/2) (cz + d)m+(µ/2)

× ϕ(v, u)θ(z; u, c2 L), (2.5.10) u∈L∗ /c2 L w∈(v+L)/L

where ϕ(v, u) =

w∈(v+L)/cL

e

1 (aS[w] + 2S(w, u) + dS[u]) c

for v ∈ L∗ /L and u ∈ L∗ /c2 L. Since up to a scalar, any element on γ ∈ SL2 (F ) can be written as a product of an element in SL2 (Z) and an upper-triangular matrix in GL2 (F ), we can compute the eﬀect of z → γ(z) by using this formula and Exercise 2.60. Lemma 2.62

Let the notation and the assumption be as above. Then ϕ(v, u) = e(−b(dS[u] + 2S(v, u)))ϕ(v + du, 0).

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In particular, ϕ(v, u) depends only on (u, v) ∈ (L∗ /L)2 . Proof

We have

ϕ(v + du, 0) =

e

#a

w∈(v+du+L)/cL

c

$ S[w] =

w∈(v+L)/cL

e

#a c

$ S[w + du] ,

which is a Gauss sum. However, aS[w + du] = aS[u] + 2adS(w, u) + ad2 S[u] = (aS[w] + 2S(w, u) + dS[u]) + c(2bS(w, u) + dbS[u]), where we used the fact that ad−bc = 1 to show the last equality. Since w ∈ v +L, e(bS(w, u)) = e(bS(v, u)), we have

$ #a 1 (aS[w] + 2S(w, u) + dS[u]) e(2bS(v, u) + dbS[u]). S[w + du] = e e c c This shows: ϕ(v + du, 0) = e(2bS(v, u) + dbS[u])

e

w∈(v+L)/cL

1 (aS[w] + 2S(w, u) + dS[u]) c

= e(b(2S(v, u) + dS[u]))ϕ(v, u), 2

which shows the formula. Combining (2.5.10) and Lemma 2.62, we get

a b Proposition 2.63 Let γ = ∈ SL2 (Z). Then we have c d 1. If c = 0, then θ(γ(z); v, L) = C,m c−n/2 (c/|c|)λ (cz + d)+(λ/2) (cz + d)m+(µ/2)

× ϕ(v, u)θ(z; u, L), u∈L∗ /L

2. If c = 0, θ(γ(z); v, L) = e(dbS[v])θ(z; v, L). Exercise 2.64 1. Prove all the statements in Example 2.59 in detail. 2. Prove that v → ΦP (v; z) is a Schwartz function on VR (that is, it is of C ∞ class and (the absolute value of ) all its derivative multiplied by a polynomial on VR decreases as v → ∞). 3. Prove that θ(z; v, L) is absolutely and locally uniformly convergent for z ∈ H and P ∈ Y. 4. Find w± ∈ VC such that S[w± ] = 0 and S(v, w± ) = ±P (v, w± ) for a positive majorant P , and specify a necessary and suﬃcient condition for the existence of nonzero w± .

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5. Prove that if η is spherical with respect to P , η ◦ g is spherical with respect to gP for g ∈ OS (R). 6. Explain why we can ﬁnd B ∈ GLn (R) as in the proof of Proposition 2.61. 2.5.3 Transformation formulas ∗ L∗ = L, Let M be the smallest integer such that M S[L ] ⊂ 2Z. If M = 1, then

0 −1 1 1 θ(z; L) is invariant under (Poisson summation formula) and 1 0 0 1 (Exercise 2.60). Since SL2 (Z) is generated by these two matrices, θ(z; L) is invariant under SL2 (Z). Thus we assume that M > 1. We are going to show

a b Theorem 2.65 Let γ ∈ ∈ Γ0 (M ) with d > 0 (by replacing γ by −γ if c d necessary). Then we have, for v ∈ L∗ ,

θ(γ(z); v, L) = χ(d)e(abS[v])(cz + d)+(λ/2) (cz + d)m+(µ/2) θ(z; av, L), where z s (z ∈ C× and s ∈ R) is an in Proposition 2.61 and 

(−1)m ∆   if n = 2m with m ∈ Z,  d χ(d) = ∆ −2c   if n = 2m + 1 for m ∈ Z. εnd d d is assumed to be equal to 1, and Here if c = 0, the Legendre symbol −2c d 1 if d ≡ 1 mod 4, εd = √ −1 if d ≡ 3 mod 4. Remark 2.66 Apply the theorem to the norm form v → N (v)/N (ai )N (aj )−1 on ai a−1 ⊂ D when D is a deﬁnite quaternion algebra. Then Θij (v ⊗ w) is a j linear combination of θ(z; 0, ai a−1 j ), and we get a direct proof of the fact that Θij (v ⊗ w) ∈ Sk (Γ0 (d(D))), and hence Θ(φ ⊗ φ ) ∈ Sk (Γ0 (d(D)) for φ, φ ∈ D (d(D)); C) as deduced in Proposition 2.51 from the Jacquet–Langlands SkD (Γ 0 correspondence. Let us give a little more detail of this elementary way of getting (at least partially) the correspondence. As before, D/Q is the ﬁxed deﬁnite quaternion algebra with a ﬁxed maximal order OD . We ﬁx a left or right OD -ideal and consider the quadratic form Q : a → Z given by Q(x) = N (x)/N (a), where N : D → Q is the reduced norm map and N (a) = [OD : a]. In the above application, a = ai a−1 j . The associated symmetric bilinear form is given by −1 S(x, y) = N (a) Tr(xy ι ) for the involution ι with xxι = xι x = N (x). We deﬁne for a left OD -ideal a, a∗ = {x ∈ D|Tr(xa) ⊂ Z}. Then a∗ is a right OD -ideal. Note that a∗p = {x ∈ Dp |Tr(xap ) ⊂ Zp } for ap = A ⊗Z Zp and Dp = ∗ = OD,p . D ⊗Q Qp . This shows if Dp ∼ = M2 (Qp ), OD,p ∼ = M2 (Zp ), and hence OD,p

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If p ramiﬁes in D, as we have already % described, taking the unramiﬁed

unique & a b a, b ∈ K , where a extension K/Qp , we can realize Dp = α(a, b) = pb a is the conjugate of a ∈ K over Qp . Then for = α(1, 1), 2 = p, and we may identify OD,p with {α(a, b)|a, b ∈ W } for the p-adic integer ring W . Then clearly, ∗ ∗ = {α(a, b/p)|a, b ∈ W }. Thus OD,p /OD,p is killed by p, and hence we have OD,p ∗ OD /OD is killed by the product d(D) of ramiﬁed primes of D. This tells us that the minimal integer M among integers N with N · S[a∗ ] ⊂ Z is given by d(D), because locally a left ideal ap of OD,p is principal given by ap = OD,p αp . In particular, θ(z; 0, a) ∈ Sk (Γ0 (d(D))) by the above theorem. There is an indeﬁnite version of this fact. If D is indeﬁnite, θ(z; v, OD ) is also a function of the majorant P . The space Y of majorants P is isomorphic to H × H, and θ(z, P ; 0, OD ) is again an automorphic form of suitable weight (depending on k and ) on Y0D (d(D)) × Y0D (d(D)). Shimura in [Sh6] showed essentially that Θ(φ ⊗ φ ) = θ(z, P ; 0, OD )(φ ⊗ φ )(P )dP Y0D (d(D))×Y0D (d(D))

for a suitable measure dP on Y. So, this direct construction using theta series also works for the indeﬁnite cases. We shall give a sketch of a proof of Theorem 2.65. Proof We start with γ ∈ SL2 (Z). We may assume that c = 0 (otherwise the formula is obvious and follows from Proposition 2.63(2)). We see easily that M L∗ ⊂ L. By replacing z by − z1 in the formula of Proposition 2.63(1) and then applying Proposition 2.61, we get

bz − a ; v, L = |c|−n/2 |D|−1 (dz − c)+(λ/2) (dz − c)m+(µ/2) θ dz − c  

 × ϕ(v, u)e(S(u, t)) θ(z; t, L). t∈L∗ /L

u∈L∗ /L

In this computation, we assumed that c < 0 when n is odd (which follows the assumption that d > 0 as we will see at the end of the proof). Note that $ #a

S[w] . ϕ(v, u) = e(−b(dS[u] + 2S(v, u)) e c w∈(v+du+L)/cL

b −a We now suppose γ = ∈ Γ0 (M ); so, d ≡ 0 mod M . d −c Since M S[L∗ ] ⊂ 2Z, we ﬁnd 12 dS[u] ∈ Z, and e(−bdS[u]) = 1. Since M L∗ ⊂ L, du ∈ L and hence v ≡ v + du mod L. Thus we have $ #a

ϕ(v, u) = e(−bS(v, u)) e S[w] . c w∈(v+L)/cL

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149

$ #a S[w] for the part of the Gauss sum. By this We put ϕ(v) = w∈(v+L)/cL e c maneuver, we reach the following expression:

bz − a θ ; v, L = |c|−n/2 |D|−1 ϕ(v)(dz − c)+(λ/2) (dz − c)m+(µ/2) dz − c  

 × e(S(u, t − bv)) θ(z; t, L). t∈L∗ /L

u∈L∗ /L

Now putting ψ(u) = e(S(u, t − bv)) for u ∈ L∗ /L, ψ is an additive character of the additive group L∗ /L. By the orthogonality relation of characters, we have

[L∗ : L] = |D| if ψ is trivial, ψ(u) = 0 otherwise. u∈L∗ /L Thus the terms which survive are those t − bv ∈ (L∗ )∗ = L. This yields

bz − a ; v, L = ϕ(v)|c|−n/2 (dz − c)+(λ/2) (dz − c)m+(µ/2) θ(z; bv, L). θ dz − c Thus we need to compute the Gauss sum

ϕ(v).

b −a a b Hereafter we rewrite γ = as γ = ∈ Γ0 (M ). Thus the d −c c d assumption c < 0 becomes d > 0. The above formula then states: θ (γ(z); v, L) = ϕ(v)|d|−n/2 (cz + d)+(λ/2) (cz + d)m+(µ/2) θ(z; av, L) b under d > 0 and c ≡ 0 mod M , where ϕ(v) = w∈(v+L)/dL e d S[w] . We modify ϕ(v) slightly. Since ad − bc = 1, ad ≡ 1 mod M and (ad − 1)v ∈ L for all v ∈ L∗ ; so, adv ≡ v mod L. Thus w ∈ (v + L)/dL satisﬁes w ≡ adv mod L and hence w = adv + u with u ∈ L/dL. Thus

b S[adv + u] ϕ(v) = e d u∈L/dL

b 2 = e ba dS[v] + 2abS(v, u) + S[u] d u

b ba2 d≡ab mod M = e(abS[v]) e S[u] . d u∈L/dL

We write W (b, d) = |d|−n/2 u∈L/dL e db S[u] . Thus the formula we are dealing with is: θ (γ(z); v, L) = e(abS[v])W (b, d)(cz + d)+(λ/2) (cz + d)m+(µ/2) θ(z; av, L). Then a standard argument from the time of Hecke proves W (b, d) = χ(d) (see [Hk] or the proof of [MFM] Lemma 4.9.2). 2

150

Automorphic forms on inner forms of GL(2)

Exercise 2.67 1. Prove M L∗ ⊂ L. 2. Prove W (b, d) = χ(d). 3. If S(x, y) = Tr(xy ι ) for x, y ∈ D of a deﬁnite quaternion algebra D, show that g → [gv, w] for v, w ∈ L(n; C) is a spherical function of g ∈ D. 2.5.4 Theta series of imaginary quadratic ﬁelds It is a well-known result of Hecke and Shimura that for a given Hecke L-function L(s, λ) of animaginary quadratic ﬁeld, we can ﬁnd a unique ∞ Hecke eigenform ∞ f = Θ(λ) = n=1 a(n, f )q n such that L(s, λ) = L(s, f ) = n=1 a(n, f )n−s . We construct such Hecke eigenforms for arithmetic Hecke characters here via theta series of quadratic forms of two variables. After doing this, we state without proof a generalization of the result in the Hilbert modular case. Let M/Q be an imaginary quadratic ﬁeld with integer ring R. We ﬁx an integral ideal and consider the quadratic form Q : a → Z given by Q(x) = N (x)/N (a), where N : M → Q is the norm map and N (a) = [R : a]. The associated symmetric bilinear form is given by S(x, y) = N (a)−1 TrM/Q (xc(y)) for the generator c ∈ Gal(M/Q). We consider the ray class group ClM (C) modulo C. We choose a complete representative set {a}a for ClF (1) made up of ideals a prime to C. Then with a pair (α, a) ∈ ((R/C)× /R× ) × ClM (1)), we associate the class αa−1 ∈ ClM (C). Exercise 2.68 Prove the following facts: 1. a∗ := {x ∈ M |S(x, a) ⊂ Z} = N (a)a−1 d−1 for the absolute diﬀerent d, and hence the discriminant D of q(x) is equal to |∆| for the discriminant ∆ of M/Q; 2. (R/C)× /R× × ClM (1) is in bijection with ClM (C) in the way described above; 3. For an integer n, M ξ → ξ n ∈ C is a spherical function of degree n with respect to S. We insert here a description of general terminology on Hecke characters of a number ﬁeld. Let L be a number ﬁeld. Deﬁne the adele ring of L by LA = L⊗Q A. and deﬁne Z[IL ] to be the additive groupof formal Let IL = Homﬁeld (L, C), linear combinations k = σ kσ σ of σ ∈ IL . Since LC = L ⊗Q C = σ∈IL C = CIL , we may regard σ ∈ IL as the projection of LC to the σ-component C. Then × × character L× for k ∈ Z[IL ], we can think of ∞ = (L ⊗Q R) → C a multiplicative × k kσ given by L∞ x → x = σ σ(x) , regarding L∞ as a subalgebra of LC . A × × (continuous) Hecke character λ : L× A /L → C is called arithmetic if there exists k k ∈ Z[IL ] such that λ(x∞ ) = x∞ for all x∞ in the identity connected component × L× ∞+ of the real Lie group L∞ . The character k is called the inﬁnity type of λ.

Theta series

151

Since λ is continuous, the restriction of λ to the ﬁnite ideles is trivial on a small . open neighborhood of the identity of L× A(∞) Exercise 2.69 Prove that Ker(λ : L× → C× ) is an open subgroup of L× . A(∞) A(∞) Writing OL for the integer ring of L, a fundamental system of neighborhoods × ∩ (1 + c) for integral ideals c, we have an is given by U (c) = O of L× L A(∞) integral ideal C maximal among the ideals c satisfying Ker(λ) ⊃ U (c), where ⊂ L×(∞) . The ideal C is called the conductor of λ. Then λ L = OL ⊗Z Z O A × induces a ﬁnite order character factoring through (O/C)× , even restricted to O L if λ is of inﬁnite order. For a given fractional ideal a prime to C, we can ﬁnd L . whose component at prime factors of C is 1 such that a = F ∩ aO a ∈ L× A(∞) Such an a is unique up to multiples by units in U (C) (on which λ is trivial), so we have a uniquely deﬁned value λ(a), for which we write λ(a). We simply put λ(a) = 0 if a has a common factor with C. i∞ → C, we We return to the imaginary quadratic ﬁeld M . Regarding M ⊂ Q −− have two embeddings IM = {1, c} for the identity inclusion “1” (which is i∞ ) and complex conjugation c. We identify Z with Z · 1 in Z[IL ]. Take a Hecke character λ of conductor C with λ((α)) = αk−1 if (α) = 1 in ClM (C) for an integer k ≥ 1. This implies that the inﬁnity type of λ is 1 − k. Indeed, we have α(∞) ∈ U (C) because α ≡ 1 mod C. Since λ((α)) = λ(α(∞) ), we have λ(α∞ ) = α1−k , because k−1 λ(α∞ ). 1 = λ(α) = λ(α(∞) )λ(α∞ ) = α k−1 Let Θα,a (λ) = ξ≡α mod Ca ξ exp(2πiN (ξ)/N (a)). Write αM : (Z/DZ)× → {±1} for the quadratic residue symbol −D . Note that Θα,a (z) = θ(z; α, Ca) for the quadratic form Q. The following fact is clear because S[α] = 2N (α)/N (a) and (Ca)∗ = C−1 a−1 d−1 . Exercise 2.70 Let m be the smallest integer such that m · S[(aC)∗ ] ⊂ Z. Prove m = N (C)D.

a b Then by Theorem 2.65, for γ = ∈ Γ0 (N (C)D), we have c d Θα,a (γ(z)) = θ(γ(z); α, Ca) = αM (d)e(2abN (α))(cz + d)k θ(z; aα, Ca) = αM (d)e(2abN (α)/N (a))Θaα,a (z).

(2.5.11)

Since α ∈ a, we have N (α)/N (a) ∈ Z, which implies e(2abN (α)/N (a)) = 1, and hence we get Θα,a (γ(z)) = αM (d)Θaα,a (z). Thus deﬁning Θ(λ) =

1

λ(a−1 )α1−k Θα,a = λ(a)q N (a) , × |R | α,a a⊂R

(2.5.12)

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Automorphic forms on inner forms of GL(2)

we have (cf. [Hk] and [Sh2]) Theorem 2.71 Take an arithmetic Hecke character λ : MA× → C× with inﬁnity type 1 − k ∈ Z such that λ ◦ c = λ for the complex conjugation c acting ˘ on MA× . Then we have a Hecke eigenform Θ(λ) ∈ Sk (Γ0 (N (C)D), λαM ) whose Hecke eigenvalue for T (n) is given by b⊂R;N (b)=n λ(b), where b runs over all ˘ is the restriction of λ to Z × (which induces a integral ideals with norm n and λ Dirichlet character modulo C ∩ Z). Proof The facts that Θ(λ) is a Hecke eigenform and its Hecke eigenvalue is as given in the theorem follow from (2.4.6) and (2.5.12) combined (cf. Exercise 2.47). ˘ Note that λ(a) = λ((a))a1−k for a ∈ Z. By (2.5.11), we ﬁnd ˘ Θ(λ)(γ(z))(cz + d)−k = αM (d)λ(d)Θ(λ) for γ ∈ Γ0 (N (C)D) as in (2.5.11), because ad ≡ 1 mod N (C)D. Thus we con˘ clude Θ(λ) ∈ Gk (Γ0 (N (C)D), α M λ). Since the Mellin transform of Θ(λ) is the Hecke L-function L(s, λ) = b⊂R λ(b)N (b)−s , it is an entire function by [BNT] VII.7 if λ ◦ c = λ. This is true for L(s, λ(χ ◦ N )) with any Dirichlet character χ, and in this case, Θ(λ) is a cusp form (because the Mellin transform of an Eisenstein series has a pole after some character twists; see, [MFM] Theorem 4.3.15). 2 We can generalize the theorem to Hilbert modular forms whose proof can be given in the same way as above. Here we describe the generalization without proof. A CM ﬁeld over F is a totally imaginary quadratic extension M/F (here F is totally real). Write again R for the integer ring of M . A CM type Σ of M is a subset of IM satisfying IM = Σ Σc for the generator c of Gal(M/F ). 2 Fix a CM type Σ. We ﬁx a weight κ = (κ1 , κ2 ) ∈ Z[I] with κ2 − κ1 ≥ I and 1 = σ∈Σc κ1,σ|F σ and κ 2 = σ∈Σ κ2,σ|F σ. We deﬁne extend it to Z[IM ] by κ κ =κ 1 + κ 2 ∈ Z[IM ]. Let λ : MA× /M × → C× be an arithmetic Hecke character with inﬁnity type − κ and of conductor C. Then we have Theorem 2.72 Let the notation be as above. Assume that λ ◦ c = λ for the generator c of Gal(M/F ). Then there exists a nonzero Hecke eigenform Θ(λ) ∈ Sκ (∆NM/F (C), ελ ; C) such that its eigenvalue for T (n) is given by b⊂R;NM/F (b)=n λ(b), where ∆ is the relative discriminant of M/F . The Neben character ελ = (ε1 , ε2 , ε+ ) is given ˘ , ε2 = 1 and ε+ = αM/F λ, ˘ where αM/F : F × /F × → {±1} by ε1 = αM/F λ| A T (Z) is the quadratic character associated to the extension M/F by class ﬁeld theory, ˘ is the restriction of λ to F × times the adelic absolute value | · |A of F × . and λ A A A representation theoretic proof of this can be found in [AFG] Section 12 (or [AAG] Section 7B), and one can give a proof using the transformation formula of theta series along the lines of the proof of Theorem 2.71. There is one more proof

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153

via Weil’s converse theorem applied to functional equations of Hecke L-functions (e.g., [Sh2], [MFM] Section 4.8 and [Y]). As we have seen in 2.3.7, if one can ﬁnd a vector with nonzero eigenvalue for T (l ) in the automorphic representation π(λ) generated by Θ(λ), the local factor πl (λ) (π(λ) = ⊗l πl (λ)) is not supercuspidal. By the eigenvalue for U (l ), we can determine the isomorphism class of πl (λ) by (2.3.36) if πl (λ) is not supercuspidal. Here is the description of the isomorphism class of πl (λ) when it is nonsupercuspidal. Corollary 2.73 Let the notation be as above. Choose a prime L of M above a prime l of F . Then we have the following classiﬁcation: 1. The local component πl (λ) is supercuspidal if and only if l does not split in M and λL ◦ c = λL for the generator c of Gal(M/F ). 2. Suppose that L is inert or ramiﬁed in M/F and that λL ◦ c = λL . Choos× × ing a Hecke character λF such that λL = λF l : Fl → C l ◦ NML /Fl , we F ∼ have πl (λ) = π(λl , αl λl ), where αl is the quadratic character giving the isomorphism Fl× /NML /Fl (ML× ) ∼ = {±1} (thus αl is the character associated to ML /Fl by local lass ﬁeld theory). 3. If l splits into the product LL in M , we have πl (λ) ∼ = π(λL , λL ) identifying ML = Fl = ML naturally. The proof of this is left to the reader. Exercise 2.74 Prove that if L is inert or ramiﬁed in M/F and λL ◦ c = λL , there exists two Hecke characters φ : Fl× → C× such that λL = φ ◦ NML /Fl and F writing λF l for one of them, the other is given by λl αl . 2.6 The basis problem of Eichler We shall give an application of Galois deformation theory to the basis problem of Eichler, which was the genesis of Theorem 2.30 Eichler initiated. The original basis problem of Eichler is to ﬁnd an explicit basis (over C) of an appropriate space of elliptic modular forms by means of theta series of the norm forms of deﬁnite quaternion algebras. He achieved this in the 1950s by comparing the traces of Hecke operators acting on the space of automorphic forms on such quaternion algebras and on elliptic modular forms (see [Ei]). This basis problem has its origin in Jacobi’s celebrated formula (in Fundamenta Nova Sections 40–42) of the number S4 (n) of ways of expressing a given integer n as a sum of four squares:

S4 (n) = 8σ1 (n) with σ1 (n) = d for odd positive integers n. 0
This formula has the following heuristic meaning in terms of the Hamilton quaternion algebra H = Q + Qi + Qj + Qk we have seen at the beginning of this section. Then the norm form on the order R2 = Z + Zi + Zj + Zk is

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given by N (x) = xx = x21 + x22 + x23 + x24 for x = x1 + x2 i + x3 j + x4 k and x = x1 − x2 i − x3 j − x4 k (quaternion conjugation). We look at the Hurwitz max/ 0 1+i+j+k imal order [Hz], OH = R2 . Recall that all right ideals of OH are 2 principal. Since [OH : R2 ] = 2, as long as n is odd, N (x) = n (for x ∈ R2 ) if and only if R2 xxR2 = nR2 (up to units in R2× = {±1, ±i, ±j, ±k}, which explains well the number 8 = |R2× | in the Jacobi formula S4 (n) = 8σ1 (n)). The number of ways of making a prime decomposition as above of the two-sided ideals pOH for a prime p into a product of a left ideal and a right ideal is given therefore by the sum σ1 (p) = 1 + p of divisors of p for odd p. We can think of the same problem for M2 (Q) in place of H. Then the norm on M2 (Z) is the determinant map:

x1 x2 , det(x) = x1 x4 − x2 x3 for x = x3 x4 which is an indeﬁnite quadratic form. As we have already remarked at the beginning in (2.0.1), the number of integer matrices x with p = det(x) for a prime p up to units in GL

2 (Z) is the number of left cosets of GL2 (Z) in the double coset 1 0 GL2 (Z) GL2 (Z): 0 p GL2 (Z)

1 0 p 0 5 1 0 GL2 (Z) = GL2 (Z) GL2 (Z) 0 p 0 1 0 p

5 5 1 1 5 1 GL2 (Z) · · · GL2 (Z) 0 p 0

p−1 p

by Lemma 2.46; so, it is given again by σ1 (p). Thus basically, we get the same formula for H and M2 (F ). Deﬁne for a subring A of C 7 ∞

n G(p; A) = f ∈ G2 (Γ0 (p))f = a(n, f )q with a(n, f ) ∈ A if n ≥ 0 n=0

& a b ∈ SL2 (Z) c ≡ 0 mod p for a prime p, where G2 (Γ0 (p)) for Γ0 (p) = c d is the space of holomorphic modular forms on Γ0 (p) of weight 2. We write S(p; A) ⊂ G(p; A) for the subspace of cusp forms. This space G(p; Z) is isomorphic to G(0,1) (p, 1; Z) with the trivial character 1 we have deﬁned in (2.3.17) by G(0,1) (p, 1; Z) f → fg ∈ G(p; Z) for g = 1 ∈ G(A). The space G(2; Q) of rational modular forms of weight 2 on Γ0 (2) is spanned by an Eisenstein series %

∞

(2) 1 + E(z) = σ (n)q n (q = exp(2πiz)), 24 n=1 1

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(2)

where σ1 (n) is the sum of odd positive divisors of n. At the same time, G(2; Q) is spanned by the theta series of the maximal order OH ⊃ R2 of H:

θ(z) = q αα ∈ G(2; Z). α∈OH × What Jacobi (basically) proved is θ(z) = 24 · E(z) (because OH = R2× % & ±1 ± i ± j ± k × | = 24; see [Hz] (5)). Thus G(2; Z) is spanned by with |OH 2 θ(z). This shows that the integral structures on G(Γ0 (2)) coming from the q-expansion and θ(z) are equal. As we have seen, this type of identity of elliptic modular forms and quaternionic automorphic forms are vastly generalized by the Jacquet–Langlands theorem (see Theorem 2.30), in terms of the identity of automorphic representations. The character (or trace) identity makes sense, because H ⊗Q Qp = M2 (Qp ) for all odd primes p; so, for such primes, local factors of automorphic representations of GL(2) and H × can be identiﬁed. However the computation of traces only yields a noncanonical identity of representations. What we would like to do in this section is to use the normalization of the Jacquet–Langlands correspondence in Proposition 2.55 in order to explore when we have a canonical identity of the two integral structures coming from theta series and q-expansions (comparing automorphic forms and modular forms deﬁned over smaller rings). Though Jacobi’s example gives the identity of the two integral structures over Z (because of the nonexistence of cusp forms on Γ0 (2)), to achieve this for cusp forms, it would be necessary to invert the Eisenstein ideal (and possibly the prime 2); that is, we need to remove maximal ideals m of the Hecke algebra for which the residual representation ρm in Theorem 2.43 is absolutely reducible. So far, the integral basis problem has been studied geometrically by using the fact that the deﬁnite quaternion algebra B with B = B ⊗Q Q ∼ = M2 (Q ) for all primes but one prime p appears as the endomorphism algebra of super-singular elliptic curves over Fp . This type of research was carried out by Ohta and Oesterle in the 1980s and yielded a good Z(p) -basis of G(p; Z(p) ) (Z(p) = Q ∩ Zp ) by means of the theta series of maximal orders (and ideals) of B, and more recently, M. Emerton determined the Z[ 12 ]-span in G(p; Z) of the theta series of the deﬁnite quaternion algebra B by reﬁning further the geometric means [Em]. We would like to present in this section a short proof in [H05a] of a result (Theorem 2.79) slightly weaker than Emerton’s theorem in [Em] Theorem 0.3, reducing the result to the original Eichler’s theorem and the method of Taylor and Wiles (see [TaW] and [Di1]), and then we will generalize the result later to quaternion algebras (unramiﬁed at all ﬁnite places) over totally real ﬁelds (Theorem 3.47), reducing it to the Jacquet–Langlands correspondence and the generalization of the work of Taylor and Wiles by Fujiwara (Theorem 3.28) to totally real base ﬁelds.

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2.6.1 The elliptic Jacquet–Langlands correspondence We interpret Theorem 2.30 in an elementary manner when D/Q is a deﬁnite quaternion algebra ramiﬁed only at p and ∞. We ﬁx a maximal order OD ⊂ D. We consider the set I of all fractional right OD -ideals of D (that is a Zlattice a of D with aOD ⊂ a). We say two such ideals a and b are equivalent if a = αb for α ∈ GD (Q). Then I/∼ = Cl is the ideal classes of D, which are ﬁnite. Take a complete representative set {ai |i = 1, . . . , h} for Cl. Then × OD,i = ai OD a−1 is another maximal order of D. We put ei = |OD,i |. Then ei is i divisible possibly only by two primes 2 and 3. If one introduces the adele ring × for DA = D ⊗Q A, D∞ = D ⊗Q R and A, then Cl ∼ = GD (Q)\GD (A)/GD (Z)D ∞ D = O p OD,p for OD,p = OD ⊗Z Zp . Suppose that 6 is invertible in A (so, e−1 ∈ A; see Lemma 3.3). We consider the space of functions i 7

−1 ei φ(ai ) = 0 . S(A) = φ : Cl → A i

Thus f ∈ S(A) can be considered as a function f : GD (Q)\GD (A) → A; similarly, a modular form can be considered as a function on GL2 (Q)\GL2 (A). Thus S(A) is a space of automorphic forms on the algebraic group GD . Assuming that 6 is invertible A (so, e−1 ∈ A), we deﬁne a pairing , : S(A) × S(A) → A i in−1 by f, g = i ei f (ai )g(ai ). Then ·, · is a perfect pairing. We can deﬁne an operator T (n) acting on S(A) for integer n > 0 as follows. If a ⊂ OD is a right or left integral ideal, we deﬁne N (a) by the index [OD : a]. If a is not integral, we deﬁne N (a) = [OD : OD ∩ a]/[a : a ∩ OD ]. For any right fractional ideal a and a right integral ideal b of norm n, we can deﬁne the product ab = { j aj bj |aj ∈ a, bj ∈ b}, which is a right fractional ideal. Thus ai b ∼ aj(i;b) for a unique j(i; b), and we may deﬁne

f |T (n)(ai ) = f (aj(i;b) ) b:N (b)=n

for b running over all integral right OD -ideals with norm n. By deﬁnition, we have f |T (n), g = f, g|T (n) (see Lemma 3.5). For simplicity, we assume that OD, ∼ = M2 (Z ) except for one prime = p and write H(A) for H2 (Γ0 (p); A). Then Theorem 2.30 boils down to the following result in our simple setting. Theorem 2.75 (Eichler) We have S(C) ∼ = S(p; C) as modules over H(C), where the action of T (n) is speciﬁed above. From this, by a descent argument, we ﬁnd Corollary 2.76 For any subring A of C, S(A) is a faithful H(A)-module, and if A is a Q-algebra, S(A) is free of rank 1 over H(A). Proof By deﬁnition, the action of H(Q) on S(Q) is faithful. Since C is faithfully ﬂat over Q, if we have two independent generators v and w in S(Q) over H(Q),

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v ⊗1 and w⊗1 in S(C) = S(Q)⊗Q C are independent over H(C), a contradiction. Thus S(Q) is free of rank 1 over H(Q). Then S(A) = S(Q) ⊗Q A is free of rank 1 over H(A). 2 As in Remark 2.66, we conclude from (2.5.2) that Θ(f ⊗ g) =

1 f (ai )g(aj )θ(ai a−1 j ) ∈ S2 (Γ0 (p); A) e e i j i,j

−1 q N (ξ)/N (ai aj ) . In [Ei] for f, g ∈ S(A) (see [Ei] II.6), where θ(ai a−1 j )= ξ∈ai a−1 j II.6, left ideals are studied instead of right ideals here; so, all the formulas there are valid after applying the involution a → a−1 to left ideals a. As we have seen in Theorem 2.30, we have the Jacquet–Langlands correspondence in a general setting. Let us restate the result, since the results can be stated in a simpler manner for elliptic modular forms. Take an open com× , and consider the pact subgroup U in GD (A) of the form U = U (p) × OD,p × ﬁnite set Y (U ) = GD (Q)\GD (A)/U D∞ . The reduced norm map N : D → Q induces N : Y (U ) → ClU = A× /Q× N (U )R× + . Taking a complete representative set {ai } for Y (U ), we deﬁne ei = |ai U a−1 ∩ GD (Q)|. Then ei is only i divisible by primes 2 and 3 (see Lemma 3.3), and if 6 is invertible in A, −1 we have a pairing φ, φ = i ei φ(ai )φ (ai ) on the space of functions on Y (U ). Deﬁne S(U ; A) ⊂ {f : Y (U ) → A} by the orthogonal complement of functions4factoring through N : Y (U ) → ClU . Decomposing a double coset U xU = y yU for x ∈ GD (A) with x∞ = 1, we deﬁne the Hecke operator (p) with [U xU ] : S(U ; A) → S(U ; A) by f |[U xU ](a) = y f (ay). Identifying O D (p) (p) (p) M2 (Z ) for OD = =p OD, and Z = =p Z , we may regard U (p) as a subgroup of GL2 (A(p∞) ). We put %

& ∗ ∗ (p) mod pM2 (Zp ) U0 (p) = U × x ∈ GL2 (Zp )|x ≡ 0 ∗

and write X(U0 (p)) for the compactiﬁed modular curve (GL2 (Q)+ \GL2 (A)+ /U0 (p)Z(R)SO2 (R)) ∪ {cusps}, where GL2 (R)+ is the identity connected component of GL2 (R), GL2 (A)+ = {x ∈ GL2 (A)|x∞ ∈ GL2 (R)+ } and GL2 (Q)+ = GL2 (Q) ∩ GL2 (A)+ in GL2 (A). We then deﬁne S(U0 (p); C) = H 0 (X(U0 (p)), ΩX(U0 (p))/C ). Exercise 2.77 Choose a complete representative set {a1 , . . . , ah } for the quotient GL2 (Q)+ \GL2 (A)+ /U0 (p)GL2 (R)+ inside GL2 (A(∞) ), and deﬁne

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Γi = ai U0 (p)GL2 (R)+ a−1 i ∩ GL2 (Q). Show the following facts: 4 1. X(U0 (p)) − {cusps} ∼ = i Γi \H as open Riemann surfaces; 2. the linear map: i ) (fi ) → i fi (z)dz ∈ S(U0 (p); C) induces an i S2 (Γ∼ isomorphism i S2 (Γi ) = S(U0 (p); C). A double coset U0 (p)xU0 (p) can be considered as an algebraic correspondence on X(U0 (p)) × X(U0 (p)), and we have a natural action of Hecke operators [U0 (p)xU0 (p)] acting on S(U0 (p); C). In this special case of weight 2, there happen to be no old form of level p, because S2 (1, 1; C) ∼ = S2 (SL2 (Z)) = 0 and hence S2 (p, 1; C) = S2new (p, 1; C). We have the following restatement of Corollary 2.31: Theorem 2.78 (Jacquet–Langlands) Let the notation and the assumption be as above. Then we have a C-linear isomorphism i : S(U ; C) ∼ = S p-new (U0 (p); C) (p∞) (p∞) satisfying i ◦ [U xU ] = ) = (DA )× and [U0 (p)xU

0 (p)] ◦ i for all x ∈ GL2 (A p 0 i ◦ [U U ] = [U0 (p) U0 (p)] ◦ i for ∈ OD,p with N () = p. Here the 0 1 superscript “p-new” indicates that the space S p-new (U0 (p); C) is spanned by new vectors at p. × , S p-new (U0 (p); C) = S(Γ 0 (p); C). As we already remarked, if U = O D 2.6.2 Eichler’s integral correspondence Take a suﬃciently large valuation ring W ﬁnite ﬂat over Z as a base ring. As we have already discussed in 1.3.4 and we will give a proof later (Theorem 3.28), Wiles proved the identity of a non-Eisenstein local ring T of H(W ) = H(Z)⊗Z W with an appropriate universal Galois deformation ring, using a limiting argument due to Wiles and R. Taylor. As an introduction to Chapter 3, we describe here brieﬂy the limiting argument without going into too much detail. All the details will be ﬁlled out in 3.2.3 and 3.2.4. Fix a local ring T with maximal ideal m, and write t(q) for the image of T (q) in T. The local ring T is called “Eisenstein” if there exists a pair of Galois characters φ, ϕ : Gal(Q/Q) → T× unramiﬁed outside p such that t(q) ≡ φ(F robq ) + ϕ(F robq ) mod m for almost all primes q outside p. In other words, T is Eisenstein if ρm in Theorem 2.43 is (absolutely) reducible. Here F robq indicates the Frobenius element at q. We assume that T is not Eisenstein. The local ring T carries the associated Galois representation ρT : Gal(Q/Q) → GL2 (T) with Tr(ρT (F robq )) = t(q) for all primes q outside p (see Corollary 2.45). The residual representation ρT = ρT mod m is absolutely irreducible (because T is not Eisenstein), and hence the isomorphism class of ρT is unique by Proposition 1.25. Take a ﬁnite set Q of primes q outside p with q ≡ 1 mod so that ρT (F robq ) has two distinct eigenvalues. Fixing a choice of an eigenvalue αq of ρT (F robq ) for q ∈ Q, we have a unique local component TQ (with maximal " ideal mQ ) of the Hecke algebra (with coeﬃcients in W ) on Γ(Q) = Γ0 (p) ∩ q∈Q Γ1 (q) covering T with u(q) ≡ αq mod mQ for the image

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u(q) of the Hecke operator U (q) (see 3.2.3 for a precise deﬁnition of TQ ). This ring TQ is written as hQ in the middle of page 127 of [MFG]. The limiting argument is done using faithful W -free modules MQ over TQ of level Γ(Q) and taking a limit as we move around inﬁnitely many Q with ﬁxed cardinality. In Theorem 3.23, a set of axiomatic properties for MQ to satisfy (when we move around the ﬁnite sets Q) is listed. Once we ﬁnd inﬁnitely many Q with MQ satisfying the axioms in Theorem 3.23, we can conclude that MQ is free of ﬁxed rank over TQ , that TQ is isomorphic to the corresponding universal Galois deformation ring RQ of ρ = ρT mod mT parameterizing deformations satisfying (D1–5) with ramiﬁcation at Q allowed, and moreover that T ∼ = R∅ is proven to be a local complete intersection over W (an assertion of Theorem 1.52). The original argument of Taylor and Wiles in [TaW] assumes the freeness of MQ over TQ , but as later pointed out by F. Diamond and K. Fujiwara, without assuming the freeness outright, a reﬁned argument also yields the freeness over TQ of the module MQ (including the initial module M∅ ; see Theorem 3.28 and Corollary 3.42). Wiles took T(H 1 (X0 (p), W )) as his initial module M∅ over T and MQ = TQ (H 1 (X(Q), W )) for each Q, where X(Q) = X(U0 (p)) for U = Γ(Q) ⊂ GD (A) deﬁned by %

& xq ≡ ∗ ∗ 0 (Q) = (x ) ∈ GD (Z) mod qOD, for all q ∈ Q Γ 0 ∗ %

& (2.6.1) ∗ ∗ D mod qOD, for all q ∈ Q Γ(Q) = (x ) ∈ G (Z) xq ≡ 0 1 in GD (A). In [MFG] 3.2.7, the local ring itself TQ is taken to be MQ (using the fact that TQ ∼ = HomW (TQ (H 0 (X(Q), ΩX(Q)/W ), W ) = TQ (H 1 (X(Q), OX(Q) )) ∼ by the Grothendieck–Serre duality; [GME] 2.1.2). We can instead take T(S(W )) W )) for Γ(Q) in (2.6.1), because as M∅ and take MQ to be the space TQ (S(Γ(Q); the Hecke algebra on Γ(Q) ⊂ SL2 (Z) over W acts on quaternionic automorphic forms in S(Γ(Q); W ) by the Jacquet–Langlands correspondence. The result is Theorem 2.79 Assume that p is an odd prime. Let be an odd prime outside 3(p − 1). Then S(Z() ) is free of rank 1 over H(Z() ), and Θ induces an isomorphism of H(Z() )-modules: S(Z() ) ⊗H(Z() ) S(Z() ) ∼ = S(p; Z() ), and H(Z() ) is a local complete intersection, where Z() =

. -a b . b

Since the linear map / / 0 0 / 0 1 1 1 1 Θ:S Z ⊗H(Z[ 3(p−1) → S p; Z S Z ]) 3(p − 1) 3(p − 1) 3(p − 1)

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0 1 is an isomorphism after localization at each maximal ideal of Z , it 3(p − 1) 0 / 1 is an isomorphism over Z . In other words, this solves Eichler’s basis / 3(p − 1) 0 / 0 1 1 problem integrally over Z , and S(p; Z ) is contained in the 3(p − 1) 3(p −/1) 0 1 subspace generated by the theta series θ(ai a−1 . ) over Z j 3(p − 1) Here is an outline of the proof in [H05a] (whose details in a more general Hilbert modular case will be given in 3.2.7). By Corollary 2.29, we need to prove that S(Z() ) is free of rank 1 over H(Z() ). Since the pairing ·, · is well deﬁned only over the ring A in which 6 is invertible, we are forced to assume that 6. Since freeness over H(Z() ) is unaﬀected by scalar extension from Z() to a valuation ring W ﬁnite ﬂat over Z , we only need to prove the freeness of M∅ over the given local ring T of the Hecke algebra H(W ) for suﬃciently large valuation ring W . Thus we may assume that T and W share the same residue ﬁeld. We show that S(W ) is free of rank 1 over H(W ) for all prime to p − 1. Since S2 (Γ0 (p)) = 0 if p ≤ 7, we may assume that p ≥ 11. A). A prime We consider the space of automorphic forms SQ (A) = S(Γ(Q); is called Eisenstein if there exists an Eisenstein local component of H(W ). If is Eisenstein, we can ﬁnd a normalized Hecke eigenform in S2 (Γ0 (p)) congruent modulo a prime above to the unique Eisenstein series on Γ0 (p) and |p − 1 (see [Mz1]). By our assumption p−1, there is no Eisenstein component, and we may apply the method of Taylor and Wiles (Theorem 3.23) component-by-component. Fix one such local component T with associated Galois representation ρT and residual representation ρ. We put MQ = TQ (S(Γ(Q); W )) on which the group × ∼ Γ0 (Q)/Γ(Q) = q∈Q (Z/qZ) acts naturally. We write ∆Q for the -Sylow sub There are ﬁve axiomatic conditions on Q, MQ , and RQ in group of Γ0 (Q)/Γ(Q). Theorem 3.23. The conditions (1)–(3) of Theorem 3.23 are about Q and RQ and hence do not depend on MQ , which are veriﬁed in [TaW] and [Di1] under the following condition (c) (as we will do in Chapter 3 in a more general setting). We therefore need to verify /

(a) The deformation problem of T is minimal; that is, either “Selmer”, “strict,” or “ﬂat” at and in Cases (A) or (B) at p in the terminology of 1.3.4. ∼ M∅ as TQ -modules, (b) MQ is free of ﬁnite rank over W [∆Q ] and MQ /aQ MQ = where aQ is the augmentation ideal of W [∆Q ] (the conditions (4) and (5) in Theorem 3.23). √ (c) ρ is absolutely irreducible over Gal(Q/Q[ ∗ ]) for ∗ = (−1)(−1)/2 . The condition (a) is a basic requirement to have a well-posed deformation problem having the universal ring. The condition (b) follows from a horizontal control theorem and is easier to verify in our setting (as we will

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do in the proof of Theorem 3.25) than the cases dealt with in [TaW] and [MFG] 3.2.7, because SQ (W ) = H 0 (Y (Q), W ) for the ﬁnite set Y (Q) = × 1 GD (Q)\GD (A)/Γ(Q)D ∞ , while the choice in [TaW] is TQ (H (X(Q), W )) for the (compactiﬁed) modular curve X(Q) and the choice in [MFG] 3.2.7 is TQ itself. We will give details of the proof of (b) in a more general setting in the proof of Theorem 3.25. We now verify the condition (a). First suppose p = . Locally at p, since the abelian variety associated to each Hecke eigenform f ∈ S2 (Γ0 (p)) is of multiρ restricted to the plicative type (see [GME] 4.2.3), the Galois representation

N ∗ decomposition group Dp at p is isomorphic to up to twists by unrami0 1 ﬁed characters. Here N is the -adic cyclotomic character. If ρ|Dp is semisimple, by the level-lowering argument of [Wi2] Chapters 2 and 3, we must have a Hecke eigenform of level 1 which gives rise to ρ. Since S2 (SL2 (Z)) = 0, this is impossible, and ρ has to be ramiﬁed nonsemisimple, and hence we are in Case (A) at p. Now we study the structure at = p. Since = p, the Galois representation ρ is associated to a ﬁnite ﬂat group scheme (by the construction of the Galois representation in [GME] 4.2.2–3), which is either in the Selmer case or in the ﬂat case (see [GME] Theorem 4.2.4). Suppose = p. If ρ is not wildly ramiﬁed, it is ﬂat, and again by level lowering combined with S2 (SL2 (Z)) = 0, this does not happen. We are in the “strict” case. In the above argument dealing with the local behavior of ρ, we have found a nontrivial unipotent element in the image of the inertia group at p under ρ, which prohibits ρ from being an induced representation from a character of Gal(Q/M ) of a quadratic ﬁeld M/Q. In particular, we conclude (c).

3 HECKE ALGEBRAS AS GALOIS DEFORMATION RINGS

This chapter is the heart of the book. So far, we have often put oﬀ detailed proofs of given statements in order to present the structure of the theory without going into technicalities. Here we give detailed proofs of the identiﬁcation of the Galois deformation ring and the corresponding Hecke algebra, assuming certain minimality conditions concerning ramiﬁcation properties of the deformations; see the conditions (h1–4). Fix a totally real number ﬁeld F . Take a holomorphic Hilbert modular Hecke eigenform on GL2 (FA ) and its p-adic Galois representation ρf : Gal(Q/F ) → GL2 (W ) for a valuation ring W ﬁnite ﬂat over Zp such that the L-function L(s, ρf ) coincides with the Mellin transform L(s, f ) (Theorem 2.43). Here we describe the method of Taylor and Wiles, and of Fujiwara of identifying the local ring of an appropriate Hecke algebra with a universal Galois deformation ring of ρf . By the Jacquet–Langlands correspondence discussed in Chapter 2, we can choose a quaternion algebra D unramiﬁed everywhere at ﬁnite places whose automorphic variety has dimension ≤ 1 on which we can ﬁnd a Hecke eigenform fD with the same eigenvalues as f . In particular, if [F : Q] is even, we can choose D with zero-dimensional automorphic variety, and hence everything becomes easier. We describe the theory of automorphic forms of deﬁnite quaternion algebras ﬁrst and later we describe how to deduce the result for a base ﬁeld of odd degree by a base-change technique. We start with an analysis of Hecke algebras acting on functions on the zerodimensional automorphic variety. More speciﬁcally, in the following section, we study the level-raising technique (ﬁrst invented by Ribet in his ICM talk in 1982; [Ri2]), following Taylor’s treatment in [Ta], which allows us to ﬁnd an automorphic new form of higher level congruent modulo an ideal to a given lower level Hecke eigenform if its eigenvalues modulo the ideal satisfy certain conditions speciﬁc to higher level forms. There are level-lowering techniques (again invented by Mazur and Ribet in the late 1980s [Ri4] for elliptic modular forms) reversing the situation, but they require more sophisticated tools to deal with and are probably premature yet in their development in the Hilbert modular case to discuss in book form, though the level lowering is crucial in the proof of Fermat’s last theorem in [Wi2]. Anyway, we do not touch in this book the level-lowering method, and basically because of this, we assume minimality to avoid such techniques (see [SW2] and the papers quoted there for level lowering). We then proceed to identify Galois deformation rings with the Hecke algebra in Section 3.2, study base-change (ascent) and descent techniques to include odd

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163

degree base ﬁeld F in Section 3.3, and give proofs of the result on the L-invariant already stated in Chapter 1 in a more general setting of Hilbert modular forms in Section 3.4. A Hecke eigenform f is called nearly p-ordinary if f |Up (p) = ap f with a p-adic unit ap (that is, |ip (ap )|p = 1). Here f is either a Hilbert modular form or a quaternionic modular form). Similarly for a prime factor p|p in F , we call f nearly p-ordinary if f |Up (p ) = ap f with a p-adic unit ap . Suppose that the Galois representation ρf restricted to the decomposition

p ∗ group Dp at p is isomorphic to an upper triangular representation 0 δp for a prime factor p|p in F (the near p-ordinarity condition). We call a nearly p-ordinary form f p-ordinary if δp is unramiﬁed and p-distinguished if δp ≡ p mod mW for the maximal ideal mW of W . Throughout this section, we ﬁx a prime p > 2 and a base totally real ﬁeld F . 3.1 Hecke algebras We look carefully into integrality of the Hilbert modular Hecke algebras by letting them act on automorphic forms on division quaternion algebras (via the Jacquet–Langlands correspondence). If a local ring of the Hecke algebra of level pN for a prime p|p outside N acts nontrivially on automorphic forms of level N, we can describe quite precisely the “new” part of the local ring (which acts faithfully on the orthogonal complement of the level N automorphic forms and their translations) in terms of the Hecke algebra of lower level N (Theorem 3.16). Though this type of result relating new and old forms is technical, it is useful later in the proof of Theorem 3.28 to deduce “freeness” of the space of automorphic forms of higher level out of the freeness of lower level, because an immediate application of the Taylor–Wiles system is such freeness for minimal levels (and we need the level-raising argument to add extra primes to the minimal level). As we have observed already, the freeness over the Hecke algebra of the space of automorphic forms is a key to solving the integral basis problem of Eichler. Our description of this level-raising argument is based on Taylor’s work [Ta] in which he applied such a technique to construct Galois representations in Theorem 2.43 to ﬁll the cases missing in Carayol’s earlier work [C1] (see also, [CPS] [68c] and [H81a]), following Wiles’ observation [Wi] of the utility of the level-raising techniques to reduce the problem to the cases already studied by Carayol. 3.1.1 Automorphic forms on deﬁnite quaternions In this section, we assume F to have even degree and D to be a totally deﬁnite quaternion algebra over F unramiﬁed at every ﬁnite place of F . By (H1) in 2.1.1, be the such a quaternion algebra exists andis unique up to isomorphisms. Let Z compact ring given by the product Z over all rational primes , and for any For the = L ⊗Z Z. lattice L of a ﬁnite-dimensional Q-vector space, we write L (∞) integer ring O of F , O is a compact ring, and FA = O + F inside the product

164

l

Hecke algebras as Galois deformation rings

Fl over all prime ideals l of F . Fix a maximal order OD of D. Identify D = M2 (O). O

(3.1.1)

(∞) (∞) ∼ Write simply G/Z for GD ) = GL2 (FA ). For /Z in this section. Thus G(A an open subgroup S of G(A(∞) ) and a commutative Z-algebra A, we deﬁne M(S; A) to be the A-module of functions f : X(S) → A, where X(S) = G(Q)\G(A(∞) )/S = G(Q)\G(A)/S · G(R). The automorphic manifold we considered in Chapter 2 is denoted by Y D (S) which is slightly diﬀerent from X(S), because

Y D (S) = G(Q)\G(A)/Z(A)S · G(R) for the center Z of G (in other words, Y D (S) = X(Z(A(∞) )S)). We pick an =O × : O-ideal N = 0 and recall the following open subgroups of G(Z) D %

& a b , c ∈ N 0 (N) = Γ ∈ G(Z) c d (3.1.2) & %

a b , 0 (N)a ≡ d ≡ 1 mod N 11 (N) = ∈Γ Γ c d = NO D . In Chapter 2, instead of the standard Γ 0 (N), we worked with where N

δ 0 0 (N)η for η = the conjugate S0 (N) = η −1 Γ for a ﬁnite idele δ with 0 1 = dO. This was to simplify the form of the q-expansion of Hilbert moduδO 0 (N)) ∼ lar forms. Since g → gη gives a Hecke equivariant isomorphism X(Γ = X(S0 (N)), identifying everything via η, we work with more standard subgroups 0 (N) in this chapter. Γ Let T = ResO/Z Gm . We identify the diagonal torus TG of ResO/Z GL(2) with 2 ⊂ G(Z) by the identiﬁcation O D = M2 (O). T in an obvious way. Then TG (Z) Until the end of this section where we become able to treat general weights κ, we ﬁx our weight to be κ = κ0 = (0, I); so, we forget about the weight. We write κ0 for the ﬁxed initial weight (0, I) if we need to distinguish the initial weight κ0 from general weights κ ∈ Z[I]2 . For a general commutative ring A, we consider the A-valued continuous ﬁnite-order character ε = → A× (of weight (0, I)) which is an A-analogue of the (ε1 , ε2 , ε+ , ε− ) : TG (Z)

a 0 Neben character we considered in Chapter 2. Thus ε = ε1 (a)ε2 (d) for 0 d → A× . Let Z ⊂ G be the center. We suppose ﬁnite-order characters εj : T (Z) (∞) × that there exists a ﬁnite-order character ε+ : Z(Q)\Z(A ) → A such that z 0 The character ε− : T (Z) → A× is a ε+ = ε1 (z)ε2 (z) for z ∈ T (Z). 0 z − ﬁnite-order character given by ε− (a) = ε−1 2 ε1 (a). Let c(ε ) be the conductor of 0 (N) → A× ε− , and we assume that c(ε− ) ⊃ N. Deﬁne a character ε : Z(A(∞) )Γ

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165

a b 0 (N) and z ∈ Z(A(∞) ). by ε(zu) = ε2 (det(u))ε (aN )ε+ (z) for u = ∈Γ c d 0 (c(ε− )). We then write M(S, ε; A) for the space Let S be an open subgroup of Γ (∞) of functions f : G(Q)\G(A ) → A satisfying −

f (zxu) = ε(uz)f (x)

for all u ∈ S and z ∈ Z(A(∞) ).

(3.1.3)

be the ring of Witt vectors with 0 (N), ε; A). Let W We put M(N, ε; A) for M(Γ as a subring of coeﬃcients in Fp which is an algebraic closure of Fp . We regard W p of an algebraic closure Qp of Qp . We ﬁx an embedding the p-adic completion Q p , and put W = i−1 (W ). We ﬁx the ip of an algebraic closure Q of Q into Q p notation B for a base ring which is a discrete valuation ring in Q. We assume that B is a suﬃciently large extension of Z(p) = Zp ∩ Q, and suppose that ε has values in B × . We write F for the residue ﬁeld of B (of characteristic p). The ring B could be p. W, and W denotes the p-adic completion of B. Let A be any B-subalgebra of Q M If A = limn An is proﬁnite with ﬁnite rings An , plainly T (p) for a large M gives ←− an idempotent acting on M(N, ε; An ). Thus the limit e = limn→∞ T (p)n! gives a projector acting on the limit M(N, ε; A). If A is a W -algebra, M(N, ε; A) = M(N, ε; W ) ⊗W A, and hence the p-ordinary projector e = limn→∞ T (p)n! is well deﬁned acting on M(N, ε; A), because it converges in EndW (M(N, ε; W )), and hence we have well-deﬁned Mn.ord (N, ε; A) = e (M(N, ε; A)). We suppose M(N, ε; B) ⊗B F ∼ = M(N, ε; F).

(sm)

Exercise 3.1 Supposing (sm), prove M(S, ε; B∞ ) is p-divisible if S is an open 0 (N), where B∞ = lim p−n B/B. subgroup of Γ −→n 4 h 0 (N). We have f (γcj uz) = Decompose G(A(∞) ) = j=1 G(Q)cj Z(A(∞) )Γ −1 0 (N)c ∩ G(Q) has to act trivially on f (cj ) ε(zu)f (cj ). Thus Γj = cj Z(A)Γ j 0 for f ∈ M(N, ε; B). Thus M(N, ε; A) = j H (Γj (ε), A(ε)), where Γj (ε) = A is a Γ -module on which Γj acts by ε. Taking a Γj / Ker(ε) and A(ε) ∼ = j generator of mB , we have a long exact sequence

H 0 (Γj (ε), B(ε)) −→ H 0 (Γj (ε), B(ε)) → H 0 (Γj (ε), F(ε)) → H 1 (Γj (ε), B(ε))

associated to the short one: 0 → B(ε) −→ B(ε) → F(ε) → 0. Thus we have Coker(H 0 (Γj (ε), B(ε)) −→ H 0 (Γj (ε), B(ε))) = H 0 (Γj (ε), B(ε)) ⊗B F, and if H 1 (Γj (ε), B(ε)) = 0, (sm) holds. In particular, if the following condition (sm0) is satisﬁed, the condition (sm) is satisﬁed: (sm0)

ε restricted to Γj := Γj /Z(Q) has order prime to p.

Since ε+ is trivial on Z(Q), ε factors through Γj = Γj /Z(Q) which is a ﬁnite group by (2.3.1). Thus if |Γj | is prime to p, (sm0) is satisﬁed by any choice of ε. For a suﬃcient condition for p |Γj |, see (sm1) in 3.1.3 and Lemma 3.3.

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Hecke algebras as Galois deformation rings

Recall the prime

elements q in Fq we ﬁxed in 2.3.2 (for each prime ideal q). 1 0 Let ηp = ∈ G(Qp ). We deﬁne [g](f )(x) = f (xg) for g ∈ G(A(∞) ) and 0 p f : G(A(∞) ) → A. Suppose that N is prime to p. Deﬁne iA : M(N, ε; A)2 → M(pN, ε; A) by iA (f1 , f2 ) = f1 + [ηp ](f2 ). Lemma 3.2 (R. Taylor)

Let K be the quotient ﬁeld of B. Then we have

M(pN, ε; B) ∩ Im(iK ) = Im(iB ) Proof

if p N.

We repeat the proof given by R. Taylor in [Ta] Lemma 4. We consider 0 (N), Y0 (N) = G(Q)\G(A(∞) )/Z(A(∞) )Γ 0 (pN). Y0 (pN) = G(Q)\G(A(∞) )/Z(A(∞) )Γ

(3.1.4)

0 (N) = Γ 0 (pN), we have two projections 0 (N)ηp−1 ∩ Γ Since ηp Γ πj : Y0 (pN) Y0 (N) given by 0 (N). 0 (pN)) = G(Q)xΓ 0 (N) and π2 (G(Q)xΓ 0 (pN)) = G(Q)xηp Γ π1 (G(Q)xΓ We need to show that if f = f1 + [ηp ](f2 ) ∈ Im(iK ) ∩ M(pN, ε; B), fi (cj ) ∈ B for all j and i. For points x, y ∈ Y0 (pN), we deﬁne x ∼ y if we can ﬁnd a sequence of points x1 , . . . , xm in Y0 (pN) such that for all i = 0, . . . , m (putting x0 = x and xm+1 = y), either π1 (xi ) = π1 (xi+1 ) or π2 (xi ) = π2 (xi+1 ). Let y1 , . . . , ys ∈ G(A(∞) ) be a complete set of representatives under this equivalence relation. Then πj (j = 1, 2) gives rise to a surjection π j : Y0 (N) {y1 , . . . , ys } sending z ∈ Y0 (N) to a unique point yk equivalent to any point z0 ∈ Y0 (pN) with πj (z0 ) = z, because any points z0 above z ∈ Y0 (N) with respect to πj are equivalent to a single yk . Then we can deﬁne a function f1 : Y0 (N) → A(ε) in M(N, ε; K) by putting f1 (z) = f ( π1 (z)). Similarly we have f2 ∈ M(N, ε; K) such that π2 (z)). Thus f1 = [ηp ](f2 ) because π2 (z0 ) = z ⇔ π1 (z0 η) = z. Then f2 (z) = f ( iK (f1 − f1 , f2 + f2 ) = f1 − f1 + [ηp ](f2 ) + [ηp ](f2 ) = iK (f1 , f2 ). Thus we may assume that f1 (z) = 0 for a given point z ∈ Y0 (N). We write [y] for the equivalence class of y ∈ Y0 (pN). For each z0 ∈ [yk ], deﬁne d(z0 ) = m to be the length of the minimal chain z0 = x0 , x1 , . . . , xm = yk with xi ∼ xi+1 (for i = 0, 1, . . . , m − 1) connecting x and yk . Pick z0 ∈ [yk ] with π1 (z0 ) = z and π2 (z0 ) = z . Since f2 (z ) = iK (f1 , f2 )(z0 ) = f (z0 ) ∈ B (by f1 (z) = 0), we ﬁnd that f2 (z ) ∈ B if d(z0 ) = 0. We now proceed by induction on d(z) to show that f1 (z) ∈ B and f2 (z ) ∈ B. Therefore suppose that fj (πj (w0 )) ∈ B if d(w0 ) < m (w0 ∈ [yk ]). Pick z with d(z) = m. Taking a minimal sequence z0 ∼ x1 ∼ · · · ∼ xm−1 ∼ yk , we ﬁnd f1 (π1 (x1 )) ∈ B and f2 (π2 (x1 )) ∈ B by the induction hypothesis. Since x1 ∼ z0 , we ﬁnd

Hecke algebras

167

fj (πj (x1 )) = fj (πj (z0 )) for one of j = 1, 2. Since f1 (z) + [ηp ](f2 )(z) = f (z0 ) ∈ B, we ﬁnd the B-integrality at z for the other one. 2 3.1.2 Hecke operators We recall the deﬁnition of the Hecke operators acting on M(N, ε; B): let % ∆0 (N) = % ∆1 (N) =

a b c d a b c d

aN ∈ O× D ∩ G(A(∞) )c ∈ N, ∈O N

&

& ∈ ∆0 (N) aN − 1 ∈ NON .

(3.1.5)

Under multiplication, ∆0 (N) is a semigroup. We extend ε to ∆0 (N) in the follow Z ing way: decompose T (A(∞) ) = T (Z)× l l choosing a prime element l for each prime ideal l in Fl , and extend ε2 to T (A(∞) ) trivially on the second factor

Z a b − ; so, ε ( ) = 1. Then we deﬁne ε(δ) = ε (det(δ))ε (a) if δ = . 2 l 2 N l l c d On Z(A(∞) ) ∩ ∆0 (N), ε coincides with ε+ (zN )ε2 (z (N) )2 , we may extend ε to Z(A(∞) )∆0 (N) by ε(zδ) = ε+ (zN )ε2 (z (N) )2 ε(δ) for z ∈ Z(A(∞) ) and δ ∈ ∆0 (N). Since ε : Z(A(∞) )∆0 (N) → B × is a character of the semigroup, it extends to the character of the subgroup of G(A(∞) ) generated by Z(A(∞) )∆0 (N), which contains ∆0 (N)ι ⊂ Z(A(∞) )∆0 (N)−1 . For each integral ideal l, we decompose, recalling the reduced norm map N : D → F which induces N : G → T , 1 (N){x ∈ ∆0 (N)|N (x) = l }Γ 11 (N) = Γ 1

5

11 (N)y Γ

y

imposing the condition det(y) = l , and deﬁne a Hecke operator T (l ) acting 1 (N) by 0 (N) ⊃ S ⊃ Γ on M(S, ε; A) for any subgroup S with Γ 1 T (l )(f )(x) =

y

ε(y)[y ι ](f )(x) = ε(l )

ε(y −ι )f (xy ι ).

y

We put T (l) = ε(l )−1 T (l ), which is independent of the choice of the representatives y. In particular, T (p) acting on M(N, ε; B) with N prime to p|p and T (p) acting on M(pN, ε; B) are diﬀerent; so, we write the latter as U (p) often. We write G1 for the derived group of G. Thus for the reduced norm map N : G → T , we have the following exact sequence: 1 → G1 → G → T . The reduced norm N is surjective onto T (Q)+ = T (Q) ∩ T (R)+ for the identity connected component T (R)+ of the Lie group T (R) (see Theorem 2.7). Thus the above sequence induces a surjection N 0 (N) − X0 (N) := G(Q)\G(A(∞) )/Γ → T (Q)+ T (Z)\T (A(∞) ).

168

Hecke algebras as Galois deformation rings

Note that T (Q)+ T (Z)\T (A(∞) ) (R)+ \T (A(∞) )T (R)+ = T (Q)+ T (Z)T ∼ (R)+ ∩ T (Q)+ T (Z)T (R)+ \T (Q)T (A(∞) )T (R)+ = T (Q)T (Z)T ∼ (R)+ \T (A) ∼ = ClF+ = T (Q)T (Z)T for the strict class group ClF+ of F . We write Iv(S, ε; A) for the subspace of all functions in M(S, ε; A) factoring through N : G(Q)\G(A(∞) ) T (Q)\T (A(∞) ). 0 (N), we write We deﬁne S(S, ε; A) := M(S, ε; A)/Iv(S, ε; A), and when S = Γ S(N, ε; A) and Iv(N, ε; a) for S(S, ε; A) and Iv(S, ε; A). Thus, under the notation in Chapter 2, we have S(S, ε; C) = SκD (S, ε; C) for κ = (0, I). Obviously, Iv(N, ε; A) is stable under Hecke operators; thus, S(N, ε; A) inherits the Hecke operator action from M(N, ε; A). We write h(N, ε; A) for the subalgebra of EndA (S(N, ε; A)) generated by T (l) for all integral ideals l over A. 0 (N) are conjugate in G(A(∞) ), the corresponding spaces of Since S0 (N) and Γ automorphic forms produce the isomorphic Hecke algebra. Thus by the Jacquet– Langlands correspondence (Corollary 2.33), we have h(N, ε; A) ∼ = hκ (N, ε; A) with κ = (0, I) for hκ (N, ε; A) deﬁned just below (2.3.24) as long as A can be embedded into C (including A = W and Qp ). Similarly we write H(N, ε; A) for the subalgebra of EndA (M(N, ε; A)) generated by T (l) for all integral ideals (p)n! if l over A. We put ep = limn→∞ T (p)n! if p N and ep = limn→∞ U p ⊃ N. Let Σp be the set of prime factors of p in F . We also put e = p|p ep and eΣ = p∈Σ ep for a subset Σ of Σp . The operators ep for level N may not induce ep of level pN, and they could diﬀer. For a subset Σ ⊂ Σp , we write X Σ-ord = eΣ (X), where X = h(N, ε; A), H(N, ε; A), S(N, ε; A) and M(N, ε; A). 3.1.3 Inner products For a given x ∈ X(S), choosing a representative, written again as x ∈ G(A(∞) ), we deﬁne ex = |Γx (S)/Z(Q)| for Γx (S) = (xSx−1 Z(A(∞) ) ∩ G(Q)). Note that (xSx−1 Z(A(∞) )∩G(Q))/Z(Q) is discrete and compact (since G(R)/Z(R) is compact), and hence, it is a ﬁnite group. Assuming e−1 x ∈ A for all x ∈ Y (S), we 0 (N) have a perfect pairing for S ⊂ Z(A(∞) )Γ (·, ·)S : M(S, ε; A) × M(S, ε−1 ; A) → A given by (f, g)S = x∈Y (S) e−1 x f (x)g(x). When S = Γ0 (N), we write the pairing (·, ·)S as (·, ·)N . We now determine the prime factors of ex . Lemma 3.3 If is an odd prime factor of ex , then we have [F (µ ) : F ] = 2. 0 (N), if the ideal N has a prime factor l inert in F (µq )/F for an odd For S = Γ rational prime q, ex is prime to q.

Hecke algebras

169

Proof If an odd prime is a factor of ex , we can ﬁnd ζ ∈ Γx (S) with order in the ﬁnite group Ex = (xSx−1 Z(A(∞) ) ∩ G(Q))/Z(Q). Then N (ζ 2 ) ∈ (F × )2 . Thus η = ζ 2 N (ζ)−1 has reduced norm 1 and has order in Ex , because is odd. Since η ex ∈ F , N (η) = 1 implies that η is a root of unity. Since F (η) is at most a quadratic extension of F and F is totally real, we ﬁnd F (µ ) ∼ = F (η) is quadratic over F . For the second assertion, consider the order %

& a b R= ∈ M2 (Ol )c ∈ lOl . c d

a b Then

→ d mod l gives a ring homomorphism π : R → O/l. If l|q, l c d cannot be inert in Ol [µq ] over Ol , since q fully ramiﬁes in Zq [µq ]/Zq . Thus we may assume that l q, and then Ol [µq ] is the l-adic integer ring of Fl [µq ]. If ζ ∈ R× has order q, π : Ol [ζ] ⊂ R → O/l is a surjective ring homomorphism, which is impossible, because Ol [ζ] modulo its maximal ideal is a quadratic extension of O/l. 2 e follows: for Let τ (N) ∈ G(A(∞) ) be as

a prime O-ideal l, if l exactly divides N 0 −1 with e > 0, we put τ (N)l = ∈ GL2 (Fl ), and if e = 0, we put τ (N)l le 0 to be the identity matrix. Then we deﬁne [τN ](f )(x) = ε+ (det(x))−1 f (xτ (N)).

Exercise 3.4 Prove that the linear operator [τN ] gives an isomorphism M(N, ε; A) ∼ = M(N, ε−1 ; A).

0 (N ) and τN a b τ −1 ≡ d Hint: use the fact that [τN ] normalizes Γ 0 c d N mod N.

∗ a

We consider the following condition (sm1) Either [F (µp ) : F ] > 2 or an inert prime l in F (µp )/F is a factor of N, under which (by Lemma 3.3) we have a well deﬁned perfect pairing (·, ·)N for a B-algebra A. Thus, assuming (sm1), we may deﬁne a perfect pairing ·, ·N : M(N, ε; A) × M(N, ε; A) → A by f, g = (f, g|[τ (N)])N . Lemma 3.5

The Hecke operator T (l) is self-adjoint under this pairing: T (l)(f ), g = f, T (l)(g).

Proof Choose a (right invariant) Haar measure dµ(x) on Q = G(Q)\G(A(∞) ). Then f, g is a nonzero constant multiple of Q f (x)([τ ]g)(x)dµ(x) for τ = τ (N). 0 (N) and decompose SyS = 4 hS. Then we have Write S = Γ h

170

Hecke algebras as Galois deformation rings

([SyS]f )(x)ε+ (det(x))−1 g(xτ )dµ(x)

Q

=

h

xh→x

=

h

= =

f (xh)ε+ (det(x))−1 g(xτ )dµ(x)

Q

h

Q

h

Q

f (x)ε+ (det(x))−1 ε+ (det(h))g(xh−1 τ )dµ(x)

Q

f (x)ε+ (det(x))−1 g(xhι τ )dµ(x) f (x)ε+ (det(x))−1 g(xτ τ −1 hι τ )dµ(x),

(3.1.6)

l 0 because ε+ (det(h))g(xh−1 ) = g(xN (h)h−1 ) = g(xhι ). Now take y = . 4 4 ι 0 1 We can choose h so that GL2 (Ol )yGL2 (Ol ) = h hGL2 (Ol ) = h h GL2 (Ol ) basically because (GL2 (Ol )hGL2 (Ol ))ι = GL2 (Ol )hGL2 (Ol ) (see [IAT] Chapter 3). Then the above computation (3.1.6) conﬁrms the desired result, l u since h commutes with τ . If l is a factor of N, take h = hu = . Then 0 1 we have by computation 5

SyS = u

mod l

5

hu S = u

τ −1 hιu τ S.

mod l

Then again by (3.1.6), we conﬁrm the desired result.

2

(A)2 \T (A), the pairings (·, ·) and By surjectivity: N : Y0 (N) T (Q)+ T (Z)T ·, · induce perfect duality on Iv(N, ε; A), and hence S(N, ε; A) ∼ = Iv(N, ε; A)⊥ as modules over Hecke operators. By Corollary 2.33, S(N, ε; A) ∼ = S(0,I) (N, ε; A) as Hecke modules. By the duality theorem (Theorem 2.28), S(0,I) (N, ε; A) ∼ = HomA (h(N, ε; A), A) as Hecke modules. The self-duality of S(N, ε; A) therefore tells us Lemma 3.6 If A is a Q-algebra, S(N, ε; A) is free of rank 1 over the Hecke algebra h(N, ε; A). 0 (N) ⊃ S ⊃ Γ 1 (N), we may More generally, for a balanced subgroup S with Γ 1 deﬁne the paring f, gS = (f, g|[τ (N)])S for f, g ∈ M(S, ε; A). Here S is called balanced if τ (N) normalizes Z(A(∞) )S. The pairings are perfect, and T (l) and 0 (N)/S are self-adjoint under ·, ·S . the action of Γ

Hecke algebras

171

We consider the adjoint map i∗A : M(pN, ε; A) → M(N, ε; A)2 under ·, · of iA (f1 , f2 ) = f1 + [ηp ](f2 ). Here is a lemma similar to [Ta] Lemma 2: Lemma 3.7

Suppose (sm1) and p N. The matrix form of i∗B ◦ iB is given by

T (p) ε+ (p )(N (p) + 1) M := , ε+ (p )T (p) ε+ (p )(N (p) + 1)

where N (p) = [O : p]. Proof

We compute the matrix

a b c d

with a, b, c, d ∈ EndA (M(N, ε; A)). We

then need to compute  f, a(g)N = f, gpN    f, b(g) = [η ](f ), g N p pN f, c(g)N = f, [ηp ](g)pN    f, d(g)N = [ηp ](f ), [ηp ](g)pN ,

1 0 ∈ G(Qp ). A computation of the Petersson inner products where ηp = 0 p analogous to the above for Hilbert modular forms has been done in [H91] (5.9). Here we shall do the computation

for quaternionic modular forms. For the second 0 −1 −1 0 (N)p , we have product, by ηp τ (p)p = ∈Γ 1 0

−1 [ηp ](f ), gpN = e−1 x f (xηp )ε+ (det(x))g(xτ (pN)) x∈Y0 (pN) xηp →x

=

ε+ (p )

−1 −1 e−1 x f (x)ε+ (det(x))g(xηp τ (pN))

x∈Y0 (pN)

= ε+ (p )(N (p) + 1)f, gN , 0 (Np)] = N (p) + 1. 0 (N) : Γ by [Γ For the third product, noting τ (p)ηp = p u for u ∈ SL2 (Op ), we have

−1 f, [ηp ](g)pN = e−1 g(xτ (pN)ηp ) x f (x)ε+ (det(x)) x∈Y0 (pN)

= ε+ (p )

−1 e−1 g(xτ (N)) x f (x)ε+ (det(x))

x∈Y0 (pN)

= ε+ (p )(N (p) + 1)f, gN . For the last one, by the above computation for the third one (replacing f by [ηp ](f )), we get, deﬁning Tr(f ) = s∈Γ0 (N)/Γ0 (pN),det(s)=1 [s](f ), [ηp ](f ), [ηp ](g)pN = ε+ (p )([ηp ](f ), [τ (N)](g))pN = ε+ (p )(Tr([ηp ](f )), [τ (N)](g))N = ε+ (p )T (p)(f ), gpN .

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For the ﬁrst product, we get

−1 f, gpN = g(xτ (p)τ (N)) e−1 x f (x)ε+ (det(x)) x∈Y0 (pN) xτ (p)→x

=

ε+ (p )

−1 e−1 )ε+ (det(x))−1 g(xτ (N)) x f (xτ (p)

x∈Y0 (pN)

= ε+ (p )(Tr([τ (p)−1 ]f ), [τ (N)](g))N . Since τ (p)−1 = p−1 ηp u with u ∈ SL2 (Op ), we conclude f, gpN = T (p)(f ), gN = f, T (p)(g)N as desired.

2

Lemma 3.8 We have Iv(N, ε; A) = 0 for any B-algebras A, unless the following condition is satisﬁed (dg) The modulo mB “Neben” character ε− = (ε− mod mB ) is trivial on T (Z) for the maximal ideal mB of B. Proof If (dg) fails, ε mod mB does not factor through N ; so, Iv(N, ε; F) = 0 which implies Iv(N, ε; A) = 0 for all A. The central character ε+ has to be totally even because of the invariance of automorphic forms in M(N, ε; A) under G(R). Thus ε+ = ψ 2 for an Fp -valued Hecke character ψ of F , because the only two torsion of the Galois group of the maximal abelian extension of F over F comes from complex conjugations (see Lemma 3.24). In particular, if ε− is trivial on since the reduced ψ ◦ N is in M(N, ε; F) if we choose ψ = ε2 on T (Z), T (Z), (∞) 2 norm N is equal to the determinant map det on G(A ). If ε− has order prime to p, (dg) is equivalent to the triviality of ε− . Suppose that ε− is trivial. Choosing a ﬁnite-order Hecke character ψ of F so that ψ|T (Z) = ε2 ,

we may identify Iv(N, ε; A) with the space of functions on φ : T (Q)+ \T (A(∞) ) → By sending A such that φ(z 2 ux) = ψ(z 2 u)φ(x) for all z ∈ T (A(∞) ) and u ∈ T (Z). −1 φ → ψ φ, we may identify Iv(N, ε; A) with the space C(A) of all functions × C(A) → A given φ : C = ClF+ /(ClF+ )2 → A. We have a pairing (·, ·)C : C(A) ∗ by (φ, φ ) = x∈C φ(x)φ (x). We thus have the adjoint jA : M(N, ε; A) C(A) ∼ of the inclusion jA : C(A) = Iv(N, ε; A) ⊂ M(N, ε; A) such that jA (φ), f N = ∗ (f ))C . (φ, jA Here is another lemma in [Ta] page 272:

Lemma 3.9 Suppose (sm) and p N. Then S(N, ε; A) is A-free of ﬁnite rank, and we have the following commutative diagram with exact rows and

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columns: 0 0 ↓ ↓ 0 → Iv(N, ε; A) → Iv(N, ε; A) → 0 ↓j ↓ ↓ 0 → Iv(N, ε; A)2 → M(N, ε; A)2 → S(N, ε; A)2 → 0 ↓ ↓ ↓ iA 0 → Iv(pN, ε; A) → M(pN, ε; A) → S(pN, ε; A) → 0 ↓ 0 Here the map j is given by j(f ) = ([ηp ](f ), −f ). Proof Under (sm), we see easily that Iv(N, ε; A) = Iv(N, ε; B) ⊗B A; so, it is an A-free direct summand of M(N, ε; A), and hence S(N, ε; A) is A-free. The commutativity is plain except for the part involving j. If f ∈ Iv(N, ε; A), we have f = φ ◦ N for φ : T (A(∞) )/T (Q)+ → A with φ(x det(u)z 2 ) = ε(uz)φ(x) 0 (N)) = det(Γ 0 (pN)), we ﬁnd that 0 (N) and z ∈ T (A(∞) ). Since det(Γ for u ∈ Γ Iv(N, ε; A) = Iv(pN, ε; A) and also [ηp ](f )(x) = φ(N (xηp ) = φ(N (x)N (ηp )) = φ(N (x)p ) still remains in Iv(N, ε; A) and [ηp ] induces an automorphism of Iv(N, ε; A). Thus j is well deﬁned, and we ﬁnd that Ker(iA |Iv(N,ε;A) ) = Im(j) by the deﬁnition of iA . Then the commutativity of the diagram is clear. We now need to prove the exactness. The exactness of the ﬁrst column follows from the above argument. The exactness of the rows follows from the deﬁnition. We shall use the notation introduced in the proof of Lemma 3.2; so, “∼” is the equivalence relation generated by x ∼ y if πj (x) = πj (y) for either j = 1, 2. Thus the equivalence class [x] of x ∈ Y0 (pN) is the union of ﬁbers of πj intersecting each other. Take (f1 , f2 ) ∈ Ker(iA ). By the proof of Lemma 3.2, we may assume that f1 (yj ) = 0 for all j. Since f1 (yj ) + [ηp ](f2 )(yj ) = iA (f1 , f2 ) = 0, we have f2 (yj ηp ) = 0. By induction on d(x), we shall show that f1 (x) = f2 (xηp ) = 0. Taking a minimal chain x ∼ x1 ∼ · · · ∼ xm ∼ yj , we ﬁnd f1 (x1 ) = f2 (x1 ηp ) = 0 by the induction hypothesis. By our deﬁnition of the equivalence, f1 (x) = f1 (x1 ) = 0 or f2 (xηp ) = f2 (x1 ηp ) = 0. Since f1 (x) + f2 (xηp ) = 0, we ﬁnd the vanishing of the other. Thus we have Ker(iA ) = {([ηp ](f1 ), −f1 )|f1 ∈ S(N, ε; A) and f1 : π1 (Y0 (pN)/ ∼) → A}. (3.1.7) Under (sm), we thus ﬁnd Ker(iA ) = Ker(iB ) ⊗B A. The validity of the exactness of the rows and columns of the diagram over K follows from the representation theory of GL2 (Fp ), because the subspace Iv(N, ε; K) is the direct sum of onedimensional representations of GL2 (Fp ). Then the exactness of the middle and

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the last column follows from Ker(iA ) = Ker(iB )⊗B A because of rankA Ker(iA ) = rankB Ker(iB ) = dimK Ker(iK ). 2 3.1.4 Ordinary Hecke algebras We ﬁx a Neben character ε = ε0 (and call it the initial Neben character). Again, as in the case of κ0 = (0, I), we usually write just ε for ε0 , but if we want to distinguish the ﬁxed one ε0 from a general Neben character, we write it as ε0 . Let m0 be a maximal ideal of hp-ord (N, ε; B) and T0 = hp-ord (N, ε; B)m0 for the localization at m0 . In other words, we ﬁx a Hecke eigenform f0 ∈ S p-ord (N, ε; B) and take a local ring T0 of hp-ord (N, ε; B) which acts nontrivially on f0 (we call the form f0 the initial Hecke eigenform). We will deform f0 (and its Galois representation) p-adically, and the data (κ0 , ε0 , f0 ) specify the origin and a coordinate system around the origin of the deformation space. This justiﬁes the terminology: “initial Hecke eigenform.” Taking the pullback M0 of m0 to H := H p-ord (N, ε; B), put M0 (A) := Mp-ord (N, ε; A) ⊗H HM0 for the localization HM0 of H at M0 . Assuming p N, the image S0old (A) ⊂ S(pN, ε; A) of iA : M0 (A)2 → M(pN, ε; A) composed with the projection M(pN, ε; A) S(pN, ε; A) is stable under U (p) and T (l) for l prime to p. Put t(l) = T (l)|S0old (B) and u(p) = U (p)|S0old (B) . Conold sider the Hecke algebra Told 0 ⊂ EndB (S0 (B)) generated by the Hecke operators u(p) and t(l) for l prime to l. We consider the following conditions (dsq ) for a prime ideal q in F , the residual Galois representation ρ = ρm0 in Theorem 2.43 restricted to the decomposition group Dq is isomorphic to εq ∗ with distinct characters εq and δ q ; 0 δq (sf) N0 = N/c(ε− ) is square-free and is prime to c(ε− ). Lemma 3.10 Let the notation and the assumptions be as above. Fix a prime ideal of q of F , and assume (sf) and (dsq ). Then the local ring T0 is generated by t(l) for l prime to q. of T0 . Here the reduced part Tred is Proof Let T be the reduced part Tred 0 0 T0 modulo its nilradical n. We prove the lemma when q = p is a factor of p, since the other case is similar and easier. Since B is a discrete valuation ring, we may assume that B is complete. Since T0 is B-free of ﬁnite rank, nQ = n ⊗Z Q is the nilradical of T0,Q = T0 ⊗Z Q and hence n = nQ ∩T0 , which is a B-free direct summand of T0 . This implies that T is B-free of ﬁnite rank and that n = ∩P P for prime ideals P with B-free quotient T0 /P . Then TQ = T ⊗Z Q ∼ = P k(P ) for the residue ﬁeld k(P ) of P . We have the Galois representation ρP as in Theorem 2.43. We consider ρ= ρP : Gal(F /F ) → GL2 (TQ ) = GL2 (k(P )). P

P

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This representation is unramiﬁed outside pc(ε1 )c(ε2 ) and satisﬁes Tr(ρ(F robl )) = t(l) for l prime to pc(ε1 )c(ε2 ). Let R be the subring of T generated by t(l) for all l prime to pc(ε1 )c(ε2 ). By the Chebotarev density theorem applied to the Galois representation ρ, we have Tr(ρ(σ)) ∈ R for all σ ∈ Gal(F /F ). Since ρP is p-ordinary, we have a character δp : Dp → TQ such that δp ([y, Fp ]) = u(y). Put εp = det(ρ)/δp . For σ ∈ Dp so that δp (σ) ≡ εp (σ) mod mR , the characteristic polynomial Pσ (X) = det(X − ρ(σ)) has coeﬃcients in R. Since Pσ (X) mod mR has two distinct roots modulo mR , by Hensel’s lemma [BCM] III.4.3, Pσ (X) has two distinct roots in R. Since the generators of ρ(DP ) can be chosen among those ρ(σ) with δp (σ) ≡ εp (σ) mod mR , the two characters have values in R× . In particular, u(y) ∈ R ⊂ T, and hence T is generated by t(l) for all primes l outside p. Let R0 be the subring of T0 generated by t(l) for l outside p. Now T0 contains t(p). Since u(p) is the unit root of X 2 − t(p)X + ε+ (p)N (p), the existence of u(p) in R = R0 /n ∩ R0 implies t(p) = ε+ (p)N (p)/u(p) + u(p) ∈ R. Again by Hensel’s lemma applied to R0 /(n ∩ R0 ) = R, we ﬁnd that t(p) ∈ R0 . Thus T0 is generated by t(l) for l outside p. 2 Remark 3.11 It is believed that T0 itself is reduced (so, n = 0) if F = Q, N0 = N/c(ε− ) is square-free and is prime to c(ε− ). This semisimplicity of T0 ⊗Z Q is equivalent to the following conjecture: for any prime factor l of N0 and any algebra homomorphism λ : h(N/l, ε; B) → Qp , X 2 − λ(T (l))X + ε+ (l)N (l) = 0 has two distinct roots if F = Q. This question is studied in [CoE] for elliptic modular forms, and the semisimplicity is proven there for elliptic modular forms of weight 2 (see [CoE] 2.1). Exercise 3.12 Prove that the projector ep of level pN annihilates Iv(pN, ε; B). By Lemma 3.10, T0 can be embedded into Told 0 sending T (l) for l prime to p to T (l) in Told 0 , because T (l) commutes with [ηp ] if l is prime to p. Lemma 3.13 Suppose (sf) and that p N, and deﬁne a linear operator U acting on (f1 , f2 ) ∈ M(N, ε; B)2 by (f1 , f2 ) → (f1 , f2 )U for U := T (p) −1 . Then we have iB ◦ U = U (p) ◦ iB . If further we suppose N (p)ε+ (p ) 0 (dsp ), then we have 2 ∼ Told 0 = T0 [X]/(X − t(p)X + ε+ (p )N (p)),

∼ where t(p) is the image of T (p) in T0 . In particular, we have ep Told = T0 as 0 B-algebras.

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Hecke algebras as Galois deformation rings

Proof Note that ε2 (p ) = 1 by our way of extending ε2 to T (A(∞) ), and hence ε(ηp ) = ε(ηpι ) = 1. Then we have

/ 1 u 0 f |T (p) = [ηp ](f ) + η ι (f ) = [ηp ](f ) + U (p)(f ). 0 1 p u

mod p

On M0 (B) ⊂ M(pN, ε; B), u(p) = t(p) − [ηp ]. Since [ηpι ηp ](f )(x) = f (xp ) = ε+ (p )f (x), we ﬁnd U (p) ◦ [ηp ] = N (p)ε+ (p ) on M(N, ε; B). On M(N, ε; B)|[ηp ], we thus have u(p) = N (p)ε+ (p )[ηp ]−1 and

T (p) −[ηp ] U (p)iB ((f1 , f2 )) = (f1 , [ηp ](f2 )) = iB ((f1 , f2 )U ) 0 N (p)ε+ (p )[ηp ]−1 for fj ∈ M(N, ε; B). If we assume (dsp ), by Lemma 3.10, t(p) ∈ T0 , and u(p) satisﬁes X 2 − t(p)X + ε+ (p )N (p) = 0. Since T (l)[ηp ] = [ηp ]T (l) for all l prime to p, we ﬁnd that Told 0 = T0 [u(p)]. By Exercise 3.12, ep of level pN annihilates Iv(pN, ε; B); thus, the image of ep ◦ iB is isomorphic to ep S0old (B) . By the Jacquet–Langlands–Shimizu correspondence (see Proposition 3.66), S0old (K) is free of rank 1 over Told 0 ⊗B K, which is isomorphic to S0old (K) ∼ = T0 [X]/(X 2 − t(p)X + ε+ (p )N (p)) ⊗B K.

(3.1.8)

From this, we conclude the ﬁrst assertion. Since X 2 − t(p)X + ε+ (p )N (p) ≡ old ∼ 2 X(X − t(p)) mod m0 and t(p) ∈ T× 0 , by Hensel’s lemma, we ﬁnd T0 = T0 and ep gives the idempotent of one of the component. Thus the last assertion follows from this fact. 2 In the above proof of Lemma 3.13, the fact that the extra level p added is a factor of p does not play an essential role, and the result of the lemma is valid for the q-old Hecke algebra for any prime ideal q pN. We have constructed the space looking at the space of p-old forms in the S0old (B) and its Hecke algebra Told 0 space of level pN. There is no particular reason for restricting ourselves to p-old q 2 forms. For any prime ideal q outside

iA : M(N, ε; A) → M(qN, ε; A) N, deﬁne 1 0 by iqA (f1 , f2 ) = f1 +f2 |[ηq ] for ηq = . Then we deﬁne the space S0q-old (A) 0 q by the image of iqA composed with the projection M(qN, ε; A) → S(qN, ε; A). for the Hecke algebra over B of S0q-old (B). Here is the precise Then we write Tq-old 0 statement we can prove by the same argument as in the proof of Lemma 3.13, whose proof is left to the reader: Lemma 3.14 Suppose (sf) and that q pN for a prime ideal q, and deﬁne a 2 linear operator Uq acting on

(f1 , f2 ) ∈ M(N, ε; B) by (f1 , f2 ) → (f1 , f2 )Uq for T (q) −1 Uq := . Then we have iqB ◦ Uq = U (q) ◦ iB . If further we N (q)ε+ (q ) 0

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177

suppose (dsq ), then we have ∼ Tq-old = T0 [X]/(X 2 − t(q)X + ε+ (q )N (q)), 0 where t(q) is the image of T (q) in T0 . Let Mp-non-ord (N, ε; A)2 be the image of M(N, ε; A)2 under 1 − e◦p for e◦p = limn→∞ T (p)n! . Here we write e◦p to emphasize the fact that the level of e◦p is N (not pN). As we have seen in the proof of Lemma 3.9, we have Iv(N, ε; A) = Iv(pN, ε; A) as spaces of function on G(A(∞) ). Lemma 3.15 The space S p-ord (pN, ε; A) is isomorphic to Mp-ord (pN, ε; A) and is A-free of ﬁnite rank. Moreover, if p N, the kernel of the composite i

ep

A M(pN, ε; A) −→ Mp-ord (pN, ε; A) Mp-ord (N, ε; A)2 −→

: f1 → ep (iA (f1 , 0)) contains Iv(N, ε; A)2 and Mp-non-ord (N, ε; A)2 , and ip-ord A induces an A-linear map S p-ord (N, ε; A) into S p-ord (pN, ε; A). Under (sm), the : S p-ord (N, ε; A) → S p-ord (pN, ε; A) is injective with A-free linear map ip-ord A cokernel. Proof

We shall use the equivalence relation “∼” introduced

in the proof of 1 u p u for u ∈ Op ; so, the Lemma 3.2. Note that x η = xp 0 1 0 1 p

p u π2 x = π1 (x). Thus by (3.1.7), Ker(iA ) is stable under right mul0 1

p u . This shows that f |U = N (p)f for the operator U in tiplication by 0 1 Lemma 3.13 if f factors through the projection: Y0 (pN) → Y0 (pN)/∼. Since functions in Iv(pN, ε; A) factor through the projection: Y0 (pN) → Y0 (pN)/∼, ep kills Iv(pN, ε; A) = Iv(N, ε; A) (cf. Exercise 3.12). Thus Mp-ord (pN, ε; A) ∼ = S p-ord (pN, ε; A). 2 Since iA kills the image of Iv(N, ε; A) in Iv(N, ε;

A) under f → ([ηp ](f ), −f ), 0 −1 ep ◦ iA kills Iv(N, ε; A)2 . Since ηp−1 τ (p) = , we have π2 = π1 ◦ [τp ]. Thus 1 0 T (p) ≡ [ηp ] = [τp ] mod mB on functions of Y0 (pN)/∼, and hence any function factoring through Y0 (pN)/∼ is killed by 1 − e◦p because [τp ] is an automorphism of M(pN, ε; A). By the expression of U in Lemma 3.13, U (p) is topologically nilpotent on iA (Mp-non-ord (N, ε; A)2 ). Thus ep ◦ iA kills Mp-non-ord (N, ε; A)2 . Since the projection π1 : Y0 (pN) → Y0 (N) is a surjection, the pullback map f1 → iA (f1 , 0) is injective. Combining this fact with the matrix expression of U , we conclude from Lemmas 3.9 and 3.13 that on the p-ordinary part, under (sm), is injective. By Lemma 3.2, the cokernel is A-free. 2 ip-ord A Let m be the unique maximal ideal of hp-ord (pN, ε; B) whose image in ep Told 0 is a nontrivial maximal ideal. By Lemma 3.13, we have the other maximal ideal

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Hecke algebras as Galois deformation rings

m of the algebra hp-ord (pN, ε; B) whose image in (1 − ep )Told is a nontrivial 0 maximal ideal. Let T = hp-ord (pN, ε; B)m and T = hp-ord (pN, ε; B)m . Let new (K) = S new (K) = Im(iK )⊥ , and for the idempotent 1T of T, we put ST new (K)). Since the image of ep and U (p) in the Hecke algebra of the p-new 1T (S part is a unit (see the argument at the beginning of the proof of the following new (K)) by theorem), T annihilates S new (K). Then we deﬁne Tnew ⊂ EndK (ST the B-subalgebra generated by u(p) and t(l) for all l prime to p. Theorem 3.16 In addition to p N, (dsp ), (sf), (sm) and (sm0), suppose that the localization S0 = T0 (S p-ord (N, ε; B)) at the maximal ideal m0 of T0 is free new (K) is nontrivial. Write u(p) of rank 1 over the localization T0 and that ST old ∼ ∼ for the image of U (p) in ep T0 = T0 , and let π : T → Told = T0 be the 0 projection taking t(q) to t(q) for all primes q outside pNc(ε). Then if U (p) has new (K), the ideal π(AnnT (Ker(π))) ⊂ T0 is generated eigenvalue ζ = ζp on ST by (u(p) − ζ), where AnnR (X) for a ring R with a subset X is the annihilator ideal {x ∈ R|xX = 0}. If further the localization S = T(S p-ord (pN, ε; B)) at the maximal ideal m of T is free of rank 1 over the localization T, we have T/(u(p) − ζ)T ∼ = Tnew . Since the automorphic representation generated by a Hecke eigenform in S new is Steinberg at p, the eigenvalue ζp is a root of unity (see Subsection 2.3.7). Proof We claim that the automorphic representation of G(A(∞) ) genernew ated by any Hecke eigenform f in ST (K) has its local component πp at p isomorphic to the Steinberg representation σ(η| · |−1 p , η) for a character η of Fp× . Indeed, by Proposition 2.44 (3), πp cannot be supercuspidal. Since −1 −1 f ⊗ ε−1 2 (g) = f (g)ε2 (N (g)) is a new vector of V (πp ⊗ ε2 )) ﬁxed by Γ0 (p)p ⊂ new GL2 (Fp ) (because f ∈ ST (K)), the only possibility is that πp is a Steinberg representation. new The eigenvalues of U (p) on the above f ∈ ST (K) are η(p ). Since the central −1 character of σ(η| · |p , η) is given by ε+ , we get η 2 = ε+ |Fp× , and the eigenvalue + of U (p) is given by ± ε+ (p ). Since the eigenvalue of U (p) modulo mT is given new (K) with by u(p) mod mT , one cannot have two distinct forms in ST +diﬀerent U (p)-eigenvalues (because p > 2). Thus we may assume that ζ = ± ε+ (p ). Since i∗B and iB commute with T (l) for all l prime to p, they commute with t(p) ∈ Told which is generated by t(l) for l prime to p. The linear map M = 0 i∗B ◦ iB |M0 (B)2 as in Lemma 3.7 is Told 0 -linear. Let 2 Mold 0 (A) = iA (M0 (A) ), MT (A) = T(M(pN, ε; A)),

MT (A) = T (M(pN, ε; A)) and M0 (A) = MT (A) + MT (A). First we assume that Iv(N, ε; B) ⊗H HM = 0 (here M is the pullback of m in H). Then by Lemma 3.9, M0 (A) ⊂ S(pN, ε; A), and M0 (A) ⊂ S(N, ε; A) and

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iA is injective on M0 (A)2 . We have the following commutative diagram: ∼

M0 (B)2 −−−−→ M0 (B) ∩ S0old (K) iB   u(p)  U ∼

M0 (B)2 −−−−→ M0 (B) ∩ S0old (K) iB

for U as in Lemma 3.13. The surjectivity of the map iB in the above diagram follows from Taylor’s lemma (Lemma 3.2). Let π old : M(pN, ε; K) → Mold (K) (resp. π new : M(pN, ε; K) → S new (K)) be the H(pN, ε; K)-equivariant projection to the old part (resp. the new part). Taking the dual, we have another commutative diagram ∼

π old (M0 (B)) −−−∗−→ S0 (B)2 iB     u(p) U ∼

π old (M0 (B)) −−−∗−→ S0 (B)2 . iB

By Lemma 3.7, we have the following identity of the congruence module C0 between π old (M0 (B)) and π new (M0 (B)) (cf. Exercise 1.54): C0 :=

S0 (B)2 ∼ π old (M0 (B)) ∼ π new (M0 (B)) = = M (S0 (B)2 ) Mnew (B) Mold 0 0 (B)

as h(pN, ε; B)-modules, where Mnew (B) = π new (M0 (K)) ∩ M0 (B) and the 0 ∗ linear map M is given by iB ◦ iB |M0 (B)2 as in Lemma 3.7. Thus the action of h(pN, ε; B) on C0 factors through the old quotient and the new quotient at the same time (see [H88a] Lemma 5.2). Since the action on C0 of h(pN, ε; B) factors through its quotient T ⊕ T , we have C0 = C0 (T) ⊕ C0 (T ) for the isotypic components C0 (T) = TC0 and C0 (T ) = T C0 . Since T annihilates the nearly p-ordinary part and a fortiori annihilates the p-new part, we actually have C0 = C0 (T), and T acts on C0 through Tnew on which u(p) = ζ. Thus u(p)−ζ annihilates C0 and hence (u(p)−ζ)MT (B) ⊂ ep (Mold 0 (B)) for MT (B) = T(M(pN, ε; B)). Since detT0 (M |S0 (B)2 ) in T0 is given by t(p)2 − ε+ (p )(1 + N (p))2 = (t(p) − ζ(1 + N (p)))(t(p) + ζ(1 + N (p))) and by Lemma 3.13, t(p) = u(p) + N (p)u(p) in Told 0 , we ﬁnd that (t(p) + ζ(1 + N (p))) ≡ (u(p) + ζ)(1 + N (p))

mod mT

is a unit in T and that (t(p) − ζ(1 + N (p))) ≡ (u(p) − ζ)(1 + N (p)) mod mT is a unit multiple of u(p) − ζ in T. Hence we conclude π old (MT (B)) π old (M0 (B)) ∼ S0 (B)2 ∼ C0 ∼ . = = = old 2 M (S0 (B) ) M0 (B) (u(p) − ζ)ep (Mold 0 (B))

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In other words, (u(p) − ζ)MT (B) = ep (Mold 0 (B)), because M acts on the old (B)) via the multiplication by detT0 (M |S0 (B)2 ), which is p-ordinary part ep (Mold 0 equal to (u(p) − ζ) times a unit. Indeed, writing u = N (p)

+ 1, by row operation, t(p)ζ 2 u ∗ we get, M |S0 (B)2 ∼ , t(p)ζ 2 u and ζ 2 (t(p) + 0 ζ 2 (t(p) + ζu)(t(p) − ζu) ζu) are in T× , and as we have seen, ζ 2 (t(p) − ζu) is a unit multiple of u(p) − ζ. ∼ Since ep (Mold 0 (B)) = T0 as T0 -modules by our assumption, we ﬁnd that π(AnnT (Ker(π))) ⊂ AnnT0 (C0 ) ⊂ (u(p) − ζ)T0 . Since u(p) acts through the multiplication by ζ on the new part, we have (u(p) − ζ)T0 ⊂ π(AnnT (Ker(π))), which proves the ﬁrst assertion. If further MT (B) ∼ = T, we ﬁnd that T/(u(p) − ζ)T ∼ = π new (M0 (B)) ∼ = Tnew . We now treat the more diﬃcult case where Iv(N, ε; B) ⊗H HM = 0, assuming (sm). By (sm) and Lemma 3.9, Ker(iA ) ∼ = Iv(N, ε; A). By Lemma 3.13, we can let U (p) act on M(N, ε; A)2 by the linear operator U . This action of U (p) deﬁnes a module structure of H(pN, ε; A) on M(N, ε; A)2 , and iA : M(N, ε; A)2 → Note that the idemM(pN, ε; A)2 is a homomorphism of H(pN, ε; A)-modules. potent ep associated to U (p) kills Iv(N, ε; A) and ep M0 (A)2 ) ∼ = S0 (A) by Lemma 3.15. Thus after applying ep , we ﬁnd that ep ◦i∗A ◦iA is just the multiplication by detT0 (M |M0 (B)2 ) projected down to T, which is equal to ζ 2 (t(p) − ζu). Then by the same argument as above, we ﬁnd (u(p) − ζ)MT (B) = ep (Mold (B)), which ﬁnishes the proof. 2 3.1.5 Automorphic forms of higher weight We now brieﬂy recall higher weight automorphic forms on G. We have given already in 2.3.5, a general deﬁnition of such automorphic forms over C. A main point here is to discuss the integral structure of the space of automorphic forms. Since our automorphic variety has dimension 0, the integrality problem is particularly easier to solve for our G (see [PAF] 4.3.2 for integral automorphic forms on indeﬁnite quaternion algebras). Recalling I = Isomﬁeld (F, Q), the weight κ of an automorphic form on G(A(∞) ) is a tuple (κ1 , κ2 ) of formal integer linear combinations κj ∈ Z[I] of embeddings in I. The automorphic forms f satisfying (3.1.3) have weight (0, I) where I = σ∈I σ ∈ Z[I]. We consider here vector valued functions on G(A(∞) ) satisfying an automorphic property (SB) dependent on κ. We hereafter assume n := κ2 − κ1 − I ≥ 0 (this means nσ ≥ 0 for all σ ∈ I, writing n = σ nσ σ). Since k ∈ Z[I] induces a morphism of algebraic groups k : T → Gm/Q given by xk = σ∈I σ(x)kσ , we can identify Z[I] with the is character group X(T ) of T . We ﬁx a maximal torus TG ⊂ G so that TG (Z) × ∼ 2 ∼ the diagonal torus in O = GL2 (O). Then TG = T over Qp , and the character D

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group X(TG ) of TG is isomorphic to X(T )2 = Z[I]2 ; so, weights of automorphic forms on G are weights of the maximal torus TG of G. To introduce the module in which automorphic forms of weight κ have values, recall κ∗ = (κ1 + I, κ2 ). We write T1 = TG ∩ G1 and identify T with T1 so that n = κ2 − κ1 − I ∈ Z[I] is the restriction of κ∗ ∈ X(TG ) to T ⊂ G1 . For a given ring A, we consider the following module L(κ∗ ; A) of the multiplicative semigroup M2 (A). Recall n ≥ 0. We consider polynomials with coeﬃcients in A The collection of (Xσ , Yσ )σ∈I homogeneous of degree nσ for each pair (Xσ , Yσ ). of all such polynomials forms an A-free module L(κ∗ ; A) of rank σ (nσ + 1). Recall the p-adic completion W of the base discrete valuation ring B ⊂ Q with residual characteristic p. Write K for the ﬁeld of fractions of W . We take a character quadruple ε = (ε1 , ε2 , ε+ , ε− ) but we allow the inﬁnite-order Hecke character ε+ : Z(A(∞) )/Z(Q) → A× as central character. We continue to assume that εj (j = 1, 2) is a ﬁnite-order character and ε− = ε−1 2 ε1 . We assume the following compatibility condition: ε+ (z) = ε1 (z)ε2 (z)zp−κ1 −κ2 +I for all z in Z(Z).

(W)

By this fact, κ1 + κ2 = [κ]I for an integer [κ] (see (2.3.10)). We assume that W contains ip (σ(Op )) for all p and all σ ∈ I and hereafter assume that A is a W -algebra. Then ip (σ(δp )) (which we write simply as σ(δp )) for δ ∈ G(A) can be regarded as an element

in M2 (A). We extend ε to ∆0 (N) by ε(δ) = a b − ε2 (N (δ))ε (a) if δc(ε− ) = . We let ∆0 (N) act on L(κ∗ ; A) as follows. c d

Xσ Xσ δ·φ = ε(δ)−1 N (δp )κ1 φ σ((δp )ι ) . (3.1.9) Yσ Yσ 0 (N), and δ ι = N (δ)δ −1 . We write L(κ∗ ε; A) for the Here z ∈ Z(A(∞) ), u ∈ Γ ∗ 0 (N))-action. module L(κ ; A) with this (∆0 (N), Z(A(∞) )Γ ∗ By computation, z ∈ Z(Z) acts on L(κ ε; A) through scalar multiplication by × ⊂ Z(A) acts trivially on L(κ∗ ε; A). If S is ε+ (zpc(ε− ) )−1 ; in particular, ∈ O+ × in G(A(∞) ), a suﬃciently small open compact subgroup so that S ∩ Z(Q) ⊂ O+ central elements in Γx = xSx−1 ∩ G(Q) act trivially on L(κ∗ ε; A). 0 (N), we think of the following For any open compact subgroup S ⊂ Γ ﬁnite set: Y (S) = Y D (S) := G(Q)\G(A(∞) )/Z(A(∞) )S. 0 (N)). We deﬁne Mκ (N, ε; A) to be the space of We write simply Y0 (N) for Y (Γ functions f on G(A(∞) ) with values in L(κ∗ , A) satisfying the following condition. The functions in Mκ (N, ε; A) are called automorphic forms on G(A) of level N with Neben character ε. (SB) We have the following automorphy f (αxuz) = ε+ (z)ε(u(∞) )u−1 p · f (x)

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0 (N). Here f (x) → up · f (x) is for all α ∈ G(Q), z ∈ Z(A(∞) ), and u ∈ Γ the action of up on L(κ∗ ; A). 0 (N)δ Γ 0 (N) for δ ∈ ∆0 (N) into 4 Γ Decomposing Γ h 0 (N)h, we shall deﬁne the 0 (N)δ Γ 0 (N)] (of the double coset Γ 0 (N)δ Γ 0 (N)) action of the Hecke operator [Γ on Mκ (N, ε; A) by

0 (N)δ Γ 0 (N)](g) = f |[Γ hι · f (ghι ). (3.1.10) h

∩ T (A(∞) ), as an operator on Mκ (N, ε; W ), We set for an integral idele y ∈ O

0 (N) 1 0 Γ 0 (N)] if the ideal yO is prime to N, T (y) = yp−κ1 [Γ 0 y (3.1.11)

1 0 −κ1 Γ0 (N)] if y ∈ ON . U (y) = yp [Γ0 (N) 0 y Strictly speaking, if yp is a nonunit, we regard U (y) and T (y) as originally deﬁned on Mκ (N, ε; K) (because of the division by ypκ1 ). Then we show that they are well-deﬁned integral operators sending W -integral automorphic forms into themselves (because of the factor N (δp )κ1 in (3.1.9)). We deﬁne in EndW (Mκ (N, ε; W )) the projectors limn→∞ U (p )n! if p | N, ep = limn→∞ T (p )n! if p N, and put e = p|p ep . Then we deﬁne Sκn.ord (N, ε; W ) = e (Mκ (N, ε; W )). The corresponding nearly ordinary Hecke algebra hn.ord (N, ε; W )) is deﬁned by the κ W -subalgebra of EndW (Sκn.ord (N, ε; W )) generated by T (y) and U (y) for all integral ideles y. If κ2 − κ1 − I ≥ 0, we call the pair (κ, ε) arithmetic. For any integral domain I ﬁnite over W [[TG (Zp )]], if a W -algebra homomorphism (P : I → W ) ∈ Spf(I)(W ) coincides with an arithmetic weight on an open subgroup of TG (Zp ), we call P an arithmetic point. We write Spf arith (I)(W ) for the collection of all arithmetic points of Spf(I) with values in W . Since the requirement (SB) coincides with (3.1.3) if κ = (0, I), we have n.ord S(0,I) (N, ε; W ) = S n.ord (N, ε; W ) and hn.ord (N, ε; W ) = hn.ord (0,I) (N, ε; W ),

and we continue to drop the subscript “(0, I)” when we deal with these spaces of weight (0, I). We write Σp for the set of prime factors of p in F . In the sequel, we often need characters κ ∈X(TG ) and εj with κj factoring through the local norm map Np : T (Zp ) → p|p Z× p given by

Np ((xp )p ) = (Np (xp ))p × p Op

(Np (xp ) = NFp /Qp (xp ))

for (xp )p ∈ = T (Zp ). We call such a pair (κ, ε) locally cyclotomic. If κj and εj factor through the global norm map NF/Q : T (Zp ) → Gm (Zp ) = Z× p , we

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call (κ, ε) cyclotomic. If κ is locally cyclotomic, κσ is independent of σ as long as ip ◦ σ induces the same p-adic place of F . Thus we can identify the submodule of locally cyclotomic characters with Z[Σp ]. Indeed, regarding X lc (T ) in X(T ) Z[Σp ] ⊂ Z[I] via p kp p → σ kσ σ given by kσ = kp if p comes from ip ◦ σ, we have X lc (T ) ∼ = Z[Σp ]. → W × given by Since (κ, ε) induces a character TG (Z)

a 0 1 2

→ ε1 (a)a−κ ε2 (d)d−κ , p p 0 d we have the associated W -algebra homomorphism πk,ε : W [[TG (Zp )]] → W induced by the restriction of this character to TG (Zp ). A W -point P of the formal spectrum Spf(W [[TG (Zp )]]) is called arithmetic if P = Ker(πκ,ε ) with κ2 − κ1 − I ≥ 0. Similarly an arithmetic point P ∈ Spec(W [[TG (Zp )]])(W ) associated with (κ, ε) is called locally cyclotomic (resp. cyclotomic) if (κ, ε) is locally cyclotomic (resp. cyclotomic). Thus locally cyclotomic (resp. cyclotomic) points are arithmetic (but the reverse implication is not always true). Take an integral domain I which is an algebra over W [[TG (Zp )]]. We call a point P ∈ Spec(I)(W ) locally cyclotomic (resp. cyclotomic) if the structure homomorphism ι : W [[TG (Zp )]] → I, ι∗ (P ) ∈ Spf(W [[TG (Zp )]]) is locally cyclotomic (resp. cyclotomic). 3.2 Galois deformation In this section, we identify a local ring of a Hecke algebra with an appropriate Galois deformation ring, following the method of Taylor and Wiles, and Fujiwara. As a by-product we also prove the freeness of the space of automorphic forms over the corresponding Hecke algebra. 3.2.1 Minimal deformation problems Write c(ε) for c(ε1 )c(ε2 ). We now exhibit a suﬃcient condition for the freeness of MT (W ) over T. We use the argument invented by Taylor and Wiles and elaborated on later by Fujiwara [Fu]. We ﬁx a Hecke eigenform f0 ∈ S(N, ε; B), whose Galois representation ρ0 = ρf0 : Gal(Q/Q) → GL2 (W ) is unramiﬁed outside c(ε)p, where as before W is the p-adic completion of B. Fixing ε and moving around N ⊂ c(ε− ), a Hecke p ) is equivalent to f0 if g and f0 have the same eigenvalue eigenform g ∈ S(N , ε; Q for T (l) for almost all primes l of F . Equivalent forms span the same automorphic representation π0 = πf0 (the strong multiplicity-one theorem; see [AAG] 5.14 and 10.10). We choose f0 having the maximal level N among Hecke eigenforms equivalent to f0 . We assume that (sf)

N0 = N/c(ε− ) is square-free and is prime to c(ε− ).

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Under this condition, by Proposition 2.44 (3), the automorphic representation π0 of G(A(∞) ) spanned by the translations of f0 is everywhere nonsupercuspidal. Recall the following fact which is a part of Proposition 2.44: Proposition 3.17 Suppose (sf) for the Hecke eigenform f0 ∈ S(N, ε; W ). Then the Galois representation ρ0 (of f0 ) restricted to the decomposition group Dq at a prime ideal q p of F is reducible. Indeed, writing π0,q ∼ = π(η1,q , η2,q ) if −1 − q N/c(ε− ) and π0,q ∼ , η ) with η = η | · | if q|N/c(ε ), we have σ(η = 1,q 2,q 2,q 1,q q   η 0  2,q ∼ π(η1,q , η2,q ) ⇔ q N/c(ε− ),  if π0,q =   0 η1,q ρ0 |Dq ∼ (3.2.1)  η1,q Nq ∗  −1 − ∼  if π0,q = σ(1, | · |q ) ⊗ η1,q ⇔ q|N/c(ε )   0 η 1,q

for the p-adic cyclotomic character Nq : Dq× → B × (⊂ W × ) (note q p). Here we have identiﬁed the local characters of Fq× with corresponding local Galois characters via the local Artin symbol. Though more general cases are treated in [Fu], in this book, at p, we assume (Oﬂ) f0 is nearly p-ordinary at all p-adic places p satisfying one of the following two conditions: (a) p is ramiﬁed over Qp , (b) p is a factor of Nc(ε). In other words, ρ0 could be ﬂat nonordinary or ﬂat nearly p-ordinary at p|p which is not listed in (Oﬂ). Often, we actually assume a slightly stronger condition eliminating the nonordinary ﬂat case: (Ord) f0 is nearly p-ordinary at all prime ideals p|p. We basically assume (Ord) in this book, though we state the theorem under weaker assumption (Oﬂ) and describe how to include the nonordinary ﬂat case at p|p (when Fp /Qp is unramiﬁed) in 3.2.5 after ﬁnishing the proof of the main theorem (Theorem 3.28) under (Ord), for the sake of completeness. By Lemma 2.40, the near p-ordinarity of f0 means that π0,p ∼ = π(η1,p , η2,p ) or π0,p ∼ = σ(η1,p , η2,p ) with η2,p = η1,p | · |−1 p , and in the two cases, we may assume η1,p (pp ) ∈ B × (⊂ W × ). Under this assumption, we have the corresponding Galois character η1,p : Dp → B × (⊂ C× ) (for j = 1) by local class ﬁeld theory (see Theorem 2.43(3), noting that κ1 = 0). Here the character ηj,p restricted to Op× is of ﬁnite-order (and hence they are continuous under the p-adic topology on W and the archimedean topology on C× ). We relate ηj and εj by imposing (nbn) ηj,l |Ol = εj,l (as characters of Fl× → B × ) for all prime ideal l of F .

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Thus the local representation type (of f0 ) determines the pair (ε1 , ε2 ) up to permutations of the two. Recall the set Σp of all prime factors of p in O. We split Σp = Σst Σn.ord Σf l Σ0 as follows: 1. p ∈ Σ0 if and only if f0 is nearly p-ordinary and ρp = ρ|Dp is associated to a character twist of the ﬁnite ﬂat group scheme (Z/pZ ⊕ µp ) ⊗Fp F over Op by one of the square roots of ε+ regarded as a Galois character (by class ﬁeld theory); 2. p ∈ Σf l if and only if p Nc(ε) and ρ|Dp is associated to a ﬁnite ﬂat group scheme and the semisimpliﬁcation (ρ|Dp )ss is absolutely irreducible; 3. p ∈ Σst if and only if ρp = ρ|Dp is not associated to a ﬁnite ﬂat group × scheme over Op and η2,p = η1,p | · |−1 p on Fp . Remark 3.18 Suppose that ρss is ﬂat at p and that p is unramiﬁed over Q. If ρss |Dp is reducible, it must be associated to (Z/pZ ⊕ µp ) ⊗Fp F over the maximal unramiﬁed extension of Op by Raynaud’s classiﬁcation of ﬁnite ﬂat group schemes (see Proposition 3.48). Thus under (Oﬂ), Im(ρss |Dp ) can never be contained in the center of GL2 (F). We put ρ := (ρ0 mod mB ); so, ρ : Gal(Q/F ) → GL2 (F). We assume the following minimality conditions on f0 : (h1) ε has order prime to p. (h2) (a) If p ∈ Σ0 , π0,p = π(η1,p , η2,p ) and p ⊃ N/c(ε− ), (b) if p ∈ Σf l , π0,p = π(η1,p , η2,p ) and p ⊃ Nc(ε), (c) if p ∈ Σst , π0,p = σ(η1,p , η2,p ) and p ⊃ N/c(ε− ), (d) if p ∈ Σn.ord , we have p ⊃ c(ε− ). (h3) For all prime ideals p ∈ Σp − Σf l in F , regarding η1,p as a Galois character (as described just below (Ord)), we have an exact sequence of Dp -modules −1

−1 ε+,p Np → ρ0 → η1,p → 0 with δ p det(ρ) = δ p , 0 → η1,p

where we put δ p = (η1,p mod mB ), the ﬁnite-order Hecke character ε+ is regarded as a global Galois character by class ﬁeld theory, and Np is the p-adic cyclotomic character restricted to Dp . The character δ p (resp. η1,p ) is called the nearly p-ordinary character of ρ (resp. ρ0 ). (h4) If a prime ideal q outside p 1 divisible by p; so, ρ|Iq ∼ 0 Iq is the inertia subgroup of

divides N/c(ε− ), ρ|Dq has ramiﬁcation index

∗ is nonsemisimple with nontrivial ∗, where 1 Dq .

We suppose that B is a discrete valuation ring with suﬃciently large ﬁnite residue ﬁeld F = Fpr so that all Hecke eigenvalues of f0 for T (l) (for primes l of O) are contained in B (and hence in its completion W ). Let CN LW be the

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category of complete noetherian local W -algebras A sharing the same residue ﬁeld with W (that is, A/mA = F for the maximal ideal mA of A). Let Q be a ﬁnite set of primes of F outside pc(ε). We consider the deformation functor ΦQ : CN LW → SET S, where ΦQ (A) is the set of (isomorphism classes of) representations ρ : Gal(Q/F ) → GL2 (A) satisfying the following conditions: (Q1) ρ mod mA ∼ = ρ. (Q2) ρ is unramiﬁed outside the prime factors of pNc(ε) and prime ideals in Q. (Q3) det(ρ) = ε+ N for the global p-adic cyclotomic character N .

p ∗ for characters p , δp : Dp → A× such that δp mod mA = (Q4) ρ|Dp ∼ 0 δp δ p and δp |Ip = ε1,p for all prime ideals p ∈ Σ0 Σst Σn.ord . (Q5) For p ∈ Σ0 , noting that δp−1 p = N on the inertia group Ip , we require that the extension class p → ρp δp gives a ﬂat cohomology class in Hf1l (Fpunr , A(1)) for the maximal unramiﬁed extension Fpunr /Fp . The ﬂat cohomology group Hf1l (Fpunr , A(1)) is the image, by Kummer’s theory, of pn ⊗Zp A in H 1 (Fpunr , A(1)). limn Opunr,× / Opunr,× ←−

∗ (Q6) ρ|Dq ∼ q for characters q , δq : Dq → A× such that δq mod mA = 0 δq δ q and δq |Iq = ε1,q for all prime ideal q|c(ε)N outside p, and ρ|Iq ⊗ ε−1 1 is unramiﬁed if q|c(ε) is outside pN. (Q7) ρ|Dp is ﬂat for all p ∈ Σf l . An A-module scheme G/Op , up to character twists “⊗χ”, ﬁts into an extension (µpr ⊗Z A) → (G ⊗ χ) (A/pr A) of ﬁnite ﬂat group schemes ´etale locally if and only if G ⊗ χ is a ﬁnite ﬂat extension of (A/pr A))by (µpr ⊗Z A) over Opunr , because Opunr is the maximal complete integral domain ´etale over Op . Indeed, if a Galois module of G ⊗ χ descends to Fp , by ´etale descent, the ﬁnite ﬂat group scheme (G ⊗χ)/Opunr descends to a ﬁnite ﬂat group scheme deﬁned over Op . Thus the isomorphism class of such a G over Opunr is determined by its ﬂat extension class in Ext1f l (A/pr A, µpr ⊗Z A) ∼ = Hf1l (Fpunr , A(1)) as described in (Q5). Thus, up to character twists, (Q5) is a ﬂatness condition for p-ordinary deformations. Let M/F be a ﬁnite extension inside Q. Recall the following conditions: (aiM ) ρ restricted to Gal(Q/M

) is absolutely irreducible; ∗ q (dsQ ) writing ρ|Dq ∼ , we have q = δ q for q ∈ Q. = 0 δq Under (aiF ), (dsQ ), and (h2–3), the deformation functor ΦQ is representable by an element RQ of CN LW (a theorem of B. Mazur; the proof is the same as the one given for Theorem 1.46; see [MFG] Theorem 2.26).

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3.2.2 Tangent spaces of local deformation functors Let F[] = F[X]/(X 2 ) with denoting the class of the indeterminate X. Since ΦQ (F[]) is the tangent space of the universal ring representing a deformation functor ΦQ (cf. Lemma 1.59), to bound the size of the deformation space, it is crucial to study the dimension of ΦQ (F[]), which we study here carefully following [Fu]. Let Gl := Gal(F l /Fl ). As before, p is a prime factor in F of p which is the characteristic of F. Lemma 3.19 we have

Let ψ : Gp → F[]× be a character. Put ψ = (ψ mod ()). Then mod ()

dimF (Ker(H 1 (ψ) −−−−−−→ H 1 (ψ))) ≤ dimF H 1 (ψ), where H q (?) = H q (Gp , ?). mod ()

Proof By the long exact sequence attached to 0 → ψ → ψ −−−−−−→ ψ → 0, we have an exact sequence: α

H 1 (ψ) → H 1 (ψ) → H 1 (ψ) → H 2 (ψ) → H 2 (ψ) − → H 2 (ψ) → 0,

(3.2.2)

because the cohomological dimension of local Galois cohomology is equal to 2 (e.g., [MFG] Theorem 4.43). From this, the estimate of the dimension follows. 2 We write Φl for the local version of the minimal deformation functor Φ∅ classifying local deformations ρl : Gl → GL2 (A) satisfying the local version of the conditions (Q1–5) for Q = ∅ depending on l. We quote the following result from [Fu] which is a generalization of [Wi2] Proposition 1.9: Proposition 3.20

Let p ∈ Σp − Σf l and αp = det(ρ)η −2 1,p on Dp . Then

1. dimF Φp (F[]) ≤ 1 + dimF H 1 (Fp , αp ) if p|p. 2. Further dimF Φp (F[]) ≤ dimF H 1 (Fp , αp ) if p ∈ Σst Σ0 . In this case, we have αp = N p := (Np mod mW ); so, H 1 (Fp , F(1)) = H 1 (Fp , αp ). We shall repeat the proof of Fujiwara in [Fu]. Proof The tangent space Φp (F[]) of Φ classiﬁes extensions p → ρ δp with coeﬃcients in F[] such that −1 • δp η1,p ≡ 1 mod (),

−1 • δp η1,p is unramiﬁed, • p δp = det(ρ0 ).

∼ F given by We consider the map πunr : Φ(F[]) → Hom(Gal(Fpunr /Fp ), F) = −1 πunr (p → ρ δp ) = φ for φ with δp η1 (σ) = 1 + φ(σ) ∈ F[]× . We need to determine the ﬁber of πunr . The ﬁber at φ is determined by ﬁxing δp to

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−1 (φ) ⊂ be δφ := (1 + φ)η1 ; so, p has to be equal to det(ρ0 )δφ−1 . Thus πunr 1 −2 −1 ∼ 1 Ext (δφ , det(ρ0 )δφ ) = H (Fp , det(ρ0 )δφ ). Since the reduction mod () of each −1 (φ) is a single element ρ, we need to compute the dimension of element of πunr the ﬁber of H 1 (det(ρ0 )δφ−2 ) over H 1 (Fp , αp ). The desired estimate follows from Lemma 3.19 applied to ψ = det(ρ0 )δφ−2 . By the local Tate duality, we have H 2 (ψ) ∼ = H 0 (ψN −1 ). If ψ = N but ψ = N , 2 2 we ﬁnd that H (ψ) ∼ = H (ψ) and by counting dimension, α in (3.2.2) has to be an isomorphism. Thus the exact sequence (3.2.2) becomes ∂

→ H 2 (ψ) → 0. H 1 (ψ) → H 1 (ψ) → H 1 (ψ) − Now suppose p ∈ Σst . Then ψ = N . Taking nontrivial φ, we ﬁnd ψ = det(ρ0 )δφ−2 = N . Thus the above sequence applies to this case ∂(c) for c ∈ H 1 (ψ) is given ﬁrst by taking a 1-chain c : Gp → ψ whose reduction mod () gives a cocycle c and then by taking its coboundary ∂c regarded as having values in ψ ⊂ ψ. If φ = 0, we thus ﬁnd that the image under ∂ of the class of ρ ∈ H 1 (ψ) in H 2 (ψ) is nontrivial. Thus only the choice φ = 0 counts, and we get the desired estimate. Now suppose p ∈ Σ0 . We write the class of ρ in H 1 (Fp , F[](1)) as cρ . We write q Hunr (?) for cohomology groups of Gal(Fpunr /Fp ). We consider the inﬂation and restriction sequence: 1 0 → Hunr (H 0 (Ip , F[](1))) → H 1 (F[](1)) Res

2 −−→ H 0 (Gal(Fpunr /Fp ), H 1 (Fpunr , F[](1))) → Hunr (H 0 (Ip , F[](1))).

By the p-distinguishedness (h3), we have µp ⊂ Fp ; so, Fp (µp )/Fp tamely ramiﬁes. Therefore H 0 (Ip , F[](1)) = 0 and the restriction map Res is an isomorphism. By the up-to-twists ﬂatness assumption (Q5), Res(cρ ) has to land in the ﬂat cohomology H 0 (Gal(Fpunr /Fp ), Hf1l (Fpunr , F[](1))), which is Opunr,× Opunr,× H 0 (Gal(Fpunr /Fp ), F[] ⊗F lim unr,× pn ) = F[] ⊗F lim unr,× pn . ←− O ←− O n

n

p

p

On the other hand, by Kummer’s theory, we have pn H 1 (F[](1)) = F[] ⊗F lim Fp× / Fp× . ←− n

Thus we lose one dimension for each choice of φ, and the desired assertion follows. 2 As for a prime ideal q outside p, we show that once ρq = ρ|Dq is reducible, any deformation ρ ∈ Φq (A) on Gq of ρq is isomorphic to an upper triangular representation. Indeed, over the splitting ﬁeld K(ρq ) of ρq , the splitting ﬁeld

Galois deformation

189

K(ρ) of ρ is tamely ramiﬁed; so, the K(ρ)/K(ρq ) is an abelian extension of degree prime to p. Since K(ρ)/K(ρq ) is a p-extension, K(ρ)/K(ρq ) is unramiﬁed. By (h1) and (h4), we have σ ∈ Iq such that either (i) ρ(σ) is nontrivial unipotent or (ii) ρ(σ) has two distinct eigenvalues. In Case (i), we have H 0 (Iq , V (ρ)) is free of rank 1 over A. In Case (ii), we have an eigenspace of ρ(σ) in V (ρ) stable under ρ which is free of rank 1 over A. This shows that ρ is isomorphic to an upper triangular representation. Let ad(ρ) be the Galois module M2 (F) on which the Galois group acts by x → ρ(σ)xρ(σ)−1 for σ ∈ Gal(Q/F ). Proposition 3.21

Let q be a prime outside p. Then we have Φq (F[]) ∼ = Hf1 (Fq , ad(ρ)), Res

where Hf1 (Fq , ?) = Ker(H 1 (Fq , ?) −−→ H 1 (Fqunr , ?)) for the maximal unramiﬁed extension Fqunr /Fq . × Proof Since ad(ρ) ∼ = ad(ρ ⊗ φ) for any Galois character φ with values in Fp ⊃ × F , if q is outside N, ad(ρ) is unramiﬁed. Writing ρ(σ) = ρ(σ) + cρ (σ)ρ(σ) for ρ ∈ Φq (F[]), we ﬁnd that cρ is a 1-cocycle with values in ad(ρ) and that ρ → cρ ∈ H 1 (Fq , ad(ρ)) gives an embedding Φq (F[]) → H 1 (Fq , ad(ρ)) (see −1 Theorem 1.57 and (1.3.6)). Since ρ ⊗ η1,q is unramiﬁed if and only if ρ ∈ Φq (F[]) 1 for q outside N, we ﬁnd Hf (Fq , ad(ρ)) ∼ = Φq (F[]) in this case. Assume that q|c(ε− ). We put η− = η2 /η1 and η − = (η− mod mW ). Note −1 −1 − that η−,q = ε− q on Iq . Since q|c(ε ), we have η −,q = N q by (h1) for the p-adic cyclotomic character Nq of Gq . Then ExtGq (η 1,q , η 2,q ) = H 1 (Fq , η −,q ) = 0 by Kummer’s theory. Thus ρ splits on the inertia group and is constant on the inertia group; so, the corresponding cocycle is trivial over the inertia group. We assume that q|N/c(ε− ). By the assumption (h4), we may assume that the restriction of ρ to the inertia group has values in the upper unipotent subgroup; (q) is of (q) (1). The p-factor of Z so, it factors through the tame inertia group ∼ =Z rank 1 isomorphic to Zp (1). Pick ρ ∈ Φq (F[]). Then ρ(Iq ) is cyclic, and therefore dimF ρ(Iq ) = 1 = dimF ρ(Iq ). Thus the deformation ρ is constant over the inertia subgroup, and hence cρ restricted to Iq is trivial. 2

3.2.3 Taylor–Wiles systems We now describe in an axiomatic way (invented by Fujiwara) the limiting process of Taylor–Wiles we have already discussed in many places. Let ΣF be the set of all prime ideals of O. For each l ∈ ΣF , we write ∆l for l the p-Sylow subgroup of (O/l)× and split (O/l)× = ∆l × ∆l . Thus ∆ is the × prime-to–p-part of (O/l) . For a ﬁnite subset Q ⊂ ΣF , we put ∆Q = q∈Q ∆q and ∆Q = q∈Q ∆q . Let W [∆Q ] be the group algebra of ∆Q with augmentation ideal aQ generated by δ − 1 for all δ ∈ ∆Q . We write ∆∅ for the trivial group with one element. Exercise 3.22 Prove that W [∆Q ] is a local ring with maximal ideal aQ + mW .

190

Hecke algebras as Galois deformation rings

Let Q be a set of ﬁnite subsets of ΣF containing an empty set ∅ as a member. A Taylor–Wiles system {RQ , MQ }Q∈Q is made up of the following data: (tw1) We have N (q) ≡ 1 mod p for all q ∈ Q, and RQ is a complete local W [∆Q ]-algebra. (tw2) We have a surjective W -algebra homomorphism RQ /aQ RQ R∅ for each Q ∈ Q. (tw3) MQ is an RQ -module for each Q ∈ Q such that (a) the RQ -action on MQ /aQ MQ factors through R∅ ; (b) MQ is free of a ﬁxed rank d over W [∆Q ]. As the notation suggests, in our application, RQ is given by the universal deformation ring for the functor ΦQ , assuming (dsQ ) for each member of Q ∈ Q. This deﬁnition of the Taylor–Wiles system is due to Fujiwara given in [Fu] 1.1.1, although a version of the system was invented earlier by A. Wiles and R. Taylor (cf. [TaW]). Here is Fujiwara’s isomorphism and freeness criterion (limited to our scope): Theorem 3.23 Let {RQ , MQ }Q∈Q be a Taylor–Wiles system. Suppose the following four conditions: 1. for any given positive integer m, there exists inﬁnitely many Q ∈ Q such that N (q) ≡ 1 mod pm for all q ∈ Q (growth control); 2. the number of elements q of Q ∈ Q is independent of Q (relation control); 3. RQ is generated by at most r elements as a complete local W -algebra for all Q ∈ Q (generator control); 4. the annihilator A of MQ /aQ MQ in R∅ is independent of Q ∈ Q (image compatibility). Then R∅ is W -free of ﬁnite rank, and we have A = 0. If we further assume 5. MQ /aQ MQ is isomorphic to a unique R∅ -module M independent of Q (module compatibility), then M is an R∅ -free module of ﬁnite rank. The proof of the criterion has been given in [Fu] 1.2. For the reader’s convenience, we repeat the proof in [MFG] Theorem 3.35 (which is based on the argument in [Fu]) adjusted to our setting. Proof We prove the theorem following Fujiwara [Fu]. Let δq be a generator of n the cyclic subgroup ∆q . If N (q) ≡ 1 mod pm , for n ≤ m, ∆q /δqp is cyclic of order pn . Thus n n n W [∆q ]/(δqp − 1) ∼ = W [∆q /δqp ] ∼ = W [[S]]/(1 + S)p − 1)

Galois deformation

191

by δq → 1+S. Pick Qm = {q1 , . . . , qr } in Q satisfying the four conditions (1)–(4) for a given positive integer m. Write simply ∆m for ∆Qm . Let In be the ideal of n n W [∆m ] generated by {pn , δqp1 − 1, . . . , δqpr − 1}. Then we have an isomorphism n n W [∆m ]/In ∼ = W [[S1 , . . . , Sr ]]/(pn , (1 + S1 )p − 1, . . . , (1 + Sr )p − 1) =: An nr

via δqj → 1 + Sj . Note that |An | = pntp for t = rankZp W . n,m Write Rn,m for the image of Rm /In Rm in EndW [∆m ]/In (Mm /In Mm ) and R for Rn,m /(δq − 1)q∈Qm . The idea is to ﬁnd an increasing sequence {m(n)|n = 1, 2, . . . } such that (Rn,m(n) , πn,n ) with appropriate surjections πn,n : Rn,m(n) Rn ,m(n ) is a projective system of W [[T1 , . . . , Tr ]]-algebras such that R∞ = limn Rn,m(n) ∼ = ←− W [[T1 , . . . , Tr ]] and the kernel of the natural projection R∞ → T is again generated by r elements, proving the complete-intersection property. Before starting making such a projective system, we remark that we have the following two algebra homomorphisms α and β:

Mm β α ∼ An −→ Rn,m −→ EndAn = Md (An ), In Mm where Md (An ) is the algebra of d × d matrices with entries in An . Since Mm Mm is W [∆m ]-free of rank d, is An -free of rank d, and hence the composite In Mm β ◦ α is injective, proving the injectivity of α (β is injective by deﬁnition of 2 Rn,m ). Thus the cardinality of Rn,m is bounded by Nn = |An |d , which is a ﬁnite number. Since Rm is generated by r elements over W and Rn,m is covered surjectively by Rm , Rn,m is generated by r elements. Pick r generators f1 , . . . , fr in the maximal n,m , (f1 , . . . , fr )). We say ideal of Rn,m . Consider triples made of ((Rn,m , α, β), R two triples n,m , (f1 , . . . , fr )) ((Rn,m , α, β), R

and

n,m , (f , . . . , f )) ((Rn,m , α , β ), R 1 r

are isomorphic if there is an isomorphism of An -algebras: ι : Rn,m → Rn,m n,m such that ι(fj ) = f for all j and the n,m ∼ inducing an isomorphism R =R j following two diagrams are commutative: α

β

α

β

An −−−−→ Rn,m −−−−→ Md (An )       ι An −−−−→ Rn,m −−−−→ Md (An )

192

Hecke algebras as Galois deformation rings

and n,m Rn,m −−−−→ R     ι n,m . Rn,m −−−−→ R Since |Rn,m | ≤ Nn (which is independent of m) and we have inﬁnitely many choices of m; even if we move around m, there are ﬁnitely many isomorphism classes of such triples. Thus starting from n = 1, we will have an inﬁnite sequence N1 ⊂ N such that 1,m , (f1 , . . . , fr )) ((R1,m , α, β), R for all m ∈ N1 are all isomorphic to each other for some choice of fj . Suppose we have constructed a sequence Nn ⊂ Nn−1 ⊂ · · · ⊂ N1 ⊂ N made of inﬁnite sets n,m , (f1 , . . . , fr )) for all m ∈ Nn are isomorphic to Nj such that ((Rn,m , α, β), R each other. When we move around n+1,m , (f1 , . . . , fr )) ((Rn+1,m , α, β), R for all m ≥ n + 1 in Nn , there are only ﬁnitely many isomorphism classes; so, we can choose an inﬁnite sequence Nn+1 ⊂ Nn such that n+1,m , (f1 , . . . , fr )) ((Rn+1,m , α, β), R for m ∈ Nn+1 are all isomorphic to each other. Let m(n) be the minimal element in Nn , and write the triple for this choice as n,m(n) , (f (n) , . . . , fr(n) )). By deﬁnition, we have a surjection ((Rn,m(n) , αn , βn ), R 1

n+1,m(n+1) Rn+1,m(n+1) R (n+1) (n+1) , , (f , . . . , f1 )mod In n+1,m(n+1) 1 In Rn+1,m(n+1) In R n,m(n+1) , (f 1 (Rn,m(n+1) , R

(n+1)

(n+1)

, . . . , f1

))

n,m(n+1) , (f (n+1) , . . . , f (n+1) )) isomorphic to for (Rn,m(n+1) , R 1 1 n,m(n) , (f , . . . , f (n) )). ((Rn,m(n) , αn , βn ), R r 1 (n)

Thus we have a projective system of triples: n,m(n) , (f (n) , . . . , f (n) ))n ∈ N . ((Rn,m(n) , αn , βn ), R r 1 Take the projective limit: ∞ , (f (∞) , . . . , fr(∞) )) ((R∞ , α∞ , β∞ ), R 1 n,m(n) , (f (n) , . . . , fr(n) )). = lim((Rn,m(n) , α, β), R 1 ←− n

Galois deformation

(∞)

Since (f1

(∞)

, . . . , fr

193

) generates R∞ , we have a surjection: W [[T1 , . . . , Tr ]] R∞

(∞)

taking Tj to fj for j = 1, 2, . . . , r. Since αn brings An injectively into Rn,m(n) , α∞ : W [[S1 , . . . , Sr ]] = limn An → R∞ is injective. Similarly by β∞ , we have ←− R∞ ⊂ Md (W [[S1 , . . . , Sr ]]). Thus R∞ is a torsion-free W [[S1 , . . . , Sr ]]-module of ﬁnite type. Thus by the validity of the going-up and down theorem (e.g., [CRT] Theorem 9.4), the Krull dimension of R∞ is r+1. Recall that the Krull dimension s of a ring A is the length of a maximal chain of prime ideals P0 ⊂ P1 ⊂ · · · ⊂ Ps in A (see [CRT] Section 5). For each prime ideal P ⊂ A, the height of P is the maximal length of sequences of prime ideals inside P . If the surjection π : W [[T1 , . . . , Tr ]] R∞ has a nontrivial kernel, then Ker(π) contains an element f = 0. Thus a minimal prime ideal P containing f is of height 1, and hence W [[T1 , . . . , Tr ]] ≤ dim W [[T1 , . . . , Tr ]]/(f ) = dim W [[T1 , . . . , Tr ]]/P = r, Ker(π) which is a contradiction. Thus Ker(π) = 0, and hence R∞ ∼ = W [[T1 , . . . , Tr ]]. By deﬁnition, we have the following exact sequence: dim

Qm n,m −→ 0, Rn,m −→ ϕRn,m −→ R where ϕ sends (aq )q∈Qm to q∈Qm aq (δq −1). Note that |Qm | = r is independent of m. Taking the projective limit of this sequence, we get another exact sequence: r ∞ = lim R R∞ −→ R∞ −→ R −→ 0. ←− n,m(n) n

∞ ∼ Thus R = R∞ /a for an ideal a generated by the r elements S1 , . . . , Sr . Since R∞ is regular, it has to be free of ﬁnite rank over W [[S1 , . . . , Sr ]] (e.g., [CRT] ∞ is free of ﬁnite rank over W and is a complete Theorem 23.1). This shows that R intersection. By the compatibility of the projective system with maps β : Rn,m → Md (An ) = EndAn (Mm /In Mm ), we have a projective system of Rn,m(n) -modules: {Mm(n) /In Mm(n) ∼ = Adn |n ∈ N}. Mm(n) Thus L = limn . Since Mm(n) /In Mm(n) is W [∆m ]/In -free of rank d, L ←− In Mm(n) is W [[S1 , . . . , Sr ]]-free of rank d. In this situation, it is known that L is free of ﬁnite rank over R∞ = W [[T1 , . . . , Tr ]] by abstract ring theory (see Remark 3.36 in [MFG]). Under (5), we have M∼ M/pn M ∼ M /(I + a)Mm(n) ∼ = lim = L/aL. = lim ←− ← − m(n) n n n

194

Hecke algebras as Galois deformation rings

∞ -module of ﬁnite rank, Since L is a free R∞ -module, M ∼ = L/aL is a free R because R∞ = R∞ /aR∞ . n n Let Jn = (pn , (1 + S1 )p − 1, . . . , (1 + Sr )p − 1) inside W [[S1 , . . . , Sr ]]. Then Mm(n) /In Mm(n) is isomorphic to L/Jn L and hence is free of ﬁnite rank over R∞ /Jn R∞ = Rn,m(n) , using freeness of L over W [[S1 , . . . , Sr ]]. From this, we also have Rn,m(n) = Rm(n) /In . ∞ . We claim that the Let T = R∅ /A. Now we want to show that T ∼ = R ∞ factors through T . By deﬁnition, T is the image of R∞ in projection R∞ R EndW (Mm(n) /(S1 , . . . , Sr )Mm(n) ). n,m(n) is the Since Mm(n) /In Mm(n) is Rn,m(n) -free as remarked above, R subalgebra of EndW (Mm(n) /(In + a)Mm(n) ) and therefore, it is a T -algebra. After taking limit with respect to n, the algebra ∞ factors through T . Since naturally R ∞ = R∞ /a homomorphism R∞ → R ∼ covers T (because Sj = 0 in T ), we have T = R∞ . From the same argument, we also see that Rn,m(n) /aRn,m(n) ∼ = T /pn T. We now prove that R∅ ∼ limn Rn,m(n) ∼ = T . Since R∞ = ← = W [[T1 , . . . , Tr ]], − we have a surjective W [[S1 , . . . , Sr ]]-algebra homomorphism W [[T1 , . . . , Tr ]] (n) Rm(n) taking Ti to the generator fi . Thus, writing m∞ for mR∞ , for each given N > 0, we ﬁnd n(N ) > 0 such that Ker(R∞ → Rn(N ),m(n(N )) ) ⊂ mN ∞ . Then to the following commutative diagram with exact rows: tensoring R∞ /mN ∞ r Rm(n(N ))  

a

−−−−→

Rm(n(N ))  

−−−−→

R∅  

−−−−→ 0

b

r n(N ) Rn(N T −−−−→ 0, ),m(n(N )) −−−−→ Rn(N ),m(n(N )) −−−−→ T /p

we get another commutative diagram with exact rows: r (Rm(n(N )) /mN ∞)   c

a

−−−−→

b

Rm(n(N )) /mN ∞   d

−−−−→ R∅ /mN R −−−−→ 0  ∅ 

r N N (Rn(N ),m(n(N )) /mN −−−−→ 0. ∞ ) −−−−→ Rn(N ),m(n(N )) /m∞ −−−−→ T /mT Here a and b take tuples (a1 , . . . , ar ) to i ai Si . Since by our choice, c and d are N ∼ surjective isomorphisms, then we have R∅ /mN R∅ = T /mT for all N , which shows ∼ that R∅ = T . This ﬁnishes the proof. 2

Galois deformation

195

Recall Σ0 ⊂ Σp , which is the set of prime factors p of p satisfying the following three conditions: 1. p is prime to N, 2. p is outside Σf l , ×

3. ρ|Dp ⊗ ψ for a Galois character ψ : Dp → Fp is associated to a ﬁnite ﬂat group scheme (Z/pZ ⊕ µp ) ⊗Z F (here ψ could be ramiﬁed). This Σ0 is the set of primes p with a bigger local ring T of level pN as in Theorem 3.16 than the Hecke algebra T∅ = T0 if we increase the level from N to pN. We pick a Hecke eigenform f0 ∈ S(N, ε; W ) satisfying (h1–4), (nbn), and (Oﬂ). For each ﬁnite set of primes Q ⊂ ΣF outside pc(ε), we recall the deformation functor ΦQ : CN LW → SET S with ΦQ (A) made of deformations ρ : Gal(Q/F ) → GL2 (A) satisfying the conditions (Q1–6). Write RQ for the universal deformation rings RQ representing ΦQ . For primes q outside pc(ε), we consider the following condition (reg) ρ(F robq ) has two distinct eigenvalues. Of course, (reg) implies (dsq ). We insert here Lemma 3.24 Let M be a number ﬁeld of ﬁnite degree and k be an algebraically × closed ﬁeld. If a continuous idele class character ψ : MA× /M × → k of ﬁnite × 2 order is trivial on M∞ , there exists an idele class character η with η = ψ. × is 2-torsion-free. If ξ 2 ∈ M × for ξ ∈ MA× , Proof We need + to prove MA× /M × M∞ 2 the extension M [ ξ ] has to split at every ﬁnite place; so, ξ ∈ M . This ﬁnishes the proof. 2

For a ﬁnite set of primes Q, we deﬁne % 11 (Q) = x ∈ GL2 (O) xq ≡ 1 Γ 0

∗ mod qOq 1

& for all q ∈ Q .

Here is another theorem of Fujiwara, though our formulation is slightly diﬀerent from [Fu] Theorem 5.1.1: Theorem 3.25 Let p be an odd prime, and suppose (h1–4) and (aiF ). If the degree [F (µp ) : F ] = 4d for an odd integer d , we assume for the unique quadratic extension M/F in F (µp ) ×

(ni) ρ is not isomorphic to IndF M δ for any character δ : Gal(Q/M ) → Fp such that Gal(F (µp )/M ) τ → δ(τ )−1 δ(στ σ −1 ) for the generator σ ∈ Gal(M/F ) is a nontrivial quadratic character.

196

Hecke algebras as Galois deformation rings

Then a Taylor–Wiles system {RQ , MQ }Q∈Q for the universal deformation ring RQ of ΦQ exists for an inﬁnite set Q of ﬁnite subsets Q ⊂ ΣF such that 1. Q is made up of ﬁnite subsets Q of primes q outside pc(ε) satisfying (tw1) and (reg); 2. MQ is the direct factor of the W -dual space HomW (S Σ0 -ord (S(Q), ε; W ), W ) 0 (N) ∩ Γ 1 (Q) under the Hecke operator action (as spefor S(Q) = Γ 1 ciﬁed in the proof ) and satisﬁes the module compatibility condition (5) in Theorem 3.23; 3. the primes q ∈ Q have residual degree 1, and the primes q in Q∈Q Q with N (q) ≡ 1 mod pn for any given n > 0 have positive density. We shall give a proof of this, now it has become standard, partially following [Fu]. Proof We ﬁrst show that Q is an inﬁnite set. Let K(ρ) be the splitting ﬁeld of ρ. For a prime q outside pc(ε), N (q) ≡ 1 mod p ⇔ F robq acts trivially on F (µp ) by class ﬁeld theory. Thus we need to ﬁnd q such that • F robq acts trivially on F (µp ), • ρ(F robq ) is not a scalar, • N (q) is a rational prime (cf. Exercise 3.26).

If we ﬁnd one such q, by the Chebotarev density theorem, we have such q with positive density. Against this expectation to have such q, suppose that for all q outside pc(ε) with F robq ﬁxing F (µp ), ρ(Frobq ) is scalar. Then ρ restricted to Gal(Q/F (µp )) δ0 0 is isomorphic to for a character δ 0 : Gal(Q/F (µp )) → F× again by 0 δ0 Chebotarev’s density. By (aiF ), ρ is a factor of IndF F (µp ) δ 0 by the Frobenius σ reciprocity law (or Shapiro’s lemma Lemma 1.17). Deﬁne δ 0 for σ ∈ Gal(Q/F ) σ by δ 0 (τ ) = δ 0 (στ σ −1 ). This action factors through Gal(F (µp )/F ), because δ 0 is a character of Gal(Q/F (µp )). Let σ

Z = {σ ∈ Gal(F (µp )/F )|δ 0 = δ 0 } and M be the ﬁxed ﬁeld of Z in F (µp ). The character δ 0 extends to a character δ : Gal(Q/M ) → F× because Gal(F (µp )/M ) is cyclic (cf. [MFG] Corollary 4.37). Moreover the stabilizer of δ in Gal(M/F ) is trivial. Thus IndF M δ is absolutely irreducible by Mackey’s theorem, and hence by Frobenius reciprocity, we have F ∼ IndF F (µp ) δ 0 = δ IndM δ, where δ runs over all extension of δ 0 to Gal(Q/M ). Then (aiF ) tells us that M/F is the unique quadratic extension of F in F (µp ) and ρ ∼ = IndF M δ for a suitable extension δ of δ 0 .

Galois deformation

197

σ−1

Since ψ = δ gets trivial over F (µp ), ψ factors through Gal(F (µp )/M ). 2 σ Since F (µp )/F is abelian over F , we ﬁnd ψ σ = ψ, which implies δ = (δ )2 , and σ−1 )σ = ψ −1 = ψ, ψ extends hence ψ is quadratic, because ψ = 1. Since ψ σ = (δ to ψ : Gal(F (µp )/F ) → µ4 . Since Gal(F (µp )/F ) is cyclic, we have the following two possibilities: 1. ψ is quadratic and M = F (µp ), or 2. ψ is of order 4. σ

The case where M = F (µp ) does not occur, because δ = δ over Gal(Q/M ) σ by (aiF ), while our assumption is that δ = δ over Gal(Q/F (µp )). Since σ 2 1−σ δδ = det(ρ), we have δ = (δ ) det(ρ) = ψ det(ρ). Since F ⊂ M F (µp ), M 2 σ is totally real. For complex conjugation c, we ﬁnd ψ(c) = δ (c)ψ(c) = δ(c)δ (c) = det(ρ(c)) = −1. Thus ψ det(ρ) is a totally even character. Any totally even Galois × ∼ character of FA× /F × F∞+ = Gal(F ab /F ) has its square root by Lemma 3.24. × Taking a Galois character η : Gal(Q/M ) → F with η 2 = ψ det(ρ) and replacp

σ−1 ing ρ by η ⊗ ρ, we may assume that δδ = det(ρ) = ψ −1 = δ . Then 2 δ = 1; so, δ is quadratic. Writing K(ξ) for the splitting ﬁeld of a Galois σ character ξ, this implies K(δ) ∩ K(δ ) = M . We thus ﬁnd that K(ρ) = σ that Gal(K(ρ)/M ) = Gal(K(δ)/M ) × Gal(K(δ σ )/M ), K(δ)K(δ ) ⊃ K(ψ), σ Gal(K(δ)/M ) ∼ )∼ )∼ = {±1} and Gal(K(ψ)/F = = Gal(K(ψ)/M = Gal(K(δ )/M ) ∼ µ4 . If 8|[F (µp ) : F ], we have ψ(c) = ψ(c) = 1 because c is of order 2 in the cyclic group Gal(F (µp )/F ) and ψ has order 2. This is impossible, since ψ(c) = det(ρ)(c) = −1 (and p > 2). Since ψ is of order 4, we conclude that 4 exactly divides [F (µp ) : F ] and that Im(ρ) is isomorphic to the dihedral group of order 8. Under (ni), we do not have this possibility even if 4 exactly divides [F (µp ) : F ]. Thus assuming (ni), we have τ0 ∈ Gal(Q/F (µp )) with nonscalar ρ(τ0 ) (which has distinct eigenvalues). Then by the Chebotarev density, we have prime ideals q of positive density such that ρ(F robq ) = ρ(τ0 ) and N (q) ≡ 1. Once this is done, n τn = τ0p ﬁxes F (µpn+1 ) and ρ(τn ) has two distinct eigenvalues. In particular, if ρ(Frobq ) = ρ(τn ), N (q) ≡ 1 mod pn . This shows the positive density of primes q with N (q) ≡ 1 mod pn for any n > 0 satisfying (dsq ) under (aiF ) and (ni). For a ﬁnite set Q made out of such primes, RQ representing ΦQ exists. " 1 n 1 (P ∞ N) = " Γ Let P = p∈Σp p and put Γ 1 n 1 (P N). Take an open subgroup 1 (P ∞ N) ∩ S(Q) for Q ∈ Q. Decompose G(A(∞) ) = S ⊃ Γ S 1 4 with Γ0 (N) ⊃ (∞) ). We consider Γj = G(Q) ∩ cj S · Z(A(∞) )c−1 and Γj = j j G(Q)cj S · Z(A Γj / Ker(ε) ∩ Γj . Since ε has order prime to p, Γj has order prime to p, and hence (sm) is satisﬁed for any S as above. −1

σ

198

Hecke algebras as Galois deformation rings

For the moment, we assume (Ord) (so, Σf l = ∅). The spaces Mn.ord (S, ε; A) and S n.ord (S, ε; A) are A-free of ﬁnite rank (see Lemma 3.15 or Lemma 3.9). Let hn.ord (S(Q), ε; A) (resp. hΣ0 -ord (S(Q), ε; A)) be the A-subalgebra of the A-linear endomorphism algebra EndA (S n.ord (S, ε; A)) (resp. EndA (S Σ0 -ord (S, ε; A)) genQ ∼ 0 (N erated over A by the action of T (l) and Γ q∈Q q)/S(Q)Z(Z) = ∆Q × ∆ . We ﬁx one eigenvalue αq of ρ(F robq ) for each q appearing in Q. By Lemma 3.14, we ﬁnd a unique maximal ideal mQ of hn.ord (S(Q), ε; W ) generated by (δ − 1) for δ ∈ ∆Q × ∆Q , U (q) − αq for q ∈ Q and T (l) − λ(T (l)) for l outside Q, where f0 |T (l) = λ(T (l))f0 and αq is the root in W of X 2 − λ(T (q)X + ε+ (q)N (q) = 0 with αq = αq mod mW . We write TQ for the localization of hn.ord (S(Q), ε; W ) at the maximal ideal mQ which is reduced by (dsq ) for q ∈ Q. We put ∗ MQ = TQ · S n.ord (S(Q), ε; W )∗ and MQ = TQ · S n.ord (S(Q), ε; W )

for S n.ord (S(Q), ε; W )∗ = HomW (S n.ord (S(Q), ε; W ), W ). This module is naturally a W [∆Q ]-module and is W -free of ﬁnite rank. Note that ∗ = TQ · S Σ0 -ord (S(Q), ε; W ), MQ = TQ · S Σ0 -ord (S(Q), ε; W )∗ and MQ

because the old–new congruence appears only when we raise the level at p ∈ Σ0 . ∗ (A) = TQ · S n.ord (S(Q), ε; A), By (sm), for any W -algebra A, deﬁning MQ ∗ ∗ (W ) ⊗W A, and hence, we also have MQ (A) = we ﬁnd that MQ (A) = MQ ∗ (A), A). Note that MQ (W ) ⊗W A for MQ (A) = HomA (MQ det(S(Q)Z(A(∞) )) = T (A(∞) )2 U (Q), where × |xq − 1 ∈ qOq for all q ∈ Q}. U (Q) = {x ∈ O Thus writing ClF+ (Q) = T (A(∞) )/T (Q)+ U (Q) for T (Q)+ = T (R)+ ∩ T (Q), the reduced norm N induces N : Y (S) → ClF+ (Q)/ClF+ (Q)2 =: Cl2+ (Q). Thus Iv(Q; W ) := Iv(S(Q), ε; W ) = 0, and we have a ﬁnite-order Hecke character √ √ ε+ : T (A(∞) )/T (Q) → W × such that ε+ ◦ N ∈ Iv(S(Q), ε; W ). Then a basis √ made of Hecke eigenforms of Iv(Q; W ) over W can be given by {(α ε+ ) ◦ N }α √ for quadratic characters α of Cl2+ (Q). The eigenvalues of T (l) for (α ε+ ) ◦ √ N is (α ε+ )(1 + N (l)) for l outside Q and c(ε)p. Thus writing MQ for the pullback of the maximal ideal mQ to the full Hecke algebra H n.ord (S(Q), ε; W ), ∗ (A) ⊂ Mn.ord (S(Q), ε; A) by (sm). the localization by MQ kills Iv(Q; W ); so, MQ ∗ 0 (N), M ∗ (A)) = M ∗ (A). We get Then by deﬁnition, H 0 (∆Q , MQ (A)) = H 0 (Γ Q ∅ by duality 0 (N), MQ (A)) = M∅ (A). MQ (A)/aQ MQ (A) = H0 (∆Q , MQ (A)) = H0 (Γ A similar identity also follows for all twisted quotients. Let us explain this in more detail, because out of this fact, freeness of MQ over W [∆Q ] follows. Let εQ be a

Galois deformation

199

×

character of ∆Q with values in Qp . Write W [εQ ] for the subring of Qp generated by the values of εQ over W , and regard it as a ∆Q -module by the character εQ . Then, we deﬁne M (εQ ) = M ⊗W W [εQ ] for any W [∆Q ]-module M and let ∆Q act diagonally, that is, δ(m⊗w) = δ(m)⊗δ(w) = εQ (δ)(δ(m)⊗w) for m ∈ M and ∼ 0 (N)/S(Q)Z(Z) w ∈ W [εQ ]. Since Γ = ∆Q ×∆Q , we may regard εQ as a character −1 0 ∗ 0 (N). Then H (∆Q , M (ε )) = TQ (S n.ord (N, εεQ ; W [εQ ])) by deﬁnition. of Γ Q Q Taking the dual, we have H0 (∆Q , MQ (εQ )) = TQ (S n.ord (N, εεQ ; W [εQ ])∗ ), which is W [εQ ]-free of ﬁnite rank. From this, freeness of MQ (A) over A[∆Q ] and constancy of its rank follows (by the lemma following this proof). This basically shows (tw3) except for the condition that the action of RQ on MQ (A)/aQ MQ (A) factors through R∅ . Write ρ for the “initial” Galois representation of f0 and ρQ for the Galois representation ρQ : Gal(Q/F ) → GL2 (TQ ) constructed as in Corollary 2.45. Since TQ is local and T∅ is a residue ring of TQ , Tr(ρQ (F robl )) ≡ t(l) ≡ Tr(ρ(F robl ))

mod mTQ

for all primes l outside Q and pc(ε) by Corollary 2.45. Then by Chebotarev density, we have Tr(ρT ) ≡ Tr(ρ) mod mTQ , which implies (Q1) by Proposition 1.25. The Galois representation ρQ satisﬁes the conditions (Q2–6) by Corollary 2.45 (as shown in [Ta2]). Here we used the condition (dsQ ) already veriﬁed. Thus we have a canonical morphism RQ → TQ which is surjective because TQ is generated by the trace of its Galois representations. Thus the action of RQ on MQ (A)/aQ MQ (A) ∼ = M∅ factors through T∅ , because M∅ is a faithful T∅ module. In particular, its factors through R∅ (and we have also veriﬁed the image compatibility and module compatibility for our Taylor–Wiles system). We now brieﬂy sketch how to prove the theorem without assuming (Ord). Thus we assume (Oﬂ). We deﬁne the maximal ideal mQ of hΣ0 -ord (S(Q), ε; W ) replacing the algebra hn.ord (S(Q), ε; W ) by hΣ0 -ord (S(Q), ε; W ) in this case. Then we write TQ for the localization of hΣ0 -ord (S(Q), ε; W ) at mQ . Since ρ|Dp for p ∈ Σf l is absolutely irreducible, again writing MQ for the pullback of the maximal ideal mQ to the full Hecke algebra H Σ0 -ord (S(Q), ε; W ), the localization by MQ kills Iv(Q; W ), because the Hecke eigenvalues on Iv(Q; W ) are associated to the Eisenstein maximal ideals of h(S(Q), ε; W ). From this fact combined with the four lemmas, Lemmas 3.9, 3.10, 3.13 and 3.15 for p ∈ Σ0 , we conclude the W ∗ = TQ · S Σ0 -ord (S(Q), ε; W ). freeness of MQ = TQ · S Σ0 -ord (S(Q), ε; W )∗ and MQ After verifying the W -freeness, all the arguments we did under (Ord) are still valid under the milder condition (Oﬂ), and we obtain the desired result. 2 Exercise 3.26 Let S be a ﬁnite set of rational primes. For any given number ﬁeld F of ﬁnite degree over Q and a ﬁnite Galois extension M/F , prove that the conjugacy classes of F robq in Gal(M/F ) for primes q outside S of F splitting over Q cover the entire Galois group Gal(M/F ). Hint: Apply the Chebotarev density to the Galois closure M gal /Q of M over Q.

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Here is a lemma we used in the above proof of the theorem. Lemma 3.27 Let H be a ﬁnite abelian p-group. Let M be a W [H]-module. For × a character ε : H → Qp , write W [ε] for the subalgebra in Qp generated by W and the values of ε. If H0 (H, M ⊗W W [ε]) for the ε-twist M (ε) is W [ε]-free of × ﬁnite rank for every character ε : H → Qp , M is W [H]-free of ﬁnite rank, and rankW [H] M = rankW [ε] (H0 (H, M ⊗W W [ε])) for any character ε. Proof If for a valuation ring W ﬁnite ﬂat over W M ⊗W W is W [H]-free, M is W [H]-free (by faithfully ﬂat descent), we may assume that all character of H has values in W . Then deﬁne the ε-twisted augmentation ideal aε of W [H] by the ideal generated by h − ε(h) for all h ∈ H. Then H0 (H, M (ε−1 )) ∼ = M/aε M canonically. Since W [H] is local, the minimal number of generators of M over W [H] is equal to dimF M/mW [H] M by Nakayama’s lemma (Lemma 1.6), which is in turn equal to the minimal number of generators of H0 (H, M (ε)) over W again by Nakayama’s lemma. Thus this number is independent of ε. Choose , and consider a W [H]-linear a lift x1 , . . . , xr ∈ M of a basis of M/mW [H] M map π : W [H]r → M given by π(a1 , . . . , ar ) = i ai xi . Then by Nakayama’s lemma applied to Coker(π), we see that π is surjective. The map π induces a πε surjection W r ∼ = W [H]r ⊗W [H] W [H]/aε −→ M/aε M after tensoring W [H]/aε over W [H]. By our assumption, M/aε M is free of rank r (because r is the minimal number of generators of M/aε M over"W ). Thus πε has to be an isomorphism; so, Ker(π) ⊂ arε , and hence Ker(π) ⊂ ε arε = 0. This shows that M is W [H]-free of rank r. 2 3.2.4 Hecke algebras are universal Let the notation be as in the previous section. We shall prove the following theorem of K. Fujiwara in [Fu] Theorem 6.1.1 (limiting ourselves to the even degree base ﬁeld F ): Theorem 3.28 Let f0 be a Hecke eigenform (nearly p-ordinary at p ∈ Σp − Σf l ) in S(N, ε; W ) satisfying (h1–4). Suppose that p is unramiﬁed over Q if p ∈ Σf l . Under (aiF (µp ) ), we have R∅ ∼ = T∅ , T∅ is a local complete intersection free of ﬁnite rank over W , and M∅∗ (W ) = T∅ (S(N, ε; W )) and M∅ (W ) = HomW (M∅∗ (W ), W ) are T∅ -free of rank 1. In this subsection, we prove this theorem assuming Σf l = ∅; so, any prime factor p of p may ramify in F/Q. In the following Subsection 3.2.5, we will discuss how to modify the proof to deal with primes in Σf l . We need to prove the existence of a subset Q ⊂ Q satisfying the conditions (1–4) in Theorem 3.23 (for Q in Theorem 3.25). Then by Theorem 3.23, all the assertions except for the T∅ -freeness of M∅∗ follow. If it is free over T∅ , it is of rank 1 by Lemma 3.6. Since a local complete intersection is a Gorenstein ring (cf. [MFG] 5.3.4), M∅∗ = HomW (M∅ , W ) ∼ = HomW (T∅ , W ) ∼ = T∅ as T∅ -modules, which shows ∗ the freeness of M∅ over T∅ . The proof verifying the conditions of Theorem 3.23

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201

is basically the same as the one given in [Wi2] in the elliptic modular case, after getting a key estimate: Proposition 3.29 on the size of the tangent space of the universal deformation space (which is a particular case of the adjoint square Selmer groups). After proving the estimate following [Fu], we will describe in detail the construction of Q as Proposition 3.33. Let F Q /F be the maximal extension in Q unramiﬁed outside pc(ε)∞ and Q. Inside F Q , we have the maximal extension FQ unramiﬁed outside pN∞ and Q. Let GQ (resp. GQ ) be the Galois group Gal(F Q /F ) (resp. Gal(FQ /F )). Let Σρ be the set of ramiﬁed primes for ρ and prime factors of p and Σρ be the set of prime factors of pN. We write SelQ (Ad(ρ)) = ΦQ (F[]) ⊂ H 1 (GQ , Ad(ρ)). Thus   H 1 (Fq , Ad(ρ)) . SelQ (Ad(ρ)) = Ker H 1 (GQ , Ad(ρ)) → Φ (F[]) Q,q ρ q∈Σ Q

Here, as before, the functor ΦQ,q is the local version at q of the deformation functor ΦQ (so, ΦQ,q = Φq if q ∈ Q). As we have seen (Proposition 3.21), cocycles in Φq (F[]) for q outside p and Q are unramiﬁed, and Ad(ρ) is unramiﬁed outside pN and Q; so,we actually have   H 1 (Fq , Ad(ρ)) . SelQ (Ad(ρ)) = Ker H 1 (GQ , Ad(ρ)) → ΦQ,q (F[]) q∈Σρ Q

1 Let Φ⊥ Q,q (F[]) ⊂ H (FQ,q , Ad(ρ)(1)) be the orthogonal complement of the subspace ΦQ,q (F[]) in H 1 (FQ,q , Ad(ρ)) under the local Tate duality (e.g., [MFG] Theorem 4.43). Then we deﬁne the dual Selmer group by   H 1 (Fq , Ad(ρ)(1)) .  1 Q Sel⊥ Q (Ad(ρ)(1)) = Ker H (G , Ad(ρ)(1)) → Φ⊥ Q,q (F[]) ρ q∈Σ Q

Since at the primes q ∈ Q, we do not impose any local condition on ΦQ , we have ΦQ,q (F[]) = H 1 (Fq , Ad(ρ)(1)), and hence Φ⊥ Q,q (F[]) = 0. Thus, writing Hq1 (Ad(ρ)(1)) for H 1 (Fq , Ad(ρ)(1)) simply, we have a slightly simpler expression of the dual Sel⊥ Q (Ad(ρ)(1))  = Ker H 1 (GQ , Ad(ρ)(1)) →

Hq1 (Ad(ρ)(1)) q∈Σρ

Φ⊥ Q,q (F[])

×

 Hq1 (Ad(ρ)(1)) .

q∈Q

(3.2.3) Proposition 3.29

Suppose (aiF (µp ) ). Then we have

dimF SelQ (Ad(ρ)) ≤ dimF Sel⊥ Q (Ad(ρ)(1)) + |Q|.

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Hecke algebras as Galois deformation rings

Proof We follow the proof of [Wi2] Proposition 1.6 with some simpliﬁcation by Fujiwara (particularly on the estimate of hp below). Note that, by (aiF (µp ) ), H 0 (GQ , Ad(ρ)(1)) = H 0 (GQ , Ad(ρ)) = 0. Indeed, by (aiF (µp ) ), we have Z(ρ) = EndF (µp ) (ρ) ∼ = F and H 0 (GQ , ad(ρ)) = 0 Z(ρ). Since ad(ρ) = Z(ρ) ⊕ Ad(ρ), we conclude H (GQ , Ad(ρ)) = 0. If 0 = φ ∈ H 0 (GQ , ad(ρ)(1)) = HomG (ρ, ρ(1)) exists, by the irreducibility of ρ, ρ ⊗ ω ∼ =ρ for the modulo p Teichm¨ uller character ω. By the lemma following this proof, this implies that F (µp ) is a quadratic extension and ρ ∼ = IndF F (µp ) ψ for a Galois character ψ of Gal(Q/F (µp )), which is prohibited by (aiF (µp ) ). By the Poitou– Tate exact sequence (cf. [MFG] Theorem 4.50 (5)) combined with local Tate duality (cf. [MFG] Theorem 4.43), the following sequence H 1 (Fq , Ad(ρ)) 0 → SelQ (Ad(ρ)) → H 1 (GQ , Ad(ρ)) → ΦQ,q (F[]) ρ q∈Σ Q

→

∗ Sel⊥ Q (Ad(ρ)(1))

→ H (G , Ad(ρ)) → 2

Q

H 0 (Fq , Ad(ρ)(1))

q∈Σρ Q

→ H 0 (GQ , Ad(ρ)(1)) = 0

(3.2.4)

is exact. This combined with the global Euler characteristic formula: dimF H 2 (GQ , Ad(ρ)) − dimF H 1 (GQ , Ad(ρ)) = −2[F : Q] tells us dimF SelQ (Ad(ρ)) − dimF Sel⊥ Q (Ad(ρ)(1)) = 2[F : Q] +

hq ,

(3.2.5)

q∈Σρ Q

where hq = dimF H 0 (Fq , Ad(ρ)(1)) − dimF H 1 (Fq , Ad(ρ)) + dimF ΦQ,q (F[]). If q ∈ Σρ is prime to p, we have, by Proposition 3.21 ΦQ,q (F[]) = Hf1 (Fq , Ad(ρ)) = Ker(H 1 (Fq , Ad(ρ)) → H 1 (Iq , Ad(ρ))) ∼ = H 1 (Gal(Fqunr /Fq ), H 0 (Iq , Ad(ρ))) ∼ = H 0 (Fq , Ad(ρ)), = H 0 (Iq , Ad(ρ))/(F robq − 1)Ad(ρ) ∼ and by the local Euler characteristic formula, hq = 0. If q ∈ Q, we do not assume any condition on ΦQ locally at q ∈ Q; so, H 1 (Fq , Ad(ρ)) = ΦQ,q (F[]), and hence, we get hq = dimF H 0 (Fq , Ad(ρ)(1)) = 1 if q ∈ Q,

(3.2.6)

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203

where the last equality follows from N (q) ≡ 1 mod p and the three eigenvalues of F robq of Ad(ρ) are distinct having 1 among them. Now we assume that q = p|p. Then we have an exact sequence: 0 → αp → Ad(ρ) → F ⊕ αp−1 → 0, −1

where αp = δ p = det(ρ)−1 η 22,p for p = p mod mW and δ p = δp mod mW as in (Q4). The above sequence is split, if αp = N p by Kummer’s theory, and in that case, we have dimF H 0 (Fp , Ad(ρ)) = 1 because αp = 1 by (h3). By the local Euler characteristic formula (cf. [MFG] Theorem 4.52), hp = dimF H 0 (Fp , Ad(ρ)(1)) − dimF H 1 (Fp , Ad(ρ)) + dimF ΦQ,p (F[]) = −(dimF Ad(ρ)(1)) dimF (Op /pOp ) − dimF H 0 (Fp , Ad(ρ)) + dimF ΦQ,p (F[]) = −3 dimF (Op /pOp ) − dimF H 0 (Fp , Ad(ρ)) + dimF ΦQ,p (F[]). By the local Euler characteristic formula, we have dimF (Op /pOp ) 1 dimF H (Fp , αp ) = dimF (Op /pOp ) − 1

if αp = N p , if αp = N p ,

since αp = 1 (by (h3)). We have by Proposition 3.20, 1 + dimF H 1 (Fp , αp ) if αp = N p, dimF ΦQ,p (F[]) ≤ 1 if αp = N p . dim H (Fp , αp ) This shows that − dimF H 0 (Fp , Ad(ρ)) + dimF ΦQ,p (F[]) ≤ dimFp Op /pOp (3.2.7) and hence hp ≤ −2 dimFp (Op /pOp ). Since p|p dimFp (Op /pOp ) = [F : Q], the desired assertion follows. 2 We now prove the lemma used in the above proof. Lemma 3.30 Suppose p > 2. Let ρ be an absolutely irreducible representation of Gal(Q/F ) into GL2 (F). If ρ ∼ = ρ ⊗ ψ for a nontrivial character × ψ : Gal(Q/F ) → Fp , ψ is a quadratic character, and for the splitting ﬁeld × M/F of ψ, we have ρ ∼ = IndF ϕ for a character ϕ : Gal(Q/M ) → F . M

p

Proof Since ρ ⊗ ψ ∼ = ρ, we have a C ∈ GL2 (Fp ) such that C(ψ(σ)ρ(σ))C −1 = ρ(σ) for all σ ∈ Gal(Q/F ). Then for the order r of ψ, we have C r ∈ Z(ρ); so, C r is a scalar by the absolute irreducibility of ρ (Schur’s lemma). Fix an eigenvalue c of C. Then all other eigenvalues of C are given by ζc for r-th roots of unity ζ. Let V [cζ] for the cζ eigenspace for C. Then by C(ψ(σ)ρ(σ))C −1 = ρ(σ), we have by computation Cρ(σ)v = ψ(σ)−1 cρ(σ)v; so, ρ(σ) brings V [c] isomorphically to V [cψ(σ)−1 ]. Since dim ρ = 2, we thus ﬁnd that r = 2 and V (ρ) = V [c] ⊕ V [−c]

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Hecke algebras as Galois deformation rings

for the representation space V (ρ) of ρ. Then V [c] is one dimensional and is stable under Gal(Q/M ) for the splitting ﬁeld M/F of ψ. Writing ϕ for the character 2 giving the action of Gal(Q/M ) on V [c], we conclude that ρ ∼ = IndF M ϕ. Here is an outline of the proof of Theorem 3.28, which we carry out in full detail after the outline. By Proposition 3.29, to show the generator control and relation control conditions in Theorem 3.23, we need to show for an inﬁnite subset Q of the set Q in Theorem 3.25 satisfying the growth control in Theorem 3.23, if Q ∈ Q 1. Sel⊥ Q (Ad(ρ)(1)) = 0, 2. r := |Sel⊥ ∅ (Ad(ρ))| = |Q|. This is the main place where we use the hypothesis (aiF (µp ) ). We now prepare some results necessary to carry out the above sketch. To prove Theorem 3.28, we proceed basically in the same way as in [Wi2] Section 2.3 and (3.9) in Chapter 3, and we repeat the proof given at the end of Chapter 3 in [MFG] incorporating the adjustment in [Fu] 6.2 to the Hilbert modular setting. We now quote the following classical result from Dickson [LGF] Section 260 on subgroups G of P GL2 (F) = GL2 (F)/F× : We recall the assumption p ≥ 3. (PL) When p|G| and G is not contained in a Borel subgroup (that is, a conjugate of the group of upper triangular matrices) in P GL2 (F), (a) if p ≥ 5, G is conjugate either to P SL2 (k) or P GL2 (k) for a subﬁeld k ⊂ Fp in P GL2 (Fp ), and (b) if p = 3, in addition to P SL2 (k) and P GL2 (k), we have one more possibility A5 (here P SL2 (F3 ) ∼ = A4 and P GL2 (F3 ) = S4 ). (SM) When p |G|, if G is neither dihedral nor cyclic, then G is isomorphic to either A4 , S4 or A5 . It is easy to see the second assertion (SM), following [Se1] Proposition 16: sup Thus we can in GL2 (F). We still have p |G|. pose p |G|. Lift G to a subgroup G lift the representation of G ⊂ GL2 (F) to G ⊂ GL2 (K) for a p-adic ﬁeld K with is a ﬁnite group, we may assume residue ﬁeld F (see [MFG] Corollary 2.7). Since G after conjugation that G ⊂ GL2 (F ) for a number ﬁeld F ⊂ K. Embed F ⊂ C, ⊂ GL2 (C). Thus G ⊂ P GL2 (C). Since G is ﬁnite, it is in a maximal and hence G compact subgroup of P GL2 (C), which is a conjugate to the 2 × 2 special unitary matrices modulo center: P SU2 (C). Thus we may assume that G ⊂ P SU2 (C). Now we invoke an exotic isomorphism P SU2 (C) ∼ = SO3 (R). We consider the symmetric second tensor representation Sym2 : SL2 (C)/{±1} → GL3 (C). Since Sym2 preserves a non-degenerate symmetric bilinear form on C3 , suitably choosing conjugation, we have an injective homomorphism P SU2 (C) → SO3 (R). By comparing dimension, we conclude that this is a surjective isomorphism. Once this isomorphism is established, G is a group of rotations acting on a Euclidean 3-space. If G is neither dihedral nor cyclic, the shape made of moving around

Galois deformation

205

one point oﬀ the origin by the action of G is a regular polyhedron. The only possibilities are, as is well known from the original Euler characteristic argument, a tetrahedron, cube, and icosahedron. The symmetric groups of the above polyhedra are isomorphic to A4 , S4 , and A5 . As for the ﬁrst assertion, it is a bit technical; so, we just refer to the book of Dickson. Exercise 3.31 1. For a polyhedron in three-dimensional Euclidean space, writing f , e, and v, respectively, for the number of faces, edges, and vertices of the polyhedron, prove f − e + v = 2 (Euler). 2. Classify all regular polyhedra up to similarity in three-dimensional Euclidean space by using the above Euler characteristic formula. 3. Prove that the symmetric group of a regular tetrahedron, cube, and icosahedron, respectively, is isomorphic to A4 , S4 , and A5 . Let Qm be the subset of the set Q of Theorem 3.25 such that Q ∈ Qm ⇔ N (q) ≡ 1 mod pm and q is of degree 1 over Q. Before showing the existence of inﬁnitely many such sets Q in Qm , we prepare one more lemma. Write H q (K /K, M ) for H q (Gal(K /K), M ) for a Galois extension K /K and a discrete Gal(K /K)-module M . Lemma 3.32 (Wiles)

Ker(ρ)

Let K = K(ρ) := Q

. Then we have

H 1 (K(µp )/F, Ad(ρ)(1)) = 0. Proof We may assume that p divides | Im(ρ)|. We have an exact sequence (from the inﬂation-restriction sequence: e.g., [MFG] 4.3.4): 0 → H 1 (K/F, Ad(ρ)(1)Gal(K(µp )/K) ) → H 1 (K(µp )/F, Ad(ρ)(1)) → H 1 (K(µp )/K, Ad(ρ)(1)) = 0. The vanishing of the last cohomology group follows from the fact that p is prime to |Gal(K(µp )/K)| (e.g., [MFG] 4.3.1). Since Gal(K(µp )/K) ﬁxes Ad(ρ) element by element, Ad(ρ)(1)Gal(K(µp )/K) = 0 if Gal(K(µp )/K) = 1, and hence H 1 (K(µp ), Ad(ρ)(1)) = 0. Thus we may suppose that K(µp ) = K. Let Z be the center of G = Gal(K/F ). By absolute irreducibility, ρ(Z) is in the center of GL2 (F) (Schur’s lemma; e.g., [MFG] Lemma 1.12); so, Z acts trivially on Ad(ρ). Then we have another exact sequence: 0 → H 1 (G/Z, Ad(ρ)(1)Z ) → H 1 (G, Ad(ρ)(1)) → H 1 (Z, Ad(ρ)(1)) = 0. The vanishing of the last cohomology again follows from the fact that p |Z|. Since Z acts trivially on Ad(ρ), if Z acts nontrivially on µp , Ad(ρ)(1)Z = 0. Thus we get H 1 (G, Ad(ρ)(1)) = 0, and hence we may suppose that Z acts on µp trivially. Then Gal(F (µp )/F ) is the quotient of G/Z, which is isomorphic

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Hecke algebras as Galois deformation rings

to either P GL2 (k) or P SL2 (k) for a subﬁeld k ⊂ Fp by (PL). Since P SL2 (k) for p ≥ 5 is known to be simple and (P GL2 (k) : P SL2 (k)) = 2 if p ≥ 5, G/Z ∼ = P SL2 (k) is impossible, because [F (µp ) : F ] ≥ 2. Writing Ad for the tracenull space {x ∈ M2 (k)|Tr(x) = 0} and letting P SL2 (k) act on Ad by conjugation, by [ClPS], we have H 1 (P SL2 (k), Ad) = 0 if |k| > 3. By the restriction and inﬂation sequence, we have ∼ H 1 (P SL2 (k), Ad) = 0 if |k| > 3. H 1 (P GL2 (k), Ad) = By Dickson, we may identify G/Z = g · P GL2 (k)g −1 for g ∈ GL2 (Fp ). Then by extending scalars to Fp and conjugating by g, we have H 1 (G/Z, Ad(ρ) ⊗F Fp ) ∼ = H 1 (P GL2 (k), Ad) ⊗k Fp . = H 1 (G/Z, Ad) ⊗k Fp ∼ This shows that H 1 (G/Z, Ad(ρ)) = 0 if G/Z ∼ = P GL2 (k) with |k| > 3 including the case where p ≥ 5. We now deal with the remaining case: p = 3. Since A5 is simple, A5 cannot have the order 2 quotient isomorphic to Gal(F (µ3 )/F ). Thus the case of A5 is done. The case where |k| > 3 and G/Z ∼ = P F L2 (k) has already been dealt with. Now assume that k = F3 . Then P SL2 (F3 ) ∼ = A4 and P GL2 (F3 ) ∼ = S4 . Let H be the normal subgroup of order 4 in P SL2 (F3 ) = A4 . Then by computation, we ﬁnd AdH = 0, and hence, H 1 (P SL2 (F3 ), Ad) ∼ = H 1 (A4 /H, AdH ) = 0. Then to using the above argument of conjugation, we need to show that any subgroup X in P GL2 (F3 ) isomorphic to A4 is conjugate to P SL2 (F3 ). This can be shown as follows. Now we denote by H the normal subgroup of X of order 4. Since H ∼ = A and B of H. Conjugating X in P GL (F ), we (Z/2Z)2 , we take two generators 2 3

1 0 may assume that A = . Since B commutes with A and are distinct, we 0 −1

× 0 −a−1 ﬁnd B = for a ∈ F3 . Then conjugating X by a diagonal matrix, we a 0 may assume that a = 1. Thus by conjugation, we can bring H into P GL2 (F3 ). Thus we may assume that H ⊂ P GL2 (F3 ). By the explicit form of A and B as above, the centralizer Z(H) of H in P GL2 (F3 ) is equal to H, and its normalizer N satisﬁes N/H ∼ = S3 which is in P GL2 (F3 ). Therefore the pullback image of the order 3 subgroup of S3 in N gives A4 = X. Then conjugating by an element in P GL2 (F3 ), we ﬁnd that H 1 (G/Z, Ad(ρ)) = 0. 2 We now ﬁnish the proof of the existence of the Taylor–Wiles system satisfying the remaining conditions (2,3) (relation control and generator control) of Theorem 3.23. We have the following commutative diagram with exact rows for V = Ad(ρ)(1): Ker(βQ )  

→

βQ

−−−−→ H 1 (GQ , V ) −−−−→   res

Hom(H, V ) −−−−→ Hom(HQ , V ) −−−−→ →

q∈Q

q∈Q

H 1 (Fqunr , V )   H 1 (Fqunr , V ),

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207

where H ⊂ G = G∅ and HQ ⊂ GQ are, respectively, the stabilizer of K(ρ)(µp ). The map res as above is injective by Lemma 3.32, and by the inﬂation map, H 1 (G, Ad(ρ)(1)) is embedded into H 1 (GQ , Ad(ρ)(1)) (an application of the inﬂation and restriction sequence). This tells us that any cohomology class in H 1 (GQ , Ad(ρ)(1)) unramiﬁed at all q ∈ Q actually comes from H 1 (G, Ad(ρ)(1)). In particular, the kernel of the restriction map: H 1 (Fq , Ad(ρ)(1)) H 1 (GQ , Ad(ρ)(1)) → q∈Q

is contained in the kernel Ker(βQ ) of the following restriction map: βQ : H 1 (G, Ad(ρ)(1)) → H 1 (Fq , Ad(ρ)(1)).

(3.2.8)

q∈Q

Them Theorem 3.28 follows from Proposition 3.33 Under the assumptions of Theorem 3.28 (in particular, under (aiM ) for the unique quadratic extension M/F inside F (µp )), there exist inﬁnitely many ﬁnite sets Q such that for a given integer m > 0 1. |Q| = r = dimF Sel⊥ ∅ (Ad(ρ)) = dimF ΦQ (F[]); 2. N (q) is a rational prime q with q ≡ 1 mod pm for all q ∈ Q; 3. βQ is injective, and hence Sel⊥ Q (Ad(ρ)(1)) = 0. Proof We follow the argument of Wiles proving a similar fact (in [Wi2] (3.8)) with some adjustment to the Hilbert modular case by Fujiwara. By the above lemma (Lemma 3.32) combined with the inﬂation-restriction sequence (e.g., [MFG] 4.3.4), we have an exact sequence: 0 = H 1 (K(µp )/F, Ad(ρ)(1)) → H 1 (G, Ad(ρ)(1)) ι → H 1 (H, Ad(ρ)(1))G ∼ − = HomG (H, Ad(ρ)(1))

for G = Gal(K(µp )/F ). The last isomorphism follows from the fact that a 1-cocycle for the trivial action is a homomorphism. First we show that by adding primes to Q, we can make βQ injective. Thus for x ∈ Ker(βQ ), we have fx : H → Ad(ρ) which is the image of x under ι. First suppose that the order of Im(ρ) is divisible by p. Then we claim, following [Wi2] Lemma 1.10 (see Lemma 3.34 following this proof), that there exists σ ∈ G∅ such that 1. ρ(σ) has order with ≥ 3 prime to p; 2. det ρ(σ) = 1 and σ ﬁxes F (µpm ). We now ﬁnish the proof of the proposition, accepting the claim, which we will prove in the lemma following this proof. We want to show that we can choose

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Hecke algebras as Galois deformation rings

σ as above satisfying one more condition: 3. fx (σ ) = 0. Let L be the minimal Galois extension of K(µp ) so that fx factors through the Galois group Gal(L/K(µp )). Then L is an abelian extension of exponent p. Thus we have an exact sequence: 1 → X = Gal(L/K(µp )) → Gal(L/F ) → G = Gal(K(µp )/F ) → 1. Then Gal(K(µp )/F ) acts on X by conjugation, and f : X → Ad(ρ)(1) is a morphism of G-modules. Let σ be an element satisfying (1)–(2), and write σ for the image of σ in Gal(K(µp )/F ). By (1)–(2), Ad(ρ)(σ ) has three distinct eigenvalues on Ad(ρ), and one of them is equal to 1. Thus we can decompose X = X[1] ⊕ X with X = 0 and X[1] = 0 so that σ − 1 is an automorphism on X and σ = 1 on X[1]. For the moment, we suppose that Ad(ρ) is absolutely irreducible; if f (X[1]) = 0, then f (X ) = f (X) is a proper invariant subspace of Ad(ρ), which is impossible. Thus we can ﬁnd τ ∈ X[1] such that f (τ ) = 0. We then deﬁne τ = τ × 1 ∈ Gal(L(µpm )/K(µp )) = Gal(K(µpm )/K(µp )) × Gal(L/K(µp )) ∼ Gal(L(µpm )/L) × Gal(L(µpm )/K(µpm )), = because L is linearly disjoint from K(µpm ) (over K(µp )) by our construction (f (X) ⊂ Ad(ρ)). Since τ ∈ X[1], τ commutes with σ in Gal(L(µpm )/F ). Then f ((τ σ ) ) = f (τ σ ) = f (τ ) + f (σ ) = f (τ ) + f (σ ) because σ ∈ X, and 0 = f (τ ) = f ((τ σ ) ) − f (σ ). This shows one of τ σ or σ satisﬁes (1), (2) and (3), which we write as σ. We then choose a prime q so that F robq = σ in Gal(L(µpm )/F ) and q splits in F/Q. Then f (F robq ) = 0 implies βQ∪{q } (x) = 0. By (2), we see N (q ) ≡ 1 mod pm . By (1) and (2), the characteristic roots α and β of ρ(σ) satisfy αβ = 1 and hence primitive -th roots. Therefore ≥ 3 (with p ) shows that α = β. This supplies us with inﬁnitely many disjoint choices of Q, by the Chebotarev density theorem. As already remarked, the injectivity of βQ implies the vanishing of the Selmer group Sel⊥ Q (Ad(ρ)(1)) since Sel⊥ Q (Ad(ρ)(1)) ⊂ Ker(βQ ). This follows from the fact that Φ⊥ Q,q (F[]) = 0 for all q ∈ Q, because by deﬁnition the functor ΦQ,q does not have any local requirements at q ∈ Q which implies ΦQ,q (F[]) is the entire local cohomology and its orthogonal complement Φ⊥ Q,q (F[]) has to vanish (see (3.2.3)). After proving the injectivity of βQ , we remove one by one elements of Q keeping injectivity of βQ . This is possible because the local contribution at q to the dimension hq is equal to 1 (see (3.2.6)). Then after a ﬁnite number of steps, we reach the equality: |Q| = dimF Sel⊥ ∅ (Ad(ρ)(1)), because in the minimal

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209

choice of Q, each q kills a one-dimensional quotient of Sel⊥ ∅ (Ad(ρ)(1)). We may ⊥ then take r to be dimF Sel∅ (Ad(ρ)(1)), which is independent of Q. We still have inﬁnitely many choices of Q, because we can start this element-removing process with another Q disjoint from Q. We now assume that Ad(ρ) is reducible. Since ρ is absolutely irreducible, Ad(ρ) is semisimple; so, Ad(ρ) ⊂ Hom(ρ, ρ) has a one-dimensional invariant subspace. Thus we have a Galois character χ such that ρ ∼ = ρ ⊗ χ. If χ is trivial, the invariant subspace is made up of scalars in End(ρ) (Schur’s lemma applied to ρ), which has nontrivial trace and is not in Ad(ρ) (the trace null space). Then χ is a nontrivial character. By Lemma 3.30, χ is of order 2, and ρ = IndF M ϕ for the − ϕ (see Exercise 5.32), splitting ﬁeld M of χ. Then we have Ad(ρ) = χ ⊕ IndF M where ϕ− (σ) = ϕ(cσc−1 σ −1 ) for an element c ∈ Gal(Q/F ) restricting to the generator of Gal(M /F ). By the absolute irreducibility of ρ, ϕ− is nontrivial. − − Thus either IndF M ϕ is irreducible, or for an extension ψ of ϕ to Gal(Q/F ), F − − ϕ = ψ ⊕ ψχ. When Ind ϕ is irreducible, the splitting ﬁeld K(Ad(ρ)) IndF M M is nonabelian over F . Since ϕ− is nontrivial, in the fully reducible case, Ad(ρ) is the sum of three distinct characters. In any case, unless M ⊂ F (µp ), for any proper nontrivial invariant subspace X ⊂ Ad(ρ), for any positive integer n, we can ﬁnd σ ∈ Gal(Q/F ) such that • • • •

σ restricts to the identity on F (µpm ) for a given positive integer m; σ acts by the identity on X; ρ(σ) has two distinct eigenvalues; fx (σ ) = 0 for the order of ρ(σ).

This is the content of [Wi2] Lemma 1.12. Arguing in the same manner as in the case where Ad(ρ) is absolutely irreducible, we ﬁnd τ ∈ Gal(K(µpm )/K(µp )) as above with fx (τ ) = 0. Then the rest of the argument proving the existence of inﬁnitely many Q is the same as in the case of absolutely irreducible Ad(ρ). The assumption M ⊂ F (µp ) follows from the irreducibility of ρ over the unique 2 quadratic extension M of F inside F (µp ). We needed to know the following claim in [Wi2] Lemma 1.10: Lemma 3.34 Suppose that p > 2. Suppose that the image of ρ in P GL2 (F) is either of order divisible by p or isomorphic to one of the following groups: A4 , S4 , and A5 . Then there exists σ ∈ Gp such that 1. ρ(σ) has order with ≥ 3 prime to p; 2. det ρ(σ) = 1 and σ ﬁxes F (µpm ) for a given positive integer m. Proof Let G be the image of ρ in GL2 (F) and Z be the center of Z. Since ρ is absolutely irreducible, the subgroup Z is made up of scalar matrices in G. Thus G/Z ⊂ P GL2 (F). We need to use the classiﬁcation of subgroups of P GL2 (F) stated in (PL): if p|G|, then G/Z ∼ = P GL2 (k) or G/Z ∼ = P SL2 (k) for a subﬁeld k ⊂ F. We can then always ﬁnd an element in the derived group D(G/Z) of

210

Hecke algebras as Galois deformation rings

order ≥ 3 prime to p and lift it to the derived group D(G) of G because of surjectivity of the projection D(G) D(G/Z). We can apply the same proof 2 when G/Z is isomorphic either to A4 , S4 , or A5 (see (SM)). This ﬁnishes the proof of Theorem 3.28 under the assumption of Σf l = ∅. 3.2.5 Flat deformations Recall that ρ0 is the Galois representation of the initial Hecke eigenform f0 ∈ S(N, ε0 ; W ). We can ease the condition (Ord) in 3.2.4 imposed to prove Theorem 3.28 allowing ﬂat nonordinary deformations: (ﬂ) At p-adic places p of F absolutely unramiﬁed outside c(ε)N, f0 could be ﬂat nonordinary though f0 is assumed to be nearly p-ordinary at other p-adic places p including all ramiﬁed places over Qp and all p-adic places in c(ε)N. Recall that f0 is said to be ﬂat if there exists a Barsotti–Tate W -module Gρ0 /Op such that the Galois representation ρ0 restricted to Dp is isomorphic to the Tate module Tp G ∼ = W 2 of G. If f0 is nonordinary at p, ρ0 |Dp is absolutely irreducible. If f0 is ﬂat at p, by deﬁnition, ρ is associated to a ﬁnite ﬂat group scheme over Op , but it could be reducible. Since for ﬂat places, Fp /Qp is unramiﬁed, and hence if ρDp is reducible, by the classiﬁcation of ﬁnite ﬂat group schemes (see Proposition 3.48), one of the two characters p and δ p , say δ p , is unramiﬁed and the other ramiﬁes. Let Σf l be the set of p-adic places, where f0 is ﬂat, satisfying the requirement described in 3.2.1. Thus we have Σp = Σ0 Σf l Σst Σn.ord . By Raynaud’s classiﬁcation of ﬁnite ﬂat F-vector spaces over an unramiﬁed p-adic valuation ring (e.g., [ARG] III.7 and Proposition 3.48 in the text), the isomorphism class of the ﬁnite ﬂat F-vector space whose generic ﬁber giving ρ is unique up to isomorphisms. We write Gρ/Op for this ﬁnite ﬂat group scheme. From this, we conclude that the Barsotti–Tate group Gρ0 is also unique up to isomorphisms. When [F : Q] were odd, by the Jacquet–Langlands correspondence, we could shift to yet another quaternion algebra B/F with B∞ ∼ = M2 (R) × Hd−1 which B B carries a Hecke eigenform f0 on Y0 (N). Then we ﬁnd a factor X of the Jacobian of Y0B (N) whose p-adic Tate module (tensored W over Zp ) contains Tp Gρ0 as a factor, and in this way, Gρ0 is basically realized inside the p-divisible group of X. Indeed, X is known to have good reduction at p ∈ Σf l by Carayol [C] (since p ∈ Σf l is outside the level c(ε)N). Since actually [F : Q] is even, we need to resort to a level-raising argument (similar to the one discussed in Section 3.1) to have an abelian scheme Xn/Op with Gρ0 [pn ] ⊂ Xn [pn ]. This has been done by Taylor in [Ta2] Theorem 1.6, and indeed, Taylor proved that ρT restricted to Dp is also associated to a Barsotti–Tate group GT over Op . A deformation ρ of ρ into GL2 (A) for an artinian local W -algebra in CLW is said to be ﬂat at p ∈ Σf l if there exists a ﬁnite ﬂat group scheme Gρ over Op on which A acts by endomorphisms such that ρ is isomorphic to the generic geometric ﬁber of Gρ . Again by classiﬁcation theory, Gρ is uniquely determined

Galois deformation

211

by ρ (up to isomorphisms). If A = limn An is a p-proﬁnite member of CLW ←− with artinian An , a deformation ρ into GL2 (A) is called ﬂat if its projection in GL2 (An ) is ﬂat for all n. We consider the deformation functor Φ of ρ satisfying (Q1–6) and (Q7)

ρ|Dp is ﬂat for all p ∈ Σf l .

As before, we write Φp for the local deformation functor of ρ|Dp , and the only conditions necessary to deﬁne Φp in this case are (Q1), (Q3), and (Q7). We will prove below (the upper bound of) the dimension of Φp (F[]) is the same as the other cases, which gives directly the p-part (p ∈ Σf l ) of the ﬁnal estimate (3.2.7) in the proof already given of Theorem 3.28. Thus all the arguments in the proof of Theorem 3.28 go through intact. We only need to prove the following identity of dimF Φp (F[]). Proposition 3.35 (Fujiwara)

We have

dimF Φp (F[]) = dimF H 0 (Fp , Ad(ρ)) + dimFp (O/p) for p ∈ Σf l . Fujiwara’s proof of this fact follows the argument of [TaW] and [R]. To give a sketch of the proof, we need to prepare a small amount of input from Fontaine’s theory. Suppose p ∈ Σf l , and write k for O/p. Since Fp is unramiﬁed over Qp , we have a unique automorphism σp of Fp inducing F robp : x → xp on k. We consider the category F LV/F of ﬁnite ﬂat F-vector spaces over Op (that is, ﬁnite ﬂat commutative group schemes with an F-action as endomorphisms) and the category MF L of triples (D, FD, ϕ) of a ﬁnite k ⊗Fp F-module D and a k ⊗Fp F∼ → D. submodule FD ⊂ D with a σp -linear isomorphism ϕ : FD ⊕ (V /FD) − Then by the result of [FoL], we have two canonical exact contravariant functors D : F LV/F → MF L and V : MF L → F LV/F with D ◦ V ∼ = idMF L and V ◦ D∼ = idF LV . We write ARTW for the category of local artinian W -algebras with residue ﬁeld F = W/mW . Lemma 3.36 Suppose p ∈ Σf l . Let A be a characteristic p object in ARTW . Then if ρ ∈ Φp (A), we have 1. D(ρ) is free of rank 2 over A ⊗Zp Op ; 2. F(D(ρ)) is free of rank 1 over A ⊗Zp Op . Proof We proceed by induction on dimF A. Suppose A = F. Then ρ = ρ, and hence it is realized on an abelian variety X/F with multiplication by the integer ring OE of a totally real ﬁeld E such that LieX ∼ = = OE ⊗Z F and ΩX/F ∼ OE ⊗Z F. We may assume that W = OE,P for a prime P of OE . Then it is known that D(ρ) = H1DR (X/F , F) with FD(ρ) ∼ = H 0 (X/F , ΩX/F ) and D(ρ)/F(D(ρ)) ∼ = Lie(X)/F and that ϕ is induced by the (divided) absolute Frobenius map (see [Od]). From this the assertions (1) and (2) follow for A = F. Since A is an artinian F-algebra, we have an ideal I ⊂ A of dimension 1. Since ρ ∈ Φp (A), the

212

Hecke algebras as Galois deformation rings

kernel of the reduction map: ρ ρ = ρ ⊗A A/I is isomorphic to ρ as Galois representation. Thus we have the exact sequence 0 → D(ρ) → D(ρ) → D(ρ ) → 0, and hence dimF D(ρ) = dimF D(ρ) + dimF D(ρ ). By the induction hypothesis, we have the assertion (1) for D(ρ ); so, dimF D(ρ) = 2 dimF A ⊗Zp Op . Since D(ρ) ⊗A F ∼ = D(ρ), choosing a lift x1 , x2 of F ⊗Fp k-basis of D(ρ) in D(ρ), we have a surjection (A⊗Zp Op )2 D(ρ) sending (a1 , a2 ) to a1 x1 +a2 x2 . Comparing the dimension, we ﬁnd that (A⊗Zp Op )2 ∼ = D(ρ). The assertion (2) can be proven similarly, and the work is left to the reader. 2 Let (D, FD) be an object of MF L satisfying the two assertions of the above lemma. Then a frame f of (D, FD) is an isomorphism f : (D, FD) → ((A ⊗Zp Op )2 , (A ⊗Zp Op )e1 ) for e1 = (1, 0). Proof of Proposition 3.35. Let G = Resk/Fp GL(2). Consider the collection of all Fantaine–Laﬀaille modules F L(A) = {(F, FD)/A } which aquire frames, and all possible frames: F r(A) = {f : (D, FD) → ((A ⊗Zp Op )2 , (A ⊗Zp Op )e1 )}. On F r(A), g ∈ G(A) = GL2 (A ⊗Fp k) acts because the trivialized F r(A) is in bijection with the collection of σp -linear automorphisms h of (A ⊗Fp k)2 by deﬁnition, and G(A) acts by composition h → gh. Thus f tρ := {(D, FD) − → ((A ⊗Zp Op )2 , (A ⊗Zp Op )e1 ) ∈ F r(F[])|D ⊗F[] F = D(ρ)} ∼ = Lie(G) = M2 (F ⊗F k). p

We have a surjection F r(A) → F L(A) by forgetting frames. Since the stabilizer of FD is an upper triangular Borel subgroup B(A ⊗Fp k), we have F r(A)/B(A ⊗Fp k) ∼ = F L(A). The action of x ∈ B(A) is multiplication by x on (A ⊗Zp Op )2 and on (FD ⊕ D/FD) = ((A ⊗Zp Op )e1 ⊕ (A ⊗Zp Op )e2 ), it is through the semisimpliﬁcation xss of x. Choose one frame f1 of (D, FD) = D(ρ) for ρ ∈ Φp (F[]). Composing (f ss )−1

ϕ

f1

1 −−→ FD ⊕ (D/FD) − → D −→ (F[] ⊗Fp k)2 , Y : (F[] ⊗Fp k) ⊕ (F[] ⊗Fp k) −−−

we get a matrix Y ∈ M2 (F[] ⊗Fp k). Writing Y = Y + y with y ∈ M2 (F ⊗Fp k) tρ gives rise to the same (and normalizing Y to be the identity matrix), f1 , f2 ∈ −1 ss (D, FD) if f1 ◦f2 ∈ 1+B for B = {σp (x)y −yx |x ∈ Lie(B)(F⊗Fp k)} for the

Galois deformation

213

matrix y ∈ M2 (F ⊗Fp k) given by our choice of the frame f1 (see [TaW] page 565 for an explicit computation). Going through a slightly demanding computation of p (F[])), (which is equal to dimF Φ the dimension of this quotient M2 (F ⊗Fp k)/B we ﬁnd p (F[]) = dimF H 0 (Fp , ad(ρ)) + dimF k. dimF Φ p Starting from ρ ∈ Φp (F[]), unramiﬁed character twists give one-dimensional deformation; thus, taking 1 oﬀ from the above formula, we get the desired amount: dimF Φp (F[]) = dimF H 0 (Fp , Ad(ρ)) + dimFp k, because Ad(ρ) ⊕ F = ad(ρ) and dimF H 0 (Fp , ad(ρ)) = dimF H 0 (Fp , Ad(ρ)) + 1. By this, we have ﬁnally ﬁnished the proof of Theorem 3.28. 2 3.2.6 Freeness over the Hecke algebra We now want to prove the freeness of the space of modular forms over the Hecke algebra in a slightly more general setting of nearly p-ordinary but possibly nonﬂat automorphic forms. We prepare some notation. Let f0 be a p-ordinary Hecke eigenform satisfying (h1–4) in S n.ord (N, ε; W ). Recall Σ0 ⊂ Σp which is the set of prime factors p of p satisfying the following three conditions: 1. p is prime to N, 2. p is outside Σf l , ×

3. ρ|Dp ⊗ ψ for a Galois character ψ : Dp → Fp is associated to a ﬁnite ﬂat group scheme (Z/pZ ⊕ µp ) ⊗Z F (here ψ could be ramiﬁed). This Σ0 is the set of primes p with a bigger local ring T of level pN as in the level from N to Theorem 3.16 than the Hecke algebra T∅ if we increase pN. For each subset P ⊂ Σ0 , deﬁning NP = N p∈P p, we have a unique Hecke eigenform fP (up to scalar multiple) in S n.ord (NP , ε; W ) inside the automorphic representation π0 generated by f0 by (3.1.8). Let TP be the local ring of hn.ord (NP , ε; W ) acting nontrivially on fP , and set SP = TP · S n.ord (NP , ε; W ) (the TP -eigenspace). Since by (h1–4), T∅ is a reduced local ring, and by Lemma 3.13, TP is also reduced. Thus by Corollary 2.45, we have the modular deformation ρP : G∅ → GL2 (TP ) of ρ. We consider the deformation functor ΦP governed by the conditions (Q1–3) and (Q6) with (Q4) for p ∈ P Σst Σn.ord , with (Q5) for p ∈ Σ0 − P and with (Q7) for p ∈ Σf l (all for Q = ∅). Then ρP ∈ ΦP (TP ). For each prime P|p, we write ΦP,p for the restriction of the functor ΦP to the local Galois group Gp = Gal(F p /Fp ). Let Wm = W/pm W , Wm [] = Wm [X]/(X 2 ) with = (X mod (X)2 ) for an indeterminate X as before and Ad(ρ0 )m = Ad(ρ0 ) ⊗W Wm , where ρ0 is the Galois representation into GL2 (W ) associated to the initial automorphic form f0 . We deﬁne Φ0P,p (Wm []) = {ρ ∈ ΦP,p (Wm [])|ρ mod () = ρ0 mod pm W }.

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Hecke algebras as Galois deformation rings

Then Φ0P,p (Wm []) can be embedded into H 1 (Fp , Ad(ρ0 ) ⊗W Wm ) in a standard manner (see Lemma 1.59 and Proposition 3.87), and we identify ΦP,p (Wm []) with its image. Then the Selmer group with respect to ΦP is deﬁned by SelP (Ad(ρ0 )m ) := Φ0P (Wm []) = {ρ ∈ ΦP (Wm [])|ρ mod () = ρ0 mod f pm }. which is therefore equal to 

 H 1 (Fq , Ad(ρ0 )m ) . Ker H 1 (G∅ , Ad(ρ0 )m ) → ΦP,q (F[]) ρ q∈Σ

We write the universal ring of ΦP as RP with the universal representation ρP : Gal(Q/F ) → GL2 (RP ). Let ρP : Gal(Q/F ) → GL2 (TP ) be the modular deformation. Then the canonical morphism πP : RP → TP satisﬁes πP ◦ ρP ∼ ρP . Let θP : TP → W given by fP |t = θP (t)fP , and we put θ P = θ P ◦ πP . Suppose that p ∈ P . Let P = P − {p}. We write t(p) for the+ image of T (p) in TP . There exists a unique root of unity ζ ∈ W of the form ± ε+ (p ) such that u(p) − ζ is a nonunit in TP (see Theorem 3.16). We write αp ∈ W for the unit root of X 2 − θP (t(p)X + ε+ (p )N (p) = 0. We regard αp as an unramiﬁed character of Gp sending F robp to αp . We deﬁne the companion twin character βp : Gp → W × given by αp βp = ε+,p Np . Lemma 3.37

Suppose that p ∈ P . Let P = P − {p}. Then we have

|SelP (Ad(ρ0 )m )|/|SelP (Ad(ρ0 )m )| ≤ |W/(αp − ζ)W | for all m > 0. Proof

By deﬁnition, we have an exact sequence: r

→ Φ0P,p (Wm [])/Φ0P ,p (Wm [)]), 0 → SelP (Ad(ρ0 )m ) → SelP (Ad(ρ0 )m ) − where r is the restriction to the local cohomology at p. Thus we need to prove |Φ0P,p (Wm [])/Φ0P ,p (Wm [)])| ≤ |W/(αp − ζ)W |. We have a unique W -free subgroup F + Ad(ρ0 ) ⊂ Ad(ρ0 ) of rank 1 such that F − Ad(ρ0 ) = αp−1 βp . Thus if the image of ρ0 (Dp ) is upper triangular, F + Ad(ρ0 ) is made of upper nilpotent matrices. Since we ﬁxed the diagonal character of the deformations (Q4) over the inertia group, we have 1 unr (F , Ad(ρ ) ) H 0 m p Φ0P,p (Wm []) = Ker H 1 (Fp , Ad(ρ0 )m ) → 1 unr + . H (Fp , F Ad(ρ0 )m ) Since ΦP ,p is made up of ﬂat deformations at p, Φ0P ,p (Wm [)]) as a subspace of Ext1Gp (βp , αp ) is brought into Hf1l (Fpunr , Wm [](1)) after restriction to the inertia subgroup Ip . Then 1 unr (F , Ad(ρ ) ) H 0 m p Φ0P ,p (Wm []) = Ker H 1 (Fp , Ad(ρ0 )m ) → 1 unr + . Hf l (Fp , F Ad(ρ0 )m )

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215

Thus we have Φ0P,p (Wm [])/Φ0P ,p (Wm [)])

→

H 1 (Fpunr , αp−1 βp )

Hf1l (Fpunr , αp−1 βp )

⊗W Wm .

Since αp−1 βp = Np on the inertia group, we have, by Kummer’s theory, H 1 (Fpunr , αp−1 βp ) = W ⊗Zp lim Fpunr,× /(Fpunr,× )p , ← − n n

Hf1l (Fpunr , αp−1 βp ) = W ⊗Zp lim Opunr,× /(Opunr,× )p . ←− n

n

Thus we have as Gal(Fpunr /Fp )-modules H 1 (Fpunr , αp−1 βp )

Hf1l (Fpunr , αp−1 βp )

Z ∼ = W ⊗Zp p p ∼ = W (αp−1 βp Np−1 ).

Since the image of the left-hand side lands in the Gal(Fpunr /Fp )-invariant subspace of the right-hand side, we conclude that the left-hand side is killed by 2 αp − ζ. This ﬁnishes the proof. For a ring B and a B-module X, we write AnnB (X) = {b ∈ B|bX = 0} (the annihilator of X). For a W -module X, we deﬁne lengthW X to be the maximal length of a sequence 0 = X0 ⊂ X1 ⊂ · · · ⊂ Xm = X with Xj /Xj−1 ∼ = F = W/mW for j < m. Here we insert without proof a ring theoretic lemma whose proof one can ﬁnd in [Lt]. Lemma 3.38 (Lenstra) Let R be a complete noetherian local W -algebra, T a ﬁnite ﬂat local W -algebra, and ϕ : R → T , π : T → W surjective W -algebra homomorphisms. Then the following three conditions are equivalent: 1. We have

lengthW

and

lengthW

Ker(π ◦ ϕ) Ker(π ◦ ϕ)2

2. We have

Ker(π ◦ ϕ) Ker(π ◦ ϕ)2

<∞

lengthW

≤ lengthW

lengthW and

Ker(π ◦ ϕ) Ker(π ◦ ϕ)2

Ker(π ◦ ϕ) Ker(π ◦ ϕ)2

.

= lengthW

W π(AnnT (Ker(π)))

<∞

W π(AnnT (Ker(π)))

.

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Hecke algebras as Galois deformation rings

3. T is a complete intersection, π(AnnT (Ker(π))) = 0, and ϕ is an isomorphism. Here a local W -algebra T ﬁnite ﬂat over W is a complete intersection if T ∼ = W [[X1 , . . . , Xr ]]/(f1 , . . . , fr ) for the formal power series ring W [[X1 , . . . , Xr ]] of r variables with r-relations fj ∈ W [[X1 , . . . , Xr ]] (see [CRT] Section 21 for complete intersection rings). If π(AnnT (Ker(π))) is principal, each generator of the ideal π(AnnT (Ker(π))) is called the congruence number of π. If the equivalent conditions of the lemma are satisﬁed, HomW (T, W ) ∼ = T as T -modules (that is, T is a Gorenstein ring; see [MFG] 5.3.4), we have a perfect W -bilinear pairing (·, ·) : T ×T → W inducing T ∼ = HomW (T, W ), and for the adjoint π ∗ : W → T of π under this pairing, the ideal π(AnnT (Ker(π))) is generated by π ◦ π ∗ . Further if T is reduced, W/π(AnnT (Ker(π))) ∼ = (T / AnnT (Ker(π))) ⊗T,π W, which is written as C0 (π, W ) in [MFG] 5.3.3 and is called the congruence module of π. This type of module was ﬁrst introduced in the study of congruence of cusp forms in the proof of Theorem 7.1 in [H81b] (see also [H88a]). Exercise 3.39 Give a proof of the following facts quoted above: 1. π(AnnT (Ker(π))) is generated by π ◦ π ∗ under the three equivalent conditions of Lemma 3.38, 2. W/π(AnnT (Ker(π))) ∼ = (T / AnnT (Ker(π))) ⊗T,π W under the reducedness of T and under the three equivalent conditions of Lemma 3.38. Proposition 3.40 Let the notation and the assumption be as above. If P = P {p} and SP is free of rank 1 over TP and RP ∼ = TP , then RP ∼ = TP . If further (sm1) is satisﬁed, SP is free of rank 1 over TP . We shall give here a proof due to K. Fujiwara for RP ∼ = TP and to T. Saito for the freeness of SP (see [Fu] Theorem 8.1.7). Proof By the near p-ordinarity, we have an exact sequence p → ρP δ p for the universal representation ρP : Gal(Q/F ) → GL2 (RP ). Then by the canonical surjection ϕP : RP TP u := δ([p , Fp ]) ∈ RP is sent to u(p). We write ∆ = u−ζ ∈ RP and we write ∆T for the image of ∆ ∈ TP . Let Sel? (Ad(ρ0 )∞ ) = limn Sel? (Ad(ρ0 )n ) and Sel∗? (Ad(ρ0 )) be the Pontryagin dual of Sel? (Ad(ρ0 )∞ ). −→ By the standard argument (see Proposition 3.87), we have Ker(θ P )/ Ker(θ P )2 ∼ = Sel∗P (Ad(ρ0 )∞ ) and by Lemma 3.37, Ker(θ P )/ Ker(θ P )2 ∼ = Ker(θP )/ Ker(θP )2 ∼ = Sel∗P (Ad(ρ0 )∞ ). Then by Lenstra’s criterion (Lemma 3.38) applied to RP TP W , we get RP ∼ = TP by Lemma 3.37 and that TP is a local complete intersection, because ϕP (u(p) − ζ) is the generator of ϕP (AnnTP (Ker(ϕP ))) for ϕP : TP TP by Theorem 3.16.

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217

We now prove the freeness of SP following an argument due to T. Saito. We have already veriﬁed in Theorem 3.16 that the congruence number for SP SP is ϕP (u(p) − ζ). In other words, i∗W ◦ iW : SP → SP is equal to multiplication by ϕP (u(p) − ζ). We have a projection πT : TP TP . Since these rings are complete intersection (in particular Gorenstein), we have the dual map πT∗ : TP TP . Then πT ◦ πT∗ has to be ϕP (u(p) − ζ) because RP ∼ = TP and d ∼ a base e of S and identify S . Lift e RP ∼ T = TP . Choose = P P i i to SP and P consider T = i TP ei . By the self-duality of SP , E = (ei , ej )i,j is a unimodular matrix. For E := (ei , ej )i,j , E mod mTP = E mod mTP , and hence E is a unimodular matrix. Thus ·, · induces the self-duality on T ⊂ SP . Since T ⊗Zp Qp = SP ⊗Zp Qp , we conclude T = SP , which is free of rank d over TP by Nakayama’s lemma. In our case, we have d = 1 by Lemma 3.6. 2 Exercise 3.41 Prove that ϕP (u(p) − ζ) is the congruence number of ϕP : TP → TP under the notation of the above proof. By induction on |P |, we get the following fact from the above lemma combined with Theorem 3.28: Corollary 3.42 Let the notation and the assumption be as above. Assume the condition (sm1). If f0 satisﬁes (h1–4), then SP is free of rank one over TP for all subsets P ⊂ Σ0 . 3.2.7 Hilbert modular basis problems As an application of Theorem 3.28, we study the integral basis problem for Hilbert modular forms. We write d(F ) for the discriminant of F/Q. Let us recall the deﬁnition of the adelic Hilbert modular forms of level 1 and of weight 2 in the traditional sense, specializing a more general deﬁnition given in 2.3.2. In our terminology of 2.3.2, it is of level N = 1 with the trivial character ε and of weight κ = (0, I). We use the same notation introduced in 2.3.2. We × write Z for the center of the algebraic group GD = ResO/Z OD/O for a quaternion algebra D and its ﬁxed maximal order as in Chapter 2. We use the same symbol Z for the center for any choice of D, because the center is canonically isomorphic to ResO/Z Gm for any choice of D. When D = M2 (F ), we choose OD = M2 (O) and write G for GD . The automorphy factor J(g, z) of weight 2 is given by det(gσ )−1 j(gσ , zσ )2 (3.2.9) J(g, z) = σ∈I

for ) ∈ G(R) = GL2 (R)I and z = (zσ ) ∈ HI . Here we put g = (gσ a b j , z = cz + d for z ∈ C. Recall the open subgroup S0 (N) of c d

δ 0 (∞) −1 in (2.3.3). Then we deﬁne G(A ) given by η Γ0 (N)η for η = 0 1

218

Hecke algebras as Galois deformation rings

S(C) = S(0,I) (1, 1; C) with the trivial character 1 to be the space of functions f : G(A) → C satisfying the following conditions (see (SA1–3) in 2.3.2): (A1) We have the following automorphy f (αxuz) = f (x)J(u∞ , i)−1 for all α√∈ G(Q), √ z ∈ Z(A), and u ∈ S0 (1)Ci for the stabilizer Ci in G(R)+ of i = ( −1, . . . , −1) ∈ Z = HI . (A2) Choosing u ∈ G(R)+ with u(i) = z for each z ∈ HI , deﬁne a function fg : HI → C by fg (z) = f (gu∞ )J(u∞ , i) for each g ∈ G(A(∞) ). Then fg is a holomorphic function on HI for all g. (A3) fg (z) is exponentially decreasing as Im(z) → ∞ for each g ∈ G(A) with g∞ = 1. Each member f of S(C) has a Fourier expansion of the following form (Proposition 2.26),

y x f = |y|A a∞ (ξy, f )eF (iξy∞ )eF (ξx). (3.2.10) 0 1 0ξ∈F

Here y → a∞ (y, f ) is a function deﬁned on y ∈ FA× only depending on its ﬁnite part y (∞) , and a∞ = ap because κ1 = 0. The function a∞ (y, f ) is supported by × F∞ ) ∩ F × of integral ideles. By (A1), f ∈ S(C) is invariant under the set (O A and therefore, a∞ (y, f ) only f (x) → f (xu) for a diagonal element u ∈ G(Z), depends on the ideal yO = y O ∩ F . In this sense, for a fractional ideal n, we take y ∈ FA× with n = yO and put a∞ (n, f ) = a∞ (y, f ). Then we have the following formula for the standard Hecke operator T (n) analogous to (2.3.25): # mn $

a∞ (m, f |T (n)) = N (d) · a∞ , f (⇒ a∞ (O, f |T (n)) = a∞ (n, f )), a2 a⊃m+n (3.2.11) where a runs over common divisors of the ideals m and n. As explained in 2.3.3, the association A → S(A) = {f ∈ S(C)|a∞ (y, f ) ∈ A} gives rise to a well deﬁned integral structure of S(C). For any ring (or even any module) A (not necessarily in C), S(A) = S(Z) ⊗Z A is therefore well deﬁned. Each element f ∈ S(A) has its q-expansion coeﬃcients a∞ (y, f ) ∈ A. By (3.2.11), the Hecke operators T (n) acts on S(A) preserving the integral structure. We write H(A) ⊂ EndA (S(A)) for the A-subalgebra generated by T (n) for all integral ideals n. Theorem 3.43 (Duality) A-bilinear pairing

For a commutative ring A with identity, deﬁne an

( , ) : H(A) × S(A) → A by (h, f ) = a∞ (1, f |h) = ap (1, f |h).

Galois deformation

219

Then ( , ) gives rise to isomorphisms HomA (S(A), A) ∼ = H(A) and HomA (H(A), A) ∼ = S(A), and the latter isomorphism is given by φ → f (φ) with a∞ (y, f (φ)) = φ(T (yO)) for all integral idele y ∈ FA× . The proof of this theorem is the same as that for Theorem 2.28. We thus have Corollary 3.44 Let H = H(A). Let V and V be H-modules free of ﬁnite rank over A with an A-bilinear pairing , : V × V → A. Deﬁne a formal q-expansion Θ(v ⊗ v ) by a∞ (y, Θ(v ⊗ v )) = v|T (yO), v for integral ideles y. Then Θ gives an H-linear map of V ⊗A V into S(A) regarding V ⊗A V as an H-module through V . If V is H-free of rank 1, HomA (V, A) ∼ = V by , and hv, v = v, hv for h ∈ H, then Θ induces an isomorphism V ⊗H V ∼ = S(A). We take a division quaternion algebra D over F unramiﬁed at every ﬁnite D with M2 (O). Let H/R place. We ﬁx a maximal order OD of D and identify O be the Hamilton quaternion algebra. Then D∞ = D ⊗Q R is isomorphic to the product of r copies of M2 (R) and d − r copies of H for some r ≡ d mod 2 (see (H1,2) in 2.1.1). For an open compact subgroup U ⊂ GD (A(∞) ), we consider the automorphic manifold Y (U ) = GD (Q)\GD (A)/U · Z(A)C ∼ = GD (Q)+ \GD (A)+ /U · Z(A)C+ and the class set Cl(U ) = GD (Q)\GD (A)/U · Z(A)GD (R)+ , where GD (R)+ is the identity connected component of GD (R), GD (A)+ = G(A(∞) ) × GD (R)+ , GD (Q)+ = GD (A)+ ∩ GD (Q), and C (resp. C+ ) is a maximal compact subgroup of GD (R) (resp. GD (R)+ )). Note that Cl(U ) = Y (U ) if r = 0. We write simply × ). Then by the approximation theorem (Theorem 2.8), Cl(U ) is a Cl for Cl(O D ﬁnite set. As is well known (Exercise 2.22), Y (U ) is a compact complex analytic space of dimension r, and if U is suﬃciently small, Y (U ) is a smooth compact complex manifold. To guarantee the W -freeness of H r (Y (U ), W ), we assume (dm)

r ≤ 1,

though our argument works well as long as we have W -freeness of the cohomology groups H r (Y (U ), W ) for all U appearing in this situation (which has been veriﬁed for Hilbert modular varieties by [G2] for quadratic F and by Dimitrov [Dim] for more general Hilbert modular varieties under some restrictive assumptions). First suppose r = 1. Choosing a complete representative set inside ﬁnite ideles of D for Cl(U ) (and writing it again as Cl(U ) by abusing the notation), we have the identity of the Betti cohomology and the group cohomology: ; H 1 (Y (U ), A) = H 1 (Γa (U ), A) a∈Cl(U )

220

Hecke algebras as Galois deformation rings

× ) ∩ GD (Q)+ . Here for Γa (U ) = Γa (U )/(Γa (U ) ∩ Z(Q)) with Γa (U ) = (aU a−1 D∞ 1 H (Γa (U ), A) is the group cohomology for the Γa (U )-module A with the trivial action. Taking a suﬃciently small U so that Y (U ) is smooth, we deﬁne

× , H1 (Y (U ), A)) (the coinvariant under the action of O × ), S(A) = H0 (O D D × , H 1 (Y (U ), A)) (the invariant under the action of O × ). S ∗ (A) = H 0 (O D D The module S(A) has the Poincar´e duality pairing ·, · : S(A) × S ∗ (A) → A, which is a perfect alternating A-duality pairing (as long as S(A) is A-free and and W , writing A∗ is canonically isomorphic to S ∗ (A); see below). If A = Z for the Pontryagin dual module of A, the above pairing ·, · induces a perfect Pontryagin duality between S ∗ (A∗ ) and S(A) (here as before W is a valuation ring ﬁnite ﬂat over Z ). By the Hochschild–Serre spectral sequence applied to H 1 (Γa (U ), W ∗ ) combined with the Poincar´e duality as above, if is prime to 6d(F ), we have ; ; × ), W ) ∼ × ), W ) ∼ S(W ) ∼ H 1 (Γa (O H1 (Γa (O (3.2.12) = = = S ∗ (W ), D D a

a

which is independent of the choice of U and is W -free of ﬁnite rank. Indeed, if H 1 (Γa (U ), W ) is Self-W -dual under the cup product, we have S(W ) ∼ = S ∗ (W ) m canonically. The self-duality follows from the W -freeness of H (Γa (U ), W ) if m ≥ 2 (⇔ H 2 (Γa (U ), W ) ∼ = W and H m (Γa (U ), W ) = 0 if m > 2), which in turn follows if Γa (U ) is -torsion free (because then Γa (U ) is isomorphic to the topological fundamental group of the curve inside Y (U ) which is a K(Γ, 1)manifold; see [CGP] Section 10, in particular, Examples on page 222). × ) (and hence Γa (U )) is -torsion free if is Lemma 3.45 The group Γa (O D prime to 6d(F ). Proof Let GD 1 = Ker(N : G → T ) for the reduced norm map N . Let Γ = × ), and put Γ1 = Γ ∩ GD (Q), Γ1 = Γ1 /(Z(Q) ∩ Γ1 ) and Γ = Γ/(Z(Q) ∩ Γ). Γa (O 1 D Then we have [Γ : Γ1 ] is a 2-power. Therefore, we need to prove that Γ1 is torsion free if is prime to 6d(F ). If ζ = 1 for ζ ∈ Γ1 , we have F [ζ] ⊂ D is at most a quadratic extension of F . If > 3, F contains the maximal real subﬁeld of Q[ζ], in which ramiﬁes; so, |d(F ). 2 Exercise 3.46 Prove that |Γ/Γ1 | = 2s for an integer s (under the notation in the proof of Lemma 3.45). Similarly to the proof of Lemma 3.45, we ﬁnd that the order of the torsion part of ∗ ) is supported by primes q which gives the torsion of Γa (O × ); S ∗ (Q/Z) = S ∗ (Z D so, S(W ) is W -free if is prime to 6d(F ). Hereafter, assuming 6d(F ), we identify S(W ) and S ∗ (W ).

Galois deformation

221

By the Jacquet–Langlands correspondence (Theorem 2.30) combined with the Eichler–Shimura isomorphism (e.g., [PAF] Theorem 4.36), S(A) is naturally a module over H(A) and so f |h, g = f, g|h for f, g ∈ S(A) and h ∈ H(A). Identifying D∞ with M2 (R) × Hd−1 , we can let GD (R)+ act on H by linear fractional transformation through the ﬁrst factor M2 (R). Under this identiﬁcation, we have GD (R)+ /C+ Z(R) ∼ = H. On the other hand, for the maximal compact subgroup C of GD (R) containing C+ , C+ is a normal subgroup of index 2 inside C, and GD (R)/C · Z(R) ∼ = GD (R)+ /C+ Z(R) ∼ = H. Thus D D + C/C+ = G (R)/G (R) acts on H, and its action is basically complex conjugation (given by z → −z if we normalize the embedding D → M2 (R) suitably). × ) and hence on S(A). Since 2 is invertible in A, we have Then C/C+ acts on Y (O D S(A) = S + (A) ⊕ S − (A) for the ± eigenspace S ± (A) of C/C+ , and the Poincare duality induces a perfect pairing ·, · : S + (A) × S − (A) → A. Since the action of C/C+ commutes with Hecke operators (Proposition 2.32), S ± (A) is a module over H(A). If A is a Q-algebra, by the same argument proving Lemma 3.6, S ± (A) is free of rank 1 over H(A). 4 (∞) ), and Now suppose r = 0. Decompose GD (A(∞) ) = i GD (Q)ai GD (Z)Z(A −1 × × write Ri = D ∩ ai OD ai and ei = |Ri /O | (which is a ﬁnite number). If 6d(F ) is invertible in A, ei for all i is invertible in A. If we take a Haar measure dx on GD (A)/Z(A) so that GD (Z)/Z( dx = 1 and the standard Haar measure on the Z) D discrete subgroup G (Q)/Z(Q) ⊂ GD (A)/Z(A), we have the quotient measure on Cl still denoted by dx. Deﬁne a pairing ·, · : H 0 (Cl, A) × H 0 (Cl, A) → A by f, g =

f (x)g(x)dx = Cl

1 f (ai )g(ai ) ei i

for f, g ∈ H 0 (Cl, A). This pairing is the Poincar´e duality on Cl and is perfect as long as 6d(F ) is invertible in A. Let ClF be the strict class group of F . Then the reduced norm map induces a map N : Cl → ClF /ClF2 , which is surjective. The space H 0 (ClF /ClF2 , A) of functions on ClF can be embedded into H 0 (Cl, A) by the pullback of N . We then deﬁne S(A) ⊂ H 0 (Cl, A) by the orthogonal complement of H 0 (ClF /ClF2 , A) ⊂ H 0 (Cl, A). The pairing ·, · on S(A) remains perfect. Again, by the Jacquet– Langlands correspondence, S(A) has a natural right action of H(A) written as f → f |h for f ∈ S(A) and h ∈ H(A), and we have f |h, g = f, g|h for all f, g ∈ S(A) and h ∈ H(A) (see Lemma 3.5). If A is a Q-algebra, by Lemma 3.6, S(A) is free of rank 1 over H(A). We then deﬁne Θ(v ⊗ v ) ∈ S(A) by a∞ (y, Θ(v ⊗ v )) = v|T (yO), v for v ⊗ v in S(A) ⊗H(A) S(A) when r = 0 and in S + (A) ⊗H(A) S − (A) when r = 1. As shown in (2.5.2) when r = 0 and F = Q, Θ(f ⊗ g) is the classical theta series of the deﬁnite quaternion algebra D/F . In the indeﬁnite case, by the analytic computation in [Sh6] II, Theorem 3.1, Θ(f ⊗ g) is the integral against Siegel’s indeﬁnite theta series (over the Shimura variety of the orthogonal group of the

222

Hecke algebras as Galois deformation rings

indeﬁnite norm form of D/F ; see 2.5.2) of f ⊗ g regarded as an automorphic form on the orthogonal similitude group isogenous to GD × GD (see [H05d] (7.9)). Theorem 3.47 Let the notationand the assumption be as above, and deﬁne a positive integer E by E = 6d(F ) ψ (the numerator of L(−1, ψ 2 )), where d(F ) is the discriminant of F and ψ runs over all unramiﬁed characters of Gal(F /F ) with values in C× . Then, for any Z[ E1 ]-algebra A, S(A) when r = 0 and S ± (A) when r = 1 are free of rank 1 over H(A), and Θ gives rise to an isomorphism of H(A)-modules: S(A) ⊗H(A) S(A) if r = 0, S(A) ∼ = + − S (A) ⊗H(A) S (A) if r = 1, and H(Z[ E1 ]) is a local complete intersection. This theorem solves the integral basis problem over Z[ E1 ] for S(Z[ E1 ]). We only complete the proof when r = 0 (though up until the point we use Theorem 3.28, we do not assume that r = 0). Indeed, we only proved Theorem 3.28 under this assumption. The general case follows from [Fu] Theorems 6.1.1–2 which gives the result stated in Theorem 3.28 for odd-degree base ﬁelds (replacing Mφ∗ in the theorem by the eigen-component of S ± (W ) for each local ring of Tφ ). We do not make the argument completely self-contained; so, we quote some results not in this book from research papers already published. Proof The proof is the same as the one given for Theorem 2.79, following Theorem 3.28 instead of [Wi2]. We shall give a sketch of the proof giving the key points of the arguments. We use the same notation as in the proof of Theorem 2.79. Thus the Galois representations in this proof (only) are -adic (contrary to the usage in other sections in this chapter). For the maximal ideal m of T, we put ρ = ρm for ρm in Theorem 2.43, and call T Eisenstein if ρ is not absolutely irreducible. In this proof only, we write k for the residue ﬁeld T/mT (so, ρ has values in GL2 (k)). Let W be a suﬃciently large valuation ring ﬁnite ﬂat over Z for primes E. We take a non-Eisenstein local component T of H(W ). By Corollary 2.45, we have a Galois representation ρT : Gal(F /F ) → GL2 (T) for an algebraic closure F of F , which is unramiﬁed outside primes of F over and characterized by the fact that ρT (F robq ) for primes of q outside is given by t(q) (the image of T (q) in T). The determinant character det(ρT ) is given by the -adic cyclotomic character N . By the solution of Iwasawa’s conjecture by Wiles [Wi1], if we have an Eisenstein component T, we claim that is irregular (with respect to F ) if 2d(F ). Thus if E, T is not Eisenstein. Here is the proof of the claim. Write ρ for ρmT in Theorem 2.43. Note that by deﬁnition, ρ is semisimple. Then ρ = ψ ⊕ ϕ for two characters ψ, ϕ : Gal(F /F ) → k × for k = Fs unramiﬁed outside . Write ψ (resp. ϕ) for the Teichim¨ uller lift of ψ (resp. ϕ). If [F : Q] is odd,

Galois deformation

223

we can ﬁnd an abelian variety factor A/F of the Jacobian of the level 1 Shimura × ) associated to D with real multiplication by the integer ring curve Y = Y (O D OE of a totally real ﬁeld E such that the semisimpliﬁcation of A[L] is isomorphic to ρ for a prime ideal L| of OE , since S(W ) is isomorphic to the cotangent space H1 (Y, W ) = H1 (J, W ) as modules over the Hecke algebra for the Jacobian J of Y and A is basically a factor corresponding to a Hecke eigenform f0 ∈ S(W ) on which T acts nontrivially (see [H81a] Theorems 4.12). (p) Suppose that [F : Q] is even. Let ρ0 : GF → GL2 (W ) be the modular Galois representation associated to a Hecke eigenform f0 ∈ S(W ) with (ρ0 mod mW )ss = ρ. Take a prime O-ideal q outside such that ρ(F robq ) = 1 by Chebotarev density. Then X 2 − t(q)X + N (q) ≡ X 2 − 2X + 1 mod mT , and by an argument similar to the proof of Theorem 3.16 (applied to q not p there), we can ﬁnd a local component Tq of the Hecke algebra h(q, 1; W ) such that • Tq /(u(q) − 1)Tq is nontrivial acting on new forms in S(q, 1; W ) and • the subalgebra in Tq generated by t(l) ∈ Tq for primes l outside q is

isomorphic to T. Therefore, by the Jacquet–Langlands correspondence (from D to M2 (F )), there exists a Hilbert modular Hecke eigen new form f1 ∈ S new (q, 1; W ) whose Galois (qp) representation ρ1 : GF → GL2 (W ) satisﬁes (ρ1 mod mTq )ss ∼ = ρ. See [Ta] for more details of+ this level-raising argument, and the eigenvalue of U (q) for forms new at q is ± ε+ (q) = ±1 (so, the requirement that X 2 − t(q)X + N (q) ≡ X 2 − 2X + 1 mod mT is necessary, and the suﬃciency is the nontrivial point in [Ta], which can be proven similarly to the proof of Theorem 3.16). Now we take one more quaternion algebra B/F with r = 1 (that is, B∞ ∼ = M2 (R) × Hd−1 ) exactly ramifying at q and unramiﬁed at all other ﬁnite places. By Corollary 2.31, f1 comes from a Hecke eigenform fB on GB (A) under the Jacquet–Langlands correspondence from B to M2 (F ), because f1 is new at q = d(B). The Shimura curve Y0B (q)/F has a Jacobian variety J, and J has a factor A as in the case of F with odd degree (whose cotangent space contains the cotangent vector associated to fB ) such that (A[L])ss ∼ = ρ. In conclusion, by the level-raising argument in [Ta], we can again ﬁnd the abelian variety A/F as above. By Carayol [C], the abelian variety A has good reduction at all places l of F above . Thus ψ (resp. ϕ) is associated to a ﬁnite locally free commutative group scheme Gψ (resp. Gϕ ) of rank s over Ol whose generic ﬁber is a k-vector space of dimension 1 (this fact also follows from Corollary 2.13 in [Dim]). Then assuming that is prime to 2d(F ) and writing κ for the composite of O/l = Fn and k = Fs , by Proposition 3.48 (following this proof), ψ([u, Fl ]) = Nκ/k (u)−ν for ν = 0, 1, where [u, Fl ] is the local Artin symbol of u ∈ Ol× and u = u mod l. Since ψϕ is the -adic Teichm¨ uller character ω, we may assume Gψ ∼ = Z/Z and ∼ Gϕ = µ . In particular, the component T is -ordinary. Thus ψ is unramiﬁed everywhere, and ψϕ = ω (see [Dim] 3.1 for an alternative argument showing the unramiﬁedness without using Proposition 3.48). This is exactly the case which

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Wiles studied in [Wi1], and the order of the ψϕ−1 -part of the class group of H(µ ) for the (strict) Hilbert class ﬁeld H/F is divisible by . Since ψϕ = ω, we have ψϕ−1 = ψ 2 ω −1 , and by the F -version of Kummer’s criterion (which follows from [Wi1]), then divides the numerator of L(−1, ψ 2 ). Thus |E, and is irregular if is prime to 2d(F ) and T is Eisenstein. As pointed out by Dimitrov, there is an alternative geometric argument to show the divisibility of the numerator of L(−1, ψ 2 ) by (for Eisenstein primes ) without recourse to Wiles’ theorem. Here is a sketch of the argument of Dimitrov. The representation ρss is unramiﬁed outside and crystalline at primes dividing of weights 0 and 1 (by Breuil and Fontaine and Laﬀaille; [Dim] Proposition 2.12). Therefore one of the two characters ψ and ϕ, say ψ as before, is unramiﬁed (Corollary 2.13 in [Dim]). Then the Eisenstein series associated to ψ and ϕ is congruent modulo m to a cusp form (on which T acts nontrivially) in the sense that they have congruent Hecke eigenvalues. By the q-expansion principle and the Andreatta–Goren determination (see [AG] Theorem 7.22) of the kernel of the q-expansion on the graded ring of all Hilbert Modular forms, one deduces that divides also the constant term of the Eisenstein series (which implies |the numerator of L(−1, ψ 2 )). Hereafter we assume that T is not Eisenstein. Let ρq be the restriction of ρ to the decomposition group Dq in Gal(F /F ) at a prime q. As in 1.3.4 (and [Wi2]), we can classify the local behavior of ρ into the following types: let l be a prime factor of in O.

l ∗ ∼ (Selmer) ρl = for characters and unramiﬁed δ of Dl and ρl is 0 δl associated to a ﬁnite ﬂat group scheme over Ol ; (Flat) ρl is irreducible and is associated to a ﬁnite ﬂat group scheme over Ol . Under the assumption d(F ), l is ramiﬁed at l; so, δ l = l . A Galois representation ρ : Gal(F /F ) → GL2 (A) for an artinian local W -algebra A with maximal ideal mA is called a -minimal deformation of ρ if the following conditions are satisﬁed: • • • •

ρ mod mA is isomorphic to ρ; ρ is unramiﬁed outside ; ρ is ﬂat or Selmer-ﬂat (i.e., nearly l-ordinary and ﬂat) at prime factors l|; det(ρ) = ι ◦ N for the W -algebra structure morphism ι : W → A.

We call ρ ﬂat at l if ρ is associated to

a ﬁnite ﬂat group scheme over Ol . We call l ∗ ∼ in GL2 (A) for characters l ≡ l mod mA ρ Selmer at l if ρl = ρ|Dl = 0 δl and unramiﬁed δl ≡ δ l mod mA . Imposing the unramiﬁedness outside and the ﬂatness or the Selmer-ﬂat condition at each prime factor l of accordingly as ρ is ﬂat at l or Selmer at l, we have the universal p-proﬁnite local ring R and a universal -minimal deformation ρ : Gal(F /F ) → GL2 (R) (this ring is Rφ

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225

∼ T by reﬁning Wiles’ for ρ in Theorem 3.28). Fujiwara proved the identity R = limiting method (taking the starting module M∅ to be T(S(W )) if r = 0 and T(S ± (W )) when r = 1) as we have done in the two cases “Selmer” and “ﬂat” in the proof of Theorem 3.28 if r = 0. His proof yields the freeness of M∅ over T. Since S(C) when r = 0 and S ± (C) when r = 1 are free of rank 1 over H(C) by the Jacquet–Langlands correspondence (Theorem 2.30) combined with the strong multiplicity-one theorem for GL(2) (see [AAG] 5.14 and 10.10), the rank of M∅ over T is equal to 1 (see Lemma 3.6). Take a ﬁnite set Q of prime ideals q outside with N (q) ≡ 1 mod such that ρ(F robq ) has two distinct eigenvalues. Fixing a choice of eigenvalues of ρ(F robq ) for each q ∈ Q, we can deﬁne the covering T. Here, as before, local ring TQ of the Hecke algebra of level Γ(Q) identifying OD with M2 (O), %

& ∗ ∗ Γ0 (Q) = x ∈ GL2 (O) xq ≡ mod qOD,q for all q ∈ Q 0 ∗ %

& × · x ∈ GL2 (O) xq ≡ 1 ∗ mod qOD,q for all q ∈ Q . Γ(Q) =O 0 1 (3.2.13) Then we choose

TQ (H 0 (Y (Γ(Q)), W )) MQ = 1 W )± ) TQ (H (Y (Γ(Q)),

if r = 0, if r = 1,

where H 1 (Y (Γ(Q)), W )± is the ±-eigenspace in H 1 (Y (Γ(Q)), W ) under the action of C/C+ . As in the case of the proof of Theorem 2.79, we need to check the following three points. 1. The deformation problem attached to T is -minimal (that is, either “Selmer-ﬂat” or “ﬂat” at l|). 2. MQ is free of ﬁnite rank over W [∆Q ] and MQ /aQ MQ ∼ = M∅ as TQ -modules, where aQ is the augmentation ideal of W [∆Q ] (condition (5) ∼ in Theorem×3.23), where ∆Q is the -Sylow subgroup of Γ0 (Q)/Γ(Q) = q∈Q (O/q) . √ 3. ρ is irreducible over Gal(F /F [ ∗ ]). This condition is necessary to ﬁnd an inﬁnite sequence of ﬁnite sets Q satisfying the properties ﬁtting well into the Taylor–Wiles system (see Proposition 3.33 and its proof, where the prime p plays the role of here). By the unramiﬁedness of in F/Q, the unique quadratic extension M/F in F (µ ) √ is given by F [ ∗ ] for ∗ = (−1)(−1)/2 . The conditions (aiM ) and (aiF (µ ) ) are equivalent as we have seen in the proof of Theorem 3.25. The condition (1) has been veriﬁed by Taylor. Indeed, by the result of [Ta2] Theorem 1.6, for non-Eisenstein T, ρT and hence ρl fall either in the Selmer-ﬂat

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case or in the ﬂat case. His argument is similar to our argument given above to show ρss is ﬂat in the Eisenstein case. In this non-Eisenstein case, roughly speaking, Taylor ﬁnds, in the same manner as in the Eisenstein case, abelian varieties An with good reduction at whose group of n -torsion points An [n ] has a factor isomorphic to ρT mod pn , and hence ρT is either in the Selmer case or in the ﬂat case. The veriﬁcation of (2) in the case where r = 0 is done in (the proof of) for a Theorem 3.28. Thus we verify (3) for primes 6d(F ). Suppose ρ ∼ = IndF M λ√ × Galois character λ : Gal(F /M ) → Fs for the quadratic extension M = Q[ ∗ ]. Here we take s with s as small as possible. Fix a prime factor l of in F . We write V for the l-adic integer ring of the l-adic completion Ml . Since ρ is ﬂat or Selmer-ﬂat at l, λ gives the action of Gal(M l /Ml ) on a ﬁnite ﬂat group scheme G/Vl which is an Fs vector space of dimension 1. Write n for the order of O/l. Again by the proposition following this proof, if > 2 is unramiﬁed in F/Q, writing m for the GCD of s and n and k for the subﬁeld of κ of order m , λ([u, Ml ]) = Nκ/k (u)−ν for ν = 0, 1, 2 and u = u mod mV , where [u, Ml ] is the local Artin symbol. We have Nκ/k (u)−2ν = λ([u, Ml ])λ([uσ , Ml ]) = det(ρ)([u, Ml ]) = N ([u, Ml ]) = NFn /F (u)−1 for the generator σ ∈ Gal(M/F ). Writing κ0 = Fn ∩ k, we have Nκ/k (u) = NFn /κ0 (u) for all u ∈ Fn , because Fn = V /mV and k is linearly disjoint over κ0 . Then the above identity implies that for all u ∈ F× n , NFn /κ0 (u)−2ν = NFn /F (u)−1 = Nκ0 /F (NFn /κ0 (u))−1 . × × Since NFn /κ0 : F× n → κ0 is surjective, rewriting x ∈ κ0 for NFn /κ0 (u), we × −2ν −1 t have x = Nκ0 /F (x) for all x ∈ κ0 . Let = |κ0 |. Since Nκ0 /F (x) = t−1 x1++···+ , we thus ﬁnd 2ν ≡ 1 + + · · · + t−1 mod t − 1. Since 0 ≤ ν ≤ 2, if ≥ 4 ≥ 2ν, this is impossible. When ν = 2, we could have 2ν = 4 = 1 + 3 for = 3 and t = 2. Thus we have veriﬁed (3) for ≥ 4. See [Dim] Lemma 3.4 for an alternative argument proving (3) for primes diﬀerent from 2k − 1 for the weight k (so in our case, k = 2 and therefore for ≥ 4). 2

The following proposition is based on the classiﬁcation theory of commutative ﬁnite ﬂat group schemes due to Oort and Tate, and to Raynaud, whose proof can be found in [Oh] Proposition 1. Proposition 3.48 Let V be a discrete valuation ring ﬁnite ﬂat over Z with residue ﬁeld Fn and quotient ﬁeld K. Let G be a ﬁnite locally free group scheme of rank s over V on which k = Fs acts by V -endomorphisms. Let m be the GCD of n and s, and regard k as the ﬁnite subﬁeld of κ with m elements. Then the action of Gal(K ab /K) for the maximal abelian extension K ab /K on

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227

the generic ﬁber of G is given by the character ϕ : Gal(K ab /K) → k × satisfying ϕ([u, K]) = Nκ/k (u)−ν ∈ k × (u = u mod mV ) for the local Artin symbol [u, K] s−1 i s − 1 for u ∈ V × , where ν ≥ 0 is an integer satisfying ν m = i=0 ci for −1 integers ci with 0 ≤ ci ≤ e for the ramiﬁcation index e of V /Z . In the circumstances of the proof of Theorem 3.47, this proposition is used in the following two cases: (1) K = Fl with e = 1 and (2) K = Ml with e = 2. In Case (1), the only possibility of ν is 0, 1, and in Case (2), if ≥ 3, the only possibility of ν is 0, 1, 2. Here is (a sketch of) the proof by Ohta of this proposition. Proof Let v : V → Z ∪ {∞} be the valuation normalized so that v() = 1 if generates the maximal ideal m of V . First we deal with the case where × of the ﬁxed ﬁeld s|n. We consider the Teichm¨ uller lift χ : k × = F× q → V inclusion k → κ. Then by [Ry] Corollary 1.5.1 (or [ARG] III.7), G is isomorphic to Spec(V [X1 , . . . , Xs ]/a), where a is the ideal generated by Xi − δi Xi+1 (i ∈ Z/Z, δi ∈ V and v(δi ) ≤ e for all i). The action of λ ∈ k × on the bialgebra i is given by [λ]Xi = χ(λ) Xi . Writing ϕG = ϕ for the character of Gal(K/K) giving the Galois action on the generic ﬁber of G, the splitting ﬁeld of ϕG is the s−1 s−2 · · · δi+s−1 . By splitting ﬁeld of the equations Xiq − ai Xi = 0 for ai = δi δi+1 the explicit formula of the tame norm residue symbol, we ﬁnd that v(t) −v(a0 )

ϕG (t) = Nκ/k ((−1)v(a0 )v(t) a0

t

mod m)

for all t ∈ K × . This shows the assertion if s|n. In general, we put N = ns/m, and take the (unique) unramiﬁed extension K inducing the residual extension κ /κ for κ = FN . Taking the valuation ring V of K with normalized valuation v and maximal ideal m , we apply the above argument to G = G ⊗V V over V . We write a0 ∈ V for the number a0 corresponding to G/V . By local class ﬁeld theory, ϕG (u) = ϕG (t) if u = NK /K (t).

Thus by the ﬁrst step of the proof, we get ϕG (u) = Nκ /k (t mod m )−v (a0 ) . Since ϕG ([u, K]) ∈ k = Fm for all u ∈ V × , we ﬁnd that v(a0 ) is divisible by (s − 1)/(m − 1). Write ν = (v(a0 )(m − 1))/(s − 1) ∈ Z. Since K /K is unramiﬁed, we have ϕG (u) = Nκ /k (t mod m )−ν = Nκ/k (u)−ν as desired.

2

Here is a brief remark on how to deal with the basis problem for higher weight. Here note that k > 2 is even. We write k = (0, (k − 1)I) for the corresponding double digit weight in 2.3.2. Let the Neben character 1 = (ε1 , ε2 , ε+ ) denote as before the identity characters εj (j = 1, 2) and ε+ = | · |2−k A , and consider S k (d(D), 1; C), for the moment assuming d(D) = 1.

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Let us generalize the deﬁnition of the spaces S(A) and S ± (A) to Sk (A) and = O × , decompose S± (A) in this setting. For an open subgroup S of GD (Z) D k 4 h GD (A(∞) ) = j=1 GD (Q)+ aj Z(A(∞) )S as before for GD (Q)+ = GD (A) ∩ (G(A(∞) )×GD (R)+ ) with the identity connected component GD (R)+ of GD (R). If D is deﬁnite, GD (R)+ = GD (R), and therefore GD (Q)+ = GD (Q); so, the decomposition is the same as the one given just above (sm). Consider in G(A(∞) ) and Γj (S) = Γj (S)/(Z(Q) ∩ Γj (S)). Γj (S) = GD (Q)+ ∩ aj Sa−1 j If A splits OD so that OD ⊗Z A → M2 (A)I , we can think of the Γj (S)-module “aj · L( k ∗ ; A)”. To make this point precise, ﬁx an extension M/F (inside Q) with integer ring R so that D ⊗F M ∼ = M2 (M ) and OD → M2 (R) under this identiﬁcation. For simplicity, we assume that M/Q is a Galois extension. Recall I = Homﬁeld (F, Q) = Homﬁeld (F, M ). Then we identify D ⊗Q M = M2 (M )I we can sending ξ ⊗ m to (ξ σ m)σ∈I ∈ M I . For an open subgroup S ⊂ GD (Z), ∗ ∗ think of the Γj (S)-module L(k ; M ) ⊂ L(k ; C) as deﬁned in 2.3.5, and the ∩ L( action of Γj (S) preserves Lj ( k ∗ ; R) k ∗ ; M )) ⊂ L( k ∗ ; M ). For k ∗ ; R) = (aj L( any ﬂat R-algebra A, we deﬁne a Γj (S)-module Lj ( k ∗ ; A) = Lj ( k ∗ ; R)⊗R A. Then D taking a normal subgroup S ⊂ G (Z) suﬃciently small so that Γj (S) is torsion free, we deﬁne Sk (S; A) = j H r (Γj (S), Lj ( k ∗ ; A)) = H r (Y (S), L( k ∗ ; A)). The locally constant sheaf L( k ∗ ; A) on Y (S) can be described as follows. Writing Yj for the connected component of Y (S) corresponding to aj , we note that π1 (Yj ) = Γj (S). Then L( k ∗ ; A)|Yj is the locally constant sheaf associated to ∗ the π1 (Yj )-module Lj (k , A). On Sk (S; A), the Hecke operators T (l ) and the group GD (Z)/S naturally act (see Section 2.4 in the text and [PAF] 4.3.3). We S (S; A)). As before, if r = 1, we deﬁne S ± (A) as the put Sk (A) = H 0 (GD (Z), k k ±-eigenspace of the action of C/C+ ∼ = GD (R)/GD (R)+ . When we deal with a higher parallel weight k > 2, we need to consider the “crystalline” condition in place of the “ﬂat” condition as discussed brieﬂy in 1.3.4. When F = Q with even [F : Q], we ﬁnd in [Ta7] the theorem corresponding to Theorem 3.28 for the crystalline deformation problem for low weights k ≤ − 1 with unramiﬁed in F/Q (when F = Q, this is done in [DiFG]). Thus strictly speaking, the statement we give could be conjectural if [F : Q] is odd. To have the universal crystalline deformation ring, we need to invert also the primes less than the weight as described in 1.3.4 for elliptic modular forms. Also if ≤ k − 2, the k ∗ , M ) is not unique (L( k ∗ , R ) isomorphism class of the GD (Z )-lattice L ⊂ L( ∗ is one choice), and L(k , R ) is not self-dual by the pairing given in (2.4.11) (so, we have trouble in the use of the Poincar´e duality). Thus primes we have to set aside to get a result similar to Theorem 3.47 for weight k ≥ 2 are × • prime factors of 6d(F ) (problem of torsion in OD ); • primes less than k and prime factors of d(F ) (problems of well-deﬁnedness

of the crystalline deformation ring and problems for Poincar´e duality);

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229

• prime factors of the numerator of L(1 − k, ψ 2 ) for an unramiﬁed character

ψ of Gal(F /F ) into C× (Eisenstein primes); • 2k − 1 if 2k − 1 is a prime; (when 2k − 1 is prime, condition (3) in the above proof could fail for = 2k − 1; for example, = 23 for k = 12 and F = Q; see [Dim] Lemma 3.4).

We describe this in a little more detail for k = 2, allowing ramiﬁcation of D at ﬁnite places, in order to present the diﬃculty which arises from adding the ramiﬁcation. For a general quaternion algebra D over a totally real ﬁeld F satisfying (dm) but ramifying at some ﬁnite places, we deﬁne S(A) and S ± (A) in exactly the same manner as above. In this weight 2 case, we do not need to extend scalar to the ﬁnite extension M/F as above, because the Γj (S)-module we deal with has trivial action. Recall d(D) for the product of ramiﬁed primes in D/F and the space of Hilbert modular new forms S2new (d(D), 1; A) on S0 (d(D)) with q-expansion coeﬃcients in A and with the trivial central character 1. Thus S2new (d(D), 1; C) is the space of new forms at d(D) as deﬁned after Theorem 2.30. Since the automorphic representation generated by elements in S2 (d(D), 1; C) is integral over Z, we have S2new (d(D), 1; C) = S2new (d(D), 1; Z) ⊗Z C for S2new (d(D), 1; Z) = S2new (d(D), 1; C) ∩ S2 (d(D), 1; Z). Then for any algebra A, we have S2new (d(D), 1; A) = S2new (d(D), 1; Z) ⊗Z A. As before, we have Θ : S(A) ⊗H(A) S(A) → S2new (d(D), 1; A), where H(A) is the Hecke algebra in EndA (S(A)). In this general case, conjecturally, assuming we have the exact level-lowering result (Mazur’s principle) for ρ (which is not yet known in full generality), we need to avoid the primes satisfying the following condition in addition to the ones excluded already: • prime factors outside of d(D) for which ρ is unramiﬁed; • prime factors > 2 of d(D) for which ρ is ﬂat (so the minimal level of ρ is

a factor d(D)/ under Mazur’s principle); 2 l|d(D) (ψ (l)N (l) − 1) for an everywhere unramiﬁed character ψ of Gal(F /F ) into C× ; strictly speaking, we need to remove prime factors of the numerator of L(d(D)) (−1, ψ 2 ) = (1 − ψ(l)N (l))L(−1, ψ 2 ).

• prime factors of

l|d(D)

Mazur’s principle dictates the minimal possible level of a Hecke eigenform f in S2 (N, 1; C) whose λ-adic Galois representation (for a prime factor λ|) lifts given ρ (as a deformation). Suppose ρ is the reduction modulo λ of the λ-adic Galois representation associated to a Hecke eigenform f in S2 (N, 1; C), and assume that N is square-free. For a prime factor q of N, we should have (conjecturally) a Hecke eigenform f in S2 (N/q, 1; C) for a prime factor q of N if either ρ is ﬂat at the prime factor q| or unramiﬁed at the prime factor q . Taking N = d(D), if

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Hecke algebras as Galois deformation rings

ρ satisﬁes the ﬂatness at q or unramiﬁedness at q as prescribed above, ρ must be associated to a Hecke eigenform of lower level than d(D), which does not exist on the quaternion side. Thus something wrong occurs, and such a prime could cause a trouble in proving freeness S(Z() ) over the Hecke algebra; so, we need to avoid them, until we ﬁnd a new method of analyzing the local Hecke module structure at such primes. 3.2.8 Locally cyclotomic deformation We assume that Σf l = ∅ in the rest of the book. Recall here that Σf l is the subset of Σp where ρ0 is ﬂat nonordinary. We continue to assume (sf) and (h2–3) in 3.2.1 for the ﬁxed Hecke eigenform f0 ∈ S(N, ε; W ) and its level N. We consider a new deformation functor Φcyc : CN LW → SET S deﬁned as follows: Φcyc (A) is the set of isomorphism classes of deformations ρ : Gal(Q/F ) → GL2 (A) satisfying (Q1–3) and (Q6) for Q = ∅ and the following condition milder than (Q4):

∗ (Q4 ) ρ|Dp ∼ p for characters p , δp : Dp → A× such that p |Ip ε−1 2,p and 0 δp ur ur δp |Ip ε−1 1,p factor through Gal(Fp (µp∞ )/Fp ) for all prime ideals p|p. We do not require the condition (Q5). Again under (h2–3), (aiF ), and (dsq ) for prime factors q|N/c(ε− ) where (h4) is not valid, the functor Φcyc is representable by a universal couple (RF , ρcyc ). Let Γp be the p-Sylow subgroup of Gal(Fpur (µp∞ )/Fpur ). Then by the cyclo. Let Γ = tomic character N , we can embed Γp into 1 + pZp ⊂ Z× F p p|p Γp .

∼ p ∗ with δ p ≡ δ p mod mR , the character δ p ε−1 factors Since ρcyc |Dp = F 1,p 0 δp × through Γp and hence, we get a character δ : ΓF → RF given by p δ p ε−1 1,p . This character induces a W [[ΓF ]]-algebra structure on RF . 1 (pr ) with T (Z/pZ)2 by 0 (pr )/Γ Identify Γ 1

a 0 r 2 0 (pr )/Γ 11 (pr ). T (Z/p Z) (a, d) → ∈Γ 0 d ∼ (Op /pr Op )× ; so, the local norm map Note that T (Z/pr Z) = (O/pr O)× = p r r × Np : Op× → Z× p gives rise to Np = p|p Np : T (Z/p Z) → p|p (Z/p Z) for each 0 (pr ) ⊃ Scyc (pr ) ⊃ Γ 1 (pr ) r > 0. Let Scyc (pr ) be a subgroup of G(A(∞) ) with Γ 1 given by 2 11 (pr ) = Ker(N 2 : T (Z/pr Z)2 → (Z/pr Z)× ). Scyc (pr )/Γ p p|p

0 (N). For m > n, we have an inclusion S(Sn , ε; A) → We put Sr = Scyc (pr ) ∩ Γ S(Sm , ε; A) which is compatible with Hecke operators T (l) and U (p) for all primes l p and primes p|p. Thus we have a W -algebra homomorphism h(Sm , ε; W ) → h(Sn , ε; W ) by restricting the Hecke operators on S(Sm , ε; W ) to the subspace

Galois deformation

231

S(Sn , ε; W ). This W -algebra homomorphism is surjective, because it sends the generators T (l) and U (p) to the generators T (l) and U (p). We take the limit lim hn.ord (Sn , ε; W ). hn.ord cyc (N, ε; W [[ΓF ]]) = ← − n

(pn )/Sn ∼ Since Im(Np ) ∼ limn Γ = = ΓF can be identiﬁed with the subgroup of ← − 0 2 n.ord × Im(Np ) , we have a character δh : ΓF → hcyc (N, ε; W [[ΓF ]]) via the action

a 0 for a ∈ ΓF given by of a := aφ(x) = ε(a)−1 φ(x a). In this way, 0 1 hn.ord cyc (N, ε; W [[ΓF ]]) becomes a W [[ΓF ]]-algebra (and the initial Hecke eigenform sits over the augmentation ideal of W [[ΓF ]]). 1 (pn ) ∩ Γ 0 (N), In [H88b] and [H89a] where we take the limit with respect to Γ 1 we exhibited that the Hecke algebra faithfully acts on a similar limit of the 1 (pn ) ∩ Γ 0 (N)). Since the limit of the Betti cohomoBetti cohomology of X(Γ 1 logy groups can be shown to be often free over the corresponding Iwasawa algebra, the faithfulness of the action tells us the torsion-freeness of the Hecke algebra over the Iwasawa algebra. In the following subsection, we will show 0 (N) and ε that hn.ord (N, ε; W [[ΓF ]]) is a torsion-free W [[ΓF ]]under (sm0) for Γ cyc module of ﬁnite type; so, here we admit this fact (see Theorem 3.53). Then hn.ord cyc (N, ε; W [[ΓF ]]) is a p-proﬁnite semilocal ring. Deﬁning a W -algebra homomorphism λ : hn.ord cyc (N, ε; W [[ΓF ]]) → W by f0 |h = λ(h)f0 , we have a unique local factor T = TF of hn.ord cyc (N, ε; W [[ΓF ]]) through which λ factors. We write t(n) (resp. u(p), t(l ), u(l )) for the image of T (n) (resp. U (p), T (l ), U (l )) in T. Proposition 3.49 Suppose (sf) and (h1–4) for f0 ∈ S(N, ε; W ) and (aiF ) for ρ. Then T is reduced, and we have a deformation (ρT : Gal(Q/F ) → GL2 (T)) ∈ Φcyc (T) satisfying the following conditions: Tr(ρT (F robl )) = t(l) for primes l pNc(ε).

(T)

ord Proof Let Tn be the unique local ring of h (Sm , ε; W ) covering T∅ . We write T0 for the local ring of level N ∩ ( p∈Σp p) covering T∅ ; so, T0 = TP for P = Σ0 under the notation of Corollary 3.42. Since u(p) is a unit in T0 , it is a unit in Tn and hence in T. Thus T = limn Tn . Let an = Ker(Tn Tn−1 ) ←− n! and Tn,Q = Tn ⊗Z Q. As is well known (see Lemma

3.51), ep = limn→∞ U (p) 0 (pn+1 )/Γ 0 (pn ) p 0 Γ 0 (pn+1 ) = Γ 0 (pn ) lowers the p-level by the fact that Γ 0 1

p 0 n n Γ0 (p )/Γ0 (p ) given in (2.3.33). Thus Tn,Q is contained in the direct 0 1 sum of direct factors Tψ,Q of hord (Nc(ψ − ), εψ; K) for the subﬁeld K[ψ] ⊂ Qp generated by the values of ψ over K, where εψ = (ε1 ψ1 , ε2 ψ2 , ε+ ) and ψ runs over characters with c(ψ)|pn and ψ+ = 1. We have Tn,Q ⊂ ψ Tψ,Q , where n × ψ = (ψ1 , ψ2 , ψ+ = 1), ψ1 running over characters of (O/p O) (note here the

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information of ψ1 determines all other data of ψ because the central character ε+ is ﬁxed, and therefore ψ2 = ψ1−1 ). The p-part (Nc(ψ − ))p is equal to the conductor c(ψ − ) except possibly for the factor of T0,Q by (h1). By (h4), if N/c(ε− ) has a prime factor q, the Galois representation cannot come from any Hecke eigenform of level N/q; so, the Hecke algebra is new at q. By these facts, Tn is reduced (see Remark 3.11); so, T is reduced. Each simple factor ﬁeld of of Tn,Q gives a prime ideal P of Tn so that the factor is the residue P . By Theorem 2.43, we have the Galois representation ρP . Let ρn = P ρP , where p runs over all prime ideals coming from the simple factors of Tn,Q . We have Tr(ρn (F robl )) = t(l) ∈ Tn for all prime l outside pc(ε), because we have Tr(ρP (F robl )) = (t(l) mod P ) by Theorem 2.43. By the Chebotarev density (and continuity of ρn ), the pseudo-representation πn of ρn has values in Tn and πn+1 ≡ πn mod an . Passing to the limit, for π∞ = limn πn : GSF → T with ←− the ramiﬁcation set S = {|pN (Nc(ε))} ∪ {∞}, (π∞ mod mT ) is the pseudorepresentation of ρ, and by (aiF ) and Proposition 1.23, we get the representation ρT : GSF → GL2 (T) with Tr(ρT (F robl )) = t(l) for all prime ideals l outside S. Then by Proposition 1.25, ρT is unique up to isomorphisms, and ρT mod mT ∼ = ρ. Since ρP satisﬁes the local conditions (Q1–6) and (Q4 ) for εψ and Q = ∅, we 2 verify ρT ∈ Φcyc (T).

∗ for characters δpT : Dp → T× and Tp : Dp → T× with δpT × δpT ≡ δ p mod mT and Tp ≡ p mod mT . Then p|p δpT ε−1 gives 1,p : ΓF → T rise to a W [[ΓF ]]-algebra structure on T, which coincides with the W [[ΓF ]]algebra structure T inherited from the Hecke algebra hn.ord cyc (N, ε; W [[ΓF ]]) (cf. Theorem 2.43 (3) and see also [H89b]). Moreover, we have u(p ) = δpT ([p , Fp ]) for the local Artin symbol [p , Fp ] ∈ Dpab . We have written (κ, ε) (with κ = (0, I)) for the weight and the Neben character of the initial Hecke eigenform f0 . Since we want to move κ in Z[I]2 and ε as variables now, we write (κ0 , ε0 ) for the weight of the initial automorphic form. Thus from now on, κ0 = (0, I), and (κ, ε) is arbitrary. For each locally cyclotomic quadruple ε = (ε1 , ε2 , ε+ , ε− ) of weight κ, we deﬁne a prime ideal Pκ,ε ⊂ W [[ΓF ]] by the ideal generated by (δ p (γ) − γ −κ1,p ε1 (γ))p|p,γ∈Γp . In the following theorem, we do not assume that [F : Q] is even, although we prove the result here assuming that 2|[F : Q]. We will give in 3.3.4 as Proposition 3.71 a proof valid under the assumption that p ≥ 5 in the case of odd degree, reducing the result to the even-degree case by a base-change argument. Write ρT |Dp ∼ =

Tp 0

Theorem 3.50 Let κ0 = (0, I) ∈ Z[I]2 = X(TG ). Suppose (aiF (µp ) ) in addi 0 (N). Then the unique W -algebra tion to (sf), (h1–4), and (sm0) for ε0 and Γ homomorphism π : RF → TF with π ◦ ρcyc ∼ = ρT induces a W [[ΓF ]]-algebra isomorphism RF ∼ = TF , and RF is a reduced complete intersection free of ﬁnite rank over W [[ΓF ]]. For any locally cyclotomic (κ, ε) satisfying

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233

1. κ2 − κ1 ≥ I, −κ −κ and j = 1, 2, 2. zp j εj (z) ≡ zp 0,j ε0,j (z) mod mW for all z ∈ T (Z) 3. ε+ = ε0,+ and εj |T (Z(p) ) = ε0,j |T (Z(p) ) for j = 1, 2, 4. c(εj )|c(ε0,j )p∞ for j = 1, 2, we have a unique local factor Tκ,ε ⊂ hn.ord (N∩c(ε− ), ε; W ) such that we have an κ ∼ ∼ isomorphism RF /Pκ,ε RF = TF /Pκ,ε TF = Tκ,ε induced by π. In particular, we have Tκ0 ,ε0 = T∅ in Theorem 3.28 if Σ0 = ∅ and Tκ0 ,ε0 = TP in Corollary 3.42 for P = Σ0 if Σ0 = ∅. Proof Since RF /Pκ,ε RF is the maximal quotient of RF on which δp induces ε1,p on Γp for all p, RF /Pκ,ε RF is the maximal quotient of RF on which δp induces [z, Fp ] → Np (z)−κ1,p ε1,p (z) (z ∈ Op× ) on the entire Ip for all p. Thus RF /Pκ,ε RF is the universal ring for Φ∅ if the initial Hecke eigemform f0 is in the space Sκ,ε (N ∩ c(ε− ), ε; W ) in place of Sκ0 ,ε0 (N, ε0 ; W ). By the universality of RF /Pκ,ε RF and T = Tκ0 ,ε0 proven in Theorem 3.28, π induces RF /Pκ0 ,ε0 RF ∼ = Tκ0 ,ε0 . Since this map factors through T/Pκ0 ,ε0 T and π is surjective (T is generated by the trace of ρT ), we get two isomorphisms RF /Pκ0 ,ε0 RF ∼ = T/Pκ0 ,ε0 T ∼ = Tκ0 ,ε0 . Writing r for rankW T, by Nakayama’s lemma (Lemma 1.6), we have a surjection πT : W [[ΓF ]]r → T which induces Wr ∼ = T modulo Pκ0 ,ε0 . As we will see in Theorem 3.53, T is W [ΓF ]]-torsion free. If Ker(πT ) is nontrivial, T will have W [[ΓF ]]-torsion, a contradiction. Thus T is W [[ΓF ]]-free. Since we have RF /Pκ0 ,ε0 RF ∼ = T, RF is generated by at most r elements as a W [[ΓF ]]-module. Since RF surjectively covers the W [[ΓF ]]-free module T of rank r, we conclude RF ∼ = T. As we will see in the following subsection, we have a unique local ring Tκ,ε as in the theorem with rankW Tκ,ε ≥ r (see Theorem 3.61 and Corollary 3.63). Since the modular Galois representation of this local ring is a deformation of ρ classiﬁed by RF , we have a W -linear surjection RF /Pκ,ε RF Tκ,ε . Since the left-hand side is generated as a W -module by at most r elements, the W -freeness of the right-hand side with at least r generators tells us that this morphism is an isomorphism. Then we get RF /Pκ,ε RF ∼ = Tκ,ε by RF ∼ = T/Pκ,ε T ∼ = T. As we have already seen, T is ∼ reduced, and hence RF = T is reduced. In the same manner as in the proof of Proposition 1.53(2), we can show by Theorem 3.28 that RF is a local complete 2 intersection over W [[ΓF ]]. 3.2.9 Locally cyclotomic Hecke algebras Fix a locally cyclotomic quadruple ε = (ε1 , ε2 , ε+ , ε− ) with values in W × and of weight κ = (0, I). Here we assume that W is free of ﬁnite rank over Zp . We consider S(Sn , ε; Wm ) for Wm = W/pm W and S(Sn , ε; W∞ ) = limm S(Sn , ε; Wm ). Then we have S(Sn , ε; W ) = limm S(Sn , ε; Wm ). Since the ←− −→ projection Y (Sn ) → Y (Sn ) is surjective for n ≥ n, we have the pullback

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inclusion S(Sn , ε; Wm ) → S(Sn , ε; Wm ) for any m ≥ m and n ≥ n. We have by

a b c d

0 (pn )Z(A(∞) )/Γ 11 (pn )Z(A(∞) ) ∼ Γ = T (Z/pn Z)

→ (a mod pn ) ∈ T (Z/pn Z) (because T 2 /Z ∼ = T ). Since

11 (pn ) = Ker(Np2 : T (Z/pn Z)2 → Scyc (pn )/Γ

2 (Z/pn Z)× ),

p|p

0 (pn )Z(A(∞) ) Γ ∼ = Im(T (Z/pn Z) → p|p (Z/pn Z)× ) =: Gn . Put G∞ = n )Z(A(∞) ) Scyc (p Im(T (Zp ) → p|p (Zp )× ) = limn Gn . Then the p-primary part Γn of Gn is a ←− ﬁnite quotient of ΓF , and we have ΓF = limn Γn . From this, a ∈ Gn acts on ←−

a 0 D n a := with Y (Sn ) and Y0 (p N) = Y (Sn )/Gn through the action of 0 1 a ∈ T (Zp ) lifting a. Thus the group Gn acts on M(Sn , ε; Wm ), S(Sn , ε; Wm ) and S n.ord (Sn , ε; Wm ) by φ(x) → ε( a)−1 φ(x a), because the action of diagonal matrices commutes with U (p ). Since the projector e kills Iv(Sn , ε; Wm ), we may identify we have

S n.ord (Sn , ε; Wm ) = Mn.ord (Sn , ε; Wm ). Let H be a closed subgroup of G∞ . Write Hn for the image of H in Gn . Then we have an open subgroup SH,n ⊂ Sn such that 0 (N) ∩ Γ 11 (pn ))Z(A(∞) ) ∼ SH,n Z(A(∞) )/(Γ = Hn ⊂ Gn . Replacing pn in the above deﬁnition by p∈Σp p, we deﬁne SH,0 (so, SH,0 = SH,1 0 (N)p . if p is unramiﬁed in F/Q). Write εp for the restriction of ε to the p-factor Γ m Since (εp mod p W ) for a given m factors through Gn for a suﬃciently large n , by deﬁnition, we have H 0 (H, M(Sn , ε; Wm )) = M(SH,n , ε; Wm ). Since k k

p 0 p 0 Sn Sn /Sn = SH,n SH,n /SH,n 0 1 0 1 for k ≥ n − n by the explicit decomposition (2.3.33) for the Iwahori subgroups Sn and SH,n , we ﬁnd that U (pn ) brings the entire space M(Sn , ε; Wm ) into H 0 (H, M(Sn , ε; Wm )) = M(SH,n , ε; Wm ) as long as (εp mod pm W ) factors through Gn . Thus we ﬁnd Lemma 3.51

If (εp mod pm W ) factors through Gn , for any n ≥ n, we have H 0 (H, S n.ord (Sn , ε; Wm )) = S n.ord (SH,n , ε; Wm ).

We consider Γx (Sn ) for x ∈ G(A(∞) ). By Exercise 3.1, supposing the condition (sm0), S n.ord (Sn , ε; W∞ ) is p-divisible. Giving it the discrete topology, its

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235

∗ Pontryagin dual Sn.ord (Sn , ε; W ) is W -free of ﬁnite type. By the dual version ∗ ∗ (Sn , ε; W )) = Sn.ord (SH,n , ε; W ) of Lemma 3.51, we get for n ≥ n H0 (H, Sn.ord which is W -free of ﬁnite rank, if εp factors through Gn . Taking H to be a closed subgroup of ΓF , we ﬁnd by Lemma 1.6 that the W [[ΓF /H]]-module ∗ Sn.ord (Sn , ε; W ) is of ﬁnite type, and its minimal set of generators has r elements for r := dimF S n.ord (SΓF ,0 , ε; F), because (ε|ΓF mod mW ) is trivial. Now twisting the action by any character ψ of ΓF /H factoring through Gn (so, the γ ) for γ ∈ ΓF ), we ﬁnd twisted action is given by γφ(x) = ψ(γ)−1 ε1 (γ)−1 φ(x that ∗ ∗ (Sn , ε; W [ψ])) ∼ (Sn , εψ ; W [ψ])), H0 (ΓF /H, Sn.ord = Sn.ord

where εψ = (ε1 ψ, ε2 ψ −1 , ε+ , ε− ψ 2 ) and W [ψ] is the subring of Qp generated over W by the values of ψ. We consider the p-torsion part ∆n of the inter 0 (Npn )x−1 Z(A(∞) ) ∩ G(Q)) for x ∈ G(A(p∞) ). Under (sm1), ∆n is section (xΓ " 2 trivial for all

n. Let ∆∞ = n ∆n . Then ζ ∈ ∆∞ is embedded into T (Zp ) by a b ζ =

→ (ap , dp ) ∈ T (Zp )2 . Since the norm of a p-power root of unity 0 d " (for odd p) over Qp is equal to 1, we ﬁnd that ∆∞ ⊂ x · S∞ x−1 = n x · Sn x−1 . Thus for any character ψ of ΓF /H factoring through on Gn , if n is suﬃciently large, εψ |Γj = ε|Γj . Here Γj is nothing to do with Γn and is deﬁned in (sm0). In other words, if (sm0) is satisﬁed for ε, it is satisﬁed by εψ . Thus assum∗ (Npn , εψ ; W [ψ]) ∼ ing (sm0) for the initial character ε = ε0 , we have Sn.ord = n.ord n n.ord n (Np , εψ ; W [ψ]) in 3.1.3, and S (Np , εψ ; W [ψ])) by the self-duality of S we ﬁnd from Lemma 3.27 Lemma 3.52 Let H be a closed subgroup of ΓF . Assume (sm0) for the initial character ε0 . If n is suﬃciently large, as W [Γn /Hn ]-modules and also as ∗ (Sn , ε; W ) are isomorphic hn.ord (Sn , ε; W )-modules, S n.ord (Sn , ε; W )) and Sn.ord each other. Moreover these two modules are W [Γn /Hn ]-free of rank r, where r is given by dimF S n.ord (SΓF ,0 , ε; F). Passing to the projective limit, we get 0 (N) Theorem 3.53 Let H be a closed subgroup of ΓF . Suppose (sm0) for Γ ∗ ∗ and ε. Then Sn.ord (SH,∞ , ε; W ) = limn Sn.ord (SH,n , ε; W ) is W [[ΓF /H]]-free of ←− ﬁnite rank r, where r is given by dimF S n.ord (SΓF ,0 , ε; F). ∗ Since hn.ord cyc (N, ε; W [[ΓF ]]) acts faithfully on Sn.ord (S∞ , ε; W ), taking H = {1}, n.ord W [[ΓF ]]we ﬁnd that the Hecke algebra hcyc (N, ε; W [[ΓF ]]) is a torsion-free module of ﬁnite type. More generally, for a subset Σ ⊂ Σp , deﬁne ΓΣ = p∈Σ Γp , and take H = ΓΣp −Σ . Then ΓF /H = ΓΣ . Deﬁning hn.ord cyc (N, ε; W [[ΓΣ ]]) to be the ∗ (SH,∞ , ε; W )) for H = ΓΣp −Σ generated W [[ΓΣ ]]-subalgebra of EndW [[ΓF ]] (Sn.ord by T (y) and U (y) forintegral ideles y, we get

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Hecke algebras as Galois deformation rings

Corollary 3.54 Let the notation be as above. Then hn.ord cyc (N, ε; W [[ΓΣ ]]) is a torsion-free W [[ΓΣ ]]-module of ﬁnite type for each subset Σ of Σp . Since S n.ord (Npn , εψ ; K[ψ]) for the ﬁeld of fractions K[ψ] of W [ψ] is free of rank 1 over hn.ord (Npn , εψ ; K[ψ]), for the prime P = Pκ,εψ of W [[ΓΣ ]] giving the ∗ (SH,∞ , ε; W )P at P satisﬁes character ψ (Lemma 3.6), the localization Sn.ord ∗ Sn.ord (SH,∞ , ε; W )P ⊗W [[ΓΣ ]] W [[ΓΣ ]]/P ∼ = Sn.ord (Npn , εψ ; K[ψ])

which is generated by a single element φ over the algebra hn.ord cyc (N, ε; W [[ΓΣ ]])P ∗ after localization at P . Take φ ∈ Sn.ord (SH,∞ , ε; W )P which lifts φ. We have by Nakayama’s lemma (Lemma 1.6) a surjective morphism ∗ hn.ord cyc (N, ε; W [[ΓΣ ]])P → Sn.ord (SH,∞ , ε; W )P

given by h → hφ. Since the action of the Hecke algebra hn.ord cyc (N, ε; W [[ΓΣ ]])P ∗ on Sn.ord (SH,∞ , ε; W )P is faithful, the above morphism is an isomorphism. Thus we ﬁnd Corollary 3.55 Let κ = (0, I) and Σ ⊂ Σp be a subset. For each ﬁnite order × character ψ : ΓΣ → Qp factoring through Γn , writing the prime ideal Pκ,εψ of W [[ΓΣ ]] given by the kernel of the W -algebra homomorphism ψ : W [[ΓΣ ]] → Qp ∼ which restricts to ψ on ΓΣ , we have (hn.ord cyc (N, ε; W [[ΓΣ ]])/Pκ,εψ ) ⊗W K = n.ord n (Np , εψ ; K[ψ]) by an isomorphism taking T (y) to T (y) and U (y) to U (y) h for all integral y. Actually the assertion of the corollary holds if κ2 − κ1 ≥ I as we will see in Chapter 4. This fact of course follows from Theorem 3.50 for the local ring T in the theorem and also from Corollary 4.21 if p ≥ 5 is unramiﬁed in F/Q. Although we stated our result for the nearly ordinary part in the above corollaries, we can obtain similar results for the partially ordinary part also in the same manner. Since this partial analogue of Corollary 3.54 is useful later proving fact. Split Σp = Σ Σ , and put Corollary 3.74, we describe this P = O ∩ ( p∈Σ pOp ) which is prime to p ∈ Σ . Then take an open subgroup (∞) so that the quotient group SΣ (Pn )Z(A(∞) )/Γ 1 n ) is SΣ (Pn ) of G(Z) 1 (P )Z(A n isomorphic to the image of the local norm map NΣ : p∈Σ (Op /p Op )× → n × 0 (Pn )/Γ 1 (Pn ) with a group. Identify Γ 1 p∈Σ (Z/p Z) . Here is how to ﬁnd such

# $2 a b n × 0 (Pn )/Γ 1 (Pn ) to the diagonal entries sending in Γ 1 p∈Σ (Op /p Op ) c d n (a, d) mod P . By this identiﬁcation, we have a norm map  2 0 (Pn )/Γ 11 (Pn ) →  :Γ NΣ,n

p∈Σ

2 (Z/pn Z)×  .

Galois deformation

237

1 (Pn ) ⊂ SΣ (Pn ) ⊂ Γ 0 (Pn ) by SΣ (Pn )/Γ 1 (Pn ) = Deﬁne an open subgroup Γ 1 1 2 n n Ker(NΣ,n ), and put SΣ,n = SΣ (P ) ∩ Γ0 (N). Using U (P ) in place of U (pn ) in the proof of Lemma 3.42 to lower the P-power level, we ﬁnd that the Pontryagin ∗ (SΣ,∞ , ε; W ) of S Σ-ord (SΣ,∞ , ε; W∞ ) = limn S Σ-ord (SΣ,n , ε; W∞ ) is dual SΣ-ord −→ W [[ΓΣ ]]-free of ﬁnite rank for ΓΣ = p∈Σ Γp . Let hΣ-ord cyc (N, ε; W [[ΓΣ ]]) be the ∗ (SΣ,∞ , ε; W )) generated by T (y) and W [[ΓΣ ]]-subalgebra of EndW [[ΓΣ ]] (SΣ-ord U (y) for integral ideles y. We record here what we can prove in this slightly more general setting: 0 (N) Proposition 3.56 Let the notation be as above, and suppose (sm0) for Γ and ε. Then we have the following assertions. 1. The Hecke algebra hΣ-ord cyc (N, ε; W [[ΓΣ ]]) is a torsion-free W [[ΓΣ ]]-module of ﬁnite type for each subset Σ of Σp . ×

2. For each ﬁnite-order character ψ : ΓΣ → Qp factoring through Γn , we ∼ Σ-ord (NPn , εψ ; K[ψ]) by an have (hΣ-ord cyc (N, ε; W [[ΓΣ ]])/Pκ,εψ ) ⊗W K = h isomorphism taking T (y) to T (y) and U (y) to U (y) for all integral y. Fix a prime factor p0 |p in F , and take Σ = Σp − {p0 }. Suppose that p0 N. We 0 (p0 N), ε; W∞ ) have an inclusion i : S Σ-ord (SΣ,∞ , ε; W∞ )2 → S Σ-ord (SΣ,∞ ∩ Γ sending (f1 , f2 ) to f1 +[ηp0 ](f2 ) as in Lemma 3.2. This i induces a W [[ΓΣ ]]-algebra homomorphism old i∗ : hn.ord , cyc (p0 N, ε; W [[ΓΣ ]]) → h 2 where hold = hn.ord cyc (N, ε; W [[ΓΣ ]])[X]/(X − T (p0 )X + ε+ (p0 )N (p0 )), sending T (y) to T (y) for integral ideles y with yp0 = 1 and U (p0 ) to X as in Lemma 3.13. Any irreducible components of Spec(hn.ord cyc (p0 N, ε; W [[ΓΣ ]])) conold tained in the image of Spec(h ) is called a p0 -old component. An irreducible component of Spec(hn.ord cyc (p0 N, ε; W [[ΓΣ ]])) is called p0 -new if it is not p0 -old. Since this map i∗ induces a ﬁnite morphism of the spectra, an irreducible component of Spec(hn.ord cyc (p0 N, ε; W [[ΓΣ ]])) is either p0 -old or p0 -new and cannot be both because the source and the target of i∗ are ´etale over W [[ΓΣ ]] at locally cyclotomic points. In other words, we have the following result:

Corollary 3.57 (Rigidity) Suppose that N is prime to a ﬁxed prime factor p0 of p and F , and put Σ = Σp −{p0 }. Then an irreducible component Spec(I) of the Hecke scheme Spec(hn.ord cyc (p0 N, ε; W [[ΓΣ ]])) is p0 -new if and only if it contains a locally cyclotomic point P such that the corresponding Hecke eigenform fP is p0 -new. This corollary is called a rigidity corollary, because if an automorphic representation is special at p0 , its (classical) deformation over Spf(W [[ΓΣ ]]) outside Spf(W [[Γp0 ]]) is rigidly special at p0 ; so, it is a local rigidity statement of the Steinberg representation. In the above study of the Hecke algebra over W [[ΓF ]], actually, the level group is not really Sn , because we used the right quotient Y (Sn ) of G(Q)\G(A(∞) ) by

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Hecke algebras as Galois deformation rings

Sn Z(A(∞) ). We can modify the above discussion taking, as level groups, smaller open subgroups Sn in place of Sn Z(A(∞) ). Here Sn is one of the following 1 (pn ) ∩ Γ 0 (N) and Sn . To state the result, we deﬁne type of groups: S11 (pn ) := Γ 1 0 ∗ H (X(S), L(κ ε; A)) to be the collection of functions f : X(S) → L(κ∗ ε; A) satisfying f (αxu) = u−1 p f (x) for all α ∈ G(Q) and u ∈ S. We write Γ for the maximal p-proﬁnite quotient of Op× and ΓZ for the maximal p-proﬁnite quotient of ClF (p∞ ) = Z(A(∞) )/Z(A(p) )Z(Q)+ . Here ClF (p∞ ) is isomorphic to the Γ acts on H 0 (X(Sn ), L(κ∗ ε; A)) and strict ray class group modulo p∞ . Then

a 0 S(Sn , ε; A) through Γ a → ∈ G(Zp ). Similarly the p-proﬁnite group 0 1 0 ∗ ΓZ acts on H (X(Sn ), L(κ ε; A)) through the central action. When A = Wm and W , these spaces have a well-deﬁned projector e = limn→∞ U (p)n! . Adding the subscript or superscript “n.ord”, we indicate the image of the idempotent e. ∗ (S∞ , ε; W ) and H 0 (X(S∞ ), L(κ∗ ε; W ))∗ by the Pontryagin We then deﬁne Sn.ord 0 dual of limn Sn.ord (S∞ , ε; W∞ ) and limn H (X(S∞ ), L(κ∗ ε; W∞ )). Then we have −→ −→ Theorem 3.58 0 (N) and ε. Then if Sn = S 1 (pn ), S ∗ 1. Suppose (sm0) for Γ 1 n.ord (S∞ , ε; W ) is W [[Γ]]-free of ﬁnite rank r, where r is given by dimF S n.ord (SΓ , ε; F) for the image Γ of ΓF in G1 . 0 2. If Sn = Sn , Hn.ord (X(S∞ ), L(κ∗ ε; W ))∗ is W [[ΓF × ΓZ ]]-free of ﬁnite rank. 0 3. If Sn = S11 (pn ), Hn.ord (X(S∞ ), L(κ∗ ε; W ))∗ is W [[Γ × ΓZ ]]-free of ﬁnite rank.

The proof of these results is essentially the same as the one for Theorem 3.53; so, we leave it to the reader. Replacing H 0 by H 1 , the same assertion holds (∞) (∞) for the quaternion algebra D over F of odd degree d with DA ∼ = M2 (FA ) d−1 (see [H88b] and [H89a]). Thus the analogous result and D∞ = M2 (R) × H holds independent of the degree condition after this modiﬁcation. We have the corresponding Hecke algebra in each of the above cases. We write q = dim X(S) (so, q = 0 if [F : Q] is even and q = 1 if [F : Q] is odd). Out of these Hecke modules, we can create the corresponding Hecke algebras. The biggest among these algebras (called the universal nearly p-ordinary Hecke algebra) is denoted by hn.ord (N, ε; W [[Γ × ΓZ ]]) which is the subalgebra q (X(S∞ ), L(κ∗ ε; W∞ ))∗ ) of the linear endomorphism algebra EndW [[Γ×ΓZ ]] (Hn.ord 1 n for Sn = S1 (p ) generated by Hecke operators T (q ) over W [[Γ × ΓZ ]] for all prime ideals q. It turns out (cf. [H89a]) that this algebra is independent of the choice of κ and the p-part of ε. Similarly we deﬁne hn.ord (N, ε; W [[Γ]]) by the q (Y (S∞ ), L(κ∗ ε; W∞ ))∗ ) for Sn = S11 (pn ) genersubalgebra of EndW [[Γ]] (Hn.ord ated by Hecke operators T (q ) over W [[Γ]] for all prime ideals q. Note here ∗ 0 (S∞ , ε; W ) = Hn.ord (Y (S∞ ), L(κ∗ ε; W∞ ))∗ if q = 0. This algebra is that Sn.ord called the universal nearly ordinary Hecke algebra with central character ε+ . We also deﬁne, in the same manner as the above two algebras, the universal locally

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cyclotomic Hecke algebra hn.ord cyc (N, ε; W [[ΓF ]]) with central character ε+ inside q (Y (S∞ ), L(κ∗ ε; W∞ ))∗ ). This algebra is already deﬁned earlier EndW [[ΓF ]] (Hn.ord when q = 0. We will revisit these algebras in Chapter 4 using geometric Hilbert modular forms in place of Betti cohomology groups used here. Assume Σf l = ∅. Throw away the local cyclotomy condition (D4), and consider the deformation functor A → Φε (A) = ΦεF (A) such that Φε (A) is made up of (the isomorphism classes of) deformations ρ of ρ with values in GL2 (A) satisfying the conditions: (cyc), (D1–3) for ν = ε+ N , (Q1–3), (Q6) for Q = ∅ and

p ∗ (Q4 ) ρ|Dp ∼ for characters p , δp : Dp → A× for all prime ideals 0 δp This functor is again representable under (aiF ) giving rise to a universal couple (RεF , ρεF ). The modular deformation of this type can be realized by the unique local ring TεF of hn.ord (N, ε; W [[Γ]]) (which covers TF under the projection: ε hn.ord (N, ε; W [[Γ]]) → hn.ord cyc (N, ε; W [[ΓF ]])) and the modular deformation ρT n.ord associated to the initial Hecke eigenform f0 ∈ Sκ0 (N, ε0 ; W ). Then in the same manner as in the proof of Theorem 3.50, we can prove Theorem 3.59 Let κ0 = (0, I) ∈ Z[I]2 = X(TG ). Suppose (aiF (µp ) ) in addi 0 (N). Then the unique W -algebra tion to (sf), (h1–4), and (sm0) for ε0 and Γ ε ε ε ∼ ε homomorphism π : RF → TF with π ◦ ρF = ρT induces a W [[Γ]]-algebra isomorphism RεF ∼ = TεF , and RεF is a reduced complete intersection free of ﬁnite rank over W [[Γ]]. For any arithmetic (κ, ε) satisfying 1. κ2 − κ1 ≥ I, −κ −κ and j = 1, 2, 2. zp j εj (z) ≡ zp 0,j ε0,j (z) mod mW for all z ∈ T (Z) 3. ε+ = ε0,+ and εj |T (Z(p) ) = ε0,j |T (Z(p) ) for j = 1, 2, 4. c(εj )|c(ε0,j )p∞ for j = 1, 2, (N ∩ c(ε− ), ε; W ) such that we have we have a unique local factor Tκ,ε ⊂ hn.ord κ ε ε ∼ ε ε ∼ an isomorphism RF /Pκ,ε RF = TF /Pκ,ε TF = Tκ,ε induced by π. In particular, Tκ,ε = T in Theorem 3.28 if (κ, ε) is locally cyclotomic. Working out a proof of this theorem (at least for the even degree base ﬁeld F ) is left to the reader, and we note that this theorem is valid without the degree condition on the base ﬁeld F . We now describe the local component Tκ,ε for κ ≥ (0, I) in an automorphic way. As seen in the proof of Theorem 3.50, we only need to establish the existence of Tκ,ε and the inequality rankW Tκ,ε ≥ r = rankW Tκ0 ,ε0 for the initial (κ0 , ε0 ); so, we may work in the characteristic p setting after reducing everything modulo p. Suppose κ > (0, I). Recall that L(κ∗ ; A) is the space of polynomials homogeneous in (Xσ , Yσ )σ∈I of degree κ2,σ − κ1,σ − 1 for each σ with coeﬃcients

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t in A. Let us consider the

evaluation at

(1, 0) of the polynomial. Thus we get X 1 L(κ∗ ε; A) → A by φ

→ φ ∈ A. Y 0 Recall the character ε = (ε1 , ε2 , ε+ ) with values in W × as in (ex1). We deﬁne the p-adic avatars ε = ( ε1 , ε2 , ε+ ) of ε by ε+ (z) = ε+ (z (∞) )zp−κ1 −κ2 +I 1 2 for z ∈ Z(A(∞) ), ε1 (a) = ε1 (a)a−κ and ε2 (d) = ε2 (d)dI−κ for a, d ∈ T (Z)G. p p

Exercise 3.60 Prove that ε+ : Z(A(∞) )/Z(Q) → W × is a continuous character. We deﬁne ε = ( ε1 , ε 2 , ε + ) with values in W × by the Teichm¨ uller lift of the ε1 , ε2 , ε+ ). The Neben character ε could reduction modulo mW of the characters ( be the initial character ε0 but could be also diﬀerent from ε0 . We prove the following theorem. Theorem 3.61 Let = p|p p . If κ2 − κ1 ≥ I, then dimF Sκn.ord (N, ε; F) depends only on ε and N, and we have an isomorphism of Hecke modules: n.ord Sκn.ord (N, ε; F) ∼ (N, ε ; F). = S(0,I)

Moreover if κ2 − κ1 ≥ 2I, we have rank Sκn.ord (N, ε; W ) = rank Sκn.ord (N, ε; W ) for N prime to p and Sκn.ord (N, ε; F) ∼ = Sκn.ord (N, ε; F) as Hecke modules. Here “an isomorphism of Hecke modules” means that the isomorphism ι satisﬁes ι ◦ T (q ) = T (q ) ◦ ι for all primes q outside pc( ε) and ι ◦ U (q ) = U (q ) ◦ ι for all prime factors of pc( ε). If the assertion of the theorem holds for the space 0 (N), ε; F) for a suﬃciently small S, the assertion basically holds for Sκn.ord (S ∩ Γ all N by the Hochschild–Serre spectral sequence applied to H 0 . Here we give a proof taking S so that Γx (S) is trivial for all x ∈ G(A(∞) ). 0 (N) so that Proof We choose an open compact subgroup S ⊂ Γ Γx (S) = G(Q) ∩ S · Z(A(∞) )/Z(Q) = {1}. 0 ()). Then we can verify by computation We write Y = Y (S) and Y = Y (S ∩ Γ ∗ ∗ using the deﬁnition of the action of ∆D 0 (N) on L(κ ε; F) that the evaluation t ∗ at (1, 0) of polynomials in L(κ ε; F) gives rise to a morphism i : L(κ∗ ε; F) → ∗ = (0, I) and ε is as above. Similarly j : L( κε , F) of ∆D 0 (N) -modules, where κ −1 ∗ κ ε; F) given by j(a) = aY κ2 −κ1 −I is a morphism of ∆D L( κε , F) → L( 0 (N)modules.

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241

0 −1 0 (N). By Choose τ ∈ M2 (Op ) to be . Thus τ ∈ G(Qp ) normalizes Γ 0 ∗ ∗ a simple computation, the endomorphism j ◦ i of the ∆D 0 (N) -module L(κ ε; F) ∗ is the action of τ , and the same for i ◦ j and L( κ ε ; F). Composing with the restriction map Res and transfer map Tr (cf. [MFG] 4.3.1), we get the following morphisms, if κ2 − κ1 ≥ 2I, i

q q q ∗ ι : Hn.ord (Y, L(κ∗ ε; F)) −−→ Hn.ord (Y , L(κ∗ ε; F)) −→ Hn.ord (Y , L( κε ; F)) Res

and q q π : Hn.ord (Y , L( κε ; F)) − → Hn.ord (Y , L( κε −1 ; F)) τ

j∗

q q −→ Hn.ord (Y , L(κ∗ ε; F)) −→ Hn.ord (Y, L(κ∗ ε; F)). Tr

Here we regard L(κ∗ ε; F) as a sheaf over Y whose stalk at x ∈ Y is H 0 (Γx (S), L(κ∗ ε; F)) = L(κ∗ ε; F) because Γx (S) = {1}. Thus we have 0 Hn.ord (Y, L(κ∗ ε; F)) = Sκn.ord (S, ε; F), 0 0 (), ε ; F). Hn.ord (Y , L( κε ; F)) = Sκn.ord (S ∩ Γ

These are morphisms of Hecke modules, since they come from the two morphisms ∗ i and j of ∆D 0 (N) -modules. For φ ∈ Sκ (S, ε; F), τ φ(x) = τ · φ(xτ ) and φ ∈ Sκ (S, ε ; F) τ φ(x) = φ(xτ ). κε ; F)) = 0, but we keep this For q > 0, we have H q (Y, L(κ∗ ε; F)) = H q (Y , L( cohomological notation, because the argument itself works well for any D with nontrivial H q for q = dim Y D (N). By the deﬁnition of the normalized Hecke operator T () by the following two commutative diagrams, τ

H q (Y , L(κ∗ ε; F)) −−−−→ H q (Y , L(κ∗ ε; F))  2   Res Tr H q (Y, L(κ∗ ε; F)) −−−−→ H q (Y, L(κ∗ ε; F)) T ()

and τp

H q (Y , L( κ∗ ε ; F)) −−−−→ H q (Y , L( κ∗ ε ; F)) 2    Res Tr H q (Y, L( κ∗ ε ; F)) −−−−→ H q (Y, L( κ∗ ε ; F)), T ()

q (Y, L(κ∗ ε; F)). Similarly we get ι ◦ π = T () we ﬁnd that π ◦ ι = T () on Hn.ord q ∗ on Hn.ord (Y , L(κ ε; F)). This shows that ι is the isomorphism we desired.

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If some of κ2,p − κ1,p = 1, the operator U (p ) lowers the level to the minimal q (Y , L( κ∗ ε ; F)) by Lemma 3.51, thus for this p, we cannot lower the level Hn.ord level to N. 2 Remark 3.62 The same result as the above theorem holds for D with D∞ ∼ = M2 (R) × Hd−1 by the same computation applied to H 1 for the curves Y and Y . See [PAF] Theorem 4.37. Corollary 3.63 Let the notation be as in the theorem. Suppose that the Tκ,ε eigenspace Tκ,ε Sκn.ord (N, ε ; F) for the local ring Tκ,ε with κ = (0, I) is free of rank 1 over Tκ,ε ⊗W F. Then there exists a unique local ring Tκ,ε of the Hecke (N, ε; W ) such that we have a surjective algebra homomorphism algebra hn.ord κ Tκ,ε ⊗W F Tκ,ε ⊗W F taking T (q ) to T (q ) for all q. Proof We just identity Tκ,ε Sκn.ord (N, ε ; F) with Tκ,ε . Then it gives rise to a direct factor of the Hecke module Sκn.ord (N, ε; F) by the isomorphism in the theorem. Thus this factor is at least isomorphic to a quotient of a unique local ring Tκ,ε of hn.ord (N, ε; W ). 2 κ In Theorem 3.61, we only discussed the embedding i : L(κ∗ ε; F) → L( κε ; F), but by replacing S ∩ Γ0 () in the deﬁnition of Y by Sn , we get Yn and have the mod pn analog in of i, which is the evaluation at t (1, 0). Take κ to be κv for v ∈ Z[Σp ] with k+2v ≥ 2I. Similarly we have an obvious mod pn -analog j = jn . Then In we get a Hecke equivariant morphism ι : H q (Y, L(κ∗ ε; Wn )) → H q (Y n , Wn ). 0 −1 the same manner as in the proof of Theorem 3.61, replacing τ by , pn 0 we can prove that ι is injective on the nearly p-ordinary part; so, we have a q Hecke equivariant embedding, still denoted by ι, of Hn.ord (Y, L(κ∗ ε; W∞ )) into q q , W∞ ) := limn Hn.ord (Yn , Wn ), whose image is the subspace where Hn.ord (Y∞ −→ 0 (Y∞ , W∞ ) = S(S∞ , ε; W∞ ). The T (Zp ) acts by the weight 2v. If q = 0, Hn.ord n.ord 0 (Y∞ , W∞ ); so, Hecke algebra h = hcyc (N, ε; W [[ΓF ]]) acts faithfully on Hn.ord by restricting the operators in h to the image of ι, we have a surjective W [[ΓF ]]h hn.ord (pN, εv ; W ), and hence h is sent by algebra κv homomorphism πv : (pN, εv ; W ). Obviously, the image of Π Π = v πv into the product v hn.ord κv is the subalgebra of the product over v generated topologically by Tp (y) and , ε; W∞ )[2v] for the image of ι which Up (y) for integral ideles y. We write S(Y∞ is the 2v-eigenspace of T (Zp ) in S(Y∞ , ε; W∞ ). Then v S(Y∞ , ε; W∞ )[2v] = S(Y∞ , ε; W∞ ) (for v running over Z[Σp ] with k + 2v ≥ 2I), because algebraic characters span a dense subspace of the space of p-adically continuous functions theorem). This shows that h can be idenof Im(Np ) with values in W (Mahler’s tiﬁed with the subalgebra of v hn.ord (pN, εv ; W ) over v generated topologically κv by Tp (y) and Up (y) for integral ideles y. This fact can also be shown when q = 1 basically by the same argument. See [H88b] Sections 8 and 9 and [H89a] for more details. We record this fact as

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Theorem 3.64 The Hecke algebra hn.ord cyc (N, ε; W [[ΓF ]]) is isomorphic to the n.ord subalgebra of v hκv (pN, εv ; W ) topologically generated over W by Tp (y) and is taken over all v ∈ Z[Σp ] with Up (y) for all integral ideles y, where the product (pN, εv ; W ) is the operator k + 2v ≥ 2I, and Tp (y) (resp. Up (y)) in v hn.ord κv whose projection to each v is given by the corresponding operator of weight κv . 3.2.10 Global deformation over a p-adic ﬁeld We continue to assume (sf) and (h1–4) in 3.2.1 for f0 (and ρ0 ). We now regard ρ0 : Gal(Q/F ) → GL2 (W ) as having values in GL2 (K) for the ﬁeld of fractions K of W . We replace the source category CN LW by the category ARTK of artinian local K-algebras with residue ﬁeld K in the deformation problem. K We write ΦK F : ARTK → SET S for the deformation functor with ΦF (A) consisting of (isomorphism classes of) deformations ρ of ρ0 satisfying the following conditions: ∼ ρ0 , and ρ : Gal(Q/F ) → GL2 (A) is p-adically continuous; (P1) ρ mod mA = (P2) ρ is unramiﬁed outside the prime factors of pc(ε) and prime ideals in Q; (P3) det(ρ) = ε+ N for the p-adic cyclotomic character N ;

p ∗ for characters p , δp : Dp → A× such that δp ≡ η1,p (P4) ρ|Dp ∼ 0 δp −1 ur ur mod mA , p |Ip ε−1 2,p and δp |Ip ε1,p factors through Gal(Fp (µp∞ )/Fp ) for all prime ideals p|p;

q ∗ for characters q , δq : Dp → A× such that δq |Iq = ε1,q (P5) ρ|Dq ∼ 0 δq for all prime ideals q|c(ε) outside p, and ρ|Iq ⊗ ε−1 1 is unramiﬁed if q|c(ε) is outside pN. Since ρ0 is absolutely irreducible and has two distinct characters δq and q for all q|pN, we can prove in the same manner as in the proof of Theorem 1.46 K K that the functor ΦK F is pro-representable over ARTK . We write (R , ρ ) for K K the universal couple. Pro-representability of ΦF tells us that R is a projective limit of objects in ARTK (that is, a pro-artinian local K-algebra with residue ﬁeld K). Theorem 3.65 Suppose (aiF ), (sf), and (h1–4) in 3.2.1. Let ϕ : RF → W be the unique W -algebra homomorphism with ϕ◦ρcyc ∼ = ρ0 in GL2 (W ), and put P = P = R F,P for the P -adic localization-completion lim RP /P n of Ker(ϕ). Write R ←−n P ) be the Galois representation ι◦ρcyc RF at P , and let ρP : Gal(Q/F ) → GL2 (R P . Then the unique K-algebra for the natural algebra homomorphism ι : RF → R K K ∼ homomorphism π : R → RP with π ◦ ρ = ρP induces a W [[ΓF ]]-algebra F,P . isomorphism RK ∼ =R

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Hecke algebras as Galois deformation rings

P , ρP ) is universal; so, it Proof Our proof shows directly that the couple (R also shows the pro-representability of ΦK . Supposing the existence of RK , we ﬁrst show that RK is (topologically) generated by the trace of ρK . Let Rtr be the subring of RK generated topologically by the trace of ρK . Then by the theory of pseudo-representation (see Proposition 1.23), we may assume that ρK has values in GL2 (Rtr ). We prove that (Rtr , ρK ) satisﬁes the universal property for ΦK . For any object A in ARTK , we have a natural restriction map ΦK (A) ∼ = HomK-alg (RK , A) → HomK-alg (Rtr , A). For any K-algebra homomorphism ϕ : RK → A, the representation ϕ ◦ ρK is determined by the restriction of ϕ to Rtr , because the trace of a given representation ρ : Gal(Q/F ) → GL2 (A) determines the isomorphism class of a representation (a result of Carayol and Serre; see Proposition 1.25). Thus ΦK (A) injects into HomK-alg (Rtr , A). If we have a K-algebra homomorphism ϕ : Rtr → A, we have a Galois representation ρ with Tr(ρ) = ϕ ◦ Tr(ρK ) by Proposition 1.23. Then ρ ∼ = ϕ ◦ ρK again by Carayol and Serre. This K shows that Φ (A) → HomK-alg (Rtr , A) is surjective. Thus HomK-alg (RK , A) ∼ = HomK-alg (Rtr , A) for all objects A in ARTK , and by Yoneda’s lemma (2.2.1) (e.g., [MFG] Lemma 4.3), we see that RK = Rtr . By the same argument, RF is generated topologically by the trace of ρcyc , P is generated by the trace of ρP . Take ρ ∈ ΦK (A). Since ρ is and hence R continuous, Im(ρ) is compact. Regarding the representation space V (ρ) ∼ = A2 as a ﬁnite-dimensional K-vector space, the compactness of Im(ρ) tells us the existence of a W -lattice L in V (ρ) stable under ρ. Thus Tr(ρ) has values in a compact subring AL = {a ∈ A|aL ⊂ L}. We consider the W -subalgebra A0 ⊂ A generated topologically over W by the trace of ρ. Then A0 ⊂ A is a compact ring. Since A0 /A0 ∩mA is a compact W -subalgebra in K, it has to be W . Since mA ∩A0 is nilpotent, any maximal ideal m of A0 contains mA ∩ A0 ; so, we have A0 /m = A0 /m+(A0 ∩mA ) = F. This shows that m+(A0 ∩mA )/A0 ∩mA = mW , and hence A0 is a local W -algebra free of ﬁnite rank over W with residue ﬁeld F = W/mW . By Proposition 1.23, we have a Galois representation ρ : Gal(Q/F ) → GL2 (A0 ) with Tr(ρ ) = Tr(ρ). Thus V (ρ ) = A20 is A0 -free. By Carayol–Serre (Proposition 1.25), we have ρ ∼ = ρ in GL2 (A). Since ρ mod mA ∼ = ρ0 , Tr(ρ ) ≡ Tr(ρ0 ) = Tr(ρ) mod mA0 . Thus ρ ≡ ρ mod mA0 . Picking an element σ in Dq with ρ(σ) having two distinct eigenvalues if q|c(ε− ), we can split V (ρ ) = V1 ⊕ V2 into two eigenspaces of ρ (σ). By a theorem of Krull and Schmidt, V1 ∼ = V2 ∼ = A0 as A0 -modules, and hence ρ restricted to the corresponding decomposition group has the upper triangular form speciﬁed in (Q4) and (Q6) for such primes. We now show that ρ ∈ Φcyc (A0 ). We need to check (Q4) and (Q6) for ρ and primes q|pc(ε− )N (because for other primes q|c(ε), these conditions follow from (P4–5)). First suppose q|c(ε− ) but q p. Since the deformation over the corresponding inertia group Iq is constant (by Proposition 3.21), ρ satisﬁes (Q6), because ρ and ρ0 satisfy the condition by (P5). For q|p, the condition (Q4) follows from (P4).

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For q|N/c(ε− ) outside p, the argument is basically the same as above. We repeat here the argument which proved Proposition 3.21 for primes q|N/c(ε− ) outside p. By the assumption (h4), we may assume that the restriction of ρ to the inertia group has values in the upper unipotent subgroup; so, its factors through (q) (1). The p-factor of Z (q) is of rank 1 isomorphic to the tame inertia group ∼ =Z

1 1 Zp (1). Writing u for the generator of Zp (1), we may normalize ρ0 (u) = 0 1 (choosing a suitable basis of ρ). By the constancy of the deformation of ρ over Iq

1 1 (Proposition 3.21), we may assume ρ (u) = . Since H0 (Iq , V (ρ ))⊗A0 F ∼ = 0 1 F by (h4), we ﬁnd that H0 (Iq , V (ρ )) ∼ ρ satisﬁes = A0 by Nakayama’s lemma; so,

Np ∗ − ∼ (Q6) for all q|N outside p. For a prime p|N/c(ε ) (p|p), since ρ0 |Dp = , 0 1

Np ∗ by Kummer’s theory, ρ |Dp ∼ . This shows that ρ satisﬁes (Q4) for = 0 1 p|N/c(ε− ). We have now veriﬁed ρ ∈ Φcyc (A0 ), and we have a unique W -algebra homomorphism ϕ : RF → A0 such that ϕ ◦ ρcyc ∼ = ρ . Since ρ mod mA0 ∼ = ρ0 (again by the trace identity), we ﬁnd that P ⊃ Ker(ϕ), and hence ϕ induces a unique P → A such that ϕ K-algebra homomorphism ϕ :R ◦ ρP ∼ is = ρ ∼ = ρ. Since ϕ P is determined by Tr(ρ ) = Tr(ρ), ϕ is uniquely determined by ρ (because R P , ρP ). 2 generated by the trace of ρP ). This shows the universality of (R 3.3 Base change We brieﬂy describe how to include general totally real base ﬁelds F (particularly those of odd degree) in the above theory via the quadratic base-change of deformation rings. In this section, G0 denotes ResO/Z GL(2). 3.3.1 p-Ordinary Jacquet–Langlands correspondence Recall the space Sκ (N, ε; C) of archimedean Hilbert modular forms deﬁned below (2.3.4) in 2.3.2. Since Sκ (N, ε; W ) in (2.3.23) is free of ﬁnite rank over W , we may deﬁne in the W -algebra EndW (Sκ (N, ε; W )) the projectors ep = lim U (p )n! and ep = lim T (p )n! n→∞

accordingly as p|N or not, and put e =

n→∞

p|p ep .

Then we deﬁne

Sκn.ord (N, ε; R) = e (Sκ (N, ε; R)) for the W -algebra R ⊂ Qp . Restricting the map ιA in Corollary 2.33 to Sκn.ord (N, ε; K) which is a subspace of SκD (N, ε; K), we get a p-ordinary version of the Jacquet–Langlands and

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Shimizu correspondence as follows: Proposition 3.66 If (κ, ε) is arithmetic and [F : Q] is even, we have a K-linear isomorphism ι : Sκn.ord (N, ε; K) ∼ = Sκn.ord (N, ε; K) satisfying ι◦T (l ) = T (l ) ◦ ι and ι ◦ U (l ) = U (l ) ◦ ι for all prime ideals l. See Chapter 2 of the text and [PAF] 4.3.2–3 for a proof of this fact. For a general base ﬁeld F , we deﬁne the Hecke algebra hn.ord (N, ε; W )) by the W -subalgebra κ of EndW (Sκn.ord (N, ε; W )) generated by T (y) and U (y) for all integral ideles y. By Proposition 3.66, this algebra coincides with the one already deﬁned for F with even degree. 3.3.2 Base ﬁelds of odd degree Suppose that [F : Q] is odd. Starting with a Hecke eigenform f0 ∈ Sκ0 (N, ε0 ; W ) (for κ0 = (0, I)) satisfying (sf), (h1–4), and (aiF (µp )) in 3.2.1, as was done in [Fu], we can again go through all the steps to prove the analogue of Theorem 3.28 and Theorem 3.50 for the odd degree F . Let us describe this in slightly more detail. The formulation of Theorem 3.23 and its proof are independent of the degree of F , and therefore, they are valid for the odd degree F (as long as we have the initial ρ0 satisfying (3.2.1), (sf), (h1–4), and (aiF ) in 3.2.1). For Q ∈ Q as in Theorem 3.23, the deformation problem for ρ governed by the conditions (Q1–6) is well posed and has the universal ring RQ . The estimates of the dimension of the tangent space of RQ given in 3.2.2 and 3.2.5 are also independent of the degree assumption. To prove the version of Theorem 3.28 over the odd-degree base ﬁeld F , therefore, we need to create good Hecke modules MQ forming a Taylor–Wiles system. For that, we take a quaternion algebra D/F unramiﬁed at every ﬁnite place and ramiﬁed at all but the one inﬁnite place corresponding to the inclusion i∞ : × F → R. Then we can think of the algebraic group GD /Q associated to D . Since GD (R) ∼ = GL2 (R)×H× ×· · ·×H× , via the projection to the GL2 (R)-component, D G (R) acts on H H = C − R through a linear fractional transformation. Write Ci for the stabilizer of i ∈ H in GD (R). Fixing a maximal order OD D with M2 (O), we have the subgroup S(Q) ⊂ GD (A(∞) ). We and identifying O deﬁne the Shimura curve Y (Q) = GD (Q)\GD (A)/Z(A)S(Q)Ci . Let TQ be the direct summand of the Hecke algebra hκ (S(Q), ε; W ) through which λ factors. Fujiwara proved the analog of Theorem 3.28 in this case taking MQ to be the TQ -eigensubspace of the cohomology group H 1 (Y (Q), Zp ) for the Shimura curve Y (Q). Strictly speaking, if Y (Q) is not smooth, in order to make sure the freeness = S(Q) ∩ S for a suitable choice of open of MQ , one has to replace S(Q) by SQ × of level prime to pNc(ε). This process adds more compact subgroup S ⊂ O D technicality to the treatment than the even-degree case. In any case, this was done in [Fu], and we have

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Theorem 3.67 (Wiles–Taylor, Fujiwara) Let f in Sκ (N, ε; W ) (with κ2 − κ1 ≥ I) be a Hecke eigenform satisfying (h1–4) and (sf) in 3.2.1. Under (aiF (µp ) ), R∅ ∼ = T∅ , T∅ is a local complete intersection. Moreover under (sm0) for ε and Γ0 (N) in addition to these conditions, TP deﬁned in Proposition 3.40 is a local complete intersection, and TP is canonically isomorphic to the universal nearly ordinary deformation ring R classifying the deformations satisfying all (Q1–6) but (Q5) for p ∈ P and Q = ∅. Out of this result, the deduction of Theorem 3.50 for the odd-degree case is the same, and therefore we have not assumed the degree condition in Theorem 3.50. As we have mentioned, the choice of MQ in [Fu] is the TQ -eigenspace of an ). There should appropriate Betti cohomology group of the Shimura curve X(SQ be more choices of MQ ; for example, in [MFG] 3.2.7, we have taken the Hecke algebra TQ itself as MQ (assuming F = Q). In the following two subsections, we indicate how to reduce by a base-change technique the proof of Theorems 3.50, 3.59 and 3.67 in the nearly p-ordinary case to the theorems proved in Section 3.2 taking a quadratic extension of F as a base ﬁeld. 3.3.3 Automorphic base change We start with a (Hilbert modular) Hecke eigenform f0 in the space Sκ0 (N, ε0 ; W ) with κ0 = (0, I) and assume (aiF (µp ) ) and the conditions (h2–3) concerning primes over p for its Galois representation ρ0 = ρf0 . We assume the following condition in place of (h1) and (h4): (h1 ) The restriction of ε0 to ∆0 (N)p is of order prime to p. (h4 ) If the local component πq (q p) of the automorphic representation π of G0 (A) = GL2 (FA ) spanned by f0 at a prime ideal q outside

p is special, 1 ∗ ρ|Dq has ramiﬁcation index divisible by p; so, ρ|Iq ∼ for nontrivial 0 1 ∗, where Iq is the inertia subgroup of Dq . These conditions are plainly weaker than (h1) and (h4). As long as F /F is a soluble Galois extension, by [BCG], we can ﬁnd a basechange lift f of a Hecke eigenform f ∈ Sκ (N, ε; W ) if κ1 < κ2 . We attach “ ” to the symbol we used for F to indicate the corresponding object deﬁned for F ; for example, O is the integer ring of F , G0 = ResO /Z GL(2) and T = ResO /Z Gm . We consider the norm map N : T → T , Z → Z and TG → TG . Thus we may deﬁne κ and ε for F by the pullback of κ and ε via the norm map. Then by ε ; W ) (for an appropriate [BCG], we have a unique base-change lift f in Sκ (N, choice of N) such that the associated Galois representation ρf : Gal(Q/F ) → GL2 (W ) is isomorphic to the restriction of ρf to Gal(Q/F ). Thus we can think of the conditions (3.2.1), (sf), and (h1–4) in 3.2.1 for ρ0 restricted to Gal(Q/F ) in terms of automorphic data with respect to G0 = ResF /Q GL(2).

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We want to take F /F in the following way: (E1) All prime factors of p split completely in F /F . (E2) (h1–4) and (3.2.1) over F are satisﬁed for all ramiﬁed primes of ρ0 = ρf0 restricted to Gal(Q/F ). (E3) ρ remains absolutely irreducible over Gal(Q/F (µp )). (E4) F /F is a soluble Galois extension of even degree. We now show that such an extension F exists. By the solution of the local Langlands conjecture for GL(2) (cf. [Kz]) and the local behavior of ρ0 (cf. [Ta] and [Ta2]) due to Carayol and Taylor, f0 satisﬁes (3.2.1) at a prime ideal q outside p if and only if ρ0 |Dq is reducible. This reducibility ρ0 |Dq can be achieved by restricting ρ0 to a quadratic extension of Fq for odd q and a biquadratic extension for even q (see [We2]). Since we can ﬁnd an extension F1 /F of a given degree with prescribed completions F1,q for ﬁnitely many places q, taking a totally real biquadratic Galois extension F1 /F linearly disjoint over F from the splitting ﬁeld of ρ, and replacing f0 by the base change of f0 to F1 , we may assume (aiF1 (µp ) ), that all nonarchimedean local components of π are either special or principal, and hence, we can achieve (E1–4) by choosing an abelian extension of F1 , because locally ρ0 over Gal(F q /F1,Q ) is either diagonal or upper triangular now. Since ε0,p has order prime to p, by choosing a totally real abelian extension F2 /F1 possibly ramifying at ramiﬁed places of ρ0 outside p, we achieve (E2). We can achieve (E3) just imposing full ramiﬁcation on F2 /F at a prime l p unramiﬁed for ρ. We may assume that 2|[F2 : Q] replacing F2 by a suitable quadratic extension if necessary. 3.3.4 Galois base change Let F /F be a cyclic Galois extension of ﬁnite degree satisfying (E1–3). We ﬁx a nearly p-ordinary Hilbert Hecke eigen-cusp form f0 on G0 and take a base change f0 to G0 . We write Φ (resp. Φ) for the deformation functor Φcyc of ρ0 (resp. ρ0 ). We suppose that f0 satisﬁes (sf), (h1–4), and (aiF (µp ) ) in 3.2.1. Then f0 satisﬁes the same conditions for F . Thus Φ (resp. Φ ) is represented by a universal couple made of a local p-proﬁnite W -algebra R (resp. R ) of (topologically) ﬁnite type and the universal representation ρ : Gal(Q/F ) → GL2 (R) (resp. ρ : Gal(Q/F ) → GL2 (R )). For ρ ∈ Φ(A), we write ρF for the restriction of ρ to Gal(Q/F ). Then we see ρF ∈ Φ (R), and hence we have a unique morphism of W -algebra π : R → R for a lifting σ to with π ◦ ρ ∼ = ρF . For a generator σ ∈ Gal(F /F ), write σ |F = σ. Gal(Q/F ); so, σ σ g σ −1 ). Then Exercise 3.68 Deﬁne the inner conjugate ρσ of ρ by ρσ (g) = ρ( σ show that the isomorphism class of ρ is uniquely determined by σ for ρ ∈ Φ(A).

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By the exercise, we have a unique W -algebra homomorphism σ : R → R σ such that ρ ∼ = σ ◦ ρ , and Gal(F /F ) acts on Φ. Since σ ◦ σ −1 is the identity map, σ is an automorphism of the W -algebra R . Since σ(P ) = P in R for the point P ∈ Spf(R ) corresponding to ρf = ρf |Gal(Q/F ) , the action of . Hereafter if Gal(F /F ) on R extends to the P -adic completion localization R confusion is unlikely, we simply write σ for σ, and in this way, Gal(F /F ) and . hence GF acts on R and R Theorem 3.69 Let σ be a generator of Gal(F /F ). Let R (σ − 1)R be the ideal of R generated by σ(r) − r for all r ∈ R . Then π induces isomorphisms ∼ /R (σ − 1)R . R∼ =R = R /R (σ − 1)R and R To prove this theorem, we admit the following well-known fact: Lemma 3.70 Suppose p > 2 and (aiF ) for ρf . Then if ρ ∈ Φ (A) is ﬁxed by Gal(F /F ), then ρ extends to a unique ρ ∈ Φ(A). See [MFG] Lemma 5.32 for a proof. , we Proof of Theorem 3.69. Since the argument is the same for R and R prove the theorem for R . Since σ n (r) − r = (σ − 1)(1 + σ + · · · + σ n−1 )(r), we ﬁnd that R (σ − 1)R is stable under the action of Gal(F /F ), and the group Gal(Fm /Fn ) acts trivially on R = R /R (σ − 1)R . For the projection = π ◦ ρ . Then ρ σ ∼ and hence ρ extends uniquely π : R R, we write ρ =ρ to a Galois representation ρ ∈ Φ(R). We only need to prove that the couple (R, ρ ) is universal for Φ. If ρ ∈ Φ(A), we restrict ρ to Gal(Q/F ), getting ρm : Gal(Q/F ) → GL2 (A). By the universality of ρ , we ﬁnd a unique morphism ϕ : R → A such that ϕ◦ρ ∼ = ρm . Since ϕ◦(ρ )σ = (ϕ◦ρ )σ ∼ = ρσm ∼ = ρm ∼ = ϕ◦ρ , we ﬁnd that ϕ ◦ σ = ϕ. Thus ϕ factors through R. By the uniqueness of the extension to Gal(Q/F ), we ﬁnd that ϕ ◦ ρ ∼ = ρ; so, (R, ρ ) is universal for Φ. Proposition 3.71 Suppose that F /F satisﬁes (E1–3) in the previous subsection and that p > [F : F ] = 2. If R is a local complete intersection free of ﬁnite rank over W [[ΓF ]], R is a local complete intersection free of ﬁnite rank over W [[ΓF ]] and is isomorphic to TF . This proposition follows from Fujiwara’s identiﬁcation of R∅ with T∅ in [Fu] when [F : Q] has odd degree (Theorem 3.67) in the same manner as in the proof of Theorem 3.50. Since we only described his result in the case of the even-degree base ﬁeld, we brieﬂy indicate how to deduce this proposition from the result over the quadratic extension F /F satisfying (E1–3), further assuming that p ≥ 5 and that F /Q is unramiﬁed at p (these assumptions are necessary for us here because of the use of Corollary 4.21 in the proof).

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Proof Write Λ (resp. Λ) for W [[ΓF ]] (resp. W [[ΓF ]]). Decomposing ΓF = − ΓF × Γ− F for the “−” eigenspace ΓF of ΓF under the action of σ, then Λ = − Λ[[y1 , . . . , ys ]] for s = rankZp ΓF , and we may assume that σ ﬁxes Λ and the action of σ on yj is given by σ(yj ) = −yj . Thus Λ /Λ (σ − 1)Λ ∼ = Λ, and Λ (σ − 1)Λ is generated by yj for all j. We take a presentation R = Λ [[x1 , . . . , xr ]]/(f1 , . . . , fr ) = Λ[[y1 , . . . , ys , x1 , . . . , xr ]]/(f1 , . . . , fr ). The group Gal(F /F ) acts on Λ through its action on ΓF . Then the Λ -algebra structure on R is compatible with the action of Gal(F /F ) on R . Decomposing R = R− ⊕ R+ as a direct sum of the ±-eigenspace of the action of σ. We may choose the presentation so that the image xj of xj for j = 1, . . . , m falls in R+ and xm+1 , . . . , xr falls in R− . Then we may extend σ to Λ [[x1 , . . . , xr ]] by σ(xj ) = ±xj accordingly as σ(xj ) = ±xj . Since (f1 , . . . , fr ) is stable under σ, we may assume that σ(fj ) = ±fj and σ(fi ) = fi ⇔ i ≤ k for a positive integer k. Under these circumstances, R (σ −1)R is generated by yi for all i = 1, . . . , s and xj with j > m, and we get R = R /R (σ − 1)R ∼ = Λ[[x1 , . . . , xm ]]/(f1 , . . . , fk ). Thus R is a local complete intersection. We now show k = m. Take a locally cyclotomic point ϕ in Spf(R )(W ) ﬁxed by σ. We have the following exact sequence of W [Gal(F /F )]-modules: (f1 , . . . , fr ) ⊗R ,ϕ W (f1 , . . . , fr )2 i

1 −→ ΩΛ [[x1 ,...,xr ]]/Λ ⊗Λ [[x1 ,...,xr ]],ϕ W → ΩR /Λ ⊗R ,ϕ W → 0.

By the automorphic base-change, P is in Spf(R)(W ). Since ΩR /Λ ⊗R ,ϕ W ∼ = ΩTκ0 ,ε0 /W ⊗Tκ0 ,ε0 W is a ﬁnite W -module, and i1 is injective (see Lemma 5.23(1)). Tensoring K over W , we ﬁnd that (f1 , . . . , fr ) ∼ ⊗R ,ϕ K −→ ΩΛ [[x1 ,...,xr ]]/Λ ⊗Λ [[x1 ,...,xr ]],ϕ K. 2 i (f1 , . . . , fr ) 1 The number m is the number of eigenvalues equal 1 of σ on the right-hand side and k is the one on the left-hand side. Thus k = m, and we conclude dim R = dim Λ and that R is a local complete intersection over Λ. In particular, the nilradical Rad(R) of R is the intersection of minimal prime ideals p with dim R/p = dim Λ (the unmixedness theorems for Cohen–Macaulay rings; see [CRT] Theorems 17.6 and 21.3). By the universality, we have the canonical surjective algebra homomorphisms ι : R → T = TF for a local ring TF of the corresponding Hecke algebra for F . By the automorphic base-change, writing the total quotient ring of a reduced algebra A as Q(A), Q(R )/Q(R )(γ − 1)Q(R ) = Q(T).

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Thus Rad(R) = Rad(Q(R)) ∩ R = 0, and ι is generically an isomorphism. Since Ker(ι) = Rad(R) = 0, ι is globally an isomorphism. Since TF is free of ﬁnite rank over Λ if p ≥ 5 (see Corollary 4.22), we conclude that RF is a local complete intersection free of ﬁnite rank over Λ. 2 3.4 L-invariants of Hilbert modular forms In this section, F is a totally real ﬁeld of ﬁnite degree (even or odd). We n.ord start with a Hilbert modular Hecke eigenform f0 ∈ S(0,I) (N, ε0 ; C) satisfying (h1–4) and (sf) in 3.2.1. Let T be the local ring of the Hecke algebra as in Theorem 3.50 associated to f0 . Pick a locally cyclotomic point P ∈ Spf(T)(W ), and let f = fP ∈ Sκn.ord (N, ε; C) be the Hecke eigenform associated to P . Then f is a locally cyclotomic Hecke eigenform of weight (κ, ε), which gives a minimal vector at primes outside p. We prove a Hilbert modular version of Theorem 1.80 (as Theorem 3.73) relating the L-invariant L(IndQ F Ad(ρf )) to the Greenberg

∂a(p) derivative determinant P (det ) ∈ K. Here, a(p) ∈ T is the image ∂xp p,p |p of Up (pp ) in T, xp ∈ T is given by δT ([γp , Fp ]) − 1, and ρ = ρf is realized on L = W 2. 3.4.1 Statement of the result Let us now formulate our theorem in a precise way. Throughout this section, we assume that p is an odd prime. The weight of a Hilbert modular form is an algebraic character κ of the dia2 gonal torus TG = ResF/Q Gm of G = ResF/Q GL(2), and therefore, κ induces a character 2 × Fp× → Qp TG (Qp ) = Fp× = p|p

again denoted by κ, where Fp is the p-adic completion of F at a prime factor p of p in F . If κ is locally cyclotomic, we normalize κ so that for xj ∈ p|p Fp× , (Np (x1,p )κ1,p Np (x2,p )κ2,p ) (x1 , x2 )κ = κ(x1 , x2 ) = p|p

for the norm map Np = NFp /Qp . We assume that the cusp form f is nearly p-ordinary and has locally cyclotomic weight κ with κ2,p − κ1,p ≥ 1. Since the central character of a Hilbert modular form of weight κ has inﬁnity type (1−[κ])I for an integer [κ] = κ1,p + κ2,p (see (2.3.10)), the sum κ1,p + κ2,p is independent of p ∈ Σp . Write F for the residue ﬁeld of W . We put V = L ⊗Zp Qp on which Gal(Q/F ) acts via ρf . Note that f is everywhere non-supercuspidal. Indeed, by near p-ordinarity, U (p ) has a nonzero eigenvalue for f ; so, the automorphic representation π generated by f cannot be supercuspidal at p|p (Lemma 2.40).

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Since the level of f outside p is equal to the level of f0 outside p and the prime-to-p part of the Neben characters ε0 and ε is the same (by the proof of Theorem 3.65), πq for a prime q p cannot be supercuspidal (see 2.3.7 in the text and [H88b] Lemma 12.2). Write c(f ) for the conductor of the irreducible automorphic representation generated by the right translations of f . The automorphic representation π generated by f has local q-factor isomorphic either to a principal π(η1,q , η2,q ) or to a special σ(η1,q , η2,q ), and the q-primary part of c(f ) is given by c(η1,q )c(η2,q ) = c(ε1,q )c(ε2,q ) if πq = π(η1,q , η2,q ) and by c(η1,q )2 q = c(ε1,q )2 q if πq = σ(η1,q , η2,q ). Thus pNc(ε) and pc(f ) have the same prime factors. The weight condition means that the motive M (f ) associated to f has the p-adic Hodge–Tate type (κ1,p , κ2,p ) at each p-adic place p (see Theorem 2.43(3)). For each embedding σ