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*Table of contents : Contents......Page 4Preface......Page 8Preface to the English edition......Page 14Synopsis......Page 16Elliptic curves......Page 28Modular forms......Page 50Galois representations......Page 96The 3–5 trick......Page 126R=T......Page 134Commutative algebra......Page 158Deformation rings......Page 174Appendix A......Page 186Bibliography......Page 204Symbol index......Page 212Subject index......Page 214*

Fermat's Last Theorem Basic Tools

Iwanami Series in Modern Mathematics

10.1090/mmono/243

Translations of

Mathematical Monographs Volume 243

Fermat's Last Theorem Basic Tools Takeshi Saito Translated from the Japanese by Masato Kuwata

American Mathematical Society Providence, Rhode Island

FERUMA YOSO (Fermat Conjecture)

by Takeshi Saito

c 2009 by Takeshi Saito First published 2009 by Iwanami Shoten, Publishers, Tokyo. This English language edition published in 2013 by the American Mathematical Society, Providence by arrangement with the author c/o Iwanami Shoten, Publishers, Tokyo Translated from the Japanese by Masato Kuwata 2010 Mathematics Subject Classiﬁcation. Primary 11D41; Secondary 11G05, 11F11, 11F80, 11G18. Library of Congress Cataloging-in-Publication Data Saito, Takeshi, 1961– Fermat’s last theorem: basic tools / Takeshi Saito ; translated by Masato Kuwata.—English language edition. pages cm.—(Translations of mathematical monographs ; volume 243) First published by Iwanami Shoten, Publishers, Tokyo, 2009. Includes bibliographical references and index. ISBN 978-0-8218-9848-2 (alk. paper) 1. Fermat’s last theorem. 2. Number theory. 3. Algebraic number theory. I. Title. II. Title: Fermat’s last theorem: basic tools. QA244.S2513 2013 512.7’4–dc23 2013023932 c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability. Information on copying and reprinting can be found in the back of this volume. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

18 17 16 15 14 13

Contents Preface

ix

Preface to the English Edition

xv

Chapter 0. Synopsis 0.1. Simple paraphrase 0.2. Elliptic curves 0.3. Elliptic curves and modular forms 0.4. Conductor of an elliptic curve and level of a modular form 0.5. -torsion points of elliptic curves and modular forms

1 1 3 5 7 9

Chapter 1. Elliptic curves 1.1. Elliptic curves over a ﬁeld 1.2. Reduction mod p 1.3. Morphisms and the Tate modules 1.4. Elliptic curves over an arbitrary scheme 1.5. Generalized elliptic curves

13 13 15 22 26 29

Chapter 2. Modular forms 2.1. The j-invariant 2.2. Moduli spaces 2.3. Modular curves and modular forms 2.4. Construction of modular curves 2.5. The genus formula 2.6. The Hecke operators 2.7. The q-expansions 2.8. Primary forms, primitive forms 2.9. Elliptic curves and modular forms 2.10. Primary forms, primitive forms, and Hecke algebras 2.11. The analytic expression 2.12. The q-expansion and analytic expression

35 35 37 40 44 52 55 58 62 65 66 70 74

v

vi

CONTENTS

2.13.

The q-expansion and Hecke operators

77

Chapter 3. Galois representations 3.1. Frobenius substitutions 3.2. Galois representations and ﬁnite group schemes 3.3. The Tate module of an elliptic curve 3.4. Modular -adic representations 3.5. Ramiﬁcation conditions 3.6. Finite ﬂat group schemes 3.7. Ramiﬁcation of the Tate module of an elliptic curve 3.8. Level of modular forms and ramiﬁcation

81 82 86 89 91 96 100 103 108

Chapter 4. The 3–5 trick 4.1. Proof of Theorem 2.54 4.2. Summary of the Proof of Theorem 0.1

111 111 116

Chapter 5. R = T 5.1. What is R = T ? 5.2. Deformation rings 5.3. Hecke algebras 5.4. Some commutative algebra 5.5. Hecke modules 5.6. Outline of the Proof of Theorem 5.22

119 119 122 126 131 135 137

Chapter 6. Commutative algebra 6.1. Proof of Theorem 5.25 6.2. Proof of Theorem 5.27

143 143 149

Chapter 7. Deformation rings 7.1. Functors and their representations 7.2. The existence theorem 7.3. Proof of Theorem 5.8 7.4. Proof of Theorem 7.7

159 159 161 162 166

Appendix A. Supplements to scheme theory A.1. Various properties of schemes A.2. Group schemes A.3. Quotient by a ﬁnite group A.4. Flat covering A.5. G-torsor A.6. Closed condition A.7. Cartier divisor

171 171 175 177 178 179 182 183

CONTENTS

A.8. Smooth commutative group scheme

vii

185

Bibliography

189

Symbol Index

197

Subject Index

199

Preface It has been more than 350 years since Pierre de Fermat wrote in the margin of his copy of Arithmetica of Diophantus: It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree; I have discovered a truly remarkable proof which this margin is too small to contain.1

This is what we call Fermat’s Last Theorem. It is certain that he has a proof in the case of cubes and biquadrates (i.e., fourth powers), but it is now widely believed that he did not have a proof in the higher degree cases. After enormous eﬀort made by a great number of mathematicians, Fermat’s Last Theorem was ﬁnally proved by Andrew Wiles and Richard Taylor in 1994. The purpose of this book is to give a comprehensive account of the proof of Fermat’s Last Theorem. Although Wiles’s proof is based on very natural ideas, its framework is quite complex, some parts of it are very technical, and it employs many diﬀerent notions in mathematics. In this book I included parts that explain the outline of what follows before introducing new notions or formulating the proof formally. Chapter 0 and §§5.1, 5.5, and 5.6 in Chapter 5 are those parts. Logically speaking, these are not necessary, but I included these in order to promote better understanding. Despite the aim of this book, I could not prove every single proposition and theorem. For the omitted proofs please consult the references indicated at the end of the book. The content of this book is as follows. We ﬁrst describe the rough outline of the proof. We relate Fermat’s Last Theorem with elliptic 1 Written originally in Latin. English translation is taken from Dickson, L. E., History of the theory of numbers. Vol. II: Diophantine analysis, Chelsea Publishing Co., New York, 1966.

ix

x

PREFACE

curves, modular forms, and Galois representations. Using these relations, we reduce Fermat’s Last Theorem to the modularity of certain -adic representations (Theorem 3.36) and a theorem on the level of mod representations (Theorem 3.55). Next, we introduce the notions of deformation rings and Hecke algebras, which are incarnations of Galois representations and modular forms, respectively. We then prove two theorems on commutative algebra. Using these theorems, we reduce Theorem 3.36 to certain properties of Selmer groups and Hecke modules, which are also incarnations of Galois representations and modular forms. We then construct fundamental objects, modular curves over Z, and the Galois representations associated with modular forms. The latter lie in the foundation of the entire proof. We also show a part of the proof of Theorem 3.55. Finally, we deﬁne the Hecke modules and the Selmer groups, and we prove Theorem 3.36, which completes the proof of Fermat’s Last Theorem. The content of each chapter is summarized at its beginning, but we introduce them here brieﬂy. In Chapter 0, we show that Fermat’s Last Theorem is derived from Theorem 0.13, which is about the connection between elliptic curves and modular forms, and Theorem 0.15, which is about the ramiﬁcation and level of -torsion points of an elliptic curve. The objective of Chapters 1–4 is to understand the content of Chapter 0 more precisely. The precise formulations of Theorems 0.13 and 0.15 will be given in Chapters 1–3. In the proof presented in Chapter 0, the leading roles are played by elliptic curves, modular forms, and Galois representations, each of which will be the main theme of Chapters 1, 2, and 3. In Chapter 3, the modularity of -adic representations will be formulated in Theorem 3.36. In Chapter 4, using Theorem 4.4 on the rational points of an elliptic curve, we deduce Theorem 0.13 from Theorem 3.36. In §4.2, we review the outline of the proof of Theorem 0.1 again. In Chapters 5–7, we describe the proof of Theorem 3.36. The principal actors in this proof are deformation rings and Hecke algebras. The roles of these rings will be explained in §5.1. In Chapter 5, using two theorems of commutative algebra, we deduce Theorem 3.36 from Theorems 5.32, 5.34, and Proposition 5.33, which concern the properties of Selmer groups and Hecke modules. The two theorems in commutative algebra will be proved in Chapter 6. In Chapter 7, we will prove the existence theorem of deformation rings.

PREFACE

xi

In Chapter 8,∗ we will deﬁne modular curves over Z and study their properties. Modular forms are deﬁned in Chapter 2 using modular curves over Q, but their arithmetic properties are often derived from the behavior of modular curves over Z at each prime number. Modular curves are known to have good reduction at primes not dividing their levels, but it is particularly important to know their precise properties at the prime factors of the level. A major factor that made it possible to prove Fermat’s Last Theorem within the twentieth century is that properties of modular curves over Z had been studied intensively. We hope the reader will appreciate this fact. In Chapter 9,∗ we construct Galois representations associated with modular forms using the results of Chapter 8, and prove a part of Theorem 3.55 which describes the relation between ramiﬁcation and the level. Unfortunately, however, we could not describe the celebrated proof of Theorem 3.55 in the case of p ≡ 1 mod by K. Ribet because it requires heavy preparations, such as the p-adic uniformization of Shimura curves and the Jacquet–Langlands–Shimizu correspondence of automorphic representations. In Chapter 10,∗ using results of Chapters 8 and 9, we construct Hecke modules as the completion of the singular homology groups of modular curves, and we then prove Theorem 5.32(2) and Proposition 5.33. In Chapter 11, we introduce the Galois cohomology groups and deﬁne the Selmer groups. Then we prove Theorems 5.32(1) and 5.34. The ﬁrst half of Chapter 11 up to §11.3 may be read independently as an introduction to Galois cohomology and the Selmer groups. Throughout the book, we assume general background in number theory, commutative algebra, and general theory of schemes. These are treated in other volumes in the Iwanami series: Number Theory 1 , 2, and 3, Commutative algebras and ﬁelds (no English translation), and Algebraic Geometry 1 and 2. For scheme theory, we give a brief supplement in Appendix A after Chapter 7. Other prerequisites are summarized in Appendices B, C, and D at the end of the volume.∗ In Appendix B, we describe algebraic curves over a discrete valuation rings and semistable curves in particular, as an algebro-geometric preparation to the study of modular curves over Z. In Appendix C, we give a linear algebraic description of ﬁnite ﬂat commutative group ∗ Chapters 8, 9, and 10 along with Appendices B, C, and D will appear in Fermat’s Last Theorem: The Proof, a forthcoming translation of the Japanese original.

xii

PREFACE

schemes over Zp , which will be important for the study of p-adic Galois representations of p-adic ﬁelds. Finally, in Appendix D, we give a summary on the Jacobian of algebraic curves and its N´eron model, which are indispensable to study the Galois representations associated with modular forms. If we gave a proof of every single theorem or proposition in Chapters 1 and 2, it would become a whole book by itself. So, we only give proofs of important or simple properties. Please consider these chapters as a summary of known facts. Reading the chapters on elliptic curves and modular forms in Number Theory 1 ,2, and 3 would also be useful to the reader. At the end of the book, we give references for the theorems and propositions for which we could not give proofs in the main text. The interested reader can consult them for further information. We regret that we did not have room to mention the history of Fermat’s Last Theorem. The reader can also refer to references at the end of the book. Due to the nature of this book, we did not cite the original paper of each theorem or proposition, and we beg the original authors for mercy. I would be extremely gratiﬁed if more people could appreciate one of the highest achievements of the twentieth century in mathematics. I would like to express sincere gratitude to Professor Kazuya Kato for proposing that I write this book. I would also thank Masato Kurihara, Masato Kuwata, and Kazuhiro Fujiwara for useful advice. Also, particularly useful were the survey articles [4], [5], and [24]. I express here special thanks to their authors. This book was based on lectures and talks at various places, including the lecture course at the University of Tokyo in the ﬁrst semester of 1996, and intensive lecture courses at Tohoku University in May 1996, at Kanazawa University in September 1996, and at Nagoya University in May 1999. I would like to thank all those who attended these lectures and took notes. I would also like to thank former and current graduate students at the University of Tokyo, Keisuke Arai, Shin Hattori, and Naoki Imai, who read the earlier manuscript carefully and pointed out many mistakes. Most of the chapters up to Chapter 7 were written while I stayed at Universit´e Paris-Nord, MaxPlanck-Institut f¨ ur Mathematik, and Universit¨at Essen. I would like to thank these universities and the Institute for their hospitality and for giving me an excellent working environment.

PREFACE

xiii

This book is the combined edition of the two books in the Iwanami series The Development of Modern Mathematics: Fermat’s Last Theorem 1 ﬁrst published in March 2000 and containing up to Chapter 7; and Fermat’s Last Theorem 2 published in February 2008. Since 1994 when the proof was ﬁrst published, the development of this subject has been remarkable: Conjecture 3.27 has been proved, and Conjecture 3.37 has almost been proved. Also, Theorem 5.22 has been generalized widely, and its proof has been simpliﬁed greatly. We should have rewritten many parts of this book to include recent developments, but we decided to wait until another opportunity arises. On the occasion of the second edition, we made corrections to known errors. However, we believe there still remain many mistakes yet to be discovered. I apologize in advance, and would be grateful if the reader could inform me. Takeshi Saito Tokyo, Japan November 2008

Preface to the English Edition This is the ﬁrst half of the English translation of Fermat’s Last Theorem in the Iwanami series, The Development of Modern Mathematics. Though the translation is based on the second combined edition of the original Japanese book published in 2008, it will be published in two volumes. The ﬁrst volume, Fermat’s Last Theorem: Basic Tools, contains Chapters 1–7 and Appendix A. The second volume, Fermat’s Last Theorem: The Proof , which will be published in a short while, contains Chapters 8–11 and Appendices B, C, and D. The author hopes that, through this edition, a wider audience of readers will appreciate one of the deepest achievements of the twentieth century in mathematics. My special thanks are due to Dr. Masato Kuwata, who not only translated the Japanese edition into English but also suggested many improvements in the text so that the present English edition is more readable than the original Japanese edition. Takeshi Saito Tokyo, Japan June 2013

xv

10.1090/mmono/243/01

CHAPTER 0

Synopsis The purpose of this book is to give a comprehensive account of the proof of the following theorem, known as Fermat’s Last Theorem: Theorem 0.1. Let n be an integer greater than or equal to 3. If integers X, Y , and Z satisfy the equation (0.1)

X n + Y n = Z n,

then at least one of X, Y , and Z must be 0. A ﬂow diagram of the proof can be drawn as follows: (0.2)

(a solution of (0.1)) =⇒ (an elliptic curve) =⇒ (a modular form) =⇒ (contradiction)

If we try to explain the meaning of this diagram in a few sentences, it goes as follows. Assume there exists a nontrivial solution to the equation (0.1), and we would like to derive a contradiction. To this end, we deﬁne an elliptic curve using such a solution. We then show that such an elliptic curve is closely associated with a modular form with certain properties. Finally, we derive a contradiction by showing that such a modular form could not exist. In this chapter we give a further explanation of the above diagram. As we can easily see, elliptic curves and modular forms play leading roles in the proof. By following the outline of the proof, the reader should familiarize her/himself with these two subjects. We indicate where the details of certain topics are treated in the main text. Skipping some unfamiliar terminology, the reader should grasp the ﬂow of the proof. 0.1. Simple paraphrase As a matter of fact, we will prove the following theorem which is stronger than Theorem 0.1. 1

2

0. SYNOPSIS

Theorem 0.2. Let be a prime number with ≥ 5, and let a be an integer with a ≥ 4. Then, the equation (0.3)

X + 2a Y = Z

has no integer solutions (X, Y, Z) such that all X, Y , and Z are odd. Let us verify that Theorem 0.1 follows from Theorem 0.2. Proof of Theorem 0.2 ⇒ Theorem 0.1. First, decomposing n into prime factors, we can see that in order to show Theorem 0.1, it suﬃces to show the cases where n = 4 and where n is a prime number greater than or equal to 3. The case n = 4 is nothing but Proposition 1.1 in Chapter 1 of Number Theory 1 . For the case of n = 3, we ﬁnd a proof in §4.1(b) in Chapter 4 of Number Theory 1 . Thus, it suﬃces to show for the case where n is a prime number with ≥ 5. The argument is a repetition of the one in §4.4 in Chapter 4 of Number Theory 1 . Let n be a prime number ≥ 5. Assume (0.1) has a nontrivial solution (X, Y, Z) = (A, B, C), and we derive a contradiction to Theorem 0.2. A solution (A, B, C) of (0.1) is called nontrivial if none of A, B, and C is 0. Dividing by their greatest common divisor, we may assume that the greatest common divisor of A, B, and C is 1. Then, considering the residue modulo 2, we see that one of A, B, and C is an even number and the others are odd. We may assume B is even by rearranging A, B, and C as follows, if necessary. If A is even, replace A with B. If C is even, replace the solution (A, B, C) with (A, −C, −B). Now that B is even, let m be the largest integer such that 2m divides B, and let B = 2m B . We have m ≥ 1 and B is odd. Then, (X, Y, Z) = (A, B , C) is a solution of (0.3) with a = m. Since we have a ≥ 5, and A, B , and C are all odd; this contradicts Theorem 0.2. In the case of n = 3, equation (0.3) deﬁnes an elliptic curve. We proved Theorem 0.1 for n = 3 by studying the rational points of this elliptic curve. In the case of n = 4, the curve deﬁned by (0.3) is not an elliptic curve. However, in this case too, we prove Theorem 0.1 by studying the rational points of an elliptic curve closely related to this curve. On the other hand, in the case of n = ≥ 5, we deﬁne an elliptic curve using a solution of (0.1), as we will see in the next section. We then prove that such an elliptic curve cannot exist. Unlike the case where n = 3, 4, it is not the existence of rational points on

0.2. ELLIPTIC CURVES

3

an elliptic curve but the existence of an elliptic curve itself that is the issue in the case of n = ≥ 5. 0.2. Elliptic curves The ﬁrst arrow in the diagram (0.2) is to paraphrase the problem in terms of elliptic curves. We will study elliptic curves in Chapter 1. For those who are interested only in the proof of Theorem 0.1, it is not too inappropriate to think that an elliptic curve is a curve deﬁned by the equation in x and y given by (0.4)

y 2 = x(x − C )(x − B ).

Here, B and C are nonzero distinct integers. In terms of elliptic curves, Theorem 0.2 is equivalent to the following. Theorem 0.3. Let ≥ 5 be a prime number. Then there does not exist an elliptic curve deﬁned over the rational number ﬁeld Q satisfying the following three conditions: (i) All 2-torsion points of E are Q-rational. (ii) E is semistable. (iii) The group of -torsion points E[] is good at all odd prime numbers p. We will explain the meaning of the terms appearing in Theorem 0.3 in Chapters 1 and 3. Note that the notion of “good at a prime number p”, appearing in (iii), is a technical term which will be deﬁned in Deﬁnition 3.31. The equivalence between Theorems 0.2 and 0.3, or in other words, the connection between Fermat’s Last Theorem and elliptic curves, is given by Proposition 0.4 below. To put it plainly, if (0.1) or (0.3) had a solution, the elliptic curve constructed from it would have too good a property to exist. Before stating the proposition, we introduce some notation. For distinct nonzero integers m and n, we denote by En,m the elliptic curve over Q deﬁned by the equation (0.5)

y 2 = x(x − n)(x − m).

Proposition 0.4. Let be an odd prime number. The following conditions on an elliptic curve E over Q are equivalent: (i) E satisﬁes all three conditions in Theorem 0.3.

4

0. SYNOPSIS

(ii) E is isomorphic to an elliptic curve En,m with n and m satisfying the following condition (0.6). (0.6)

The integers n and m are nonzero, distinct, and relatively prime. In addition, n ≡ −1 mod 4, n and n − m are both -th powers, and m is of the form 2a b with a ≥ 4.

By Proposition 0.4 we can show that Theorems 0.2 and 0.3 are equivalent. Here, we prove the fact that Theorem 0.2 follows from Theorem 0.3 and Proposition 0.4. In fact, the converse is not necessary for the proof of Theorem 0.1. Proof of Theorem 0.3 + Proposition 0.4 ⇒ Theorem 0.2. It suﬃces to show that a counterexample of Theorem 0.2 gives a counterexample of Theorem 0.3. By Proposition 0.4, if there are integers n and m satisfying (0.6), the elliptic curve En,m is a counterexample of Theorem 0.3. Let ≥ 5 be a prime number, and let a ≥ 4 be an integer. Suppose (0.3) has a solution (X, Y, Z) = (A, B, C) such that A, B, and C are all odd. From this we would like to ﬁnd integers n and m satisfying (0.6). Dividing by their greatest common divisor, we may assume that the greatest common divisor of A, B, and C is 1. Notice that (X, Y, Z) = (−A, −B, −C) is again a solution of (0.3) such that all members are odd. Since either C or −C is congruent to −1 modulo 4, we may assume C ≡ −1 mod 4 by replacing C by −C, if necessary. Let n = C and m = 2a B . We show that these satisfy (0.6). Both n and m are nonzero, they are distinct, and they are relatively prime. Since C ≡ −1 mod 4, we have n ≡ −1 mod 4. Moreover, both n = C and n−m = A are -th powers, and m = 2a B is the product of an -th power and 2 to the power a ≥ 4. Thus, n and m satisfy the condition (0.6). This gives a counterexample of Theorem 0.3. Question 1. Verify the fact that Theorem 0.3 follows from Theorem 0.2 and Proposition 0.4. Fermat’s Last Theorem (Theorem 0.1) is thus reduced to Theorem 0.3, which is about elliptic curves. We can summarize as follows. Let n = be a prime number greater than or equal to 5. Suppose there exists a nontrivial solution (X, Y, Z) = (A, B, C) to equation (0.1). Rearranging suitably, we may assume A, B, and C are relatively prime, B is even, and C ≡ −1 mod 4. Then, the elliptic curve EC ,B

0.3. ELLIPTIC CURVES AND MODULAR FORMS

5

deﬁned by y 2 = x(x − C )(x − B ) gives a counterexample of Theorem 0.3. The proof of Theorem 0.3 is given by studying the relation between elliptic curves and modular forms. We will see its outline in the following sections. 0.3. Elliptic curves and modular forms The second arrow in diagram (0.2) is the connection between elliptic curves and modular forms. It is often through this connection to modular forms that a profound arithmetic property of elliptic curves reveal themselves. The proof of Fermat’s Last Theorem is an example of this. We will study modular forms in detail in Chapter 2. Here, we only introduce necessary terminology in order to explain the outline. In Chapter 2, we will deﬁne a ﬁnite dimensional complex vector space S(N )C called the space of modular forms of level N for each integer N ≥ 1. This is a subspace of the space of formal power series C[[q]]. An important property of the space of modular forms is that an endomorphism Tn : S(N )C → S(N )C called the Hecke operator is deﬁned for each positive integer n. Among the most important modular forms are normalized cusp forms that are simultaneous eigenforms of all Hecke operators. Since such a form appears many times, we call it in this book a “primary form” for short. Its formal deﬁnition is as follows. Definition 0.5. A modular form of level N ∞ am (f )q m ∈ S(N )C (0.7) f= m=1

is called a primary form if it is nonzero, and it satisﬁes (0.8)

Tn f = an (f )f

for all integers n ≥ 1. A modular form f is determined by the coeﬃcients am (f ), m = 1, 2, 3, . . . , as in (0.7). As a matter of fact, a primary form can be determined only by the coeﬃcients ap (f ), where p = 2, 3, 5, 7, . . . are prime numbers. To formulate the connection between elliptic curves and modular forms, we deﬁne a sequence ap (E) for an elliptic curve E. Its precise deﬁnition will be given in Chapter 1, but it is roughly as follows.

6

0. SYNOPSIS

Let E be an elliptic curve. Consider an equation with integer coeﬃcients that deﬁnes E, such as (0.4) for example. Considering this equation modulo p for each prime number p, we obtain an equation with coeﬃcients in the ﬁnite ﬁeld Fp . Except for a ﬁnite number of p, this equation deﬁnes an elliptic curve over Fp , which we denote by EFp . Since the set EFp (Fp ) of all Fp -valued points of EFp is a ﬁnite set, we deﬁne (0.9)

ap (E) = p + 1 − EFp (Fp ),

where stands for the number of elements. Example 0.6. Let n and m be distinct nonzero integers relatively prime to each other such that (0.10)

n ≡ −1 mod 4,

m ≡ 0 mod 16.

Let E = En,m . By Proposition 0.4, E is a semistable elliptic curve. For a prime number p, the equation y 2 = x(x − n)(x − m) with Fp coeﬃcients deﬁnes an elliptic curve over Fp if and only if p does not divide nm(n − m). For such a prime number p, we have (0.11) EFp (Fp ) = {(x, y) ∈ Fp × Fp | y 2 = x(x − n)(x − m)} ∪ {∞}. ap (E) equals the diﬀerence A − B of the number A (resp. B) of elements x = 0, n, m in Fp such that x(x − n)(x − m) is nonsquare (resp. square). Using the quadratic residue symbol, we have x(x − n)(x − m) ap (E) = − . p x∈Fp ,x=0,n,m

Definition 0.7. An elliptic curve E over the rational number ∞ ﬁeld Q is modular if there exists a primary form f = m=1 am (f )q m such that (0.12)

ap (E) = ap (f )

for all but a ﬁnite number of prime numbers p. The contents of this book are, in substance, an account of the proof of Theorem 0.8, or of Theorem 0.13, which is a partial but reﬁned result of it. Theorem 0.8. Any elliptic curve E over the rational number ﬁeld In other words, there exists a primary form f = ∞ Q is modular. m a (f )q such that ap (E) = ap (f ) for all but a ﬁnite number m=1 m of prime numbers p.

0.4. CONDUCTOR OF ELLIPTIC CURVE, LEVEL OF MODULAR FORM

7

This theorem is the second arrow of the diagram (0.2). In this book we explain the proof of Theorem 0.8 in the case where elliptic curve E is semistable. On the one hand, the proof for the general case is much more complicated, on the other hand this special case is already enough to prove Theorem 0.3. Using Theorem 0.8, we can prove Theorem 0.3 as follows. Suppose an elliptic curve E over Q satisﬁes the conditions (i) to (iii) in Theorem 0.3. Then, by Theorem 0.8, there exists a primary form f such that ap (E) = ap (f ) for all but a ﬁnite number of p. Now, it suﬃces to show that the existence of such a primary form contradicts the conditions (i) to (iii) in Theorem 0.3. This is the last arrow of the diagram (0.2). 0.4. Conductor of an elliptic curve and level of a modular form Since what the last arrow of the diagram (0.2) means is rather complicated, we will illustrate it using its easier analogue, the case of quadratic number ﬁelds. Proposition 0.9. There exists no quadratic extension of Q that is unramiﬁed at every prime number. √ If we write a quadratic extension in the form Q( a), it is not so diﬃcult to prove Proposition 0.9 directly. Here, however, we would like to think it as an analogue of Theorem 0.3, and we derive it from the following proposition. Proposition 0.10. (1) Any quadratic extension of Q is a subﬁeld of a cyclotomic ﬁeld. (2) Let N be an integer, and let p be a prime number. Let pe be the largest power of p that divides N , and let N = pe M . If a subﬁeld K of the cyclotomic ﬁeld Q(ζN ) is unramiﬁed at p, then K is a subﬁeld of Q(ζM ). Proposition 0.10 is a part of class ﬁeld theory. Proposition 0.10(1) and (2) are Theorem 5.10(1) and (3) in Chapter 5 of Number Theory 2 , respectively. We can derive Proposition 0.9 from Proposition 0.10 as follows. Proof of Proposition 0.10 ⇒ Proposition 0.9. Let K be a quadratic extension unramiﬁed at all prime numbers. By Proposition 0.10(1), there exists a cyclotomic ﬁeld Q(ζN ) that contains K as

8

0. SYNOPSIS

a subﬁeld. Applying Proposition 0.10(2) for each prime factor of N repeatedly, we conclude that K is a subﬁeld of Q = Q(ζ1 ), which is a contradiction. We would like to prove Theorem 0.3 using a similar argument. In other words, we would like to replace quadratic extensions by elliptic curves, and cyclotomic ﬁelds by modular forms. Proposition 0.10(1) corresponds to Theorem 0.8. Just as quadratic extensions are contained in cyclotomic ﬁelds, elliptic curves are associated with modular forms. However, we still do not have a statement corresponding to Proposition 0.10(2), namely, the relation between the conductor of an elliptic curve and the level of a modular form. In the above proof, it is important to know in which cyclotomic ﬁeld a quadratic ﬁeld is contained. Similarly, it is important to know what level of modular form an elliptic curve is related to. So, we state a reﬁnement of Deﬁnition 0.7. Definition 0.11. An elliptic curve E over ∞Q is called modular of level N , if there exists a primary form f = m=1 am (f )q m ∈ S(N )C of level N such that ap (E) = ap (f ) for any prime number p not dividing N . To make the story simple, we focus only on semistable elliptic curves in the following. For a semistable elliptic curve, its conductor is deﬁned as the product of all prime numbers at which E has bad reduction. The conductor N is thus square-free. Example 0.12. Let E = En,m be the semistable elliptic curve in Example 0.6. In this case, the conductor N of E is the product of all 1 nm(n − m). the primes dividing 16 This is Proposition 1.9(2) in Chapter 1. We have the following reﬁnement of Theorem 0.8, which describes the relation between the conductor and the level. Theorem 0.13. Let E be a semistable elliptic curve over Q, and let N be the conductor of E. Then, E is modular of level N . Unfortunately, this theorem is not enough to prove Theorem 0.3. To the elliptic curve E = En,m in Example 0.12, there exists a primary form of level N , which leads to no contradiction. In order to derive a contradiction, we use condition (iii) in Theorem 0.3, but we are not ready for it yet. To do so, we use not the equality ap (E) = ap (f ),

-TORSION POINTS AND MODULAR FORMS

9

but the congruence relation ap (E) ≡ ap (f ) mod as a condition on f . In fancier terms, we look at not only the elliptic curve E but also at the group of -torsion points E[] as a representation of the absolute Galois group GQ = Gal(Q/Q), and we study its relation to the modular forms. 0.5. -torsion points of elliptic curves and modular forms In order to formulate the relation between the group of -torsion points of E and the modular forms more precisely, we would like to study modular forms a little more. In §0.3, we introduced the space S(N )C of modular forms of level N as a ﬁnite dimensional complex vector space. However, it turns out that we can deﬁne more naturally a ﬁnite dimensional Qvector space S(N )Q of modular forms of level N with Q-coeﬃcients. The C-vector space S(N )C is the extension of coeﬃcients of S(N )Q to the complex number ﬁeld. The Hecke operators are also deﬁned m ∈ as endomorphisms Tn : S(N )Q → S(N )Q . If f = ∞ m=1 am (f )q S(N )C is a primary form, then each coeﬃcient am (f ) is an algebraic number, and we can see from this fact that the ﬁeld Q(f ) = Q(am (f ), m ∈ N) generated by all the coeﬃcients am (f ), m = 1, 2, 3, . . . , is a ﬁnite extension. As a matter of fact, each coeﬃcient am (f ) is an algebraic integer of Q(f ). These facts will be treated in Chapters 2 and 9. Once we know that each am (f ) is an algebraic integer of Q(f ), we can formulate the relation between the group of -torsion points of E and the modular forms as follows. Definition 0.14. Let E be an elliptic curve over Q. Suppose is a prime number such that the group of -torsion points E[] is irreducible as a representation of the absolute Galois group GQ = Gal(Q/Q). Then we say that E[] is modular of level N if there m of level dividing N and a exists a primary form f = ∞ m=1 am (f )q prime ideal λ of the integer ring of Q(f ) containing such that ap (E) ≡ ap (f )

mod λ

for all primes p not dividing N . The terminology appearing in Deﬁnition 0.14 will be deﬁned in Chapter 3. If an elliptic curve E is modular of level N , and the subgroup of -torsion points E[] is irreducible as a GQ -representation,

10

0. SYNOPSIS

then E[] is modular of level N . The meaning of the last arrow in (0.2) is the following two theorems. Theorem 0.15. Let E be an elliptic curve over Q, let N be a positive integer, and let and p be odd prime numbers. Suppose the group of -torsion points E[] satisﬁes the following conditions: (i) E[] is irreducible as a GQ -representation. (ii) E[] is modular of level N . (iii) E[] is good at p. Then, if p divides N = pM once and only once, E[] is modular of level M . The meaning of these three conditions will be explained in Chapter 3. The bulk of the proof of Theorem 0.15 will be given in Chapter 9. The following theorem gives a suﬃcient condition for condition (i) in Theorem 0.15. Theorem 0.16. Let E be a semistable elliptic curve over Q, all of whose 2-torsion points are Q-rational. Let be a prime number with ≥ 5. Then, the group of -torsion points E[] is irreducible as a representation of the absolute Galois group GQ = Gal(Q/Q). Theorem 0.16 says that if an elliptic curve over Q satisﬁes conditions (i) and (ii) of Theorem 0.3, it also satisﬁes condition (i) of Theorem 0.15. Unfortunately, we can only show a small part of the proof of Theorem 0.16 in this book. Proof of Theorems 0.13, 0.15, and 0.16 ⇒ Theorem 0.3. Let be a prime number greater than or equal to 5, and let E be an elliptic curve over Q satisfying all the conditions (i)–(iii) in Theorem 0.3. First we show that the subgroup of -torsion points E[] is modular of level 2, and then we derive a contradiction from it. Let N be the conductor of E. N is a square-free positive integer. We show that E satisﬁes conditions (i)–(iii) in Theorem 0.15 for all primes p greater than or equal to 3. Since E satisﬁes conditions (i)–(ii) in Theorem 0.3, E satisﬁes condition (i) in Theorem 0.15 by Theorem 0.16. By Theorem 0.13, E is modular of level N , and thus E satisﬁes condition (ii) in Theorem 0.15. If p is an odd prime number, then p satisﬁes condition (iii) in Theorem 0.3. This implies that E satisﬁes condition (iii) in Theorem 0.15. Since N is square-free, by applying Theorem 0.15 to each odd prime factor p of N repeatedly, we see that the subgroup of -torsion

-TORSION POINTS AND MODULAR FORMS

11

points E[] is modular of level 2. However, this contradicts the following facts. Proposition 0.17. The spaces of modular forms S(1)C of level 1 and S(2)C of level 2 are both 0. This proposition will be proved in Chapter 2. We now come back to the proof of Theorem 0.3. Since the group of -torsion points E[] is modular of level 2, there must exist a primary form of the level dividing 2. However, the primary form is nonzero, and this contradicts Proposition 0.17. To conclude this chapter, we review the outline of the proof by following the diagram (0.2) again. Suppose the equation (0.1) X n + Y n = Z n has a nontrivial integral solution (X, Y, Z) = (A, B, C). We may assume that n is a prime number ≥ 5, the greatest common divisor of A, B and C is 1, C ≡ −1 mod 4, and B is even. Consider the elliptic curve EC ,B deﬁned by the equation (0.4) y 2 = x(x − C )(x − B ). Then, by Theorem 0.13, E = EC ,B is modular. Furthermore, by Theorems 0.15 and 0.16, the group of -torsion points E[] is modular of level 2. However there does not exist a nonzero modular form of level 1 or 2, which is a contradiction. We hope the reader has grasped the outline. We begin to see the details from the next chapter. Let us review what we have to prove. If Theorem 0.3 and Proposition 0.4 hold, then Theorem 0.2 holds, and so does Theorem 0.1. This has been proved in §§0.1– 0.2. We also explained in §§0.4–0.5 that Theorem 0.3 follows from Theorems 0.13, 0.15, 0.16, and Proposition 0.17. Therefore, what we really have to show are Propositions 0.4 and 0.17, and Theorems 0.13, 0.15, and 0.16. Proposition 0.4 will be proved in Chapters 1 and 3, and Proposition 0.17 will be proved in Chapter 2. Theorem 0.13 will be reduced to Theorem 3.36 in Chapter 4. The outline of the proof of Theorem 3.36 will be illustrated in Chapter 5. Theorem 0.15 will be dealt with in Chapter 9. We will show a part of the proof of Theorem 0.16 in Chapter 4. The proof of Theorem 0.1 will be reviewed in §4.2 again.

12

0. SYNOPSIS

Notation and terminology. N, Z, Q, R, and C represent, as usual, the set of nonnegative integers, the ring of rational integers, the rational number ﬁeld, the real number ﬁeld, and the complex number ﬁeld, respectively. A ring always contains a unit 1, and a ring homomorphism maps 1 to 1. The characteristic of a ﬁeld K is denoted by char(K). If a property for prime numbers holds for all but a ﬁnite number of primes, we say that this property holds for almost all primes.

10.1090/mmono/243/02

CHAPTER 1

Elliptic curves We discuss elliptic curves over a ﬁeld in the ﬁrst half of this chapter, and those over an arbitrary scheme in the second half. In §1.1, we review the deﬁnition of elliptic curves over a ﬁeld, and we study their 2-torsion points. In §1.2, we introduce the terminology concerning the reduction modulo a prime number p of an elliptic curve E over Q, and we deﬁne an integer ap (E) for each p. We will give an equivalent condition to each of conditions (i) and (ii) in Theorem 0.3. In §1.3 we deﬁne the Tate modules, and we use them to study the integer ap (E). The contents of §1.4 and §1.5 will be used in the deﬁnition of modular curves in Chapter 2. The meaning of reduction modulo a prime p in terms of algebraic geometry will become clear there, too. We also make preparations for completing the proof of Proposition 0.4 in Chapter 3. 1.1. Elliptic curves over a ﬁeld In this section we will review the deﬁnition of elliptic curves over a ﬁeld, and we study their 2-torsion points. While the deﬁnition of an elliptic curve over a ﬁeld is found in §1.1(b) in Chapter 1 of Number Theory 1 , we review it here once again. For simplicity, we assume the characteristic of the base ﬁeld is diﬀerent from 2. The deﬁnition of the general case will be given in Deﬁnition 1.11. Definition 1.1. Let K be a ﬁeld with char(K) = 2. An algebraic curve E over K deﬁned by an equation of the following form is called an elliptic curve over K: (1.1)

y 2 = ax3 + bx2 + cx + d,

where a, b, c, d ∈ K, a = 0, and the cubic polynomial ax3 +bx2 +cx+d does not have a multiple root. 13

14

1. ELLIPTIC CURVES

In the case a = 4, b = 0, c = −g2 , and d = −g3 , the cubic 4x3 − g2 x − g3 of the right-hand side does not have a multiple root if and only if its discriminant Δ = g23 − 27g32 is nonzero. An algebraic curve deﬁned by (1.1) is, to be more precise, a subvariety of the projective plane P2K deﬁned by the homogeneous equation Y 2 Z = aX 3 + bX 2 Z + cXZ 2 + dZ 3 . This is a subvariety of the aﬃne plane A2K deﬁned by y 2 = ax3 + bx2 + cx + d together with a point at inﬁnity O = (0 : 1 : 0). An elliptic curve deﬁned by the equation y 2 = ax3 + bx2 + cx + d is often written simply as “an elliptic curve y 2 = ax3 + bx2 + cx + d”. The meaning of Deﬁnition 1.1 in terms of algebraic geometry will be explained in the next section. Let u, v, and w be elements of K, and suppose u and v are nonzero. If E is an elliptic curve deﬁned by y 2 = ax3 + bx2 + cx + d, then the elliptic curve obtained by the change of coordinates x = ux + v, y = wy : (1.2) E : y 2 = w−2 a(ux + v)3 + b(ux + v)2 + c(ux + v) + d is an elliptic curve isomorphic to E. We often identify E with E through the isomorphism (x , y ) → (ux + v, wy ). Example 1.2. The equation y 2 = 4x3 − 4x2 − 40x − 79

(1.3)

deﬁnes an elliptic curve over Q. By the change of coordinates x = x + 13 , the equation (1.3) becomes y 2 = 4x3 −

(1.4) and we have

124 3 3

− 27

2501 2 27

124 2501 x − , 3 27 = −115 = 0.

Let L be an extension of K. An elliptic curve over K deﬁnes an elliptic curve over L by regarding the coeﬃcients of the deﬁning equation as elements of L. The set of all rational points in an elliptic curve forms an additive group. Its deﬁnition is given in §1.2 in Chapter 1 of Number Theory 1 , but we review it here also. Let E be an elliptic curve over K deﬁned by y 2 = ax3 + bx2 + cx + d. A K-rational point of E is a solution (x, y) ∈ K × K to the equation or the point at inﬁnity O. The set of all K-rational points of E is denoted by E(K). We have (1.5) E(K) = {(x, y) ∈ K × K | y 2 = ax3 + bx2 + cx + d} ∪ {O}.

1.2. REDUCTION MOD p

15

Recall that the group law of E(K) is characterized by the condition that the three points P , Q, and R in E(K) satisfy P +Q+R = O if and only if P , Q, and R are collinear. If L is an extension K, E(K) is a subgroup of E(L). Let us study the points of order 2 of an elliptic curve. We will give a condition equivalent to condition (i) in Theorem 0.3. Definition 1.3. Let E be an elliptic curve deﬁned over a ﬁeld K with char(K) = 2. We say that all the 2-torsion points are K-rational if for any extension L of K, the set of points of order 2 in E(L) is contained in E(K). Proposition 1.4. Let K be a ﬁeld with char(K) = 2, and let E be an elliptic curve over K. The following conditions are equivalent. (i) All 2-torsion points of E are K-rational. (ii) There exist distinct nonzero elements n and m in K such that E is isomorphic to the elliptic curve deﬁned by (1.6)

y 2 = x(x − n)(x − m).

Proof. Suppose E is deﬁned by the equation y 2 = f (x). Let L be an extension of K. We prove that a point P = (s, t) = O in E(L) is of order 2 if and only if f (s) = t = 0. By the deﬁnition of the group law, P is of order 2 if and only if the tangent line at P passes through O = (0 : 1 : 0). Since a line in P2 passes through (0 : 1 : 0) if and only if it is parallel to the y-axis, the tangent line must be the line x = s. This is equivalent to the fact that the system of equation x = s, y 2 = f (x) has a multiple root at (x, y) = (s, t). This, in turn, is equivalent to f (s) = t = 0. Thus, all the 2-torsion points of E are K-rational if and only if f (x) is decomposed into the linear factors over K. If f (x) is decomposed as f (x) = a(x − α)(x − β)(x − γ), α, β, γ ∈ K, it now suﬃces to make a change of coordinates x = a(x − α), y = ay. 1.2. Reduction mod p In this section, we deﬁne fundamentals on the reduction modulo p of an elliptic curve E over Q. We examine condition (ii) in Theorem 0.3. For a good prime p, we also deﬁne an elliptic curve obtained by the reduction modulo p of E. Counting the number of its rational points, we deﬁne the integer ap (E).

16

1. ELLIPTIC CURVES

For a prime p, recall that m Z(p) = ∈ Q | m, n ∈ Z, is relatively prime to p n is called the localization of Z at p. For an element a = m n in Z(p) , its reduction modulo p is deﬁned as an element of Fp by a mod p = (m mod p)(n mod p)−1 . The mapping Z(p) → Fp ; a → a mod p is a ring homomorphism. Let E be an elliptic curve over Q deﬁned by the equation y 2 = 3 ax + bx2 + cx + d. We can make a change of coordinates such that each coeﬃcient of the equation belongs to Z(p) . There are many ways to do so, and the reduction modulo p of E depends on how good we can make it. Definition 1.5. Let E be an elliptic curve over Q. (1) Let p be an odd prime. E has good reduction modulo p if we can choose a deﬁning equation y 2 = ax3 + bx2 + cx + d of E satisfying the following condition: (1.7)

3 2 a ∈ Z× (p) , b, c, d ∈ Z(p) and the cubic ax + bx + cx + d mod p ∈ Fp [x] obtained by the reduction modulo p of the right-hand side does not have a multiple root.

(2) Let p be an odd prime. E has stable reduction modulo p if we can choose a deﬁning equation y 2 = ax3 + bx2 + cx + d of E satisfying the following condition: (1.8)

3 2 a ∈ Z× (p) , b, c, d ∈ Z(p) and the cubic ax + bx + cx + d mod p ∈ Fp [x] obtained by the reduction modulo p of the right-hand side does not have a triple root.

(3) E has stable reduction modulo 2 if we can choose a deﬁning equation y 2 = ax3 + bx2 + cx + d of E satisfying the following condition: (1.9) a, b, c, d ∈ Z(2) ,

a ∈ Z× (2) and 4 3 ax + bx2 + cx + d ≡ (bx + d)2 ≡ 0 mod 4.

(4) E has good reduction modulo 2 if we can choose a deﬁning equation y 2 = ax3 + bx2 + cx + d of E satisfying the following condition:

1.2. REDUCTION MOD p

(1.10)

17

In addition to (1.9), if b ≡ 1 mod 2, the following condition is satisﬁed. Let x = bx + d and deﬁne a cubic in x with Z(2) coeﬃcients by

(1.11) a x3 + b x2 + c x + d =

1 3 ax + bx2 + cx + d − (bx + d)2 , 4

then c ≡ d mod 2. (5) If E has stable reduction modulo p but does not have good reduction, then E is said to have multiplicative reduction modulo p. If E does not have stable reduction, E is said to have additive reduction modulo p. Suppose E has stable reduction modulo an odd prime p. Among the deﬁning equations of E satisfying (1.8), one of them satisﬁes (1.7) if and only if all such equations satisfy (1.7). Similarly, if E has stable reduction modulo 2, among the deﬁning equations satisfying (1.9), one of them satisﬁes (1.10) if and only if all such equations satisfy (1.10). The deﬁnition in the case p = 2 may not seem natural, but we will explain its meaning later in this section. Proposition 1.6. Let E be an elliptic curve over Q. Then E has good reduction modulo almost all primes p. Proof. Suppose E is deﬁned by the equation y 2 = ax3 + bx2 + cx + d (a, b, c, d ∈ Q, a = 0). We may assume a = 4, b = 0, c, d ∈ Z. Let c = −g2 , d = −g3 . Then, since the cubic 4x3 − g2 x − g3 does not have a multiple root, we have g23 − 27g32 = 0. Any odd prime p that does not divide g23 − 27g32 satisﬁes the condition (1.7), and thus E has good reduction modulo p for such p. Definition 1.7. If an elliptic curve E over Q has stable reduction modulo all primes, we say that E is semistable. For a semistable elliptic curve E over Q, the product NE of all the primes at which E has bad reduction is called the conductor of E. By Proposition 1.6, there are only ﬁnitely many primes at which a semistable elliptic curve E has bad reduction, and thus the deﬁnition of the conductor makes sense. The conductor of a semistable elliptic curve does not have a square factor.

18

1. ELLIPTIC CURVES

Example 1.8. The elliptic curve E over Q in Example 1.2 deﬁned by (1.3) as y 2 = 4x3 − 4x2 − 40x − 79 has good reduction modulo all primes except 11. It has multiplicative reduction at p = 11. Thus, E is a semistable elliptic curve with the conductor 11. These facts can be veriﬁed as follows. If p = 2, we have a4 = 44 = 1 ∈ Z× (2) and 4x3 − 4x2 − 40x − 79 ≡ 1 ≡ 0 mod 4. If p = 3, the cubic 4x3 − 4x2 − 40x − 79 ≡ x3 − x2 − x − 1 mod 3 does not have a root in F3 , and it does not have a multiple root. If p = 11, we have 4x3 − 4x2 − 40x − 79 ≡ 4(x − 5)2 (x − 2) mod 11. If p = 2, 3, 11, we 2 124 3 2501 have 4x3 − 124 − 27 2501 = −115 ≡ 3 x − 27 ∈ Z(p) [x ] and 3 27 0 mod p. The elliptic curve of conductor 11 is, as a matter of fact, a semistable elliptic curve of the smallest conductor. This can be seen from Theorem 2.54 and Example 2.17 in Chapter 2. We study condition (ii) in Theorem 0.3 when condition (i) holds. Proposition 1.9. (1) Let E be an elliptic curve over Q. The following conditions are equivalent. (i) E is semistable and all of its 2-torsion points are Qrational. (ii) E is deﬁned by equation (1.6) y 2 = x(x − n)(x − m), where n and m are integers satisfying the following condition: (1.12)

n and m are distinct, relatively prime, nonzero integers such that n ≡ −1 mod 4 and m ≡ 0 mod 16. (2) Let E be a semistable elliptic curve over Q deﬁned by the equation (1.6) with some n and m satisfying (1.12). Then, the con1 ductor of E is the product of all the prime factors of 16 nm(n − m).

Proof of Proposition 1.9(1)(ii) ⇒ (i) and (2). Suppose E is a semistable elliptic curve deﬁned by equation (1.6) with some n and m satisfying (1.12). By Proposition 1.4 all the 2-torsion points of E are Q-rational. Let p be an odd prime. Since n and m are relatively prime, equation (1.6) satisﬁes condition (1.8). Let n = −1+4n , m = 16m . By the change of coordinates x = 4x , y = 4y , equation (1.6) becomes y 2 = x (4x +1−4n )(x −4m ). The leading coeﬃcient of the right-hand side is 4, and x (4x +1−4n )(x −4m ) ≡ x2 mod 4,

1.2. REDUCTION MOD p

19

and thus it satisﬁes condition (1.9). Thus, E has stable reduction at all p, which means E is semistable. If p ≥ 3, equation (1.6) satisﬁes condition (1.7) if and only if p does not divide nm(n − m). If p = 2, then we have 14 x (4x + 1 − 4n )(x − 4m ) − x2 ≡ x (x2 − n x − m ) mod 2. This satisﬁes condition (1.10) if and only 1 m. if 2 does not divide m = 16 Proof of (i) ⇒ (ii) in (1). We do not need this fact to prove Theorem 0.1, so, we only indicate an outline of the proof brieﬂy. The details will be left to the interested reader as an exercise. An elliptic curve E satisfying (i) can be deﬁned by an equation of the form y 2 = ax(x − n)(x − m), where a, n, m are integers satisfying the following conditions: a is a square-free positive integer; n, m are distinct, relatively prime nonzero integers; n is odd; and m is even. To show this, we may proceed similarly as the proof of Proposition 1.4. We show that a = 1 or 2. This can be done as follows. Let p be an odd prime. By the change of coordinates x = ux − v, y = wy , the equation becomes au3

v v + n v + m x − x − . y 2 = 2 x − w u u u If this satisﬁes (1.8), then v/u, (v+n)/u, and (v+n)/u are all elements in Z(p) , and they do not become all equal modulo p. Thus, we may assume v = 0, and we see that u is invertible modulo p. Since au3 /w2 is also invertible, a is also invertible modulo p. Suppose p = 2. Similarly to the case of an odd prime, two of v/u, (v +n)/u and (v +m)/u are in Z(2) , and 4 times the other is invertible in F2 . Thus, we may assume v = 0, u/4 is invertible at 2, and m is divisible by 8. Since a3 /w2 is invertible at 2, we may assume a = 1, u = w = 4, and the equation becomes y 2 = x (4x − m)(x − m/4). Since the right-hand side must be congruent to x2 modulo 4, we have n ≡ −1 mod 4, m ≡ 0 mod 16. We now clarify the algebro-geometric meaning of Deﬁnition 1.1 and the case of p = 2 in Deﬁnition 1.5. The reader who does not care about such things can go directly to Deﬁnition 1.3 for the moment. The following lemma permits us to rephrase Deﬁnition 1.1 in another form below. Lemma 1.10. Let K be a ﬁeld with char(K) = 2.

20

1. ELLIPTIC CURVES

(1) Suppose a ∈ K × and b, c, d ∈ K. A subvariety of P2K deﬁned by the homogeneous equation of the form Y 2 Z = aX 3 + bX 2 Z + cXZ 2 + dZ 3 is smooth over K if and only if the cubic ax3 + bx2 + cx + d does not have a multiple root. (2) The genus of an elliptic curve over K is 1. (3) Let E be a proper smooth connected curve of genus 1 over K and O its K-rational point. Then, there is a closed immersion of E in P2K such that its image is given by an equation of the form Y 2 Z = aX 3 + bX 2 Z + cXZ 2 + dZ 3 (a ∈ K × , b, c, d ∈ K), and the image of O is (0 : 1 : 0). We omit the proof of this lemma. Definition 1.11. An elliptic curve E over a ﬁeld K is a proper smooth connected curve of genus 1 over K together with a K-rational point O. This deﬁnition is valid even if the characteristic of K is 2. The following lemma shows that Deﬁnition 1.5(4) is a natural deﬁnition. By the change of coordinates y = 2y + bx + d, the equation y 2 = ax3 + bx2 + cx + d becomes 1 (1.13) y 2 + bxy + dy = (ax3 + bx2 + cx + d) − (bx + d)2 . 4 Lemma 1.12. (1) Let K be a ﬁeld with char(K) = 2, and let E be a proper algebraic curve over K. The following conditions are equivalent. (i) E is an elliptic curve over K. (ii) E is isomorphic to a plane curve deﬁned by the equation of the form (1.14)

y 2 + (bx + d)y = x3 + b1 x2 + c1 x + d1 (b, d, b1 , c1 , d1 ∈ K),

and E is smooth. (2) Suppose K = F2 . The following are equivalent. (i) The algebraic curve over F2 deﬁned by the equation (1.14) is smooth. (ii) One of the following holds: (a) b = 0, d = 1. (b) b = 1, and if we deﬁne b , c , d so that (x + d)3 + b (x + d)2 + c (x + d) + d = x3 + b1 x2 + c1 x + d1 holds, then we have c = d

1.2. REDUCTION MOD p

21

We omit the proof of this lemma, too. We use the equation that appeared in Deﬁnition 1.5 to deﬁne the reduction modulo p of an elliptic curve. Definition 1.13. Let E be an elliptic curve over Q. (1) Let p be an odd prime. Suppose E has good reduction modulo p, and choose a deﬁning equation of E so that it satisﬁes condition (1.7). The elliptic curve over Fp deﬁned by y 2 = ax3 + bx2 + cx + d mod p is called the reduction of E modulo p, and it is denoted by EFp . Using the number of elements of the set of rational points of EFp : EFp (Fp ) = {(x, y) ∈ Fp × Fp | y 2 = ax3 + bx2 + cx + d} ∪ {O}, deﬁne ap (E) = p + 1 − EFp (Fp ).

(1.15)

(2) Let p be an odd prime. Suppose E has multiplicative reduction at p. Take a deﬁning equation of E so that it satisﬁes condition (1.8). The right-hand side becomes a(x − α)2 (x − β) (α = β ∈ Fp ) by the reduction modulo p. Deﬁne ap (E) = 1 if a(α − β) is a square in F× p , and ap (E) = −1 otherwise. (3) Suppose E has good reduction modulo 2, and take a deﬁning equation of E so that it satisﬁes the condition (1.10). Then, the curve over F2 deﬁned by (1.13) y 2 + bxy + dy = algebraic 1 3 2 2 with F2 coeﬃcients is an 4 (ax + bx + cx + d) − (bx + d) elliptic curve over F2 . Here, we regard the right-hand side as a polynomial with F2 coeﬃcients. This elliptic curve over F2 is called the reduction of E modulo 2. Using the number of the elements of the set of rational points of EF2 , EF2 (F2 ) = {(x, y) ∈ F2 × F2 | solutions of (1.13)} ∪ {O}, deﬁne (1.16)

a2 (E) = 2 + 1 − EF2 (F2 ).

(4) Suppose E has multiplicative reduction modulo 2. Choose a deﬁning equation y 2 = ax3 + bx2 + cx + d of E so that it satisﬁes the condition (1.9). This does not satisfy the condition (1.10).

22

1. ELLIPTIC CURVES

Under the notation of (1.10), deﬁne a2 (E) = 1 if b ≡ 0 mod 2, and a2 (E) = −1 if b ≡ 1 mod 2. (5) If E has additive reduction modulo p, deﬁne ap (E) = 0. If E has good reduction modulo p, the isomorphism class of the reduction of E at p does not depend on the deﬁning equation that satisﬁes the condition (1.7) or (1.10). Also, if E has multiplicative reduction modulo p, ap (E) does not depend on the deﬁning equation that satisﬁes condition (1.8) or (1.9). Example 1.14. For the semistable elliptic curve of conductor 11 in Example 1.8, the number of elements Np of EFp (Fp ) and ap (E) are as follows. p 2 3 Np 5 5 ap (E) −2 −1

5 7 11 13 17 19 23 29 31 37 · · · 5 10 − 10 20 20 25 30 25 35 · · · 1 −2 1 4 −2 0 −1 0 7 3 · · ·

The holomorphic function in s deﬁned by the inﬁnite product −1 (1.17) 1 − ap (E)p−s + p1−2s 1 − ap (E)p−s )−1 × p:good

p:bad

is called the L-function of E and is denoted by L(E, s). By the following theorem, the inﬁnite product (1.17) converges absolutely for Re s > 32 . Theorem 1.15. Let p be a prime, and let E be an elliptic curve over the ﬁnite ﬁeld Fp . Then, we have the inequality

p + 1 − E(Fp ) < 2√p. This is a special case of the Weil conjectures. The proof will be given at the end of the next section. 1.3. Morphisms and the Tate modules From now on we use the terminology of algebraic geometry freely, such as the terms, ﬁnite, ﬂat, ´etale, etc. We give a brief summary of the theory of schemes in Appendix A. Let K be a ﬁeld, and let E and E be elliptic curves over K. A morphism of algebraic curves f : E → E is called a morphism of elliptic curves if f sends the 0-section of E to the 0-section of E . If f : E → E is a morphism of elliptic curves over K and L is an extension of K, then the mapping E(L) → E (L) induced by f is a group homomorphism.

1.3. MORPHISMS AND THE TATE MODULES

23

Suppose f, g : E → E are two morphisms of elliptic curves, we deﬁne the sum f + g : E → E to be the composition diagonal mapping

(f,g)

+

E −−−−−−−−−−−→ E × E −−−−→ E × E −−−−→ E . The set of all morphisms of elliptic curves E → E , denoted by Hom(E, E ), forms an additive group with respect to this addition. If E = E , a morphism f : E → E of elliptic curves is called an endomorphism. The set of all endomorphisms EndK (E) forms a ring with respect to the multiplication deﬁned by the composition of morphisms, whose identity element is the identity morphism. For an integer N we can deﬁne the multiplication-by-N morphism [N ] : E → E. If L is an extension of K, the endomorphism of abelian group E(L) induced by the multiplication-by-N morphism is nothing but the multiplication by N . If K is a ﬁnite ﬁeld Fp , an elliptic curve E over Fp has an endomorphism called the geometric Frobenius. If E is an elliptic curve over Fp , the endomorphism of E deﬁned by raising the pth power of all the elements of the coordinate ring is called the geometric Frobenius of E and denoted by Frp . For example if E is given by the equation y 2 = ax3 +bx2 +cx+d, then Frp is the morphism of algebraic varieties deﬁned by (x, y) → (xp , y p ). If f : E → E is a nonzero morphism of elliptic curves, then f is ﬁnite and ﬂat. The degree of f as a morphism of algebraic curves is called the degree of morphism f , and denoted by deg f . If f equals 0, we deﬁne deg f = 0. The degree of the geometric Frobenius Frp of an elliptic curve over Fp is p. If the degree of f is not divisible by the characteristic of K, then f is ﬁnite and ´etale. In this case, the degree of f equals the order of the ﬁnite abelian group Ker(f : E(K) → E (K)). Proposition 1.16. Let E be an elliptic curve over K, and let N be a positive integer. The degree of the multiplication-by-N morphism [N ] : E → E is N 2 . This proposition can be proved from the relation between the dual of a morphism and its degree, but we do not prove it in this book. Corollary 1.17. Let E be an elliptic curve over a ﬁeld K, and let N be a positive integer relatively prime to the characteristic of K. Then the multiplication-by-N morphism [N ] : E → E is ﬁnite and

24

1. ELLIPTIC CURVES

´etale. The kernel E[N ](K) = Ker([N ] : E(K) → E(K)) is a ﬁnite abelian group isomorphic to (Z/N Z)2 . Proof of Corollary. The multiplication-by-N morphism [N ] is ﬁnite and ´etale since its degree, N 2 , is relatively prime to the characteristic of K. The order of the ﬁnite abelian group E[N ](K) is N 2 . By the structure theorem of ﬁnite abelian groups, r there exist positive integers N1 , . . . , Nr such that E[N ](K) i=1 Z/Ni Z, r 1 = Nr | · · · |N1 |N , N 2 = i=1 Ni . We have E[Nr ](K) (Z/Nr Z)r . Since Nr is relatively prime to the characteristic of K, we have r = 2, and N1 = N2 = N . If f is an endomorphism of an elliptic curve E, f induces an endomorphism of the subgroup of N -torsion points E[N ](K) (Z/N Z)2 . We have the following property with regard to the degree of f and the action on the N -torsion points. Take a basis of E[N ](K), and express the action of f on E[N ](K) by a square matrix of degree 2 with Z/N Z coeﬃcients. Since the determinant of this matrix does not depend on the choice of the basis, we denote it by det(f : E[N ](K)). It is deﬁned as an element of Z/N Z. Proposition 1.18. Let E be an elliptic curve over a ﬁeld K, let f be an endomorphism of E, and let N be a positive integer with char(K) N . Then, the determinant det(f : E[N ](K)) of the action of f on E[N ](K) (Z/N Z)2 is congruent to the degree of f modulo N : (1.18)

deg f ≡ det(f : E[N ](K)) mod N.

The proof is given using the eN -pairing, but we do not give it in this book. Rather than the ﬁnite abelian group E[N ](K), it is more convenient to consider the free Z -module obtained as its limit. This is the Tate module, which is the title of this section. Let be a prime with = char(K). For a positive integers n ≥ m, deﬁne ϕm,n : E[n ](K) → E[m ](K) to be the multiplication-byn−m mapping. Then, (E[n ](K), ϕm,n ) is a projective system of ﬁnite abelian group. We denote by T E its projective limit limn E[n ](K), ←− and we call it the Tate module. The Tate module T E has a structure of Z -module in a natural way.

1.3. MORPHISMS AND THE TATE MODULES

25

Proposition 1.19. Let E be an elliptic curve over a ﬁeld K, and let be a prime with = char(K). Then the Tate module T E is isomorphic to Z2 . Proof. Let M1 ⊂ · · · ⊂ Mn ⊂ Mn+1 ⊂ · · · be a chain of ﬁnite abelian groups such that Mn (Z/n Z)2 . We deﬁne ϕm,n : Mn → Mm to be the multiplication-by-n−m mapping, and we show M = limn Mn is isomorphic to Z2 . Take a basis a1 , b1 of M1 over F . Since ←− ϕn−1,n : Mn → Mn−1 is surjective, choose an and bn inductively by the relation ϕn−1,n (an ) = an−1 , ϕn−1,n (bn ) = bn−1 . Let e1 , e2 be a standard basis of (Z/n Z)2 , and deﬁne gn : (Z/n Z)2 → Mn by e1 → an , e2 → bn . Then (gn )n deﬁnes an isomorphism of projective system. Thus, it induces an isomorphism Z2 → M of the projective limits. If f is an endomorphism of an elliptic curveE, then f induces an endomorphism of projective system E[m ](K) n . By taking a basis of T E over Z , we deﬁne the determinant det(f : T E) of f similarly as above. Corollary 1.20. Let E be an elliptic curve over a ﬁeld K, let f be an endomorphism of E, and let = char(K) be a prime. Then, the determinant det(f : T E) of the action of f on T (E) Z2 equals the degree of f : (1.19)

deg f ≡ det(f : T E).

Proof. Let N = n in (1.18), and take the projective limit.

We now prove Theorem 1.15. To do so, we ﬁrst show the following proposition. Let E be an elliptic curve over Fp , and let be a prime diﬀerent from p. Taking a matrix expression of the action of the geometric Frobenius Frp on the Tate module T E, we deﬁne the polynomial det(X − Frp · Y : T E) ∈ Z [X, Y ]. Proposition 1.21. Let E be an elliptic curve over the ﬁnite ﬁeld Fp , and let be a prime diﬀerent from p. If a = p + 1 − E(Fp), then we have (1.20)

det(X − Frp · Y : T E) = X 2 − aXY + pY 2 .

Proof. It suﬃces to show both sides are equal when we plug in (X, Y ) = (1, 0), (0, 1) and (1, 1). By Corollary 1.20, we have

26

1. ELLIPTIC CURVES

det(n − m Frp : T E) = deg(n − m Frp ), and thus it suﬃces to show (1.21)

deg 1 = 1,

deg Frp = p,

deg(1 − Frp ) = E(Fp ). The ﬁrst two are clear. We show the last identity. Since the mapping induced by the endomorphism 1 − Frp on the tangent space is the identity, 1 − Frp is ﬁnite and ´etale. Thus, the degree of the kernel Ker(1 − Frp : E(Fp ) → E(Fp )) is the number of elements. Since this kernel is nothing but E(Fp ), we have deg(1 − Frp ) = E(Fp ). Proof of Theorem 1.15. By Proposition 1.21, we have n2 − anm − pm2 = deg(n − m Frp ) for any integers n, m. Since the righthand side is nonnegative, the discriminant a2 − 4p of the quadratic polynomial X 2 − aX + p is less than or equal to 0. 1.4. Elliptic curves over an arbitrary scheme As in the previous section, we continue to use the terminology of algebraic geometry freely in this section and the next. Readers should consult Appendix A as necessary. Definition 1.22. An elliptic curve over a scheme S is a proper smooth scheme f : E → S with a section 0 : S → E satisfying the following condition: (1.22)

Every geometric ﬁber Es¯ of f : E → S is a connected algebraic curve of genus 1 over the algebraic closed ﬁeld k(¯ s).

An elliptic curve over S is, precisely speaking, a pair (f : E → S, 0) consisting of a scheme f : E → S and its section 0. However, we often denote it simply by E. If E is an elliptic curve over S, then by Proposition 1.30 below, there exists a unique structure of commutative group schemes on E such that 0 is its 0-section. Therefore, to equip E, a structure of commutative group schemes over S is equivalent to giving a 0-section to E. Appendix A.2 gives a brief summary of group schemes. Explicitly, an elliptic curve is given as follows. We deﬁne a ﬁvevariable polynomial with Z-coeﬃcients Δ ∈ Z[a1 , a2 , a3 , a4 , a6 ] as follows. First, deﬁne b2 , b4 , b6 , b8 ∈ Z[a1 , a2 , a3 , a4 , a6 ] by b2 = a21 + 4a2 ,

b4 = 2a4 + a1 a3 ,

b6 = a23 + 4a6 ,

b8 = a21 a6 + 4a2 a6 − a1 a3 a4 + a2 a23 − a24 .

1.4. ELLIPTIC CURVES OVER AN ARBITRARY SCHEME

27

Then, deﬁne (1.23)

Δ = 9b2 b4 b6 − b22 b8 − 8b34 − 27b26 .

We call Δ the discriminant of E. Lemma 1.23. Let S = Spec A be an aﬃne scheme, and let a1 , . . . , a6 ∈ A. The closed subscheme E of P2A deﬁned by Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 is smooth over S if and only if Δ(a1 , . . . , a6 ) is an invertible element of A. If this is the case, the proper smooth subscheme E satisﬁes condition (1.22). Moreover, the closed subscheme O of E deﬁned by X = Z = 0 is a section of E over S. Thus, the pair (E, O) is an elliptic curve over S. We omit the proof of this lemma. Conversely, let E be an elliptic curve over an arbitrary scheme S. Then, we can show there exists an aﬃne open covering S = λ∈Λ Uλ such that the restriction of E to each Uλ is isomorphic to a scheme deﬁned as in Lemma 1.23. In Deﬁnition 1.22 we gave an abstract deﬁnition, but locally on S, we can think it is given explicitly as in Lemma 1.23. In particular, if K is a ﬁeld and S = Spec K, then Deﬁnition 1.22 coincides with Deﬁnition 1.11. If, further, char(K) = 2, then these deﬁnitions coincide with Deﬁnition 1.1. Let E be an elliptic curve over S, and let S → S be a morphism of schemes. Then, the base change ES → S is an elliptic curve over S . If E is an elliptic curve over K and L is an extension of K, then, as we stated in §1.1, E deﬁnes an elliptic curve over L. This is a special case of the base change. In terms of the notion of elliptic curves over a general base, Deﬁnitions 1.5 and 1.13 signify the following. Proposition 1.24. Let E be an elliptic curve over Q, and let p be a prime number. (1) The following are equivalent: (i) E has good reduction modulo p. (ii) There exists an elliptic curve EZ(p) over the ring Z(p) such that EZ(p) ⊗Z(p) Q equals E. This elliptic curve EZ(p) over Z(p) is called a smooth model of E over Z(p) . (2) A smooth model is unique up to a unique isomorphism. (3) If EZ(p) is a smooth model of E, the reduction EFp at p coincides with the elliptic curve EZ(p) ⊗Z(p) Fp .

28

1. ELLIPTIC CURVES

We omit the proof of this proposition. Proposition 1.6 may be interpreted as follows. If E is an elliptic curve over Q, then there exists a positive integer N and an elliptic curve EZ[1/N ] over Z[1/N ] such that E equals EZ[1/N ] ⊗Z[1/N ] Q. Such an elliptic curve EZ[1/N ] is called a smooth model of E over Z[1/N ]. A smooth model over Z[1/N ] is unique up to a unique isomorphism. Let EZ[1/N ] be a smooth model of E over Z[1/N ], and p a prime with p N . Then EZ[1/N ] ⊗Z[1/N ] Z(p) is a smooth model of E over Z(p) , and E has good reduction modulo p. Example 1.25. A smooth model over Z[1/11] of the elliptic curve in Example 1.8 is given by y 2 + y = x3 − x2 − 10x − 20. We have Δ = −115 . As we stated at the end of Example 1.8, there does not exist a semistable elliptic curve with conductor 1. This means there is no elliptic curve over Z. Let E be an elliptic curve over S, and let N be a positive integer. Since E is a commutative group scheme over S, we can deﬁne the multiplication-by-N morphism [N ] : E → E, as in the case of elliptic curve over a ﬁeld. This morphism is characterized by the condition that for any scheme T over S, the induced mapping [N ] : E(T ) → E(T ) is the multiplication-by-N mapping of the commutative group E(T ). The morphism [N ] has the following properties. Proposition 1.26. Let E be an elliptic curve over a scheme S, and let N be a positive integer. The multiplication-by-N morphism [N ] : E → E is ﬁnite and ﬂat, and its degree is N 2 . If, furthermore, N is invertible in S, then [N ] : E → E is ´etale. Proof. Since E is proper and ﬂat over S, by Proposition A.5 and Corollaries A.6 and A.12, it suﬃces to show it for each geometric ﬁber. For an elliptic curve over a ﬁeld, the statement is nothing but Proposition 1.16 and Corollary 1.17. Let E be an elliptic curve over a scheme S, and let N be a positive integer. The kernel E[N ] = [N ]−1 0 of the multiplication-by-N morphism [N ] : E → E is the ﬁbered product over E of [N ] : E → E and the 0-section : S → E. Thus, we have the following corollary of Proposition 1.16. Corollary 1.27. Let E be an elliptic curve over a scheme S, and let N be a positive integer. The kernel E[N ] of the multiplicationby-N morphism [N ] : E → E is a ﬁnite and ﬂat commutative group

1.5. GENERALIZED ELLIPTIC CURVES

29

scheme over S, and its degree equals N 2 . If N is invertible in S, E[N ] is a ﬁnite ´etale commutative group scheme. Corollary 1.28. Let E be an elliptic curve over Q. Suppose that E has good reduction modulo p, and let EZ(p) be its smooth model. If N is a positive integer, the kernel EZ(p) [N ] of the multiplication-byN morphism [N ] : EZ(p) → EZ(p) is a ﬁnite ﬂat commutative group scheme over Z(p) . If p N , then EZ(p) ,N is a ﬁnite ´etale commutative group scheme over Z(p) .

1.5. Generalized elliptic curves In order to study elliptic curves with stable reduction, it is convenient to generalize the deﬁnition of elliptic curves. It is also useful when we deﬁne the compactiﬁcation of the modular curves in the next chapter. First, we deﬁne something to be added. Let K be a ﬁeld, and let n be a positive integer. In the case n ≥ 2, the N´eron n-gon Pn,K = i∈Z/nZ Ui is deﬁned by gluing n schemes Ui = Spec K[Xi , Yi ]/(Xi Yi ), i ∈ Z/nZ, in the following way. For each i ∈ Z/nZ, the open subscheme Vi = Ui [Xi−1 ] = Spec K[Xi , Xi−1 ] −1 of Ui is identiﬁed with the open subscheme Vi−i = Ui−1 [Yi−1 ] = −1 −1 Spec K[Yi−1 , Yi−1 ] of Ui−1 by Xi → Yi−1 . In the case n = 1, P1,K is deﬁned as the subscheme of P2 deﬁned by the polynomial Y (Y + X)Z − X 3 . TheN´eron n-gon is a proper ﬂat scheme over K. sm = i∈Z/N Z Vi is called the smooth part of the If n ≥ 2, Pn,K sm N´eron n-gon Pn,K , and if n = 1, P1,K = P1,K {(0 : 0 : 1)} is called sm the smooth part. Pn,K is the largest open subscheme of Pn,K that sm is smooth over K. To the smooth part Pn,K , deﬁne a structure of sm sm sm commutative group scheme Pn,K ×K Pn,K → Pn,K as follows. In the case n ≥ 2, identify each Vi with the multiplicativegroup Gm,K = sm = i∈Z/nZ Vi with Spec K[X, X −1 ] by Xi → X, and identify Pn,K the product group scheme Gm,K ×(Z/nZ). In the case n = 1, identify sm with Gm,K = P1 {(0 : 1), (1 : 0)} by the isomorphism deﬁned P1,K by (X, Y, Z) → (Y + X, Y ). sm sm If n ≥ 2, deﬁne the action of Pn,K on Pn,K , + : Pn,K ×K Pn,K → Pn,K , by Vi × Uj → Ui+j ;

(xi , (xj , yj )) → (xi xj , x−1 i yj ).

30

1. ELLIPTIC CURVES

If n = 1, deﬁne Gm,K ×K P1,K : (t, (x, y)) →

t(x + y)y t(x + y)y 2 , (tx + (t − 1)y)2 (tx + (t − 1)y)3

.

In the following, we consider the N´eron n-gon as the pair (Pn,K , +). sm sm ×K Pn,K is nothing but the The restriction of the action + to Pn,K sm group operation of Pn,K . Definition 1.29. A generalized elliptic curve over a scheme S is a proper ﬂat scheme f : E → S together with a morphism + : E sm ×S E → E that satisﬁes conditions (1.24) and (1.25) below. Here, the smooth part E sm is the largest open subscheme U of E such that f |U : U → S is smooth. (1.24)

The restriction of + to E sm deﬁnes a structure of commutative group scheme + : E sm ×S E sm → E sm . Moreover, + : E sm ×S E → E is an action of the group scheme E sm on the scheme E.

(1.25)

Any geometric ﬁber (Es¯, +s¯) of the pair (f : E → S, +) is either an elliptic curve over the algebraic closed ﬁeld κ(¯ s) or isomorphic to a N´eron n-gon Pn,κ(¯s) for some positive integer n.

A generalized elliptic curve is by deﬁnition a pair (f : E → S, +) of a scheme f : E → S and a morphism + : E sm ×S E → E, but we simply denote it by E. Proposition 1.30. Let S be an arbitrary scheme, and let f : E → S be a proper ﬂat scheme over S such that each geometric ﬁber is isomorphic to either an elliptic curve or the N´eron 1-gon. Denote by E sm the largest open subscheme of E such that f is smooth. If 0 : S → E sm is a section, then there exists a unique morphism + : E sm ×S E → E such that (f : E → S, +) is a generalized elliptic curve, and 0 is the 0-section of the commutative group scheme E sm . Hence, for such a scheme f : E → S as in Proposition 1.30, to give a section S → E sm is equivalent to regarding E as a generalized elliptic curve. We omit the proof of this proposition. Deﬁnition 1.5 has the following meaning. Proposition 1.31. Let E be an elliptic curve over Q, and let p be a prime number.

1.5. GENERALIZED ELLIPTIC CURVES

31

(1) The following are equivalent: (i) E has stable reduction modulo p. (ii) There exists a generalized elliptic curve EZ(p) over the ring Z(p) such that EZ(p) ⊗Z(p) Q is isomorphic to E. (iii) There exists a regular generalized elliptic curve EZ(p) over the ring Z(p) such that EZ(p) ⊗Z(p) Q is isomorphic to E. (2) The regular generalized elliptic curve satisfying the condition (iii) in (1) above is unique up to a canonical isomorphism. We also omit the proof of this proposition. The regular generalized elliptic curve satisfying the condition (iii) in (1) is called the semistable model of E over Z(p) . Suppose E has multiplicative reduction modulo p. If EZ(p) is the semistable model of E over Z(p) , the identity component of the geometric closed ﬁber EZsm(p) ⊗Z(p) Fp of the smooth part E sm is isomorphic to Gm . This is the reason for the term multiplicative reduction. We will study condition (iii) of Theorem 0.3 in Chapter 3. Here, we give some preparation for it. Namely, we study the number of connected components of a geometric closed ﬁber of the semistable model. Proposition 1.32. Let p be an odd prime, and let a, b, c be elements of Z(p) such that a2 + ba + c ≡ 0 mod p, 0 = b2 − 4c ≡ 0 mod p. Let E be the elliptic curve over Q deﬁned by the equation y 2 = (x − a)(x2 + bx + c), and let d be the p-adic valuation of the discriminant b2 − 4c. Then, E has multiplicative reduction modulo p. If EZ(p) is its semistable model, the number of irreducible components of the geometric closed ﬁber EZ(p) ⊗Z(p) Fp is d. We omit the proof. The semistable model can be obtained as follows. By some change of coordinates, we assume b = 0. The semistable model EZ(p) is obtained by blowing up [ d2 ] times the singular point (x, y) = (0, 0). Proposition 1.33. Let be an odd prime, and let E be an elliptic curve over Q. The following conditions are equivalent. (i) E satisﬁes conditions (i) and (ii) in Theorem 0.3 together with the following condition (iii ): (iii ) For each odd prime p, one of the following conditions (a) or (b) holds.

32

1. ELLIPTIC CURVES

(a) E has good reduction modulo p. (b) E has multiplicative reduction modulo p, and the number of irreducible components of the geometric closed ﬁber of the semistable model of E over Z(p) is divisible by . (ii) E satisﬁes condition (ii) in Proposition 0.4. Proof. By Proposition 1.9(1), it suﬃces to show the following. Let n and m be integers satisfying condition (1.12), and let E be the elliptic curve deﬁned by equation (1.6). Then, E satisﬁes condition (iii ) if and only if n and m satisfy (1.26). (1.26)

n, m, and n − m are all th power times a power of 2.

Condition (1.26) means that the p-adic valuation of n, m, and n − m are all divisible by for all odd prime p. Let p be an odd prime. At least two of the p-adic valuations are 0, and let ep be the valuation of the remaining one. If ep = 0, E has good reduction modulo p. If ep > 0, then by Proposition 1.32, E has multiplicative reduction modulo p and the number of irreducible components of the geometric closed ﬁber of the semistable model of E over Z(p) is 2ep . Since = 2, condition (iii ) implies that ep is divisible by for any odd prime p. To show Proposition 0.4, it suﬃces to show that condition (iii ) in Proposition 1.33(1) and condition (iii) in Theorem 0.3 are equivalent. We will prove this in Proposition 3.48 in Chapter 3. There, we use the relation between the number of irreducible components of the geometric closed ﬁber and condition (iii) in Theorem 0.3. We will describe this relation in Corollary 1.36 below. To state the corollary, we need some preparation. Let S be a scheme, and let f : E → S be a general elliptic curve. Deﬁne an open subscheme E (1) of E by E (1) = {x ∈ E sm | x belongs to the identity component of the ﬁber Efsm (x) }. This is an open subgroup scheme of E sm . Let N be a positive integer, and let E be a generalized elliptic curve over S such that each geometric ﬁber is either an elliptic curve or a N´eron n-gon Pn,κ(¯s) , where n is a multiple of N . Deﬁne E (N ) to be the inverse image of E (1) by the multiplication-by-N morphism [N ] : E sm → E sm . E (N ) is an open subscheme of E sm . Its geometric ﬁber is either an elliptic curve

1.5. GENERALIZED ELLIPTIC CURVES

33

or is isomorphic to the product Gm × (Z/N Z). The restriction of the multiplication-by-N morphism to E (N ) has the following property. Proposition 1.34. Let E be a generalized elliptic curve over S, and let N be a positive integer. Suppose each geometric ﬁber of E is either an elliptic curve or a N´eron n-gon Pn,κ(¯s) , where n is a multiple of N . Let E (N ) be the inverse image of E (1) by the multiplicationby-N morphism [N ] : E sm → E sm . Then, the restriction of the multiplication-by-N morphism to E (N ) , [N ] : E (N ) → E (1) , is ﬁnite and ﬂat of degree N 2 . Moreover, if N is invertible in S, then [N ] : E (N ) → E (1) is ´etale. (N )

(1)

Proof. Each geometric ﬁber Es¯ → Es¯ of [N ] : E (N ) → E (1) is ﬁnite and ﬂat of degree N 2 . Moreover, it is ´etale if N is invertible. Since E (N ) and E (1) are ﬂat over S, it follows from Proposition A.5 in Appendix A that [N ] : E (N ) → E (1) is also ﬂat. Moreover, by Corollary A.6, it is ´etale if N is invertible. [N ] : E (N ) → E (1) is quasiﬁnite and the degree of each ﬁber is N 2 . Thus, by Corollary A.10, [N ] : E (N ) → E (1) is ﬁnite and ﬂat of degree N 2 . Corollary 1.35. Let E, S, N , and E (N ) be as in Proposition 1.34. The kernel E (N ) [N ] of the restriction of the multiplicationby-N morphism, [N ] : E (N ) → E (1) , is a ﬁnite ﬂat commutative group scheme over S of degree N 2 . Moreover, if N is invertible in S, E (N ) [N ] is a ﬁnite ´etale commutative group scheme over S. Corollary 1.36. Let E be an elliptic over Q. Suppose E has multiplicative reduction modulo a prime p, and let EZ(p) be its semistable model. If N divides the number of irreducible components of (N ) the geometric closed ﬁber of EZ(p) , then, the kernel EZ(p) [N ] of the (N )

(1)

multiplication-by-N morphism [N ] : EZ(p) → EZ(p) is a ﬁnite ﬂat commutative group scheme over Z(p) . If p does not divide N , then (N ) EZ(p) [N ] is a ﬁnite ´etale commutative group scheme over Z(p) .

10.1090/mmono/243/03

CHAPTER 2

Modular forms In this chapter we deﬁne modular forms and study their fundamental properties. We ﬁrst deﬁne modular forms as regular diﬀerentials on a modular curve. A modular curve is deﬁned as the moduli space of elliptic curves with certain level structure. Our ﬁrst goal is to study the structure of modular curves. Among the properties of modular forms, the important ones are the action of the Hecke operators (§2.6) and the q-expansions (§2.7). The relation between elliptic curves and modular forms is established through these properties. It is fundamental to regard the space of modular forms as a module over the Hecke algebra. In practice, it is often convenient to consider modular forms as complex functions and study them analytically. In the last sections of this chapter, we describe a meaning of the q-expansions and the Hecke operators using the analytic expression of modular forms. Everything in this chapter can be done over Z, and it may be more natural to do so. However, in order to avoid unnecessary complications, we consider everything over Q here. We will come back again and discuss them over Z in Chapter 9. 1

2.1. The j-invariant In Chapter 1, we studied properties of individual elliptic curves. Now, we may ask ourselves, “How many diﬀerent elliptic curves are there?” Moduli problems formulate such questions geometrically, and modular curves are the solutions to such problems. 1 Please recall that Chapters 8, 9, and 10 along with Appendices B, C, and D will appear in Fermat’s Last Theorem: The Proof, a forthcoming translation of the Japanese original.

35

36

2. MODULAR FORMS

Arithmetically, the solution to the same problem is the link between elliptic curves and modular forms. It is a coincidence, I suppose, that the same object, modular curves, appears in both aspects. As a ﬁrst example of modular curves, we introduce the j-line. Definition 2.1. Let S be a scheme over Q, and let E be an elliptic curve over S. The j-invariant jE ∈ Γ(S, O) is a regular function satisfying the following condition. If, on an aﬃne open set U = Spec A of S, E is deﬁned by the equation (2.1)

y 2 = 4x3 − g2 x − g3

(g2 , g3 ∈ A),

then we have (2.2)

jE |U = 123

g23 . g23 − 27g32

Proposition 2.2. Let S be any scheme over Q. (1) For any elliptic curve E over S, there exists a unique regular function j ∈ Γ(S, O) satisfying (2.2). (2) If two elliptic curves E and E over S are isomorphic, then jE = jE . Proof. It is suﬃcient to show that the right-hand side of (2.2) is invariant under the change of coordinates. This is just a simple calculation, and we leave it to the reader. Proposition 2.2 says that for a given elliptic curve, the j-invariant is uniquely determined. Conversely, for a given j ∈ Γ(S, O), is there a unique elliptic curve E such that j = jE ? The answer is “no” in general, but under some circumstances the answer can become “yes”. First of all, in order to answer such questions, we need to deal with the isomorphism classes of elliptic curves instead of the elliptic curves themselves by Proposition 2.2(2). And yet, there still remains the following problem: Let K be a ﬁeld with char(K) = 2, and let E be an elliptic curve deﬁned by the equation y 2 = ax3 + bx2 + cx + d. Take e ∈ K × , and let E be the elliptic curve deﬁned by y 2 = aex3 + bex2 + cex + de. Then, we can show that jE = jE , but E and E are not isomorphic unless e is a square in K × . On the other hand, if K is algebraically closed, we will show later that E ∼ = E if and only if jE = jE .

2.2. MODULI SPACES

37

Consider the above situation geometrically. Let A1Q = Spec Q[j] be the aﬃne line over Q. Let E be an elliptic curve over an arbitrary scheme S. The mapping that sends f ∗ j ∈ Γ(S, O) to a morphism f : S → A1Q is a bijection. Thus, the j-invariant jE of an elliptic curve E over S deﬁnes a morphism of schemes jE : S → A1 . Moreover, if S = Spec K is the spectrum of an algebraic closed ﬁeld K, then the mapping that sends the isomorphism class of an elliptic curve E over K to the morphism jE : Spec K → A1Q deﬁned by j-invariant jE to an elliptic curve E: (2.3)

class of E ∈ {isomorphism classes of elliptic curves over K} ↓ ↓ ∈ {morphisms of schemes Spec K → A1Q }. jE

is a bijection. This means that the geometric points of A1Q are in one-to-one correspondence with the isomorphism classes of elliptic curves. Thus, A1Q may be regarded as the set of isomorphism classes equipped with a suitable algebraic geometric structure. In general, the set of isomorphism classes of certain objects equipped with a geometric structure is called a moduli space. When we regard A1Q as the moduli space of elliptic curves, we call it the j-line. In the following, in order to deﬁne modular forms, we will deﬁne so-called modular curves as a moduli space similar to the j-line. First, in §2.2 we present the rudiments of moduli spaces. 2.2. Moduli spaces In order to introduce a geometric structure on a set of isomorphism classes, it is convenient to use the notion of functors. Definition 2.3. Let S be a scheme. (1) M is a functor over S if (i) for any scheme T over S, a set M(T ) is given, and (ii) for any morphism f : T → T of schemes over S, a morphism f ∗ : M(T ) → M(T ) is given and satisﬁes the two conditions that (a) for any scheme T over S, id∗T = idM(T ) , and (b) for any morphisms f : T → T and g : T → T of schemes over S, (g ◦ f )∗ = f ∗ ◦ g ∗ . (2) Let M and M be functors over S. a : M → M is a morphism of functors if

38

2. MODULAR FORMS

(i) for any scheme T over S, a mapping aT : M(T ) → M (T ) is given and satisﬁes the condition that (a) for any morphism f : T → T of schemes over S, the following diagram is commutative. a

a

M(T ) −−−T−→ M (T ) ⏐ ⏐ ⏐ ∗ ⏐ f ∗ f M(T ) −−−T−→ M (T ) (3) A morphism of functors a : M → M is an isomorphism if for any scheme T over S, the mapping aT : M(T ) → M (T ) is bijective. If S = Spec A is an aﬃne scheme, a functor over S is called a functor over A. What we call a “functor over S” in this book is called more precisely a contravariant functor from the category of schemes over S to the category of sets. It is often customary to deﬁne a functor M by just giving the set M(T ) for each scheme T without giving f ∗ explicitly, which is often obvious from the deﬁnition of M(T ). Example 2.4. (1) Deﬁne a functor M over Q as follows. (i) For any scheme T over Q, deﬁne (2.4)

M(T ) = {isomorphism classes of elliptic curves}. (ii) For any morphism f : T → T of schemes over Q, deﬁne f ∗ : M(T ) → M(T ) by f ∗ (isomorphism class of E) = (isomorphism class of E ×T T ).

(2) Let M be a scheme over S. For any scheme T over S, let M (T ) = {morphisms of schemes T → M over S}, and for any morphism f : T → T of schemes over S, deﬁne a mapping f ∗ : M (T ) → M (T ) by f ∗ (g) = f ◦ g. This is a functor over S. This is called the functor represented by M and, by an abuse of notation, we denote it also by M . Let M be a functor over S, and M a scheme over S. A morphism from the functor M to the functor represented by the scheme M is called a morphism from the functor M to the scheme M . Conversely, a morphism from the functor M represented by the scheme M to the functor M is called a morphism from the scheme M to the functor M.

2.2. MODULI SPACES

39

Proposition 2.5. Let M be a functor over S, and let M be a scheme over S. Then, the mapping {morphisms from M to M} → M(M ) ∪ ∪ f → fM (idM ) is bijective. Proof. The inverse mapping is given as follows. Suppose a ∈ M(M ). Deﬁne a morphism of functors a : M → M by deﬁning aT : M (T ) → M(T ) by f → f ∗ a for any scheme T over S. Example 2.6. A morphism j from the functor M in Example 2.4(1) to the aﬃne line A1Q = Spec Q[j] is deﬁned as follows. Let T be a scheme over Q. Deﬁne a mapping jT : {isomorphism classes of elliptic curves over T} → A1Q (T ) = Γ(T, O) by (2.5)

jT (isomorphism class of E) = (j-invariant jE of E)

for any elliptic curve E over T . Definition 2.7. (1) Let M be a functor over S. A scheme M over S is a ﬁne moduli scheme of the functor M if there is an isomorphism of functors a : M → M. We call aM (idM ) ∈ M(M ) the universal element of the functor M. (2) Let M be a functor over S. A scheme M over S is a coarse moduli scheme of M if there is a morphism of functors a : M → M satisfying the following conditions: (i) For an arbitrary scheme M over S, the mapping that sends a morphism of schemes g : M → M to the morphism g ◦ a : M → M from the functor M to the scheme M , {morphisms M → M of schemes over S} the functor , → morphisms over S from M to the scheme M is bijective. s) → (ii) For any geometric point s¯ of S, the mapping as¯ : M(¯ M (¯ s) is bijective.

40

2. MODULAR FORMS

As in Deﬁnition A.2 of Appendix A, a geometric point of a scheme S is a morphism from the spectrum s¯ = Spec K of an algebraic closed ﬁeld K to S. If a scheme M over S is a ﬁne moduli scheme of a functor M, we say that M is represented by M . By Proposition 2.5, a bijection aT : M (T ) → M(T ) for any scheme T over S is given by g → g ∗ u, where u ∈ M(M ) is the universal element of the functor M. A ﬁne moduli scheme M of a functor M is a coarse moduli scheme of M. Let f : M → M be a morphism of functors over S, and let a : M → M and a : M → M be coarse moduli schemes of M and M , respectively. Then, there is a unique morphism g : M → M of schemes over S that makes the following diagram commutative. f

M −−−−→ ⏐ ⏐ a

M ⏐ ⏐ a

M −−−−→ M g

In particular, a coarse moduli scheme of a functor M is unique up to a canonical isomorphism. We will show, in Proposition 2.15(1), that the aﬃne line A1Q is a coarse moduli scheme of the functor M in Example 2.4(1) through the morphism j. This is the ﬁrst example of a modular curve. 2.3. Modular curves and modular forms In this section we deﬁne modular curves as the moduli scheme of functors by the set of isomorphism classes of elliptic curves with various level structures. Furthermore, we deﬁne modular forms as diﬀerential forms on modular curves. As we deﬁned in Deﬁnition A.19(2), a ﬁnite ´etale commutative group scheme C over S is a cyclic group scheme of order N if for any geometric point s¯, the ﬁnite commutative group C(¯ s) is isomorphic to the cyclic group Z/N Z. Definition 2.8. Let N be a positive integer. (1) Deﬁne a functor M0 (N ) as follows. (i) For any scheme T over Q, deﬁne ⎧ ⎫ isomorphism classes of pairs (E, C), ⎪ ⎪ ⎨ ⎬ (2.6) M0 (N )(T ) = where E is an elliptic curve over T , and . ⎪ ⎩ C is its cyclic subgroup scheme of order ⎪ ⎭ N

2.3. MODULAR CURVES AND MODULAR FORMS

41

(ii) For any morphism of schemes f : T → T over Q, deﬁne f ∗ (M0 (N ))(T ) → M0 (N )(T ) by f ∗ (isomorphism class of (E, C)) = (isomorphism class of (E ×T T, C ×T T )). (2) Let E be a generalized elliptic curve over S. A Γ0 (N ) structure of E is a cyclic subgroup scheme C of order N of the smooth part E sm such that for any geometric points s¯ of S, C ×S s¯ intersects with all the irreducible components of Es¯. (3) Deﬁne a functor M0 (N ) over Q as follows. (i) For any scheme T over Q, (2.7) ⎫ ⎧ ⎨ isomorphism classes of pairs (E, C) of gen- ⎬ M0 (N )(T ) = eralized elliptic curve E over T and a . ⎭ ⎩ Γ0 (N )-structure C of E (ii) For any morphism of schemes f : T → T over Q, deﬁne f ∗ : M0 (N )(T ) → M0 (T ) by f ∗ (isomorphism class of (E, C)) = (isomorphism class of (E ×T T, C ×T T )). If E is an elliptic curve over a scheme S, a Γ0 (N )-structure of E is just a cyclic subgroup scheme C of E of order N . Example 2.9. (1) If N = 1, the functor M0 (1) is nothing but the functor M in Example 2.4(1). (2) Let P1 be a N´eron 1-gon. The cyclic subgroup μN ⊂ Gm = P1sm is a Γ0 (N )-structure of P1 . Theorem 2.10. Let N be a positive integer. (1) There exists a coarse moduli scheme of each of the functors M0 (N ) and M0 (N ). Let X0 (N ) and Y0 (N ) be the coarse moduli schemes of M 0 (N ) and M0 (N ), respectively. (2) X0 (N ) is a proper smooth curve over Q and Y0 (N ) is a dense aﬃne open subscheme of X0 (N ). (3) X0 (N ) is connected and its ﬁeld of constants is Q. We will prove (1) and (2) in the next section. Part (3) will be proved in §2.11 using the analytic expression of the modular curves.

42

2. MODULAR FORMS

Corollary 2.11. The space S(N ) = Γ(X0 (N ), Ω1X0 (N )/Q ) of regular diﬀerential forms on X0 (N ) is a ﬁnite dimensional Q-vector space. Definition 2.12. Let N be a positive integer. (1) The algebraic curve X0 (N ) is called the modular curve of level N . (2) A regular diﬀerential form on X0 (N ) is called a modular form of level N with Q-coeﬃcients. The space S(N ) of regular differential forms is called the space of modular forms with Qcoeﬃcients. (3) Let E be a generalized elliptic curve over a scheme S. Suppose C is a Γ0 (N )-structure of E. The image of the isomorphism class of (E, C) by the mapping M0 (N )(S) → X0 (N )(S) is a morphism S → X0 (N ) of schemes. This morphism is called the morphism deﬁned by (E, C). (4) A point in the complement X0 (N ) Y0 (N ) is called a cusp. The cusp of X0 (N ) deﬁned by the pair (Γ1 , μN ) in Example 2.9(2) is called the ∞-cusp. Usually, X0 (N ) is called the modular curve of level Γ0 (N ) over Q, and an element of S(N ) is called a cusp form of weight 2, level N , and the trivial character with Q-coeﬃcients. However, until Chapter 5, there will not appear any other modular curves and modular forms, and so we call them as such for simplicity. The reader should be careful when reading other books. This S(N ) is the space S(N )Q in Chapter 0. Suppose M divides N . Then, the natural morphism of functors M0 (N ) → M0 (M ) induces a morphism of coarse moduli schemes Y0 (N ) → Y0 (M ) and X0 (N ) → X0 (M ). As a pullback by this morphism, an injection of Q-vector spaces S(M ) → S(N )

(2.8)

is deﬁned. The dimension of S(N ) is calculated as follows. Definition 2.13. Deﬁne functions of positive integers ψ, ϕ∞ , ϕ4 , ϕ6 : N {0} → N {0} as follows. Here, ϕ(N ) stands for the number of elements in the ﬁnite group (Z/N Z)× . (1) ψ(N ) = P1 (Z/N Z) = {cyclic subgroups of order N of (Z/N Z)2 }.

2.3. MODULAR CURVES AND MODULAR FORMS

(2) ϕ∞ (N ) =

43

ϕ gcd(d, N/d) . d|N

(3) ϕ6 (N ) = {a ∈ Z/N Z | a2 + a + 1 = 0}, ϕ4 (N ) = {a ∈ Z/N Z | a2 + 1 = 0}. Note that we have P1 (Z/N Z) = {(a, b) ∈ (Z/N Z)2 | Z/N Z is generated by a, b}/(Z/N Z)× , and P1 (Z/N Z) = Z/N Z {∞} unless N is a prime number. ψ(N ), ϕ∞ (N ), ϕ4 (N ), ϕ6 (N ) can be calculated as follows. Lemma 2.14. Let “?” denote either ∞, 6, or 4. (1) ψ(1) = ϕ? (1) = 1. (2) If N and M are relatively prime positive integers, then we have ψ(N M ) = ψ(N )ψ(M ), ϕ? (N M ) = ϕ? (N )ϕ? (M ). (3) If N = pe with e ≥ 1, then ψ(N ) = (p + 1)pe−1 , 2p(e−1)/2 ϕ∞ (N ) = (p + 1)pe/2−1

(2.9) (2.10)

if e is odd, if e is even.

(4) If 9 | N or N has a prime factor p with p ≡ −1 mod 3, then ϕ6 (N ) = 0. Otherwise, ϕ6 (N ) = 2n , where n is the number of prime factors p with p ≡ 1 mod 3. (5) If 4 | N or N has a prime factor p with p ≡ −1 mod 4, then ϕ4 (N ) = 0. Otherwise, ϕ4 (N ) = 2n , where n is the number of prime factors p with p ≡ 1 mod 4. Proposition 2.15. (1) The morphism j : M0 (1) → A1Q induces an isomorphism Y0 (1) → A1Q . This morphism is uniquely extended to X0 (N ) → P1Q . (2) The genus g0 (N ) of X0 (N ) is given by the formula (2.11)

g0 (N ) = 1 +

1 1 1 1 ψ(N ) − ϕ∞ (N ) − ϕ6 (N ) − ϕ4 (N ). 12 2 3 4

Corollary 2.16. The dimension of the Q-vector space S(N ) = Γ X0 (N ), Ω1X0 (N )/Q equals g0 (N ).

44

2. MODULAR FORMS

Example 2.17. N ψ(N )

1 2 3 4 5

6

7

8

9

10 11 12 13 14

1 3 4 6 6 12

8 12 12 18 12 24 14 24

ϕ∞ (N )

1 2 2 3 2

4

2

4

4

4

2

6

2

4

ϕ6 (N )

1 0 1 0 0

0

2

0

0

0

0

0

2

0

ϕ4 (N )

1 1 0 0 2

0

0

0

0

2

0

0

2

0

g0 (N )

0 0 0 0 0

0

0

0

0

0

1

0

0

1

N g0 (N )

15 16 17 18 19 20 21 22 23 24 25

26 · · ·

1

2

0

1

0

1

1

1

2

2

1

0

···

In particular, this proves Proposition 0.17. Corollary 2.18. The spaces of modular forms S(1) of level 1 and S(2) of level 2 are 0. Example 2.19. (1) If g0 (N ) = 0, then X0 (N ) is a proper smooth geometrically connected curve over Q of genus 0. Its ∞-cusp is a rational point, and thus it is isomorphic to P1Q . In this case, the space of modular forms S(N ) is 0. There are ﬁfteen N with g0 (N ) = 0, namely N = 1, 2, . . . , 10, 12, 13, 16, 18, and 25. (2) If g0 (N ) = 1, then X0 (N ) is a proper smooth connected curve over Q of genus 1 whose ∞-cusp is a rational point. Thus, it is an elliptic curve over Q. In this case, the space of modular forms S(N ) has dimension 1. There are twelve N with g0 (N ) = 1, namely N = 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, and 49. (3) The elliptic curve X0 (11) over Q is isomorphic to the elliptic curve y 2 = 4x3 −4x2 −40x−79 in Example 1.2. The space of modular forms S(11) has dimension 1, and the diﬀerential form f11 = dx/y is a basis of this space. 2.4. Construction of modular curves To show Theorem 2.10, it suﬃces to show Theorem 2.21 and Proposition 2.23 below. We deﬁne some other kinds of modular curves. The modular curve X(N ) we are about to deﬁne is easier to deal with in that it has a group action, and it is a ﬁne moduli scheme if N ≥ 3. However, it is X0 (N ) that is related directly to elliptic curves and Galois representations.

2.4. CONSTRUCTION OF MODULAR CURVES

45

Definition 2.20. (1) Deﬁne a functor M(N ) as follows. (i) For any scheme T over Q, deﬁne (2.12)

⎧ ⎫ isomorphism classes of pairs (E, α), where ⎪ ⎪ ⎨ ⎬ E is an elliptic curve over T and α : M(N )(T ) = . ⎪ (Z/N Z)2 → E[N ] is an isomorphism of ⎪ ⎩ ⎭ group schemes (ii) For any morphism of schemes f : T → T over Q, deﬁne f ∗ (M(N ))(T ) → M(N )(T ) by f ∗ (isomorphism class of (E, α)) = (isomorphism class of (E ×T T, αT )).

(2) Let E be a generalized elliptic curve over S. A Γ(N )-structure of E is a closed immersion α : (Z/N Z)2 → E sm of group schemes such that for any geometric points s¯ of S, the image of αs¯ intersects with all the irreducible components of the geometric ﬁber Es¯. (3) Deﬁne a functor M(N ) over Q as follows. (i) For any scheme T over Q, (2.13) ⎧ ⎫ ⎨isomorphism classes of pairs (E, α) of a⎬ M(N )(T ) = generalized elliptic curve E over T and a . ⎩ ⎭ Γ(N )-structure α of E (ii) For any morphism of schemes f : T → T over Q, deﬁne f ∗ : M(N )(T ) → M(N )(T ) by f ∗ (isomorphism class of (E, α)) = (isomorphism class of (E ×T T, αT )). Theorem 2.21. Let N be a positive integer. (1) There exist coarse moduli schemes of the functor M(N ) and that of M(N ), respectively. If N ≥ 3, then they are ﬁne moduli schemes. (2) X(N ) is a proper smooth curve over Q. Y (N ) is a dense aﬃne open subschema of X(N ). Proposition 2.22. (1) X(N ) is connected and its ﬁeld of constants is the cyclotomic ﬁeld Q(ζN ).

46

2. MODULAR FORMS

(2) Let g(N ) be the genus of X(N ) as an algebraic curve over Q(ζN ). Then we have (2.14) 1 N ϕ(N )ψ(N ) − 14 ϕ(N )ψ(N ) + 1 if N ≥ 3, g(N ) = 24 0 if N = 1 or 2. We omit the proof of this proposition. We can prove (1) by using the eN -pairing and the analytic expression of modular curves. The proof of (2) is similar to that of Proposition 2.15(2). Deﬁne a natural right action of the group GL2 (Z/N Z) on the functors M(N ) and M(N ) as follows. For σ ∈ GL2 (Z/N Z) and an arbitrary scheme T over Q, deﬁne σ ∗ : M(N )(T ) → M(N )(T ) by σ ∗ (isomorphism class of (E, α)) = (isomorphism class of (E, α ◦ σ)). Deﬁne similarly for M(N ). These actions induce a GL2 (Z/N Z) action on the coarse moduli schemes X(N ) and Y (N ), respectively. Deﬁne a subgroup B(Z/N Z) of GL2 (Z/N Z) by a b (2.15) B(Z/N Z) = ∈ GL2 (Z/N Z) . 0 d Proposition 2.23. Let N be a positive integer. The quotient of X(N ) by B(Z/N Z) is a coarse moduli scheme of the functor M0 (N ), and the quotient of Y (N ) by B(Z/N Z) is a coarse moduli scheme of the functor M0 (N ). See §A.3 in Appendix A for a brief summary of the quotient of a scheme by a ﬁnite group. Proof of Theorem 2.21 + Proposition 2.23 ⇒ Theorem 2.10(1)(2). (1) is contained in Proposition 2.23. We show (2). By Theorem 2.21(2) and Proposition 2.23, X0 (N ) is the quotient of a proper smooth curve X(N ) over Q by the action of B(Z/N Z). Thus, X0 (N ) is also a proper smooth curve over Q. Since Y0 (N ) is the quotient of a dense aﬃne open subscheme Y (N ) of X(N ) by the action of B(Z/N Z), Y0 (N ) is also a dense aﬃne open subscheme X0 (N ). Since the proofs of Theorem 2.21 and Proposition 2.23 are rather tedious, we just indicate an outline and point out the essence of the proof. First, we show Theorem 2.21 in the case N = 3.

2.4. CONSTRUCTION OF MODULAR CURVES

47

Proof of Theorem 2.21 for N = 3. Let X(3) = P1Q(ζ3 ) , and let μ be its nonhomogenous coordinate. Write ω = ζ3 . We deﬁne a generalized elliptic curve EX(3) over X(3) and its Γ(3) structure αX(3) . Deﬁne EX(3) as a plane curve given by the homogenous equation (2.16)

X 3 + Y 3 + Z 3 − 3μXY Z = 0.

Deﬁne a section O of EX(3) by X = 0, Y + Z = 0. Let Y (3) = X(3) {1, ω, ω 2 , ∞}. Over Y (3), EX(3) is a smooth plane cubic curve with a 0-section, and thus it is an elliptic curve. If x ∈ {1, ω, ω 2 , ∞}, the ﬁber Ex at x is a N´eron 3-gon. E∞ is deﬁned by the equation XY Z = 0, and E1 is deﬁned by the equation (2.17)

X 3 + Y 3 + Z 3 − 3XY Z = (X + Y + Z)(X + ωY + ω 2 Z)(X + ω 2 Y + ωZ) = 0.

The ﬁbers at x = ω and ω 2 are similar. EX(3) has the structure of a generalized elliptic curve over X(3). Deﬁne sections P and Q of the sm by (X = 0, Y + ωZ) and (Y = 0, X + Z = 0), smooth part EX(3) respectively. Let e1 , e2 be the standard basis of (Z/3Z)2 , and deﬁne sm a homomorphism of group schemes αX(3) : (Z/3Z)2 → EX(3) by αX(3) (e1 ) = P and αX(3) (e2 ) = Q. This is a Γ(3) structure of EX(3) . Now that we have deﬁned the pair (EX(3) , α) of a generalized elliptic curve over X(3) and its Γ(3) structure, we have a morphism from the scheme X(3) to the functor M(3). We show this is an isomorphism. To do so, it suﬃces to show that for a pair (E, α) of a generalized elliptic curve over an arbitrary S and its Γ(3) structure α, there is a unique morphism f : S → X(3) satisfying the condition: (2.18) the pair (E, α) is isomorphic to the pullback of (EX(3) , α) by f . To ease the notation, we restrict ourselves to the case S = Spec K for a ﬁeld K. The general case is quite similar to this case. For further simplicity, we assume that if E is a N´eron 3-gon, P = α(e1 ) is on an irreducible component diﬀerent from O. If this is not the case, it suﬃces to replace it by Q = α(e2 ). All we have to show is the following. Lemma 2.24. Let E be a generalized elliptic curve over a ﬁeld of characteristic diﬀerent from 3, and let α be a Γ(3) structure of E. If

48

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E is a N´eron 3-gon, assume P is a point on an irreducible component that does not contain the 0-section O. Then, there exist a unique μ ∈ K and a primitive root of unity ω ∈ K satisfying the following condition: (2.19)

E is deﬁned by the equation X 3 +Y 3 +Z 3 −3μXY Z = 0. P and Q = α(e2 ) are determined by (X = 0, Y +ωZ = 0) and (Y = 0, X + Z = 0), respectively.

The idea of the proof is as follows. Let E be a plane cubic curve. Choose a coordinate system such that the three lines passing through (O, P, 2P ), (Q, Q + P, 2P + Q), and (2Q, P + 2Q, 2P + 2Q) become (X = 0), (Y = 0), and (Z = 0), respectively. Scaling each coordinate if necessary, we may assume that the three lines passing through (O, Q, 2Q), (P, P + Q, P + 2Q), and (2P, 2P + Q, 2P + 2Q) are given by (X + Y + Z = 0), (X + ω 2 Y + ωZ = 0), and (X + ωY + ω 2 Z = 0), respectively. Then, the function X 3 + Y 3 + Z 3 − 3XY Z = (X + Y + Z)(X + ω 2 Y + ωZ)(X + ωY + ω 2 Z) becomes 0 at the nine points generated by P and Q, and thus it must be a constant multiple of XY Z. It now suﬃces to let this constant equal 3(μ − 1). Proof. The subgroup P of E sm generated by P is a very ample divisor. Let L = O(P ), and V = Γ(E, L). Then V is a three dimensional K-vector space, and we have a closed immersion E → PK (V ) P2K . Choose a basis of V as follows. Here, for a K-rational point R ∈ E(K), the automorphism of E given by adding the point R is denoted by +R : E → E. If we let (2.20)

L0 = Γ(E, O),

L1 = Γ(E, O(P − (+Q)∗ P )),

L2 = Γ(E, O(P − (+2Q)∗ P )), then V is decomposed into the direct sum L0 ⊕ L1 ⊕ L2 of one dimensional subspaces. The natural action on V of the cyclic subgroup P of order 3 preserves this decomposition. The action of P on L1 is the multiplication by a primitive third root of unity, dentoed by ω. Then, P acts on L2 by the multiplication by ω 2 . If L is the kernel of V → L|Q , then the composition L → V → Li , i = 0, 1, 2, are all isomorphisms. Let Y and Z be the elements of L1 and L2 , respectively, that correspond to the basis X = 1 of L0 through these isomorphisms.

2.4. CONSTRUCTION OF MODULAR CURVES

49

We show that X 3 + Y 3 + Z 3 − 3XY Z ∈ Γ(E, L⊗3 ) is a multiple of XY Z. The kernel L of Γ(E, L⊗3 ) → L⊗3 |E[3] is a one dimensional Klinear space, and XY Z is a basis of this space. Since X +Y +Z equals 0 at Q, X + ωY + ω 2 Z = (+P )∗ (X + Y + Z) and X + ω 2 Y + ωZ = (+2P )∗ (X + Y + Z) are 0 at (+P )∗ Q and (+2P )∗ Q, respectively. Thus, (X + Y + Z)(X + ωY + ω 2 Z)(X + ω 2 Y + ωZ) = X 3 + Y 3 + Z 3 − 3XY Z is an element of L . Hence, it is a multiple of XY Z. Deﬁne μ ∈ K such that X 3 +Y 3 +Z 3 −3XY Z = 3(μ−1)XY Z. Then, E is deﬁned by the equation X 3 + Y 3 + Z 3 − 3μXY Z = 0. P satisﬁes X = 0 and X + ωY + ω 2 Z = 0, and Q satisﬁes Y = 0 and X + Y + Z = 0. We thus obtained ω, μ ∈ K satisfying the required conditions. The uniqueness is clear by following the above argument backward. The proof of Theorem 2.21 and Proposition 2.23 still continues. From now on, the arguments become more abstract, so the reader who is not familiar with this type of argument may skip it for the moment, and may come back here later on. The case of general N is reduced to the case N = 3 by Lemma 2.26 below. Lemma 2.25. Let N be a positive integer, let S be a scheme where N is invertible, and let E be an elliptic curve over S. Then, the functor over S given by Γ(N )E : T → {isomorphisms : (Z/N Z)2 → ET [N ]} can be represented by an open and closed subscheme of E[N ] ×S E[N ]. Γ(N )E is ﬁnite and ´etale over S. Proof. Let T be a scheme over S. Let e1 , e2 be the standard basis of (Z/N Z)2 . Since the isomorphism α : (Z/N Z)2 → E[N ] is determined by α(e1 ), α(e2 ) ∈ E[N ](T ), we obtain an injective morphism of functors Γ(N )E → E[N ] ×S E[N ]. Let P, Q ∈ E[N ](T ), and deﬁne a morphism αP,Q : (Z/N Z)2 → ET [N ] of group schemes over T . Let T = E[N ] ×S E[N ] and P, Q = pr1 , pr2 ∈ E[N ](E[N ] ×S E[N ]). Then, Γ(N )E is represented by the open and closed subscheme of E[N ] ×S E[N ] determined by the condition that αP,Q is an isomorphism. Since Γ(N )E is an open and closed subscheme of a ﬁnite and ´etale scheme over S, it is ﬁnite and ´etale over S. Lemma 2.26. Let N be a positive integer. Suppose a ﬁne moduli scheme X(N ) of the functor M(N ) over Q exists. Let E = EX(N ) be the universal generalized elliptic curve over X(N ).

50

2. MODULAR FORMS

(1) Let Y (N ) be the open subscheme of X(N ) where E is smooth. Then, Y (N ) is a ﬁne moduli scheme of the functor M(N ). (2) Let M be a multiple of N . Let ΓN,EY (N ) and ΓM,EY (N ) be ﬁnite ´etale schemes over Y (N ) that represent functors over Y (N ), T → {isomorphisms (Z/N Z)2T → ET [N ]} and {isomorphisms (Z/M Z)2T → ET [M ]}, respectively. Let Y (N ) → ΓN,EY (N ) be the section determined by the universal isomorphism (Z/N Z)2 → E[N ], and let ΓN,EY (N ) → ΓM,EY (N ) be the natural morphism. Then, the functor M(M ) is represented by the ﬁbered product ΓM,EY (N ) ×ΓN,EY (N ) Y (N ). The integral closure X(M ) of Y (M ) in X(N ) is a ﬁne moduli scheme of the functor M(M ). (3) Suppose M divides N , and let G be the kernel of the natural morphism GL2 (Z/N Z) → GL2 (Z/M Z). Then, the quotients X(N )/G and Y (N )/G are coarse moduli schemes of the functors M(M ) and M(M ), respectively. If M ≥ 3, then they are ﬁne moduli schemes. Proof of Lemma 2.26 ⇒ Theorem 2.21. By Lemma 2.26(2), Theorem 2.21 holds for any multiple of 3. Then, by Lemma 2.26(3), Theorem 2.21 holds for any N since N divides 3N . Except for the last assertion of (2), we see Lemma 2.26(1) and (2) easily from the deﬁnition. The proof of (3) is similar to that of Proposition 2.23 below, and we omit it here. Proof of Proposition 2.23. For simplicity, we show only for Y0 (N ). We show in the case N ≥ 3, where Y (N ) is a ﬁne moduli scheme of M(N ). The case N ≤ 2 can be shown similarly by taking a multiple of N . Let Y0 (N ) be the quotient Y (N )/B(Z/N Z), and we show this is a coarse moduli scheme of M0 (N ). By the deﬁnition of the quotient, Deﬁnition A.25, and the assumption that Y (N ) is a ﬁne moduli scheme of the functor M(N ), it suﬃces to show the following lemma. Lemma 2.27. Let N be a positive integer, and let e1 = (1, 0) ∈ (Z/N Z)2 . Let ϕ : M(N ) → M0 (N ) be the morphism of functors determined by sending an isomorphism class (E, α) to the isomorphism class (E, α(e1 )). (1) Let T be any scheme over Q. To a morphism g from the functor M0 (N ) to the scheme T , the mapping that sends the B(Z/N Z)

2.4. CONSTRUCTION OF MODULAR CURVES

51

invariant morphism g ◦ϕ from the functor M(N ) to the scheme, {morphisms from the functor M0 (N ) to the scheme T } B(Z/N Z) invariant morphisms from the → , functor M(N ) to the scheme T is a bijection. (2) For any algebraically closed ﬁeld K of characteristic 0, the mapping determined by ϕ M(N )(K)/B(Z/N Z) → M0 (N )(K) is a bijection. Proof. (1) We deﬁne the inverse mapping. Let f : M(N ) → T be a B(Z/N Z)-invariant morphism from the functor M(N ) to the scheme T . We construct a morphism g from the functor M0 (N ) to the scheme T satisfying f = g ◦ ϕ as follows. For any scheme U over Q, we deﬁne gU : M0 (N )(U ) → T (U ). Let E be an elliptic curve over U , and let C be its cyclic subgroup of order N . Let x = (isomorphism class of (E, C)) ∈ M0 (N )(U ), and deﬁne gU (x) ∈ T (U ) as follows. Consider the functor over U given by W → {isomorphism α : (Z/N Z)2 → EW [N ] such that α(e1 ) = CW }. This can be represented by a ﬁnite ´etale scheme V over U . Its proof is similar to that of Lemma 2.25, and we omit it here. The group B(Z/N Z) acts naturally on V , and we have V /B(Z/N Z) = U . The isomorphism class of the pair (EV , αV ) of the pullback EV and the universal isomorphism αV : (Z/N Z)2 → EV [N ] over V determines an element y in M(N )(V ). The element y commutes with the action of B(Z/N Z) on V and M(N ). Consider the following diagram. x ∈ M0 (N )(U ) −−−−→ M0 (N )(V ) ←−−−− M(N )(V ) y ⏐ ⏐ ⏐ ⏐ ⏐ ⏐f gU gV V (2.21) T (U )

−−−−→ h

T (V )

T (V )

Here, gU and gV are yet to be deﬁned. Or rather, we want to deﬁne them so that the diagram (2.21) commutates. Since U is the quotient of V by B(Z/N Z), h is a bijection from T (U ) to the B(Z/N Z)invariant part of T (V ). Since the morphism M(N ) → T is B(Z/N Z) invariant, fV (y) ∈ T (V ) belongs to the B(Z/N Z)-invariant part.

52

2. MODULAR FORMS

Thus, there exists a unique z ∈ T (U ) satisfying h(z) = fV (u) ∈ T (V ). In order to make (2.21) commutative, we cannot but deﬁne gU (x) = z, and so we deﬁne gU : M0 (N )(U ) → T (U ) by letting z = gU (x). We omit the veriﬁcation that this determines a morphism g : M0 (T ) → T , and it satisﬁes f = g ◦ ϕ. The uniqueness of g is clear form the construction. (2) Let π : M(N ) → M and π0 : M0 (N ) → M be natural morphisms that send the isomorphism class of (E, α) and of (E, C) to the isomorphism class of E, respectively. If E is an elliptic curve over an algebraically closed ﬁeld K, the inverse image π ∗ ([E]) in M(N )(K) of [E] = (isomorphism class of E) ∈ M(K) is given by {isomorphism classes of (E, α) | α : (Z/N Z)2 → E[N ](K) isomorphism}. Similarly, the inverse image π0∗ ([E]) in M0 (N )(K) is {isomorphism classes of (E, C) | C cyclic subgroup of order N of E[N ](K)}. Choose any isomorphism α0 : (Z/N Z)2 → E[N ](K). Then, the set {isomorphisms α : (Z/N Z)2 → E[N ](K)} may be identiﬁed with GL2 (Z/N Z) through the bijection g → α0 ◦ g. Under this identiﬁcation, we have (2.22)

π ∗ ([E]) = (image of Aut(E))\GL2 (Z/N Z).

Similarly, the set {cyclic subgroups of order N of E[N ](K)} may be identiﬁed with the quotient GL2 (Z/N Z)/B(Z/N Z) through the bijection g → α0 (ge1 ), and we have (2.23)

π0∗ ([E]) = (image of Aut(E))\GL2 (Z/N Z)/B(Z/N Z).

Thus, π ∗ ([E])/B(Z/N Z) → π0∗ ([E]) is bijective.

2.5. The genus formula Proof of Proposition 2.15(1). By the deﬁnition of coarse moduli schemes, the morphism of functors j : M → A1Q deﬁnes a morphism of schemes j : Y0 (1) → A1Q . By Theorem 2.10(2), it sufﬁces to show that the morphism j : Y0 (1) → A1Q of smooth aﬃne curves is an isomorphism. First, we show that j is birational. Let 1 U = A1Q {0, 123 } = Spec Q j, j(j−12 3 ) , and we deﬁne the inverse i : U → Y0 (1). Deﬁne an elliptic curve E over U by the equation (2.24)

y 2 = 4x3 −

123 j 243 x− . 3 j − 12 j − 123

2.5. THE GENUS FORMULA

53

Since its discriminant 3 2 123 j 243 j 1212 j 2 (2.25) Δ= − 27 = 3 3 j − 12 j − 12 (j − 123 )3 is invertible in U , E is an elliptic curve over U . Deﬁne i : U → Y0 (1) by the image of (isomorphism class of E) ∈ M(U ) in Y0 (1)(U ). The 123 j 3 1 j-invariant of the elliptic curve E equals 123 j−2 13 Δ = j. Thus, the composition j ◦ i is the identity mapping of U , and j is birational. The fact that j is surjective follows from above, together with the fact that the j-invariants of the elliptic curves y 2 = x3 − 1 and y 2 = x3 − x are 0 and 123 , respectively. To show Proposition 2.15(2), we apply the Riemann–Hurwitz formula to the ﬁnite morphism of algebraic curves π0 : X0 (N ) → X0 (1) = P1Q . We count the number of points in the inverse image of each point. Lemma 2.28. For j ∈ X0 (1)(Q) = P1 (Q), the number of points in π0−1 (j) is as follows. (1) If j = 0, 123 , ∞, then ψ(N ). (2) If j = 0, then 13 ψ(N ) + 23 ϕ6 (N ). (3) If j = 123 , then 12 ψ(N ) + 12 ϕ4 (N ). (4) If j = ∞, then ϕ∞ (N ). Proof of Proposition 2.15(2). By Lemma 2.28(1), the degree of π0 : X0 (N ) → X0 (1) equals ψ(N ). Applying the Riemann– Hurwitz formula to π0 : X0 (N ) → X0 (1) = P1Q , we have (2.26) 2g0 (N ) − 2 + (π0−1 (∞)) + (π0−1 (0)) + (π0−1 (123 )) = ψ(N )(2 · 0 − 2 + 3). It now suﬃces to replace each term by the results of Lemma 2.28.

Proof of Lemma 2.28. If j = ∞, let E = Ej be an elliptic curve over Q with j-invariant equal to j. If j = ∞, let E = E∞ = PN be the N´eron N -gon. Let Aut(E) be the automorphism group of a generalized elliptic curve E. As we have shown in Lemma 2.27(2), the inverse image π0−1 (j) of j ∈ X0 (1)(Q) is in one-to-one correspondence with (2.27) (image of Aut(E))\GL2 (Z/N Z)/B(Z/N Z) = (image of Aut(E))\P1 (Z/N Z).

54

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We count the number of elements of the quotient set on the right. First, we compute Aut(E). If j = 0, 123 , ∞, then Aut(E) is a cyclic group of order 2 generated by σ2 : (x, y) → (x, −y). If j = 0, then E is deﬁned by the equation y 2 = x3 − 1. Denoting by ω a primitive third root of unity, Aut(E) is a cyclic group of order 6 generated by σ6 : (x, y) → (ωx, −y). If j = 123 , then E is deﬁned by the equation y 2 = x3 − x. Denoting by i a primitive fourth root of unity, Aut(E) is a cyclic group of order 4 generated by σ4 : (x, y) → (−x, iy). If j = ∞, then the automorphism group of the N´eron N -gon PN is the semidirect product {±1} μN . If j = 0, 123 , ∞, then the action of the generator σ2 of Aut(E) on 1 P (Z/N Z) is trivial, and thus the number of elements equals ψ(N ). Suppose j = 0. The third power of the action of the generator σ6 of Aut(E) is trivial. Thus, if we denote by Fix6 ⊂ P1 (Z/N Z) the set of ﬁxed points of σ6 , then we have (2.28)

1 (P1 (Z/N Z) Fix6 ) + Fix6 3 2 1 = ψ(N ) + Fix6 . 3 3

(Aut(E)\P1 (Z/N Z)) =

Since Fix6 = {a ∈ Z/N Z | a2 + a + 1 = 0} ⊂ P1 (Z/N Z), its number of elements equals ϕ6 (N ). Similarly, suppose j = 123 . The square of the action of the generator σ4 of Aut(E) is trivial. Thus, if we denote by Fix4 ⊂ P1 (Z/N Z) the set of ﬁxed points of σ4 , then we have (2.29)

1 (P1 (Z/N Z) Fix4 ) + Fix4 2 1 1 = ψ(N ) + Fix4 . 2 2

(Aut(E)\P1 (Z/N Z)) =

Since Fix4 = {a ∈ Z/N Z | a2 + 1 = 0} ⊂ P1 (Z/N Z), its number of elements equals ϕ4 (N ). Finally, suppose j = ∞. For a divisor d of N , let Pd = {(a : b) ∈ is stable of P1 (Z/N Z) | (a, N ) = d}. The subset Pd under the action 1 b

the group IN = (image of Aut(PN )) = b ∈ Z/N Z . The 0 1 quotient IN \Pd can be identiﬁed with

×

(Z/ Nd Z)× × (Z/dZ)× /(Z/N Z)× Z/(d, Nd )Z

2.6. THE HECKE OPERATORS

55

by the mapping (a : b) → ( ad , b). Thus, the number of elements of the set IN \P 1 (Z/N Z) equals ϕ∞ (N ). 2.6. The Hecke operators In order to deﬁne Hecke operators, we deﬁne some more kinds of modular curves. Definition 2.29. Let N and n be positive integers. (1) Deﬁne a functor M0 (N, n) over Q as follows. (i) For any scheme T , ⎧ deﬁne ⎫ isomorphism classes of ⎪ ⎪ ⎪ ⎪ ⎪ (E, C, E , C , f ), where E and ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ E are elliptic curves over T , ⎪ M0 (N, n)(T ) = C, and C are their cyclic . ⎪ ⎪ ⎪ group subschemes of order N , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ and f : E → E is a morphism ⎪ ⎪ ⎪ ⎩ ⎭ of order n such that f (C) = C (ii) For any morphism of schemes f : T → T over Q, deﬁne f ∗ : M0 (N, n)(T ) → M0 (N, n)(T ) by f ∗ (isomorphism class of (E, C, E , C , f )) = (isomorphism class of (ET , CT , ET , CT , fT )). (2) Deﬁne morphisms of functors s, t : M0 (N, n) → M0 (N ) as follows. For an arbitrary scheme T over Q, deﬁne sT , tT : M0 (N, n)(T ) → M0 (N )(T ), respectively, by sT (isomorphism class of (E, C, E , C , f )) = (isomorphism class of (E, C)), tT (isomorphism class of (E, C, E , C , f )) = (isomorphism class of (E , C )). For positive integers N and n, deﬁne a ﬁnite set SN,n by ⎧

⎫

C is a cyclic subgroup of order N of ⎬ ⎨

SN,n = (C, H)

(Z/N nZ)2 , H is a subgroup of order n of . ⎩ ⎭

(Z/nZ)2 such that C ∩ H = 0 The right action of the group GL2 (Z/N Z) on SN,n is deﬁned by the natural left action of the inverse element.

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Proposition 2.30. Let N and n be positive integers. (1) The quotient of Y (N n) × SN,n by GL2 (Z/N Z) Y0 (N, n) = (Y (N n) × SN,n )/GL2 (Z/N Z) is a coarse moduli scheme of the functor M0 (N, n) over Q. (2) Y0 (N, n) is a smooth aﬃne curve over Q. (3) The morphism of functors s, t induce ﬁnite ﬂat morphisms s, t : Y0 (N, n) → Y0 (N ), respectively. The proof of Proposition 2.30 is similar to those of Theorems 2.10 and 2.21 and Proposition 2.23, and thus we do not give it here. We can prove it using the fact that for any scheme T over Q, M0 (N, n)(T ) may be identiﬁed with isomorphism classes of (E, C, Cn), where E is an elliptic curve over T , C is its cyclic group subscheme of order N , and Cn is a . subscheme of order n such that C ∩ Cn = 0 If n is relatively prime to N and is square-free, then we have Y0 (N, n) = Y0 (N n). Deﬁne X0 (N, n) as a smooth projective curve over Q containing Y0 (N, n) as its dense open subscheme. X0 (N, n) is uniquely determined up to isomorphism. X0 (N, n) and Y0 (N, n) may not be connected. The morphisms s, t : Y0 (N, n) → Y0 (N ) can be extended to ﬁnite ﬂat morphisms s, t : X0 (N, n) → X0 (N ), respectively, in a unique way. In order to deﬁne Hecke operators, we deﬁne the trace mapping of diﬀerential forms. In general, for smooth projective curves X and Y over a ﬁeld K and a ﬁnite ﬂat morphism f : X → Y , two K-linear maps f ∗ : Γ(Y, Ω) → Γ(X, Ω) and f∗ : Γ(X, Ω) → Γ(Y, Ω) are deﬁned. f ∗ is nothing but the pullback. The trace mapping f∗ is deﬁned as follows. For simplicity, we assume there exists a dense open subset V in Y such that f |U : U = f −1 (V ) → V is ´etale. Take a dense aﬃne open subset V = Spec A of Y small enough so that the following condition is satisﬁed: f |U : U = f −1 (V ) = Spec B → V is ´etale and the A-module Γ(V, Ω) has a basis ω. Then f ∗ ω is a basis of B-module Γ(U, Ω). Since B is a ﬁnitely generated projective A-module, the trace mapping TrB/A : B → A is deﬁned. Using this, deﬁne the trace mapping (f |U )∗ : Γ(U, Ω) → Γ(V, Ω) by (f |U )∗ (g · f ∗ ω) = TrB/A b · ω. This is independent of the choice of a basis ω. This trace mapping sends the subspace Γ(X, Ω) ⊂ Γ(U, Ω) into the subspace Γ(Y, Ω) ⊂ Γ(V, Ω). The mapping induced as such is f∗ : Γ(X, Ω) → Γ(Y, Ω).

2.6. THE HECKE OPERATORS

57

Definition 2.31. Let N and n be positive integers, and deﬁne s, t : X0 (N, n) → X0 (N ) as above. Deﬁne the Hecke operator Tn : S(N ) → S(N ) by the composition t∗

s

∗ S(N ) = Γ(X0 (N ), Ω) −→ Γ(X0 (N, n), Ω) −→ Γ(X0 (N ), Ω) = S(N ).

Proposition 2.32. Let N be a positive integer. (1) The Hecke operator T1 is the identity mapping on S(N ). If n and m are relatively prime, then the Hecke operators Tn and Tm commute each other: Tn Tm = Tm Tn . (2) Suppose M divides N . If n and N are relatively prime, then the action of the Hecke operator Tn commutes with the injective morphism S(M ) → S(N ) in (2.8). We omit the proof of this proposition, too. Though it is not diﬃcult to prove it directly, it may also be proved by using the qexpansion Proposition 2.41. If n and N are not relatively prime, then Tn and S(M ) → S(N ) are not commutative in general, and so we have to be careful. Definition 2.33. The subring of EndQ (S(N )) generated by all the Hecke operators Tn , n ≥ 1, T (N ) = Q[Tn , n ≥ 1] ⊂ EndQ (S(N )) is called the Hecke algebra of level N . By Proposition 2.32, T (N ) is a commutative ring with the unit T1 , and it is a ﬁnite dimensional Q-vector space. Later in Proposition 2.55, we will show that the dimension of T (N ) is equal to that of S(N ). The following proposition shows that the Hecke algebra T (N ) is generated by the operators Tp for prime numbers p. Proposition 2.34. Let N be a positive integer. (1) If positive integers n and m are relatively prime, then we have Tmn = Tn Tm . (2) Let p be a prime. For any positive integer e ≥ 0, we have (2.30)

Tpe+2 = Tp Tpe+1 − pTpe

if p |N and Tpe = Tpe if p|N .

58

2. MODULAR FORMS

Once again we omit the proof. Proposition 2.34 can be expressed in the following form of formal inﬁnite product. (2.31) ∞ Tn n−s = (1 − Tp p−s + pp−2s )−1 × (1 − Tp p−s )−1 . n=1

p:prime pN

p:prime p|N

The meaning of (2.31) is as follows. Bringing the right-hand side to the left, we have (2.32)

(1−Tp p−s +pp−2s )×

p:prime pN

(1−Tp p−s )×

∞

Tn n−s = 1.

n=1

p:prime p|N

∞ Expand this formula and collect them in the form n=1 fn n−s , fn ∈ T (N ), using the identity m−s n−s = (mn)−s . Then, (2.31) signiﬁes that f1 = id and fn = 0 for n > 1. If we regard s as a complex number, (2.31) may be considered as an equality of functions in s in the region where both sides converge. We can show that this region is Re s > 32 using Theorem 2.47 below. Using the q-expansion, which we introduce in the next section, we can treat the Hecke operators more concretely. 2.7. The q-expansions In §2.3, a modular form was deﬁned as a diﬀerential form on a modular curve. We can study it by expressing it as a power series. This is the q-expansion, which is the theme of this section. Let N be a positive integer. We ﬁrst deﬁne a morphism e : Spec Q[[q]] → X0 (N ), and then deﬁne an injective morphism e∗ : S(N ) → Q[[q]]. By the deﬁnition of X0 (N ), in order to deﬁne the morphism e, it suﬃces to deﬁne a generalized elliptic curve E over the power series ring Q[[q]] and a Γ0 (N )-structure on it. We begin with the deﬁnition of this generalized elliptic curve E. As we have seen in Corollary 3.19 in Chapter 3 of Number Theory 2 , if k is a positive even integer, the value of Riemann’s ζ function (k − 1)! √ ζ(k), is a rational at a negative odd integer, ζ(1 − k) = 2 (2π −1)k number. For example, (2.33)

ζ(−1) = −

1 , 12

ζ(−3) =

1 , 120

ζ(−5) = −

1 , 252

···

2.7. THE q-EXPANSIONS

59

(see also Exercise 7.7 in Chapter 7 of Number Theory 2 ). As in §9.2(b) in Chapter 9 of Number Theory 3 , let σk−1 (n) = d|n dk−1 for positive integers k and n. For an even integer k ≥ 2, deﬁne a power series Ek (q) ∈ Q[[q]] by (2.34)

Ek (q) = 1 +

∞ 2 σk−1 (n)q n . ζ(1 − k) n=1

Proposition 2.35. The equation 1 1 (2.35) y 2 = 4x3 − E4 (q)x + E6 (q) 12 216 deﬁnes a generalized elliptic curve over the power series ring Q[[q]]. E is an elliptic curve over the ﬁeld of power series Q((q)). Proof. By Theorem 9.5 in Chapter 9 of Number Theory 3 , the discriminant of the cubic of the right-hand side of (2.35) is ∞

1 3

1 2 E4 (q) − 27 E6 (q) = q (2.36) Δ(q) = (1 − q n )24 = 0. 12 216 n=1 Thus, equation (2.35) deﬁnes an elliptic curve over Q((q)). If we let q = 0, the right-hand side of (2.35) becomes

1 1

1 2 1 = x+ 2x − (2.37) 4x3 − x + , 12 216 6 6 which does not have a triple root. Thus, at q = 0, it is a N´eron 1-gon. Since the point at inﬁnity is a section of the smooth part E sm , it follows from Proposition 1.30 that E is a generalized elliptic curve. For a positive integer N , we deﬁne a Γ0 (N )-structure on the generalized elliptic curve E. Deﬁne power series x(q, t), y(q, t) ∈ 1 ][[q]] by Q[t, t(1−t) x(q, t) = (2.38)

∞ t 1 d −d + d(t + t ) q n + E2 (q), (1 − t)2 n=1 12 d|n

y(q, t) = t

∞ ∂ t(1 + t) 2 d −d + d (t − t ) qn . x(q, t) = ∂t (1 − t)3 n=1 d|n

The ﬁrst term of each of the formula is h2 (t) and h3 (t), respectively, in Deﬁnition 3.2 in Chapter 3 of Number Theory 1 . These power

60

2. MODULAR FORMS

series satisfy the equation (2.39)

y(q, t)2 = 4x(q, t)3 −

1 1 E4 (q)x(q, t) + E6 (q). 12 216

The relation (2.39) will be proved in §2.12 using the analytic expression of elliptic curve (2.35). By (2.39), we can deﬁne a ring homomorphism ! 2 y − 4x3 −

(2.40) α∗ : Q[[q]][x, y]

1 12 E4 (q)x

+

1 216 E6 (q)

1 [[q]] → Q t, t(t−1)

by letting x → x(q, t) and y → y(q, t). Using the ring homomorphism α∗ , we deﬁne a closed immersion α : Spec Q[[q]][t]/(tN − 1) → E as follows. Since Q[[q]][t]/(tN − 1) = N −1 i closed immersions Q[[q]] × Q[[q]][t]/ i=0 t , it suﬃces to deﬁne N −1 i → E in order α0 : Spec Q[[q]] → E and α1 : Spec Q[[q]][t]/ i=0 t to deﬁne α. Let α0 be the 0-section of the generalized elliptic curve E. Deﬁne α1 by the homomorphism ! 1 1 E4 (q)x + 216 E6 (q) (2.41) Q[[q]][x, y] y 2 − 4x3 − 12 "N −1 # "N −1 # α∗ 1 [[q]] → Q[t]/ → Q t, t(t−1) ti [[q]] = Q[[q]][t]/ ti . i=0

i=0

Proposition 2.36. α : Spec Q[[q]][t]/(t − 1) = μN,Q[[q]] → E is a closed immersion, and it deﬁnes a Γ0 (N )-structure on the generalized elliptic curve E. N

Proposition 2.36 may be derived from Proposition 2.69(2) using analytic expressions, but we omit the proof here. In the following, we denote by e the morphism Spec Q[[q]] → X0 (N ) determined by the pair (E, α : μN → E) ∈ M0 (Q[[q]]) of the generalized elliptic curve E over Q[[q]] and its Γ0 (N )-structure α : μn → E. In case we want to specify N , we denote it by eN . If M divides N , the composition of EN : Spec Q[[q]] → X0 (N ) with the natural morphism X0 (N ) → X0 (M ) coincides with eM : Spec Q[[q]] → X0 (M ). At q = 0, the generalized elliptic curve E is the N´eron 1-gon, and thus the image of the closed point q = 0 is the ∞-cusp in X0 (N ).

2.7. THE q-EXPANSIONS

61

Example 2.37. If N = 1, the morphism e1 : Spec Q[[q]] → X0 (1) = P1Q is given by (2.42)

j(q) =

E4 (q)3 1 = + 744 + 196884q 2 + 21493760q 2 + · · · . Δ(q) q

Using the morphism e, we deﬁne a linear mapping e∗ : S(N ) → Q[[q]]. As its preparation, we deﬁne a mapping ΩQ[[q]]/Q → Q[[q]] ﬁrst. For each positive integer n, we deﬁne a Q[[q]]/q n Q[[q]]-linear mapping Ω(Q[[q]]/qn Q[[q]])/Q → Q[[q]]/q n Q[[q]] by dq → q. As the limit of the composition ΩQ[[q]]/Q → Ω(Q[[q]]/qn Q[[q]])/Q → Q[[q]]/q n Q[[q]], we deﬁne the mapping ΩQ[[q]]/Q → limn Q[[q]]/q n Q[[q]] = Q[[q]]. ←− Definition 2.38. Deﬁne the q-expansion mapping e∗ : S(N ) → Q[[q]] as the composition of the pullback e∗ : S(N ) = Γ(X0 , Ω) → ΩQ[[q]]/Q of the morphism e : Spec Q[[q]] → X0 (N ) and the mapping ΩQ[[q]]/Q → Q[[q]] above. For a modular form f ∈ S(N ), e∗ f = ∞ n n=1 an (f )q ∈ Q[[q]] is called the q-expansion of f . By the deﬁnition of the mapping ΩQ[[q]]/Q → Q[[q]], the constant term of the q-expansion of a modular form is always 0. Example 2.39. Let N = 11. As in Example 2.19(3), identify X0 (11) with the elliptic curve y 2 = 4x3 − 4x2 − 40x − 79. The qexpansion of f11 = dx/y ∈ S(11) is ±1 times (2.43) q

∞

(1 − q n )2 (1 − q 11n )2

n=1 2

= q − 2q − q 3 + 2q 4 + q 5 + 2q 6 − 2q 7 − 2q 9 − 2q 10 + q 11 + · · · . In the following, replacing y by −y if necessary, we assume f11 is given by the formula (2.43). Later in §2.12, we will give an analytic interpretation of modular forms, and it will become clear why Deﬁnition 2.38 is a natural deﬁnition. If M divides N , the composition of eN : Spec Q[[q]] → X0 (N ) with the natural morphism X0 (N ) → X0 (M ) is eM : Spec Q[[q]] → X0 (M ), the composition of the natural mapping with the q-expansion e∗

N S(M ) → S(N ) → Q[[q]] is the q-expansion e∗M : S(M ) → Q[[q]]. The following proposition states that a modular form is determined by its q-expansion, which is often referred to as the q-expansion principle.

Proposition 2.40. The q-expansion mapping e∗ : S(N ) → Q[[q]] is injective.

62

2. MODULAR FORMS

Using this injection e∗ : S(N ) → Q[[q]], we often identify S(N ) with the subspace ofnQ[[q]], and a modular form f with its q-expansion e∗ f = ∞ n=1 an (f )q ∈ Q[[q]]. We will prove Proposition 2.40 in §2.12 using the analytic expression. With the q-expansion, the Hecke operators may be expressed in the following concrete way. Proposition ∞ 2.41.m Let f ∈ S(N ), and suppose its q-expansion is given by ∈ Q[[q]]. For a positive integer n, the qm=1 am q expansion of Tn f is given by (2.44)

∞

m=1

a|(m,n),(a,N )=1

amn/a2 aq m .

In particular, we have a1 (Tn f ) = an (f ). Proposition 2.41 will be proved in §2.13 using the analytic expression. 2.8. Primary forms, primitive forms As we indicated in Chapter 0, modular forms that are particularly important are the eigenvectors of all the Hecke operators. Definition 2.42. Let N be a positive integer, and let K be a ﬁeld of characteristic 0. (1) Let S(N )K = S(N ) ⊗Q K. We call an element of S(N )K a modular form of level N with K coeﬃcients. (2) The q-expansion mapping e∗ : S(N ) → Q[[q]] induces a mapping e∗K : S(N )K → K[[q]], which is also called the q-expansion mapping. The image of f ∈ S(N )K by this mapping is also called the q-expansion of f . (3) A modular form f ∈ S(N )K is called a primary form if the ∞ m q-expansion of f is m=1 am (f )q , then for any n ∈ N, we have (2.45)

Tn f = an (f )f,

and f = 0. The term primary form is used only in this book; it is usually called a normalized simultaneous eigen-cuspform. If K = C, S(N )C has already appeared in §0.3.

2.8. PRIMARY FORMS, PRIMITIVE FORMS

63

Proposition 2.43. Let K be a ﬁeld of characteristic 0, and let form of N be a positive integer. Suppose f ∈ S(N )K is a modular a (f )q m . level N with K coeﬃcients, and its q-expansion is ∞ m m=1 Then, the following conditions are equivalent. (i) f is a primary form. (ii) a1 = 1, and f is an eigenvector of Tn for any n ≥ 1. (iii) If we deﬁne a Q-linear mapping ϕf : T (N ) → K by T → a1 (T f ), then ϕf is a ring homomorphism. Proof. (1) ⇒ (2). If f is a primary form, then for any n ≥ 1, f is an eigenvector of Tn . Since T1 = id, we have f = T1 f = a1 (f )f . Since f = 0, we have a1 (f ) = 1. (2) ⇒ (3). Suppose f satisﬁes the assumptions in (2). Since T (N ) is generated by Tn , n ≥ 1, f is an eigenvector of any T ∈ T (N ). If we let T f = λT f , we have ϕf (T ) = a1 (T f ) = a1 (λT f ) = λT a1 (f ) = λT . Thus, for any T, T ∈ T (N ), we have ϕf (T T ) = λT T = λT λT = ϕf (T )ϕf (T ), and ϕf (1) = a1 (f ) = 1. (3) ⇒ (1). Suppose f satisﬁes the assumption in (3). Then, since a1 (f ) = ϕf (1) = 1, we see f = 0. We show Tn f = an (f )f for any n ∈ N. By Proposition 2.40, the q-expansion mapping is injective, it suﬃces to show am (Tn f ) = an (f )am (f ) for any m. By Proposition 2.41, we have am (Tn f ) = a1 (Tm Tn f ) = ϕf (Tm Tn ), an (f )am (f ) = a1 (Tn f )a1 (Tm f ) = ϕf (Tn )ϕf (Tm ). By assumptions, we have ϕf (Tm Tn ) = ϕf (Tn )ϕf (Tm ), and thus we have am (Tn f ) = an (f )am (f ). Corollary 2.44. There are only a ﬁnite number of primary forms of level N with K coeﬃcients. Proof. Since the Hecke algebra T (N ) is a ﬁnite dimensional Qvector space, there exist only a ﬁnite number of ring homomorphisms ϕ : T (N ) → K. Thus, the assertion follows immediately from Proposition 2.43 (i) ⇔ (iii). Later, in Corollary 2.65(3), we show the number of primary forms is at most g0 (N ) = dimQ S(N ). If f is a primary form, then an (f ) satisﬁes the following relation. Proposition 2.45. Let K be a ﬁeld of characteristic 0, and let N be a positive integer. Suppose f ∈ S(N )K is a modular form of n level N with K coeﬃcients, and its q-expansion is ∞ n=1 an (f )q .

64

2. MODULAR FORMS

(1) If n and m are relatively prime, then anm (f ) = an (f )am (f ). (2) Let p be a prime number. If p N , then for any integer e ≥ 0, (2.46)

ape+2 (f ) = ap (f )ape+1 (f ) − pape (f ).

If p | N , then for any integer e ≥ 0, ape (f ) = ap (f )e .

Proof. Clear form Proposition 2.34.

We see from Proposition 2.45 that for a primary form f , the coefﬁcients an (f ) are determined by the coeﬃcients of prime degree ap (f ). Thus, the coeﬃcients ap (f )are of particular importance. ∞ The complex function n=1 an (f )n−s is called the L-function of a modular form f and is denoted by L(f, s). If f is a primary form of level N , then by Proposition 2.45, L(f, s) can be expressed as the inﬁnite product −1 −1 1 − ap (f )p−s + p1−2s 1 − ap (f )p−s (2.47) L(f, s) = × . pN

p|N

Theorem 2.47 below shows that the L-function of a primary form f converges absolutely in Re s > 32 . The following proposition and theorem will be proved in Corollary 9.3 and Theorem 9.1 in Chapter 9. Proposition 2.46. Let K be a ﬁeld of characteristic 0, and let form of N be a positive integer. Suppose f ∈ S(N )K is a modular ∞ level N with K coeﬃcients, and its q-expansion is m=1 am (f )q m . Then, for any integer n ≥ 1, an (f ) is an algebraic integer. Theorem 2.47. Let K be a ﬁeld of characteristic 0, and let N be a positive integer. Suppose f ∈ S(N )K is amodular form of level N ∞ with K coeﬃcients, and its q-expansion is m=1 am (f )q m . If p is a prime number with p N , then any conjugate of the algebraic integer ap (f ) is a real number, and its absolute value satisﬁes the inequality √ (2.48) |ap (f )| ≤ 2 p. The primitive forms, which we deﬁne next, is more fundamental than the primary forms. Definition 2.48. Let K be a ﬁeld of characteristic 0. A modular form f ∈ S(N )K of level N with K coeﬃcients is a primitive form if f is a primary form and it satisﬁes the following condition. If g is a modular form of level N such that a1 (g) = 1 and Tp (g) = ap (f )g for almost all prime numbers p, then g = f .

2.9. ELLIPTIC CURVES AND MODULAR FORMS

65

The term primitive form, too, is used only in this book, and it is usually called a normalized simultaneous eigen-newform. The following theorem, which is called the strong multiplicity-one theorem, is very important, but its proof cannot be given in this book. Theorem 2.49. Let K be a ﬁeld of characteristic 0, and let N be a positive integer. (1) Let f be a primary form of level N with K coeﬃcients. Then there exists a unique pair (g, M ), where M is an integer dividing N , and g is a primitive form of level M with K coeﬃcients such that ap (f ) = ap (g) for almost all prime numbers p. Furthermore, the equality an (f ) = an (g) holds in fact for any positive integer n relatively prime to N . (2) Let f be a primitive form of level N with K coeﬃcients. If a modular form g ∈ S(N )K satisﬁes Tp g = ap (f )g for almost all prime numbers p, then g is a constant multiple of f . Example 2.50. Let N = 11. The modular form f11 ∈ S(11) in Examples 2.19(3) and 2.39 is the unique primary form of level 11, and n it is a primitive form. If the q-expansion of f11 is ∞ n=1 an (f11 )q , then the action of the Hecke operator Tn on S(11) is the multiplication by an (f11 ). For some small primes p, the value of ap (f11 ) is as follows. 2 3 p ap (f11 ) −2 −1

5 7 1 −2

11 13 17 19 23 29 31 37 · · · 1 4 −2 0 −1 0 7 3 · · ·

2.9. Elliptic curves and modular forms We are now ready to deﬁne the notion that an elliptic curve is modular. Definition 2.51. An elliptic curve E over Q is modular if there exists a primary form f with Q coeﬃcients that satisﬁes (2.49)

ap (E) = ap (f )

for almost all prime numbers p. Let N be a positive integer. An elliptic curve E is said to be modular of level N if we can take the primary form f in the above deﬁnition in such a way that its level divides N and the relation (2.49) holds for all prime numbers not dividing N . If we can take as f a primitive form of level N , E is said to be modular exactly of level N .

66

2. MODULAR FORMS

By Theorem 2.49, we can replace the primary form in the above deﬁnition by a primitive form. The condition of Deﬁnition 2.51 implies that the inﬁnite product expansion (1.17) of L-function L(E, s) and the inﬁnite product expansion (2.47) of L(f, s) coincide except for a ﬁnite number of factors. As a matter of fact, the condition in Deﬁnition 2.51 turns out to be equivalent to the equality L(E, s) = L(f, s). Example 2.52. Let E = X0 (11). The unique primitive form f11 ∈ S(11) of level 11 satisﬁes ap (E) = ap (11) for all primes p. This will be proved in Theorem 9.13 (see also Example 9.15). Thus, X0 (11) is a modular elliptic curve of exact level 11. Compare Examples 1.14 and 2.50. Theorem 2.53. Any elliptic curve E over Q is modular. In other words, there exists a primitive form f that satisﬁes ap (E) = ap (f ) for almost all prime numbers p. This is Theorem 0.8 in Chapter 0. The proof of Theorem 2.53 is beyond the scope of this book; however, we prove the following theorem that covers only semistable elliptic curves. Theorem 2.54. Let E be a semistable elliptic curve over Q. If N is the conductor of E, then E is modular exactly of level N . In other words, there exists a primitive form f satisfying ap (E) = ap (f ) for all prime numbers p not dividing N . This is a precise version of Theorem 0.13. Theorem 2.54 will be derived from Theorem 3.36 in Chapter 4. 2.10. Primary forms, primitive forms, and Hecke algebras The important step of the proof of Theorem 2.54 is to reformulate the statement in terms of commutative algebra. Here in this section, we reformulate the part related to modular forms. The space of modular forms S(N ) of level N is naturally a T (N )-module. Corollary 2.56(2) below gives an interpretation of primary forms in terms of commutative rings. Let T (N )∨ be the dual of T (N ) as a Q-vector space, namely, T (N )∨ = HomQ (T (N ), Q). Deﬁne a structure of T (N )-module on T (N )∨ as follows. For T ∈ T (N ) and ϕ ∈ T (N )∨ , deﬁne T ϕ ∈ T (N )∨ by (T ϕ)(T ) = ϕ(T T ).

PRIMARY AND PRIMITIVE FORMS, AND HECKE ALGEBRAS

67

Proposition 2.55. For f ∈ S(N ), deﬁne ϕf ∈ T (N )∨ by ϕf (T ) = a1 (T f ). Then, the mapping (2.50)

a : S(N ) → T (N )∨

that sends f ∈ S(N ) to ϕf ∈ T (N )∨ is an isomorphism of T (N )modules. In particular, we have dimQ S(N ) = dimQ T (N ). Later in Corollary 9.8, we will prove that S(N ) is isomorphic to T (N ) as a T (N )-module using the Eichler–Shimura isomorphism. Proof. It is easy to verify that the mapping a is a homomorphism of T (N )-modules. In order to show that a is an isomorphism, it suﬃces to show that the bilinear form (2.51)

S(N ) × T (N ) → Q; (f, T ) → a1 (T f )

is nondegenerate. This can be derived from Proposition 2.40 (qexpansion principle) as follows. We ﬁrst show that a : S(N ) → T (N )∨ is injective. By Proposition 2.40, if a modular form satisfy a1 (Tn f ) = an (f ) = 0 for all n, then f = 0, which implies a is injective. Next we show that the dual a∨ : T (N ) → S(N )∨ is injective. Suppose T ∈ T (N ) satisﬁes a1 (T f ) = 0 for any f , and we show T = 0. If we choose any g ∈ S(N ) and n ≥ 1, we have an (T g) = a1 (Tn T g) = a1 (T Tn g) = 0, which implies T = 0. Thus, we have shown a∨ is injective. For a ﬁeld K of characteristic 0, we call T (N )K = T (N ) ⊗Q K the Hecke algebra of level N with K coeﬃcients. Deﬁne T (N )∨ K = HomQ (T (N ), K). Corollary 2.56. Let K be a ﬁeld of characteristic 0. For f ∈ S(N )K , deﬁne a Q-linear mapping ϕf : T (N ) → K by ϕf (T ) = a1 (T f ). (1) The mapping (2.52)

aK : S(N )K → T (N )∨ K

that maps f ∈ S(N )K to ϕf ∈ T (N )∨ K is an isomorphism of T (N )K -modules. (2) The mapping aK induces a bijection (2.53) ring homomorphisms primary forms of level N → T (N ) → K with K coeﬃcients ∪ ∪ f → ϕf

68

2. MODULAR FORMS

(3) The number of primary forms of level N with K coeﬃcients is at most g0 (N ) = dimQ S(N ). Proof. (1) It suﬃces to take the extension of coeﬃcients to K of the isomorphism a in Proposition 2.55. (2) By (1), it suﬃces to show that for a modular form with K coeﬃcients, f is a primary form if and only if ϕf : T (N ) → K is a ring homomorphism. This is nothing but Proposition 2.43 (1) ⇒ (3). (3) Since dimK T (N )K = dimK S(N )K , it follows form (2). ∞ If a primary form f has the q-expansion f = n=1 an (f )q n , then the corresponding ring homomorphism ϕf : T (N ) → K is given by ϕf (Tn ) = an (f ). As for primitive forms, we can also give a similar interpretation in terms of commutative algebra, as in the case of primary forms. Definition 2.57. The subring of EndQ (S(N )) generated by the Hecke operators Tn with (n, N ) = 1: (2.54)

T (N ) = Q([Tn , (n, N ) = 1]) ⊂ EndQ (S(N ))

is called the reduced Hecke algebra of level N . For a ﬁeld of characteristic 0, the reduced Hecke algebra of level N with K coeﬃcients is deﬁned by T (N )K = T (N ) ⊗Q K. The reduced Hecke algebra T (N ) is a subring of the Hecke algebra T (N ). The following proposition shows that a reduced Hecke algebra is a reduced ring, as its name suggests. The proof will be given in Corollary 9.11. Proposition 2.58. The reduced Hecke algebra T (N ) is a reduced ring. If M is a divisor of N , then by Proposition 2.32(2), the injection S(M ) → S(N ) in (2.8) commutes with the action of the Hecke operators Tn with (n, N ) = 1. Thus, this injection induces a ring homomorphism T (N ) → T (M ); Tn → Tn . As we will show later in Corollary 2.60, this ring homomorphism is surjective. Note that for the full Hecke algebras, there does not exist a natural homomorphism T (N ) → T (M ); Tn → Tn . As an analogue of Corollary 2.56(2), we have the following. Proposition 2.59. Let K be a ﬁeld of characteristic 0, and let N be a positive integer. For a primitive form f with K coeﬃcients of

PRIMARY AND PRIMITIVE FORMS, AND HECKE ALGEBRAS

69

level M dividing N , deﬁne a ring homomorphism ϕf : T (N ) → K as ϕf

the composition T (N ) → T (M ) ⊂ T (M ) −→ K. Then, the mapping (2.55) ring homomorphisms primitive forms of level → T (N ) → K M with K coeﬃcients M |N ∪ ∪ f → ϕf is bijective. Proof. The essential part of this proposition is Theorem 2.49. It suﬃces to show that for a ring homomorphism ϕ : T (N ) → K, there exists a unique pair (M, g), where M is a divisor of N , and g is a primitive form of level M with K coeﬃcients satisfying ϕ = ϕg . Since T (N ) ⊂ T (N ), there exist a ﬁnite Galois extension L of K and a ring homomorphism T (N ) → L that extends ϕ. Let f be the corresponding primary form of level N with L coeﬃcients. By Theorem 2.49, there exists a unique pair (M, g) of a divisor M of N and a primitive form g such that an (f ) = an (g) for all n relatively prime to N . We have ϕ(Tn ) = an (f ) = an (g) = ϕg (Tn ) for all n relatively prime to N . All we have to show now is that g has K coeﬃcients. A conjugate g of g over K is a primitive form of level M with L coeﬃcients satisfying an (f ) = an (g ) for all n relatively prime to N . Thus, by the uniqueness, g must be equal to g. Thus, g has K coeﬃcients. Corollary 2.60. Let M be a multiple of N . Then, for any n with (n, M ) = 1, the ring homomorphism T (M ) → T (N ); Tn → Tn is surjective. Proof. Since T (M ) and T (N ) are reduced, it suﬃces to show that the mapping {ring homomorphisms T (N ) → Q} → {ring homomorphisms T (M ) → Q} induced by the homomorphism T (M ) → T (N ) is injective. But, the mapping corresponding to this by the bijection (2.55) is an inclusion. Corollary 2.61. Let K be a ﬁeld of characteristic 0, and let f be a primitive form of level N with K coeﬃcients. The subﬁeld Q(f ) of K generated over Q by an (f ), (n, N ) = 1, is a ﬁnite extension of Q, and f is a primitive form with Q(f ) coeﬃcients. If M is a multiple of N , then Q(f ) is generated by an (f ), (n, M ) = 1.

70

2. MODULAR FORMS

The ring homomorphism T (N ) ⊗T (N ) K → K deﬁned by f is an isomorphism. Proof. The subﬁeld Q(f ) is the image of the ring homomorphism ϕf : T (N ) → K corresponding to f . Since T (N ) is ﬁnite dimensional over Q, Q(f ) is a ﬁnite extension of Q. The primitive form corresponding to ϕf : T (N ) → Q(f ) has Q(f ) coeﬃcients. If we con sider it as K coeﬃcients, it is the original f . Since T (M ) → T (N ) is surjective, we have Q(f ) = Q an (f ), (n, M ) = 1 . It then follows form Theorem 2.49(2) and Corollary 2.56(1) that T (N ) ⊗T (N ) K → K is an isomorphism. ∞ Suppose a primitive form f has the q-expansion f = n=1 an (f )q n . Then, the corresponding ring homomorphism ϕf : T (N ) → K is determined by ϕf (Tn ) = an (f ) with (n, N ) = 1. Definition 2.62. Let Φ(N )K = Spec T (N )K . An element of Φ(N )K is called a primitive form over K. If f ∈ Φ(N )K is a primitive form over K, then its residue ﬁeld is a ﬁnite extension of K. It is somewhat confusing between primitive forms over K and primitive forms with K coeﬃcients, but among the primitive forms f over K, those satisfying Kf = K are primitive forms with K coeﬃcients. Since T (N )K is reduced, we have Kf . (2.56) T (N )K = f ∈Φ(N )K

If f ∈ Φ(N )K , the primitive form with Kf coeﬃcients, corresponding to the natural surjection ∞ T (N )K n→ Kf , is also denoted by f . If the q-expansion of f is n=1 an (f )q , then Kf is generated over K by an (f ), (n, N ) = 1. 2.11. The analytic expression The contents of the remaining part of this chapter will not be used until Chapter 8, and so the reader may skip it for now. Classically, modular forms are deﬁned as holomorphic functions on the upper half-plane that satisfy certain conditions. We now explain the relation between this description and our earlier deﬁnition. Analytically, an elliptic curve over C can be regarded as the quotient of a one dimensional C-vector space by a lattice. A lattice in a one dimensional C-vector space L, is a Z-submodule T generated by

2.11. THE ANALYTIC EXPRESSION

71

a basis of L regarded as an R-vector space. If T is a lattice in L, the quotient L/T has a structure of a compact Riemann surface. Theorem 2.63. Let E be an elliptic curve over C. Let Lie(E) be the dual vector space of the one dimensional C-vector space Γ(E, Ω), and let T (E) be the free Z-module H1 (E(C), Z) of rank 2. (1) The linear mapping $ T (E) → Lie(E) that maps γ ∈ T (E) to the linear form ω → γ ω on Γ(E, Ω) is injective, and its image is a lattice. (2) Use the injective mapping in (1) to identify T (E) with a lattice in Lie(E). For a point P ∈ E(C), the integral along a path from the 0-section O ∈ E(C) to P deﬁnes a linear form on Γ(E, Ω). This linear form is determined uniquely up to the image of T (E), and the mapping % P

ω mod T (E) (2.57) E(C) → Lie(E)/T (E); P → ω → O

is an isomorphism of compact Riemann surfaces. An elliptic curve as a compact Riemann surface may be identiﬁed with Lie(E)/T (E) naturally. We call this identiﬁcation the analytic expression of an elliptic curve. Theorem 2.64. Let L be a one dimensional C-vector space, let T be its lattice, and let z be a coordinate of L. 1 3 2 (1) Let Gk = ω∈T,=0 z(ω)k . Then, (60G4 ) − 27(140G6 ) = 0, and the equation y 2 = 4x3 − 60G4 · x − 140G6

(2.58)

deﬁnes an elliptic curve E over C. (2) Deﬁne a meromorphic function ℘ on L by 1 1 1 − (2.59) ℘= 2 + , z (z + z(ω))2 z(ω)2 ω∈L,=0

Then, (℘, ℘ ) : L T → A2 (C) induces an and let ℘ = isomorphism L/T → E(C) of compact Riemann surfaces. d℘ dz .

We omit the proof of these two theorems. Corollary 2.65. Let M be the set of isomorphism classes of pairs (E, α) of an elliptic curve E and an isomorphism of Z-module α : Z2 → T (E). For an imaginary number τ ∈ C R, let Eτ

72

2. MODULAR FORMS

be the elliptic curve corresponding to the lattice Z + Zτ in C. Let ατ : Z2 → T (Eτ ) = Z+Zτ be the isomorphism that maps the standard basis e1 , e2 to 1, τ . Then, the mapping u : C R → M that maps τ ∈ C R to the pair (Eτ , ατ ) is bijective. Proof. We deﬁne the inverse of u. Let E be an elliptic curve, and α : Z2 → T (E) an isomorphism of Z-modules. As in Theorem 2.63(1), we regard T (E) = H1 (E(C), Z) as a lattice in Lie(E). For (E, α), let τ be the unique complex number such that α(e2 ) = τ α(e1 ). Deﬁne a mapping v : M → C R by sending this τ to the isomorphism class (E, α). A C-linear isomorphism C → Lie(E); 1 → α(e1 ) maps the lattice Z+Zτ to the lattice T (E) in Lie(E). Thus, we have (E, α) (Eτ , ατ ), and v is the inverse of u. Using the bijection u, we regard the Riemann surface Y0 (N )(C) as the quotient of the upper half-plane H = {τ ∈ C | Im τ > 0} by a subgroup of SL2(Z). Deﬁne the left action of SL2 (Z) on the upper +b half-plane H by ac db · τ = aτ cτ +d . If τ ∈ H, then the class of 1/N is a point of order N in the elliptic curve Eτ , and the subgroup generated by it CN,τ = 1/N is a cyclic subgroup of order N of Eτ . Thus, the isomorphism class of the pair (Eτ , CN,τ ) deﬁnes a C-valued point on the modular curve Y0 (N ). Corollary 2.66. Deﬁne a (2.60) Γ0 (N ) = c

a subgroup Γ0 (N ) of SL2 (Z) by

b ∈ SL2 (Z)

c ≡ 0 mod N . d

Then, the bijection u induces an isomorphism of Riemann surfaces (2.61)

Γ0 (N )\H → Y0 (N )(C);

τ → (Eτ , CN,τ ).

As a Riemann surface, the modular curve Y0 (N )C may be identiﬁed naturally with Γ0 (N )\H. We call it the analytic expression of the modular curve Y0 (N ). Proof. The group GL2 (Z) acts on M naturally on the right by (isomorphism class of (E, α)) · γ = (isomorphism class of (E, α ◦ γ)) for γ ∈ GL2 (Z). If we deﬁne an anti-isomorphism ι of GL2 (Z) by ι ac db = ( dc ab ), then u satisﬁes u(γτ ) = u(τ )ι(γ). If we deﬁne a subgroup GΓ0 (N ) of GL2 (Z/N Z) by

a b

∈ GL2 (Z) c ≡ 0 mod N , (2.62) GΓ0 (N ) = c d

2.11. THE ANALYTIC EXPRESSION

73

then by Corollary 2.65, u induces an isomorphism of Riemann surfaces Γ0 (N )\H → M/GΓ0 (N ). The group GL2 (Z) acts naturally on (Z/N Z)2 . Deﬁne a right action of GL2 (Z) on the set SN = {cyclic subgroups of order N in (Z/N Z)2 } as the action induced by the left action of the inverse element on (Z/N Z)2 . Denote by M ×GL2 (Z) SN the quotient (M × SN )/GL2 (Z). GL2 (Z) acts on SN transitively, and GΓ0 (N ) is the stabilizer of the cyclic group generated by e1 = (1, 0) with respect to this action. Thus, we have M/GΓ0 (N ) = M ×GL2 (Z) SN . If E is an elliptic curve over C, then the isomorphism Z2 → T (E) induces an isomorphism (Z/N Z)2 → E[N ](C). GL2 (Z) acts on the set {isomorphisms Z2 → T (E)} transitively and freely. Thus, the mapping M ×GL2 (Z) SN → M0 (N )(C) = Y0 (N )(C) is bijective, and gives an isomorphism of Riemann surfaces. Proof of Theorem 2.10(3). It suﬃces to show that Y0 (N )⊗Q C is connected. Since the Riemann surface Y0 (N )(C) is connected by the analytic expression, the algebraic curve Y0 (N ) ⊗Q C over C is also connected. By the analytic expression, we see that Deﬁnition 2.12(2) of modular forms in this book coincides with the usual deﬁnition. For a holomorphic function f on the upper half-plane H and γ = ac db ∈ SL2 (Z), deﬁne a holomorphic function γ ∗ f by γ ∗ f (τ ) = 1 (cτ +d)2 f

aτ +b cτ +d

.

Corollary 2.67. The pullback of S(N )C by u : H → X0 (N )(C) (2.63)

S(N )C = Γ(X0 (N )(C), Ω) → ∪ ω →

Γ(H, Ω) = Γ(H, O) dτ ∪ f (τ ) dτ

is injective, and its image is the space of holomorphic functions satisfying the following two conditions. (i) γ ∗ f = f for γ ∈ Γ0 (N ). (ii) limτ →√−1∞ γ ∗ f (τ ) = 0 for γ ∈ SL2 (Z). Writing a modular form as a holomorphic function on the upper half-plane H in this way is called the analytic expression of a modular form. We omit the proof of Corollary 2.67, but we now explain the meaning of the conditions. Since γ ∗ (f dτ ) = (γ ∗ f ) dτ , condition (i) says that the diﬀerential form f dτ on H is invariant under the action

74

2. MODULAR FORMS

of Γ0 (N ). If we let γ = ( 10 11√ ) in (i), we have f (τ + 1) = f (τ ). Thus, if we write q = exp(2π −1τ ), f is a function of q. Since ∗ √ dq = 2π −1q dτ , condition (ii) says that γ (fdqdτ ) is holomorphic at q = 0. 2.12. The q-expansion and analytic expression We now explain the analytic meaning of q-expansion using the analytic expression of the modular curves. Let Δ be the unit disk {q ∈ C | |q| < 1}, and let Δ∗ = {q ∈ C | 0 < |q| < 1}. Identify Z with the subgroup { ( 10 1b )| b ∈ Z} of SL2 (Z), and deﬁne the action of the group Z on the upper half-plane H by b · τ = τ + b for b ∈ Z. Identify ∗ ∗ the quotient Z\H √ with Δ by using the isomorphism Z\H → Δ ; τ → q = exp(2π −1τ ). Proposition 2.68. Let N be a positive integer. (1) The mapping Δ∗ → X0 (N )(C) deﬁned by the composition (2.64)

∼

∼

Δ∗ ← Z\H → Γ0 (N )\H → Y0 (N )(C) ⊂ X0 (N )(C)

is extended to a holomorphic mapping e : Δ → X0 (N )(C) of Riemann surfaces by deﬁning the image of 0 to be the ∞-cusp. (2) The restriction of the holomorphic mapping e : Δ → X0 (N )(C) to the set {q ∈ Δ | |q| < e−2π } is an open immersion. (3) The q-expansion mapping e∗ : S(N )C → C[[q]] is the composition of the pullback by the holomorphic mapping e and the inω jective mapping Γ(Δ, Ω) → C[[q]]; ω → q dq (2.65)

e∗

S(N )C = Γ(X0 (N )(C), Ω) → Γ(Δ, Ω) → C[[q]].

Proof of Proposition 2.68 ⇒ Proposition 2.40. By Propoe∗ sition 2.68(3), it suﬃces to show that the mapping Γ(X0 (N )(C), Ω) → Γ(Δ, Ω) is injective. Since X0 (N )(C) is connected, this follows from Proposition 2.68(2). Proposition 2.68 follows from the modular interpretation of the mapping e. While in §2.7, we regarded Ek (q), x(q, t) and y(q, t) as formal power series; here we regard them as holomorphic functions on q ∈ Δ, (q, t) ∈ Δ × C× deﬁned by the same formula. Proposition 2.69. Let q ∈ Δ. (1) If q = 0, then the equation 1 1 (2.66) y 2 = 4x3 − E4 (q)x + E6 (q) 12 216

2.12. THE q-EXPANSION AND ANALYTIC EXPRESSION

75

deﬁnes an elliptic curve E(q) over C. If q = 0, this is a N´eron 1-gon. In the following, suppose q ∈ Δ∗ . (2) Let q Z be the cyclic subgroup of C× generated by q. If t ∈ C× q Z , then we have (x(q, t), y(q, t)) ∈ E(q)(C)

(2.67)

{O}.

The mapping C× q Z → E(q)(C); t → (x(q, t), y(q, t)) can be extended to a holomorphic mapping C× → E(q)(C) by sending t ∈ q Z to O ∈ E(q)(C). This is a surjective homomorphism of groups, and its kernel equals q Z . For a positive integer N , (2.68) CN (q) = {(x(q, t), y(q, t)) ∈ E(q)(C) | tN = 1, t = 1} {O} is a cyclic subgroup of E(q) of order N . (3) Let N be a positive integer. The image e(q) ∈ X0 (N )(C) of q ∈ Δ∗ is the point deﬁned by the isomorphism class on (E(q), CN (q)). We omit the details, but Proposition 2.68 is derived from Proposition 2.69 as follows. (1) By Proposition 2.69(3), it suﬃces to show that, as q tends to 0, the isomorphism class of (E(q), CN (q)) converges to the pair (P1 , μN ) of a N´eron 1-gon P1 and its Γ0 (N )-structure μN . This follows from the fact that E(q) is deﬁned by the same equation, including the case q = 0 in Proposition 2.69(1). (2) It suﬃces to show that the holomorphic mapping {q ∈ Δ | 0 < |q| < e−2π } → Γ0 (N )\H is injective. Suppose τ ∈ H and γ ∈ SL2 (Z). It suﬃces to show that if Im τ, Im γτ > 1, there exists b ∈ Z such that τ Im τ γ = ± ( 10 1b ). Since Im γτ = |cτIm+d| 2 , we have Im γτ ≤ c2 | Im τ |2 < 1 if c = 0. This implies c = 0, and we obtain a = d = ±1. (3) The assertion follows from the following three facts, but we omit the detail. (1) The elliptic curve in Propositions 2.35 and 2.36 appeared in the deﬁnition of the q-expansion and its cyclic subgroup α : μN → E is deﬁned by the same equation as the elliptic curve E(q) in Proposition 2.69(1). (2) The mapping e : Spec Q[[q]] → X0 (N ) is deﬁned by the isomorphism classes of (E, CN ).

76

2. MODULAR FORMS

(3) By Proposition 2.69(3), the mapping e : Δ → X0 (N )(C) is deﬁned by sending q to the isomorphism class of (E(q), CN (q)). Proof of equality (2.39). (2.67) means that the equality (2.39) holds as an equality of complex functions in (q, t). Therefore, (2.39) holds as an equality of formal power series. We now prove Proposition 2.69. Let H = {τ ∈ C | Im τ > 0} be the upper half-plane, and τ ∈ H. Let T = Z + Zτ be a lattice in L = C. Then, Theorem 2.64 says as follows. For an even number k ≥ 4, deﬁne a holomorphic function on the upper half-plane H by 1 , (2.69) Gk (τ ) = (m + nτ )k 2 (m,n)∈Z ,=(0,0)

and deﬁne a holomorphic function ℘ on H × C by (2.70) 1 1 1 ℘(τ, z) = 2 + − . z (z + m + nτ )2 (m + nτ )2 2 (m,n)∈Z ,=(0,0)

Then, the elliptic curve Eτ corresponding to the lattice Z + Zτ in C is deﬁned by the equation (2.71)

y 2 = 4x3 − 60G4 (τ )x − 140G6 (τ ),

and the isomorphism C/(Z + Zτ ) → Eτ (C) is given by z → ℘(τ, z), ∂ ∂z ℘(τ, z) . We use the following formulas. √ Lemma 2.70. √ Suppose τ ∈ H and z ∈ C. Let q = exp(2π −1τ ) and t = exp(2π −1z). Then, for even number k ≥ 4, the following formulas hold: √ (2π −1)k ζ(1 − k)Ek (q), (2.72) Gk (τ ) = (k − 1)! √ ℘(τ, z) = (2π −1)2 x(q, t), (2.73) √ ∂ ℘(τ, z) = (2π −1)3 y(q, t). (2.74) ∂z

Proof of Proposition 2.69. (1) In view √of Lemma 2.70, if we √ make the change of coordinates x = (2π −1)2 x and y = (2π −1)3 y, equation (2.71) of the elliptic curve Eτ becomes equation (2.66) of E(q).

2.13. THE q-EXPANSION AND HECKE OPERATORS

77

(2) The composition C× /q Z → E(q)(C) of the following isomorphisms of topological groups √ C× /q Z ← C/(Z + Zτ ); exp(2π −1z) ← z, ∂ C/(Z + Zτ ) → Eτ (C); z → ℘(τ, z), ∂z ℘(τ, z) ,

Eτ (C) → E(q)(C); (x, y) → (2π√x−1)2 , (2π√y−1)3 , is an isomorphism of topological groups. By Lemma 2.70, this isomorphism is given by t → (x(q, t), y(q, t)). (3) e(q) is the isomorphism class of (Eτ , CN,τ ), which is isomor phic to (E(q), CN (q)) by (2). Proof of Lemma 2.70. The formula (2.72) is proved in §9.2(b) in Chapter 9 of Number Theory 3 in a slightly diﬀerent notation, using the formula (9.14) there. We can prove (2.73) and (2.74) similarly, using Proposition 3.3(2) in Chapter 3 of Number Theory 1 . 2.13. The q-expansion and Hecke operators In this section, we prove Proposition 2.41 using an analytic expression. First, we give an analytic description of the trace mapping. Let f : X → Y be a ﬁnite morphism of compact Riemann surfaces. Take an open subset V intersecting all the connected components of Y small enough such that f |U : U = f −1 (V ) → V is ´etale and Γ(V, O)-module Γ(V, Ω) has a basis ω. For a holomorphic function g ∈ Γ(U, O), deﬁne a holomorphic function f∗ g ∈ Γ(V, O) by f∗ g(y) = x∈f −1 (y) g(x). The mapping fU/V,∗ : Γ(U, Ω) → Γ(V, Ω) is deﬁned by g · ω → f∗ g · ω. This mapping sends the subspace Γ(X, Ω) ⊂ Γ(U, Ω) into the subspace Γ(Y, Ω) ⊂ Γ(V, Ω). The mapping thus induced coincides with the mapping f∗ : Γ(X, Ω) → Γ(Y, Ω) deﬁned in §2.6. Let e : Δ∗ → X0 (N )(C) be the mapping in Proposition 2.68(1). By Proposition 2.68(2), the restriction of e to V = {q ∈ Δ | 0 < |q| < e−2π } is an open immersion. Applying the analytic description above to the open immersion e : V → X0 (N )(C), we calculate the Hecke operator Tn = s∗ ◦ t∗ . To do so, we compute the inverse image s−1 (e(q)) of the image of q ∈ V in X0 (N )(C). For q ∈ Δ∗ , let E(q) C× /q Z be the elliptic curve in Proposition 2.69(1). The image of q ∈ Δ∗ in e(q) ∈ X0 (N )(C) is the isomorphism class of (E(q), CN (q)).

78

2. MODULAR FORMS

For positive integers a, b, the homomorphism C× /q Z → C× /q aZ induced by the ath power mapping C× → C× is denoted by a, and the homomorphism E(q b ) → E(q) corresponding to the homomorphism C× /q bZ → C× /q Z induced by the identity mapping C× → C× is denoted by b. Lemma 2.71. Let q ∈ Δ∗ , and let E(q) be the elliptic curve corresponding to C× /q Z . Let n be a positive integer. (1) If f : E(q) → E is a homomorphism of degree n, then there exists a pair (a, q ), where a is an integer with a = n/b for some integer b and q is a root of the equation q a = q b , that satisﬁes the following condition. There exists an isomorphism E(q ) → E such that f coincides with the composition (2.75)

b∨

E(q) → E(q a ) = E(q b ) → E(q ) → E . a

(2) Let a and b be positive integers, and let q be a root of the equation q a = q b . For a positive integer N , let CN (q ) = μN be a a cyclic subgroup of E(q) of order N . The composition E(q) → b∨

E(q a ) = E(q b ) → E(q ) maps CN (q ) to its image isomorphically if and only if a and N are relatively prime. Furthermore, if this is the case, the image of CN (q) is CN (q ). Corollary 2.72. Suppose q ∈ V = {q ∈ Δ∗ | |q| < e−2π }. Then, we have (2.76)

s−1 (e(q)) =

&

{q ∈ Δ∗ | q a = q b }.

ab=n,(a,N )=1

If we regard E(q) = C× /q Z and CN (q) = μN ⊂ E(q), then the proof of Lemma 2.71 is straightforward, so we omit it here. Proof of Proposition 2.41. Let N and n be positive integers. For a divisor a of n relatively prime to N , deﬁne Ua = {(q, q ) ∈ V × Δ∗ | q a = q b }, and deﬁne s : Ua → V and t : Ua → Δ∗ by s(q, q ) = q and t(q, q ) = q , respectively. Deﬁne e˜ : Ua → X0 (N, n)(C) by (q, q ) → (isomorphism class of (E(q), CN (q), E(q ), CN (q ), b∨ ◦ a)).

2.13. THE q-EXPANSION AND HECKE OPERATORS

Then the diagram

t

Ua −−−−→

Δ∗

ab=n,(a,N )=1

⏐ ⏐

(2.77)

s

←−−−−

V

79

⏐ ⏐e

⏐ ⏐

e|V

e˜

X0 (N )(C) ←−−−− X0 (N, n)(C) −−−−→ X0 (N )(C) s

t

is commutative. In the left-side of the diagram (2.77), & Ua ab=n,(a,N )=1

is isomorphic to the inverse image of e(V ) by the mapping s

X0 (N, n)(C) → X0 (N )(C), by Corollary 2.72. Thus, by the analytic description of the trace mapping given at the beginning of this section, we have the following commutative diagram. s∗ t∗ −− Γ(Ua , Ω) ←−−−− Γ(Δ∗ , Ω) Γ(V, Ω) ←−− (2.78)

e|∗ V

ab=n,(a,N )=1

' ⏐ ⏐

' ⏐∗ ⏐e

' ⏐

e˜∗ ⏐

S(N ) ←−−−− s∗

Γ(X0 (N, n), Ω)

←−− −− ∗ t

S(N )

∗ We calculate S(N ), Let f ∈ ∞ Tn = ns∗ ◦ t using the diagram∗ (2.78). n a q d log q, and f = n=1 an q its q-expansion. Since e f = ∞ n=1 n we have ∞ (2.79) e|∗V ◦Tn f = s∗ ◦t∗ ◦e∗ f = am q m d log q . ab=n,(a,N )=1 q b =q a m=1 b

a

Since q = q , we have b d log q = ad log q. If we make the substitution bq am/b if b | m, m q = (2.80) 0 if b m, b a q =q

then, the right-hand side of (2.79) becomes

∞

am aq am/b d log q.

ab=n,(a,N )=1 m=1,b|m

It now suﬃces to replace am/b by m.

10.1090/mmono/243/04

CHAPTER 3

Galois representations The proof of Theorem 0.13 is given by comparing the Tate modules of elliptic curves and Galois representations associated with modular forms. Elliptic curves over Q are algebro-geometric objects, and modular forms over Q are representation theoretical objects, while representations of the absolute Galois group GQ are objects in linear algebra over Q. As linear algebra is easier than algebraic geometry or representation theory, Galois representations are easier to handle compared to elliptic curves and modular forms. The biggest advantage is that we can take the reduction of -adic representations and consider mod representations. In the ﬁrst part of this chapter, up to §3.4, we introduce the notion of Galois representations. As preparation, we give a summary on Frobenius substitutions in §3.1. From an elliptic curve E or a modular form f over Q, we obtain ap (E) or ap (f ). From a Galois representation, we obtain the trace of a Frobenius substitution for each prime p. In §3.4 we state the modularity of an elliptic curve in terms of a Galois representation (Proposition 3.23). The modularity of a mod representation is also deﬁned there using these numbers. The meaning of conditions (i) to (iii) of Theorem 0.15 will be explained in §§3.3–3.5. In the second part, starting from §3.5, we study the ramiﬁcation of a Galois representation at each prime p. This corresponds to the bad reduction of an elliptic curve or the level of a modular form. We formulate a theorem about the modularity of a Galois representation (Theorem 3.36) in §3.5. This is the main theorem to which we give a proof in this book. In Chapter 4, we deduce Theorem 0.13 from Theorem 3.36. The subsequent chapters will be devoted to proving Theorem 3.36. In the last two sections, §§3.7, 3.8, we study the ramiﬁcations of Galois representation on the Tate module of an elliptic curve and that associated with a modular form. The conductor of an elliptic curve or 81

82

3. GALOIS REPRESENTATIONS

the level of a modular curve reﬂects the ramiﬁcation of the associated Galois representation. This relation between the level and the ramiﬁcation will play a very important role in the proof of Theorem 3.36 which starts from Chapter 5.

3.1. Frobenius substitutions Though the deﬁnition of the Frobenius substitution can be found in §6.3(a) in Chapter 6 of Number Theory 2 , we review it here since it is very important. Let p be a prime number, and let L be a ﬁnite Galois extension of Q that is unramiﬁed at p. Suppose q is a prime ideal of OL lying above p, and let Dq = {σ ∈ Gal(L/Q) | σq = q} be the decomposition group of q. If Fq is the residue ﬁeld of q, the natural mapping Dq → Gal(Fq /Fp ) is an isomorphism. Gal(Fq /Fp ) is a cyclic group generated by the pth power mapping ϕq : Fq → Fq . The inverse image of the pth power mapping ϕq ∈ Gal(Fq /Fp ) in Dq ⊂ Gal(L/Q) is called the Frobenius substitution at q. Since all the prime ideals of OL lying above p are conjugate to q, the conjugacy class of ϕq depends only on p. This class is called the Frobenius conjugacy class at p and is denoted by ϕp . The reason that the Frobenius substitutions play an important role in Galois representation theory lies in the following theorem. This is Theorem 8.7 in Chapter 8 of Number Theory 2 . Theorem 3.1. Let L be a ﬁnite Galois extension of Q. For any conjugacy class c of the Galois group Gal(L/Q), there exist inﬁnitely many prime numbers p such that p is unramiﬁed in L and c = ϕp . An inﬁnite Galois extension L over Q is unramiﬁed at p if L is the compositum of all ﬁnite Galois extensions Lλ (λ ∈ Λ) unramiﬁed at p. In this case we have Gal(L/Q) = limλ Gal(Lλ /Q). The limit ←− of the Frobenius conjugacy class ϕp of Gal(Lλ /Q) is also denoted by ϕp and is called the Frobenius conjugacy class of p. In the following, we denote by GQ the absolute Galois group Gal(Q/Q). Definition 3.2. Let R be a complete noetherian local ring, let n be a positive integer, and let p be a prime number. Suppose ρ : GQ → GLn (R) is a continuous homomorphism. (1) We say ρ : GQ → GLn (R) is unramiﬁed at p if the ﬁeld Lρ corresponding to the kernel of ρ is unramiﬁed at p.

3.1. FROBENIUS SUBSTITUTIONS

83

(2) Suppose ρ is unramiﬁed at p. The characteristic polynomial det(T − ρ(σ)) ∈ R[T ] of the image of an element σ of the Frobenius conjugacy class ϕp ⊂ Gal(Lρ /Q) does not depend on the choice of σ, but depends only on p. This is called the characteristic polynomial of the Frobenius conjugacy class ϕp and is denoted by det(T − ρ(ϕp )). The trace Tr ρ(ϕp ) and the determinant det ρ(ϕp ) are deﬁned similarly. (3) Let be a prime number. Take as R the ring of integer O of a ﬁnite extension of Q . If a continuous homomorphism ρ : GQ → GLn (O) is unramiﬁed except at a ﬁnite number of primes, ρ is called an -adic representation of GQ . If we take a ﬁnite extension F of F as R, a continuous homomorphism ρ¯ : GQ → GLn (F) is called a mod representation of GQ . The deﬁnition goes similarly for local ﬁelds instead of a complete noetherian local ring. Let K be a ﬁnite extension of Q , and let n be a positive integer. A continuous homomorphism ρ : GQ → GLn (K) is unramiﬁed at p if the ﬁeld Lρ corresponding to the kernel of ρ is unramiﬁed at p. If ρ is unramiﬁed at p, then the characteristic polynomial det(T − ρ(ϕp )), the trace Tr ρ(ϕp ), and the determinant det ρ(ϕp ) are also deﬁned similarly. As an important example of Galois representations, we study the Galois action on the roots of unity. Let N be a positive integer, × and let μN (Q) = {ζ ∈ Q | ζ N = 1} be the multiplicative group formed by the N th roots of unity in Q. As an abelian group μN (Q) is isomorphic to Z/N Z. The absolute Galois group GQ = Gal(Q/Q) acts naturally on μN (Q). Definition 3.3. Let N be a positive integer. The character determined by the natural action of the absolute Galois group GQ on μN (Q) χ ¯N : GQ → (Z/N Z)× is called the mod N cyclotomic character . Let be a prime number. The character χ : GQ → Z× determined by the projective limit of the character χ ¯m is called the -adic cyclotomic character .

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By Theorem 5.4 and Proposition 5.12 in Chapter 5 of Number Theory 2 , the mod N cyclotomic character χ ¯N is surjective and satisﬁes the following condition. (3.1)

χ ¯N is unramiﬁed at all primes p not dividing N , and satisﬁes χ ¯N (ϕp ) = p mod N .

Thus, the -adic cyclotomic character χ is also surjective and satisﬁes the following condition. (3.2)

χ is unramiﬁed at all primes p diﬀerent from , and it satisﬁes χ (ϕp ) = p.

By Theorem 3.1, the mod N cyclotomic character and the adic cyclotomic character are characterized by the following weaker conditions, respectively. (3.3)

χ ¯N (ϕp ) is unramiﬁed at almost all primes p, and satisﬁes χ ¯N (ϕp ) = p mod N .

(3.4)

χ (ϕp ) is unramiﬁed at almost all primes p, and satisﬁes χ (ϕp ) = p.

Irreducible representations of general degree are also characterized by the Frobenius substitutions. Let R be either a complete noetherian ring with a ﬁnite residue ﬁeld or a ﬁnite extension of Q . In general, two continuous representations of a topological group G, ρ, ρ : G → GLn (R) are said to be isomorphic if there exists an invertible matrix P ∈ GLn (R) such that ρ = P ρP −1 . If R is a local ﬁeld K or a ﬁnite ﬁeld F, ρ : G → GLn (R) is said to be irreducible if there are exactly two subspaces of Rn , Rn and 0, that are stable under the G-action. If ρ is not irreducible, we say it is reducible. Proposition 3.4. Let be a prime number. (1) Let K be a ﬁnite extension of Q , and let ρ, ρ : GQ → GLn (K) be two continuous homomorphism unramiﬁed at almost all primes. If ρ is irreducible, and Tr ρ(ϕp ) = Tr ρ (ϕp ) for almost all primes, then ρ and ρ are isomorphic. (2) Let O be the ring of integers of a ﬁnite extension K of Q , let F be its residue ﬁeld, and let ρ, ρ : GQ → GLn (K) be two continuous homomorphisms unramiﬁed at almost all primes. If the composition ρ¯ : GQ → GLn (O) → GLn (F) is irreducible and Tr ρ(ϕp ) = Tr ρ (ϕp ) for almost all primes, then ρ and ρ are isomorphic.

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85

(3) Let F be a ﬁnite extension of F , and let ρ¯, ρ¯ : GQ → GLn (F) be continuous homomorphisms. If ρ¯ is irreducible, and Tr ρ¯(ϕp ) = Tr ρ¯ (ϕp ) for almost all primes, then ρ¯ and ρ¯ are isomorphic. Proof. We just indicate the outline brieﬂy. (1) Since ρ is irreducible, it suﬃces to show that the semisimpliﬁcation of ρ is isomorphic to ρ. Thus, we may assume ρ is semisimple. Let A be the K-subalgebra of Mn (K) × Mn (K) generated by (ρ × ρ )(GQ ) over K. Then, A is semisimple. We want to show that the A-modules M and M , corresponding respectively to the representations ρ and ρ , are isomorphic as A-modules. By the assumption and Theorem 3.1, we have Tr(a : M ) = Tr(a : M ) for any A. Since A is semisimple, we have M M . We can show (3) similarly using the fact that a ﬁnite skew ﬁeld is commutative. (2) Since ρ¯ is irreducible, ρK : GQ → GLn (O) ⊂ GLn (K) is also irreducible. By (1), we have ρK ρK . Thus, there exists P ∈ GLn (K) such that ρK = P ρK P −1 . Since ρ¯ is irreducible, the only nonzero ﬁnitely generated O-submodules of K n that is stable under the action of ρ(GQ ) are constant multiples of On . Since P On is stable under the action of ρ(GQ ), it is a constant multiple of On . Therefore, a suitable multiple P of P belongs to GLn (O). Since ρK = P ρK P −1 , we conclude ρ ρ .

We deal with extensions of local ﬁelds or ﬁnite ﬁelds and introduce a stronger condition than the irreducibility, called the absolute irreducibility. Let R be a local ﬁeld or a ﬁnite ﬁeld, and ρ : G → GLn (R) a continuous representation of a topological group G. We say ρ is absolutely irreducible, if for any ﬁnite extension R of R, the composite ρ representation ρR : G → GLn (R) → GLn (R ) is irreducible. A necessary and suﬃcient condition for ρ to be absolutely irreducible is that the matrix ring Mn (R) is generated over R by the image ρ(G). For the representations that will appear frequently from now on namely degree 2 representations whose determinant is the cyclotomic character, the irreducibility and absolute irreducibility are equivalent. Proposition 3.5. Let R be a ﬁnite extension of either Q , or F with = 2. Then, an irreducible continuous representation ρ : G → GL2 (R) is absolutely irreducible if det ρ : GQ → R× is the cyclotomic character.

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Proof. If σ ∈ GQ is the complex conjugate, then we have ρ(σ)2 = 1, and det(σ) = −1 = 1. If R2 has a one dimensional subspace stable under the action of GQ , then it must be an eigenspace of ρ(σ). This subspace must be deﬁned over R, which contradicts the assumption that ρ is irreducible. 3.2. Galois representations and ﬁnite group schemes Representations of the absolute Galois group of a ﬁeld of characteristic 0 and ﬁnite group scheme over the same ﬁeld are the same objects of study. This important fact will appear over and over again in this book, and so we give a summary in this section. We also give a brief summary on the group schemes in Appendix A Let K be a ﬁeld, let K be its separable closure, and let GK = Gal(K/K) be the absolute Galois group of K. For a ﬁnite abelian group M , its automorphism group is denoted by Aut(M ). Definition 3.6. An abelian group M is a ﬁnite GK -module if a continuous homomorphism ρ : GK → Aut(M ) of topological groups is given. If this is the case, ρ is called a representation of GK to M . If A is a ﬁnite ´etale commutative group scheme over K, then the group of K-valued points A(K) is a ﬁnite GK -module through the natural GK -action. Proposition 3.7. Let K be a ﬁeld. The functor that sends a ﬁnite ´etale commutative group scheme to a ﬁnite GK -module A(K) (3.5) (ﬁnite ´etale commutative group scheme over K) → (ﬁnite GK -module) is an equivalence of categories. Example 3.8. Let N be a prime number with N = char(K). × Then, μN (K) = {x ∈ K | xN = 1} is a ﬁnite abelian group isomorphic to Z/N Z with a natural action of the absolute Galois group GK . Through this action we consider μN (K) as a ﬁnite GK module. The character determined by the action of GK on μN (K), GK → (Z/N Z)× , is called the mod N cyclotomic character. If K = Q, this deﬁnition coincides with Deﬁnition 3.3(1). The ﬁnite ´etale group scheme over K corresponding to the ﬁnite GK -module μN (K) is μN = Spec K[X]/(X N − 1).

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Proposition 3.7 is a consequence of the following proposition which characterizes the absolute Galois group GK . A ﬁnite set X is called a ﬁnite GK -set if a continuous homomorphism ρ : GK → {bijection from X to X} is given. If Z is a ﬁnite ´etale scheme, then the set of K-valued points Z(K) is a ﬁnite GK -set through the natural action of GK . Proposition 3.9. Let K be a ﬁeld. The functor that sends a ﬁnite ´etale scheme Z over K to the GK -set Z(K) (3.6)

(ﬁnite ´etale scheme over K) → (ﬁnite GK -set)

is an equivalence of categories. Proof. The functor that sends a ﬁnite ´etale scheme over K to a ﬁnite ´etale K-algebra Γ(Z, O) (3.7)

(ﬁnite ´etale scheme over K) → (ﬁnite ´etale K-algebra)

is a contravariant equivalence of categories. Thus, it suﬃces to show that the functor that sends the ﬁnite ´etale K-algebra R to the ﬁnite GK -set X (R) = {K-algebra homomorphisms R → K}: (3.8)

X : (ﬁnite ´etale K-algebra) → (ﬁnite GK -set)

is a contravariant equivalence of categories. We construct the inverse functor. For a ﬁnite GK -set X, deﬁne R(X) = {GK -equivariant mappings X → K}. This set naturally has a structure of K-algebra. Since R(X) ⊗K K = X K , R(X) is ﬁnite ´etale over K. We omit the veriﬁcation that the functor (3.9)

R : (ﬁnite GK -set) → (ﬁnite ´etale K-algebra)

is in fact the inverse of X . As a matter of fact, this is nothing but Galois theory once we use the following fact: a ﬁnite ´etale K-algebra is isomorphic to the product of a ﬁnite number of ﬁnite separable extensions of K. In the case K = Q, for the equivalence of categories in Proposition 3.7, we consider the condition for the ramiﬁcation at each prime. Definition 3.10. A ﬁnite GQ -module M is unramiﬁed at p if the extension ﬁeld corresponding to the kernel of ρ : GQ → Aut(M ) is unramiﬁed at p.

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Let A be a ﬁnite ´etale commutative group scheme over Q, and let p be a prime number. Consider a ﬁnite ´etale commutative group scheme AZ(p) over Z(p) such that AZ(p) ⊗Z(p) Q A. Since an ´etale scheme over an integrally closed ring is normal, AZ(p) is the integral closure of Z(p) in A. The addition of AZ(p) is given by the restriction of the mapping OA → OA ⊗Q OA that deﬁnes the operation of A to OAZ(p) . Thus, if such a group scheme exists, it is unique. The condition for the existence is that the corresponding Galois representation is unramiﬁed, as shown in the following proposition. Proposition 3.11. Let p be a prime number, and let A be a ﬁnite ´etale commutative group scheme over Q. Then, the following conditions are equivalent. (i) There exists a ﬁnite ´etale group scheme AZ(p) over Z(p) such that AZ(p) ⊗Z(p) Q A. (ii) The ﬁnite GQ -module A(Q) corresponding to A is unramiﬁed at p. Example 3.12. Let N be a positive integer, and let p be an odd prime. Then, the following conditions are all equivalent. (i) μN,Z(p) = Spec Z(p) [X]/(X N − 1) is a ﬁnite ´etale commutative group scheme over Z(p) . (ii) The ﬁnite GQ -module μN (Q) is unramiﬁed at p. (iii) p N . Proposition 3.11 follows from the following proposition. Proposition 3.13. Let p be a prime number, and let X be a ﬁnite ´etale scheme over Q. Then, the following conditions are equivalent. (i) The integral closure of Z(p) in X is ﬁnite ´etale over Z(p) . (ii) The extension corresponding to the kernel of the action of GQ on X(Q) is unramiﬁed at p. Proof. It suﬃces to show when X = Spec K, where K is a ﬁnite extension over Q. Part (i) means K is unramiﬁed at p, and part (ii) means the Galois closure L of K is unramiﬁed at p. Since K ⊂ L, (i) ⇒ (ii) is clear. Since L is a compositum of all the conjugates of K, and the compositum of unramiﬁed extension is unramiﬁed, (ii) ⇒ (i) is also proved.

3.3. THE TATE MODULE OF AN ELLIPTIC CURVE

89

3.3. The Tate module of an elliptic curve For an elliptic curve E over K, we consider the Galois representations on the group of N -torsion points and the Tate modules. Let E be an elliptic curve over a ﬁeld K. Let N be a positive integer with char(K) N . By Corollary 1.17, the ﬁnite abelian group E[N ](K) = Ker([N ] : E(K) → E(K)) is isomorphic to (Z/N Z)2 . The ﬁnite abelian group E[N ](K) is a ﬁnite GK -set through the natural GK -action. If K = Q and N = is a prime, then the ﬁnite GQ -module E[](Q) is what we denoted by E[] in Chapter 0. Let be a prime number with = char(K). By Proposition 1.19, the Tate module T E = limn E[n ](K) is isomorphic to Z2 as Z ←− modules. The Tate module T E also has a natural GK -action. Definition 3.14. Let E be an elliptic curve over a ﬁeld K. (1) Let N be a positive integer with char(K) N . The continuous homomorphism, obtained by taking a Z/N Z-basis of E[N ](K) and expressing matricially ρ¯E,N : GK → GLZ/N Z (E[N ](K)) GL2 (Z/N Z), is called the Galois representation deﬁned by the N -torsion points of E. (2) Let be a prime with = char(K). The continuous homomorphism, obtained by taking a Z -basis of T E and expressing matricially ρE, : GK → GLZ (T E) GL2 (Z ), is called the Galois representation on the -adic Tate module of E or the -adic representation deﬁned by E of GK . Condition (i) in Theorem 0.15 is that the Galois representation determined by the -torsion points ρ¯E, : GQ → GL2 (F ) is irreducible. In the correspondence in Proposition 3.7, the ﬁnite ´etale group scheme over K of degree N 2 corresponding to the ﬁnite GK -module E[N ](K) is the kernel of the multiplication-by-N mapping E[N ]. If K = Q, the characteristic polynomial of the Frobenius substitution is determined by ap (E) as follows. Proposition 3.15. Let E be an elliptic curve over Q, and suppose E has good reduction modulo a prime p.

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(1) Let N be a positive integer with p N , and let ρ¯E,N be the Galois representation on the N -torsion points of E. Then, ρ¯E,N is unramiﬁed at p, and we have (3.10) det 1 − ρ¯E,N (ϕp )t ≡ 1 − ap (E)t + pt2 mod N. (2) Let be a prime diﬀerent from p, and let ρE, be the Galois representation on the Tate module T E. Then, ρE, is unramiﬁed at p, and we have (3.11) det 1 − ρE, (ϕp )t = 1 − ap (E)t + pt2 . Proof. Once (1) is proved, (2) is obtained by taking the limit. Since E has good reduction at p, a smooth model EZ(p) of E over Z(p) is an elliptic curve over Z(p) satisfying EZ(p) ⊗Z(p) Q E. Let EFp denote the elliptic curve EZ(p) ⊗Z(p) Fp over Fp . Since p N , the kernel EZ(p) [N ] of the multiplication-by-N mapping [N ] : EZ(p) → EZ(p) is a ﬁnite ´etale commutative group scheme by Corollary 1.28. Thus, by Proposition 3.11, ρ¯E,N is unramiﬁed at p. Take a place q lying above p of the Galois extension L corresponding to the kernel of ρ¯E,N . Since EZ(p) [N ] is a ﬁnite ´etale over Z(p) , we identify E[N ](Q) = EZ(p) [N ](L) EZ(p) [N ](Fq ) = EFp [N ](Fp ). By the deﬁnition of the Frobenius substitution, the action of ϕp on E[N ](Q) is to raise to the pth power of the coordinates of point of EFp [N ](Fp ). This is the same as the action of the geometric Frobenius Frp : EFp → EFp on EFp [N ]. Thus, we see det 1 − ρ¯E,N (ϕp )t = det 1 − Frp · t : EFp [N ](Fp ) . Similar to Proposition 1.21, we can see det 1 − Frp · t : EFp [N ](Fp ) = 1 − ap (E)t + pt2 mod N. Thus, we see (3.10), too.

Corollary 3.16. Let E be an elliptic curve over Q. If N is a positive integer, the character of the absolute Galois group GQ , det ρ¯E,N : GQ → (Z/N Z)× , is the mod N cyclotomic character. If is a prime, det ρE, : GQ → Z× is the -adic cyclotomic character. Proof. By Proposition 1.6, E has good reduction except for a ﬁnite number of primes. If E has good reduction modulo p and p N , then we have det ρ¯E,N (ϕp ) = p. The mod N cyclotomic character is characterized by condition (3.3), and det ρ¯E,N satisﬁes this condition.

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91

Thus, det ρ¯E,N is the mod N cyclotomic character. It is similar for the case det ρE, . As for the irreducibility of the -adic representation and mod representation deﬁned by an elliptic curve over Q, the following is known, though we do not have enough space to give its proof in ρE, this book. We also call the composite mapping ρE,Q : GQ −→ GL2 (Z ) → GL2 (Q ) the -adic representation determined by E. Theorem 3.17. Let E be an elliptic curve over Q. (1) For all prime numbers , the -adic representation ρE,Q : GQ → GL2 (Q ) deﬁned by E is irreducible. (2) For any prime number with possible exceptions of the twelve primes 2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67, and 163, the mod representation ρ¯E, : GQ → GL2 (F ), deﬁned by the -torsion points of E, is irreducible. By Corollary 3.16 and Proposition 3.5, ρE,Q is always absolutely irreducible. Similarly, if = 2 and ρ¯E, is irreducible, then it is also absolutely irreducible. 3.4. Modular -adic representations As an elliptic curve deﬁnes an -adic representation, a primary form also deﬁnes an -adic representation. Theorem 3.18. Let be a prime number, and let O be the ring of integers of a ﬁnite extension K of Q . Suppose f is a primary form of level N with K coeﬃcients. (1) There exists a continuous homomorphism ρ : GQ → GL2 (O) that satisﬁes the following condition. (3.12)

For any prime p N , ρ is unramiﬁed at p, and we have det 1 − ρ(ϕp )t = 1 − ap (f )t + pt2 .

(2) Let ρ : GQ → GL2 (O) be a continuous homomorphism that satisﬁes (3.12). Then, the composite representation ρK : GQ → GL2 (O) ⊂ GL2 (K) is irreducible. We will prove (1) in Theorem 9.13 using the Tate module of the Jacobian of a modular curve. The proof of (2) will not be given in this book.

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Corollary 3.19. Let the notation be the same as in the theorem. (1) The representation ρK : GQ → GL2 (K) is characterized by the following condition uniquely up to an isomorphism. (3.13)

For almost all primes p, ρ is unramiﬁed at p, and we have Tr ρ(ϕp ) = ap (f ). ρ

(2) Let F be the residue ﬁeld of O. If the composition ρ¯ : GQ → GL2 (O) → GL2 (F) is irreducible, then ρ is also determined by the condition (3.13) uniquely up to an isomorphism. Proof. (1) follows from Theorem 3.18(2) and Proposition 3.4(1). (2) follows from Proposition 3.4(2). Definition 3.20. Let be a prime, and let O be the ring of integers of a ﬁnite extension K of Q . Suppose ρ : GQ → GL2 (O) is an -adic representation. (1) Let f be a primary form with K coeﬃcients. An -adic representation ρ is called the -adic representation associated with f if it satisﬁes condition (3.12). (2) We say ρ is modular if there is a primitive form f with K coefﬁcients such that ρ is the -adic representation associated with f. (3) Let N be a positive integer. We say ρ is modular of level N if there is a primitive form f of level N with K coeﬃcients such that ρ is an -adic representation associated with f . We also call the composition of -adic cyclotomic characters × × GQ → Z× and Z → O the -adic cyclotomic character. Lemma 3.21. Let ρ : GQ → GL2 (O) be a modular -adic representation. (1) The character det ρ : GQ → O× is the -adic cyclotomic character. (2) The representation ρK : GQ → GL2 (K) is absolutely irreducible. Proof. (1) follows from condition (3.12) and the characterization of the -adic cyclotomic characters (3.4). (2) follows from (1), Theorem 3.18(2), and Proposition 3.5. The primitive form f in Deﬁnition 3.20(3) is determined uniquely by ρ.

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Proposition 3.22. Let O be the ring of integers of a ﬁnite extension of Q , and let ρ : GQ → GL2 (O) be a modular -adic representation. A primitive form f with K coeﬃcients such that ρ is the -adic representation associated with f is unique. Proof. Suppose both f and g are primitive forms that satisfy condition (3.12). Then, for almost all primes p, we have ap (f ) = ap (g). Thus, by Theorem 2.49, we have f = g. That an elliptic curve E over Q is modular is equivalent to that the Galois representation ρE, on the -adic Tate module T E is modular. More precisely, we have the following. Proposition 3.23. Let E be an elliptic curve over Q. The following conditions are equivalent. (i) E is modular. (i ) There exists a primitive form f with Q coeﬃcients such that ap (E) = ap (f ) for all primes p at which E has good reduction. (ii) There exists a prime such that the Galois representation ρE, on the -adic Tate module T E is modular. (ii ) For any prime , the Galois representation ρE, on the -adic Tate module T E is modular. Proof. By Proposition 3.15(2) and Corollary 3.19(1) we see that (i) ⇒ (ii ). (ii ) ⇒ (ii) is clear. We show (ii) ⇒ (i). By assumption, there exists a primitive form f with Q coeﬃcients such that ap (E) = ap (f ) for almost all primes p. Since ap (f ) ∈ Q for almost all primes p, f is a primitive form with Q coeﬃcients. This proves (i). By Proposition 1.6, we see that (i ) ⇒ (i). We will prove (i) ⇒ (i ) in §3.8. According to Proposition 3.23, in order to show E is modular, it suﬃces to show ρE, is modular for one prime . This fact will be used when we derive Theorem 0.13 from Theorem 3.36 in Chapter 3. In the proof of Theorem 0.13, it is crucial to consider the modularity of mod representations. We now deﬁne the modularity of a mod representation similarly to an -adic representation. In this book, we restrict ourselves to the case = 2 for simplicity. Definition 3.24. Let be an odd prime, and F a ﬁnite extension of F . (1) Let N be a positive integer. We call an irreducible continuous representation of the Galois group ρ¯ : GQ → GL2 (F) modular of

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level N if there exist a ﬁnite extension K, a primitive form f with K coeﬃcients of level dividing N and the ﬁeld homomorphism from F to the residue ﬁeld FK of K that satisfy the following condition. (3.14)

For any prime p with p N , ρ¯ is unramiﬁed at p, and we have an equality in FK [t]: i det 1 − ρ¯(ϕp )t = 1 − ap (f )t + pt2 mod mK .

(2) If ρ¯ is modular with some level N , ρ¯ is called modular. The equality in condition (3.14) is an equality as quadratic polynomials with FK coeﬃcients. The left-hand side is the image of a quadratic polynomial with F coeﬃcients by the homomorphism i : F → FK . The right-hand side is the image of a quadratic polynomial with coeﬃcients in the ring of integers O of K by the homomorphism O → FK . By Proposition 3.15(1), condition (ii) of Theorem 0.15 is that the mod representation ρ¯E, : GQ → GL2 (F ) is modular. We call the composition of mod cyclotomic character GQ → F× × and F× → F also the mod cyclotomic character. Lemma 3.25. Let be an odd prime, and let F be a ﬁnite extension of F . Suppose ρ¯ : GQ → GL2 (F) is a modular irreducible continuous representation. (1) The character det ρ¯ is the mod cyclotomic character. (2) The representation ρ¯ : GQ → GL2 (F) is absolutely irreducible. Lemma 3.26. Let the notation be the same as Deﬁnition 3.24. (1) Suppose ρ¯ : GQ → GL2 (F) is an irreducible continuous representation. Then the condition (3.14) is equivalent to the following condition. (3.15)

For almost all primes p, ρ¯ is unramiﬁed at p, and we have an equality in FK [t]: i Tr ρ¯(ϕp ) = ap (f ) mod mK .

(2) An irreducible continuous representation ρ¯ : GQ → GL2 (F) that satisﬁes (3.15) is, if it exists, unique up to an isomorphism. Proof. (1) Let ρf : GQ → GL2 (OK ) be the -adic represenρf tation associated with f . The composite representation ρ¯f : GQ → the equality in (3.14) with the leftGL2 (OK ) → GL2 (FK ) satisﬁes hand side replaced by det 1 − ρ¯f (ϕp )t . Let ρ¯ : GQ → GL2 (F) be an

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95

irreducible continuous representation satisfying the condition (3.15). If the composite representation i ◦ ρ¯ is irreducible, then by Proposition 3.4(3), i ◦ ρ¯ and ρ¯f are isomorphic. Thus, ρ¯ satisﬁes condition (3.14). Assuming i ◦ ρ¯ is reducible, we obtain a contradiction as follows. Using Theorem 3.1 and the assumption = 2, we can show that the semisimpliﬁcation of i ◦ ρ¯ and ρ¯f are isomorphic in a similar way as Proposition 3.4(3). We omit the proof of this fact. Since we have 1 2 2 ¯ = det ρ¯f is the 2 (Tr ρ(σ)) −Tr ρ(σ ) = det ρ(σ), the character det ρ cyclotomic character. By Proposition 3.5 ρ¯ is absolutely irreducible, and thus i ◦ ρ¯ is irreducible. (2) follows easily from Proposition 3.4(3). In general, Lemma 3.25(1) is considered as a suﬃcient condition for ρ¯ to be modular. Conjecture 3.27. 1 Let be an odd prime number, and let F be a ﬁnite extension of F . An irreducible continuous representation ρ¯ : GQ → GL2 (F) is modular if det ρ¯ is the mod cyclotomic character. It is known that Conjecture 3.27 is true for F = F3 . Theorem 3.28. An irreducible continuous representation ρ¯ : GQ → GL2 (F3 ) is modular if det ρ¯ is the mod 3 cyclotomic character. This theorem plays a crucial role in the proof of Theorem 2.54, and thus that of Theorem 0.13. Unfortunately, we do not have enough space in this book to describe its proof. In 1999, the proof of Theorem 0.8 was completed by proving Conjecture 3.27 in the case of F = F5 . Theorem 3.29. An irreducible continuous representation ρ¯ : GQ → GL2 (F5 ) is modular if det ρ¯ is the mod 5 cyclotomic character. The following lemma is clear. Lemma 3.30. Let be an odd prime number, and let O be the ring of integers of a ﬁnite extension K of Q . Let ρ : GQ → GL2 (O) be a continuous representation. Let F be the residue ﬁeld of O, and ρ let ρ¯ be the composition GQ → GL2 (O) → GL2 (F). If ρ is modular of level N and ρ¯ is irreducible, then ρ¯ is also modular of level N . 1 Conjecture

3.27 was proved in 2007.

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3.5. Ramiﬁcation conditions In this section we deﬁne the notion that a representation of GQ is good, or semistable at p. Then, we formulate Theorem 3.36, which is the main theorem on the modularity of an -adic representation. Recall that we deﬁned the notion that an elliptic curve E over Q has good reduction or is semistable at p. Suppose E has good reduction at p and N is a positive integer. If EZ(p) is a smooth model of E over Z(p) , then the kernel EZ(p) [N ] of the multiplication-by-N mapping [N ] : EZ(p) → EZ(p) is, by Corollary 1.28, a ﬁnite ﬂat commutative group scheme over Z(p) . Thus, we deﬁne as follows. Definition 3.31. Let M be a ﬁnite GQ -module, and let p be a prime number. We say M is good if there exists a ﬁnite ﬂat group scheme A over Z(p) such that A(Q) is isomorphic to M as a ﬁnite GQ -module. Condition (iii) in Theorem 0.3 says that the ﬁnite GQ -module E[](Q) satisﬁes the condition in Deﬁnition 3.31 for all p except 2. The condition (iii) in Theorem 0.15 is similar. Proposition 3.32. Let M be a ﬁnite GQ -module, and let p be a prime number. In regard to the following two conditions, (ii) ⇒ (i) always holds, and (i) ⇒ (ii) holds if p does not divide the order of M . (i) M is good at p. (ii) M is unramiﬁed at p. Proof. By Proposition 3.11, condition (ii) is equivalent to the following condition. There exists a ﬁnite ´etale commutative group scheme A over Z(p) such that A(Q) is isomorphic to M as a ﬁnite GQ -module. Since p does not divide the order of M , this is equivalent to (i) by Proposition A.21 in Appendix A. The group of roots of unity μN (Q) is a ﬁnite GQ -module that is good at every prime p. The following proposition, which we do not prove in this book, will be used in Chapter 4. Proposition 3.33. Let be a prime, and let χ : GQ → F× be a power of the mod cyclotomic character χ ¯i , i ∈ Z. The following conditions are equivalent. (i) The ﬁnite GQ -module of order determined by χ : GQ → F× is good at .

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(ii) The character χ : GQ → F× is either trivial or is the mod cyclotomic character χ ¯ . To deﬁne the notion of semistability of a Galois representation, we summarize properties of the inertia group. Let L be a ﬁnite Galois extension of Q, and q a place of L lying above p. The kernel of the natural mapping from the decomposition group Dq = {σ ∈ Gal(L/Q) | σ(q) = q} to the Galois group of the residue ﬁeld Gal(Fq /Fp ) is called the inertia group of q. Since any place of L lying above p is conjugate to each other, the inertia group Iq depends only on p up to conjugate. By an abuse of notation, we denote by Ip this subgroup determined up to conjugate, and we call it the inertia group of p. Express an algebraic closure of Q as the compositum of ﬁnite Galois extensions, Q = λ∈Λ Lλ , and take the places qλ of Lλ lying above p so that they form a projective system. Then, the projective limit limλ Iqλ ⊂ limλ Gal(Lλ /Q) = GQ is determined up to conju←− ←− gate. This is also denoted by Ip , and we call it the inertia group of p. Proposition 3.34. Let M be a ﬁnite GQ -module. The following conditions are equivalent. (i) M is unramiﬁed at p. (ii) The action of the inertia group Ip on M is trivial. Proof. Let L be the ﬁnite Galois extension corresponding to the kernel of the action of GQ on M . Both (i) and (ii) amount to saying that L is unramiﬁed at p. If a ﬁnite ring R satisﬁes the condition that N · 1R = 0 for a positive integer N , the composition GQ → R× of the mod N cyclotomic character GQ → (Z/N Z)× and the natural mapping (Z/N Z)× → R× are also called cyclotomic characters. If the residue ﬁeld F of a complete noetherian local ring R is a ﬁnite extension of F , then the quotient R/mn of R by a power of the maximal ideal m is a ﬁnite ring of order a power of . The composition GQ → R× of the -adic × × cyclotomic character GQ → Z× and the natural mapping Z → R are also called cyclotomic characters. Definition 3.35. Let R be either a ﬁnite ring or a complete noetherian local ring whose residue ﬁeld is a ﬁnite ﬁeld. Let ρ : GQ → GL2 (R) be a continuous homomorphism of topological groups.

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(1) If R is a ﬁnite ring, we say ρ is good at a prime p if the GQ module R2 is good at p. If R is a complete noetherian local ring whose residue ﬁeld is a ﬁnite ﬁeld, we say ρ is good at a prime p if for any positive ρ integer n the composition GQ → GL2 (R) → GL2 (R/mn ) is good at p. (2) We say ρ is ordinary at a prime p if there exist P ∈ GL2 (R) and a function b : Ip → R such that the restriction of the conjugate of ρ by P to the inertia group Ip satisﬁes the following condition. (3.16)

If χ : Ip → R× is the restriction of the cyclotomic character to the inertia group, then for any σ ∈ Ip , we have χ(σ) b(σ) −1 P ρ(σ)P = . 0 1

(3) We say ρ is semistable at a prime p if ρ is either good or ordinary at p. (4) We say ρ is semistable if ρ is unramiﬁed for almost all primes p and semistable at all primes p. (5) If ρ is semistable, the product of all the primes at which ρ is not good is called the conductor of ρ. Condition (iii) in Theorem 0.3 is that the mod representation ρ¯E, : GQ → GL2 (F ) is good at all primes p except 2. The same is true for condition (iii) in Theorem 0.15. In some of the literature, the deﬁnitions of these notions are slightly diﬀerent from ours. So, please be careful while reading such references. The conductor of a semistable representation ρ is a square-free positive integer. Condition (3.16) may be paraphrased as follows. (3.17)

There exists a direct summand L of R2 isomorphic to R and stable under the action of Ip such that Ip acts on L as the restriction of the cyclotomic character and acts trivially on R2 /L.

The next theorem expresses a strong statement that if ρ is semistable, the converse of Lemma 3.30 holds. Theorem 3.36. Let be an odd prime, let O be the ring of integers of a ﬁnite extension K of Q , and let F be the residue ﬁeld of O. Suppose a semistable -adic representation ρ : GQ → GL2 (O) satisﬁes the following conditions.

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99

(i) The determinant det ρ : GQ → O× is the cyclotomic character. ρ (ii) The composition ρ¯ : GQ → GL2 (O) → GL2 (F) is irreducible and modular. Then, ρ is modular of level Nρ . In other words, there exists a primitive form of level dividing Nρ with K coeﬃcients such that for any prime p Nρ we have det 1 − ρ(ϕp ) = 1 − ap (f )t + pt2 . We will see that the level of the primitive form f is exactly Nρ by Corollary 3.53 in §3.8. In Chapter 4, we will show that Theorem 0.13 follows from Theorem 3.36. The chapters starting from Chapter 5 will be devoted to the proof of Theorem 3.36. By the deﬁnition of the modularity, condition (i) is a necessary condition. It is conjectured that the following weaker condition than (ii) may be suﬃcient for ρ to be modular. Conjecture 3.37. 2 Let be an odd prime, and let O be the ring of integers of a ﬁnite extension K of Q . If a semistable continuous representation ρ : GQ → GL2 (O) satisﬁes condition (i) in Theorem 3.36 and the composite representation ρK : GQ → GL2 (K) is irreducible, then ρ is modular of level Nρ . We are now going to explain the meaning of Deﬁnition 3.35 for the rest of this section. Proposition 3.38. Let p be a prime, let N be a positive integer not divisible by p, and let be a prime diﬀerent from p. Suppose R is a ﬁnite ring satisfying N · 1R = 0 or a complete noetherian local ring whose residue ﬁeld is F . Let ρ : GQ → GL2 (R) be a continuous homomorphism. Then, in each of the following (1) and (2), conditions (i) and (ii) are equivalent. (1) (i) ρ is good at p. (ii) The restriction of ρ to the inertia group Ip is trivial. (2) (i) ρ is ordinary at p. (ii) There exist P ∈ GL2 (R) and a continuous representation b : Ip → R such that the restriction of the conjugate of ρ by P to the inertia group Ip satisﬁes the following condition. (3.18)

2 As

For any σ ∈ Ip , we have 1 b(σ) −1 P ρ(σ)P = . 0 1 of 2008, this conjecture has also been proved in a fairly general cases.

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Proof. (1) It is enough to show the case where R is ﬁnite. In this case, the assertion follows from Propositions 3.32 and 3.34. (2) Under the assumption on R, the cyclotomic character is unramiﬁed at p. Thus, the assertion is clear from the deﬁnition. Whether a Galois representation ρ is good or ordinary is a property that depends only on the restriction of ρ to the inertia group Ip . If p N or p = , this follows from Proposition 3.38. The deﬁnition that ρ is ordinary itself depends only on its restriction to Ip . Proposition 3.40 shows that the property that ρ is good depends only on its restriction to Ip even if p | N or p = . If p N or p = , then that ρ is good at p implies that ρ is ordinary by Proposition 3.38. On the other hand, if p | N or p = , ρ may not be ordinary at p even if ρ is good at p. Under a certain condition, however, that ρ is ordinary implies ρ is good. We will show this in Proposition 7.11 in Chapter 7. 3.6. Finite ﬂat group schemes Let Qp be an algebraic closure of Qp , and Qnr p be the maximal unramiﬁed extension of Qp in Qp . The inertia group Ip may be as follows. identiﬁed with the absolute Galois group GQnr p Proposition 3.39. Let i : Q → Qp be a morphism of ﬁelds. The restriction of the morphism GQp → GQ induced by i deﬁnes an → Ip . isomorphism of the subgroups GQnr p Proof. If L is a ﬁnite extension of Q, a place q of L lying above p is determined by i. If L is a Galois extension of Q, the completion Lq of L at q is a ﬁnite Galois extension of Qp . The image of the natural injection Gal(Lq /Qp ) → Gal(L/Q) is the decomposition group Dq . Let Mq be the maximal unramiﬁed extension of Qp in Lq . Then, this injection induces an isomorphism Gal(Lq /Mq ) → Iq . Since Qp is the compositum of Qp and i(Q), by letting L run, we obtain an isomorphism GQnr → Ip as the limit. p Proposition 3.40. Let M be a ﬁnite GQ -module, and let p be a prime. Denote by M |Ip the ﬁnite Ip -module obtained by restricting . Then, the the action of GQ on M to the inertia group Ip = GQnr p following conditions are equivalent. (i) M is good at p.

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101

(ii) There exists a ﬁnite ﬂat commutative group scheme A over the nr ring of integers Znr p of Qp such that A(Qp ) is isomorphic to M |Ip as a ﬁnite Ip -module. Hence, that M is good at p depends only on M |Ip . Proof. (i) ⇒ (ii) is clear. We show the outline of (ii) ⇒ (i). In general, for a ﬁnite dimensional Q-vector space E, there is a natural one-to-one correspondence, {Z(p) -module generated by a basis of E} nr Zp -module generated by a basis of E⊗Q Qnr p . −→ that is stable under GQp -action We use this bijection. The mapping from left to right is given by L → L ⊗Z(p) Znr p , and that from right to left is R → R ∩ E. Let A0,Q be the ﬁnite ´etale commutative group scheme over Q corresponding to M , and let A be a ﬁnite ﬂat commutative group scheme over Znr p satisfying condition (ii). We would like to apply the above correspondence to E = Γ(A0,Q , O) and R = Γ(A, O). To do so, it suﬃces to show that we can make R stable under the GQp -action by replacing A if necessary. The number of conjugates of R under the GQp -action is ﬁnite, and the subring R of E ⊗Q Qnr p generated by these conjugates is stable under the GQp -action. We now proceed as follows. We ﬁrst verify that A = Spec R is nr a ﬁnite ﬂat group scheme that satisﬁes A ⊗Znr Qnr p = A0,Q ⊗ Qp . p We then let R0 = R ∩ E and A0 = Spec R0 , and show that this A0 is a ﬁnite ﬂat commutative group scheme over Z(p) such that A0 ⊗Z(p) Q = A0,Q . We omit the detail. We now give a summary of properties of ﬁnite ﬂat commutative group schemes which we use in Chapters 5 and 7. Proposition 3.41. Let OK be a discrete valuation ring, and let K be its ﬁeld of fractions. Let A be a ﬁnite ﬂat commutative group scheme over OK such that AK = A ⊗OK K is ´etale, and let M be a ﬁnite GK -module A(K). If N is a GK -submodule of M , then there exists a ﬁnite ﬂat commutative closed subgroup scheme A such that N = A (K). The same is true for the quotient GK -modules. Corollary 3.42. Let p be a prime number. (1) If M is a ﬁnite GQ -module good at p, then GQ -submodules and quotient GQ -modules of M are ﬁnite GQ -modules good at p.

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(2) If M1 and M2 are ﬁnite GQ -modules good at p, then the direct product M1 × M2 is also a ﬁnite GQ -module good at p. Proof. (1) can be seen by applying Proposition 3.41 to OK = Z(p) . (2) is clear.

Proof of Proposition 3.41. For the quotient modules, we can reduce it to the case of submodules by the Cartier duality and Proposition A.22, and thus we omit it. We show the assertion for GK -submodules. Let A be a ﬁnite ﬂat commutative group scheme over OK , and let AK be a closed subscheme of AK such that AK (K) = N . Let OA be the image of the homomorphism of rings OA = Γ(A, O) → OAK = Γ(AK , O), and let A = Spec OA . We show A satisﬁes the condition. The structure of group scheme is deﬁned on A as follows. The ring homomorphisms +A and +AK , which deﬁne the addition of group schemes, make the following diagram commutative. OA ⏐ ⏐ +A

−−−−→

OAK ⏐ ⏐+A K

OA ⊗OK OA −−−−→ OAK ⊗OK OAK Thus, a ring homomorphism +A : OA → OA ⊗OK OA is induced to deﬁne the structure of a group scheme. By this, A becomes a commutative group scheme. Since OA is a torsion-free ﬁnitely generated OK -module, it is ﬁnite ﬂat over OK . Since OAK → OAK is surjective, we have OA ⊗OK K = OAK . Hence, AK = A ⊗OK K. Theorem 3.43. Let OK be a discrete valuation ring, and let K be its ﬁeld of fractions. Suppose an odd prime p is a prime element in OK . Let A and A be ﬁnite ﬂat commutative group schemes over OK . If fK : AK → AK is an isomorphism of group schemes over K, then there exists a unique isomorphism f : A → A of ﬁnite ﬂat commutative group schemes over OK that induces fK .

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Corollary 3.44. Let OK , K, p, A, and A be as in Theorem 3.43. The natural mapping morphisms A → A of ﬁnite ﬂat commutative f ∈ group schemes over OK ↓ ↓ homomorphisms fK : A(K) → A (K) of fK ∈ ﬁnite GK -modules is bijective. Under the assumptions of Theorem 3.43 and Corollary 3.44, a ﬁnite ﬂat commutative group scheme A over OK can be identiﬁed with the ﬁnite GK -module A(K). Proposition 3.45. Let OK be a Henselian discrete valuation ring, and let A be a ﬁnite ﬂat commutative group scheme over OK . (1) Let 0 be the origin of the closed ﬁber of A, and let A0 = Spec OA,0 be the connected component of A containing 0. Then, A0 is a ﬁnite ﬂat closed subgroup scheme of A. ´ et be the maximal ´etale OK -submodule of OA = Γ(A, O). (2) Let OA Then, the group structure of A induces a group structure of ´ et . A´et is a ﬁnite ´etale commutative group scheme A´et = Spec OA over OK . (3) The sequence 0 → A0 (K) → A(K) → A´et (K) → 0 is an exact sequence of ﬁnite GK -modules. A → A´et is faithfully ﬂat, and its kernel equals A0 . A0 is called the connected component of A containing 0, and A´et is called the maximal ´etale quotient. We do not give a proof of Theorem 3.43, Corollary 3.44, and Proposition 3.45. 3.7. Ramiﬁcation of the Tate module of an elliptic curve On the ramiﬁcation of the Tate module of an elliptic curve, we have the following. Proposition 3.46. Let E be an elliptic curve over Q, and let be a prime number. Let ρE, : GQ → GL2 (Z ) be the -adic representation deﬁned by the Tate module T E. Then, in each of the following (1) and (2), conditions (i) and (ii) are equivalent.

104

(1) (2)

3. GALOIS REPRESENTATIONS

(i) (ii) (i) (ii)

E has good reduction modulo p. The Galois representation ρE, is good at p. E has stable reduction modulo p. The Galois representation ρE, is semistable at p.

If E has good reduction EFp modulo p, then ρE,p is ordinary if and only if EFp (Fp ) has an element of order p. This fact will not be used later, though. Corollary 3.47. Let E be an elliptic curve over Q, and let be a prime number. That E is semistable is equivalent to that the -adic representation ρE, is semistable. Moreover, in this case, the conductor of E equals the conductor of ρE, . For the mod N representation ρ¯E,N , we have the following. This proposition completes the proof of Proposition 0.4. Proposition 3.48. Let E be an elliptic curve over Q, let p be a prime number, and let N be a positive integer. Let ρ¯E,N be the representation deﬁned by the N -torsion points of E. (1) Consider the following conditions. (i) E has good reduction modulo p. (i ) E has multiplicative reduction modulo p and N divides the number of connected components of the geometric closed ﬁber of the semistable model of E over Z(p) . (ii) The Galois representation ρ¯E,N is good at p. If (i) or (i ) holds, then (ii) holds. Conversely, if one of the following conditions is satisﬁed, (ii) implies (i) or (i ). (a) N ≥ 3 and p N . (b) E has stable reduction modulo p. (2) Consider the following conditions. (i) E has stable reduction modulo p. (ii) The Galois representation ρ¯E,N is semistable at p. If (i) holds, then (ii) holds. Conversely, if the following condition is satisﬁed, (ii) implies (i). (a) N ≥ 5 and p N . Proof of Proposition 0.4. By Proposition 1.33, it suﬃces to show that condition (iii ) in Proposition 1.33 and (iii) in Theorem 0.3 are equivalent. Condition (iii ) in Proposition 1.33 states that one of the conditions (i) and (i ) of Proposition 3.48(1) in the case N = holds for all odd primes p. On the other hand, condition (iii) in Theorem 0.3 is that condition (ii) in Proposition 3.48(1) in the case

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N = holds. Since E is semistable, the assertion follows from Proposition 3.48(1). Example 3.49. (1) Let E be the semistable elliptic curve X0 (11), and let N be a positive integer. The mod N representation ρ¯E,N determined by the N -torsion points of E is good at all primes except 11. The number of connected components of the geometric ﬁber of the semistable model of E at p = 11 is 5. By Proposition 3.48(1), ρ¯E,N is good at p = 11 if and only if N = 5 or 1. (2) Let E be the elliptic curve y 2 = x3 − 5x. Then, ρ¯E,5 is good at p = 5, but E has additive reduction modulo p = 5. Thus, in the statement (ii) ⇒ (i) or (i ) in Proposition 3.48, we cannot drop condition (a) or (b). Proof of (i), (i ) ⇒ (ii) in Proposition 3.46 and Proposition 3.48. The implication (i) ⇒ (ii) in Proposition 3.48(1) is contained in Proposition 3.15(1). (i) ⇒ (ii) in Proposition 3.46(1) is also contained in Proposition 3.15(2). (i ) ⇒ (ii) in Proposition 3.48(1) follows from Corollary 1.36. Thus, it suﬃces to show (i) ⇒ (ii) in Proposition 3.48(2) and in Proposition 3.46(2) assuming E has multiplicative reduction. Proof of (i) ⇒ (ii) in Proposition 3.48(2). Let EZ(p) be a (1)

semistable model of E, and deﬁne EZ(p) as in Proposition 1.34 us(1)

ing the connected component containing 0. Let E = EZ(p) ⊗Z(p) Znr (p) . Fp is isomorphic to the multiplicative The geometric ﬁber E ⊗Znr (p) group Gm,Fp . The multiplication-by-N mapping [N ] : E → E is quasi-ﬁnite and ﬂat, and its kernel E[N ] is a quasi-ﬁnite ﬂat commutative group scheme over Znr p . By Corollary A.11 in Appendix A, E[N ] has a unique open ﬁnite subgroup scheme E[N ]f over Znr p such nr that E[N ]f ⊗Znr F = E[N ] ⊗ F . By (3.17), it suﬃces to show the p Zp p p following statement. The ﬁnite Ip -module E[N ]f (Qp ) is isomorphic to μN (Qp ), and the quotient Ip -module E[N ](Qp )/E[N ]f (Qp ) is isomorphic to Z/N Z. Since the determinant of the action of Ip on E[N ](Qp ) is the cyclotomic character, it suﬃces to show the ﬁrst part E[N ]f (Qp ) μN (Qp ). We show that the group scheme E[N ]f is isomorphic to μN .

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Since E ⊗Znr Fp Gm,Fp , we have E[N ]f ⊗Znr Fp = E[N ] ⊗Znr Fp p p (p) f μN,Fp . Thus, by Corollary A.24, we have E[N ] μN . Proof of (i) ⇒ (ii) in Proposition 3.46(2). limn E[n ]f (Qp ) ←− is a direct summand of T (E) of rank 1, and the inertia group Ip acts on limn E[N ]f (Qp ) as the -adic cyclotomic character. Thus, the ←− proof goes similarly as above. Proof of (ii) ⇒ (i) or (i ) in Propositions 3.46 and 3.48. Here, we show the case p = in Proposition 3.46, and p N in Proposition 3.48(1). The case p | N can be proved using the theory of Tate’s elliptic curve and the following proposition, but we omit it here. Note that, in order to derive Theorem 0.1 from Theorem 0.3, we need not to use this implication (ii) ⇒ (i). Proposition 3.50. Let K be a complete discrete valuation ﬁeld, and let N be a positive integer with char(K) N . For q ∈ K × O× , × deﬁne a ﬁnite GK -module Mq of order N 2 to be Ker(N : K /q Z → × Z K /q ). Then, the following conditions are equivalent. (i) There exists a ﬁnite ﬂat commutative group scheme A over the valuation ring OK such that the GK -module Mq is isomorphic to A(K). (ii) The valuation of q is a multiple of N . In what follows, we assume p = and p N . First, we derive (ii) ⇒ (i) in Proposition 3.46(1) from (ii) ⇒ (i) or (i ) in Proposition 3.48(1). If ρE, is good at p, then for any n, ρ¯E,n is good at p. Suppose n ≥ 3. From the implication (ii) ⇒ (i) or (i ) in Proposition 3.48(1), if E does not have good reduction modulo p, E has multiplicative reduction modulo p, and the number of irreducible components of its geometric closed ﬁber is divisible by n . Since we can take n as large as we like, we conclude that E must have good reduction modulo p. Similarly, we can derive (ii) ⇒ (i) in Proposition 3.46(2) from (ii) ⇒ (i) or (i ) in Proposition 3.48(2). We now show (ii) ⇒ (i) or (i ) in Proposition 3.48. We use the following theorem. Theorem 3.51. Let E be an elliptic curve over a discrete valuation ﬁeld K. (1) Up to an isomorphism, there exists a unique separated smooth commutative group scheme EOK over the valuation ring OK

3.7. RAMIFICATION OF THE TATE MODULE

107

such that E = EOK ⊗OK K and EOK satisﬁes the following condition: For any smooth scheme X over OK , the natural mapping {morphisms X → EOK over OK } → {morphisms X ⊗OK K → E over K} is bijective. This EOK is called the N´eron model of E. Let F be an algebraic closure of the residue ﬁeld of OK . (2) If E has good reduction, then the N´eron model of E is a smooth model of E. The geometric closed ﬁber EF = EOK ⊗OK F is an elliptic curve over F . (3) If E has multiplicative reduction, then the N´eron model of E is the smooth part of the semistable model of E. The geometric closed ﬁber EF is isomorphic to Gm,F × (Z/N Z), where N is the number of its connected components. (4) If E has additive reduction, then the connected component EF0 of the geometric closed ﬁber EF is isomorphic to the additive group Ga,F . The order of the ﬁnite abelian group C formed by the connected components of EF is less than or equal to 4. (5) Let K be an extension of K with a discrete valuation extending that of K. Suppose a prime element of K is a prime element of K . Then, the N´eron model EOK of the base change E ⊗K K is the base change of E ⊗K OK . We do not give a proof of this theorem. Proof of Proposition 3.48(1)(ii) ⇒ (i) or (i ). Let N be a positive integer with N ≥ 3. Let A be a ﬁnite ´etale commutative group scheme over Z(p) such that A(Q) is isomorphic to E[N ](Q) as GQ -modules. This means that A ⊗Z(p) Q is isomorphic to E[N ] as a group scheme over Q. Let EZ(p) be the N´eron model of E over Z(p) . Then, by deﬁnition, there exists a homomorphism of group schemes f : A → EZ(p) that extends the isomorphism A ⊗Z(p) Q → E[N ]. Let A be the image of A → EZ(p) endowed with the structure of reduced scheme. A is a ﬁnite ﬂat commutative group scheme of degree equal to N 2 , the degree of A. Since p N by assumption, it follows from Proposition A.21 that A is also ´etale. Thus, A → A is isomorphic. This implies that f : A → EZ(p) is a closed immersion.

108

3. GALOIS REPRESENTATIONS

A closed immersion f : A → EZ(p) of commutative group schemes over Z(p) induces an injective homomorphism A(Fp ) → EZ(p) (Fp ). We also denote it by f . Let f¯ be the composition of f and the natural mapping from EZ(p) (Fp ) to the ﬁnite abelian group C of the connected components of EFp = EZ(p) ⊗Z(p) Fp . Since A is ﬁnite ´etale, A(Fp ) is isomorphic to A(Q) (Z/N Z)2 , and its order N 2 is at least 9 and relatively prime to p. Assume E has additive reduction, and we derive a contradiction. 0 of EFp is isomorphic to Ga,Fp . Since The connected component EF p

the order of a nonzero element of Ga (Fp ) = Fp is p and the order of 0 (Fp ) ∩ A(Fp ) = 0. Thus, A(Fp ) is relatively prime to p, we have EF p ¯ f : A(Fp ) → C is injective. Since the order of C is at most 4 and the order of A(Fp ) is at least 9, we have a contradiction. Assume E has multiplicative reduction, and we show that the 0 of EFp is isoorder of C divides N . The connected component EF p

×

morphic to Gm,Fp . Since a ﬁnite subgroup of Gm (Fp ) = Fp is a 0 (Fp ) ∩ A(Fp ) cyclic group and A(Fp ) (Z/N Z)2 , the order d of EF p divides N . Thus, the order of the image f¯ : A(Fp ) → C, N 2 /d and hence the order of C are divisible by N . Proof of (ii) ⇒ (i) in Proposition 3.48(2). We have already proved Proposition 3.48(1), so we only have to show the assertion nr assuming ρ¯E,N is ordinary. Let K = Qnr p , and let OK = Zp . By assumption, E[N ](Qp ) contains a submodule isomorphic to μN (Qp ) as an Ip -module. This means that we have a closed immersion μN,K → EK . Let EOK be the N´eron model of EK . By the assumption that p does not divide N , μN,OK is a ﬁnite ´etale commutative group scheme over OK . Thus, similarly as above, it is extended to a closed immersion μN,OK → EOK . By assumption, the order N of μN,OK (Fp ) is at least 5 and relatively prime to p. Thus, if E has additive reduction, we derive a contradiction in the same way as above. 3.8. Level of modular forms and ramiﬁcation With regard to the ramiﬁcation and level of the -adic representation associated with a modular form, we have the following. Theorem 3.52. Let O be the ring of integers of a ﬁnite extension K of Q , and let f be a primitive form of level N with K coeﬃcients.

3.8. LEVEL OF MODULAR FORMS AND RAMIFICATION

109

Let ρf : GQ → GL2 (O) be the -adic representation associated with f . For a prime number p, the following conditions (i) and (ii) in each of (1) and (2) are equivalent. (1) (2)

(i) (ii) (i) (ii)

p N. ρf is good at p. p2 N . ρf is semistable at p.

Theorem 3.52 will be proved in §9.4. Corollary 3.53. Let f be a primitive form of level N , and let ρ be the -adic representation associated with f . Then ρ is semistable if and only if N is square-free. If this is the case, the conductor of ρ equals the level of N . Proof. The equivalence of the two conditions follow from (2). It then follows from (1) that the prime factors of N coincide with the primes at which ρ is not good. Proof Proposition 3.23(i) ⇒ (i ). Let f be a primitive form of level N with Q coeﬃcients such that ap (E) = ap (f ) for almost all prime numbers. Let be a prime number, and ρf,Q : GQ → GL2 (Q ) the -adic representation associated with f . By Proposition 3.15(2), ρE,Q and ρf,Q are unramiﬁed, and we have Tr ρE,Q (ϕp ) = ap (E) = ap (f ) = Tr ρf,Q (ϕp ) for almost all primes p. Thus, by Proposition 3.4(1) and Theorems 3.17(1) and 3.18(2), ρE,Q and ρf,Q are isomorphic. Let p be a prime where E has good reduction, and let be a prime diﬀerent from p. By Proposition 3.15(2), ρE,Q is unramiﬁed at p, and thus ρf,Q is also unramiﬁed at p. Thus, by the implication (ii) ⇒ (i) of Theorem 3.52(1), we have p N . Thus, we have ap (E) = Tr ρE,Q (ϕp ) = Tr ρf,Q (ϕp ) = ap (f ).

By Proposition 3.23 and Corollaries 3.47 and 3.53, in order to show Theorem 2.54, and in turn Theorem 0.13, it suﬃces to show that ρE, is modular for one prime . Corollary 3.54. Let E be a semistable elliptic curve, and let N be its conductor. If E is modular, then E is modular exactly of level N .

110

3. GALOIS REPRESENTATIONS

Proof. Let f be a primitive form with Q coeﬃcients such that ap (E) = ap (f ) for almost all prime numbers. By Corollary 3.47, the conductor of the semistable -adic representation ρE,Q ρf,Q is N . Thus, by Corollary 3.53, the level of f is also N . Then, by the implication (i) ⇒ (i ) in Proposition 3.23, we have the equality ap (E) = ap (f ) for all primes p with p N . Also, with regard to the ramiﬁcation of modular mod representation also, we have the following. For a positive integer N and a prime number p, the largest divisor of N relatively prime to p is called the prime to p part of N . Theorem 3.55. Let be an odd prime, let F be a ﬁnite extension of F , and let ρ¯ : GQ → GL2 (F) be an irreducible continuous representation. Suppose ρ¯ is modular of level N and M is the prime to p part of N . Then, the following conditions (i) and (ii) in each of (1) and (2) are equivalent. (1) (i) ρ¯ is modular of level M . (ii) ρ¯ is good at p. (2) (i) ρ¯ is modular of level pM . (ii) ρ¯ is semistable at p. Theorem 0.15 in Chapter 0 is obtained by applying Theorem 3.55(1) (ii) ⇒ (i) to ρ¯ = ρ¯E, . The implication (i) ⇒ (ii) and a part of (ii) ⇒ (i) of (1) in Theorem 3.55 will be proved in §9.7. Corollary 3.56. Let be an odd prime, and let F be a ﬁnite extension of F . Suppose ρ¯ : GQ → GL2 (F) is a modular irreducible semistable continuous representation. If Nρ is the conductor of ρ¯, then ρ¯ is modular of level Nρ . Proof. Let N be the smallest positive integer such that ρ¯ is modular of level N . Applying Theorem 3.55(2) (ii) ⇒ (i) repeatedly with each prime factor of N , we see that N is square-free. Furthermore, applying Theorem 3.55(1) (ii) ⇒ (i) repeatedly, we see that N divides Nρ .

10.1090/mmono/243/05

CHAPTER 4

The 3–5 trick In §4.1, we derive Theorem 2.54, and in turn Theorem 0.13, from Theorem 3.36. If the representation on the group of 3-torsion points of a semistable elliptic curve E is irreducible, we apply Theorem 3.36 directly to the 3-adic representation. If it is reducible, then we use the 5-torsion points. Here, we need a deep result (Theorem 4.4) on the rational points of an elliptic curve. In §4.2, we give a summary of the proof of Fermat’s Last Theorem (Theorem 0.1). 4.1. Proof of Theorem 2.54 For now we assume Theorem 3.36, and we show the following lemma. Lemma 4.1. Let E be a semistable elliptic curve, and let be an odd prime. Let N be the conductor of E. Suppose the mod representation ρ¯E, : GQ → GL2 (F ) associated with the -torsion points of E is irreducible and modular. Then, E is modular exactly of level N . In other words, there exists a primitive form of level N with Q coeﬃcients such that ap (E) = ap (f ) for any prime number p not dividing N . Proof. First, we show that the -adic representation ρE, : GQ → GL2 (Z ) is modular. It is enough to verify that ρE, satisﬁes the conditions of Theorem 3.36. By Corollary 3.47, ρE, is semistable, and its conductor is equal to the conductor N of E. By Corollary 3.16, det ρE, is the cyclotomic character, and thus condition (i) in Theorem 3.36 is satisﬁed. Since ρ¯E, is modular by assumption, condition (ii) is also satisﬁed. So, applying Theorem 3.36 to ρE, , we see that ρE, is modular of level N . It now suﬃces to apply Proposition 3.23 (ii) ⇒ (i ) and Corollary 3.54. 111

112

4. THE 3–5 TRICK

Proof of Theorem 3.36 ⇒ Theorem 2.54. Let E be a semistable elliptic curve over Q. By Lemma 4.1, it suﬃces to show that the mod representation ρ¯E, is modular for at least one of = 3 or 5. Suppose the representation ρ¯E,3 determined by the 3-torsion points is irreducible. Then, since det ρ¯E,3 is the cyclotomic character, ρ¯E,3 is modular by Theorem 3.28. Thus, in this case, it is enough to apply Lemma 4.1 with = 3. Suppose now the mod 3 representation ρ¯E,3 is reducible. In this case, we want to apply Lemma 4.1 with = 5. This is made possible by the following proposition. Proposition 4.2. Let E be a semistable elliptic curve over Q. (1) If the mod 3 representation ρ¯E,3 determined by the 3-torsion points of E is reducible, then the mod 5 representation ρ¯E,5 is irreducible. (2) If the mod 5 representation ρ¯E,5 is irreducible, then there exists a semistable elliptic curve E such that the mod 3 representation ρ¯E ,3 is irreducible and ρ¯E,5 ρ¯E ,5 . Admitting this proposition for the moment, we continue the proof. Suppose E is a semistable elliptic curve whose mod 3 representation ρ¯E,3 is reducible. Then, by Proposition 4.2(1), its mod 5 representation ρ¯E,5 is irreducible. So, we take a semistable elliptic curve E that satisﬁes the condition in Proposition 4.2(2). Then, since ρ¯E ,3 is irreducible, E is modular, as we show above. Since ρ¯E ,5 is irreducible, it follows from Proposition 3.23 (i) ⇒ (ii) and Lemma 3.30 that ρ¯E ,5 is modular. Hence, ρ¯E,5 is also modular. Now, it suﬃces to apply Lemma 4.1 with = 5. In the above proof, if ρ¯E,5 is irreducible, some readers might think it enough to apply Theorem 3.29. However, the proof of Theorem 3.29 consists of the above argument and the reﬁned version of the remaining part of this book, and so we would fall into a circular argument. The rest of this section is devoted to prove Proposition 4.2. Proposition 4.2(1) is the case = 5, and = 3 in Proposition 4.3(i). Proposition 4.3. Let E be a semistable elliptic curve over Q, and let be an odd prime. If one of the following three conditions is satisﬁed, then the representation ρ¯E, determined by the -torsion points of E is irreducible.

4.1. PROOF OF THEOREM 2.54

113

(i) There exists a prime number diﬀerent from such that ρ¯E , is reducible. (ii) is greater than or equal to 5, all the 2-torsion points of E are Q-rational. (iii) is greater than or equal to 11. Theorem 0.16 in Chapter 0 is case (ii) in Proposition 4.3. Here, we derive Proposition 4.3 from Theorem 4.4 below. The proof of Theorem 4.4 is based on deep arithmetic properties of modular curves, and we cannot present them in this book. Theorem 4.4. The torsion subgroup of the group of rational points E(Q) of an elliptic curve E over Q is isomorphic to one of the following ﬁfteen groups. Z/N Z (1 ≤ N ≤ 10 or N = 12), Z/2N Z × Z/2Z (1 ≤ N ≤ 4). To prove Proposition 4.3, we ﬁrst show the following lemma. Lemma 4.5. Let be a prime number, let F be a ﬁnite extension of F , and let ρ¯ : GQ → GL2 (F) be a reducible semistable mod representation. If C is a one dimensional subrepresentation of ρ¯, then the character χ : GQ → F× determined by C is either trivial or the mod cyclotomic character χ ¯ . Proof. Since ρ¯ is semistable by assumption, the character χ is unramiﬁed for all prime p except . By a theorem of class ﬁeld theory (Theorem 5.10(1) in Number Theory 2 ), there exists a positive integer n such that the abelian extension of Q corresponding to the kernel of χ is contained in the cyclotomic ﬁeld Q(ζn ). Since the order of χ is relatively prime to , we may assume n = 1. Thus, χ is a power of mod cyclotomic character χ ¯i . If ρ¯ is good at p = , then by Corollary 3.42, GQ -module C is also good at p = . Thus, in this case we apply Proposition 3.33. If ρ¯ is ordinary at p = , then the restriction of χ to the inertia ¯ . Since group I is either trivial or the mod cyclotomic character χ I → Gal(Q(ζ )/Q) is surjective, this case is also proved. Proof of Proposition 4.3. Since each case is similar, we only show (i). Let C be a sub-GQ -module of E[ ](Q) of order . Assume there exists a sub-GQ -module of E[](Q) of order , and derive a × contradiction. Let χ : GQ → F× and χ : GQ → F be characters determined by C and C , respectively. By Proposition 3.48(2) (i) ⇒ (ii), ρ¯E, is semistable. By Lemma 4.5, these are either trivial or

114

4. THE 3–5 TRICK

mod or cyclotomic character. If χ and χ are both trivial, then E(Q) has a subgroup of order , which contradicts Theorem 4.4. If either χ or χ is nontrivial, then we deﬁne the quotient E of E as follows. If χ = 1 and χ = 1, then let E = E/C . Since det ρ¯E, is the mod cyclotomic character, E (Q) has a subgroup E[]/C Z/Z of order . Since E (Q) has a subgroup of order which is the image of C , it contradicts Theorem 4.4 as above. Similarly, if χ = 1 and χ = 1, let E = E/C , and if χ = 1 and χ = 1, let E = E/(C × C ). Then, we can derive a contradiction similarly. Proof of Proposition 4.2(2). We show the following lemma ﬁrst. Lemma 4.6. Let M(E[5]) be the functor over Q that sends a scheme T over Q to the set {isomorphism classes of the pair (E , α), where E is an elliptic curve over T , and α : E[5]T → E [5] is an isomorphism of group schemes }. Then, the functor M(E[5]) is represented by an aﬃne smooth curve Y over Q. Y ⊗Q Q = YQ is isomorphic to Y (5)Q . Proof. The proof is similar to that of Theorems 2.10 and 2.21 and Proposition 2.23. We only illustrate how to construct the ﬁne moduli scheme Y . The functor over Q that sends a scheme T over Q to the set {morphisms of group schemes (Z/5Z)2T → E[5]T } is represented by an open and closed subscheme of E[5] × E[5]. We denote this ﬁnite ´etale scheme over Q by Isom((Z/5Z)2 , E[5]). It has a natural action of GL2 (Z/5Z). The ﬁne moduli scheme Y of the functor M(E[5]) is the quotient of the product Y (5) ×Q Isom((Z/5Z)2 , E[5]) by GL2 (Z/5Z). The rational points of Y are in one-to-one correspondence with the pairs (E , α) of elliptic curve E over Q and the isomorphism α : ρ¯E,5 → ρ¯E ,5 of GQ representations. Let P0 be the rational point of Y corresponding to the pair (E, id). Let Y0 be the connected component of Y containing P0 . Y0 is a connected smooth aﬃne curve whose ﬁeld of constants is Q. By Proposition 2.22, the genus of Y (5) is 0. Thus, by Lemma 4.6, the genus of Y0 is also 0. Since Y0 has a rational point P0 , it is isomorphic to an open subscheme of P1Q . Identify Y0 with an open subscheme of P1Q , and let t be a nonhomogeneous coordinate of P1Q . Then, except for a ﬁnite number

4.1. PROOF OF THEOREM 2.54

115

of rational numbers, the point whose coordinate t is a rational number determines a rational point of Y0 . We consider the conditions for an elliptic curve E corresponding to a rational point of Y0 to satisfy the conditions in Proposition 4.2(2): (i) E is semistable. (ii) The representation ρ¯E ,3 is irreducible. (iii) The representation ρ¯E ,5 is isomorphic to the representation ρ¯E,5 . By the deﬁnition of Y , condition (iii) is always satisﬁed. We study condition (ii). We have the following lemma similar to Lemma 4.6. The proof is omitted. Lemma 4.7. Let M(E[5], 3) be the functor over Q that sends a scheme T over Q to the set ⎧ ⎫ isomorphism classes of the triple (E , α, C), where E ⎪ ⎪ ⎨ ⎬ is an elliptic curve over T , α : E[5]T → E [5] is an iso. ⎪ ⎩ morphism of group schemes, and C is a cyclic subgroup ⎪ ⎭ scheme of order 3 Let Y (5, 3) be the quotient of the modular curve Y (15) by the group B(Z/3Z) ⊂ GL2 (Z/15Z). Then, the functor M(E[5], 3) is represented by an aﬃne smooth curve Y over Q. Y ⊗Q Q = YQ is isomorphic to Y (5, 3)Q . The rational points of Y are in one-to-one correspondence with the triples (E , α, χ) of elliptic curve E over Q, the isomorphism α : ρ¯E,5 → ρ¯E ,5 of GQ representations, and one dimensional subrepresentation ρ¯E ,3 . Thus, the (representation associated with) 3torsion points of E corresponding to a rational point P in Y is irreducible if and only if P is not the image of a rational point of Y . We show Y (Q) is a ﬁnite set. Let X be a (not connected) proper smooth curve over Q such that X contains Y as a dense open subscheme. Let C be a connected component of X whose ﬁeld of constants is Q. By the following theorem, it is enough to show that the genus of C is at least 2. Theorem 4.8. If X is an algebraic curve over Q of genus at least 2, then the set of rational points X(Q) is a ﬁnite set. Let X(5, 3) be the quotient of the modular curve X(15) by B(Z/3Z) ⊂ GL2 (Z/15Z). X(5, 3) is a proper smooth connected curve over the ﬁeld of constants Q(ζ5 ). C ⊗Q Q is isomorphic to

116

4. THE 3–5 TRICK

X(5, 3) ⊗Q(ζ5 ) Q. It is enough to show that the genus g of the connected curve X(5, 3) over Q(ζ5 ) is 9. As in the proof of Proposition 2.15, we apply the Riemann–Hurwitz formula to the ﬁnite morphism X(5, 3) → X(5). Its degree is 4, and it is ´etale over Y (5). Over Q(ζ5 ), there are |GL2 (Z/5Z)|/(2 · 5 · [Q(ζ5 ) : Q]) = 12 cusps in X(5), and at each cusp there are two points of X(5, 3) on it. Since X(5) P1Q(ζ5 ) , we have 2g − 2 + 2 · 12 = 4(2 · 0 − 2 + 12) = 40, which shows g = 9. Finally, we study the condition for E to be semistable. Since ρ¯E,5 ρ¯E ,5 , and it is semistable, E is also semistable at any primes other than 5 by Proposition 3.48(2) (ii) ⇒ (i). We show that if a point P is 5-adically suﬃciently close to P0 , then corresponding elliptic curve E is semistable at p = 5. Identify Y0 with an open subscheme of P1Q , and take a nonhomogeneous coordinate t of P1Q in such a way that the coordinate of P0 is 0. Let c(t) and d(t) be rational functions of t such that the universal elliptic curve EY0 over Y0 is given by y 2 = 4x3 − c(t)x − d(t). Since E is semistable at 5, we may assume c(0), d(0) ∈ Z(5) , and c(0) ≡ 0 or d(0) ≡ 0 mod 5 by making a change of coordinates of EY0 if necessary. Then, if t is suﬃciently close to 0 5-adically, then c(t), d(t) ∈ Z(5) , and c(0) ≡ 0 or d(0) ≡ 0 mod 5. An elliptic curve E corresponding to such a t is also semistable at 5. To put everything so far together, if a rational number t is 5adically suﬃciently close to 0, and a rational point P of Y0 whose coordinate is t is not in the image of the ﬁnite set Y (0), then the corresponding elliptic curve E satisﬁes all the conditions (i), (ii), and (iii). Since there are inﬁnitely many such rational numbers, we have proved the proposition. 4.2. Summary of the Proof of Theorem 0.1 Since the proof of Theorem 0.1 is rather complicated, let us go over again the diagram (0.2). We have not proved Theorems 3.36 and 3.55, but we admit them here. For an integer n greater than or equal to 3, assume the equation (0.1)

Xn + Y n = Zn

admits a nontrivial integer solution (X, Y, Z) = (A, B, C). As we have seen in §§0.1–0.2, we may assume is a prime number at least

4.2. SUMMARY OF THE PROOF OF THEOREM 0.1

117

5, A, B, C are relatively prime, B is even and C ≡ −1 mod 4. Let E be an elliptic curve deﬁned by (0.4)

y 2 = x(x − C )(x − B ).

By Proposition 1.9(1) (ii) ⇒ (i), E is semistable, and its 2-torsion points are all rational. Let N be the conductor of E. By Proposi1 A B C . tion 1.9(2), N equals the product of all the prime factors of 16 Since ≥ 5, N is the product of all prime factors of ABC, and it is even. This is the precise content of the ﬁrst arrow of (0.2). The fact that E is modular of level N can be proved as follows. In general, for a semistable elliptic curve E , the -adic representation ρE , and the mod representation ρ¯E , are semistable by Proposition 3.46(2) (i) ⇒ (ii) and Proposition 3.48(2) (i) ⇒ (ii), respectively. Their determinants are the cyclotomic characters by Corollary 3.16. By Proposition 4.3(ii), the mod 5 representation ρ¯E,5 : GQ → GL2 (F5 ) determined by the 5-torsion points of E is irreducible. Then, by Proposition 4.2(2), there exist a semistable elliptic curve E such that its mod 3 representation ρ¯E ,3 is irreducible, and ρ¯E,5 ρ¯E ,5 . By Theorem 3.28, the mod 3 representation ρ¯E ,3 is modular. Applying Theorem 3.36, the semistable 3-adic representation ρE ,3 is modular. Then, by Proposition 3.23 (ii) ⇒ (ii ), the 5-adic representation ρE ,5 is also modular. Since the mod 5 representation ρ¯E,5 ρ¯E ,5 is irreducible, this is again modular. This time, applying Theorem 3.36 to the semistable 5-adic representation ρE,5 , we see that ρE,5 is also modular. By Proposition 3.23 (ii) ⇒ (i), E is modular, and there exists a primitive form with Q coeﬃcients such that ap (E) = ap (f ) for almost all primes. By Corollary 3.54, the conductor of this primitive form is N . This is the precise content of the second arrow of (0.2). From the existence of such a modular form f , a contradiction is derived as follows. Since ≥ 5, and all the 2-torsion points of E are rational, the mod representation ρ¯E, : GQ → GL2 (F ) determined by the -torsion points of E is irreducible by Proposition 4.3(ii). Since E is modular of level N , ρ¯E, is modular of level N . We show ρ¯E, is modular of level 2. Let p be an odd prime factor of N . By Proposition 1.33 (ii) ⇒ (i), E has multiplicative reduction at p, and the number of irreducible components of the geometric closed ﬁber EZ(p) ⊗Z(p) Fp of its semistable model EZ(p) is a multiple of . By Proposition 3.48(1) (i ) ⇒ (ii), the mod representation ρ¯E, is good at p. This holds for all odd prime factors of N , and thus we apply

118

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Theorem 3.55(1) (ii) ⇒ (i) repeatedly to see that ρ¯E, is modular of level 2. Note that we use Theorem 3.55(1) only in the case where N is divisible by p only once. We will prove Theorem 3.55 in §9.7, but we prove only a part of this case. Back to the proof. Since ρ¯E, is modular of level 2, there exists a primitive form g of level 1 or 2 such that ap (g) ≡ Tr ρ¯E, (ϕp ) = ap (f ) for all p 2. However, by Corollary 2.18, the space of modular forms of level 1 and 2, S(1) and S(2), are both 0. By the deﬁnition of primitive form, g is a nonzero element of S(1) or S(2), which is a contradiction. This is the precise content of the third arrow of (0.2). This is the proof of Theorem 0.1. In Chapter 5 we begin describing the proof of Theorem 3.36, which is used in the above proof.

10.1090/mmono/243/06

CHAPTER 5

R=T In this chapter we describe the outline of the proof of Theorem 3.36. While Theorem 3.36 is written as a theorem on the -adic representation ρ, we can state it as a theorem on the mod representation ρ¯. This change of viewpoint is more signiﬁcant than it appears. Let be an odd prime, and let F be a ﬁnite extension of F . Let ρ¯ : GQ → GL2 (F) be a mod representation. Let O be the ring of integers of a ﬁnite extension of Q whose residue ﬁeld is F. We say that an -adic representation ρ : GQ → GL2 (O) is a lifting of ρ¯ to O ρ if the composition GQ → GL2 (O) → GL2 (F) coincides with ρ¯. Theorem 5.1. Let be an odd prime, let O be the ring of integers of a ﬁnite extension K of Q , and let F be its residue ﬁeld. Suppose ρ¯ : GQ → GL2 (F) is a semistable modular representation. Then, a semistable lifting ρ : GQ → GL2 (O) of ρ¯ to O is modular of level Nρ if its determinant det ρ : GQ → O× is the cyclotomic character. From §5.2 on, we prove this theorem using the deformation rings and the Hecke algebras. Before that, we explain in §5.1 why we use such rings. Most of the content of this chapter is valid for = 2. However, for the sake of simplicity, we always assume that stands for an odd prime in this chapter. 5.1. What is R = T ? In this section we sacriﬁce some accuracy in order to make it easier to grasp the idea. The strict formulation will be given in §§5.2–5.3. Logically speaking, we may dispense with this section. Theorem 5.1 is about modular forms and Galois representations. Elliptic curves, which played a principal role, have already exited from the scene. Now the principal roles are played by the deformation 119

120

5. R = T

rings, which are incarnations of Galois representations, and the Hecke algebras, which are incarnations of modular forms. Theorem 5.1 may be paraphrased as follows. First, deﬁne

ρ is a semistable lifting of ρ¯ to O

R = ρ : GQ → GL2 (O) whose determinant det ρ is the cy- ,

clotomic character.

a (f ) ≡ Tr ρ¯(ϕp ) for almost T = primitive forms f

p . with K coeﬃcients all primes p. We can deﬁne an injective mapping ϕ : T → R by sending a primitive form f to the -adic representation ρ associated with f . Theorem 5.1 states that this is bijective. As they are, these sets are inﬁnite sets which are not easy to handle. So, we consider the following subsets. Let Σ be a ﬁnite set of primes.

ramiﬁes to the same extent as ρ¯ , RΣ = ρ ∈ R ρ except at the primes in Σ

ρ ramiﬁes to the same extent as ρ¯ TΣ = f ∈ T f . except at the primes in Σ

We can also deﬁne an injective mapping ϕΣ : TΣ → RΣ . Since R = Σ RΣ , it suﬃces to show that ϕΣ is bijective for each Σ in order to show that ϕ is bijective. An important step in showing that ϕΣ : TΣ → RΣ is bijective is to interpret it in terms of commutative algebra. Deﬁne O-algebras RΣ , TΣ , and a morphism of O-algebras fΣ : RΣ → TΣ that satisﬁes the following three conditions: (5.1)

RΣ = {morphisms of O-algebras RΣ → O},

(5.2)

TΣ = {morphisms of O-algebras TΣ → O},

(5.3)

The mapping ϕΣ is given by g ↓ g ◦ fΣ

∈ ∈

{morphisms of O-algebras TΣ → O} ↓ {morphisms of O-algebras RΣ → O}.

To show that the mapping ϕΣ : TΣ → RΣ is bijective is reduced to showing that the ring homomorphism fΣ : RΣ → TΣ is an isomorphism. We deﬁne RΣ , TΣ , fΣ : RΣ → TΣ in §§5.2–5.3. After some preparations in §§5.4–5.5, we show the outline of proof that fΣ is an isomorphism in §5.6. Here, we indicate the idea of the deﬁnition of RΣ , TΣ and fΣ : RΣ → TΣ .

5.1. WHAT IS R = T ?

121

The property the RΣ must satisfy may be written as {morphisms of O-algebras RΣ → O}

ρ is a lifting of ρ¯ to O such that det ρ is the

= ρ cyclotomic character and ρ ramiﬁes to the

same extent as ρ¯ except at the primes in Σ

.

We deﬁne RΣ by imposing this property not only to the homomorphisms to O but also to the homomorphisms to any O-algebras. Namely, RΣ is characterized by the following property. (5.4) For any O-algebra A, RΣ satisﬁes {morphisms of O-algebras RΣ → A}

ρ is a lifting of ρ¯ to A such that det ρ is the

= ρ cyclotomic character and ρ ramiﬁes to the

same extent as ρ¯ except at the primes in Σ

.

By an analogy to the deformation theory in algebraic geometry, we call RΣ the deformation ring. The O-algebra TΣ is deﬁned as an O-subalgebra of the reduced Hecke algebra. Deﬁne NΣ suitably such that NΣ is a multiple of the conductor Nρ¯ and satisﬁes TΣ = {f ∈ T | the level of f is a divisor of NΣ }. Let Φ(NΣ )K,¯ρ be the ﬁnite set consisting of primitive forms f over K ap (f ) ≡ Tr ρ¯(ϕp ) for almost whose level is a divisor of NΣ such that ∞ all primes p. For a primitive form f = n=1 an (f )q n ∈ Φ(NΣ )K,¯ρ , let Kf be the ﬁeld generated by an (f ), n ∈ N, over K, and let Of be its ring of integers. We deﬁne the morphism of O-algebras RΣ → Of . An -adic representation ρf : GQ → GL2 (Of ) associated with f is a lifting of ρ¯ to Of such that det ρf is the cyclotomic character and ρf ramiﬁes to the same extent as ρ¯ except for the primes p in Σ. Thus, by the deﬁnition of RΣ , there is an O-algebra homomorphism ψf : RΣ→ Of . TΣ is the image of the product of these maps ΨΣ : RΣ → f ∈Φ(NΣ )K,ρ¯ Of . In §5.3, we will give a more straightforward deﬁnition. Since TΣ is deﬁned using the reduced Hecke algebra, TΣ itself is also called the reduced Hecke algebra or simply the Hecke algebra. The morphism of O-algebras fΣ : RΣ → TΣ is deﬁned as the surjective homomorphism induced by ΨΣ : RΣ → f ∈Φ(NΣ )K,ρ¯ Of . This is how we deﬁne RΣ , TΣ and fΣ . Let us look into the precise deﬁnition beginning in the next section.

122

5. R = T

5.2. Deformation rings In this section we deﬁne the deformation ring precisely. The existence theorem of the deformation ring will be given as Theorem 5.8, which will be proved in Chapter 7. First, we deﬁne the liftings of a representation. Let O be a complete discrete valuation ring, and F its residue ﬁeld. Definition 5.2. We say that ring A is a proﬁnitely generated complete local O-algebra if A is a complete local noetherian ring with a local homomorphism O → A such that the residue ﬁeld A/mA of A is a ﬁnite extension of the residue ﬁeld F . For a proﬁnitely generated complete local O-algebra A, let pA : A → FA be the natural surjection to the residue ﬁeld FA of A, and let iA : F → FA be the homomorphism induced by the local morphism O → A. A local morphism f : A → A of proﬁnitely generated complete local O-algebras induces a continuous homomorphism of topological groups GLn (f ) : GLn (A) → GLn (A ). For a continuous representation ρ : G → GLn (A) of a proﬁnite group G, the composite GLn (f )

ρ

representation G → GLn (A) −−−−−→ GLn (A ) is denoted by f∗ (ρ). The adjoint ad(P )(ρ) : G → GLn (A) of ρ : G → GLn (A) by P ∈ GLn (A) is deﬁned by ad(P )(ρ)(σ) = P ρ(σ)P −1 . Deﬁne U 1 GLn (A) = Ker(GLn (A) → GLn (FA )) = {M ∈ GLn (A) | M ≡ 1 mod mA }. Definition 5.3. Let G be a proﬁnite group, and ρ¯ : G → GLn (F ) a continuous representation. (1) Let A be a proﬁnitely generated complete local O-algebra. A continuous homomorphism ρ : G → GLn (A) is a lifting of ρ¯ to A if the following diagram is commutative.

(5.5)

G ⏐ ⏐ ρ

ρ¯

−−−−→

GLn (F ) ⏐ ⏐GL (i ) n A

GLn (A) −−−−−−→ GLn (FA ) GLn (pA )

(2) Let A be a proﬁnitely generated complete local O-algebra, and let ρ, ρ : G → GLn (A) be two liftings of ρ¯ to A. We say ρ and ρ are equivalent if there exists a matrix P ∈ U 1 GLn (A) such that ρ = ad(P )(ρ), and we denote it by ρ ∼ ρ .

5.2. DEFORMATION RINGS

123

The commutativity of diagram (5.5) means pA ∗ (ρ) = iA∗ (¯ ρ). If ρ : G → GLn (A) is a lifting of ρ¯ and P ∈ U 1 GLn (A), then the adjoint ad(P )(ρ) is also a lifting of ρ¯. If f : A → B is a local morphism of proﬁnitely generated complete local O-algebras, then f∗ preserves the equivalence of liftings. Let ρ, ρ : G → GLn (A) be liftings of ρ¯ to A. Recall that ρ and ρ are isomorphic if there is a matrix P ∈ GLn (A) satisfying ρ = ad(P )(ρ). If ρ and ρ are isomorphic, we write ρ ρ . Proposition 5.4. Let G be a proﬁnite group, and ρ¯ : G → GLn (F ) a continuous representation. Let A be a proﬁnitely generated complete local O-algebra, and let ρ, ρ : G → GLn (A) be two liftings of ρ¯ to A. If ρ¯ is absolutely irreducible and ρ and ρ are isomorphic, then ρ and ρ are equivalent. Proof. Suppose there exists P ∈ GLn (A) such that ρ = ad(P )(ρ). It suﬃces to show that there exists P1 ∈ U 1 GLn (A) such that ρ = ad(P1 )(ρ). Let FA be the residue ﬁeld of A, and let P be the image of P in GLn (FA ). Since ρ¯ is absolutely irreducible, P is a scalar matrix. Take a ∈ A× such that P = a ¯. Then, it suﬃces to let P1 = a−1 P . In the following, let G = GQ , let be an odd prime, let F be a ﬁnite extension of F , and let ρ¯ : GQ → GL2 (F) be a semistable irreducible mod representation. If det ρ¯ is the mod cyclotomic character, in particular if ρ¯ is modular, then by Proposition 3.5 ρ¯ is absolutely irreducible. We now give a precise formulation of the notion that a lifting ρ of ρ¯ “ramiﬁes to the same extent” as ρ¯ outside Σ. Definition 5.5. Let be an odd prime. Let O be the ring of integers of a ﬁnite extension of Q , and let F be its residue ﬁeld. Suppose ρ¯ : GQ → GL2 (F) is a semistable irreducible mod representation such that det ρ¯ : GQ → F× is the cyclotomic character. Deﬁne a ﬁnite set Sρ¯ as (5.6)

Sρ¯ = {p:prime | ρ¯ is not good at p}.

Let Σ be a ﬁnite set of primes satisfying the condition (5.7)

Σ ∩ Sρ¯ = ∅, and if ∈ Σ, ρ¯ is good and ordinary at .

Let ρ : GQ → GL2 (A) be a lifting of ρ¯ to a proﬁnitely generated complete local O-algebra A. We say ρ is of type DΣ if the following three conditions are satisﬁed.

124

5. R = T

(i) If p ∈ Sρ¯ ∪ Σ, then ρ is good at p. (ii) If p ∈ Sρ¯ or p = , then ρ is semistable at p. (iii) det ρ is the cyclotomic character. The conductor Nρ¯ of ρ¯ is p∈Sρ¯ p. The following is clear from the deﬁnition. Proposition 5.6. Let the notation be the same as in Deﬁnition 5.5. Let ρ : GQ → GL2 (O) be a semistable lifting of ρ¯ such that det ρ : GQ → O× is the cyclotomic character. Deﬁne Σ(ρ) = {p | ρ¯ is good at p, but ρ is not good at p}. Then Σ(p) satisﬁes the condition (5.7), and ρ is a lifting of ρ¯ of type DΣ(p) . Proposition 5.7. Let ρ¯ : GQ → GL2 (F) be a semistable irreducible mod representation such that det ρ¯ is the cyclotomic character. Let A be a proﬁnitely generated complete local O-algebra, and let ρ be a lifting of ρ¯ to A. (1) Let B be a proﬁnitely generated complete local O-algebra, and let f : A → B be a local morphism of local O-algebras. If ρ is a lifting of type DΣ , then so is f∗ (ρ). (2) Let P ∈ U 1 GLn (A). If ρ is a lifting of type DΣ , then so is its adjoint ad(P )(ρ). Proof. What we have to show is that if ρ is good at p = , then f∗ (ρ) is good at p = . Taking the quotient of A and B, respectively, by a power of the maximal ideal, we may assume that A and B are of ﬁnite order. Since B is ﬁnitely generated as an A-module, the ﬁnite GQ -module B 2 is a quotient of a ﬁnite direct sum of the ﬁnite GQ module A2 . Thus, the assertion follows from Corollary 3.42(1). If ρ is a lifting of ρ¯ to A of type DΣ , the adjoint of ρ by U 1 GLn (A) is also of type DΣ by Proposition 5.7(2). A U 1 GLn (A) adjoint class of liftings of ρ¯ of type DΣ is called a deformation of ρ¯ of type DΣ . The set of all the deformations of ρ¯ to A of type DΣ is denoted by Def ρ,D ¯ Σ (A). By Proposition 5.7(1), we have a mapping f∗ : Def ρ,D ¯ Σ (A) → Def ρ,D ¯ Σ (B). Theorem 5.8. Let be an odd prime, let O be the ring of integers of a ﬁnite extension of Q , and let F be the residue ﬁeld of O. Suppose ρ¯ : GQ → GL2 (F) is a semistable irreducible mod representation such that det ρ is the cyclotomic character. Let Σ be a ﬁnite set satisfying condition (5.7).

5.2. DEFORMATION RINGS

125

(1) There exist a proﬁnitely generated complete local O-algebra RΣ and a lifting of ρ¯ to RΣ of type DΣ that satisfy the following condition: For any proﬁnitely generated complete local O-algebra A, the mapping local morphism of local → Def ρ,D ¯ Σ (A) O-algebras RΣ → A (5.8) ∪ ∪ f → class of f∗ (ρΣ ) is bijective. (2) The subring of RΣ generated over O by the set ( ) Tr ρΣ (ϕp ) ∈ RΣ | p is prime, p ∈ Sρ ∪ Σ ∪ {} is dense in RΣ . (3) The residue ﬁeld of the local ring RΣ is F. The O-algebra RΣ is called the deformation ring over O of ρ¯ of type DΣ . RΣ satisﬁes condition (5.1). Corollary 5.9. If ρ is a lifting of ρ¯ to O, the following conditions are equivalent. (i) ρ is of type DΣ . (ii) There exists a local morphism of local O-algebras π : RΣ → O such that ρ and π∗ (ρΣ ) are equivalent. Corollary 5.10. Let O be the ring of integers of a ﬁnite extension of the ﬁeld of fractions K of O, and let F be its residue ﬁeld. If RΣ,O is the deformation ring over O of type DΣ of the composiρ¯

tion GQ → GL2 (F) → GL2 (F ), then there is a natural isomorphism RΣ,O ⊗O O RΣ,O . Proof. Since a proﬁnitely generated complete local O -algebra A is also a proﬁnitely generated complete local O-algebra, the maps {local morphism of local O-algebras : RΣ,O → A} → Def ρ,D ¯ Σ (A), {local morphism of local O -algebras : RΣ,O → A} → Def ρ,D ¯ Σ (A) are both bijective. By Theorem 5.8(3), RΣ,O ⊗O O is also a local ring. Since we have {local morphism of local O-algebras : RΣ,O → A} = {local morphism of local O -algebras : RΣ,O ⊗O O → A}, we obtain a natural isomorphism RΣ,O ⊗O O RΣ,O .

126

5. R = T

5.3. Hecke algebras In this section we deﬁne the Hecke algebra TΣ and a natural surjective homomorphism fΣ : RΣ → TΣ . Then, we formulate a theorem stating that fΣ : RΣ → TΣ is an isomorphism. We continue to assume is an odd prime and F is a ﬁnite extension of F . In the following, we assume that ρ¯ : GQ → GL2 (F) is a semistable irreducible modular mod representation. By Lemma 3.25(2), ρ¯ is absolutely irreducible. Definition 5.11. Let be an odd prime, and ρ¯ : GQ → GL2 (F) a semistable irreducible modular mod representation. Suppose a ﬁnite set of primes Σ satisﬁes condition (5.7). Deﬁne a positive integer NΣ by the formula 1 if ∈ Σ, (5.9) NΣ = Nρ × p2 × if ∈ Σ, p∈Σ,p= where Nρ¯ = p∈Sρ¯ p is the conductor of ρ¯. The reason of the deﬁnition of NΣ lies in the following proposition. Proposition 5.12. Let be an odd prime, let O be the ring of integers of a ﬁnite extension of Q , and let F be its residue ﬁeld. Suppose ρ¯ : GQ → GL2 (F) is a semistable irreducible modular mod representation. Then, the following conditions on a modular lifting ρ of ρ¯ to O are equivalent. (i) ρ is of type DΣ . (ii) ρ is modular of level NΣ . The implication (i) ⇒ (ii) explains the reason for the deﬁnition of NΣ , but it is not necessary for the proof of Theorem 5.1. This implication is a consequence of the compatibility of the global and local Langlands correspondences, and it is beyond the scope of this book. We show (ii) ⇒ (i) only. Proof of (ii) ⇒ (i). Let f be a primitive form with K coeﬃcients such that ρ is the -adic representation associated with f , and let N be its level. Then, N divides NΣ . Suppose p ∈ Sρ¯ ∪ Σ. Since p does not divide NΣ , p does not divide N either. Thus, by Theorem 3.52(1) (i) ⇒ (ii), ρ is good at p. Next, suppose p ∈ Sρ¯ or p = ∈ Σ. Then, since p2 does not divide NΣ , p2 does not divide N either. Thus, by Theorem 3.52(2) (ii) ⇒ (i), ρ is semistable at p.

5.3. HECKE ALGEBRAS

127

There is nothing to prove when p = , p ∈ Σ. Finally, since det ρ is the cyclotomic character, ρ is a lifting of type DΣ . Let O be the ring of integers of a ﬁnite extension K of Q , and let F be its residue ﬁeld. We deﬁned Φ(NΣ )K = Spec T (NΣ )K in Deﬁnition 2.62. As a set, it consists of all primitive forms over K whose level is a divisor of NΣ , and it is a ﬁnite set. For f ∈ Φ(NΣ )K , let Kf be its residue ﬁeld. Then, we have T (NΣ ) = f ∈Φ(NΣ )K Kf , and it is a reduced ring. For f ∈ Φ(NΣ )K , let Of be the ring of integers of Kf , and Ff = Of /λf its residue ﬁeld. Ff is a ﬁnite extension of the residue ﬁeld F of O. Definition 5.13. Deﬁne a subset Φ(NΣ )K,¯ρ of Φ(NΣ )K by (5.10)

{f ∈ Φ(NΣ )K | Tr ρ¯(ϕp ) ≡ ap (f ) mod λf if p NΣ }.

The congruence relation Tr ρ¯(ϕp ) ≡ ap (f ) mod λf is an equality as elements in Ff . The left-hand side is an element of F, and the right-hand side is the image of an element of Of in Ff . Proposition 5.14. Let Σ be a ﬁnite set of primes that satisﬁes condition (5.7). (1) Let f ∈ Φ(NΣ )K be a primitive form over K. The following conditions are equivalent. (i) f ∈ Φ(NΣ )K,¯ρ . (ii) If we take a suitable O-basis of the -adic representation associated with f , then ρf : GQ → GL2 (Of ) is a lifting of ρ¯. (2) Φ(NΣ )K,¯ρ is nonempty. Proof. (1) Condition (ii) implies that ρ¯ : GQ → GL2 (Ff ) ρf and ρ¯f : GQ −→ GL2 (Of ) → GL2 (Ff ) are isomorphic. Since ρ¯ is absolutely irreducible, by Proposition 3.4(3), it is equivalent to say that Trρ¯(ϕp ) = Trρ¯f (ϕp ) for all primes p NΣ . This is nothing but condition (i). (2) By Corollary 3.56, ρ¯ is modular of level Nρ¯. The assertion follows from the fact that NΣ is a multiple of Nρ¯. The next proposition will not be used for the proof of Theorem 5.1, and we omit the proof. Still, together with Corollary 5.18 below, we see that TΣ satisﬁes condition (5.2).

128

5. R = T

Proposition 5.15. Suppose ρ : GQ → GL2 (O) is a modular lifting of ρ¯, and f = fρ is a primitive form with K coeﬃcients such that ρ ρf . (1) The following conditions are equivalent. (i) ρ is a lifting of ρ¯ of type DΣ . (ii) f ∈ Φ(NΣ )K,¯ρ (K). (2) The mapping that sends the equivalence class of ρ to fρ equivalence classes of modular → Φ(NΣ )K,¯ρ (K) liftings of ρ¯ to O of type DΣ is bijective. For any positive integer n relatively prime to NΣ , the image of the Hecke operator Tn by the natural surjection T (NΣ )K → f ∈Φ(NΣ )K,ρ¯ Kf is also denoted by Tn . By Proposition 2.46, for each Φ(NΣ )K , the image an (f ) of Tn in Kf is an algebraic integer. Thus, it is contained in the ringof integers Of of Kf . Also, Tn ∈ f ∈Φ(NΣ )K,ρ¯ Kf is contained in Φ(NΣ )K,ρ¯ Of . Definition 5.16. The O-subalgebra of f ∈Φ(NΣ )K,ρ¯ Of generated over O by Tp for all primes p with p NΣ Of O Tp : p NΣ ⊂ f ∈Φ(NΣ )K,ρ¯

is called the reduced Hecke algebra over O of level NΣ , and denoted by TΣ . If we want to indicate O explicitly, we write TΣ,O . TΣ is often called the Hecke algebra simply. The Hecke algebra TΣ is a ﬁnitely generated free O-module. By Proposition 5.14(2), TΣ is not 0. As its name suggests, TΣ is a reduced ring. TΣ,K = TΣ ⊗O K has the following simple expression. Proposition 5.17. TΣ ⊗O K = f ∈Φ(NΣ )K,ρ¯ Kf . Proof. The left-hand side is a subring of the right-hand side that is generated over K by Tp for all primes p with p N Σ . Thus, it is the image of the composition T (NΣ )K → T (NΣ )K = f ∈Φ(NΣ )K Kf → The assertion now follows from the fact that f ∈Φ(NΣ )K,ρ¯ Kf . T (NΣ ) → T (NΣ ) is surjective, which is shown in Corollary 2.60. For a primitive form f ∈ Φ(NΣ )K,ρ (K) over K, that is, a primitive form f ∈ Φ(NΣ )K,ρ with K coeﬃcients, deﬁne a morphism of O-algebras πf : TΣ → O as the composition of the inclusion

5.3. HECKE ALGEBRAS

129

TΣ → g Og and the projection to the f component g Og → Of = O. We have πf (Tp ) = ap (f ) for p NΣ . Corollary 5.18. The mapping that sends f ∈ Φ(NΣ )K,¯ρ (K) to π f : TΣ → O Φ(NΣ )K,¯ρ (K) ∪ f

→ {morphisms of O-algebras TΣ → O} ∪ → πf

is bijective. Proof. Since TΣ is ﬁnitely generated as an O-module, morphisms of O-algebras TΣ → O are in one-to-one correspondence with morphisms of K-algebras TΣ,K → K. It follows from Proposition 5.17 that morphisms of K-algebras TΣ,K → K are in one-to-one correspondence with primitive forms f ∈ Φ(NΣ )K,¯ρ (K) with K-coeﬃcients. The morphism corresponding to f is the projection to the f component. Corollary 5.19. Let O be the ring of integers of a ﬁnite extension of K. Suppose TΣ,O is the reduced Hecke algebra over O of level NΣ . Then the natural mapping TΣ,O ⊗O O → TΣ,O is an isomorphism. are, respectively, O -subalgebras Proof. TΣ,O ⊗O O and TΣ,O of f ∈Φ(NΣ )K,ρ¯ Kf ⊗K K and of f ∈Φ(NΣ )K ,ρ¯ Kf generated by Tp , p NΣ . By Deﬁnition 5.13, we have Φ(NΣ )K,¯ρ ⊗K K = Φ(NΣ )K ,¯ρ , from which the assertion follows.

We deﬁne a surjection fΣ : RΣ → TΣ . Suppose f ∈ Φ(NΣ )K,¯ρ . By Proposition 5.14(1), ρf : GQ → GL2 (Of ) is a lifting of ρ¯ if we take a suitable basis of the representation associated with f . Then, by Proposition 5.12 (ii) ⇒ (i), ρf is a lifting of type DΣ . Since ρ¯ is absolutely irreducible, it follows from Proposition 5.4 that the equivalence class of the lifting ρf of ρ¯ does not depend on the choice of the basis. Thus, by the deﬁnition of RΣ , there is a unique local morphism of local O-algebras ψf : RΣ → Of such that ρf ∼ ψf ∗ (ρΣ ). Proposition 5.20. Let ΨΣ : RΣ → f ∈Φ(NΣ )K,ρ¯ Of be the product of ψf : RΣ → Of . Then, the image of ΨΣ is TΣ . Proof. By Theorem 5.8(2), The O-subalgebra of RΣ generated by the traces of Frobenius substitutions {Tr(ρΣ (ϕp )) | p ∈ Sρ¯∪Σ∪{}} is dense. By the deﬁnition of NΣ , p ∈ Sρ¯ ∪ Σ ∪ {} is equivalent to

130

5. R = T

p NΣ for any prime p. For a prime p NΣ and a primitive form f ∈ Φ(NΣ )K,¯ρ , we have Tr(ρf (ϕp )) = ap (f ), and thus the image of ΨΣ by Tr(ρΣ (ϕp )) is Tp . Thus, the image of RΣ is TΣ . Let fΣ : RΣ → TΣ be the surjective homomorphism induced by ΨΣ : RΣ → f ∈Φ(NΣ )K,ρ¯ Of . RΣ is a local ring whose residue ﬁeld is F. Since TΣ is a nonzero ring, TΣ is also a local ring whose residue ﬁeld is F. : GQ → GL2 (TΣ ) by fΣ ∗ (ρΣ ). Let f ∈ Deﬁne a lifting ρmod Σ Φ(NΣ )K,¯ρ be a primitive form over K, and let πf : TΣ → Of be a homomorphism determined by the projection to f component. Then πf ∗ (ρmod Σ ) = ψf ∗ (ρΣ ) is equivalent to the -adic representation ρf : GQ → GL2 (Of ) associated with f . Proposition 5.21. Let ρ be a lifting of ρ¯ to O of type DΣ . Then the following conditions are equivalent. (i) ρ is modular of level NΣ . (i ) ρ is modular. (ii) There exists a morphism of O-algebras π : TΣ → O such that ρ and π∗ (ρmod Σ ) are equivalent. By Proposition 5.12, (i) and (i ) are equivalent. The implication (i ) ⇒ (i) is not used for the proof of Theorem 5.1.

Proof. (i) ⇒ (ii). Let ρ be a lifting of ρ¯ to O that is modular of level NΣ . If we take a primitive form f with K coeﬃcients such that ρ ρf , then by Proposition 5.14 (ii) ⇒ (i), we have f ∈ Φ(NΣ )K,¯ρ (K). If we take the morphism of O-algebras πf as π : TΣ → O in (ii), then ρ and π∗ (ρmod Σ ) ∼ ρf are equivalent. (ii) ⇒ (i). Let π : TΣ → O be a morphism of O-algebras. By Corollary 5.18, there exists a primitive form f ∈ Φ(NΣ )K,¯ρ such that π = πf . Since πf ∗ (ρmod Σ ) = ψf ∗ (ρΣ ) ∼ ρf is an -adic representation associated with f , ρ is modular of level NΣ Theorem 5.22. Let be an odd prime, let O be the ring of integers of a ﬁnite extension K of Q , and let F be the residue ﬁeld of O. Suppose ρ¯ : GQ → GL2 (F) is a semistable irreducible modular mod representation. If Σ is a ﬁnite set of primes that satisﬁes condition (5.7), then the natural surjection fΣ : RΣ → TΣ is an isomorphism.

5.4. SOME COMMUTATIVE ALGEBRA

131

After some preparations in §§5.4–5.5, we will show an outline of proof of the fact that fΣ is an isomorphism in §5.6. Here, we deduce Theorem 5.1 from Theorem 5.22. Proof of Theorem 5.22 ⇒ Theorem 5.1. Suppose ρ : GQ → GL2 (O) is a semistable lifting of ρ¯ whose determinant det ρ is the cyclotomic character. As in Proposition 5.6, deﬁne Σ = Σ(ρ) = {p | ρ¯ is good at p, but ρ is not good at p}. Then, ρ is a lifting of ρ¯ of type DΣ(ρ) . By Corollary 5.9, there exists a unique local morphism of local O-algebras π : RΣ → O such that ρ is equivalent to π∗ (ρΣ ). By Theorem 5.22, f : RΣ → TΣ is an isomorphism, and thus ρ is equivalent to (π ◦ fΣ−1 )∗ (ρmod Σ ). Then, by Proposition 5.21(ii) ⇒ (i), ρ is modular of level NΣ . Furthermore, by Corollary 3.53, ρ is modular of level Nρ . 5.4. Some commutative algebra Theorem 5.22 will be proved using some theorems of commutative algebra. In this section we state two theorems of commutative algebra, which will be proved then in Chapter 6. We begin with some necessary deﬁnitions. Definition 5.23. Let O be a complete discrete valuation ring, and let T be a local O-algebra. Suppose T is a ﬁnitely generated free O-module, and the residue ﬁeld of T is the same as that of O. We say T is a complete intersection if there exist a positive integer n and f1 , . . . , fn ∈ O[[T1 , . . . , Tn ]] such that T is isomorphic to the quotient ring O[[T1 , . . . , Tn ]]/(f1 , . . . , fn ) as an O-algebra. The important point of this deﬁnition is that the number of variables is the same as the number of the generators of the ideal. Definition 5.24. Let O be a complete discrete valuation ring. Let R be a proﬁnitely generated complete local O-algebra whose residue ﬁeld equals that of O, and let M be an R-module. Suppose O is a proﬁnitely generated complete local O-algebra with a local morphism of local O-algebras o : O → O given. A lifting of the pair (R, M ) along (O , o ) is a quintuple R = (R , M , a : O → R , r : R → R, m : M → M ) satisfying the following four conditions. (i) R is a proﬁnitely generated complete local O-algebra. (ii) M is an R -module.

132

5. R = T

(iii) a : O → R and r : R → R are local morphisms of local O-algebras that satisfy the following condition. (5.11)

The diagram

a

O −−−−→ ⏐ ⏐ o

R ⏐ ⏐ r

O −−−−→ R is commutative, and the morphism R ⊗O O → R is an isomorphism. (iv) m : M → M is a morphism of additive groups that satisﬁes the following condition. (5.12)

For a ∈ R and x ∈ M , m (ax) = r (a)m (x), and M ⊗R R → M is an isomorphism.

For a ﬁnite group G, we denote by O[G] the group algebra of G over O. The morphism of O-algebras O[G] → O; σ → 1 (σ ∈ G) is called the augmentation. If O is a complete discrete valuation ring whose residue ﬁeld is of characteristic , and G is a ﬁnite abelian group of order a power of , then O[G] is a local O-algebra and the augmentation O[G] → O is a local morphism of local O-algebras. Theorem 5.25. Let be a prime, and let O be the ring of integers of a ﬁnite extension of Q . Suppose R is a proﬁnitely generated complete local O-algebra whose residue ﬁeld is equal to the residue ﬁeld F = O/mO and M is a nonzero R-module. Then the following conditions are equivalent. (i) R is a ﬁnitely generate free O-module. The ring R is a complete intersection. Moreover, the R-module M is ﬁnitely generated and free. (ii) Let mR be the maximal ideal of R, and r the dimension of the Fvector space mR /(mO R + m2R ). For any positive integer n, there exists a lifting Rn = (Rn , Mn , an , rn , mn ) of the pair (R, M ) along the pair (On , on ) of the group algebra On = O[(Z/n Z)r ] and the augmentation on that satisﬁes the following two conditions. (5.13)

If mRn is the maximal ideal of Rn , then we have

dimF mRn /(mO Rn + m2Rn ) = r = dimF mR /(mO R + m2R ). (5.14)

Mn is a ﬁnitely generated free On -module.

5.4. SOME COMMUTATIVE ALGEBRA

133

The important part of the conditions of the theorem is that the dimension r of mRn /(mO Rn + m2Rn ) is the same r that appears in the deﬁnition of the group algebra On . This corresponds to the condition of Deﬁnition 5.23 in which the number of variables is the same as the number of generators of the ideal. Definition 5.26. Let O be a complete discrete valuation ring, and let n be a positive integer. (1) A quintuple (R, T, M, f, π) consisting of the following data is called in this book an RTM-triple over O of rank n. (i) R is a proﬁnitely generated complete local O-algebra. (ii) T is a local O-algebra that is ﬁnitely generated and free as an O-module. (iii) M is a T -module that is ﬁnitely generated and free as an O-module. MK = M ⊗O K is a free TK = T ⊗O K-module of rank n. (iv) f : R → T is a surjective local morphism of local Oalgebras. (v) π : T → O is a surjective local morphism of local Oalgebras. If we let pT = Ker π, then the O-module pT /p2T = pT ⊗T O is of ﬁnite length. (2) An RTM-triple (R, T, M, f, π) over O of rank n is complete if the following conditions are satisﬁed. (i) f : R → T is an isomorphism. (ii) R T is a complete intersection. (iii) M is a free T -module (of rank n). (3) Let R = (R, T, M, f, π) and R = (R , T , M , f , π ) be RTMtriples over O of rank n. A surjection of RTM-triples F : R → R is a quadruple F = (r : R → R , t : T → T , m : M → M , m∨ : M → M ) satisfying the following conditions. (i) r : R → R is a surjective local morphism of local Oalgebras. (ii) t : T → T is a surjective local morphism of local Oalgebras that makes the following diagram commutative. f

(5.15)

R −−−−→ ⏐ ⏐ r

π

T −−−−→ O ⏐ ⏐ || t

R −−−− → T −−−− → O f

π

134

5. R = T

(iii) m : M → M and m∨ : M → M are morphisms of Rmodules satisfying the following condition. (5.16)

If we regard M as a T -module via t, then m and m are T -linear. m is surjective, m∨ is injective, and the cokernel of m∨ is a ﬁnitely generated free O-module.

Let F = (r, t, m, m∨ ) be a surjection of RTM-triples from R = (R, T, M, f, π) to R = (R , T , M , f , π ). An element Δ ∈ T is a multiplier of F if the composition m ◦ m∨ : M → M → M is the multiplication-by-Δ mapping. Theorem 5.27. Let O be a complete discrete valuation ring, and n a positive integer. Let R = (R, T, M, f, π),

R = (R , T , M , f , π )

be RTM-triples over O of rank n. Let F = (r, t, m, m∨ ) : R → R be a surjection of RTM-triples, and let Δ ∈ T be its multiplier. Let pR = Ker π ◦ f , pR = Ker π ◦ f . If R is complete, π (Δ) = 0 and the inequality (5.17)

lengthO (pR /p2R ) − lengthO (pR /p2R ) ≤ ordO π (Δ)

holds, then R is also complete. We introduce two propositions that will be used in the proof of Theorem 5.22. Proposition 5.28. Let O be a complete discrete valuation ring, and let R = (R, T, M, f, π) be an RTM-triple of rank n > 0. If R is a complete intersection and M is a free R-module, then R is complete. Proof. It suﬃces to show that f : R → T is injective. Since f M is nonzero and it is a free R-module, the composition R → T → EndO (M ) is injective. Thus, f : R → T is also injective. The next proposition will be used to show that the full Hecke algebra, which will be deﬁned in Deﬁnition 10.2 in §10.1 in Chapter 10,1 is isomorphic to the reduced Hecke algebra. This is also an important step of the proof of Theorem 5.22. 1 Please recall that Chapters 8, 9, and 10 along with Appendices B, C, and D will appear in Fermat’s Last Theorem: The Proof, a forthcoming translation of the Japanese original.

5.5. HECKE MODULES

135

Proposition 5.29. Let O be a complete discrete valuation ring, and let R = (R, T, M, f, π) be an RTM-triple of rank n > 0. Let T be a local O-algebra that is a ﬁnitely generated free O-module. If a local morphism of local O-algebras i : T → T satisﬁes the following condition, then it is an isomorphism. (5.18)

= T ⊗O K The mapping iK : TK = T ⊗O K → TK induced by i is an isomorphism. M is a T -module, and if x ∈ M , t ∈ T , then tx = i(t)x.

Proof. By assumption, M is a free T -module of rank n, and T is a ﬁnitely generated free O-module. Thus, EndT (M ) Mn (T ) is also a ﬁnitely generated free O-module. Let j be the homomorphism of rings T → EndT (M ); t → (x → t x). Since MK is a ﬁnitely is an isomorphism, generated free TK -module, and iK : TK → TK jK : TK → EndTK (MK ) is an isomorphism to the subring TK of EndTK (MK ). Since T is also a free O-module, j : T → EndT (M ) is injective, and its image is contained in TK ∩ EndT (M ) = T . Thus, i : T → T is an isomorphism. 5.5. Hecke modules In the remaining sections of this chapter, we will illustrate how to deduce Theorem 5.22 from Theorems 5.25 and 5.27. More detail will be provided in Chapter 10. Logically speaking, the contents of the remaining two sections should be included in Chapter 10. What is described here will not be used until Chapter 9, and there is no fear of a circular argument. In the case Σ = ∅, Theorem 5.22 is proved using Theorem 5.25. A lifting ρ of ρ¯ is of type D∅ if the ramiﬁcation of ρ is the same extent as that of ρ¯. In other words, the ramiﬁcation of ρ is so good that it cannot be improved any further. Thus, we call this case the minimal case. For a general Σ, we use Theorem 5.27 to prove by induction on the number of elements of Σ. In any case, the ﬁrst thing we do is to construct, for each Σ, an RTM-triple of rank 2, RΣ = (RΣ , TΣ , MΣ , fΣ , πΣ ), and for each Σ = Σ {p}, a surjection of RTM-triples FΣ ,Σ : RΣ → RΣ . We have already deﬁned RΣ , TΣ , fΣ in §§5.2–5.3. We need to show that fΣ is an isomorphism, but, by introducing TΣ -module MΣ and showing a stronger Theorem 5.31, which will be introduced in the next section, the induction rolls smoothly.

136

5. R = T

We begin by deﬁning the ring homomorphism πΣ : TΣ → O. As we showed in Proposition 5.14(2), the subset Φ(N∅ )K,¯ρ ⊂ Φ(NΣ )K,¯ρ is not empty, and so we take a primitive form f ∈ Φ(N∅ )K,¯ρ over K. Take an extension of scalars so that f has K coeﬃcients, and deﬁne πΣ as the ring homomorphism πf : TΣ → O corresponding to f by Corollary 5.18. By Corollaries 5.10 and 5.19, we may replace K by its ﬁnite extension in order to show that fΣ is an isomorphism. The TΣ -module MΣ is deﬁned using the singular cohomology group H 1 (X0 (NΣ )(C), Z) of the modular curve X0 (NΣ )(C) over C as follows. We deﬁne a Z-structure T (NΣ )Z of the Hecke algebra T (NΣ )Q of level NΣ . Corresponding to a mod representation ρ¯, a maximal ideal m = mρ¯ of the tensor product T (NΣ )Z ⊗Z O. The completion with respect to this maximal ideal is denoted by T (NΣ )m , and we call it a full Hecke algebra. T (NΣ )m is a ﬁnitely generated free O-module. Using a theory of new forms, we deﬁne the local morphism of local O-algebras iΣ : TΣ → T (NΣ )m . If we take the extension of scalars of iΣ to K, then iΣ,K : TΣ ⊗O K → T (NΣ )m ⊗O K is an isomorphism. Once Theorem 5.22 is proved, it follows from Proposition 5.29 that iΣ : TΣ → T (NΣ )m is an isomorphism. The singular homology group H1 (X0 (XΣ )(C), Z) of the modular curve X0 (NΣ )(C) is naturally a T (NΣ )Z -module. Thus, the completion of the tensor product H1 (X0 (XΣ )(C), Z) ⊗Z O at the maximal ideal m is a module over the full Hecke algebra T (NΣ )m . We denote it by MΣ . By the ring homomorphism iΣ : TΣ → T (NΣ )m , we regard MΣ as a TΣ -module. MΣ is called the Hecke module. The fact that MΣ,K = MΣ ⊗O K is a free TΣ,K = TΣ ⊗O K-module of rank 2 follows from the Eichler–Shimura isomorphism and the isomorphism iΣ,K : TΣ ⊗O K → T (NΣ )m ⊗O K. An RTM-triple RΣ of rank 2 is thus constructed. Suppose Σ ⊃ Σ . A surjection of RTM-triples FΣ ,Σ = (rΣ ,Σ , tΣ ,Σ , mΣ ,Σ , m∨ Σ ,Σ ) : RΣ → RΣ is deﬁned as follows. We deﬁne rΣ ,Σ : RΣ → RΣ . ρΣ is a lifting of ρ¯ of type DΣ . Since a lifting of type DΣ is a lifting of type DΣ , a local morphism of local O-algebra rΣ ,Σ : RΣ → RΣ such that ρΣ ∼ rΣ ,Σ ∗ (ρΣ ) is determined by the deﬁnition of RΣ . tΣ ,Σ : TΣ → TΣ is deﬁned as follows. We have Φ(NΣ )K,¯ρ ⊃ Φ(NΣ )K,¯ρ . The natural surjection Of → Of f ∈Φ(NΣ )K,ρ¯

induces tΣ ,Σ : TΣ → TΣ .

f ∈Φ(NΣ )K,ρ¯

5.6. OUTLINE OF THE PROOF OF THEOREM 5.22

137

mΣ ,Σ : MΣ → MΣ is deﬁned using the maps on the homology groups induced by some natural maps of modular curves X0 (NΣ ) → X0 (NΣ ). m∨ Σ ,Σ : MΣ → MΣ is deﬁned as the dual of mΣ ,Σ : MΣ → MΣ . By the Poincar´e duality, H1 (X0 (NΣ )(C), Z) is self-dual, and so MΣ is also self-dual. The surjectivity of rΣ ,Σ and tΣ ,Σ and the commutativity of the diagram (5.15) follow easily from the deﬁnition of rΣ ,Σ and tΣ ,Σ and Proposition 5.20. The TΣ -linearity of mΣ ,Σ and m∨ Σ ,Σ follows from the deﬁnition. ∨ Since m∨ Σ ,Σ is the dual of mΣ ,Σ , the fact that mΣ ,Σ is injective and its cokernel is a ﬁnitely generated free O-module is equivalent to the fact that mΣ ,Σ is surjective. Thus, the fact that FΣ ,Σ is a surjection of RTM-triples follows from the following proposition. Proposition 5.30. mΣ ,Σ : MΣ → MΣ is surjective. We will prove this in §10.3 by relating the singular homology group H1 (X0 (NΣ )(C), Z) to the abelianization of the congruence subgroup Γ(NΣ ) ⊂ SL2 (Z). 5.6. Outline of the Proof of Theorem 5.22 With the preparation we have done so far, let us see how Theorem 5.22 is proved. As a matter of fact, it is the following theorem that we will prove. Theorem 5.31. Let the notation and assumptions be the same as in Theorem 5.22. Suppose, furthermore, that there exists a primitive form f ∈ Φ(NΣ )K,¯ρ (K) with K coeﬃcients. Let Σ be a ﬁnite set satisfying the condition (5.7), and deﬁne the RTM-triple RΣ = (RΣ , TΣ , MΣ , fΣ , πΣ ) as in the previous section. Then the RTM-triple RΣ is complete. If an RTM-triple RΣ = (RΣ , TΣ , MΣ , fΣ , πΣ ) is complete, then fΣ : RΣ → TΣ is an isomorphism. As we noted in the previous section, we may replace K by its ﬁnite extension. Thus, Theorem 5.22 follows from Theorem 5.31. By Proposition 5.29, the fact that the full Hecke algebra is equal to the reduced Hecke algebra follows from Theorem 5.31. We will prove Theorem 5.31 in §10.7. In the minimal case, we prove Theorem 5.31 by applying Theorem 5.25(ii) ⇒ (i) to the pair (R∅ , M∅ ). Since TΣ is reduced, if

138

5. R = T

the pair (R∅ , M∅ ) satisﬁes condition (i) in Theorem 5.25, then the RTM-triple R∅ is complete by Proposition 5.28. In order to use Theorem 5.25, we construct a lifting of the pair (R∅ , M∅ ) along (On , on ) for each positive integer n ≥ 1 as follows. Let Q = {q1 , . . . , qr } be a set of r primes. Let ni be the -adic valuation of qi − 1, and ΔQ = ri=1 Z/ni Z. Let OQ = O[ΔQ ] be the group algebra, and let oQ : OQ → O be the augmentation. Suppose each element qi ∈ Q satisﬁes the condition (5.19)

qi ∈ Sρ¯,

qi ≡ 1 mod ,

and

Tr ρ¯(ϕqi ) = ±2.

Replacing O by a suitable unramiﬁed quadratic extension, we may assume (Tr ρ¯(ϕqi ))2 − 4 ∈ (F× )2 . By condition (5.19), we have (Sρ¯ ∪{})∩Q = ∅. Studying the ramiﬁcation at each qi , we construct a local morphism of local O-algebras aQ : OQ → RQ such that the diagram aQ

OQ −−−−→ ⏐ ⏐ oQ

RQ ⏐ ⏐r∅,Q

O −−−−→ R∅ satisﬁes condition (iii). Condition (5.12) follows from Proposition 5.30, (5.16), and Theorem 5.32(2) below. The quintuple RQ = (RQ , MQ , aQ , rQ , mQ ) is a lifting of (R∅ , M∅ ) along (OQ , oQ ). Let n ≥ 1 be a positive integer, and suppose each element qi in Q satisﬁes the condition (5.19)n

qi ∈ Sρ¯,

qi ≡ 1 mod n ,

and

Tr ρ¯(ϕqi ) = ±2.

Let OQ→ On be the mapping induced by the natural surjection ΔQ = ri=1 Z/ni Z → (Z/n Z)r . Deﬁne RQ,n = (RQ,n , MQ,n , aQ,n , rQ,n , mQ,n ) to be R ⊗OQ On . In other words, we let RQ,n = RQ ⊗OQ On , MQ,n = MQ ⊗OQ On , and we let aQ,n : On → RQ,n , rQ,n : RQ,n → R, and mQ,n : MQ,n → M , the maps induced by aQ , rQ and mQ , respectively. The quintuple RQ,n is a lifting (R∅ , M∅ ) along (On , on ). In order to apply Theorem 5.25, it suﬃces to ﬁnd a set of r prime numbers Qn satisfying (5.19)n for each integer n ≥ 1 such that the lifting RQn ,n satisﬁes conditions (5.13) and (5.14). The existence of such Qn follows from the next theorem. The proof of Theorem 5.32 is the heart of the proof of the minimal case of Theorem 5.31. We will prove Theorem 5.32 in §11.6 and §10.6.

5.6. OUTLINE OF THE PROOF OF THEOREM 5.22

139

Theorem 5.32. (1) Let n be a positive integer. There exists a set of r primes Qn that satisﬁes (5.19)n and the condition (5.20)

F-vector space mRQn /(mO RQn +m2RQn ) is r dimensional.

(2) Let Q be a set of r prime numbers satisfying (5.19). Regard the RQ -module MQ as an OQ -module via the ring homomorphism aQ . Then, MQ is a free OQ -module. For each positive integer n ≥ 1, take Qn as in Theorem 5.32(1). Then the complete local ring RQn ,n satisﬁes condition (5.13) in Theorem 5.25. Moreover, by Theorem 5.32(2), the OQn ,n -module MQn ,n satisﬁes (5.14). Thus, condition (ii) in Theorem 5.25 is satisﬁed. It now follows from Theorem 5.25 that R∅ is complete. Thus, the minimal case is proved. Next we describe the outline of Theorem 5.32. This is the content of Chapters 10 and 11. It follows from the deﬁnition of the deformation ring that the F-vector space mRQn /(mO RQn + m2RQn ) can be identiﬁed with the dual of the group deﬁned by the Galois cohomology of Q called the Selmer group of ad0 ρ¯. Theorem 5.32(1) is proved using the properties of Galois cohomology of Q derived from class ﬁeld theory and Theorem 3.1, and a part of p-adic Hodge theory related to ﬁnite ﬂat commutative group schemes. Theorem 5.32(2) is proved by studying the homology of modular deﬁned curves. In the actual proof, we use the Hecke module MQ using a modular curve diﬀerent from the ones we described in the previous section. We will deﬁne it in §10.5. We prove it by expressing the action of the Galois group on the homology by a geometric operator on a modular curve called the diamond operator, similar to the Hecke operator. We prove it by translating an arithmetic problem of Galois action to a problem in topology. For a general Σ, we prove it by induction on the number of elements in Σ using Theorem 5.27. Assuming Theorem 5.31 is proved for Σ , it suﬃces to show in the case Σ = Σ {p}. The strategy of proof may be summarized as follows. Suppose RΣ TΣ . RΣ and TΣ are obtained by inﬂating RΣ and TΣ respectively. We prove that RΣ TΣ by showing that the extent of inﬂation is the same. The actual proof goes as follows. As with the induction hypothesis, we assume the RTM-triple RΣ is complete. In view of Theorem 5.27, it suﬃces to show that the surjection of RTM-triples FΣ ,Σ : RΣ → RΣ satisﬁes the assumptions of Theorem 5.27. By

140

5. R = T

Proposition 5.29, we have T (NΣ )m = TΣ . If ∈ Sρ¯ ∪ Σ , then T ∈ T Σ . Proposition 5.33. Suppose p ∈ Sρ¯ ∪ Σ . Deﬁne Σ = Σ ∪ {p}, and (5.21)

Δ = (p − 1) Tp2 − (p + 1)2 ∈ TΣ .

Then, there exists an invertible element u ∈ TΣ× such that the composition mΣ ,Σ ◦m∨ Σ ,Σ : MΣ → MΣ → MΣ is the multiplication-by-u·Δ mapping. We prove this in Proposition 10.14 in §10.2. The proof is an explicit calculation of the mapping. This proposition says the Δ in (5.21) is the multiplier of FΣ ,Σ . We can verify the hypothesis of Theorem 5.27 as follows. Since πΣ : TΣ → O is determined by f with K coeﬃcients, we have πΣ (Δ) = the primitive form (p − 1) ap (f )2 − (p + 1)2 . This is nonzero by Theorem 2.47. Thus, what remains to show is inequality (5.17). Since the natural mapping pRΣ /p2RΣ → pRΣ /p2RΣ is surjective, what we need to show is reduced to the following theorem. Theorem 5.34. We have the inequality (5.22)

lengthO Ker(pRΣ /p2RΣ → pRΣ /p2RΣ ) ≤ ordO πΣ (Δ).

Theorem 5.34 will be proved in §11.5. As in the proof of Theorem 5.32(1), the modules pRΣ /p2RΣ and pRΣ /p2RΣ are identiﬁed with the dual of the group deﬁned by the Galois cohomology group of Q called the Selmer group of ad0 ρ. The group appearing in the lefthand side of (5.22) can be expressed by the Galois cohomology group of Qp . The proof of (5.22) is easier if p = , but in the case of p = , the proof uses a part of p-adic Hodge theory as we did for the proof of Theorem 5.32(1). This is the sketch of the proof of Theorem 5.22. We now summarize the proof of Theorem 5.1. To show the relation between Galois representations and modular forms (Theorem 5.1) it suﬃces to show that the natural homomorphism fΣ : RΣ → TΣ from the deformation ring to the Hecke algebra is an isomorphism. This is reduced to showing some properties of (5.23) and (5.24) below by introducing the Hecke module MΣ , and applying theorems in

5.6. OUTLINE OF THE PROOF OF THEOREM 5.22

141

commutative algebra (Theorems 5.25 and 5.27): (5.23)

F-vector space mRΣ /(mO RΣ + m2RΣ ), O-module Ker(pRΣ /p2RΣ → pRΣ /p2RΣ ), etc.

(5.24)

O[ΔQ ]-module MQ ,

the composition mΣ ,Σ ◦ m∨ Σ ,Σ : MΣ → MΣ → MΣ , etc. The objects in (5.23) are deﬁned by the deformation rings, and they are incarnations of Galois representations. These can be expressed in terms of the group called the Selmer group, and grasped as Galois cohomology. The key ingredients to the proof in the minimal case are global number theory such as class ﬁeld theory, and p-adic Hodge theory. Local number theory suﬃces to show the inductive step. The objects in (5.24) are deﬁned by the singular homology group of modular curves, and they are incarnations of modular forms. They are studied by using geometric operators on the homology group of modular curves, such as Hecke operators and diamond operators. The proof is done by reducing it to a problem of topology in the end. Last, we mention the similarity and diﬀerence between Iwasawa theory and the proof of Theorem 5.25, which is used in the minimal case of Theorem 5.22 and will be proved in the next chapter. An apparent similarity is the use of the limit of group algebra and the formal power series rings. Then, by using some good properties as a commutative ring of the formal power series rings, the arguments go smoothly. On the other hand, while in the case of Iwasawa theory, the group algebras form a projective system naturally, which is not the case in the proof of Theorem 5.25. In Iwasawa theory, a Zp -extension is given in the ﬁrst place and the groups Z/pn Z form a projective system naturally. To the contrary, for the set of primes Qn obtained from Theorem 5.32(1), there is no reason for the group algebras OQn to form a projective system. In fact, if we take the projective limit of this sequence of group algebras, things do not go as well. Nevertheless, we can still construct a formal power series ring by taking a suitable projective limit, and prove Theorem 5.25.

10.1090/mmono/243/07

CHAPTER 6

Commutative algebra In this chapter we prove Theorems 5.25 and 5.27. Let n be a positive integer, let O be a complete discrete valuation ring, and let F be its residue ﬁeld. 6.1. Proof of Theorem 5.25 Proposition 6.1. Let O be a complete discrete valuation ring, and let F be its residue ﬁeld. Let R be a proﬁnitely generated local O-algebra. Suppose its residue ﬁeld R/mR equals F . (1) There exists a surjective homomorphism O[[X1 , . . . , Xr ]] → R, where r = dimF mR /(mO R + m2R ). (2) Suppose R is a ﬁnitely generated free O-module and is a complete intersection. If O[[X1 , . . . , Xn ]] → R is a surjective Ohomomorphism, the kernel is generated by n elements. It is easy to see (1) follows from Nakayama’s lemma. We will use (2) only for the proof of Theorem 5.25 (i) ⇒ (ii), and we omit the proof. Using Proposition 6.1, we show Theorem 5.25 (i) ⇒ (ii). As a matter of fact, (i) ⇒ (ii) holds for a general complete discrete valuation ring. Proof of Theorem 5.25(i) ⇒ (ii). Let R be a proﬁnitely generated local O-algebra whose residue ﬁeld equals F . Suppose R is a ﬁnitely generated free O-module and is a complete intersection. By Proposition 6.1(1), there exists a surjective homomorphism r˜ : O[[X1 , . . . , Xr ]] → R. Moreover, by (2), its kernel is generated * = O[[S1 , . . . , Sr ]] by r elements f1 , . . . , fr ∈ O[[X1 , . . . , Xr ]]. Let O * * * and R = O[[X1 , . . . , Xr ]], and deﬁne O → R by Si → fi . Since O = * * O = O[[X1 , . . . , Xr ]]/(f1 , . . . , fr ) R. O/(S 1 , . . . , Sr ), we have R ⊗O Let M be a free O-module of rank N , and identify M with RN . Let + = R *N , and deﬁne m + = R *N → M = RN as r*N . For any M * : M 143

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* → On = O[(Z/n Z)r ] by Si → σi − 1, positive integer n, deﬁne O * ⊗O On , where σi , i = 1, . . . , r, is a basis of (Z/n Z)r . Let Rn = R + ⊗O On , an : On → Rn the natural mapping, and rn , mn Mn = M the surjections induced by r*. Then, (Rn ) = (Rn , Mn , an , rn , mn ) is a lifting along (On , on ) that satisﬁes conditions (5.13) and (5.14). We show Theorem 5.25(ii) ⇒ (i) using a theorem of commutative algebra (Theorem 6.7), and taking a suitable projective limit. Before going into the proof, we give a summary on completed group algebras of the group Zr . Completed group algebras are treated in §10.1(d) in Chapter 10 of Number Theory 3 . Let O be the ring of integers of a ﬁnite extension of Q , and let r be a positive integer. The projective limit limn O[(Z/n Z)r ] of the group algebras O[(Z/n Z)r ] ←− is called the completed group algebra of Zr , and denoted by O[[Zr ]]. The projective limit of the augmentations O[(Z/n Z)r ] → O determines the augmentation O[[Zr ]] → O. We also have O[[Zr ]] = limn (O/n O)[(Z/n Z)r ]. ←− Proposition 6.2. Let σi ∈ Zr ⊂ O[[Zr ]] be a basis of Zr . Then, the morphism O[[S1 , . . . , Sr ]] → O[[Zr ]];

Si → σi − 1

is an isomorphism of complete local O-algebras. Proof. Since the morphism O[[S1 , . . . , Sr ]] → O[(Z/n Z)r ]; Si → σi − 1 is surjective, its projective limit O[[S1 , . . . , Sr ]] → O[[Zr ]]; Si → σi − 1 is also surjective. To show the injectivity, it suﬃces to use the following lemma. Lemma 6.3. The kernel I of O[[S1 , . . . , Sr ]]→ (O/n O)[(Z/n Z)r ]; Si → σi − 1 is contained in mn , where m is the maximal ideal of O[[S1 , . . . , Sr ]]. n

Proof. The kernel I is generated by σi −1 (i = 1, . . . , r). Since n n ∈ mn is clear, it is enough to show σ − 1 ∈ (, σ − 1)n ⊂ mn for any σ ∈ Zr by induction on n. The case n = 1 is clear. Now the −1 n ·i n+1 n and − 1 = (σ − 1) assertion follows from the facts σ i=0 σ −1 n ·i ≡ mod (σ − 1). i=0 σ Proof of Theorem 5.25(ii) ⇒ (i). Let O be the ring of integers of a ﬁnite extension of Q . For any positive integer n, a lifting (Rn ) = (Rn , Mn , an , rn , mn ) of R along (On , on ) satisﬁes conditions (5.13) and (5.14). The strategy of proof is as follows.

6.1. PROOF OF THEOREM 5.25

145

* and M +. (a) Modify Rn , and take projective limits R * and M + have some good properties using a theorem of (b) Show R commutative algebra. * M + and R, M . (c) Relate R, * and M +. (d) Derive good properties of R and M using those of R By conditions (5.12) and (5.14), M is a ﬁnitely generated free Omodule, and its rank as an On -module is the same as that of Mn . Let N > 0 be its rank. For any positive integer n, let On = On /n On , and let M n = Mn ⊗ On = Mn /n Mn . M n is a free On -module of rank N . Let Rn be the image of the ring homomorphism Rn → EndOn (M n ). Instead of taking the projective limit of Rn , consider taking the projective limit of (Rn , M n ). On -module M n can be considered as a projective system in the following way. Take a basis x1 , . . . , xN of the free O-module M of rank N . Then, take a basis x1n , . . . , xN n of free On -module M n of rank N such that their images in M/n M = M n ⊗On (O/n O) coincide with x1 , . . . , xN . For m ≥ n, deﬁne mn,m : M m → M n by xim → xin , i = 1, . . . , N . Then the modules M n form a projective + = lim M n . Then, this is a free O * = system of modules. Let M ←−n limn On -module of rank N . ←− Unfortunately, however, the subrings Rn of EndOn (M n ) do not form a projective system. So, we make the following modiﬁcation. By condition (5.13) and Proposition 6.1(1), there exists a surjective homomorphism of complete local rings O[[X1 , . . . , Xr ]] → Rn for each n. So, choose a surjective homomorphism f˜n : O[[X1 , . . . , Xr ]] → Rn for each n arbitrarily. Rn is the image of fn : O[[X1 , . . . , Xr ]] → Rn → EndOn (M n ). Thus, for the subring Rn to form a projective system, it is enough for the homomorphism fn : O[[X1 , . . . , Xr ]] → EndOn (M n ) to form a projective system. We now apply the following lemma. Lemma 6.4. Let (Fn , ϕn,m ) be a projective system of ﬁnite sets, and let (fn )n ∈ n Fn . Then, for each n, there exists a sequence (mn )n with mn ≥ n such that (ϕn,mn (fmn ))n ∈ limn Fn . ←− Proof. Let Fn = {ϕn,m (fm ) | m ≥ n}. This is a projective system of ﬁnite sets. Since the projective limit limn Fn is nonempty, ←− take an element (fn )n . By the deﬁnition of Fn , there exists an m with m ≥ n such that fn = ϕn,m (fm ). So, take one and call it mn .

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Consider the projective system of ﬁnite sets Fn = {continuous O-homomorphism O[[X1 , . . . , Xr ]] → EndOn (M n )}. Let(mn )n be the sequence obtained by applying Lemma 6.4 to (fn )n ∈ n Fn . For each n, deﬁne a lifting of R along (On , on ) to be Rmn ⊗Omn On = (Rmn ⊗Omn On , Mmn ⊗Omn On , . . . ). By the deﬁnition of (mn )n , the homomorphisms fn : O[[X1 , . . . , Xr ]] → EndOn (M n ) form a projective system. Thus, we replace (Rn ) by (Rn ) instead, and assume that ring homomorphisms O[[X1 , . . . , Xr ]] → EndOn (M n ) form a projective system. The ring homomorphism EndOm (M m ) → EndOn (M n ) induced by mn,m : M m → M n maps the image Rn of fm : O[[X1 , . . . , Xr ]] → * = lim Rn be the projective limit of this EndOm (M m ) to Rn . Let R ←−n +) containing O, * * is a subring of End (M projective system of rings. R O * and it is also a quotient ring of O[[X1 , . . . , Xr ]]. Note that both O and O[[X1 , . . . , Xr ]] are formal power series rings of r variables. * and M + have good properties. Next we show that R * is a ﬁnitely generated free O-module. * Lemma 6.5. (1) R * is an isomorphism of rings. (2) O[[X1 , . . . , Xr ]] → R + * (3) M is a ﬁnitely generated free R-module. We use the following facts in commutative algebra. Proposition 6.6. Let A be a ring such that the dimension dim A is deﬁned. Let f : A → B be a ring homomorphism, and suppose B is ﬁnitely generated as an A-module. (1) The dimension dim B is also deﬁned and dim A ≥ dim B. Moreover, if f is injective, we have dim A = dim B. (2) Suppose A is an integral domain. Then the equality dim A = dim B implies that f : A → B is injective. Theorem 6.7. Let A and B be regular local noetherian rings, and let f : A → B be an injective local homomorphism. Let M be a ﬁnitely generated B-module, and suppose B is ﬁnitely generated as an A-module. (1) If M is a free A-module, then it is also a free B-module. (2) B is a free A-module. We omit the proofs.

6.1. PROOF OF THEOREM 5.25

147

* ⊂ End (M +), R * is ﬁnitely genProof of Lemma 6.5. Since R O * erated as an O-module. Applying Proposition 6.6(1) to the inclusion * * * * = r + 1. Apply Proposition 6.6(2) O → R, we obtain dim O = dim R * Since dim O[[X1 , . . . , Xr ]] = to the surjection O[[X1 , . . . , Xr ]] → R. * = r +1, the surjection O[[X1 , . . . , Xr ]] → R * is an isomorphism. dim R * * * +. Apply Theorem 6.7 to the inclusion O → R and R-module M + * + Since M is a nonzero ﬁnitely generated free O-module, M is a ﬁnitely * * is a ﬁnitely generated free O-module. * generated free R-module and R * M +) and (R, M ). We deﬁne We now relate the projective limit (R, * → R. We must be careful because the composia homomorphism R * → Rn → R depends on n. For a local ring A, we denote by tion R A/mn the quotient ring by the nth power of the maximal ideal. * → Rn → R → R/mn form a Lemma 6.8. The compositions R projective system. Proof. Let Tn be the image of the ring homomorphism R → EndO/n O (M/n M ). The diagram

(6.1)

* −−−−→ Rn −−−a−→ R ⏐ ⏐ b

⊂

Rn −−−−→ ⏐ ⏐

EndOn (M n ) ⏐ ⏐

⊂

R −−−−→ Tn −−−−→ EndO/n O (M/n M ) is commutative. Since Rn → R is surjective, so is Rn → Tn . By * the morphisms R * → Rn form a surjective projecthe deﬁnition of R, * tive system. Thus, R → Tn also form a surjective projective system. + → M n → M/n M induces an isomorphism Since the morphism M n + ⊗ (O/ O) = M n ⊗ (O/n O) → M/n M , we can identify Tn M O O * → EndO/n O (M + ⊗ (O/n O)). with the image of the composition R O Through this identiﬁcation, we show that Tn coincides with the sub+ ⊗ (O/n O)). To do so, we need * ⊗O (O/n O) ⊂ EndO/n O (M ring R O the following lemma. * → O be a surjective homomorphism. Lemma 6.9. Let π : O * ⊗ O → End(M + ⊗ O ) induced by π is The ring homomorphism R O O injective, and its image coincides with the image of the ring homo* → EndO (M + ⊗ O ). morphism R O

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6. COMMUTATIVE ALGEBRA

+ is a free R-module, * + ⊗ O is a ﬁnitely genProof. Since M M O * ⊗ O -module. Thus, R * ⊗ O → EndO (M + ⊗ O ) is erated free R O O O *→R * ⊗ O is surjective, its image is the same as injective. Since R O + ⊗ O ). * → EndO (M that of R O

*⊗ Applying Lemma 6.9 to O = O/n O, we identify Tn with R O n n (O/ O). Similarly, applying to O = On = On / On , we identify * ⊗ On . Rn with R O * → Tn forms a projective system, We show Lemma 6.8. Since R n ¯ it suﬃces to show that b : R/m → Tn /mn is an isomorphism. By * → On is contained in mn , the nth power Lemma 6.3, the kernel of O O * Thus, R/m * n → Rn /mn = (R * ⊗ On )/mn of the maximal ideal of O. O is an isomorphism. This isomorphism is the composition of the maps induced by the homomorphism of the upper line of (6.1), that is, a ¯ * n → Rn /mn → Rn /mn . Since both of these homomorphisms are R/m n surjective, a ¯ : Rn /m → Rn /mn is also an isomorphism. We show that the mapping b : R → Tn in the lower line of (6.1) is obtained by tensoring idO to the mapping a : Rn → Rn in the upper line. By condition (iii) in Deﬁnition 5.24, Rn ⊗On O → R is * ⊗ On and Tn = R * ⊗ (O/n O), an isomorphism. Since Rn = R O O Rn ⊗On O → Tn is also an isomorphism. Thus, we have b = a ⊗ idO . Since a ¯ : Rn /mn → Rn /mn is an isomorphism, ¯b : R/mn → Tn /mn ∼ * → Tn → Tn /mn ← is also an isomorphism. Thus, the composition R Rn /mn forms a projective system. By the commutative diagram (6.1), this is equal to the composition mapping in Lemma 6.8. Now that Lemma 6.8 is proved, we take the projective limit of * * → R. We show R * ⊗ O → R R → Rn → R → R/mn to deﬁne R O * n→ is an isomorphism. As we show in the proof of Lemma 6.8, R/m n n n n Rn /m , R/m → Tn /m and Rn ⊗On (O/ O) → Tn are all iso* n ⊗ O → R/mn , and its projective limit morphisms. Thus, R/m O * ⊗ O → R. R O + → M as the projective limit of M n → Deﬁne a linear mapping M n M/ M induced by the mapping mn : Mn → M . We show that this + → M is compatible with the ring homomorphism R *→R mapping M deﬁned above. The mapping M n /mn M n → M/(n M + mn M ) is

6.2. PROOF OF THEOREM 5.27

149

* n → R/mn = compatible with the natural ring homomorphism R/m n n n n Tn /m . Since M/m M = M/( M + m M ), its projective limit + → M is compatible with R * → R. M + → M induces an isomorphism We show the linear mapping M + M ⊗ R → M . The linear mapping above M n /mn M n → M/mn M R

induces M n /mn M n ⊗On O → M/mn M . Thus, its projective limit + ⊗ R = M + ⊗ O → M is an isomorphism. M R O * We show R and M satisfy condition (i) in Theorem 5.25. Since R * * ⊗ O, is a ﬁnitely generated free O-module, and R is isomorphic to R O R is a ﬁnitely generated free O-module. Let fi , i = 1, . . . , r, be * in R * = O[[X1 , . . . , Xr ]]. Since the ring R is the image of σi − 1 ∈ O * isomorphic to R ⊗O O = O[[X1 , . . . , Xr ]]/(f1 , . . . , fr ), it is a complete + is a ﬁnitely generated free R-module * intersection. Since M and M is + isomorphic to M ⊗R R, M is a ﬁnitely generated free R-module. 6.2. Proof of Theorem 5.27 We show the following proposition and theorem, and prove Theorem 5.27. Let n be a positive integer, and we denote the RTM-triple of rank n (O, O, On , id, id) by R0 . Proposition 6.10. Let O be a complete discrete valuation ring, and let n be a positive integer. Let R0 be the RTM-triple of rank n (O, O, On , id, id). If R = (R, T, M, f, π) is an RTM-triple over O of rank n, then there exists a surjection of RTM-triples F = (π ◦ f, π, m, m∨ ) : R → R0 with the multiplier Δ ∈ O such that (6.2)

lengthO (pR /p2R ) = ordO Δ.

Theorem 6.11. Let O be a complete discrete valuation ring, and let n be a positive integer. Let R0 be the RTM-triple of rank n (O, O, On , id, id). Let R = (R, T, M, f, π) be an RTM-triple rank n, and F = (π ◦ f, π, m, m∨ ) : R → R0 a surjection of RTM-triples with the multiplier Δ ∈ O. (1) We have Δ = 0, and the following inequality holds. (6.3)

lengthO (pR /p2R ) ≥ ordO Δ.

(2) The following are equivalent. (i) R is complete. (ii) The following equality holds. (6.4)

lengthO (pR /p2R ) = ordO Δ.

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Proof of Proposition 6.10 + Theorem 6.11 ⇒ Theorem 5.27. Let R = (R, T, M, f, π) and R = (R , T , M , f , π ) be two RTM-triples n > 0. Suppose F = (r, t, m, m∨ ) : R → R is a surjection of RTM-triples with multiplier Δ ∈ T that satisﬁes the inequality (6.5)

lengthO (pR /p2R ) − lengthO (pR /p2R ) ≤ ordO π (Δ).

Assume R is complete. By Proposition 6.10, there exists a surjection of RTM-triples F1 = (π ◦ f , π , m1 , m∨ 1 ) : R → R0 with multiplier Δ1 ∈ O such that (6.6)

lengthO (pR /p2R ) = ordO Δ1 .

∨ ∨ Let m0 = m1 ◦ m and m∨ 0 = m ◦ m1 . We show that F0 = ∨ (π ◦ f, π, m0 , m0 ) is a surjection of RTM-triples R → R0 . We verify the conditions in Deﬁnition 5.26(3). Conditions (i) and (ii) are clear. We verify (iii). Since the maps m, m1 are surjective, so is m0 . By n assumption, m∨ : M → M and m∨ 1 : O → M are injective, and their cokernels are both ﬁnitely generated free O-modules. Thus, the composition m∨ 0 is also injective, and its cokernel is a ﬁnitely generated free O-module. Thus, F0 is a surjection of RTM-triples. The multiplier of F0 is Δ0 = π (Δ)Δ1 ∈ O. By (6.5) and (6.6), we have the inequality

lengthO (pR /p2R ) ≤ ordO Δ0 . By Theorem 6.11(1), this inequality is in fact an equality. Moreover, by Theorem 6.11(2) (ii) ⇒ (i), R is complete. We deduce Proposition 6.10 from the following proposition. To do so, we introduce some notation. Let T be an O-algebra that is a ﬁnitely generated free O-module. Let T ∨ = HomO (T, O). For a ∈ T and ϕ ∈ T ∨ , deﬁne aϕ ∈ T ∨ by aϕ(b) = ϕ(ab), and T ∨ becomes a T -module by this operation. Let T = O[[X1 , . . . , Xr ]]/(f1 , . . . , fr ), and xi ∈ T the image of Xi . The image of fi by the ring homomorphism O[[X1 , . . . , Xr ]] → T [[X1 , . . . , Xr ]]; Xi → Xi is also denoted by fi . The kernel of the homomorphism T [[X1 , . . . , Xr ]] → T ; Xi → xi is (X1 − x1 , . . . , Xr − xr ). Since fi is sent to 0 by this homomorphism, we have fi ∈ (X1 − x1 , . . . , Xr − xr ). So, we take gij ∈ T [[X1 , . . . , Xr ]] such that fi = j gij · (Xj − xj ). For an O-linear mapping ϕ : T → O, the O[[X1 , . . . , X r ]]-linear mapping T [[X1 , . . . , Xr ]] → O[[X1 , . . . , Xr ]]; i i a X → ˜ i i i ϕ(ai )X is denoted by ϕ.

6.2. PROOF OF THEOREM 5.27

151

Proposition 6.12. Let O be a complete discrete valuation ring, let r be a positive integer, and let f1 , . . . , fr ∈ O[[X1 , . . . , Xr ]]. Suppose the quotient ring T = O[[X1 , . . . , Xr ]]/(f1 , . . . , fr ) is ﬁnitely generated and free as an O-module. Take gij ∈ T [[X1 , . . . , Xr ]] as above, and let G = (gij ) ∈ Mr (T [[X1 , . . . , Xr ]]). Deﬁne a mapping a : T ∨ → T by (6.7)

a(ϕ) ≡ ϕ(det ˜ G) mod (f1 , . . . , fr ) ∈ T

for ϕ : T → O. Then, a is an isomorphism of T -modules. The proof of this proposition is given using the Koszul complex, but we do not present it in this book. Proof of Proposition 6.10. Suppose R = (R, T, M, f, π) is complete, and identify it with T = O[[X1 , . . . , Xr ]]/(f1 , . . . , fr ). Let a : T ∨ → T be the isomorphism of Proposition 6.12, and let Δ = π ◦ a(π) ∈ O. We construct a surjection of RTM-triples F : R → R0 with multiplier Δ. We may assume R = T , M = T n and f = id. Also, by considering the direct sum, we may assume M = T . Let π ∨ : O → T be the composition of the dual O = O∨ → T ∨ of π : T → O as an O-linear mapping and the isomorphism a : T ∨ → T . We show F = (π, π, π, π ∨ ) is a surjection of RTM-triples. Since π : T → O is a surjection of ﬁnitely generated free O-modules, its dual O = O∨ → T ∨ is injective, and its cokernel is also a ﬁnitely generated free O-module. Thus, F is a surjection of RTM-triples. Since π ∨ (1) = a(π) ∈ T , Δ = π ◦ π ∨ (1) ∈ O is the multiplier of F. We show ordO Δ = lengthO (pT /p2T ). Let ai be the image of Xi π by the composition O[[X1 , . . . , Xr ]] → T → O. Replacing Xi by Xi − ai if necessary, we may assume ai = 0. Then, the kernel p of the composition O[[X1 , . . . , Xr ]] → O is the ideal (X1 , . . . , Xr ). Let ∂fi (0) ∈ Mr (O). We show the following. J = ∂X j (i) pT /p2T is isomorphic to the cokernel of J : Or → Or . (ii) Δ = det J. We show (i). pT /p2T is the cokernel of I = (f1 , . . . , fr ) → p/p2 . p/p is a free O-module with a basis consisting of the images of Xj , ∂fi j = 1, . . . , r, and the image of fi is j ∂X (0)Xj . Thus, we have (i). j We show (ii). By the deﬁnition of a, Δ = π ◦ a(π) is the image of π ˜ det G by the ring homomorphism T [[X1 , . . . , Xr ]] → O[[X1 , . . . , Xr ]] → π T → O. Since Xi and xi are all mapped to 0 by this homomorphism, 2

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6. COMMUTATIVE ALGEBRA

∂fi gij is mapped to ∂X (0) ∈ O. Thus, we have Δ = det J, and we have j proved (ii). We have ordO Δ = lengthO (pT /p2T ) by the theory of modules over a principal ideal domain (PID).

We show Theorem 6.11. We divide it into a theorem and a proposition by deﬁning an ideal ηT of O. In general, for a proﬁnitely generated complete local O-algebra R and a local O-homomorphism πR : R → O, deﬁne pR = Ker(πR : R → O), (6.8)

IR = AnnR pR = {a ∈ R | apR = 0} = HomR (O, R), ηR = πR (IR ).

In the second formula, O is regarded as an R-module through the homomorphism πR : R → O. pR and IR are ideals of R. Since πR is surjective, ηR is an ideal of O. We have pR /p2R = pR ⊗R O. Theorem 6.13. Let O be a complete discrete valuation ring, let R and T be proﬁnitely generated complete local O-algebras, and let f π R → T → O be a surjective local O-homomorphism. (1) We have the inequality (6.9)

lengthO (pR /p2R ) ≥ lengthO (pT /p2T ) ≥ lengthO (O/ηT ).

(2) Suppose T is a ﬁnitely generated free O-module such that pT /p2T is an O-module of ﬁnite length. Then, the following two conditions are equivalent. (i) f : R → T is an isomorphism, and T is a complete intersection. (ii) The following inequality holds. (6.10)

lengthO (pR /p2R ) = lengthO (O/ηT ).

We only prove Theorem 6.13(1) and (2) (i) ⇒ (ii) at the end of this chapter. Proposition 6.14. Let T = (T, T, M, id, π) be an RTM-triple of rank n, and F : T → R0 a surjection of RTM-triples with multiplier Δ ∈ O. (1) The following inclusion holds. (6.11)

ΔO ⊃ ηT = 0.

6.2. PROOF OF THEOREM 5.27

153

(2) Suppose T is a complete intersection. Then, the following conditions are equivalent. (i) T is complete. (ii) The following equality holds. (6.12)

ΔO = ηT .

Condition (i) in (2) means M is a free T -module. We will prove Proposition 6.14 shortly afterward. For the time being, we admit Theorem 6.13 and Proposition 6.14 and prove Theorem 6.11. Proof of Theorem 6.11. (1) Let F : R → R0 be a surjection of RTM-triples with multiplier Δ ∈ O. Combining inequality (6.9) in Theorem 6.13(1) and inclusion (6.11) in Proposition 6.14(2), we obtain Δ = 0 and the inequality (6.13)

lengthO (pR /p2R ) ≥ lengthO (O/ηT ) ≥ ordO Δ.

(2) (i) ⇒ (ii). Suppose R is complete. Then, both (i) in Theorem 6.13(2) and (i) in Proposition 6.14(2) are satisﬁed. Thus, applying (i) ⇒ (ii) in each statement, we obtain equalities (6.10) and (6.12). We obtain (6.4) by combining these two. (ii) ⇒ (i). Suppose (6.4) holds. Then, all the inequalities are in fact equalities. First, by Theorem 6.13(2) (ii) ⇒ (i), f : R → T is an isomorphism, and T is a complete intersection. Now, the assumptions of Proposition 6.14(2) being satisﬁed, we deduce R is complete from (ii) ⇒ (i). We prove Proposition 6.14. Proof of Proposition 6.14. Let F = (π, π, m, m∨ ); T → R0 be a surjection of RTM-triples, and Δ ∈ O its multiplier. Let L = On . We regard L as a T -module through the ring homomorphism π : T → O. We ﬁrst show the following lemma. Lemma 6.15. If we deﬁne M [pT ] = {x ∈ M | pT x = 0}, then we have m∨ L = M [pT ]. Proof. Using the isomorphism π ⊗ id : TK /pT TK → K, we identify TK /pT TK with K. We ﬁrst show that TK = T ⊗O K is decomposed into the direct product of rings (6.14)

TK = K × A.

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Since TK is ﬁnite dimensional over K, it is a direct product of local rings. If we write Spec TK = {p0 , p1 , . . . , pk }, p0 = pT TK , then we have a decomposition TK,p0 × ki=1 TK,pi . We show TK,p0 = K. By assumption, pT /p2T is of ﬁnite length, thus, we have p0 /p20 = (pT /p2T ) ⊗O K = 0. Hence, we have TK,p0 , and we see TK = K × A. We show m∨ L = M [pT ]. Since pT L = 0, we have m∨ L ⊂ M [pT ]. We ﬁrst show m∨ L⊗O K = MK [pT TK ]. By the decomposition (6.14), MK = M ⊗O K and LK = L ⊗O K are decomposed into MK = M ⊗T K × M ⊗T A and LK = L ⊗T K × L ⊗T A, respectively. Since L is originally an O-module that is regarded as a T -module, we have LK = L ⊗T K. By the direct product decomposition of M , MK [pT TK ] → M ⊗T K is an isomorphism. By condition (5.16), m : M → L is surjective. Thus, the K-linear mapping M ⊗T K → LK induced by m is also surjective. Since they have the same dimension, this is an isomorphism. By (5.16), m∨ : L → M is injective, its localization at the prime ideal pT , LK → M ⊗T K, is also injective. So, the composition (6.15)

m∨

∼

∼

LK → MK [pT TK ] → M ⊗T K → LK

is an injective endomorphism of the ﬁnite dimensional K-vector space m∨

LK , and thus it is an isomorphism. Hence, L ⊗O K → MK [pT TK ] is an isomorphism. We thus proved m∨ L ⊗O K = MK [pT TK ]. Since M/m∨ L is a free O-module by (5.16), we have m∨ L = M ∩ (m∨ L ⊗O K). Thus, we have shown the inclusion of the opposite direction m∨ L = M ∩ (m∨ L ⊗O K) = M ∩ MK [pT TK ] ⊃ M [pT ]. We prove Proposition 6.14(1). We have IT M ⊂ M [pT ] = m∨ L. Taking the image of the surjection m : M → L, we have ηT L ⊂ ΔL. Since L is a nonzero free O-module, we have ηT ⊂ ΔO. Since pT TK → A is surjective by (6.14), IT is the kernel T → A. Thus, IT TK → K is surjective. This means that the image ηT of IT → O is nonzero. Proof of Proposition 6.14(i) ⇒ (ii). By Proposition 6.10, there exists a surjective morphism T → R0 of RTM-triples with multiplier Δ such that lengthO (pT /p2T ) = ordO Δ. Since M is a free T -module, we have M [pT ] = IT M . Thus, in the same way as the proof of (1), we see ηT L = ΔL. Since L is a nonzero free O-module, we have ηT = ΔO.

6.2. PROOF OF THEOREM 5.27

155

Proof of Proposition 6.14 (ii) ⇒ (i). Let F = (π, π, m, m∨ ) : R = (T, T, M, id, π) → (O) = (O, O, On , id, id) be a surjection of RTM-triples with multiplier Δ such that ΔO = ηT . It suﬃces to show M is a free T -module. Let R1 = (T, T, T n , id, π) be a complete RTM-triple of rank n. We write On = L and T n = N . Since T is a complete intersection, there exists a surjective homomorn phism F1 = (π, π, π n , π ∨ ) : R1 → R0 of RTM-triples with multiplier Δ1 such that Δ1 O = ηT by Proposition 6.10. By comparing this with F, we show M is a free T module. First, we construct a commutative diagram. π ∗n

(6.16)

L −−−−→ ⏐ ⏐ a m∗

πn

N −−−−→ ⏐ ⏐ b

L , , ,

m

L −−−−→ M −−−−→ L Since N is a free T -module, we take any T -linear mapping b : N → M that makes the right square commutative. We deﬁne a. Since pT L = n 0, we have b ◦ π ∨ (L) ⊂ M [pT ] = m∨ (L) by Lemma 6.15. Since the multiplier Δ is nonzero, m∨ is injective. Thus, there exists a unique a : L → L that makes the diagram commutative. We show both a : L → L and b : N → M are isomorphisms. First n we consider a. The compositions m ◦ m∨ : L → L and π n ◦ π ∨ are the multiplication by Δ and Δ1 , respectively. Thus, both maps are isomorphisms to ηT L. Since the diagram is commutative, a : L → L is an isomorphism. Next, we show b : N → M is an isomorphism. Since both N and M are free O-modules of rank n × rankO T , it is enough to show the dual b∨ : M ∨ = HomO (M, O) → N ∨ = Hom(N, O) is surjective. Consider the dual of the left rectangle of the diagram (6.16). The mapping m∨ : L → M is injective, and its cokernel Coker(m∨ : L → M ) is a free O-module by assumption. Thus, its dual (m∨ )∨ ◦ b∨ : M ∨ → N ∨ → L∨ is surjective. Since the diagram is commutative and n a is an isomorphism, the composition (π ∨ )∨ ◦b∨ : M ∨ → N ∨ → L∨ is n also surjective. On the other hand, by the deﬁnition of π ∨ : L → N , ∨ ∨ ∨n ∨ the mapping N ⊗T O → L induced by its dual (π ) : N ∨ → L∨ is an isomorphism. Thus, by Nakayama’s lemma, b∨ is surjective. Hence, b : N → M is an isomorphism of T -modules. Thus, M is a free T -module, and the RTM-triple T is complete.

156

6. COMMUTATIVE ALGEBRA

Proof of Theorem 6.13(1). Since f : R → T is surjective, pR → pT is also surjective. Thus, pR /p2R → pT /p2T is also surjective. Hence, we have lengthO (pR /p2R ) ≥ lengthO (pT /p2T ). To show the second inequality, we use a property of the Fitting ideal. Definition 6.16. Let A be a ring, and let M be a ﬁnitely generated A-module. The Fitting ideal FA (M ) is an ideal of A generated by

∞

P ∈ Mn (A) such that there exists a

det P ∈ A surjection to M from the cokernel of .

n=0 P : An → An Proposition 6.17. Let A be a ring, and let M be a ﬁnitely generated A-module. (1) The Fitting ideal FA (M ) is contained in the annihilator AnnA (M ). (2) Let P ∈ Mn,m (A). The Fitting ideal of FA (M ) of the cokernel M = Coker(P : Am → An ) is generated by all the minors of order n of P . (3) If M is ﬁnitely presented, then for a homomorphism of rings A → B, we have FB (M ⊗A B) = FA (M )B. (4) If A is a discrete valuation ring, and M is an A-module of ﬁnite length l, then the Fitting ideal FA (M ) is ml , where m is the maximal ideal of A. Proof. (1) Considering the matrix of cofactors, we have det P · An ⊂ Im(P : An → An ). Thus, det P annihilates the cokernel of P : An → An . Therefore, if there exists a surjection from the cokernel of P to M , det P annihilates M . (2) This is an exercise in linear algebra, so we omit the proof. (3) If M = Coker(P : Am → An ), then M ⊗A B = Coker(P : B m → B n ). Thus, the assertion follows from (2). (4) We may assume M = Coker(P : An → An ). Then, the assertion follows from (2) and the theory of modules over a PID. We show the second inequality in Theorem 6.13(1). Let FT (pT ) be the Fitting ideal of pT . By Proposition 6.17(1), we have FT (pT ) ⊂ IT . Then, by Proposition 6.17(3) and the fact pT /p2T = pT ⊗T O, we have FO (pT /p2T ) = πT (FT (pT )) ⊂ πt (IT ) = ηT .

6.2. PROOF OF THEOREM 5.27

157

Thus, by Proposition 6.17(4), we have lengthO (pT /p2T ) = lengthO O/FO (pT /p2T ) ≥ lengthO (O/ηT ).

Proof of Theorem 6.13(2) (i) ⇒ (ii). Suppose f : R → T is an isomorphism and T is a complete intersection. Since the RTMtriple R = (R, T, T, f, π) is complete, by Proposition 6.10, there exists a surjection of RTM-triples R → R0 with multiplier Δ ∈ O such that ordO Δ = lengthO (pR /p2R ). By Proposition 6.14(2), we have ΔO = ηT . This shows (6.10).

10.1090/mmono/243/08

CHAPTER 7

Deformation rings We prove Theorem 5.8 in this chapter. First, we present an axiomatic treatment of the deformation rings and formulate the existence theorem (Theorem 7.7). In §7.3, we derive Theorem 7.7 from Theorem 5.8. We prove Theorem 7.7 in §7.4. Let O be a complete discrete valuation ring, and let F be its residue ﬁeld. 7.1. Functors and their representations Definition 7.1. (1) F is a functor over O if (i) for any proﬁnitely generated complete local O-algebra A, a set F(T ) is given, and (ii) for any local homomorphism f : A → A over O, a mapping f ∗ : F(A) → F(A ) is given to satisfy the following two conditions: (a) For any proﬁnitely generated complete local algebra A over O, id∗A = idF (A) . (b) For any local homomorphisms f : A → A and g : A → A over O, (f ◦ g)∗ = f∗ ◦ g∗ . (2) Let F and G be functors over O. a : F → G is a morphism of functors if (i) for any proﬁnitely generated complete local O-algebra A, a mapping aA : F(A) → G(A) is given to satisfy the following condition: (a) For any local morphism of O-algebras f : A → A , the following diagram is commutative. a

F(A) −−−A−→ G(A) ⏐ ⏐ ⏐ ⏐f f∗ ∗ a

F(A ) −−−A−→ G(A ) 159

160

7. DEFORMATION RINGS

For any proﬁnitely generated complete local O-algebra A, if F(A) is a subset G(A) and the mapping aA : F(A) → G(A) is the inclusion, we call F a subfunctor of G. (3) Let F be a functor over O. F is represented by a proﬁnitely generated complete local O-algebra R if there exists r ∈ F(R) satisfying the following condition. (i) For any proﬁnitely generated complete local O-algebra A, the mapping {local morphisms of O-algebras R → A} → FA ∪ ∪ f → f∗ (r) is bijective. In this case r ∈ F(R) is called the universal element of the functor F. Example 7.2. (1) Let n be a natural number. For any proﬁnitely generated complete local O-algebra A, let mA be its maximal ideal, and let Fn (A) be the direct product (mA )n . For any local morphism of O-algebras f : A → A , deﬁne f∗ : Fn (A) → Fn (A ) by f n : (mA )n → (mA )n . This is a functor over O. The functor Fn is represented by the formal power series ring R = O[[X1 , . . . , Xn ]]. The universal element is r ∈ Fn = (mR )n is r = (X1 , . . . , Xn ). (2) Let G be a proﬁnite group, and ρ¯ : G → GLn (F ) a continuous homomorphism. Deﬁne the functor Liftρ¯ over O as follows. (i) For a proﬁnitely generated complete local O-algebra A, let Liftρ¯(A) be the set of lifts of ρ to A Liftρ¯(A) = {ρ : G → GLn (A) | ρ is continuous and satisﬁes pA ∗ (ρ) = iA∗ (¯ ρ)}, where pA : A → FA is the natural surjection to the residue ﬁeld of A, and iA : F → FA is the natural injection. (ii) For a local morphism of O-algebras f : A → A , let f∗ : Liftρ¯(A) → Liftρ¯(A ) be the mapping that sends ρ : G → GLn (A) to its composition f∗ (ρ) : G → GLn (A ) with the homomorphism GLn (A) → GLn (A ) induced by f .

7.2. THE EXISTENCE THEOREM

161

7.2. The existence theorem Definition 7.3. A ring A is a local O-algebra of ﬁnite length if A is a local ring with a local homomorphism O → A such that A is of ﬁnite length as an O-module. If A is a proﬁnitely generated complete local O-algebra and mA is its maximal ideal, then A/mnA is local O-algebra of ﬁnite length. Definition 7.4. Let ρ¯ : G → GLn (F ) be a continuous representation of a proﬁnite group G. Γ is a type of the lift of ρ¯ if for any local O-algebra A of ﬁnite length, there is a subset Liftρ,Γ ¯ (A) of Liftρ¯(A) satisfying axioms (1)–(5) below. (i) ρ¯ ∈ Liftρ,Γ ¯ (F ). (ii) If f : A → A is a local morphism of O-algebras, f∗ : Liftρ¯(A) → Liftρ¯(A ) maps Liftρ,Γ ¯ (A) to Liftρ,Γ ¯ (A ). (iii) Let I1 , I2 be ideals of A satisfying I1 ∩ I2 = 0, and let πi : A → A/Ii be a natural surjection. If a lift ρ ∈ Liftρ¯(A) satisﬁes πi,∗ (ρ) ∈ Liftρ,Γ ¯ (A/Ii ) for i = 1, 2, then ρ ∈ Liftρ,Γ ¯ (A). (iv) If f : A → A is an injective local morphism of O-algebras, then the inverse image of Liftρ,Γ ¯ (A ) by f∗ : Liftρ¯(A) → Liftρ¯(A ) is contained in Liftρ,Γ ¯ (A). (v) If ρ ∈ Liftρ,Γ ¯ (A), and if ρ and ρ are equivalent, then ρ ∈ Liftρ,Γ ¯ (A). Proposition 7.5. Let ρ¯ : G → GLn (F ) be a continuous homomorphism, and Γ a type of the lift of ρ. (1) For a proﬁnitely generated complete local O-algebra A, deﬁne n Liftρ,Γ ¯ (A) = lim Liftρ,Γ ¯ (A/mA ). ← − n

is a subfunctor of the functor Liftρ¯. Then, Liftρ,Γ ¯ (2) For any proﬁnitely generated complete local O-algebra and its local morphism of O-algebras, the axioms (i), (ii), (iv), and (v) in Deﬁnition 7.4 and the following condition (iii ) are satisﬁed. . (iii ) Let Iλ (λ ∈ Λ) be a family of ideals of A satisfying λ Iλ = 0, and let πλ : A → A/Iλ be a natural surjection. If a lift ρ ∈ Liftρ¯(A) satisﬁes πλ,∗ (ρ) ∈ Liftρ,Γ ¯ (A/Iλ ) for any (A). λ ∈ Λ, then ρ ∈ Liftρ,Γ ¯ We omit the proof.

162

7. DEFORMATION RINGS

Definition 7.6. Let D be a type of liftings of ρ¯. (1) For a proﬁnitely generated complete local O-algebra A, if a lifting ρ of ρ¯ is an element of Liftρ,D ¯ (A), we say that ρ is a lifting of type D. over O as follows. (2) Deﬁne a functor Def ρ,D ¯ (i) For a proﬁnitely generated complete local O-algebra A, Def ρ,D ¯ (A) is the set of equivalence classes of Liftρ,D ¯ . (ii) For a local morphism of O-algebras f : A → A , deﬁne f∗ : Def ρ,D ¯ (A) → Def ρ,D ¯ (A ) to be the induced mapping of f∗ : Liftρ,D ¯ (A) → Liftρ,D ¯ (A ).

Theorem 7.7. Let O be a complete discrete valuation ring, and let F be its residue ﬁeld. Let ρ¯ : G → GLn (F ) be an absolute irreducible continuous representation of a proﬁnite group G, and let D be a type of lifting of ρ¯. Suppose there exists a quotient group G that satisﬁes the following two conditions. (i) For any proﬁnitely generated complete local O-algebra A, any lifting ρ of ρ¯ of type D factors through the quotient group G. (ii) There exists a ﬁnitely generated dense subgroup of G. Then, we have the following. (1) There exists a proﬁnitely generated complete local O-algebra R over O. that represents the functor Def ρ,D ¯ (2) Let ρR ∈ Def ρ,D (R) be the universal element. Then, the subring of R generated over O by the subset {Tr ρR ∈ R | σ ∈ G} is dense in R. (3) The residue ﬁeld of R is equal to that of O. We will prove Theorem 7.7 in §7.4. 7.3. Proof of Theorem 5.8 Keep the notation the same as in Theorem 5.8. Let be an odd prime, let O be the ring of integers of a ﬁnite extension of Q , and let F be the residue ﬁeld of O. Suppose ρ¯ : GQ → GL2 (F) is a semistable irreducible mod representation such that det ρ is the cyclotomic character. Let Σ be a set of prime numbers satisfying condition (5.7). For a proﬁnitely generated complete local O-algebra A, the set of liftings of ρ¯ of type DΣ is denoted by Liftρ,D ¯ Σ (A).

7.3. PROOF OF THEOREM 5.8

163

We show Theorem 5.8 by applying Theorem 7.7 in the case G = GQ , ρ¯ = ρ¯, and D = DΣ . Let us compare the conclusions. Theorem 5.8(1) and (3) are nothing but Theorem 7.7(1) and (3), respectively. So, we deduce Theorem 5.8(2) by Theorem 7.7(2). It suﬃces to show that {Tr ρR (ϕp ) | p NΣ } is dense in {Tr ρR (σ) | σ ∈ G}. This follows from Theorem 3.1 Therefore, in order to show Theorem 5.8, it suﬃces to verify that the assumptions of Theorem 7.7 are satisﬁed. By Proposition 3.5, ρ¯ is absolutely irreducible. What we have to show now is reduced to the following two propositions. Proposition 7.8. Let the notation and assumptions be the same as in Theorem 5.8. (1) For a local O-algebra of ﬁnite length, the subset Liftρ,D ¯ Σ (A) of Liftρ¯(A) satisﬁes all the axioms (i)–(v) in Deﬁnition 7.4. (2) For a proﬁnitely generated complete local O-algebra A, the forn mula Liftρ,D ¯ Σ (A) = limn Liftρ,D ¯ Σ (A/mA ) in Proposition 7.5 ←− holds. Proposition 7.9. Let the notation and assumptions be the same as in Theorem 5.8. There exists a quotient group G of ΓQ that satisﬁes conditions (i) and (ii) of Theorem 7.7. Proposition 7.8 is a local property, while Proposition 7.9 is a global property. Proof of Proposition 7.8. It suﬃces to verify that the conditions in Proposition 7.8(1) and (2) are satisﬁed in each case of conditions (i)–(iii) in Deﬁnition 5.5. It is easy to verify Deﬁnition 5.5(iii), “det ρ is the cyclotomic character”. We verify Deﬁnition 5.5(i) and (ii) in each case. Axioms (i) and (v) in Deﬁnition 7.4 are clear by deﬁnition. Axiom (ii) is nothing but Proposition 5.7(1). We verify that axioms (iii), (iv), and the condition in Proposition 7.8(2) are satisﬁed on a case-by-case basis. The case p ∈ Sρ¯ ∪ Σ. In this case the condition at p is “good at p”. Thus, (2) is nothing but the deﬁnition. By Corollary 3.42, axioms (iii) and (iv) follow from the fact that A is a subring of A/I1 × A/I2 and A , respectively. The case p ∈ Sρ¯. In this case, ρ¯ is not good at p, and condition at p is “ordinary at p”. Then, each condition of Proposition 7.8 can be easily veriﬁed using the following condition.

164

7. DEFORMATION RINGS

Lemma 7.10. Let ρ¯ be ordinary at p. If furthermore p = , suppose ρ¯ ramiﬁes at p. Let ρ be a lifting of ρ¯ to a proﬁnitely generated complete local O-algebra. Then the following conditions are equivalent. (i) ρ is ordinary at p. to (ii) If χ : Ip → A× is the restriction of the cyclotomic character the inertia group at p, then for any σ, τ ∈ I we have ρ(σ) − p χ(σ) ρ(τ ) − 1 = 0. Proof. (i) ⇒ (ii) is clear. We show (ii) ⇒ (i). If p = , then choose τ ∈ Ip such that ρ¯(τ ) = 1. If p = , choose τ ∈ Ip such that χ(τ ¯ ) = 1. Deﬁne N ⊂ M = A2 as N = (ρ(τ ) − 1)M . Then, N is a direct summand of M , and Ip acts on N as χ. Since we also have N = Ker ρ(τ ) − χ(τ ) : M → M , Ip acts trivially on M/N . It now suﬃces to express ρ using matrices by taking a basis of M extending a basis of N . There is nothing to show if p ∈ Σ and p = . If p = ∈ Σ, the assertion follows from the following proposition and Lemma 7.10. Proposition 7.11. Suppose ρ¯ is ordinary at p = ≥ 3. A lifting ρ of ρ¯ to a local O-algebra A of ﬁnite length that is good at p is ordinary at p. ¯ that is good at p. Proof. Let K = Qnr p . Suppose ρ is a lifting of ρ By Theorem 3.43, we identify a ﬁnite ﬂat commutative group scheme over the ring of integers OK with the corresponding Ip -module. Let M be the Ip -module A2 corresponding to ρ. Let N be the Ip -submodule deﬁned by the connected component of M as a ﬁnite ﬂat commutative group scheme. Since M/N is ´etale, it suﬃces to show that N is a direct summand of rank 1 of M and Ip acts on N as the cyclotomic character restricted to Ip . We use the following lemma to show it. Lemma 7.12. Let A be a local ring of ﬁnite length, and M a ﬁnitely generated free A-module. Let I = {a ∈ A | am = 0} be the annihilator of the maximal ideal m of A. An A-submodule N is a direct summand of M if and only if IM ∩ N = IN . Proof. It is enough to show the “if” part. Let L be the free A-module generated by a lifting of a basis of the image of N in M = M/mM . It suﬃces to show that L is equal to N . Considering the

7.3. PROOF OF THEOREM 5.8

165

quotient by L, we can reduce it to the case N ⊂ mM . In this case, we have {x ∈ N | mx = 0} = IM ∩ N = IN = 0, and thus N = 0. We come back to the proof of Proposition 7.11. By assumption, suppose N is a connected component of M = M/mM . Since IM = M ⊗A I, its connected component IM ∩ N is equal to IN = N ⊗A I. Thus, by Lemma 7.12, N is a direct summand of M . Since N is a one dimensional (A/m)-vector space, the rank of N is 1. We show that the action of Ip on N is the restriction of the cyclotomic character restricted to Ip . The action of Ip on N is the restriction of the cyclotomic character to Ip . Thus, the Cartier dual of N is ´etale, and so the Cartier dual of N is also ´etale. Therefore, the action of Ip on N is the restriction of the cyclotomic character restricted to Ip . Proof of Proposition 7.9. Let L be the number ﬁeld corresponding to the kernel of ρ¯, and let M be the compositum of all ﬁnite Galois extensions L ⊂ Q of L that satisﬁes the following two conditions. (a) Let T be the ﬁnite set of prime ideals in OL lying above Sρ¯ ∪ Σ ∪ {}. L is unramiﬁed outside T . (b) The order of Gal(L /L) is a power of . M is a Galois extension of Q. We show that G = Gal(M/Q) satisﬁes conditions (i) and (ii) in Theorem 7.7. First, we show (i). Let A be a local O-algebra of ﬁnite length, and let ρ : GQ → GL2 (A) be a lifting of ρ¯ of type DΣ . It suﬃces to show that the ﬁeld corresponding to the kernel of ρ satisﬁes the conditions (a) and (b). Since ρ is unramiﬁed outside Sρ¯ ∪ Σ ∪ {}, (a) is satisﬁed. Since Gal(L /L) is isomorphic to a subgroup of U 1 GL2 (A) through ρ, its order is a power of . Next, we show (ii). It suﬃces to show that there exists a dense ﬁnitely generated subgroup of G1 = Gal(M/L). This group is a projective limit of ﬁnite -groups. We use the following proposition from group theory. Lemma 7.13. Let be a prime number, and let G be a ﬁnite -group. Let G = Gab /(Gab ) . If the image of a subset S of G generates G , then, G is generated by S.

166

7. DEFORMATION RINGS

ab By Lemma 7.13, it suﬃces to show that G1 = Gab 1 /(G1 ) is a ﬁnite group. Let M1 be the ﬁeld obtained by modifying the condition (ii) in the deﬁnition of M by the following. (ii ) Gal(L /L) is isomorphic to a ﬁnite direct sum of Z/Z. Since G1 = Gal(M1 /L), it suﬃces to show that M1 is a ﬁnite extension of L. Let a be the ideal q∈T q of OL . We use the notation of §5.3 in Chapter 5 of Number Theory 2 . Let L be a ﬁnite abelian extension of L satisfying conditions (i) and (ii ). Then, by Theorem 5.21 in Chapter 5 of Number Theory 2 , there exists a positive integer n such that L ⊂ L(an ). Take n0 suﬃciently large so that 1 + qn0 Oq ⊂ (Oq× ) for all q ∈ T , and we show L ⊂ L(an0 ). We have Gal(L(an )/L) Cl (L, an ) and Gal(L /L) = 1. Thus, it suﬃces to show that if n ≥ n0 , the natural surjection α : Cl (L, an ) → Cl (L, an0 ) induces an isomorphism α ¯ : Cl (L, an )/ → Cl (L, an0 )/. The kernel n of Cl (L, a ) → Cl (L, an0 ) is generated by the image of the surjection n0 ¯ is an isomorphism. Thus, q∈T (1 + q Oq ). By the choice of n0 , α M1 is contained in L(an0 ). Since Gal(L(an0 )/L) Cl (L, an0 ) is a ﬁnite group, so is Gal(M1 /L).

7.4. Proof of Theorem 7.7 If there exists a dense ﬁnitely generated subgroup of G, we say G is topologically ﬁnitely generated. To prove Theorem 7.7, we may assume G is topologically ﬁnitely generated by replacing G by G. We ﬁrst show the following proposition. Proposition 7.14. Let G be a topologically ﬁnitely generated proﬁnite group, let O be a complete discrete valuation ring, and let F be its residue ﬁeld. Let ρ¯ : G → GLn (F ) be a continuous representation, and let D be a type of liftings of ρ¯. Then, there exists a proﬁnitely generated complete local O-algebra R that represents the over O. functor Liftρ,D ¯ Proposition 7.15. Let G be a proﬁnite group, let O be a complete discrete valuation ring, and let F be its residue ﬁeld. Let ρ¯ : G → GLn (F ) be an absolutely irreducible continuous representation. Let A be a proﬁnitely generated complete local O-algebra whose residue ﬁeld is F , and let ρ : G → GLn (A) be a lifting of ρ¯ to A. Let A0 be a subring containing O[Tr(ρ(σ)) | σ ∈ G] that is a proﬁnitely generated complete local O-algebra. Let i : A0 → A be the inclusion. Then, there exists a lifting ρ0 : G → GLn (A0 ) of ρ to A0 such that i∗ (ρ0 ) is equivalent to ρ.

7.4. PROOF OF THEOREM 7.7

167

Proof of Proposition 7.14. Suppose that the subgroup generated by a subset S = {σ1 , . . . , σm } ⊂ G is dense in G. Deﬁne a over O by functor Liftρ(S) ¯ (A) = {(Pk ) ∈ GLn (A)m | pA (Pk ) = iA (¯ ρ(σk )), k = 1, . . . , m}. Liftρ(S) ¯ We show that this functor is represented by the power series ring R0 = O[[Xijk , 1 ≤ i ≤ n, 1 ≤ j ≤ n, 1 ≤ k ≤ m]] of N = n2 m variables. Let FN : A → (mA )N be the functor in Example 7.2(1). The functor FN : A → (mA )N is represented by the ring R0 . For each 1 ≤ k ≤ m, choose a lifting Qk ∈ GLn (O) of ρ¯(σk ) ∈ GLn (F ) to GLn (O). The mapping Liftρ(S) (A) → FN (A) = (mA )N ; (each component of Pk − Qk )k ¯ deﬁnes an isomorphism of functors Liftρ(S) → FN . Thus, the functor ¯ is also represented by the same ring R0 . The universal element Liftρ(S) ¯ is P0 = (Qk + (Xijk )ij )k ∈ GLn (R0 )m . → Liftρ(S) by deﬁning Deﬁne a morphism of functor Liftρ,D ¯ ¯ Liftρ,D (A) for each A by ρ → (ρ(σk ))k . Since the sub¯ (A) → Liftρ(S) ¯ group generated by S is dense in G, this morphism is injective. So, as a subfunctor in Liftρ(S) . through this injection, we regard Liftρ,D ¯ ¯ We show that the subfunctor Liftρ,D is represented by a quotient ring ¯ R0 of R0 . Let I be the set of all the open ideals I in R0 satisfying the following condition. If πI : R0 → R0 /I is the natural surjection, πI ∗ (P0 ) ∈ (R0 /I) is contained in Liftρ,D Liftρ(S) ¯ (R0 /I). ¯ Corresponding to axioms (i)–(iii) in Deﬁnition 7.4, I satisﬁes the following conditions. (i) mR0 ∈ I = ∅. (ii) If I ∈ I, J ⊃ I, then J ∈ I. (iii) If I1 , I2 ∈ I, then I1 ∩ I2 ∈ I. . Let R = R0 / I∈I I = limI∈I R0 /I. We show that R represents ←− the functor Liftρ,D ¯ . Let A be a proﬁnitely generated complete local O-algebra. It suﬃces to show that the bijection (A) {local morphisms of O-algebras R0 → A} → Liftρ(S) ¯ induces a bijection between subsets (7.1)

{local morphisms of O-algebras R → A} → Liftρ,D ¯ (A).

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Since each of {local morphisms of O-algebras R → A} → limn {local morphisms of O-algebras R → A/mn }, ←− n Liftρ,D ¯ (A) → lim Liftρ,D ¯ (A/m ) ← − n

is a bijection, respectively, it suﬃces to show bijection (7.1) when A is a local O-algebra of ﬁnite length. Let f : R → A be a local morphism of O-algebras . Choose an ideal I ∈ I contained in the kernel of the composition R0 → R → A. Since πI ∗ (ρ0 ) ∈ Liftρ,D ¯ (R0 /I), it follows from axiom (ii) that (A). (f ◦ πI )∗ (P0 ) ∈ Liftρ,D ¯ Let ρ ∈ Liftρ,D ¯ (A). Let I be the kernel of the corresponding homomorphism R0 → A. Since the ring homomorphism i : R0 /I → A is injective, we obtain a continuous homomorphism ρI : G → GLn (R0 /I) satisfying ρ = i∗ (ρI ). By axiom (iv), ρI is a lifting of ρ¯ of type D. Thus, we have I ∈ I, and the composite homomorphism R0 → R0 /I → A factors through the quotient ring R . Thus, the mapping in (7.1), {local morphisms of O-algebras R → A} → Liftρ,D ¯ (A), is proved to be bijective. Proof of Proposition 7.15. First, we show that Mn (A) is generated by ρ(G) over A. By assumption, ρ¯ is absolutely irreducible, and we have F [¯ ρ(G)] = Mn (F ). By Nakayama’s lemma, we have Mn (A) = A[ρ(G)]. Let E0 = A0 [ρ(G)] ⊂ Mn (A) be the subring generated by ρ(G) over A0 . We show that E0 is a ﬁnitely generated free A0 -module, and the natural mapping E0 ⊗A0 A → Mn (A) is an isomorphism. By assumption, if x ∈ E0 , then Tr x ∈ A0 . Since E0 → Mn (F ) is surjective, take a lifting X = {xij | 1 ≤ i, j ≤ n} of the standard basis {eij | 1 ≤ i, j ≤ n} of Mn (F ) to E0 . Since X is a basis over A of Mn (A), the A0 -submodule E0 of Mn (A) generated by X is a free A0 -module and E0 ⊗A0 A → Mn (A) is an isomorphism. We show E0 = E0 . For x ∈ E0 , we show that there exists x ∈ E0 ∨ such that Tr(x − x )y = 0 for any y ∈ E0 . Since E0 → E0 = HomA0 (E0 , A0 ); x → (y → Tr xy) is an isomorphism after tensoring ⊗A0 A, it must be an isomorphism itself. Thus, an element x ∈ E0 as above exists. Since Tr(x − x )y = 0 for any y ∈ Mn (A) = E0 ⊗A0 A, we have x = x ∈ E0 . Thus, E0 = E0 is a ﬁnitely generated

7.4. PROOF OF THEOREM 7.7

169

free A0 -module, and the natural mapping E0 ⊗A0 A → Mn (A) is an isomorphism. We show that there exists an isomorphism of A0 -algebra f : E0 → Mn (A0 ) that induces the identity Mn (F ) → Mn (F ). By Hensel’s lemma, there exists e1 ∈ E0 such that e1 ≡ e11 mod mA0 and e21 = e1 . Let M0 be the left ideal of E0 given by M0 = E0 e1 . Since M0 is the image of a projector, it is a direct summand of E0 as an A0 module. The homomorphism E0 → EndA0 (M0 ); x → (y → xy) is an isomorphism by Nakayama’s lemma. Take a basis of M0 as an A0 -module that is a lifting of the basis {e11 , . . . , en1 } of M0 ⊗A0 F , and identify EndA0 (M0 ) = Mn (A0 ). Deﬁne ρ0 : G → GLn (A0 ) by ρ0 = f ◦ ρ. This is a lifting of ρ¯, and i∗ (ρ0 ) is equivalent to ρ. Proof of Theorem 7.7. Let R be the ring that represents the functor Liftρ,D constructed in Proposition 7.14, and ρ : G → GLn (R ) ¯ the universal element. Let R be the closure of the O-subalgebra of R generated by {Tr ρ (σ) ∈ R | σ ∈ G}. We show that R is a proﬁnitely generated complete local O-algebra. For n a positive integer, let Rn = R /mnR , and let ρn be the image of ρ . Let Rn be the subring of Rn generated by Tr ρn (σ), σ ∈ G. Rn is a local O-algebra of ﬁnite length. Rn is the image of the composition of the inclusion i : R → R and the natural surjection R → R /mnR , and R = limn Rn . Since the residue ﬁeld of Rn is contained in R , ←− it coincides with the residue ﬁeld of O. We show R is a complete local noetherian ring. By Nakayama’s lemma, it suﬃces to show that there exists n0 such that the surjection mRn /m2Rn → mRn0 /m2Rn 0 is an isomorphism as long as n ≥ n0 . It is enough to show that dim mRn /m2Rn ≤ dim mRn0 /m2Rn . 0 Apply Proposition 7.15 to in : Rn ⊂ Rn . There exists a lifting ρn ∈ Liftρ,D ¯ (Rn ) such that in∗ (ρn ) is equivalent to ρn . Let gn : R → Rn be the ring homomorphism corresponding to ρn . If σ ∈ G, then we have Tr ρn (σ) = gn (Tr ρ (σ)), and thus gn is surjective. Thus, we have shown dim mRn /m2Rn ≤ dim mRn0 /m2Rn . Hence, R is a proﬁnitely 0 generated complete local O-algebra, and the residue ﬁeld of R equals that of O. Applying Proposition 7.15 to the inclusion i : R → R , we obtain a lifting ρR of ρ¯ to R such that i∗ (ρR ) is equivalent to ρ . By axioms (iv) and (v), ρR is a lifting of ρ¯ of type D. We show that R represents the functor Def ρ,D ¯ . Let A be any proﬁnitely generated complete local

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O-algebra, and consider the diagram (7.2) a

{local morphisms of O-algebras R1 → A} −−→ Liftρ,D ¯ (A) ⏐ ⏐ ⏐c ⏐ b {local morphisms of O-algebras R → A} −−→ Def ρ,D ¯ (A) a ⏐ ⏐ d {continuous mappings G → A}.

Here, a , a, b, c, and d are given, respectively, by a (f ) = f∗ (ρ ), a(g) = (class of g∗ (ρ)), b(f ) = f ◦ i, c natural mapping, and d(ρ) = (σ → Tr(ρ(σ))). What we need to show is that a is bijective. We show that the diagram (7.2) is commutative. If f : R → A is a local Ohomomorphism, a◦b(f ) = (class of (f ◦i)∗ (ρ)) equals c◦a (f ) = (class of f∗ (ρ )) since i∗ and ρ are equivalent. Thus, (7.2) is commutative. We show a is bijective. Since a is bijective and c is surjective, a is surjective. Since the O-subalgebra of R generated by Tr(ρR (σ)) (σ ∈ G) is dense in R, d ◦ a is injective. Thus, we have shown a is bijective. Hence, we proved Theorem 7.7(1). Since ρR is a universal element, Theorem 7.7(2) is also satisﬁed. (3) has already been proved.

APPENDIX A

Supplements to scheme theory A.1. Various properties of schemes In this appendix we give a summary of properties of schemes that were used in the main text. Definition A.1. Let f : X → Y be a morphism of schemes. (1) Let d ≥ 0 be a natural number. A morphism f is smooth of relative dimension d if for any x ∈ X, there exist an aﬃne open neighborhood U Spec B of x and an aﬃne open subset V Spec A of Y containing f (U ) satisfying the following condition. The ring B is a ﬁnitely generated algebra over A, and if A[X1 , . . . , Xn ] → B is a surjective homomorphism over A, its kernel I is a ﬁnitely generated ideal of A[X1 , . . . , Xn ]. If p is the prime ideal of A[X1 , . . . , Xn ] corresponding to x, the localization Ip is generated by n − d elements f1 , . . . , fn−d as an ideal of A[X1, . . . ,Xn ]p . Furthermore, the rank of the (n − d) × n ma∂fi mod p with the coeﬃcients in the residue ﬁeld κ(x) trix ∂X j equals n − d. (2) A morphism f is ´etale if f is smooth of relative dimension 0. Definition A.2. A geometric point of a scheme S is a morphism s¯ → S from the spectrum s¯ of an algebraically closed ﬁeld to S. A geometric ﬁber of a morphism of schemes f : X → S is the ﬁbered product Xs¯ = X ×S s¯ with respect to a geometric point s¯ → S. The base change Xs¯ to an algebraically closed ﬁeld loses its arithmetic features coming from the base ﬁeld κ(s), and it keeps only geometric properties. This is why Xs¯ is called a geometric ﬁber. Proposition A.3. (1) Let f : X → S be a morphism of schemes. The following conditions are equivalent. 171

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(i) f is ´etale. (ii) f is ﬂat, locally of ﬁnite presentation, and any geometric ﬁber Xs¯ → s¯ is ´etale. (2) Let S be a spectrum of an algebraically closed ﬁeld K, and let f : X → S be a morphism of schemes. The following conditions are equivalent. (i) f is ´etale. (ii) X is discrete as a topological space, and the local ring at each point in X is K. Smooth morphisms have similar properties. Proposition A.4. (1) Let f : X → S be a smooth morphism. The following conditions are equivalent. (i) f is smooth. (ii) f is locally of ﬁnite presentation, ﬂat, and any geometric ﬁber Xs¯ → s¯ is smooth. (2) Let S be a spectrum of an algebraically closed ﬁeld, and let f : X → S be a morphism of schemes. The following conditions are equivalent. (i) f is smooth. (ii) X is locally of ﬁnite type over S and is regular. Some properties of a morphism of ﬂat schemes can be veriﬁed just by looking at their geometric ﬁbers. Proposition A.5. Let X and Y be schemes over S that are ﬂat and locally of ﬁnite presentation, and let f : X → Y be a morphism of schemes over S. Then the following conditions are equivalent. (i) f is ﬂat. (ii) fs¯ : Xs¯ → Ys¯ is ﬂat for any geometric point s¯ of S. Corollary A.6. Let X and Y be schemes over S that are ﬂat and locally of ﬁnite presentation, and let f : X → Y be a morphism of schemes over S. Then the following conditions are equivalent. (i) f is ´etale. (ii) fs¯ : Xs¯ → Ys¯ is ´etale for any geometric point s¯ in S. Proof of Corollary A.6. It suﬃces to show (ii)⇒(i). By Proposition A.5, f : X → Y is ﬂat. Since X and Y are locally of ﬁnite presentation, so is f . Since a geometric ﬁber of f is a geometric ﬁber of fs¯ for some geometric point in S, now the assertion follows from Proposition A.3.

A.1. VARIOUS PROPERTIES OF SCHEMES

173

Definition A.7. Let f : X → Y be a morphism of schemes. (1) We say f is ﬁnite if for any aﬃne open set V Spec A in Y , XV = X ×Y V is aﬃne and Γ(XV , O) is ﬁnitely generated as an A-module. (2) We say f is quasi-ﬁnite if f is of ﬁnite type, and every geometric ﬁber is discrete as a topological space. Theorem A.8. The following two conditions on a scheme X over an aﬃne scheme Y are equivalent. (i) X is isomorphic over Y to an open subscheme of a scheme X ﬁnite over Y . (ii) X is quasi-ﬁnite and separated over Y . Corollary A.9. The following two conditions on a morphism of schemes are equivalent. (i) f is ﬁnite. (ii) f is quasi-ﬁnite and proper. Corollary A.10. Let f : X → Y be a morphism of schemes that is quasi-ﬁnite, ﬂat, and of ﬁnite presentation. The following two conditions are equivalent. (i) f is ﬁnite. (ii) The degree of any ﬁber of f is constant. Proof of Corollary A.10. We show (ii) ⇒ (i). By Corollary A.9, it suﬃces to show that f is proper. By the valuation criterion, we may assume Y is the spectrum of a discrete valuation ring. By Theorem A.8, X is an open subscheme of a scheme X ﬁnite over Y . We may assume that X is ﬂat over Y , and X is dense in X. Let y and η be the closed point and the generic point of Y , respectively. Then, since deg X y − deg Xy = deg X η − deg Xη = 0, we have Xy = X y . Thus, X = X is ﬁnite. Corollary A.11. Let A be a Henselian discrete valuation ring, and let X be a scheme over A. The following two conditions are equivalent. (i) X is quasi-ﬁnite over A. (ii) X is the disjoint union of a ﬁnite scheme X1 over A and a ﬁnite scheme over the ﬁeld of fractions K of A. Proof. We show (i)⇒(ii). By Theorem A.8, there exists a ﬁnite scheme X over A containing X as an open subscheme. Since A is

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Henselian, X is a ﬁnite disjoint union of local schemes. Thus, we may assume X is a local scheme. The closed ﬁber of X consists of only / X, one point x. If x ∈ X, then X = X, and X is ﬁnite over A. If x ∈ then X ⊂ X K , and X is ﬁnite over K. Whether or not a morphism of proper schemes is ﬁnite can be determined by the geometric ﬁbers. Corollary A.12. Let X and Y be proper schemes over S, and let f : X → Y be a morphism over S. The following two conditions are equivalent. (i) f is ﬁnite. (ii) fs¯ : Xs¯ → Ys¯ is quasi-ﬁnite for any geometric point s¯. Proof. It suﬃces to show (ii)⇒(i). A geometric ﬁber of f is the geometric ﬁber of fs¯ for some geometric point s¯ of S. Thus, if f satisﬁes (ii), then f is quasi-ﬁnite. Since f : X → Y is proper, the assertion follows from Corollary A.9. Proposition A.13. Let f : X → S be a morphism of noetherian schemes. (1) If f : X → S is faithfully ﬂat and X is regular, then S is regular. (2) Suppose X is normal, S is regular, both X and S are connected, and f : X → S is a ﬁnite surjection. If either S is of dimension 2 or X is regular, then f is faithfully ﬂat. (3) If f : X → S is smooth and S is regular, then X is also regular. Corollary A.14. Let Y → S be a morphism of schemes of ﬁnite presentation, and let X → Y be a morphism of schemes that is faithfully ﬂat of ﬁnite presentation. If the composition X → S is smooth, so is Y → S. Proof. Since X → S is ﬂat and X → Y is faithfully ﬂat, Y → S is also ﬂat. For any geometric point s¯ of S, the ﬁber Xs¯ is regular by Proposition A.4. Moreover, by Propositions A.5 and A.13(1), the ﬁber Ys¯ is regular. Thus, by Proposition A.4, Y → S is smooth. Theorem A.15. Let X → S be a ﬁnite morphism of connected normal noetherian schemes such that at the generic point η, the ﬁber X ×S η → η is ´etale. If the number of points on the geometric ﬁber at each geometric point s¯ → S is constant, then X → S is ´etale.

A.2. GROUP SCHEMES

175

Theorem A.16. Let S be a noetherian scheme, and let f : X → S be a proper morphism. If OS → f∗ OX is an isomorphism, then for any geometric point s¯ → S, X ×S s¯ is connected. A.2. Group schemes Definition A.17. A scheme A over a scheme S is a commutative group scheme if a morphism + : A ×S A → A is given and satisﬁes the following condition. For any scheme T over S, the mapping + : A(T ) × A(T ) → A(T ) induced by + deﬁnes a structure of commutative group on A(T ) = HomS (T, A). If S = Spec R is the spectrum of a ring R and if A is a commutative group scheme over S, we say that A is a group scheme over R. Example A.18. (1) Additive group. Let Ga = Spec Z[X], and let + : Ga ×Z Ga → Ga be the morphism induced by the ring homomorphism Z[X] → Z[X] ⊗Z Z[X]; X → X ⊗ 1 + 1 ⊗ X. Then, for any scheme T , we have Ga (T ) = Γ(T, O), and Γ(T, O) × Γ(T, O) → Γ(T, O) is the addition. Thus, Ga is a group scheme over Z. We call it the additive group. For an arbitrary scheme S, we denote the base change Ga ×Z S by Ga,S and we call it the additive group over S. (2) Multiplicative group. Let Gm = Spec Z[X, X −1 ], and let × : Gm ×Z Gm → Gm be the morphism induced by the homomorphism Z[X, X −1 ] → Z[X, X −1 ] ⊗Z Z[X, X −1 ]; X → X ⊗ X. For any scheme T , we have Gm (T ) = Γ(T, O)× , and Γ(T, O)× × Γ(T, O)× → Γ(T, O)× is the multiplication. Thus, Gm is a commutative group scheme over Z. We call it the multiplicative group. For an arbitrary scheme S, the multiplicative group over S is deﬁned similarly by the base change. (3) Constant group. Let C be a ﬁnite abelian group. Deﬁne RC to be the direct product ring c∈C Z, and let GC = Spec RC . Deﬁne a ring homomorphism RC → RC ⊗Z RC by ec → c1 +c2 =c ec1 ⊗ ec2 . The morphism GC ×Z GC → GC induced by this ring homomorphism deﬁnes a structure of a commutative group scheme. This is the constant group scheme determined by C. Usually, GC is simply written as C. If S is a connected scheme, then we have GC (S) = C, and GC (S) × GC (S) → GC (S) coincides with the original operation of C. For an arbitrary scheme S, the constant group scheme over S is deﬁned as the base change.

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(4) Roots of unity. Let N be a positive integer. Let μN = Spec Z[X]/(X N − 1), and let × : μN ×Z μN → μN be the morphism determined by the ring homomorphism Z[X]/(X N − 1) ∪ X

→ Z[X]/(X N − 1) ⊗Z Z[X]/(X N − 1) ∪ → X ⊗ X.

For any scheme T , we have μN (T ) = {x ∈ Γ(T, O)× | X N = 1}, and × : μN (T ) × μN (T ) → μN (T ) is the multiplication. Thus, μN is a commutative group scheme over Z. We call it the group scheme of the N th roots of unity. For an arbitrary scheme S, μN,S is deﬁned as the base change. The base change μN,Z[ N1 ,μN ] is naturally isomorphic to the constant group scheme determined by the cyclic group μN (Z[ N1 , μN ]) Z/N Z of order N . Definition A.19. Let S be a scheme, and let A be a commutative group scheme over S. (1) If A is ﬁnite and ﬂat as a scheme over S, A is called a ﬁnite ﬂat commutative group scheme. Similarly, if A is ﬁnite and ´etale as a scheme over S, A is called a ﬁnite ´etale commutative group scheme. (2) Let N be a positive integer. If A is a ﬁnite ´etale commutative group scheme over S, and for any geometric point s¯, the ﬁnite abelian group A(¯ s) is a cyclic group of order N , then A is called a cyclic group scheme of order N . Example A.20. (1) Let C be a ﬁnite abelian group. The constant group scheme C is ﬁnite ´etale. If S is nonempty, then a ﬁnite abelian group C is a cyclic group of order N if and only if a constant group scheme C is a cyclic group scheme of order N . (2) Let N be a positive integer. The group scheme μN is ﬁnite ﬂat over S. If N is invertible in S, then μN is ﬁnite ´etale over S, and it is a cyclic group scheme of order N . Proposition A.21. Let S be a scheme, and let G be a ﬁnite ﬂat commutative group scheme over S. If there is a positive integer N invertible on S such that the multiplication-by-N mapping [N ] : G → G coincides with the 0 mapping G → G, then G is ´etale over S. Proof. By Proposition A.3, it suﬃces to show when S is the spectrum of an algebraically closed ﬁeld. Let m be the maximal ideal

A.3. QUOTIENT BY A FINITE GROUP

177

of the local ring OG,0 . By Nakayama’s lemma, it suﬃces to show that the K-vector space m/m2 is 0. The mapping induced on m/m2 by the multiplication-by-N mapping [N ] : G → G is the multiplication by N and at the same time 0. Since N is invertible by assumption, 0 is an isomorphism, and thus m/m2 = 0. Proposition A.22. Let R be a commutative ring, and let A be a ﬁnite ﬂat commutative group scheme over R. Let a : OA → OA ⊗R OA be the ring homomorphism that deﬁnes the group operation of A, and ∨ OA the R-module {R-linear mapping OA → R}. If we deﬁne the mul∨ is a ﬁnite ﬂat commutatiplication of OA∨ as the dual of a, then OA tive ring. Let a : OA∨ → OA∨ ⊗R OA∨ be the dual of the multiplication of OA . Then A∨ = Spec OA∨ is a ﬁnite ﬂat commutative group scheme over R. The ﬁnite ﬂat commutative group scheme A∨ is called the Cartier dual of A. Contravariant functor A → A∨ is an isomorphism of the category of ﬁnite ﬂat group schemes over R to itself. Example A.23. The Cartier dual of the constant group scheme Z/N Z is μN , and the Cartier dual of μN is Z/N Z. Corollary A.24. Let OK be a Henselian discrete valuation ring with residue ﬁeld F , and let G be a ﬁnite ﬂat commutative group scheme over OK . If G ⊗OK F μN , then G μN . Proof. By Proposition A.22 and Example A.23, the assertion follows from the fact that G ⊗OK F Z/N Z implies G Z/N Z. A.3. Quotient by a ﬁnite group Definition A.25. Let X be a scheme over S, and let G be a ﬁnite group. Suppose G acts on X on the right as automorphisms over S. A scheme Y over S is a quotient of X by G if there is a morphism π : X → Y over S that is G-invariant and satisﬁes the following two conditions. (1) For any schemes T over S, the mapping g ∈ {morphism Y → T over S} ↓ ↓ g ◦ π ∈ {G-invariant morphism X → T over S} is a bijection. (2) For any geometric point s¯ of S, the mapping X(¯ s)/G → Y (¯ s) induced by π is a bijection.

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For the existence of a quotient, the following is known. Proposition A.26. Let X be a scheme over S, and let G be a ﬁnite group. Suppose G acts on X on the right as automorphisms over S. Suppose X has an aﬃne covering that is stable under the G-action. Let Y = X/G be the quotient topological space, and let π : X → Y be the natural surjection. Deﬁne the sheaf of rings OY as the G-invariant part of the direct image π∗ OX ; OY = (π∗ OX )G . Then, the ringed space (Y, OY ) is a scheme over S, and π : X → Y is a ﬁnite morphism of schemes. Y is a quotient of X by G. The hypotheses of X are satisﬁed if X is quasi-projective over S. A.4. Flat covering For a scheme T , a family of ﬂat morphisms locally of ﬁnite pre sentation (Ui → T )i∈I such that T = i∈I Im(Ui → T ) is called a ﬂat covering ofT . Similarly, a family of ´etale morphisms (Ui → T ) such that T = i∈I Im(Ui → T ) is called an ´etale covering of T . If U = (Ui → T )i∈I and V = (Vj → T )j∈J are ﬂat coverings of T , a pair of a mapping ϕ : J → I and a family of morphisms Vj → Uϕ(j) (j ∈ J) over T is called a morphism from V to U. Let S be a scheme, and let F be a functor over S. For a ﬂat covering U = (Ui → T )i∈I of a scheme T over S, deﬁne

(A.1) F (U) = (fi )i ∈ F (Ui ) pr∗1 (fi ) = pr∗2 (fj ) i∈I

on F (Ui ×T Uj ) for any i, j ∈ I .

A morphism V → U of ﬂat coverings induces a morphism F (U) → F (V). Let T be an arbitrary scheme over S, and let U = (Ui → T )i∈I be any ﬂat covering of T . If the natural mapping F (T ) → F (U) is a bijection, we say that F is a ﬂat sheaf over S. For a functor F over S, deﬁne a functor F over S by (A.2)

F (T ) = lim F (U), −→ U

where U runs all ﬂat coverings of T , and the limit is an inductive limit. For a functor F over S, F a = F is a ﬂat sheaf over S. This is called the ﬂat sheaﬁﬁcation of F . For any geometric point s¯, the s) is an isomorphism. If a functor F natural mapping F (¯ s) → F a (¯ over S is representable, then F is a ﬂat sheaf.

A.5. G-TORSOR

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Replacing ﬂat coverings by ´etale coverings, we deﬁne ´etale sheaf and ´etale sheaﬁﬁcation. Definition A.27. Let S be a scheme. Let P be a condition on ﬂat schemes locally of ﬁnite presentation over S satisfying P(T ) = P(T ) for any ﬂat morphism T → T of ﬂat schemes locally of ﬁnite presentation over S. (1) We say that a condition P is ﬂat local over S if for any ﬂat covering (Ui → T )i∈I of any ﬂat schemes T locally of ﬁnite presentation over S, the condition that P(Ui ) holds for any i ∈ I implies that P(T ) holds. (2) Let T be a ﬂat scheme locally of ﬁnite presentation over S. We say that a condition P holds ﬂat locally over T if there exists a ﬂat covering (Ui → T )i∈I such that P(Ui ) holds for all i ∈ I. The term ´etale local is deﬁned similarly. For example, the condition that a morphism X → S of schemes is smooth is ﬂat local over S and ´etale local over X. In other words, if P(T ) is the condition that X ×S T → T is smooth, then P is ﬂat local over S. If Q(U ) is the condition that the composition U → X → S is smooth, then Q is ´etale local over X. Definition A.28. Let P be a condition on open subschemes of S such that P(U ) holds if P(U ) holds for some U ⊃ U . If the presheaf FP on a topological space S deﬁned by singleton {∗} if P holds, FP (U ) = ∅ if P does not hold is a sheaf with respect to Zariski topology, we say that the condition P is local over S. If P is local over S, the maximal open subscheme U of S satisfying FP = {∗} is called an open subscheme deﬁned by P. A.5. G-torsor Definition A.29. Let S be a scheme, let G be a group scheme over S, and let X be a group scheme over S. Suppose an action of G on X over S, G ×S X → X, is given. X is a G-torsor over S if X is isomorphic to G ﬂat locally over S. In other words, there exist a ﬂat covering U = (Ui → S)i∈I of S and isomorphisms G ×S Ui → X ×S Ui for all Ui compatible with the G-action.

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The set of isomorphism classes of G-torsors over S is denoted by H 1 (S, G). If G is a ﬁnite group, a torsor of constant group scheme deﬁned by G is called a G-torsor. We will give a condition that a scheme with an action of a ﬁnite group G is a G-torsor. Definition A.30. Let X be a scheme. Suppose an action G × X → X of a ﬁnite group G on X is given. (1) Let x ∈ X. The subgroup of G given by Ix = {g ∈ G | g(x) = x and the action of g on the residue ﬁeld κ(x) is trivial} is called the inertia group. (2) If Ix is trivial for any x ∈ X, then the action of G on X is called free. Lemma A.31. Let X be a scheme. Suppose an action G×X → X of a ﬁnite group G on X is given. (1) If X is a G-torsor over Y , then X is ﬁnite ´etale over Y and Y is a quotient of X by G. (2) Let S be a scheme, and let X be a ﬁnite scheme of ﬁnite presentation over S. Suppose the action of G on X is an action over S. Then, there exists a quotient Y = X/G, and Y is a ﬁnite scheme of ﬁnite presentation over S, and the natural morphism X → Y is ﬁnite and of ﬁnite presentation. Furthermore, the following three conditions are equivalent. (i) X is a G-torsor over Y . (ii) For any geometric point y¯, Xy¯ is a G-torsor over y¯. (iii) The action of G on X is free. Proof. (1) X is ﬁnite ´etale, ﬂat locally over Y . Thus, X is ﬁnite ´etale over Y . There exists a quotient X/G, and the natural morphism X/G → Y , which is an isomorphism ´etale locally over Y , is an isomorphism. (2) We may assume that S = Spec R, where R is a noetherian ring. Since X is ﬁnite over S, we may assume X = Spec B, where B is a ﬁnitely generated R-algebra. Let A = B G , the G-invariant part of B. Then, we have X/G = Spec A, and A is also ﬁnitely generated as an R-algebra. Thus, B is ﬁnitely generated as an A-algebra. (i)⇒(ii)⇒(iii) is clear. We show (iii)⇒(i). First, we show that X → Y is ´etale. Assuming S = Y , and considering the completion of

A.5. G-TORSOR

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the local ring at each point, we may assume A is a complete noetherian local ring. Let y be the closed point of Y . For x ∈ X ×Y y, let OX,x . Then Bx is a complete local ring that is ﬁnite over A, and Bx = X = x∈X×Y y Spec Bx . Moreover, replacing A by some unramiﬁed extension, we may assume that the residue ﬁeld at each point x in X ×Y y is a purely inseparable extension of κ(y). Since the inertia group Ix at each point is trivial, the ﬁnite G-set X ×Y y is a G G B , A → Bx is torsor. Since A is the G-invariant part x x∈X×Y y an isomorphism for any x ∈ X ×Y y. Thus, X → Y is ´etale. Since X → Y is an ´etale covering of Y and the morphism G × X → X ×Y X; (g, x) → (gx, x) is an isomorphism, X is a G-torsor over Y . If X is a G-torsor over Y , the quotient Y = X/G has the following functorial description. Definition A.32. Let S be a scheme, and let M be a functor over S. Suppose an action of a ﬁnite group G on M is given. Deﬁne the functor [M/G] over S by sending a scheme T over S to the set ⎫ ⎧ ⎨The isomorphism class of a pair (P, α), where⎬ [M/G](T ) = P is a G-torsor over T and α is a G-invariant . ⎭ ⎩ element α ∈ M(P ) If M is the functor deﬁned by a scheme X over S, deﬁne the functor [X/G] similarly by ⎧ ⎫ ⎨The isomorphism class of a pair (P, f : P → X),⎬ [X/G](T ) = where P is a G-torsor over T and f : P → X is . ⎩ ⎭ a morphism over S compatible with the action Lemma A.33. Let S be a scheme, and let G be a ﬁnite group. Suppose Y is a scheme over S, and X is a G-torsor over Y . Then, Y represents the functor [X/G]. Proof. Deﬁne a morphism [X/G] → Y of functors over S as follows. Let T be a scheme over S, and let (P, f : P → X) ∈ [X/G](T ). Then f : P → X induces a morphism T = P/G → Y = X/G. Thus, we have a morphism [X/G] → Y of functors. Next, we deﬁne its inverse. If T is a scheme over S and g : T → Y is a morphism over S, then X ×Y T is a G-torsor over T and the projection X ×Y T → X is a morphism over S compatible with the G-action. Thus, we have a morphism Y → [X/G]. The fact that these are inverse to each

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other follows from the fact that a morphism of G-torsors is an isomorphism. Lemma A.34. Let S be a normal noetherian scheme, and let G be a ﬁnite group. Suppose X is a normal ﬁnite scheme over S with a G-action over S. If each geometric ﬁber of X → S is a G-torsor and X → S is ´etale at the generic point of each irreducible component of S, then X is a G-torsor over S. Proof. We may assume S is connected. Let X be the union of all connected components of X containing a point in the inverse image of the generic point η of S. Since the image of X is a closed set containing η, X → S is surjective. Since each geometric ﬁber of X is a G-torsor, we have X = X . Thus, by Theorem A.15, X is ﬁnite ´etale over S. The morphism G × X → X ×S X; (g, x) → (gx, x) of ﬁnite ´etale schemes over X is bijective at each geometric ﬁber, and thus it is an isomorphism. A.6. Closed condition Definition A.35. Let S be a scheme. Let P be a condition on schemes over S satisfying the following property: (A.3) For a morphism T → T of schemes over S, P(T ) implies P(T ). Deﬁne a functor FP over S as follows. For a scheme T over S deﬁne singleton {∗} if P(T ) holds, (A.4) FP = ∅ if P(T ) does not hold. If the functor FP is representable by a closed subscheme P of S, then we say P is a closed condition over S. P is called the closed subscheme deﬁned by P. Lemma A.36. Let S be a scheme, let E be a locally ﬁnitely generated free OS -module, let F be a quasi-coherent OS -module, and let f : E → F be a surjective morphism of OS -module. Then the condition P that fT : ET → FT is an isomorphism for a scheme T is a closed condition over S. Proof. Since the assertion is local on S, we may assume S = Spec A, E is a quasi-coherent sheaf deﬁned by A-module An , and F is one that deﬁned by A-module An /N . In this case the functor FP is represented by the closed subscheme of S determined by the ideal I = (xi ; x = (x1 , . . . , xn ) ∈ N, i = 1, . . . , n) ⊂ A.

A.7. CARTIER DIVISOR

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Corollary A.37. Let S be a scheme, and let X be a scheme over S. Let D and D be closed subschemes of X, and suppose that D is ﬁnite ﬂat of ﬁnite presentation over S. (1) The condition that DT is a closed subscheme of DT for a scheme T over S is a closed condition. (2) If D is also ﬁnite ﬂat of ﬁnite presentation over S, then the condition DT = DT for a scheme T over S is a closed condition. Proof. (1) It suﬃces to apply Lemma A.36 to the surjective morphism of OS -modules OD → OD×X D . (2) Clear from (1). Corollary A.38. Let S be a scheme, and let X be a ﬁnite scheme over S. Let n be a positive integer, and for each geometric point s¯ → S, suppose the degree of the geometric ﬁber Xs¯ is at most n. Then, the condition that XT is ﬁnite ﬂat locally of ﬁnite presentation over T of degree n for a scheme T over S is a closed condition. Proof. The assertion is local on S. By the assumption and Nakayama’s lemma, we may assume there exists a surjective morphism OSn → OX . Then the assertion follows easily from Lemma A.36. A.7. Cartier divisor Definition A.39. Let X be a scheme. (1) An invertible sheaf on X is an OX -module that is isomorphic to OX locally on X. (2) A closed subscheme D of X is called a Cartier divisor if its deﬁning ideal sheaf ID is an invertible sheaf on X. If D1 and D2 are Cartier divisors, the product ID1 ID2 of ideal sheaves is again an invertible sheaf, which deﬁnes the sum D1 + D2 of Cartier divisors. Lemma A.40. Let X → S be a ﬂat morphism of schemes. Let D be a closed subscheme of X locally deﬁned by a principal ideal, and let T be a Cartier divisor of S. If the diagram D −−−−→ X ⏐ ⏐ ⏐ ⏐ T −−−−→ S is commutative, then D is also a Cartier divisor of X.

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Proof. Since the assertion is local, we may assume S = Spec A, T = Spec A/f A, X = Spec B, and D = Spec B/gB. By assumption, the image of f in B is not a zero divisor and f ∈ gB. Thus, g is not a zero divisor of B. Lemma A.41. Let A be a noetherian ring, and let f ∈ A. Suppose A[ f1 ] is normal and A/f A is reduced. Furthermore, the integral closure A of A in A[ f1 ] is ﬁnitely generated as an A-module, and for any minimal prime ideal p of A/f A, the localization Ap at its inverse image p in A is a discrete valuation ring. Then, f is not a zero divisor, and A is also normal. Proof. First, we show A/f A → A/f A is injective. By assumption, for each minimal prime ideal p, Ap → Ap is an isomorphism, and so is (A/f A)p → (A/f A)p . Since A/f A is reduced, A/f A → p (A/f A)p = p (A/f A)p is injective, and so is A/f A → A/f A. Next, we show A → A is injective. Let I = Ker(A → A). Since A/f A → A/f A is injective, so is A/f A → A/(I + f A). Since the image of f in A/I ⊂ A ⊂ A[ f1 ] is not a zero divisor, we have I/f I = 0 by the snake lemma. Since the support of the A-module I is contained in V (f ) ⊂ Spec A, Nakayama’s lemma implies that I = 0. Thus, A → A is injective. Finally, we show A = A. Let N = Coker(A → A). We have Ass(N ) ⊂ Supp(N ) ⊂ V (f ). Since A/f A → A/f A is injective, we see that the multiplication-by-f mapping N → N is injective by the snake lemma. Thus, we have Ass(N ) = ∅. Therefore, we have N = 0, and A is normal. Corollary A.42. Let X be a noetherian scheme, and let F ⊂ D be closed subschemes of X. Suppose D is deﬁned locally by a principal ideal and D F is dense in D. Suppose, furthermore, both X F and D are normal, and the integral closure X of X in X F is ﬁnite over X. If D F is a Cartier divisor of X F , then D is a Cartier divisor in X, and X is normal. Proof. We may assume X = Spec A. Suppose D is deﬁned by f ∈ A. Then, the hypotheses of Lemma A.41 are satisﬁed. Thus, D is a Cartier divisor in X and X is normal. Lemma A.43. Let S be a scheme, let D be a Cartier divisor in S, and let U = S D be complementary open subscheme. Let X be a ﬁnite scheme over S, and suppose XU is ﬂat of ﬁnite presentation over

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U of degree N . If for each geometric point s¯ → D, the degree of the geometric ﬁber Xs¯ is at most N , then X is ﬂat of ﬁnite presentation of degree N over S. Proof. By Corollary A.38, the condition that X is ﬁnite ﬂat of ﬁnite presentation deﬁnes a closed subscheme T of S. Since T contains U as a subscheme, we have S = T . Corollary A.44. Let S be a scheme, let D be a Cartier divisor in S, and let U = S D be the complementary open subscheme. Let X be a scheme over S, and let A and B be closed subschemes of X. If B is ﬁnite ﬂat of ﬁnite presentation over S, then we have the following. (1) If A is also ﬁnite over S, and for each geometric point s¯ of S we have det As¯ ≤ deg Bs¯, then AU = BU implies A = B. (2) If AU contains BU as a closed subscheme, then A contains B as a closed subscheme. Proof. (1) By Lemma A.43, A is also ﬁnite ﬂat of ﬁnite presentation. Thus, the assertion follows from the hypothesis AU = BU and Corollary A.37. (2) It suﬃces to apply (1) to the ﬁnite scheme A = A ×X B over S. Lemma A.45. Let S be a scheme, let D be a Cartier divisor in S, let U = S D be the complementary open subscheme. Let h : X → S be a faithfully ﬂat scheme over S, let Y be a scheme over S, and let f : X → Y be a morphism over S. Let gU : U → Y ×S U be a section on U satisfying f |X×S U = g ◦ h|X×S U . Then, there exists a unique section g : S → Y extending gU and satisfying f = g ◦ h. Proof. Since the assertion is local, we may assume that S = Spec R, X = Spec A, Y = Spec B, and D is deﬁned by a nonzero divisor a in R. Since A is faithfully ﬂat over R, we have A∩R[ a1 ] = R. Thus, the image of the ring homomorphism B → A corresponding to f : X → Y is contained in A ∩ R[ a1 ] = R. A.8. Smooth commutative group scheme Definition A.46. Let S be a scheme. (1) A proper smooth commutative group scheme over S such that each geometric ﬁber is connected is called an abelian scheme

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over S. In the case where k is a ﬁeld and S = Spec k, it is called an abelian variety over k. (2) A smooth commutative group scheme over S that is isomorphic ´etale locally over S to the product of a ﬁnite number of Gm is called a torus over S. (3) If a smooth commutative group scheme G is a successive extension of ﬁnite number of Ga , then G is called unipotent. (4) A ﬁnite ﬂat commutative group scheme G is called multiplicative if its Cartier dual G∨ is ´etale. An elliptic curve is an abelian scheme of relative dimension 1. Lemma A.47. Let S be a scheme, and let N be a positive integer. (1) Let A be an abelian scheme over S of relative dimension g. The multiplication-by-N mapping [N ] : A → A is ﬁnite ﬂat of ﬁnite presentation of degree N 2g . The kernel A[N ] = Ker([N ] : A → A) is a ﬁnite ﬂat commutative group scheme of ﬁnite presentation over S. (2) Let G be a torus of relative dimension g. The ´etale sheaf X = Hom(G, Gm ) on S is ´etale locally isomorphic to Zg . The multiplication-by-N mapping [N ] : G → G is ﬁnite ﬂat of ﬁnite presentation of degree N g . The kernel G[N ] = Ker([N ] : G → G) is a ﬁnite ﬂat commutative group scheme of ﬁnite presentation over S and it is isomorphic to Hom(X, μN ). We do not prove Lemma A.47(1). (2) is clear from the deﬁnition. For a torus over S, the ´etale sheaf X = Hom(C, Gm ) over S is called the character group of G. For a smooth connected separated commutative group scheme G of ﬁnite type over k and a prime number = char(k), limn G[n ] is ←− called the -adic Tate module of G and is denoted by T G. The Tate module T G is an -adic representation of the absolute Galois group ¯ Gk = Gal(k/k) of k. The -adic representation T Gm = limn μn is ←− denoted by Z (1), and Q ⊗Z Z (1) is denoted by Q (1). Gk acts on Z (1) and Q (1) by the -adic cyclotomic character. If k = C, the exponential mapping Lie G → Gan is a universal covering of Gan , and it deﬁnes an isomorphism Lie G/H1 (Gan , Z) → Gan of complex manifolds. From this we obtain a natural isomorphism Z ⊗Z H1 (Gan , Z) → T G. Corollary A.48. Let k be a ﬁeld, let N be a positive integer invertible in k, and let = char(k) be a prime number.

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¯ (1) Let A be an abelian variety of dimension g over k. Then A[N ](k) is a free Z/N Z-module of rank 2g. The Tate module T A = ¯ is a free Z -module of rank 2g, and the dimension limn A[n ](k) ←− of the Q -vector space V A = Q ×Z T A is 2g. (2) Let G be a torus of dimension g over k. The character group X = Hom(G, Gm ) is a free Z-module of rank g with a continuous ¯ action of the absolute Galois group Gk = Gal(k/k). The ﬁnite ¯ ¯ As a ﬁGk -module G[N ](k) is isomorphic to Hom(X, μN (k)). nite abelian group, it is a free Z/N Z-module of rank g. The Tate ¯ is isomorphic to Hom(X, Z (1)) as module T G = limn G(n )(k) ←− an -adic representation, and as a Z -module, it is a free Z module of rank g. The Q -vector space V G = Q ⊗Z T G is g dimensional, and as an -adic representation of Gk , it is isomorphic to Hom(X, Q (1)). In general, for a smooth connected commutative group scheme over a perfect ﬁeld, we have the following. Theorem A.49. Let F be a perfect ﬁeld, and let G be a smooth connected separated commutative group scheme of ﬁnite type over F . (1) There exist, unique up to isomorphism, an abelian variety A over F and a surjective morphism G → A whose kernel H is a smooth connected aﬃne commutative group scheme. For an abelian variety B over F , a morphism G → B of commutative group schemes restricts to 0 on H, and induces a morphism A → B of abelian varieties. (2) A smooth connected aﬃne commutative group scheme over F is decomposed uniquely into the product of a torus T over F and a unipotent smooth connected aﬃne commutative group scheme U . For a smooth connected separated commutative group scheme of ﬁnite type over F , the abelian variety A deﬁned by Theorem A.49(1) is called the abelian part of G. Moreover, for the smooth connected aﬃne commutative group H deﬁned by Theorem A.49(1), the torus T deﬁned by Theorem A.49(2) is called the torus part of G, and U is called the unipotent part of G. Corollary A.50. Let F be a perfect ﬁeld, and let = char(F ) be a prime number. Let G be a smooth connected separated commutative group scheme of ﬁnite type over F . Let A be the abelian part of G, and let T be its torus part.

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(1) We have an exact sequence (A.5) 0 −→ T T −→ T G −→ T A −→ 0. T G is a free Z -module of rank 2a + t, where a = dim A and t = dim T . (2) Let G → H be a morphism of smooth connected separated commutative group schemes of ﬁnite type, and let B be the abelian part of H. If T G → T H is injective, so is T A → T B. Proof. (1) Replacing G by its quotient by the unipotent part if necessary, we may assume G is an extension of A by T . Since [n ] : T → T is surjective, we obtain the exact sequence (A.5). It is clear from (A.5) and Corollary A.48 that rank T G = 2a + t. (2) Let S be the torus part of H. As in the proof of (1), we may assume that G is an extension of A by T , and H is an extension of B by S. Since T G → T H is injective, Ker(G → H) is ﬁnite. Since Ker(A → B) is proper and Coker(T → S) is aﬃne, the image of the connected morphism Ker(A → B) → Coker(T → S) is ﬁnite. Thus, by the snake lemma, Ker(A → B) is ﬁnite and T A → T B is injective. Proposition A.51. Let A be an abelian variety over k. (1) The endomorphism ring End A is a ﬁnitely generated free Zmodule. (2) If = char(k) is a prime number, then Z ⊗ End A → End T (A) is injective. (3) If k = C, then End A → End H1 (Aan , Z) is injective. (4) If char(k) = 0, the natural homomorphism End A → End Γ(A, Ω) is injective.

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Symbol Index

∼, 122 , 123 an (f ), 61 ad(P )(ρ), 122 ap (E), 21 Aut(E), 53 B(Z/N Z), 46 CN (q), 75 CN,τ , 72 χ , 83 χ ¯N , 83 Def ρ,D ¯ Σ , 124 deg f , 23 Δ, 14, 26, 74, 134, 140 Δ∗ , 74 Δ(q), 59 det(T − ρ(ϕp )), 83 det ρ(ϕp ), 83 e, 60 e∗ , 61 eN , 60 EFp , 21 E(K), 14 E[N ], 28 E[N ](K), 24 E (N ) , 32 Ek (q), 59 E sm , 30 Eτ , 71 EZ(p) , 27 f∗ (ρ), 122 f11 , 44 Frp , 23

Ga , 175 Gm , 175 GK , 86 GQ , 82 Γ0 (N ), 72 Γ0 (N ), 41 g0 (N ), 43 g(N ), 46 j, 39 jE , 36 Kf , 70 L(E, s), 22 L(f, s), 64 Liftρ¯, 160 M0 (N ), 40 M0 (N ), 41 M0 (N, n), 55 M(N ), 45 M(N ), 45 μN (K), 86 μN , 176 μN (Q), 83 [N ], 23, 28 NΣ , 121 ℘, 71 ℘(τ, z), 76 Φ(N )K , 70 Φ(NΣ )K,ρ¯, 121 ϕ4 (N ), 43 ϕ6 (N ), 43 ϕ(N ), 42 ϕ∞ (N ), 43 ϕf , 63 197

198

ϕp , 82 Pn,K , 29 ψ(N ), 42 Q(f ), 69 Qnr p , 100 RΣ , 120 RΣ , 120 ρ¯E,N , 89 ρE, , 89 , 130 ρmod Σ S(N ), 42 s, t, 56 Sρ¯, 123 Σ, 120 Σ(ρ), 124 T (N ), 57 T (N )K , 67 T (N )∨ , 66 T (N ), 68 T (N )K , 68 TΣ , 120 T E, 24 Tn , 57 TΣ , 120 Tr ρ(ϕp ), 83 U 1 GLn (A), 122 X0 (11), 44 X0 (N ), 41 X0 (N, n), 56 X(N ), 45 x(q, t), 59 Y0 (N ), 41 Y0 (N, n), 56 Y (N ), 45 y(q, t), 59 Z(p) , 16 Znr p , 101

SYMBOL INDEX

Subject Index

elliptic curve over a ﬁeld, 13 equivalent of liftings, 122 ´ etale covering, 178 ´ etale local, 179 ´ etale sheaf, 179 ´ etale sheaﬁﬁcation, 179

abelian part, 187 abelian scheme, 185 abelian variety, 186 absolute Galois group, 82 absolutely irreducible, 85 additive group, 175 additive reduction, 17 analytic expression elliptic curve, 71 modular curve, 72 modular form, 73 augmentation, 132

ﬁne moduli scheme, 39 ﬁnite GK -module, 86 ﬁnite ´ etale commutative group scheme, 176 ﬁnite ﬂat commutative group scheme, 176 ﬁnite morphism, 173 Fitting ideal, 156 ﬂat covering, 178 ﬂat local, 179 ﬂat sheaf, 178 free, 180 Frobenius conjugacy class, 82 Frobenius substitution, 82 full Hecke algebra, 136 functor over O, 159 over S, 37

Cartier divisor, 183 Cartier dual, 177 character group, 186 closed condition, 182 coarse moduli scheme, 39 commutative group scheme, 175 complete, 133 complete intersection, 131 completed group algebra, 144 conductor, 17, 98 connected component, 103 constant group scheme, 175 cyclic group scheme, 176 cyclotomic character, 83

G-torsor, 179 Galois representation deﬁned by the N -torsion points of E, 89 Γ0 (N )-structure, 41 Γ(N )-structure, 45 geometric Frobenius, 23 good, 3, 10

deformation ring, 125 degree morphism of elliptic curves, 23 discriminant, 14, 27 elliptic curve, 26 199

200

SUBJECT INDEX

GQ -module, 96 Galois representation, 98 good reduction, 16 Hecke algebra over O, 128 over Q, 57 Hecke module, 136 Hecke operator, 57 inertia group, 97, 180 irreducible, 84 j-invariant, 36 -adic cyclotomic character, 83 -adic representation, 83 associated with f , 92 determined by E, 89 L-function of a modular form, 64 of an elliptic curve, 22 lifting, 122, 131 local over S, 179 modular, 6 -adic representation, 92 elliptic curve, 65 mod representation, 93 of level N , 8 mod representation, 83 morphism of elliptic curves, 22 multiplication-by-N morphism, 23, 28 multiplicative, 186 multiplicative group, 175 multiplicative reduction, 17 multiplier, 134 N´ eron model, 107 open subscheme deﬁned by P, 179 ordinary, 98 primary form, 5, 62 primitive form, 64 over K, 70 proﬁnitely generated complete local O-algebra, 122

q-expansion, 61 q-expansion principle, 61 quasi-ﬁnite morphism, 173 quotient, 177 reduced Hecke algebra over O, 128 over Q, 68 reducible, 84 RTM-triple, 133 Selmer group, 139 semistable elliptic curve, 17 Galois representation, 98 semistable model, 31 smooth model, 27 smooth of relative dimension d, 171 stable reduction, 16 subfunctor, 160 surjection of RTM-triples, 133 Tate module, 24, 89, 103, 186 torus, 186 torus part, 187 type, 161 type DΣ , 123 unipotent, 186 universal element, 39, 160 unramiﬁed, 82