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*English*
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*Table of contents : Contents......Page 4Preface......Page 8Modular curves over Z......Page 16Modular forms and Galois representations......Page 76Hecke modules......Page 122Selmer groups......Page 158Curves over discrete valuation rings......Page 194Finite commutative group scheme over Z_p......Page 206Jacobian of a curve and its Néron model......Page 214Bibliography......Page 228Symbol index......Page 232Subject index......Page 236*

Fermat's Last Theorem The Proof

IWANAMI SERIES IN MODERN MATHEMATICS

10.1090/mmono/245

Translations of

MATHEMATICAL MONOGRAPHS Volume 245

Fermat's Last Theorem The Proof 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 2014 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 11F11, 11F80, 11G05, 11G18. Library of Congress Cataloging-in-Publication Data ISBN 978-0-8218-9849-9 Fermat’s last theorem: the proof (Translations of mathematical monographs ; volume 245) The ﬁrst volume was catalogued as follows: 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 2014 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

19 18 17 16 15 14

Contents Basic Tools Preface

xi

Preface to the English Edition

xvii

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

35 35 37 40 44 52 55 58 62 65

v

vi

CONTENTS

2.10. 2.11. 2.12. 2.13.

Primary forms, primitive forms, and Hecke algebras The analytic expression The q-expansion and analytic expression The q-expansion and Hecke operators

66 70 74 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

171 171 175 177 178

CONTENTS

A.5. G-torsor A.6. Closed condition A.7. Cartier divisor A.8. Smooth commutative group scheme

vii

179 182 183 185

Bibliography

189

Symbol Index

197

Subject Index

199

The Proof Preface

ix

Preface to the English Edition

xv

Chapter 8. Modular curves over Z 8.1. Elliptic curves in characteristic p > 0 8.2. Cyclic group schemes 8.3. Drinfeld level structure 8.4. Modular curves over Z 8.5. Modular curve Y (r)Z[ r1 ] 8.6. Igusa curves 8.7. Modular curve Y1 (N )Z 8.8. Modular curve Y0 (N )Z 8.9. Compactiﬁcations

1 1 6 12 20 25 32 37 41 48

Chapter 9. Modular forms and Galois representations 9.1. Hecke algebras with Z coeﬃcients 9.2. Congruence relations 9.3. Modular mod representations and non-Eisenstein ideals 9.4. Level of modular forms and ramiﬁcation of -adic representations 9.5. Old part 9.6. N´eron model of the Jacobian J0 (M p) 9.7. Level of modular forms and ramiﬁcation of mod representations

61 61 70 76 81 90 97 102

viii

CONTENTS

Chapter 10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7.

10. Hecke modules Full Hecke algebras Hecke modules Proof of Proposition 10.11 Deformation rings and group rings Family of liftings Proof of Proposition 10.37 Proof of Theorem 5.22

Chapter 11.1. 11.2. 11.3. 11.4. 11.5.

11. Selmer groups Cohomology of groups Galois cohomology Selmer groups Selmer groups and deformation rings Calculation of local conditions and proof of Proposition 11.38 11.6. Proof of Theorem 11.37

107 108 113 118 125 129 136 140 143 143 149 157 161 165 169

Appendix B. Curves over discrete valuation rings B.1. Curves B.2. Semistable curve over a discrete valuation ring B.3. Dual chain complex of curves over a discrete valuation ring

179 179 182

Appendix C. Finite commutative group scheme over Zp C.1. Finite ﬂat commutative group scheme over Fp C.2. Finite ﬂat commutative group scheme over Zp

191 191 192

Appendix D. D.1. The D.2. The D.3. The D.4. The

199 199 201 205 209

Jacobian of a curve and its N´eron model divisor class group of a curve Jacobian of a curve N´eron model of an abelian variety N´eron model of the Jacobian of a curve

187

Bibliography

213

Symbol Index

217

Subject Index

221

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 ∗ Chapters 0–7 along with Appendix A appeared in Fermat’s Last Theorem: Basic Tools, a translation of the Japanese original.

PREFACE

xi

in commutative algebra will be proved in Chapter 6. In Chapter 7, we will prove the existence theorem of deformation rings. 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 schemes over Zp , which will be important for the study of p-adic

xii

PREFACE

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 second 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 0–7 and Appendix A. The second volume, Fermat’s Last Theorem: The Proof , contains Chapters 8–11 and Appendices B, C, and D. This second volume of the book on the proof of Fermat’s Last Theorem by Wiles and Taylor presents a full account of the proof started in the ﬁrst volume. As well as the proof itself, basic materials behind the proof, including the Galois representations associated with modular forms, the integral models of modular curves, the Hecke modules, the Selmer groups, etc., are studied in detail. 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 October 2014

xv

10.1090/mmono/245/01

CHAPTER 8

Modular curves over Z In Chapter 2, we used modular curves over Q to deﬁne modular forms with Q coeﬃcients. A modular curve over Q is the ﬁber over the generic point of a modular curve over Z. In this chapter we will deﬁne modular curves over Z, and prove their fundamental properties. In the next chapter we will derive various properties of Galois representations associated with modular forms by examining the properties of modular curves over Z at each prime number. In §8.1, we classify elliptic curves in positive characteristics into ordinary elliptic curves and supersingular elliptic curves. We will deﬁne modular curves over Z using the Drinfeld level structure, which will be introduced in §8.3 after some preparations in §8.2. The Drinfeld level structure plays an important role when we study the structure of modular curves at a prime number dividing the level. In §8.5, we will deﬁne modular curves that play a complementary role, and we study reduction of these curves modulo p in §8.6. Using the results of §8.6, we will prove fundamental properties, Theorems 8.34 and 8.32 in §8.7 and §8.8, respectively. Since the modular curves deﬁned in §8.4 are aﬃne curves, we will compactify them and prove fundamental properties, Theorems 8.63 and 8.66, in §8.9. 8.1. Elliptic curves in characteristic p > 0 Let p be a prime number, let S be a scheme over Fp , and let X be a scheme over S. Let FS : S → S be the absolute Frobenius morphism, which is deﬁned by the pth power mapping of the coordinate rings. We denote by X (p) the ﬁber product X ×S S by FS : S → S. The morphism X → X (p) deﬁned by the commutative diagram F

X −−−X−→ ⏐ ⏐ F

X ⏐ ⏐

S −−−S−→ S 1

2

8. MODULAR CURVES OVER Z

of the absolute Frobenius morphisms is denoted by F and is called the relative Frobenius morphism. If X = E is an elliptic curve, the morphism F : E → E (p) is a morphism of elliptic curves over S of degree p. Let V : E (p) → E be the dual morphism of F : E → E (p) . V is also a morphism of elliptic curves of degree p, and the compositions V ◦ F and F ◦ V are the multiplication-by-p mappings [p] : E → E and [p] : E (p) → E (p) , respectively. For example, if S = Spec A and E is an elliptic curve given by y 2 +a1 xy+a3 y = x3 +a2 x2 +a4 x+a6 , then E (p) is deﬁned by the equation y 2 +ap1 xy +ap3 y = x3 +ap2 x2 +ap4 x+ap6 , and F : E → E (p) is deﬁned by (x, y) → (xp , y p ). e For a nonnegative integer e, we denote by F e : E → E (p ) the i i+1 composition of F : E (p ) → E (p ) , i = 0, . . . , e − 1, and let V e : e E (p ) → E be the dual of F e . Definition 8.1. Let p be a prime number, let S be a scheme over Fp , and let E be an elliptic curve over S. (1) If V : E (p) → E is ´etale, we say E is ordinary. (2) If E[p] = Ker F 2 , we say E is supersingular . A supersingular elliptic curve over a ﬁeld is smooth and thus nonsingular. The term “supersingular” suggests it is very special in some sense. However, it is not directly related to the term “singular” in the sense that the local ring is not regular. Later in Theorem 8.32(4), we will prove that the points on the modular curve Y0 (M p)Fp (p M ) corresponding to supersingular elliptic curves coincide with the singular points of this modular curve. By Lemma 8.44, there exists an ordinary elliptic curve for each prime number p. We also prove that there exists a supersingular curve for each p in Corollary 8.64. We also calculate the number of isomorphism classes of supersingular elliptic curves over an algebraically closed ﬁeld of characteristic p > 0. Let p be a prime number, let S be a scheme over Fp , and let E be an elliptic curve over S. Since the condition that V : E (p) → E is ´etale is an open condition on S, the condition that E is ordinary is also an open condition. We denote by S ord the maximal open subscheme U of S such that the restriction EU is ordinary. Meanwhile, if we apply Corollary A.37(2) to Ker F 2 and E[p], the condition P that ET = E ×S T is supersingular for an S-scheme T is a closed condition on S. We denote by S ss the closed subscheme of S deﬁned by the closed condition P. We show that S ord is the complementary open subscheme of S ss . More precisely, we have the following.

8.1. ELLIPTIC CURVES IN CHARACTERISTIC p > 0

3

Proposition 8.2. Let k be a ﬁeld of characteristic p > 0, and let E be an elliptic curve over k. Let k¯ be an algebraic closure of k. Then, the following hold. ¯ is either p or 1. (1) The order of the abelian group E[p](k) (2) The following conditions (i)–(iv) are equivalent. ¯ is p. (i) The order of the abelian group E[p](k) (ii) E is ordinary. (iii) For any integer e ≥ 1, Ker V e is ´etale and the abelian ¯ is isomorphic to Z/pe Z. group Ker V e (k) (iv) For any integer e ≥ 1, the group scheme E[pe ]k¯ is isomorphic to Z/pe Z × μpe . (3) The following conditions (i)–(iii) are equivalent. ¯ is 1. (i) The order of the abelian group E[p](k) (ii) E is supersingular. (iii) For any integer e ≥ 1, the only closed subgroup scheme of E of order pe is Ker F e . ¯ Proof. It suﬃces to prove the case where k = k. 0 (1) As in Proposition 3.45, let E[p] the connected component of E[p] containing 0, and let E[p]´et be the maximal ´etale quotient. Consider the exact sequence 0 −→ E[p]0 −→ E[p] −→ E[p]´et −→ 0. ¯ the exact sequence (8.1) gives an isomorSince we assumed k = k, phism of ﬁnite groups E[p](k) → E[p]´et (k). Consider the Cartier dual of (8.1). By the Weil pairing, the Cartier dual of E[p] is E[p] itself, and the Cartier dual of E[p]´et is connected. Thus, the Cartier dual (E[p]´et )∨ is a closed subgroup scheme of E[p]0 . Hence, (E[p](k))2 = (deg E[p]´et )(deg(E[p]´et )∨ ) divides (deg E[p]´et )(deg E[p]0 ) = deg E[p] = p2 . (2) (i) ⇒ (ii). Since [p] = V ◦ F , we obtain the exact sequence (8.2) F 0 −→ Ker(F : E → E (p) ) −→ E[p] −→ Ker(V : E (p) → E) −→ 0. (8.1)

Since Ker F (k) = 0, E[p](k) → Ker V (k) is an isomorphism of ﬁnite groups. By (i), the order of Ker V (k) is p, and thus the isogeny V : E (p) → E of degree p is ´etale. e−1 (ii) ⇒ (iii). Since V e = V ◦ V (p) ◦ · · · ◦ V (p ) , if V is ´etale, e e e V is also ´etale of degree p . Thus, Ker V (k) is an abelian group of order pe . Moreover, since the order of a p-torsion point of Ker V e ⊂ e E (p ) is p, Ker V e is isomorphic to Z/pe Z.

4

8. MODULAR CURVES OVER Z

(iii) ⇒ (iv). In the exact sequence (8.3) Fe

e

e

0 → Ker(F e : E → E (p ) ) −→ E[pe ] −→ Ker(V e : E (p

)

→ E) → 0,

Ker F e is connected and Ker V e is ´etale. Let E[pe ]red be the reduced part of E[pe ]. Then, the composition E[pe ]red → E[pe ] → Ker V e is an isomorphism since k is algebraically closed. Since Ker V e is isomorphic to Z/pe Z, Ker F e is isomorphic to its Cartier dual μpe . (iv) ⇒ (i). This is clear from μp (k) = {1}. (3) (i) ⇒ (iii). If G is a closed subgroup scheme of E[pe ] of degree pe , then G is connected. Therefore, if m0 is the maximal ideal e of the local ring OE,0 , we have G = Spec OE,0 /mp0 = Ker F e . (iii) ⇒ (ii). Since E[p] is a closed subgroup scheme of degree p2 , we have E[p] = Ker F 2 . ¯ = 0. ¯ = Ker F 2 (k) (ii) ⇒ (i). We have E[p](k) Corollary 8.3. Let p be a prime number, let S be a scheme over Fp , and let E be an elliptic curve over S. Then, we have (8.4)

S ord = S

S ss .

Proof. It is clear from Proposition 8.2.

Lemma 8.4. Let p be a prime number, let S be a scheme over Fp , and let E be an elliptic curve over S. Let e ≥ 1 be an integer. e

(1) If E is ordinary, then V e : E (p ) → E is ´etale, and Ker V e is isomorphic to Z/pe Z ´etale locally. (2) If f : E → E is an ´etale isogeny of degree pe , then both E and E are ordinary, and there exists a unique isomorphism g : E → e E (p ) satisfying f = V e ◦ g. Proof. (1) Similar to the proof of Proposition 8.2(2)(ii) ⇒ (iii). (2) Let tf : E → E be the dual of f . Since f ◦ tf = [pe ], Ker tf is an open and closed subgroup scheme of E[pe ] of degree pe , and its underlying set is equal to that of the 0-section of E . Thus, Ker tf is a closed subgroup scheme. Hence, we have Ker tf = Ker F e , and an e isomorphism g : E → E (p ) satisfying F e = g ◦ tf is induced. Clearly, g satisﬁes the required condition and g is unique. e Since V e : E (p ) → E is ´etale, E is ordinary. Moreover, since e E and E (p ) are isomorphic, E is also ordinary.

8.1. ELLIPTIC CURVES IN CHARACTERISTIC p > 0

5

Proposition 8.5. Let p be a prime number, and let E be an elliptic curve over Fp . Let a = 1 + p − E(Fp ). Then, the following conditions (i)–(iii) are equivalent. (i) E is ordinary. (ii) p a. (iii) If p = 2, a = ±1. If p = 3, a = ±1, ±2. If p ≥ 5, a = 0. Proof. As in Proposition 1.21, we have 1 − at + pt2 = det(1−F t : D(E)) by Theorem C.1(4). By Proposition 8.2(2)(iv)⇔(ii) and Theorem C.1(2), E is ordinary if and only if one of the eigenvalues of the action of F on D(E) is a p-adic unit. This is equivalent to condition (ii). The equivalence of conditions (ii) and (iii) follows from the fact √ |a| < 2 p and Theorem 1.15. Example 8.6. Let p be an odd prime number, and let E be the elliptic curve over Fp given by y 2 = x3 − x. Then, E is ordinary if p ≡ 1 mod 4, and E is supersingular if p ≡ −1 mod 4. Indeed, since E[2] = {∞, (0, 0), (±1, 0)} is a subgroup of E(Fp ), we have E(Fp ) = p + 1 − a ≡ 0 mod 4. Thus, if p ≡ 1 mod 4, we have a ≡ 2 mod 4, which means a = 0. If p ≡ −1 mod 4, then −1 is not a quadratic residue mod p. Hence, if x = 0, ±1, either x3 − x or (−x)3 − (−x) = −(x3 − x) is a quadratic residue, and not both. Thus, we have E(Fp ) = p + 1, which means a = 0. Similarly, for a prime number p ≥ 5, let E be the elliptic curve over Fp deﬁned by y 2 = x3 − 1. Then, E is ordinary if p ≡ 1 mod 3, and its supersingular if p ≡ −1 mod 3. Corollary 8.7. Let p be an odd prime number, and let E be an elliptic curve over Qp . Then the following conditions (i) and (ii) are equivalent. (i) The p-adic representation Vp E of GQp is ordinary. (ii) Either E has good reduction and EFp is ordinary or E has multiplicative reduction. Proof. First, we assume E has good reduction, and we show Vp E is ordinary if and only if EFp is ordinary. By Theorem C.6(3), the subspace D (E) ⊂ D(E) is one dimensional. Thus, by Corollary C.8, Vp E is ordinary if and only if there exist p-adic units α and β such that 1 − at + pt2 = deg(1 − F t : D(E)) decomposes into (1 − αt)(1 − pβt). This is in turn equivalent to that EFp is ordinary by Proposition 8.5.

6

8. MODULAR CURVES OVER Z

Furthermore, by Proposition 3.46(2), E has stable reduction if Vp E is ordinary. This shows (i)⇒(ii). Suppose E has multiplicative reduction. In this case, we have already proved that Vp E is ordinary in the proof of Proposition 3.46(2) (i)⇒(ii). This shows (ii)⇒(i). 8.2. Cyclic group schemes In this section we deﬁne cyclic group scheme as a preparation for the deﬁnition of modular curves over Z. Definition 8.8. Let S be a scheme, let N ≥ 1 be an integer, and let X be a ﬁnite ﬂat scheme of ﬁnite presentation over S of degree N . A family of sections P1 , . . . , PN : S → X is called a full set of sections of X if it satisﬁes (8.5)

NXR /R (f ) =

N

f (Pi )

i=1

for any commutative ring R, any morphism Spec R → S, and any element f ∈ Γ(X ×S Spec R, O). Lemma 8.9. Let S be a scheme, let N ≥ 1 be an integer, and let X be a ﬁnite ﬂat scheme of ﬁnite presentation over S of degree N . If a family of sections P1 , . . . , PN : S → X is a full set of sections, the morphism (8.6)

N

Pi : S

···

S→X

i=1

is surjective. Proof. It suﬃces to show it when S = Spec k, where k is an algebraically closed ﬁeld, but it is clear in this case. Even if the morphism (8.6) is surjective, P1 , . . . , PN may not be a full set of sections of X. For example, let k be a ﬁeld, let S = Spec k[]/(2 ), and let X = Spec k[, ](2 , 2 ). Deﬁne sec tions P1 , P2: S → X by → 0 and → , respectively. Then, P1 P2 : S S → X is surjective. However, if we let f = 1 + , we have NX/S (f ) = 1 = f (P1 )f (P2 ) = 1 + . This means P1 and P2 do not form a full set of sections of X. If X is ´etale, the condition in Lemma 8.9 is a necessary and suﬃcient condition.

8.2. CYCLIC GROUP SCHEMES

7

Corollary 8.10. If X is ´etale over S in Lemma 8.9, then the following conditions are equivalent. (i) P1 , . . . , PN : S → X form a full set of sections of X. Pi : S · · · S → X is an isomorphism. (ii) N i=1 N (iii) i=1 Pi : S · · · S → X is surjective. (ii) ⇒ (i) is clear. (i) ⇒ (iii) holds by Lemma 8.9. Since Proof. S · · · S and X are both ﬁnite ´etale of degree N , (ii) and (iii) are equivalent. Proposition 8.11. Let S be a scheme, let N ≥ 1 be an integer, and let X be a ﬁnite ﬂat scheme of ﬁnite presentation over S of degree N . Let P1 , . . . , PN : S → X be a family of sections of X. The condition P that P1 , . . . , PN form a full set of sections of X is a closed condition on S. The ideal of OS that deﬁnes the closed subscheme T of S deﬁned by the closed condition P is locally of ﬁnite type. Proof. Since the assertion is local on S, it suﬃces to show the cases where S = Spec A and X = Spec B with B a free A-module of rank N . Let g1 , . . . , gN be a basis of the A-module B. The equality (8.5) holds for any R and f if and only if (8.5) holds for the polynomial ring R = A[T1 , . . . , TN ] and f = N j=1 gj Tj ∈ B[T1 , . . . , TN ]. For such R and f , (8.5) becomes N N N

(8.7) NB[T1 ,...,TN ]/A[T1 ,...,TN ] gj Tj = gj (Pi )Tj . j=1

i=1 j=1

If I ⊂ A is the ideal generated by the coeﬃcients of the diﬀerence of the both sides of (8.7), the closed subscheme T of S deﬁned by I represents the functor FP . Since each side of (8.7) is a homogeneous polynomial of degree N in T1 , . . . , TN with A coeﬃcients, I is ﬁnitely generated. If X is a closed subscheme of a smooth curve, we have the proposition below. Note that if E is a smooth curve over S and X is a closed subscheme of E that is ﬁnite ﬂat of ﬁnite presentation over S, then X is a Cartier divisor of E by Lemma B.2(1). In particular, a section P : S → E deﬁnes a Cartier divisor of E. Proposition 8.12. Let S be a scheme, let E be a smooth curve over S, and let N ≥ 1 be an integer. Suppose X is a closed subscheme of E that is ﬁnite ﬂat of ﬁnite presentation over S of degree N . For sections P1 , . . . , PN : S → X, the following are equivalent.

8

8. MODULAR CURVES OVER Z

(1) P1 , . . . , PN form a full set of sections of X. (2) The following equality of Cartier divisors holds:

X=

N

[Pi ].

i=1

Proof. (ii) ⇒ (i). Let Spec R → S be a morphism of schemes. We show NXR /R (f ) = N i=1 f (Pi ) for f ∈ Γ(XR , O). Replacing S by Spec R, we may assume S = Spec R. For i = 1, . . . , N , let Ii be the deﬁning ideal sheaf of the Cartier divisor [Pi ] of E. By the equality of N divisors X = i=1 [Pi ], the ﬁnitely generated free OS -module OX is a i−1 i successive extension of the invertible OS -modules j=1 Ij / j=1 Ij . Since the multiplication-by-f map of OX induces the multiplicationi−1 i N by-f (Pi ) map of j=1 Ij / j=1 Ij , we have NX/S (f ) = i=1 f (Pi ). N (i) ⇒ (ii). Both X and i=1 [Pi ] are ﬁnite ﬂat of ﬁnite presentation over S of degree N . Thus, it suﬃces to show X is a closed N subscheme of i=1 [Pi ]. Let s ∈ S. We may replace S by Spec OS,s . By Lemma 8.9, we have X = N i=1 Pi (S). Since A = OS,s is a local P (s) implies P (S) ∩ Pj (S) = ∅ for i, j = 1, . . . , N . ring, Pi (s) = j i Thus X = x→s Pi (s)=x Pi (S). Hence, for an inverse image x of s, we have Spec OX,x = Pi (s)=x Pi (S) and X = x→s Spec OX,x . Therefore, it suﬃces to show the assertion assuming X = Spec OX,x . Replacing E by an open neighborhood of x, we may assume E is also aﬃne. Let E = Spec B and X = Spec B. Replacing E by an open neighborhood of x again if necessary, we may assume the divisor [P1 ] of E is deﬁned at t ∈ B. For i = 2, . . . , N , t − t(Pi ) ∈ B is also 0 on Pi . Since the divisor [x] = [Pi (s)] of Es is deﬁned by t − t(Pi ) on a neighborhood of x, it follows from Nakayama’s lemma that a divisor [Pi ] of E is deﬁned by t − t(Pi ) ∈ B on a neighborhood of x. Replacing E once again by an open neighborhood of x if necessary, we may assume the divisor [Pi ] of E is deﬁned by t − t(Pi ) ∈ B for i = 1, . . . , N . Let Φ(T ) = NB/A (T − t|X ) ∈ A[T ]. Applying (8.5) to f = T − t ∈ B ⊗A A[T ], we obtain Φ(T ) = N i=1 (T − t(Pi )). Thus, N Φ(t) = i=1 (t − t(Pi )) ∈ B is a generator of the deﬁning ideal of the divisor N i=1 [Pi ] of E. On the other hand, by the theorem of Cayley and Hamilton, Φ(t|X ) ∈ B is 0. Thus, Φ(t) ∈ B is contained

8.2. CYCLIC GROUP SCHEMES

9

in the deﬁning ideal Ker(B → B) of X. This shows that X is a closed N subscheme of i=1 [Pi ] at each point of X. Definition 8.13. Let S be a scheme, let N ≥ 1 be an integer, and let G be a ﬁnite ﬂat commutative group scheme of ﬁnite presentation over S of degree N . (1) Let P : S → G be a section of G. If the family 0, P , 2P, . . . , (N − 1)P is a full set of sections, we call P a generator of G. (2) G is called a cyclic group scheme if there exists a generator of G ﬂat locally on S. The degree of a cyclic group scheme is sometimes called the order. If N is invertible on S, Lemma 8.15 below shows this coincides with the ordinary deﬁnition. We show the following proposition ﬁrst. Proposition 8.14. Let S be a scheme, and let G be a ﬁnite ﬂat commutative group scheme of ﬁnite presentation of degree M . Then, the multiplication-by-M mapping of G is the 0 mapping. Proof. It suﬃces to show g M = 1 for any scheme T over S and any section g ∈ G(T ). Replacing S by T , we may assume S = T . Since the assertion is local on S, we may assume S = Spec R, G = Spec A, and A is a free R-module of rank M . Let G∨ = Spec A∨ , where A∨ = HomR (A, R), be the Cartier dual of G. The multiplication of A∨ is the dual of the ring morphism A → A ⊗R A that deﬁnes the group operation μ : G ×S G → G. Let g ∈ G(S), let μg : G → G be the translation by g, and let μg : A → A be the corresponding isomorphism of rings. We identify g ∈ G(S) = HomR-alg (A, R) ⊂ HomR (A, R) with an element of A∨ = HomR (A, R). Furthermore, we identify μg ∈ G(G) = HomR-alg (A, A) ⊂ EndR A with an element of EndR A = A ⊗R A∨ . Since G(G) is a subgroup of (A ⊗R A∨ )× , μg ∈ A ⊗R A∨ is invertible. Moreover, since μg : G → G is the product of id = μ1 : G → G g and G → S → G, we have μg = μ1 · (1 ⊗ g) ∈ A ⊗R A∨ . Thus, taking the norm N = NA⊗R A∨ /A∨ of both sides, we obtain N (μg ) = N (μ1 ) · g M ∈ A∨× . On the other hand, since μg (μ1 ) = μ1 ◦ μg = μg , the homomorphism of A∨ -algebras (μg ⊗ 1) : A ⊗R A∨ → A ⊗R A∨ maps μ1 to μg . Thus, we have N (μg ) = N (μ1 ) ∈ A∨× . This shows g M = 1 ∈ G(S) ⊂ A∨ .

10

8. MODULAR CURVES OVER Z

Lemma 8.15. Let S be a scheme, let N ≥ 1 be an integer invertible on S, and let G be a ﬁnite ﬂat commutative group scheme of ﬁnite presentation over S of degree N . (1) For a section P : G → S, the following are equivalent. (i) P is a generator of G. s) is a generator (ii) For any geometric point s¯ in S, Ps¯ ∈ G(¯ of G(¯ s). (iii) There exists an isomorphism of commutative group schemes Z/N Z → G that maps 1 to P . (2) The following are equivalent. (i) G is a cyclic group scheme. (ii) For any geometric point s¯ in S, G(¯ s) is a cyclic group. (iii) There exits an isomorphism of commutative group schemes Z/N Z → G ´etale locally on S. Proof. Since G is ´etale over S by Proposition 8.14 and Deﬁnition A.17, the assertions are clear from Corollary 8.10. Lemma 8.16. Let S be a scheme, let N ≥ 1 be an integer, and let G be a ﬁnite ﬂat commutative group scheme over S of ﬁnite presentation of degree N . (1) Let P : S → G be a section of D. The condition that P is a generator of G is a closed condition on S. The ideal of OS deﬁning the closed subscheme T of S deﬁned by this condition is locally of ﬁnite type. (2) The functor that associates the set (8.8)

{P ∈ G(T ) | P is a generator of GT } to a scheme T over S is represented by a closed subgroup scheme G× of G.

Proof. (1) It suﬃces to apply Proposition 8.11 to the sections 0, P, . . . , (N − 1)P of G. (2) It suﬃces to apply (1) to the pullback section of G to G deﬁned by the diagonal map G → G ×S G. The closed subgroup scheme G× of G is called the scheme of generators of G. If G× is faithfully ﬂat over S, then G× is a ﬂat covering of S, and thus G is a cyclic group scheme. Later in Corollary 8.53(1), we will show a partial converse of this property. Lemma 8.17. Let S be a scheme, let 0 → G → G → G → 0 be an exact sequence of ﬁnite ﬂat commutative group schemes of ﬁnite

8.2. CYCLIC GROUP SCHEMES

11

presentation over S, and let P be a section of G. Suppose G is ´etale of degree M . Then, the following are equivalent. (i) P is a generator of G. (ii) The image P of P is a generator of G , and M P is a generator of G . Proof. (i) ⇒ (ii). Let P be a generator of G, and let N be −1 : S · · · S → G → G the degree of G. By Lemma 8.9, N i=0 iP N −1 is surjective. Thus, by Proposition 8.14, i=0 iP : S · · · S → G is also surjective. Therefore, by Corollary 8.10, P deﬁnes an isomorphism Z/M Z → G , and P is a generator of G . Moreover, M −1 −1 since M i=0 (+iP ) : i=0 G → G is an isomorphism of schemes, P is a generator of G if and only if M P is a generator of G . (ii) ⇒ (i). P deﬁnes an isomorphism Z/M Z → G , by Corollary 8.10. The remaining part is similar to the proof of (i) ⇒ (ii). Lemma 8.18. Let S be a scheme, and let E be a commutative group scheme over S that is a smooth curve over S. Let N ≥ 1 be an integer. (1) Let P : S → E be a section of E. Then, the following are equivalent. (i) There exists a closed subgroup scheme of E of order N such that P is its generator. −1 (ii) The Cartier divisor N i=0 [iP ] of E is a closed subgroup scheme of E. (2) If one of the equivalent conditions of (1) holds, then N P = 0, N −1 and G = i=0 [iP ]. (3) Let G be a closed subgroup scheme of E ﬁnite ﬂat of ﬁnite presentation of degree N over S. Let P be a section of E over G deﬁned by the diagonal map G → E ×S G. The scheme G× of generators of G is a closed subscheme deﬁned by the closed condition on G that for any scheme T over G, the two closed N −1 subschemes of ET , GT and i=0 [iP ]T are equal. Proof. (1) Clear from Proposition 8.12. N −1 (2) Since the cyclic subgroup scheme i=0 [iP ] is a ﬁnite ﬂat commutative group scheme of ﬁnite presentation of degree N , the N −1 image of the multiplication-by-N map of i=0 [iP ] is 0 by PropoN −1 sition 8.14. Thus, we have N P = 0. G = i=0 [iP ] is clear from Proposition 8.12. (3) Clear from (2).

12

8. MODULAR CURVES OVER Z

Lemma 8.19. Let N ≥ 1 be an integer. The group scheme μN = Spec Z[X]/(X N − 1) consisting of N th roots of unity is a cyclic group scheme. If ΦN (X) ∈ Z[X] is the N th cyclotomic polynomial, the scheme μ× N of generators of μN is Spec Z[X]/(ΦN (X)). Proof. Spec Z[X]/(ΦN (X)) is a ﬂat covering of Spec Z. Thus, if we show μ× N = Spec Z[X]/(ΦN (X)), then it follows that μN is a cyclic group scheme. Clearly, μ× = Spec Z[ N1 ][X]/(ΦN (X)) N,Spec Z[ 1 ] N

over Spec Z[ N1 ] by Lemma 8.15. Thus, applying Corollary A.44(1) to S = Spec Z, X = Gm,S , A = μ× N,S , and B = Spec Z[X]/(ΦN (X)), it suﬃces to show the inequality deg μ× N,k ≤ ϕ(N ) for any algebraically closed ﬁeld k. By Lemma 8.17, it suﬃces to show the case where the characteristic of k is p > 0 and N = pe . By changing coordinates, let Gm = Spec k[X, (1 + X)−1 ] and G = μN = Spec k[X]/(X N ). Let P : G → Gm ×k G be the diagonal section. By Lemma 8.18(3), the closed subscheme G× of G is deﬁned by the closed condition that the two closed subschemes of Gm ×k G = Spec k[X, (1 + X)−1 , T ]/(T N ), G ×k G = Spec k[X, T ]/(X N , T N ) and the pullback of

N −1 N −1 −1 i N [iP ] = Spec k[X, (1 + X) , T ]/ (1 + X − (1 + T ) ), T i=0

i=0

are equal. Thus, if we let N −1

N −1 1 + X − (1 + T )i = X N − aj (T )X j ,

i=0 ×

j=0 N

we have G = Spec k[T ]/(T , a0 (T ), . . . , aN −1 (T )). Since we have (1 + T )i − 1 ≡ iT mod T 2 , the T -adic valuation ord((1 + T )i − 1) is 1 if p i, and at least 2 if p | i. Thus, we have ord aN/p (T ) = {i | p i, 0 ≤ i < N } = N − N/p = ϕ(N ), and we have deg G× = min(N, ord a0 (T ), . . . , ord aN −1 (T )) ≤ ϕ(N ). 8.3. Drinfeld level structure In Chapter 2, we deﬁned modular curves over Q using a cyclic subgroup of order N of an elliptic curve. However, for a supersingular elliptic curve over a scheme over Fp , there is no cyclic subgroup scheme of order p in a usual sense. In order to deﬁne modular curves over Z, we use Deﬁnition 8.13 as the deﬁnition of a cyclic subgroup

8.3. DRINFELD LEVEL STRUCTURE

13

scheme. The level structure deﬁned in such a way is called the Drinfeld level structure. Definition 8.20. Let S be a scheme, and let E be a commutative group scheme over S that is a smooth curve over S. Let N ≥ 1 be an integer. (1) A section P : S → E has exact order N if the Cartier divisor N −1 i=0 [iP ] is a closed subgroup scheme of E. If P has exact order N , we call [iP ] (8.9) P = i∈Z/N Z

the cyclic subgroup scheme of order N generated by P . (2) The functor M0 (N )E over S is deﬁned by associating to a scheme T over S the set (8.10) M0 (N )E (T ) = {cyclic subgroup scheme of ET of order N }. (3) The functor M1 (N )E over S is deﬁned by associating to a scheme T over S the set (8.11)

M1 (N )E (T ) = {section of ET of exact order N }.

By Lemma 8.18, the cyclic subgroup scheme in Deﬁnition 8.20(1) is a cyclic group scheme in the sense of Deﬁnition 8.13(2). To a section P ∈ M1 (N )E (T ) of exact order N , associate the cyclic subgroup scheme P ∈ M0 (N )E (T ), and we obtain a natural morphism of functors M1 (N )E → M0 (N )E . If N is invertible on S, Deﬁnition 8.20 is a standard one. Lemma 8.21. Let S be a scheme, and let E be a commutative group scheme over S that is a smooth curve over S. Let P : S → E be a section of E, and let N ≥ 1 be an integer. If N is invertible on S, the following conditions (i)–(iii) are equivalent. (i) P has exact order N . (ii) There exists a closed immersion Z/N Z → E of commutative group schemes over S such that P : S → E is deﬁned by 1 ∈ Z/N Z. (iii) N P = 0, and for any geometric point s¯ in S, the element Ps¯ of the abelian group E(¯ s) has exact order N . Proof. Clear from Lemmas 8.18(2) and 8.15.

14

8. MODULAR CURVES OVER Z

For a scheme over Fp unusual phenomena occur unlike schemes over Q. Lemma 8.22. Let S be a scheme, and let E be a commutative group scheme over S that is a smooth curve over S. Let p be a prime number, and let e ≥ 1 be an integer. Then, the following conditions (i)–(iii) are equivalent. (i) S is a scheme over Fp . (ii) The 0-section of E has exact order pe . (iii) The Cartier divisor G = pe [0] of E is a cyclic subgroup scheme of order pe . If one and hence all of these conditions hold, we have G = Ker F e . Proof. (i) ⇒ (ii), (iii). If S is a scheme over Fp , pe [0] = Ker F e is a cyclic subgroup scheme of order pe , and the 0-section has exact order pe . (ii) ⇒ (iii) is clear. (iii) ⇒ (i). Since the assertion is local on S, we may assume S = Spec A. Furthermore, we may assume that the Cartier divisor [0] of E is deﬁned by a section T of OE on a neighborhood of [0]. Then, e we have G = Spec A[T ]/(T p ). Let F (T, S) be the image of T by e e e the ring homomorphism A[T ]/(T p ) → A[T ]/(T p ) ⊗A A[T ]/(T p ) = e e A[T, S]/(T p , S p ) corresponding to the group operation G × G → G. Here, we identify T = T ⊗ 1, and let S = 1 ⊗ T . e e e F (T, S)p equals 0 as an element of A[T, S]/(T p , S p ). Since e e F (T, 0) = F (0, T ) = T , there exists f (S, T ) ∈ A[T, S]/(T p , S p ) such that F (T, S) = T + S + ST f (S, T ). Looking at the homogee e neous degree pe part of F (T, S)p , all the coeﬃcients of (T + S)p − e e e e e p −1 p p −i i (T p + S p ) = S are 0 as elements of A. Since i=1 i T pe e is p, A is an Fp the greatest common divisor of p1 and pe−1 algebra. Lemma 8.23. Let S be a scheme, and let E be a commutative group scheme over S that is a smooth curve over S. Let N ≥ 1 be an integer, and let P : S → E be a section of E. For a scheme T over S, the condition P that PT is a section of exact order N of ET is a closed condition on S. N −1 Proof. Let G = i=0 [iP ]. By Deﬁnition 8.20(1), the condition P is that GT is a closed subgroup scheme of ET . This condition is in turn equivalent to the following. The closed subscheme GT ×T GT

8.3. DRINFELD LEVEL STRUCTURE

15

of ET ×T ET is a closed subscheme of the inverse image of GT by the addition + : ET ×T ET , and GT is equal to the inverse image of GT by the multiplication-by-(−1) morphism ET → ET . Thus, by Corollary A.37, the condition P is a closed condition. Corollary 8.24. Let S be a scheme, and let E be an elliptic curve over S. Let N ≥ 1 be an integer. The functor M1 (N )E over S is represented by a scheme M1 (N )E ﬁnite of ﬁnite presentation over S. If N is invertible on S, M1 (N )E is ´etale over S. Proof. Applying Lemma 8.23 to the diagonal section E[N ] → E ×S E[N ] over E[N ], M1 (N )E is represented by a closed subscheme of E[N ]. If N is invertible on S, E[N ] is ﬁnite ´etale over S by Corollary 1.27. Since the assertion is ´etale local on S, we may assume that E[N ] is isomorphic to the constant group scheme (Z/N Z)2 . But, in this case M1 (N )E is isomorphic to a∈(Z/N Z)n ,ord(a)=N S. Lemma 8.25. Let k be a ﬁeld of characteristic p, and let E be an elliptic curve over k. Let N ≥ 1 be an integer. (1) If p > 0 and G is a closed subgroup scheme of E of degree pe , then G is a cyclic subgroup scheme of order pe . (2) If G is a cyclic subgroup scheme of E of order N , then we have (8.12)

deg G× ≤ ϕ(N ).

The equality holds unless p | N and E is supersingular. (3) We have an equality (8.13)

deg M1 (N )E = ϕ(N )ψ(N )

unless p | N and E is supersingular. Later in Proposition 8.52 and Corollary 8.53, we will show that the equality holds even if E is supersingular. Question. In case p | N and E is supersingular, can we prove directly the equality (8.12) as in the proof below? Proof. We may assume k is algebraically closed. (1) If E is supersingular, a closed subgroup scheme of degree pe is Ker F e by Proposition 8.2(3), and 0 is a generator of this. If E is ordinary, E[pe ] is isomorphic to Z/pe Z × μpe by Proposition 8.2(2). Let G be a closed subgroup scheme of E[pe ] of degree pe . Since G ∩ μpe is a closed subgroup scheme of μpe , we have G ∩ μpe = μpb for some b ≥ e. Since k is algebraically closed, G is

16

8. MODULAR CURVES OVER Z

isomorphic to Z/pa Z × μpb , a + b = e. By Lemma 8.17, (1, 1) is a generator of Z/pa Z × μpb , and this is a cyclic subgroup scheme. (2) By Lemma 8.17 it suﬃces to show it when p > 0 and N = pe . We show inequality (8.12) when E is supersingular. The proof goes similarly to that of Lemma 8.19. By Proposition 8.2(3), we have E,0 , we identify G = Ker F e . Choosing an isomorphism k[[X]] → O E,0 . Then, we have G = Spec k[[X]]/(X N ). For an integer k[[X]] = O i, we denote by [i]∗ the ring homomorphism k[[X]] → k[[X]] induced by the multiplication-by-i mapping [i] : E → E. G× is a closed subscheme of G = Spec k[T ]/(T N ) deﬁned by the condition that the N −1 ideal ( i=0 (X − [i]∗ T )) is equal to the ideal (X N ), and we have [i]∗ (T ) ≡ iT mod T 2 . After this point, the proof goes in the same way as the proof of Lemma 8.19. We show the equality in (8.12) when E is ordinary. By the proof of (1), we may assume G = Z/pa × μpb , a + b = e. If a = 0, then the equality follows from Lemma 8.19. Suppose a > 0. By Lemma 8.17, a section P of G = Z/pa Z × μpb is a generator if and only if the projection of P to Z/pa Z is a generator of Z/pa Z and pa P is a generator of μpb . By the assumption a > 0 and Lemma 8.19, pa P is a generator of μpb for any P . Thus, G× is equal to (Z/pa Z)× × μpb , and the equality holds. (3) By Lemma 8.17, it suﬃces to show it when p > 0 and N = pe . Suppose E is ordinary, and we show (8.13). As above, we may identify E[N ] with G = Z/N Z × μN . M1 (N )E is the closed subgroup scheme all the sections of G of exact order N . Decompose consisting of G = i∈Z/N Z Gi = i∈Z/N Z μN , and M1 (N )E = i∈Z/N Z M1 (N )iE . If i ∈ Z/N Z has order pa , and a ≤ e = a + b, then by Lemma 8.17, M1 (N )iE is the inverse image of μ× by the multiplication-by-pa mappb i ping Gi = μpe → μpb . Thus, M1 (N )iE = μ× pe if b > 0, and M1 (N )E = i G if b = 0. The equality (8.13) is clear. As a preparation for studying the compactiﬁcation of modular curves in §8.9, we deﬁne and study the Drinfeld level structure of a commutative group scheme. Let N ≥ 1 be an integer. Deﬁne a morphism Z → Z × Gm of commutative group schemes over Z[q, q −1 ] by sending 1 to (N, q), and deﬁne T (N ) to be the cokernel of this homomorphism. T (N ) is an extension of Z/N Z by Gm , and the kernel T [N ] of the multiplication-by-N map T (N ) → T (N ) is an extension of Z/N Z by μN . For i ∈ Z/N Z, let T (N )i and T [N ]i be the inverse

8.3. DRINFELD LEVEL STRUCTURE

17

images of the natural morphisms to Z/N Z. We have T

(N )

=

N −1

T

(N )i

=

i=0

N −1

Spec Z[q, q −1 ][T, T −1 ]

i=0

and (8.14)

T [N ] =

N −1 i=0

T [N ]i =

N −1

Spec Z[q, q −1 ][T ]/(T N − q i ).

i=0

Proposition 8.26. Let N ≥ 1 be an integer. (1) The functor M0 (N )T (N ) over Z[q, q −1 ] is represented by Spec Z[ζd ][q, q −1 ][T ]/(T d1 − ζd q d1 ). dd =N

Here, for d and d satisfying dd = N , d is the greatest common divisor of d and d , and d1 = d/d , d1 = d /d . (2) The functor M1 (N )T (N ) over Z[q, q −1 ] is represented by N −1

Spec Z[ζd ][q, q −1 ][T ]/(T d − ζd q i ).

i=0

Here, for 0 ≤ i < N , d is the greatest common divisor of N and i, and d = N/d , i = i/d . Proof. (1) Let S be a scheme Z[q, q −1 ]. For a ﬁnite ﬂat closed (N ) (N ) subgroup scheme G ⊂ TS over S and i ∈ Z, let Gi = G ∩ TS i . (N ) G0 is the kernel of G → TS → Z/N Z and is a closed subgroup (N ) 0 = Gm,s that is ﬁnite ﬂat over S. scheme of TS Let d be a divisor of N , and let d = N/d. Deﬁne the subfunctor (d) M0 (N )T (N ) of M0 (N )T (N ) by associating the set M0 (N )T (N ) (S) = {G ∈ M0 (N )T (N ) (S) | G0 is degree d over S} (d)

to a scheme S over Z[q, q −1 ]. It suﬃces to show that the functor (d) M0 (N )T (N ) is represented by Spec Z[ζd ][q, q −1 ][T ]/(T d1 − ζd q d1 ). By Proposition 8.14, G0 has degree d over S if and only if G0 = μd . d−1 In this case, we also have G = i=0 Gd i . We show the following lemma. Lemma 8.27. Let S be a scheme over Z[q, q −1 ], and let N = dd ≥ 1 be an integer. Let d be the greatest common divisor of d and d , and let d = d d1 .

18

8. MODULAR CURVES OVER Z (N )

(1) Let G be a closed subgroup scheme of TS that is ﬁnite ﬂat of ﬁnite presentation over S of degree N and such that G0 = μd ,S . Then, there exists a unique section s : S → T (N )d such that for each 0 ≤ i < N , the diagram

⊂

si

2

(N )d

(N )d2

Gd i −−−−→ T (N )d i ⏐ ⏐ ⏐ ⏐ [d ]

(8.15)

S −−−−→ T (N )d i is Cartesian. Moreover, s satisﬁes sd = 1. 2 (2) Conversely, suppose s : S → T (N )d is a section that satisﬁes d−1 (N ) sd = 1. Deﬁne a closed subgroup scheme G = i=0 Gd i of TS by the condition that diagram (8.15) is Cartesian for each i. Then, G is a ﬁnite ﬂat scheme of ﬁnite presentation over S of (N ) degree N , and TS is a closed subgroup scheme. (3) Let G and s be as above. Then, G is a cyclic subgroup scheme if and only if sd1 is a generator of μd ,S . Proof. (1) Since [d ] : TS → TS is faithfully ﬂat, a (N )d2 that satisﬁes the condition is unique once section s : S → T it exists. Thus, the assertion is ﬂat local on S. Since Gd is a ﬂat covering of S, we may assume Gd has a section t : S → Gd . For each i, the section t : S → Gd deﬁnes the vertical isomorphisms in the diagram μd −−−−→ ⏐ ⏐ ×ti

[d ]

−−−−→

Gm ⏐ ⏐ ×ti (N )d i

Gd i −−−−→ TS

[d ]

Gm ⏐ ⏐ d i ×t (N )d2 i

−−−−→ TS

d

.

Thus, by letting s = t , diagram (8.15) is Cartesian for each i. We have sd = tN = 1 by Proposition 8.14. (2) Since [d ] : Gm → Gm is ﬁnite ﬂat of degree d , G is a ﬁnite ﬂat scheme of ﬁnite presentation of degree N . If sd = 1, it is easy to (N ) see that G is a closed subgroup scheme of TS . (3) By Lemma 8.17, G is a cyclic subgroup scheme if and only if there exists a section P of Gd such that P d is a generator of μd ﬂat locally on S, or equivalently, Gd × μ× d → S has a section ﬂat [d] Gm

locally on S.

8.3. DRINFELD LEVEL STRUCTURE

We have Gd

×

[d] Gm

μ× d = S

19

×

μ× d by

×

T (N )d

s T (N )d2 [d ]

[d] Gm

(8.15). By Lemma 8.19, all the vertical morphisms in the diagram [d]

T (N )d −−−−→ ⏐ ⏐ [d ] 2

T (N )d

Gm ←−−−− ⏐ ⏐[d ] 1

μ× d ⏐ ⏐

−−−−→ Gm ←−−−− μ× d [d1 ]

×

are faithfully ﬂat. Thus, Gd

[d] Gm

μ× d = S

×

T (N )d

s T (N )d2 [d ]

μ× d is faithfully ﬂat over the closed subscheme S

×

d μ× d of S. Therefore, the condition that G

×

[d] Gm

T (N )d

×

[d1 ] Gm

×

[d1 ] Gm

μ× d → S has a

section ﬂat locally on S is equivalent to the equality S = S 2

2

T (N )d

s T (N )d2

×

[d] Gm

×

s T (N )d2

d1 μ× being a generator d , and it is also equivalent to s

of μd ,S .

Back to the proof of Proposition 8.26. By Lemma 8.27(1), we 2 (d) deﬁne an injection of functor M0 (N )T (N ) → T (N )d by associating s to G. Since d = d d1 , if sd1 is a generator of μd ,S , then sd = 1. (d) Furthermore, by Lemma 8.27(2) and (3), the functor M0 (N )T (N ) is 2 (N )d2 represented by the closed subscheme T (N )d × μ× . d of T [d1 ] Gm

2

By Lemma 8.19, T (N )d

×

[d1 ] Gm

μ× d is the spectrum of

Z[q, q −1 ][T, T −1 ] ⊗Z[q,q−1 ][T,T −1 ] Z[ζd ][q, q −1 ]

= Z[ζd ][q, q −1 ][T ]/(T d1 − ζd q d1 ). Here, the tensor product is taken with respect to the homomorphism 2 sending T to T d1 /q d1 d /N = T d1 /q d1 and the homomorphism sending T to ζd . (2) Suppose 0 ≤ i < N . Let d | N be the order of i ∈ Z/N Z, and let dd = N . By Lemma 8.17, a section P of T (N )i over S has exact order N if and only if P d is a generator of μd . Thus, we have (8.16)

M1 (N )T (N ) =

N −1 i=0

T (N )i

×

[d] Gm

μ× d .

20

8. MODULAR CURVES OVER Z

T (N )i

×

[d] Gm

μ× d is the spectrum of

Z[q, q −1 ][T, T −1 ] ⊗Z[q,q−1 ][T,T −1 ] Z[ζd ][q, q −1 ]

= Z[ζd ][q, q −1 ][T ]/(T d − ζd q i ). Here, the tensor product is taken with respect to the homomorphism sending T to T d /q di/N and the homomorphism sending T to ζd . 8.4. Modular curves over Z Definition 8.28. Let N ≥ 1 be an integer. (1) Deﬁne a functor M0 (N ) over Z by associating to a scheme T the set ⎧ ⎫ isomorphism classes of pairs (E, C), ⎪ ⎪ ⎬ ⎨ (8.17) M0 (N )(T ) = where E is an elliptic curve over T , . ⎪ ⎩ and C is its cyclic subgroup scheme of ⎪ ⎭ order N (2) Deﬁne a functor M1 (N ) over Z by associating to a scheme T the set isomorphism classes of pairs (E, P ), (8.18) M1 (N )(T ) = where E is an elliptic curve over T , . and P is its section of exact order N By associating to an isomorphism class of (E, P ) the isomorphism class of (E, P ), we deﬁne a morphism of functors (8.19)

M1 (N ) −→ M0 (N ).

Lemma 8.29. Let N = 4. The restriction of the functor M1 (4) to Z[ 41 ] is represented by 1 1 . (8.20) Y1 (4)Z[ 14 ] = Spec Z , d, 4 d(d − 4) The universal elliptic curve E and the universal section P of order 4 are given by (8.21)

E : dy 2 = x3 + (d − 2)x2 + x,

P = (1, 1).

let f : E → S be an elliptic Proof. Let S be a scheme over curve over S, and let P be a section of exact order 4. Let O be the 0section, and let Q = 2P and R = 3P . Let x be the inverse image of 1 by the isomorphism f∗ O(2[O] − 2[Q]) → OP , and let y be the inverse image of 1 by the isomorphism f∗ O(3[O]−(2[P ]+[Q])) → OP . Deﬁne the immersion E → P2S by the basis x, y, 1 of f∗ O(3[O]). Then x and Z[ 14 ],

8.4. MODULAR CURVES OVER Z

21

y satisfy an equation of the form y 2 + a1 xy + a3 y = a0 x3 + a2 x2 + a4 x + a6 (ai ∈ OS ), which deﬁnes E. Since the three points O, P, R are collinear, the coordinates of O, P, Q, R are (0 : 1 : 0), (1 : 0 : 1), (0 : 0 : 1), and (1 : 1 : 1), respectively. In the inhomogeneous coordinates, E is tangent to the line x = 0 at Q = (0, 0), intersects with the line y = 0 at Q, is tangent to it at P = (1, 0), and passes through R = (1, 1). Thus, we have a6 = a3 = 0, a0 x3 + a2 x2 + a4 x + a6 = a0 x(x − 1)2 , and a1 = −1. Hence, the elliptic curve E is deﬁned by the equation y 2 − xy = a0 x(x − 1)2 , where a0 ∈ OS× . Let a0 = d/4, and substitute x − 2y by y. Then, the equation 4d(y 2 − xy) = x(x − 1)2 becomes dy 2 = dx2 + x(x − 1)2 . The condition that the right-hand side has a triple root is d = 0, 4. If N = 1, we have M0 (1) = M1 (1). We will denote them simply by M. For a scheme T , we have (8.22) M(T ) = { isomorphism classes of elliptic curves E over T }. Associating E to the isomorphism class of (E, C) or that of (E, P ), we obtain morphisms of functors (8.23)

M0 (N ) −→ M and M1 (N ) −→ M.

The functor M deﬁned in Example 2.4 of Chapter 2 is MQ ; i.e., the restriction to Q of the functor M just deﬁned. In Example 2.6, we deﬁned the morphism of functors j : MQ → A1Q over Q. This is extended naturally to a morphism of functors j : M → A1Z over Z. Lemma 8.30. (1) There exists a unique morphism of functors j : M → A1Z over Z that extends the morphism of functors j : MQ → A1Q over Q. (2) For any algebraically closed ﬁeld k, the mapping j : M(k) → A1Z (k) = k is a bijection. Question. Prove Lemma 8.30. (Hint: To show the uniqueness, it suﬃces to consider the elliptic curve E over 1 A = Z[a1 , a2 , a3 , a4 , a6 ] Δ deﬁned by the equation y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 . To show the existence, verify j(E) ∈ A, and it is invariant under the change of coordinates.)

22

8. MODULAR CURVES OVER Z

Example 8.31. The j-invariant of the elliptic curve y 2 = (x − α)(x − β)(x − γ) over Z[ 21 , α, β, γ][Δ−1 ], where Δ = ((α − β)(β − γ)(γ − α))2 , is given by (α2 + β 2 + γ 2 − αβ − βγ − γα)3 . Δ Thus the j-invariant of the elliptic curve y 2 = x(x2 + ax + b) over Z[ 12 , a, b][b−1 , (a2 − 4b)−1 ] is given by (8.24)

(8.25)

28 ·

28 ·

(a2 − 3b)3 . b2 (a2 − 4b)

The j-invariant of the universal elliptic curve dy 2 = x(x2 +(d−2)x+1) over Y1 (4)Z[ 12 ] is given by (8.26)

28 ·

(d2 − 4d + 1)3 . d(d − 4)

Suppose a ∈ (Z/N Z)× . If P is a section of exact order N , then aP also has exact order N . Thus, by associating to the isomorphism of (E, P ) the isomorphism class of (E, aP ), we obtain an isomorphism of functors (8.27)

a : M1 (N ) −→ M1 (N ).

This is called the diamond operator . The diamond operator a : M1 (N ) → M1 (N ) is an automorphism over M0 (N ). Since (E, P ) is isomorphic to (E, −P ), we have −1 = 1. If N = N N with (N , N ) = 1, we have an identiﬁcation ⎫ ⎧ isomorphism classes of triples (E, C , C ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ where E is an elliptic curve over T , C is ⎪ M0 (N )(T ) = its cyclic subgroup scheme of order N , . ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ and C is its cyclic subgroup scheme of ⎪ order N Similarly, ⎧ ⎫ ⎪ isomorphism classes of triples (E, P , P ), ⎬ ⎪ ⎨ M1 (N )(T ) = where E is an elliptic curve over T , P is . ⎪ ⎩ its section of exact order N , and P is its ⎪ ⎭ section of exact order N Associating to the isomorphism class of triples (E, C , C ) the isomorphism class of triples (E/C , E[N ]/C , (C + C )/C ), we obtain a morphism of functors (8.28)

wN : M0 (N ) −→ M0 (N ),

8.4. MODULAR CURVES OVER Z

23

2 called the Atkin–Lehner involution. It is easy to see wN is the identity. Let N ≥ 1 be an integer, and let p be a prime number. We denote by M0 (N )Fp the restriction of the functor M0 (N ) to schemes over Fp . To the isomorphism class of a pair (E, C), we associate the isomorphism class of (E (p) , C (p) ) to obtain a morphism of functors F : M0 (N )Fp → M0 (N )Fp . We call it the Frobenius morphism. Deﬁne a subfunctor M0 (N )ss Fp of M0 (N )Fp by associating to a scheme T over Fp the set ⎧ ⎫ isomorphism classes of pairs (E, C), ⎪ ⎪ ⎨ ⎬ where E is a supersingular elliptic . (8.29) M0 (N )ss Fp (T ) = ⎪ ⎩ curve over T , and C is its cyclic sub- ⎪ ⎭ group scheme of order N

Deﬁne the restriction of functor M1 (N )Fp and the Frobenius morphism M1 (N )Fp → M1 (N )Fp similarly. Let p be a prime number, and let M ≥ 1 be an integer relatively prime to p. Let N = M p. Deﬁne a morphism of functors over Fp j0 : M0 (M )Fp → M0 (N )Fp by associating to the isomorphism class of a pair (E, C) the isomorphism class of (E, C, Ker F ). Also, deﬁne j1 : M0 (M )Fp → M0 (N )Fp by associating to the isomorphism class of a pair (E, C) the isomorphism class of (E (p) , C (p) , Ker V ). In this chapter we show the following theorem. Theorem 8.32. Let N ≥ 1 be an integer. (1) There exists a coarse moduli scheme Y0 (N )Z of M0 (N ) over Z. Y0 (N )Z is a normal connected aﬃne curve over Z. (2) The morphism (8.30)

Y0 (N )Z −→ Y (1)Z

induced by the morphism of functors M0 (N ) → M (8.23) is ﬁnite ﬂat of degree ψ(N ). (3) Y0 (N )Z is smooth over Z[ N1 ]. For a prime number p N , Y0 (N )Fp = Y0 (N )Z ⊗Z Fp is the coarse moduli scheme of the restriction M0 (N )Fp . (4) Let p be a prime number, and assume N = M p for an integer M prime to p. Then, Y0 (N )Z is weakly semistable at p. The morphisms j0 , j1 : M0 (M )Fp → M0 (N )Fp over Fp induce closed immersions j0 , j1 : Y0 (M )Fp → Y0 (N )Z . The ﬁber Y0 (N )Fp is the union of the image C0 of j0 and the image C1 of j1 . The intersection of C0 = Y0 (M )Fp and C1 is the

24

8. MODULAR CURVES OVER Z ss coarse moduli Y0 (M )ss Fp of M0 (M )Fp . The index ex of the ordinary double point x = [(E, C)] ∈ Y0 (M )ss Fp is the order of Aut(EFp , CFp )/{±1}.

The ﬁber Y0 (N )Q = Y0 (N )Z ⊗Z Q of Y0 (N )Z at the generic point is the modular curve Y0 (N )Q over Q that we constructed in Theorem 2.10. Example 8.33. Replacing y by 1+2y, the deﬁning equation (1.3) y 2 = 4x3 − 4x2 − 40x − 79 of X0 (11)Q becomes (8.31)

y 2 + y = x3 − x2 − 10x − 20.

The homogeneous equation of (8.31) deﬁnes an elliptic curve over y−60 1 ], and the ﬁber of F11 is the N´eron 1-gon. If we let s = x−16 , Z[ 11 2 2 (8.31) becomes s (x − 16) + 121s = x + 15x + 230. The morphism Spec Z[x, s]/(s2 (x − 16) + 121s − (x2 + 15x + 230)) → Spec Z[x, y]/(y 2 + y − (x3 − x2 − 10x − 20)) 1 is an open immersion over Z[ 11 ]. Through this immersion, we identify

Y0 (11)Z = Spec Z[x, s]/(s2 (x − 16) + 121s − (x2 + 15x + 230)). We have Y0 (11)F11 = Spec F11 [x, s]/((s2 − (x − 2))(x − 5)), and the curve Y0 (11)Z over Z is weakly semistable at p = 11. It has two ordinary double points, (x, s) = (5, 5) and (5, 6). Their indices are 2 and 3, respectively. Theorem 8.34. Let N ≥ 1 be an integer. (1) There exists a coarse moduli scheme Y1 (N )Z of M1 (N ) over Z. If N ≥ 4, Y1 (N )Z[ N1 ] is a ﬁne moduli scheme. Y1 (N )Z is a normal connected aﬃne curve over Z. Y1 (N )Q = Y1 (N )Z ⊗Z Q is a smooth connected aﬃne curve over Q. (2) The morphism (8.32)

Y1 (N )Z −→ Y (1)Z

induced by the morphism of functors M1 (N ) → M (8.23) is ﬁnite ﬂat of degree ψ(N )φ(N )/2 if N ≥ 3, and of degree 3 if N = 2. (3) Y1 (N )Z is smooth over Z[ N1 ]. For a prime number p N , Y1 (N )Fp = Y1 (N )Z ⊗Z Fp is a coarse moduli scheme of the restriction M1 (N )Fp .

8.5. MODULAR CURVE Y (r)Z[ 1 ]

25

r

Remark. The modular curves Y0 (N )Z and Y1 (N )Z are the integral closures of Y0 (N )Z[ N1 ] and Y1 (N )Z[ N1 ] , respectively, by Theorems 8.32 and 8.34. However, if we deﬁned Y0 (N )Z and Y1 (N )Z as the integral closures, their detailed structures could not be studied any further. The deﬁnition using the Drinfeld level structures is the only way for us to study their detailed structures, which we will do starting from §8.6. 8.5. Modular curve Y (r)Z[ r1 ] In order to prove Theorems 8.32 and 8.34, we deﬁne auxiliary functors. Definition 8.35. Let N ≥ 1 be an integer, and let r ≥ 1 be an integer relatively prime to N . (1) Deﬁne the functor M0,∗ (N, r)Z[ r1 ] over Z[ r1 ] by associating to a scheme T over Z[ r1 ] the set (8.33) ⎧ ⎫ isomorphism classes of triples (E, C, α), ⎪ ⎪ ⎨ ⎬ where E is an elliptic curve over T , C is its . M0,∗ (N, r)Z[ r1 ] (T ) = scheme of order N , and ⎪ ⎪ ⎩ cyclic subgroup ⎭ α : (Z/rZ)2 → E[r] is an isomorphism (2) Deﬁne the functor M1,∗ (N, r)Z[ r1 ] over Z[ r1 ] by associating to a scheme T over Z[ r1 ] the set (8.34) ⎧ ⎫ ⎪ isomorphism classes of triples (E, P, α), ⎪ ⎨ ⎬ where E is an elliptic curve over T , . M1,∗ (N, r)Z[ r1 ] (T ) = of exact order N , and ⎪ ⎪ ⎩ P is a section ⎭ α : (Z/rZ)2 → E[r] is an isomorphism The natural action of GL(2, Z/rZ) on (Z/rZ)2 induces actions of GL(2, Z/rZ) on M0,∗ (1, r)Z[ r1 ] and on M1,∗ (1, r)Z[ r1 ] . If N = 1, we have M0,∗ (1, r)Z[ r1 ] = M1,∗ (1, r)Z[ r1 ] , which we denote simply by M(r)Z[ r1 ] . by

Example 8.36. Let r = 3. The functor M(3)Z[ 13 ] is represented

(8.35)

Y (3)Z[ 13 ] = Spec Z

1 3

, ζ3 , μ,

1 . μ3 − 1

26

8. MODULAR CURVES OVER Z

The universal curve (E, O) and the universal base α are given respectively by (8.36)

E : X 3 + Y 3 + Z 3 − 3μXY Z = 0,

O : (0, 1, −1),

α : (1, 0) → (0, ζ3 , −1), (0, 1) → (1, 0, −1).

Question. Verify Example 8.36. (Hint: Use the proof of Theorem 2.21 for the case N = 3.) Let r, s ≥ 1 be integers. Let H be the kernel Ker GL2 (Z/rsZ) → GL2 (Z/rZ) . 1 ] Deﬁne a morphism of functors over Z[ rs

(8.37)

1 1 /H] M(r)Z[ rs ] −→ [M(rs)Z[ rs ]

1 ], let E be an elliptic curve as follows. Let S be a scheme over Z[ rs 2 over S, and let α : (Z/rZ) → E[r] be an isomorphism. Deﬁne a functor Isom α ((Z/rsZ)2 , E[rs]) over S by associating to a scheme T over S the set Isom α (Z/rsZ)2 , E[rs] (T )

= {isomorphism β : (Z/rsZ)2 → ET [rs] | β induces α}. The functor Isom α (Z/rsZ)2 , E[rs] is represented by an H-torsor 1 (P ) deﬁned by the pair (EP , β) P over S. The element of M(rs)Z[ rs ] of the pullback EP of E to P , and the universal isomorphism β over P is compatible with the action of H. Thus, it deﬁnes an ele1 /H](S). We thus have a morphism M(r) 1 ment of [M(rs)Z[ rs ] Z[ rs ] → 1 /H]. [M(rs)Z[ rs ] Similarly, for relatively prime integers N, r ≥ 1, we deﬁne morphisms of functors over Z[ r1 ] (8.38)

M0 (N )Z[ r1 ] −→ [M0,∗ (N, r)Z[ r1 ] /GL2 (Z/rZ)], M1 (N )Z[ r1 ] −→ [M1,∗ (N, r)Z[ r1 ] /GL2 (Z/rZ)].

Lemma 8.37. Let r ≥ 3 be an integer. The functor M(r)Z[ r1 ] over Z[ r1 ] is represented by a smooth aﬃne connected curve Y (r)Z[ r1 ] over Z[ 1r ]. The ﬁeld of constants of Y (r)Q = Y (r)Z[ r1 ] ⊗Z[ r1 ] Q is Q(ζr ). Y (r)Z[ r1 ] ⊗Z[ r1 ] Q is the modular curve Y (r) constructed in Theorem 2.21. Lemma 8.37 is proved in the same way as in §2.4.

8.5. MODULAR CURVE Y (r)Z[ 1 ]

27

r

Proof. If r = 3, it is given by Example 8.36. If r = 4, take (E, P ) where E is the universal elliptic curve over Y1 (4)Z[ 14 ] and P is the universal element of order 4. Let Q be the diagonal section of E over E[4]. Let Y (4)Z[ 14 ] be the closed and open subscheme of E[4] deﬁned by the condition that (P, Q) gives a basis of E[4]. Then M(4)Z[ 14 ] is represented by Y (4)Z[ 14 ] . If r is a multiple of 3, M(r)Z[ r1 ] is represented by the ﬁnite ´etale scheme Y (r)Z[ r1 ] over Y (3)Z[ r1 ] . The case where r is a multiple of 4 is similar. We show the general case. To do so, we ﬁrst show the following lemma. Lemma 8.38. Let S be a scheme, let f : E → S be an elliptic curve, and let g be an automorphism of E over S. (1) Let r ≥ 3 be an integer, and let S be a scheme over Z[ 1r ]. If the restriction g|E[r] is the identity morphism, then so is g. (2) Let N ≥ 4 be an integer, and let S be a scheme over Z[ N1 ]. Let C be a cyclic subgroup scheme of E of order N . If the restriction g|C is the identity morphism, then so is g. Proof. The invertible sheaf L = O(3[O]) on E deﬁnes a closed immersion E → P(f∗ L). (1) Let D ≥ 0 be the Cartier divisor deﬁned by D = [E[r]] − [O]. Since r is invertible on S, we have D ∩ O = ∅, and L|D = O|D . Moreover, since r ≥ 3, we have deg D = r 2 − 1 > 3, and f∗ L → L|D is injective. Thus, if g|E[r] is the identity, the action of g on f∗ L ⊂ L|D = OD is trivial, and so is the action on P(f∗ L). Hence, the action of g on E is also trivial. (2) Let D = C−[O]. If N ≥ 5, we have deg D = N −1 > 3. Then, the proof goes similarly to (1). If N = 4, let P ∈ C be the section of exact order 2. Then, by Example D.4, we have L(−D) O([P ]−[O]) locally on S. Thus f∗ L → L|D is injective, and the rest is similar to the proof of (1). Corollary 8.39. (1) Let s ≥ 1 be an integer, and let H = Ker GL2 (Z/rsZ) → GL2 (Z/rZ) . If r ≥ 3, the morphism of 1 1 1 /H] (8.37) over Z[ functors M(r)Z[ rs ] → [M(rs)Z[ rs ] rs ] is an isomorphism. (2) Let N ≥ 4 be an integer relatively prime to r. The morphism of functors M1 (N )Z[ N1r ] → [M1,∗ (N, r)Z[ N1r ] /GL2 (Z/rZ)] (8.38) over Z[ N1r ] is an isomorphism.

28

8. MODULAR CURVES OVER Z

Proof. (1) We construct the inverse morphism. Let S be a 1 ]. Let P be an H-torsor over S, and let (E, β) ∈ scheme over Z[ rs 1 (P ) be an H-invariant pair of an elliptic curve over P and M(rs)Z[ rs ] an isomorphism β : (Z/rsZ)2 → E[rs]. Let α : (Z/rZ)2 → E[r] be the isomorphism induced by β. Let g ∈ H. Then, by the assumption r ≥ 3 and Lemma 8.38(1), there exists a unique isomorphism g ∗ (E, β) = P × (E, β) → (E, β ◦ g) over P . Thus, the action of H g P

on P extends uniquely to a free action on E, the quotient ES = E/H is an elliptic curve over S, and the natural morphism ES ×S P → E is an isomorphism. Moreover, the isomorphism α : (Z/rZ)2 → E[r] is the pullback of an isomorphism αS : (Z/rZ)2 → ES [r]. Sending 1 (E, β) to (ES , αS ), we obtain the inverse morphism M(r)Z[ rs ] → 1 /H]. [M(rs)Z[ rs ] (2) We construct the inverse morphism. Let S be a scheme over Z[ N1r ], and let Q be a GL2 (Z/rZ)-torsor over S. Let (E, P, α) ∈ M1,∗ (N, r)Z[ N1r ] (Q) be a GL2 (Z/rZ)-invariant triple of an elliptic curve E over Q, a section P of exact order N , and an isomorphism α : (Z/rZ)2 → E[r]. The section P deﬁnes an isomorphism Z/N Z → P ⊂ E. Suppose g ∈ GL2 (Z/rZ). Then, by the assumption N ≥ 4 and Lemma 8.38(2), there exists a unique isomorphism g ∗ (E, P ) → (E, P ) over Q. The rest is similar to the proof of (1). We show Lemma 8.37 when r is general. Suppose s = 3 or 4. The 1 1 . By Lemma 8.38(1), functor M(rs)Z[ rs ] is represented by Y (rs)Z[ rs ] the natural action of H = Ker GL2 (Z/rsZ) → GL2 (Z/rZ) on 1 Y (rs)Z[ rs ] is free. Thus, by Lemmas A.31 and A.33, the natural 1 1 /H is ﬁnite and ´ morphism Y (rs)Z[ rs etale, and the ] → Y (rs)Z[ rs ] 1 /H represents the functor [M(rs) 1 /H] over quotient Y (rs)Z[ rs ] Z[ rs ] 1 1 1 /H Z[ rs ]. By Corollary 8.39(1), the quotient Y (r)Z[ rs = Y (rs) ] Z[ rs ] 1 . Moreover, Y (r) 1 represents the functor M(r)Z[ rs is a smooth ] Z[ rs ] 1 1 ]. Y (r)Z[ r1 ] is obtained by gluing Y (r)Z[ 3r aﬃne curve over Z[ rs ] and Y (r)Z[ 4r1 ] on Y (r)Z[ 6r1 ] . Let ( , )E[r] : E[r] × E[r] → μr be the Weil pairing. Associating to the pair (E, α) the root of unity (α(1, 0), α(0, 1))E[r], we obtain Y (r)Z[ r1 ] → Z[ 1r , ζr ]. To show that the ﬁeld of constants of Y (r)Q = Y (r)Z[ r1 ] ⊗Z[ r1 ] Q is Q(ζr ), it suﬃces to show that the Riemann surface

8.5. MODULAR CURVE Y (r)Z[ 1 ]

29

r

Y (r)an deﬁned by Y (r)C = Y (r)Q ⊗Q(ζr ) C is connected. Let Γ(r) be the subgroup of SL2 (Z) deﬁned by (8.39) Γ(r) = Ker SL2 (Z) → SL2 (Z/rZ) , and consider the natural action of Γ(r) on the upper half-plane H = {τ ∈ C | Im τ > 0}. As in Corollary 2.66, we obtain an isomorphism of Riemann surfaces (8.40)

Γ(r)\H −→ Y (r)an .

Thus Y (r)an is connected and Y (r)Q(ζr ) is a smooth connected aﬃne curve over Q(ζr ). Corollary 8.40. (1) There exists a coarse moduli scheme Y (1)Z of the functor M. (2) The morphism of functors M → A1Z deﬁned by the j-invariant induces an isomorphism (8.41)

j : Y (1)Z −→ A1Z .

(3) Let r ≥ 3 be an integer. The restriction of the morphism natural 1 ⊂ A1Z j : Y (r)Z[ r1 ] → Y (1)Z = A1Z to U = Spec Z j, j(j−12 3) Y (r)Z[ r1 ] ×A1Z U −→ U is a GL2 (Z/rZ)/{±1}-torsor. Proof. (1) As in the proof of Lemmas 2.27 and 8.37, the coarse moduli scheme Y (1)Z of M is obtained by gluing the quotient of Y (3)Z[ 13 ] by GL2 (Z/3Z) and the quotient of Y (4)Z[ 14 ] by GL2 (Z/4Z). (2) By the construction in (1), Y (1)Z is a normal aﬃne curve over Z. Since j : Y (1)Z → A1Z is an isomorphism over Q, it is a birational morphism. Moreover, by Lemma 8.30(2), the morphism of normal schemes j : Y (1)Z → A1Z induces a bijection on each geometric ﬁber, and thus it is an isomorphism. (3) The natural action of GL2 (Z/rZ) on Y (r)Z[ r1 ] is an action as an automorphism over Y (1)Z . Since the multiplication-by-(−1) morphism is an automorphism of the universal elliptic curve, the action of −1 ∈ GL2 (Z/rZ) on Y (r)Z[ r1 ] is trivial. Since Y (r)Z[ r1 ] → Y (1)Z[ r1 ] is a ﬁnite morphism of regular schemes, it suﬃces to show, by 1 Lemma A.34, that each geometric ﬁber over U = Spec Z[ 1r ][j, j(j−12 3) ] is a GL2 (Z/rZ)/{±1}-torsor. Let k be an algebraically closed ﬁeld with r ∈ k× , and let E be an elliptic curve over k with j(E) = 0, 123 . Since Y (r)Z[ r1 ] is a ﬁne moduli scheme, the ﬁber of the morphism

30

8. MODULAR CURVES OVER Z

Y (r)Z[ r1 ] → Y (1) at j(E) ∈ A1 (k) = Y (1)Z[ r1 ] (k) is identiﬁed with Isom((Z/rZ)2 , E[r])/{±1}. This is a GL2 (Z/rZ)/{±1}-torsor since we have Aut(E) = {±1} by Lemma 8.41 below. Lemma 8.41. Let k be an algebraically closed ﬁeld of characteristic p ≥ 0, and let E be an elliptic curve over k. (1) The automorphism group Aut(E) is ﬁnite, and the order of g ∈ Aut(E) is either a divisor of 4 or a divisor of 6. (2) If j(E) = 0, 123 , then Aut(E) = {±1}. (3) If p = 2, 3 and j(E) = 0, then Aut(E) = μ6 . (4) If p = 2, 3 and j(E) = 123 , then Aut(E) = μ4 . (5) If p = 3 and j(E) = 0 = 123 , then Aut(E) = 12, and 1 → {±1} → Aut(E) → Aut(E[2]) → 1 is an exact sequence. (6) If p = 2 and j(E) = 0 = 123 , then Aut(E) = 24, and the natural mapping Aut(E) → Aut(E[3], ( , )3 ) SL2 (F3 ) is an isomorphism. Proof. (1) Let r ≥ 3 be an integer invertible in k. Since Aut(E) → Aut(E[r]) is injective by Lemma 8.38(1), Aut(E) is a ﬁnite group. If g ∈ Aut(E), then the order of g is ﬁnite. The characteristic polynomial det(T − g) ∈ Z[T ] is of degree 2, and its leading coeﬃcient and constant term are both 1. Thus, the coeﬃcient of T must be one of 0, ±1, ±2, and the order of g is one of 1, 2, 3, 4, and 6. (2) (3) (4) We show it only in the case where the characteristic of k is diﬀerent from 2 and 3. In this case we may assume E is deﬁned by y 2 = x3 + ax + b, a, b ∈ k. An automorphism of E is then given by (x, y) → (u2 x, u3 y) with u ∈ k× satisfying u4 a = a, u6 b = b. If j = 0, 1728, then we have a = 0, b = 0, and thus u = ±1. If j = 0, then we have a = 0, b = 0, and thus u is a 6th root of unity. If j = 1728, then we have a = 0, b = 0, and thus u is a 4th root of unity. We omit the proof of (5) and (6). Example 8.42. an elliptic curve E over the open scheme Deﬁne 1 ⊂ Y (1)Z = A1Z = Spec Z[j] by UZ[ 16 ] = Spec Z[j] 16 , j(j−12 3) (2.24)

y 2 = 4x3 −

123 j 243 j x − . j − 123 j − 123

As we showed in Proposition 2.15(1), the j-invariant of E equals j. Let r ≥ 3 be an integer. The functor associating to a scheme T over 2 1 UZ[ 6r ] the set {isomorphisms (Z/rZ) → E[r]T of group schemes over T } is represented by a GL2 (Z/rZ)-torsor M (r)E,UZ[ 1 ] . The 6r

8.5. MODULAR CURVE Y (r)Z[ 1 ]

31

r

morphism M (r)E,UZ[

1 ] 6r

1 1 → Y (r)Z[ r1 ] ×Y (1)Z UZ[ 6r ] over UZ[ 6r ] deﬁned

by the universal isomorphism (Z/rZ)2 → E[r] is compatible with the action of GL2 (Z/rZ). This induces an isomorphism M (r)E,UZ[

1 ] 6r

/{±1} −→ Y (r)Z[ r1 ] ×Y (1)Z UZ[ 6r1 ]

1 . Y (r) of GL2 (Z/rZ)-torsors over UZ[ 6r ] Z[ r1 ] is isomorphic to the integral closure of Y (1)Z[ r1 ] in M (r)E,UZ[ 1 ] /{±1}. 6r

1 Question. Let E be the elliptic curve over Spec Z[ 12 , λ, λ(1−λ) ] 2 2 deﬁned by y = x(x − 1)(x − λ), and let α : (Z/2Z) → E[2] be the isomorphism deﬁned basis (0, 0), (1, 0). Show that by the 1 1 the pair (E, α) ∈ M(2)Z[ 12 ] Spec Z 2 , λ, λ(1−λ) deﬁnes an isomor1 ] → Y (2)Z[ 12 ] , where Y (2)Z[ 12 ] is the coarse phism Spec Z[ 21 , λ, λ(1−λ) moduli scheme of M(2)Z[ 12 ] .

Corollary 8.43. Let N ≥ 1 be an integer, and let r ≥ 3 be an integer relatively prime to N . The functor M1,∗ (N, r)Z[ r1 ] over Z[ 1r ] is represented by a ﬁnite scheme Y1,∗ (N, r)Z[ r1 ] over Y (r)Z[ r1 ] . 1 1 Y1,∗ (N, r)Z[ rN ] is smooth over Z[ rN ]. The ﬁeld of constants of Y1,∗ (N, r)Q = Y1,∗ (N, r)Z[ r1 ] ⊗Z[ r1 ] Q is Q(ζr ). Proof. Let E be the universal elliptic curve over Y (r)Z[ r1 ] . By Corollary 8.24, the functor M1,∗ (N, r)Z[ r1 ] is represented by the ﬁnite scheme M1 (N )E = Y1,∗ (N, r)Z[ r1 ] over Y (r)Z[ r1 ] . Since Y1,∗ (N, r)Z[ N1r ] → Y (r)Z[ N1r ] is ´etale and Y (r)Z[ N1r ] is a smooth aﬃne curve over Z[ N1r ], Y1,∗ (N, r)Z[ N1r ] is also a smooth aﬃne curve over Z[ N1r ]. The proof of the fact that Q(ζr ) is the ﬁeld of constants of Y1,∗ (N, r)Q = Y1,∗ (N, r)Z[ r1 ] ⊗Z[ r1 ] Q is similar to that of Lemma 8.37, and we omit it. The proofs of Theorems 8.32 and 8.34 go as follows. We ﬁrst deﬁne Igusa curves and study their properties in §8.6. Then in §8.7 we study the modular curve Y1,∗ (N, r)Z[ r1 ] using Igusa curves and prove Theorem 8.34. Finally in §8.8, we study the modular curve Y0,∗ (N, r)Z[ r1 ] and prove Theorem 8.32.

32

8. MODULAR CURVES OVER Z

8.6. Igusa curves Let p be a prime number, and let r ≥ 3 be an integer relatively prime to p. Let E be the universal elliptic curve over Y (r)Fp = Y (r)Z[ r1 ] ⊗ Fp , and let Y (r)ss Fp be the closed subscheme of Y (r)Fp deﬁned by the condition that E is supersingular. Lemma 8.44. Let p be a prime number, and let r ≥ 3 be an integer relatively prime to p. Y (r)ss Fp is a Cartier divisor of Y (r)Fp and is ﬁnite ´etale over Fp . Proof. We show S = Y (r)ss etale over Fp . It suﬃces to show Fp is ´ that the absolute Frobenius morphism F : S → S is an automorphism of S of ﬁnite order. F : S → S is the endomorphism deﬁned by sending the isomorphism class [(E, α)] of a pair of a supersingular elliptic curve E and a basis α of E[r] to the isomorphism class [(E (p) , α(p) )]. Since E is supersingular, we have Ker[p] = Ker F 2 . Thus, we have an 2 isomorphism fE : E → E (p ) satisfying F 2 = fE ◦ [p]. This implies 2 2 that F 2 maps [(E, α)] to [(E (p ) , α(p ) )] = [(E, α ◦ p)]. Hence, if n is the order of p ∈ (Z/rZ)× , F 2n is the identity morphism of S. Y (r)ss etale over Fp of the smooth Fp is a closed subscheme that is ´ curve Y (r)Fp over Fp . Thus, Y (r)ss is a Cartier divisor of Y (r)Fp Fp and is ﬁnite over Fp . Corollary 8.45. Let p be a prime number, let S be a scheme over Fp , and let E be an elliptic curve over S. E is supersingular if and only if there exists an integer a > 0 such that Ker[pa ] = Ker F 2a . Proof. Since the “if” part is clear from the deﬁnition, we show the “only if” part. Since the assertion is ´etale local on S, we may assume there exists an isomorphism α : (Z/rZ)2 → E[r], where r ≥ 3 is an integer relatively prime to p. Let S → Y (r)Fp be the morphism deﬁned by (E, α). Since S → Y (r)Fp factors through the closed subscheme of Y (r)Fp deﬁned by the closed condition Ker[pa ] = Ker F 2a , it suﬃces to show the assertion assuming S to be the closed subscheme of Y (r)Fp deﬁned by the condition Ker[pa ] = Ker F 2a . Then, similarly to the proof of Lemma 8.44, the absolute Frobenius morphism of S is an automorphism of S of ﬁnite order, and thus S is ´etale over Fp . The assertion now follows from Proposition 8.2(3) (i) ⇒ (ii). Lemma 8.46. Let p be a prime number, let S be a scheme over Fp , and let E be an elliptic curve over S. Let e = a + b ≥ g ≥ 0 be

8.6. IGUSA CURVES

an integer, and deﬁne G(a,b) by b−a Ker(V a F b : E → E (p ) ) (8.42) G(a,b) = a−b Ker(V a F b : E (p ) → E)

33

if a ≤ b, if a ≥ b.

(1) G× (a,b) is a ﬁnite ﬂat scheme of ﬁnite presentation over S of degree ϕ(pe ). (2) G(a,b) is a cyclic subgroup scheme of order pe . (3) If P is a generator of G(a,0) = Ker V a , P has exact order pe as a a section of E (p ) . Proof. (1) Let r ≥ 3 be an integer relatively prime to p. Since the assertion is ﬂat local on S, we may assume there is a basis α of E[r] over S. Since α deﬁnes a morphism S → Y (r)Fp , it suﬃces to show it assuming S = Y (r)Fp . By Lemma 8.15(1), G = G(a,b) is a cyclic subgroup scheme of order pe at each point of S. By Lemma 8.25(2), G× has degree ϕ(pe ) at each point of S ord , and degree ≤ ϕ(pe ) at each point of S ss . By Lemma 8.44, G× is a ﬁnite ﬂat scheme of degree ϕ(pe ) over a dense open subscheme U ⊂ S ord of S. Thus, by Lemma A.43, G× is a ﬁnite ﬂat scheme of degree ϕ(pe ) at every point of S. (2) By (1), G× (a,b) is a ﬂat covering of S and G(a,b) has a universal × generator on G(a,b) . (3) As in (1), the assertion is ﬂat local on S. Thus, it suﬃces to prove, assuming that r ≥ 3 is an integer relatively prime to p, S is G× (a,0) over Y (r)Fp and P is the universal generator of G(a,0) . Since 0 is a generator of Ker F b , the assertion on S ord follows by applying Lemma 8.17 to the exact sequence 0 → Ker F b → G(a,b) → Ker V a → 0. Since S is ﬂat over Y (r)Fp , there is no closed subscheme of S other than S itself that contains S ord as an open subscheme. Thus, the assertion follows from Lemma 8.23. Definition 8.47. Let p be a prime number, and let a ≥ 0, M ≥ 1, r ≥ 3 be integers. Assume M , r and p are pairwise relatively prime. Let E be the universal elliptic curve over Y1,∗ (M, r)Fp . The ﬁnite ﬂat scheme a (p G× (a,0) = (Ker V : E

a

)

→ E)×

of degree ϕ(pa ) over Y1,∗ (M, r)Fp is called the Igusa curve and is denoted by Ig(M pa , r)Fp .

34

8. MODULAR CURVES OVER Z

If a = 0, we have Ig(M, r)Fp = Y1,∗ (M, r)Fp . The Igusa curve Ig(M pa , r)Fp represents the functor that associates to a scheme T over Fp the set ⎫ ⎧ isomorphism classes of quadruples (E, P, P , α), where⎪ ⎪ ⎪ ⎪ ⎬ ⎨E is an elliptic curve over T , P is a generator of G (a,0) = . (8.43) a a (p ) ⎪ Ker V ⊂ E , P is a section of E of exact order M ,⎪ ⎪ ⎪ ⎭ ⎩ 2 and α : (Z/rZ) → E[r] is an isomorphism. If P is the universal generator of the cyclic subgroup scheme G(a,0) =0 over the Igusa curve Ig(M pa , r)Fp , we denote by Ig(M pa , r)P the Fp a closed subscheme of Ig(M p , r)Fp deﬁned by the closed condition P = 0. Lemma 8.48. Let p be a prime number, and let a ≥ 0, M ≥ 1, r ≥ 3 be integers. Suppose M , r, p are pairwise relatively prime. (1) The Igusa curve Ig(M pa , r)Fp is a smooth aﬃne curve over Fp . (2) The natural morphism Ig(M pa , r)Fp → Y1,∗ (M, r)Fp is ´etale over Y1,∗ (M, r)ord Fp . =0 is the (3) Suppose a ≥ 1. The closed subscheme Ig(M pa , r)P Fp a ss a =0 → reduced part of Ig(M p , r)Fp . The morphism Ig(M p , r)P Fp Y1,∗ (M, r)ss is an isomorphism. Fp Proof. The morphism Ig(M pa , r)Fp → Y1,∗ (M, r)Fp is the base change of the morphism Ig(pa , r)Fp → Y (r)Fp by the ´etale morphism Y1,∗ (M, r)Fp → Y (r)Fp . Thus, it suﬃces to show them assuming M = 1. a (2) Since Ker V a is ´etale over Y (r)ord Fp , the morphism Ig(p , r)Fp → ord Y (r)Fp is ´etale over Y (r)Fp . =0 ⊂ Ig(pa , r)ss (3) We show Ig(pa , r)P Fp Fp . To do so, it suﬃces =0 is to show that the universal elliptic curve E over S = Ig(pa , r)P Fp a (pa ) over S, supersingular. Since P = 0 is a generator of Ker V ⊂ E we have Ker V a = Ker F a , and thus Ker[pa ] = Ker F 2a ⊂ E. Hence, =0 ⊂ by Corollary 8.45, E is supersingular. Since we have Ig(pa , r)P Fp a ss a P =0 ss Ig(p , r)Fp , we obtain a morphism Ig(p , r)Fp → Y (r)Fp . This is a P =0 an isomorphism since the inverse morphism Y (r)ss Fp → Ig(p , r)Fp is deﬁned by sending (E, α) to (E, 0, α). For a supersingular elliptic curve over a ﬁeld k of characteristic p, the only section P ∈ E(k) that generates Ker V a is P = 0. Thus,

8.6. IGUSA CURVES

35

=0 a ss a P =0 ss Ig(pa , r)P Fp → Ig(p , r)Fp is surjective. Since Ig(p , r)Fp = Y (r)Fp a ss is reduced, this is the reduced part of Ig(p , r)Fp . (1) It suﬃces to show it assuming a ≥ 1. Let E be the universal elliptic curve over Ig(pa , r)Fp . By Lemma B.2(1), the 0-section of E =0 is a Cartier divisor of E. Thus, the closed subscheme Ig(pa , r)P Fp ⊂ a a Ig(p , r)Fp is deﬁned locally by a principal ideal. Since Ig(p , r)Fp =0 is ﬂat over Y (r)Fp , Ig(pa , r)P is a Cartier divisor of Ig(pa , r)Fp by Fp =0 is ´etale over Fp . Thus, by Lemma B.2(2), Lemma A.40. Ig(pa , r)P Fp a =0 Ig(p , r)Fp is smooth over Fp on a neighborhood of Ig(pa , r)P Fp . a P =0 Since the complementary open subscheme Ig(pa , r)ord Fp of Ig(p , r)Fp ord a is ´etale over Y (r)Fp , it is smooth over Fp . Thus, Ig(p , r)Fp is smooth everywhere over Fp .

Let 0 ≤ a ≤ e be integers, and let N = M pe . Let r ≥ 3 be an integer, and suppose M , r and p are pairwise relatively prime. For an elliptic curve E over a scheme T over Fp , its section P a a of exact order M and a basis α of E[r], P (p ) and α(p ) deﬁne a (pa ) (pa ) section of E of exact order M and a basis of E [r], respectively. Thus, by Lemma 8.46(3), a morphism ja : Ig(M pa , r)Fp → Y1,∗ (N, r)Fp ⊂ Y1,∗ (N, r)Z[ r1 ] is deﬁned by sending the isomorphism a a a class [(E, P, P , α)] to the isomorphism class [(E (p ), (P, P (p ) ), α(p ) )]. Proposition 8.49. Let p be a prime, and let e ≥ 0, M ≥ 1, r ≥ 3 be integers. Suppose M , r and p are pairwise relatively prime. Let N = M pa . (1) For 0 ≤ a ≤ e, the morphism (8.44)

ja : Ig(M pa , r)Fp −→ Y1,∗ (N, r)Fp

is a closed immersion. (2) For 0 ≤ a ≤ e, if we denote by Ca the image of the closed immersion ja : Ig(M pa , r)Fp → Y1,∗ (N, r)Fp , then we have Y1,∗ (N, r)Fp = ea=0 Ca . For each 0 ≤ a ≤ e, the inclusion Cass → Y1,∗ (N, r)ss Fp is bijective. (3) For 0 ≤ a < a ≤ e, the intersection Ca ×Y1,∗ (N,r)Fp Ca is Cass . Proof. Similarly to the proof of Lemma 8.48, it suﬃces to show it in the case where M = 1, N = pe , e ≥ 1. (1) Let 0 ≤ a ≤ e be integers. In general, if S is a scheme over Fp , E is an elliptic curve over S, and P is a section of E over S, then by Lemma 8.23, the condition that P has exact order pa is a closed

36

8. MODULAR CURVES OVER Z

condition. Let T be the closed subscheme of S deﬁned by this closed pa −1 condition, and let P pa = i=0 [iP ] over T . The condition that the kernel of the dual of ET → E = ET /P pa equals Ker(F a : E → E (p ) ) a

is a closed condition on T . Deﬁne a closed subscheme Ca of Y1,∗ (N, r)Fp by the closed condition: (8.45)

The universal generator P has exact order pa , and the kernel of the dual of E → E/P pa equals Ker F a .

The morphism ja : Ig(pa , r)Fp → Y1,∗ (N, r)Fp deﬁnes a morphism Ig(pa , r)Fp → Ca . Deﬁne a morphism Ca → Ig(pa , r)Fp by sending (E, P, α) to the isomorphism class of (E , P, the image of α ◦ p−a ). a Since E = E (p ) over Ca , and V a ◦ F a = [pa ], this is the inverse morphism of Ig(pa , r)Fp → Ca . Thus, the morphism Ig(pa , r)Fp → Ca is an isomorphism, and ja : Ig(pa , r)Fp → Y1,∗ (N, r)Fp is a closed immersion. (2) For a rational point [(E, P, α)] ∈ Y1,∗ (N, r)(k) over an algebraically closed ﬁeld of characteristic p, let pa be the order of P ∈ E(k). Then we have [(E, P, α)] ∈ Ca (k). Thus, we have Y1,∗ (N, r)Fp = ea=0 Ca . If E is supersingular, then P = 0. In this case, for any 0 ≤ a ≤ e, we have [(E, P, α)] ∈ Ca (k), and thus ss the mapping Ig(pa , r)ss Fp → Y1,∗ (N, r)Fp is bijective. (3) Let (E, P, α) be an elliptic curve with the universal level structure over the intersection Ca ×Y1,∗ (N,r)Fp Ca . Consider E → E = E/P pa → E = E/P pa . Since pa P = 0, we have Ker(E → E ) = Ker F a −a . Since the kernel of its dual is also Ker F a −a , we have E [pa −a ] = Ker F 2(a −a) . Thus, by Corollary 8.45, (pa ) E is supersingular. Hence, E E is also supersingular. The intersection Ca ∩ Ca is a closed subscheme of Cass . We now show Cass ⊂ Ca ∩ Ca . Let (E, P, α) be the universal elliptic curve with level structure over Cass . Since Ker[pa ] = Ker F 2a , the kernel of E → E = E/P pa equals Ker F a . Since pa P = 0, pa −1 P pa = i=0 [iP ] is the inverse image of Ker F a −a by E → E . Since P pa = Ker F a and Ker[pa ] = Ker F 2a , we have Cass ⊂ C a .

8.7. MODULAR CURVE Y1 (N )Z

37

8.7. Modular curve Y1 (N )Z Proposition 8.50. Let N ≥ 1 be an integer, and let r ≥ 3 be an integer relatively prime to N . The ﬁne moduli scheme Y1,∗ (N, r)Z[ r1 ] is a regular aﬃne curve over Z[ 1r ]. The natural morphism Y1,∗ (N, r)Z[ r1 ] → Y1 (r)Z[ r1 ] is ﬁnite ﬂat of degree ϕ(N )ψ(N ). Proof. Let p be a prime number. We ﬁrst show Y1,∗ (pe , r)Z[ r1 ] is 1 regular in the case where N = pe > 1. Since Y1,∗ (pe , r)Z[ pr ] is smooth 1 over Z[ pr ], it suﬃces to examine a neighborhood of Y1,∗ (pe , r)Fp . Let S = Y1,∗ (pe , r)Z[ r1 ] . For 0 ≤ a ≤ e, deﬁne a closed subscheme Da of S = Y1,∗ (pe , r)Z[ r1 ] as follows. Let P be a universal element over S of exact order pe . If a < e, let Da be the closed subscheme deﬁned by the condition pa P = 0, and deﬁne De = Y1,∗ (pe , r)Fp . If a < e, Da is the pullback of the 0-section by the morphism S → E deﬁned by pa P . Thus, by Lemma B.2(1), Da is deﬁned locally by a principal ideal. If a = e, De is the principal ideal (p). We study the relation between Da and the image Ca of the closed immersion ja : Ig(pa , r) → S (8.44). Let Da´et be the open subscheme pa −1 of Da deﬁned by the condition that the divisor i=0 [iP ] is ´etale over Da .

Lemma 8.51. (1) C0 = D0 . (2) If 0 < a ≤ e, we have Caord = Da´et . Proof. (1) C0 is the closed subscheme of Y1,∗ (pe , r)Z[ r1 ] deﬁned by the condition p = 0 and P = 0, and D0 is the closed subscheme deﬁned by the condition P = 0. Since D0 is a scheme over Fp by Lemma 8.22, we have C0 = D0 . (2) On Caord , we have pa P = 0, and P pa = Ker(V a : E = E (p

a

)

→ E )

is ´etale. Thus, we have Caord ⊂ Da´et . Conversely, on Da´et , P pa = pa −1 etale closed subgroup scheme. By Lemma 8.17, i=0 [iP ] is an ´ pa P = 0 has exact order pe−a . We have p = 0 on Da´et , by Lemma 8.22 if a < e, and by deﬁnition if a = e. Since P pa is ´etale, the kernel of the dual E/P pa → E equals Ker F a by Lemma 8.4, and P pa = Ker V a . Thus, Da´et ⊂ Caord .

38

8. MODULAR CURVES OVER Z

ss 1 We show S SF is regular. SZ[ p1 ] → Y (r)Z[ pr etale ] is ﬁnite ´ p of degree ϕ(N )ψ(N ). Thus, by Lemmas 8.25 and A.43, we see that ss → Y (r)Z[ r1 ] Y (r)ss S SF Fp is ﬁnite ﬂat. Since Y (r)Z[ r1 ] is smooth p 1 ss over Z[ r ], S SFp is also ﬁnite ﬂat over Z[ 1r ]. Since Da is deﬁned by ss a principal ideal locally on S, Caord is a Cartier divisor of S SF by p Lemmas 8.51 and A.40. By Lemma 8.48, Ca is a regular subscheme of ss is regular. S. Since SFp = ea=0 Ca by Proposition 8.49(2), S SF p ss Applying Corollary A.42 to the closed immersion SFp ⊂ D0 = C0 → ss S, we see that S is regular on a neighborhood of SF . p ss . S → Y (r)Z[ r1 ] is ﬁnite ﬂat of degree ϕ(N )ψ(N ) except for SF p Since both S and Y (r)Z[ r1 ] are regular, S → Y (r)Z[ r1 ] is ﬁnite ﬂat everywhere and of degree ϕ(N )ψ(N ) by Proposition A.13(2). This proves that Y1,∗ (N, r)Z[ r1 ] is a regular aﬃne curve over Z[ r1 ] in the case N = pe . We now show it for general N . If N = p|N pep , Y1,∗ (N, r)Z[ r1 ] has an open covering p|N Y1,∗ (N, r)Z[ pep ] . Since the natural morrN

phism Y1,∗ (N, r)Z[ pep ] → Y1,∗ (pep , r)Z[ pep ] is ﬁnite ´etale of degree rN

rN

ϕ(N/pep )ψ(N/pep ), the proof is reduced to the case N = pe .

Proof of Theorem 8.34. (1) Let r ≥ 3 be an integer relatively prime to N . Consider the action of the group GL2 (Z/rZ) on Y1,∗ (N, r)Z[ r1 ] . As in Proposition 2.23, the quotient Y1 (N )Z[ r1 ] of Y1,∗ (N, r)Z[ r1 ] by the action of the group GL2 (Z/rZ) is the coarse moduli scheme of the restriction of the functor M1 (N ) over Z to Z[ 1r ]. By Proposition 8.50, Y1 (N )Z[ r1 ] is a normal aﬃne curve over Z[ 1r ]. Since the coarse moduli scheme Y1 (N )Z is obtained by patching together Y1 (N )Z[ r1 ] with (N, r) = 1, it is a normal aﬃne curve over Z. If N ≥ 4, then by Lemma 8.38(2), the action of GL2 (Z/rZ) on Y1,∗ (N, r)Z[ r1 ] is free. Similarly to the proof of Lemma 8.37, Y1 (N )Z[ N1r ] represents the functor [M1,∗ (N, r)Z[ N1 r] /GL2 (Z/rZ)] by Lemmas A.31, and A.33. Thus, by Corollary 8.39(2), Y1 (N )Z[ N1r ] is a ﬁne moduli scheme of M1 (N )Z[ N1r ] . We now show Y1 (N )Q is a smooth connected aﬃne curve over Q. It suﬃces to show the Riemann surface Y1 (N )an deﬁned by Y1 (N )Q is connected. Let Γ1 (N ) be the subgroup of SL2 (Z) deﬁned by $ !" # % $ a b $ ∈ SL2 (Z)$ a ≡ 1 mod N, c ≡ 0 mod N , (8.46) Γ1 (N ) = c d

8.7. MODULAR CURVE Y1 (N )Z

39

and consider the action of Γ1 (N ) on the upper half-plane H = {τ ∈ C | Im τ > 0}. As in Corollary 2.66, we obtain an isomorphism of Riemann surfaces (8.47)

Γ1 (N )\H −→ Y1 (N )an .

This implies Y1 (N )an is connected, and Y1 (N )Q is a smooth connected aﬃne curve over Q. Hence, Y1 (N )Q is connected, and so is Y1 (N )Z . (2) Y1,∗ (N, r)Z[ r1 ] → Y1 (r)Z[ r1 ] is a ﬁnite surjective morphism of two-dimensional normal schemes by Proposition 8.50. Thus, the induced morphism Y1 (N )Z → Y (1)Z is also a ﬁnite surjective morphism of two-dimensional normal schemes. Since Y (1)Z is isomorphic to A1Z , it is thus regular. Hence, Y1 (N )Z → Y (1)Z is ﬁnite ﬂat by Proposition A.13(2). Y1,∗ (N, r)Z[ r1 ] → Y1 (N )Z[ r1 ] is a Galois covering whose Galois group equals GL2 (Z/rZ) for N ≥ 3 and GL2 (Z/rZ)/{±1} for N = 1, 2 by Lemma 8.41(2). Since the degree of Y1,∗ (N, r)Z[ r1 ] → Y1 (r)Z[ r1 ] is ψ(N )ϕ(N ) by Proposition 8.50, the degree of Y1 (N )Z[ r1 ] → Y (1)Z[ r1 ] equals ψ(N )ϕ(N )/2 for N ≥ 3, and ψ(N )ϕ(N ) for N = 1, 2. (3) Since Y1,∗ (N, r)Z[ r1 ] is a smooth aﬃne curve over Z[ 1r ], its quotient Y1 (N )Z[ r1 ] is a smooth aﬃne curve by Proposition B.10(1). Suppose p N . If N ≥ 4, Y1 (N )Z[ N1 ] is a ﬁne moduli scheme, and thus Y1 (N )Z ⊗Z Fp is a ﬁne moduli scheme of the restriction M1 (N )Fp . As for the action of GL2 (Z/rZ) on Y0,∗ (N, r)Z[ r1 ] , the inertia group at the generic point of each irreducible component of each ﬁber is a subgroup of {±1} by Lemma 8.41(2). If N > 2, we have −P = P for an element P of order N , and thus the inertia group equals 1. If N ≤ 2, the inertia group is {±1}. Hence, by Corollary B.11(1), Y1 (N )Z ⊗Z Fp is the quotient of Y1,∗ (N, r)Fp by GL2 (Z/rZ), and it is a coarse moduli scheme of the restriction of the functor M1 (N )Fp . We present some consequences of Proposition 8.50. Proposition 8.52. Let S be a scheme, and let E be an elliptic curve over S. Let N ≥ 1 be an integer. (1) The ﬁnite scheme M1 (N )E over S is ﬂat of ﬁnite presentation over S of degree ϕ(N )ψ(N ). The scheme M1 (N )E is a Cartier divisor of E. If N = N N with (N , N ) = 1, then we have M1 (N )E = M1 (N )E ×S M1 (N )E . If N = pe with e ≥ 1, then we have an

40

8. MODULAR CURVES OVER Z

equality of Cartier divisors (8.48) M1 (N )E = Ker[pe ] − Ker[pe−1 ] = E[pe ]

×

[pe−1 ] E[p]

(E[p] − [0]).

(2) Let order N . Let G = P = P be a section of E of exact × [aP ]. Then, we have G = a∈Z/N Z a∈(Z/N Z)× [aP ]. Proof. (1) It suﬃces to show the following case: N = pe , r ≥ 3 an integer relatively prime to N , S = Y (r)Z[ r1 ] , and E the universal elliptic curve over S. In this case, we have M1 (N )E = Y1,∗ (N, r)Z[ r1 ] . By Proposition 8.50, M1 (N )E → S is ﬁnite ﬂat of degree ϕ(N )ψ(N ). We show (8.48). Since S is ﬂat over Z, itsuﬃces to apply Corol lary A.44 to U = S[1/p], A = M1 (N )E , B = Ker[pe ] − Ker[pe−1 ] , × (E[p] − [0]). and E[pe ] [pe−1 ] E[p]

(2) We may assume S and E as above. Let P be the universal generator over T = M1 (N )E . Since T = M1 (N )E is ﬂat over Z, it × suﬃces to apply Corollary A.44 to the closed subscheme A = G of ET and B = a∈(Z/N Z)× [aP ]. Corollary 8.53. Let S be a scheme, and let E be an elliptic curve over S. Let G be a closed subgroup scheme of E ﬁnite ﬂat over S of degree N . (1) The following conditions (i) and (ii) are equivalent. (i) G is a cyclic subgroup scheme of order N . (ii) The scheme G× of generators of G is a ﬁnite ﬂat scheme over S of degree ϕ(N ). (2) The condition for a scheme T over S that GT is a cyclic subgroup scheme of ET is a closed condition on S. Proof. (1) The proof of (ii) ⇒ (i) is similar to the proof of Lemma 8.46(2). We show (i) ⇒ (ii). Since the assertion is ﬂat local, we may assume there exists a section of exact order N satisfying G = P by Deﬁnition 8.13. In this case the assertion follows easily from Proposition 8.52(2). (2) By (1), the condition that GT is a cyclic subgroup scheme of ET is equivalent to the condition that G× T is a ﬁnite ﬂat scheme on T of degree ϕ(N ). Moreover, by Proposition 8.52(2) and Nakayama’s lemma, the quasi-coherent sheaf OG× over S is generated by ϕ(N ) sections locally on S. Hence, by Corollary A.38, this condition is a closed condition.

8.8. MODULAR CURVE Y0 (N )Z

41

8.8. Modular curve Y0 (N )Z Proposition 8.54. Let S be a scheme, and let E be an elliptic curve over S. Let N ≥ 1 an integer. The functor M0 (N )E over S is represented by a ﬁnite ﬂat scheme of ﬁnite presentation M0 (N )E over S of degree ψ(N ). The natural morphism M1 (N )E → M0 (N )E is ﬁnite ﬂat of ﬁnite presentation of degree ϕ(N ). If N is invertible on S, M0 (N )E is ´etale over S. We ﬁrst show the following lemma. Lemma 8.55. Let S be a scheme, let E be an elliptic curve over S. Let N ≥ 1 be an integer. The functor N -IsogE deﬁned by associating to a scheme T over S the set closed subgroup scheme of ET that is a ﬁnite N - IsogE (T ) = ﬂat scheme of ﬁnite presentation of degree N as a scheme over T . is represented by a ﬁnite scheme TN,E over S. Proof. The functor that associates to a scheme T over S the set {locally free quotient OT -module of rank N of the locally free OT -module OE[N ]×T } is represented by the Grassmannian scheme denoted by Grass(OE[N ] , N ), which is a proper scheme over S. If G ⊂ ET is a closed subgroup scheme ﬂat ﬁnite of ﬁnite presentation of degree N as a scheme over T , then OG is a locally free quotient OT modules of rank N of OE[N ]×S T . Thus, as in the proof of Lemma 8.23, the functor N -IsogE is represented by a closed subscheme TN,E of the Grassmannian scheme Grass(OE[N ] , N ). We show TN,E is ﬁnite over S. Since TN,E is proper over S, it suﬃces to show, by Corollary A.9, that each geometric ﬁber is ﬁnite. We may assume S = Spec k, k is an algebraically closed ﬁeld of characteristic p > 0, and N = pe . If E is supersingular, then we have G = Ker F e by Proposition 8.2. If E is ordinary, we have E[N ] Z/pe Z × μpe , and G is of the form Z/pa Z × μpb , a + b = e, and there are e + 1 of such G. Proof of Proposition 8.54. We show M0 (N )E is representable. Let GM0 (N )E ⊂ EM0 (N )E be the universal closed subgroup scheme. Applying Corollary 8.53(2) to GTN,E ⊂ ETN ,E , we see that M0 (N )E is represented by a closed subscheme M0 (N )E of TN,E . We now show M1 (N )E → M0 (N )E is ﬁnite ﬂat of degree ϕ(N ). Let GM0 (N )E ⊂ EM0 (N )E be the universal cyclic subgroup scheme of

42

8. MODULAR CURVES OVER Z

degree N . Then, since M1 (N )E = G× M0 (N )E , M1 (N )E → M0 (N )E is ﬁnite ﬂat of degree ϕ(N ) by Corollary 8.53(1). Since M1 (N )E is ﬁnite ﬂat over S of degree ϕ(N )ψ(N ) by Proposition 8.52(1), M0 (N )E is also ﬁnite ﬂat over S of degree ψ(N ). Assuming N is invertible on S, we show M0 (N )E is ´etale over S. Since the assertion is ´etale local on S, we may assume E[N ] is isomor2 phic to (Z/N Z) . In this case, we have, by Corollary 8.10, M0 (N )E = C:cyclic subgroup of(Z/N Z)2 of order N S, and the assertion is now clear. Corollary 8.56. Let N ≥ 1 be an integer, and let r ≥ 3 be an integer relatively prime to N . The functor M0,∗ (N, r)Z[ r1 ] over Z[ 1r ] is represented by a regular ﬁnite ﬂat scheme Y0,∗ (N, r)Z[ r1 ] over 1 1 Y (r)Z[ r1 ] of degree ψ(N ). Y0,∗ (N, r)Z[ rN ] is smooth over Z[ rN ]. The ﬁeld of constants of Y0,∗ (N, r)Q = Y0,∗ (N, r)Z[ r1 ] ⊗Z[ r1 ] Q is Q(ζr ). Proof. Let E be the universal elliptic curve over Y (r)Z[ r1 ] . The functor M0,∗ (N, r)Z[ r1 ] is represented by M0 (N )E = Y0,∗ (N, r)Z[ r1 ] . This is a ﬁnite ﬂat scheme over Y (r)Z[ r1 ] of degree ψ(N ). By Proposition 8.54, Y1,∗ (N, r)Z[ r1 ] = M1 (N )E → Y0,∗ (N, r)Z[ r1 ] = M0 (N )E is faithfully ﬂat. Since Y1,∗ (N, r)Z[ r1 ] is regular, so is Y0,∗ (N, r)Z[ r1 ] by 1 1 , it Proposition A.13(1). Since Y0,∗ (N, r)Z[ rN etale over Y (r)Z[ rN ] is ´ ] 1 ]. is smooth over Z[ rN We omit the proof that the ﬁeld of constants of Y0,∗ (N, r)Q = Y0,∗ (N, r)Z[ r1 ] ⊗Z[ r1 ] Q equals Q(ζr ) because it is similar to that of Lemma 8.37. Let p be a prime number, N = M pe with (p, M ) = 1, e ≥ 1 and integer, and let r ≥ 3 be an integer relatively prime to N . Let S be a scheme over Fp , and let E be an elliptic curve over S. Then, by Lemma 8.46(2), for e = a + b ≥ a ≥ 0, b−a

G(a,b) = Ker(V a F b : E → E (p

)

)

is a cyclic subgroup scheme of E of order p if a ≤ b, and e

a−b

G(a,b) = Ker(V a F b : E (p

a−b

)

→ E)

is a cyclic subgroup scheme of E (p ) of order pe if a ≥ b. Deﬁne a morphism of schemes ja : Y0,∗ (M, r)Fp → Y0,∗ (N, r)Z[ r1 ] by

8.8. MODULAR CURVE Y0 (N )Z

43

[(E, C, α)] → [(E, (G(a,b) , C), α)] if a ≤ b, and by [(E, C, α)] → a−b a−b a−b [(E (p ) , (G(a,b) , C (p ) ), α(p ) )] if a ≥ b. Proposition 8.57. Let p be a prime number, let N = M pe with (p, M ) = 1 an integer, and let r ≥ 3 be an integer relatively prime to N . (1) For 0 ≤ a ≤ e, the morphism (8.49)

ja : Y0,∗ (M, r)Fp −→ Y0,∗ (N, r)Fp

is a closed immersion. (2) For 0 ≤ a ≤ e, let Ca be the image of the closed immersion ja : Y0,∗ (M, r)Fp → Y0,∗ (N, r)Fp . Then, we have Y0,∗ (N, r)Fp = e ss ss a=0 Ca , and the inclusion Ca → Y0,∗ (N, r)Fp is a bijection. The multiplicities of C0 and of Ce in Y0,∗ (N, r)Fp are 1. (3) If 0 ≤ a ≤ e and 0 ≤ a ≤ e with a = a , then the intersection Ca ∩ Ca = Ca ×Y0,∗ (N,r)Fp Ca equals Cass = Cass . Proof. As in the proof of Proposition 8.49, it suﬃces to show the assertions when M = 1 and N = pe ≥ 1. (1) We ﬁrst show it in the case a ≤ b = e − a. Let G be the universal cyclic subgroup scheme of order pe over Y0,∗ (N, r)Fp . For an integer 0 ≤ a ≤ b = e − a, deﬁne a closed subscheme of Y0,∗ (N, r)Fp by the closed condition b−a (8.50) G = Ker V a F b : E → E (p ) . The morphism ja : Y (r)Fp → Y0,∗ (N, r)Z[ r1 ] deﬁnes an isomorphism Y (r)Fp → Ca by deﬁnition. Thus, ja : Y (r)Fp → Y0,∗ (N, r)Z[ r1 ] is a closed immersion. If b = e − a ≤ a, a similar proof works if we deﬁne Ca by the closed condition (8.51)

the kernel Ker(E → E) of the dual of E → E = E/G a−b equals Ker(V b F a : E → E (p ) ).

(2) Let k be an algebraically closed ﬁeld of characteristic p, let [(E, G, α)] ∈ Y0,∗ (N, r)(k) be a k-rational point, and let pa be the order of G(k). If E is supersingular, then we have a = 0 and [(E, G, α)] ∈ C0 (k). If E is ordinary, we also have [(E, G, α)] ∈ Ca (k). Thus, we have Y0,∗ (N, r)Fp = ea=0 Ca . If E is supersingular, we have G = Ker F e and [(E, G, α)] ∈ Ca (k) for all 0 ≤ a ≤ e. Thus, the ss mapping ja : Y (r)ss Fp → Y0,∗ (N, r)Fp is a bijection.

44

8. MODULAR CURVES OVER Z

By Lemma 8.4, C0ord equals the open subscheme of Y0,∗ (N, r)Fp deﬁned by the condition that the dual of E → E/G is ´etale. Thus, the multiplicity of C0 equals 1. Similarly, Ceord equals the open subscheme of Y0,∗ (N, r)Fp deﬁned by the condition that G is ´etale, and thus its multiplicity equals 1. (3) Let (E, G, α) be an elliptic curve with universal level structure over the intersection Ca ∩ Ca . Let b = e − a and b = e − a . If a < a ≤ e/2, then since we have G = Ker pa F b −a = Ker pa F b−a , we have Ker pa −a = Ker F 2(a −a) , and thus E is supersingular. The case e/2 ≤ a < a is similar. Suppose a < e/2 < a . Let E → E be the dual of E → E = E/G. The composition E → E → E equals pb F a −b ◦ pa F b−a = pe . Thus, we have Ker pa −a = Ker F 2(a −a) , and E is supersingular in this case, too. The case a < e/2 < a is similar. This concludes the proof that the intersection Ca ∩ Ca is a closed subscheme of Cass . We now show Cass ⊂ Ca ∩ Ca . Let G be the universal cyclic subgroup scheme over Cass . If a, a ≤ e/2, then G = Ker pa F b−a = Ker pa F b −a , and thus Cass ⊂ Ca . Similarly, if a, a ≥ e/2, we also ss have Ca ⊂ Ca . If a < e/2 < a , then the kernel of the dual of E → E = E/G equals Ker pa V b−a = Ker pa F b −a , and again we have Cass ⊂ Ca . The case a < e/2 < a is also similar. Corollary 8.58. If e = 1, the regular curve Y0,∗ (N, r)Z[ r1 ] over is semistable at p. The ﬁber Y0,∗ (N, r)Fp is the union of C0 and C1 . Z[ 1r ]

Proof. It follows easily from Corollary 8.56, Proposition 8.57 and Lemma B.8. Proof of Theorem 8.32. We omit the proof of (1)–(3) since they are similar to the proof of Theorem 8.34. (4) Let r ≥ 3 be an integer relatively prime to p. As for the action of GL2 (Z/rZ) on Y0,∗ (M p, r)Z[ r1 ] , the inertia group at the generic point of each irreducible component of each ﬁber is {±1}. Thus, by Corollaries 8.58 and B.11(2), Y0 (M p) is weakly semistable, and j0 , j1 : Y0 (M )Fp → Y0 (M p)Fp are closed immersions. The intersection of the image C0 of j0 and that of C1 of j1 is Y0 (M )ss Fp = /GL (Z/rZ). Y0,∗ (M, r)ss 2 Fp Let x = [(E, C)] ∈ Y0 (M )ss Fp be an ordinary double point of Y0 (M p)Fp , let x = [(E, C, α)] ∈ Y0,∗ (M, r)ss Fp be a point in the inverse image of x, and let η = [(E0 , C0 , α0 )] be the generic point of

8.8. MODULAR CURVE Y0 (N )Z

45

Y0,∗ (N, r)Z . Then, the inertia group Ix is the image of the injection Aut(EFp , CFp ) → GL2 (Z/rZ), and by Lemma 8.41, the inertia group Iη is {±1} ⊂ GL2 (Z/rZ). Thus, by Corollary B.11(2), the index of x equals [Ix : Iη ] = Aut(EFp , CFp )/{±1}. We deﬁne morphisms between modular curves. Proposition 8.59. Let S be a scheme, and let E be an elliptic curve over S. Let N = M dM ≥ 1 be an integer. (1) Let P be a section of E of exact order N . Then, P = (N/d)P has exact order d. Let H = d−1 i=0 [iP ]. Then the image P of M P in E = E/H is a section of exact order M . (2) Let C be a cyclic subgroup scheme of E of order N . Then, there exists a unique cyclic subgroup scheme H of E of order d such that ﬂat locally on S, N/d times of a generator of C is a generator. Moreover, there exists a unique cyclic subgroup scheme C of E = E/H of order M such that ﬂat locally on S, M times of a generator of C is a generator. Proof. (1) It suﬃces to show the following case: r ≥ 3 is an integer relatively prime to N , S = Y1,∗ (N, r)Z[ r1 ] , E is the universal elliptic curve over S, and P is the universal section of exact order N . The assertion is clear on S[ N1 ]. Thus, it suﬃces to apply Corollary A.44(2) to the closed subscheme M1 (d)E of E and the section P , and the closed subscheme M1 (d)E of E and the section P . (2) Let X = C × , S = M0 (d)E , and let H be the universal cyclic subgroup scheme of ES of order d. By (1), we obtain a morphism d−1 N f : X → Y deﬁned by P → M P ⊂ EX / i=0 [i d P ] . It suﬃces to show that there exists a section g : S → Y such that f : X → Y is the composition of h : X → S and g : S → Y . Since h : X → S is faithfully ﬂat, g : S → Y is unique if it exists. We show the existence. Let r ≥ 3 be an integer relatively prime to N . Since the assertion is ﬂat local on S, we may assume there exists a basis α for E[r]. Then, (E, C, α) deﬁnes a morphism S → Y0,∗ (N, r)Z[ r1 ] . Thus, it suﬃces to show the case S = Y0,∗ (N, r)Z[ r1 ] . On SZ[ N1 ] the assertion is clear. It now suﬃces to apply Lemma A.45. We deﬁne a morphism of functors (8.52)

sd : M1 (N ) −→ M1 (M )

46

8. MODULAR CURVES OVER Z

by sending the isomorphism class of (E, P ) to the isomorphism class (E , P ). Similarly, we deﬁne a morphism of functors (8.53)

sd : M0 (N ) −→ M0 (M )

by sending the isomorphism class of (E, C) to the isomorphism class (E , C ). Lemma 8.60. Let M d | N ≥ 1 be integers. The morphisms of modular curves deﬁned by the morphism of functors sd (8.54)

sd : Y1 (N )Z −→ Y1 (M )Z , sd : Y0 (N )Z −→ Y0 (M )Z

are ﬁnite. Proof. We show the morphism sd : Y1 (N )Z → Y1 (M )Z is ﬁnite. Let r ≥ 3 be an integer relatively prime to N . Deﬁne sd : Y1,∗ (N, r)Z[ r1 ] → Y1,∗ (M, r)Z[ r1 ] in the same way as sd : Y1 (N )Z → Y1 (M )Z . We show this morphism is ﬁnite. Let (E, P, α) be the universal elliptic curve with level structure over S = Y1,∗ (M, r)Z[ r1 ] . Let A = M0 (d)E , and let G ⊂ EA be the universal cyclic subgroup scheme over A of order d. Let E = EA /G, and let g : E → EA be the dual of EA → E = EA /G. Let B = M1 (N )E , and let P : B → EB be the N universal section of exact order N . The condition that g( M d P ) = P and Nd P is a generator of the kernel of g : EB → EB is a closed condition on B. Let C be the closed subscheme deﬁned by this closed condition. C is ﬁnite over S by Proposition 8.54 and Corollary 8.24. : (Z/rZ)2 → EC [r] be the composition of α and the inverse Let αC [r] → EC [r]. Then, the triple (EC , PC , α (X)) of the isomorphism EC deﬁnes a morphism C → Y1,∗ (N, r)Z[ r1 ] . We show that the morphism C → Y1,∗ (N, r)Z[ r1 ] is an isomorphism and that the composition of the inverse of this and the natural morphism C → Y1,∗ (M, r)Z[ r1 ] is sd : Y1,∗ (N, r)Z[ r1 ] → Y1,∗ (M, r)Z[ r1 ] . Let (E , P , α ) be the universal elliptic curve over Y1,∗ (N, r)Z[ r1 ] with level structure. The dual E → E of g : E → E = E / Nd P deﬁnes a morphism Y1,∗ (N, r)Z[ r1 ] → A that extends sd . The universal section P deﬁnes Y1,∗ (N, r)Z[ r1 ] → C ⊂ B. It is easy to see that this is the inverse. Thus, sd : Y1,∗ (N, r)Z[ r1 ] → Y1,∗ (M, r)Z[ r1 ] is ﬁnite. Letting r ≥ 3 run integers relatively prime to N , we obtain a ﬁnite morphism sd : Y1 (N )Z → Y1 (M )Z by taking the quotients and patching them. Similarly, sd : Y0 (N )Z → Y0 (M )Z is ﬁnite.

8.8. MODULAR CURVE Y0 (N )Z

47

Example 8.61. Since Y0 (4)Z is the quotient of Y1 (4)Z by the diamond operator (Z/4Z)× = {±1}, we have Y0 (4)Z = Y1 (4)Z . We show Y1 (4)Z = Y0 (4)Z = Spec Z[s, t, u]/(st−28 , u(s+24 )−24 t, u(t+24 )−t2 ). By the Remark after Theorem 8.34, Y1 (4)Z is the integral closure of Y (1)Z = Spec Z[j] in Y1 (4)Z[ 14 ] . Let A = Z[s, t, u]/(st − 28 , u(s + 24 ) − 24 t, u(t + 24 ) − t2 ). 1 1 1 We have A[ 21 ] = Z[ 12 ][s, s(s+2 4 ) ]. This is isomorphic to Z[ 4 , d, d(d−4) ] by s → 4(d − 4). Through this isomorphism we identify Y1 (4)Z[ 14 ] = Spec A[ 12 ]. 1 We show A is an integrally closed domain. A[ 12 ] = Z[ 21 ][s, s(s+2 4) ] is an integrally closed domain. A/2A = F2 [s, t, u]/(st, us, (u − t)t) = F2 [s, t, u − t]/(st, (u − t)s, (u − t)t) is isomorphic to the subring of F2 [t] × F2 [u] × F2 [s] given by {(f, g, h) ∈ F2 [t] × F2 [u] × F2 [s] | f (0) = g(0) = h(0)} by the mapping s → (0, 0, s), t → (t, 0, 0), u → (t, u, 0). Thus, A/2A is reduced. Spec A is smooth over Z except at the maximal ideal m = (2, s, t, u). Thus, by Lemma A.41, A is an integrally closed domain. If we let k = s(s + 24 ) = 24 d(d − 4), we have ku = 212 . By (8.26), the morphism j : Y1 (4)Z[ 14 ] → Y (1)Z deﬁned by the j-invariant is deﬁned by

Z[j] → j

→ 28 ·

⊃

A[ 21 ] (d2 −4d+1)3 d(d−4)

=

(k+24 )3 k

A

= k2 + 3 · 24 k + 3 · 28 + u.

We show that the integral closure of Z[j] in A[ 12 ] is A. Since A is integrally closed, it suﬃces to show that the generators s, t, u and k are integral over Z[j]. Since we have s(s+24 ) = k and (k+24 )3 = jk, k and s are integral over Z[j]. Moreover, since u = j−(k2 +3·24 k+3·28 ) and t2 = u(t + 24 ), u and t are also integral over Z[j]. This concludes the proof of Y1 (4)Z = Spec A. The intermediate covering Y1 (2)Z equals Spec Z[k, u]/(ku − 212 ). The surjective morphisms of rings A/2A → F2 [t], A/2A → F2 [u] and A/2A → F2 [s] deﬁne closed subschemes C0 , C1 and C2 ⊂ Y0 (4)F2 , respectively. We deﬁne isomorphisms ji : Y (1)F2 → Ci (i = 0, 1, 2) by t → j, u → j and s → j. The Atkin–Lehner involution w4 : Y0 (4)Z → Y0 (4)Z is deﬁned by s → t, t → s, u → v = s + t − u − 24 . It suﬃces show that

48

8. MODULAR CURVES OVER Z

4d w4 : Y0 (4)Z[ 14 ] → Y0 (4)Z[ 14 ] is deﬁned by d → d−4 . Extending √ 1 the coeﬃcients to Z[ −1, 4 ], we compute w4 : Y1 (4)Z[√−1, 1 ] → 4 Y1 (4)Z[√−1, 14 ] . Let P = (1, 1) ∈ E be the universal section of order 4. The quotient E = E/2P of the universal elliptic curve E : dy 2 = x(x2 + (d − 2)x + 1) is given by dy 2 = x (x + d)(x + 4), and E → E is given by x = x + x1 − 2, y = xy (x − x1 ). Moreover, E = E/P is given by dy 2 = x (x2 − 2(d + 4)x + (d − 4)2 ). , y = xy x − 4d E → E is given by x = x + d + 4 + 4d x √ x . If we let x = −(d − 4)x1 and y = (d − 4)2 2 −1y1 , E is given by 2 2(d+4) 4d 2 d−4 y1 = x1 x1 + d−4 x1 + 1 . Since the universal section of E is 4d . given by (1, 1), we have w4∗ (d) = d−4 4 If we let l = t(t+2 ) and v = s+t−u−24 , s4 = s0 ◦w4 : Y0 (4)Z → Y (1)Z is deﬁned by j → l2 + 3 · 24 l + 3 · 28 + v. Since the j-invariant 2 3 8 3 ) of E is j(E ) = 24 (dd−4d+16) = (k+2 = k + 3 · 28 + 3 · 24 u + u2 , 2 (d−4)2 k2 s2 : Y0 (4)Z → Y (1)Z is deﬁned by j → k + 3 · 28 + 3 · 24 u + u2 . The image of j = s∗1 j, s∗2 j, s∗4 j in F2 [t] × F2 [u] × F2 [s] is given by (t, u, s4 ), (t2 , u2 , s2 ), (t4 , u, s), respectively.

8.9. Compactiﬁcations In this book we deﬁne the compactiﬁcation X0 (N )Z and X1 (N )Z of modular curves Y0 (N )Z and Y1 (N )Z as the integral closure of the j-line. The meaning of these curves as moduli schemes has been studied, but we do not mention it here. Definition 8.62. Let N ≥ 1 be an integer. (1) Deﬁne X0 (N )Z as the integral closure of P1Z with respect to j : Y0 (N )Z → A1Z . (2) Deﬁne X1 (N )Z as the integral closure of P1Z with respect to j : Y1 (N )Z → A1Z . In this section we prove the following fundamental properties of X0 (N )Z and X1 (N )Z . Theorem 8.63. Let N ≥ 1 be an integer. (1) X0 (N )Z is a normal projective curve over Z, and its each geometric ﬁber is connected. (2) Let p N be a prime number. Then, X0 (N )Z is smooth at p. The ﬁber X0 (N )Fp = X0 (N )Z ⊗Z Fp is a smooth compactiﬁcation of Y0 (N )Fp .

8.9. COMPACTIFICATIONS

49

(3) Let N = M p with p M . Then, X0 (N )Z is weakly semistable at p. The closed immersions j0 : Y0 (M )Fp → Y0 (N )Fp and j1 : Y0 (M )Fp → Y0 (N )Fp extend to closed immersions j0 : X0 (M )Fp → X0 (N )Fp and j1 : X0 (M )Fp → X0 (N )Fp . The ﬁber X0 (N )Fp is the union of the image C 0 of j0 and the image C 1 of j1 , and the intersection of C 0 and C 1 is the coarse ss moduli scheme Y0 (M )ss Fp of M0 (M )Fp . The index of the ordinary double point x = [(E, C)] ∈ Y0 (M )ss Fp is the order of Aut(EF¯ p , CF¯ p )/{±1}. Using Theorem 8.63, we compute the number of isomorphism classes of supersingular elliptic curves, which is equal to deg Y (1)ss Fp . Corollary 8.64. Let p be a prime number. Then, we have Y (1)ss Fp = ∅. The number of isomorphism classes of supersingular elliptic curves over Fp equals deg Y (1)ss Fp , which equals p−a (if p ≡ a = 2, 3, 5, 7, −1, 13 mod 12). 12 Proof. By Theorem 8.63, X0 (p)Fp = C 0 ∪ C 1 is connected. Thus, C 0 ∩C 1 = Y (1)ss Fp is nonempty. Since the coarse moduli scheme ss ss Y (1)Fp is reduced, we have deg Y (1)ss Fp = Y (1)Fp (Fp ). We have {isomorphism classes of supersingular elliptic curves over ss ¯ Fp } = M(1)ss Fp (Fp ) = Y (1)Fp (Fp ). By Corollary D.21(1) and the fact that g0 (1) = 0, we have g0 (p) = deg Y (1)ss Fp − 1. By Proposition 2.15 and Lemma 2.14, we have 1 + g0 (p) = 1 +

g0 (p) = 1 +

1 1 (p + 1) − · 2 12 2 ⎧ ⎧ ⎪0 (p ≡ 2 mod 3) ⎪0 (p ≡ 3 mod 4) ⎨ 1 1⎨ − − 1 (p = 3) 1 (p = 2) 3⎪ 4⎪ ⎩ ⎩ 2 (p ≡ 1 mod 3) 2 (p ≡ 1 mod 4).

Example 8.65. By Example 8.6, the elliptic curves E over Fp whose j-invariant equals 1728 is supersingular if p ≡ −1 mod 4, and the elliptic curves E over Fp whose j-invariant equals 0 is supersingular if p ≡ −1 mod 3. Thus, by Lemma 8.41 and Corollary 8.64, we obtain p−1 1 = . Aut E 24 isomorphism classes of supersingular elliptic curves E

50

8. MODULAR CURVES OVER Z

For a prime number p, the number of supersingular elliptic curves over Fp and their j-invariants are as follows. p of s.s. curvesa s.s. j-invariantb indexc

2 1 0 12

3 1 0 6

5 7 11 13 17 1 1 2 1 2 0 −1 0, 1 6 0, 8 3 2 3, 2 1 3, 1

19 2 −1, 7 2, 1

··· ··· ··· ···

a number

of isomorphism classes of supersingular elliptic curves of supersingular elliptic curve c index at each point = 1 Aut(E) 2 b j-invariant

Theorem 8.66. Let N ≥ 1 be an integer. (1) X1 (N )Z is a normal projective curve over Z and each geometric ﬁber is connected. (2) Let p N . Then, X1 (N )Z is smooth at p. The ﬁber X1 (N )Fp = X1 (N )Z ⊗Z Fp is a smooth compactiﬁcation of Y1 (N )Fp . In order to describe the compactiﬁcation, we deﬁne the Tate curves. In Chapter 2, (2.34), we deﬁned the power series ∞ 2 Ek (q) = 1 + σk−1 (n)q n ∈ Q[[q]], ζ(1 − k) n=1

where σk−1 (n) = d|n dk−1 . We then deﬁned an elliptic curve over the ﬁeld of power series Q((q)) by (2.35)

y 2 = 4x3 −

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

By the change of coordinates x = x + (2.35) becomes

1 12 ,

y = 2y + x , the equation

1 (E4 (q) − 1)x 48 1 1 (E6 (q) − 1). − (E4 (q) − 1) + 4 · 122 4 · 216

(8.55) y 2 + x y = x3 −

∞ n = ζ(1−k) (Ek (q) − 1) ∈ Z[[q]]. Since Let sk (q) = n=1 σk−1 (n)q 2 1 1 ζ(−3) = 120 and ζ(−5) = − 252 , the coeﬃcient of the degree 1 term of the right-hand side of (8.55) equals −5s4 (q), and the constant term 1 (5s4 (q) + 7s6 (q)). is − 12

8.9. COMPACTIFICATIONS

51

1 Question. Verify that 12 (5s4 (q) + 7s6 (q)) ∈ Z[[q]]. Let Z((q)) = −1 Z[[q]][q ]. Show that the equation 1 (8.56) y 2 + xy = x3 − 5s4 (q)x − (5s4 (q) + 7s6 (q)) 12 deﬁnes an elliptic curve over Z((q)).

Definition 8.67. The elliptic curve Eq over Z((q)) = Z[[q]][q −1 ] deﬁned by (8.56) is called the Tate curve. Lemma 8.68. The morphism e : Spec Z((q)) → Y (1)Z = A1Z deﬁned by the Tate curve Eq extends uniquely to e¯ : Spec Z[[q]] → 1 X(1)Z = P1Z . Let X(1)Z |∧ ∞ be the completion of X(1)Z = PZ along ∧ ∞. Then e¯ induces an isomorphism eˆ : Spec Z[[q]] → X(1)Z |∞ . Proof. As we have seen Example 2.37 in Chapter 2, the jinvariant of the Tate curve is 1 E4 (q)3 = + 744 + 196884q + 21493760q 2 + · · · . j(q) = Δ(q) q n 24 ∈ qZ[[q]]× and E4 (q) ∈ Since we have Δ(q) = q ∞ n=1 (1 − q ) 1 × 1 + Z[[q]] ⊂ Z[[q]] , we have j(q) ∈ q · Z[[q]]× . The assertion now follows easily. The following is a proposition concerning torsion points of the Tate curve, for which we omit the proof. Proposition 8.69. Let N ≥ 1 be an integer. The group scheme Eq [N ] of N -torsion points of the Tate curve Eq is isomorphic to the pullback of T [N ] (8.14) by the inclusion Z[q, q −1 ] → Z((q)) of rings. From now on, we identify Eq [N ] and T [N ] through the isomorphism in Proposition 8.69. Let r ≥ 1 be an integer. Deﬁne a ring homomorphism Z((q)) → Z[ 1r , ζr ]((qr )) by q → qrr . Let Eqrr be the pullback of the Tate curve Eq over Z[ 1r , ζr ]((qr )) by this ring homomorphism. We deﬁne an isomorphism αr : (Z/rZ)2 → Eqrr [r] = T [r] ⊗Z[q,q−1 ] Z[ 1r , ζr ]((qr )) of group schemes over Z[ 1r , ζr ]((qr )) by αr ((1, 0)) = (0, ζr ) and αr ((0, 1)) = (1, qr ). The morphism er : Spec Z[ 1r , ζr ]((qr )) → Y (r)Z[ r1 ] deﬁned by the pair (Eqrr , αr ) is called the morphism deﬁned by the Tate curve. We have e1 = e. For a ∈ (Z/rZ)× and b ∈ Z/rZ, let σa,b = a0 1b ∈ GL2 (Z/rZ). Deﬁne a subgroup V (Z/rZ) = {σa,b | a ∈ (Z/rZ)× , b ∈ Z/rZ} ⊂ GL2 (Z/rZ), and deﬁne an action of V (Z/rZ) on Z[ r1 , ζr ]((qr )) by

52

8. MODULAR CURVES OVER Z

σa,b (ζr ) = ζra and σa,b (qr ) = ζrb qr . We deﬁne an action of −1 ∈ GL2 (Z/rZ) as the trivial action. Corollary 8.70. Let r ≥ 1 be an integer. The morphism deﬁned by the Tate curve 1 er : Spec Z ζr , ((qr )) → Y (r)Z[ r1 ] r is compatible with the action of V (Z/rZ) · {±1} ⊂ GL2 (Z/rZ). The commutative diagram induced by er Spec Z[ζr , 1r ]((qr )) −−−−→ Y (r)Z[ r1 ] σ∈GL2 (Z/rZ)/V (Z/rZ)·{±1}

(8.57)

⏐ ⏐

⏐ ⏐

Spec Z[ 1r ]((q))

e

−−−1−→ Y (1)Z[ r1 ]

is Cartesian. The morphism er : Spec Z[ζr , 1r ]((qr )) → Y (r)Z[ r1 ] uniquely extends to e¯r : Spec Z[ζr , 1r ][[qr ]] → X(r)Z[ r1 ] . Proof. It is easy to see that er : Spec Z[ζr , 1r ]((qr )) → Y (r)Z[ r1 ] is compatible with the action of the subgroup V (Z/rZ) · {±1} ⊂ GL2 (Z/rZ). The morphism deﬁned by the diagram (8.57) 1 Spec Z ζr , ((qr )) r σ∈GL2 (Z/rZ)/V (Z/rZ)·{±1} 1 [[q]] ×Y (1)Z[ 1 ] Y (r)Z[ r1 ] → Spec Z r r is, by Corollary 8.40(3), a morphism of GL2 (Z/rZ)/{±1}-torsors over Spec Z[ 1r ]((q)), and thus it is an isomorphism. The integral closure of Z[ 1r ][[q]] in Z[ζr , 1r ]((qr )) is Z[ζr , 1r ][[qr ]]. Thus, the morphism er : Spec Z[ζr , 1r ]((qr )) → Y (r)Z[ r1 ] uniquely ex tends to e¯r : Spec Z[ζr , 1r ][[qr ]] → X(r)Z[ r1 ] . The ﬁbered product Spec Z[ζr , 1r ]((qr ))

×

er Y (r)Z[ 1 ]

Y0,∗ (N, r)Z[ r1 ] is

r

isomorphic to the spectrum of the ring 1 Z ζr , ((qr )) ⊗Z((q)) Z[ζd ]((q))[T ]/(T d1 − ζd q d1 ) (8.58) r dd =N

by Proposition 8.26(1). The integral closure of Z[ζr , 1r ][[qr ]] in this ring is calculated as follows.

8.9. COMPACTIFICATIONS

53

Lemma 8.71. Let m ≥ 1 and r ≥ 1 be integers, and let a, b ≥ 1 be relatively prime integers. (1) If we deﬁne a ring homomorphism Z[q, q −1 ] → Z[ζr , 1r ]((qr )) by q → qrr , the tensor product (8.59)

1 Z ζr , ((qr )) ⊗Z[q,q−1 ] Z[ζm ][q, q −1 ][T ]/(T a − ζm q b ) r

is isomorphic to g∈Gal(Q(ζm )/Q) e|d2 h∈Gal(Q(ζmd )/Q(ζm ))

1 Z ζmdr/m d , ((qr ))[T ]/(T a − ζmd qrbr ). r

Here, m = (m, r), d = (a, r), d = (d, r/m ), and we let a = a d, r = r d. Moreover, d = d1 d2 , where the prime factors of d1 are prime factors of m and (m, d2 ) = 1. For a divisor e of d2 , 1/d s is the greatest common divisor of r/m and d1 e. ζmd12 is an 1/d

md1 th root of unity satisfying (ζmd12 )d2 = ζmd1 . (2) Let n ≥ 1, and let r | mn. The integral closure of Z[ζr , 1r ][[qr ]] in Z[ζmn , 1r ]((qr ))[T ]/(T a −ζm qrb ) is isomorphic to Z[ζmn , 1r ][[S]]. The homomorphism Z[ζr , 1r ][[qr ]] → Z[ζmn , 1r ][[S]] is given by −c a qr → ζm S for some positive integer c relatively prime to a. Proof. (1) Since Z[ζr , 1r ] ⊗Z Z[ζm ] is the integral closure of in Q(ζr ) ⊗Q Q(ζm ) = g∈Gal(Q(ζm )/Q) Q(ζmr/m ), it equals 1 g∈Gal(Q(ζm )/Q) Z[ζmr/m , r ]. Thus, the ring (8.59) equals

Z[ 1r ]

g∈Gal(Q(ζm )/Q)

1 Z ζmr/m , ((qr ))[T ]/(T a − ζm qrbr ). r

Moreover, we have 1 Z ζmr/m , ((qr ))[T ]/(T a − ζm qrbr ) r

1 = Z ζmr/m , ((qr ))[U ]/(U d − ζm ) [T ]/(T a − U qrbr ). r

54

8. MODULAR CURVES OVER Z

Factorize d = d1 d2 , where prime factors of d1 are prime factors of m and (m, d2 ) = 1. Then we have Q(ζmr/m )[U ]/(U d − ζm ) = Q(ζmr/m ) ⊗Q(ζm ) Q(ζm )[U ]/(U d − ζm ) = Q(ζmr/m ) ⊗Q(ζm ) Q(ζmd1 )[U ]/(U d2 − ζmd1 ) Q(ζmd1 e ) = Q(ζmr/m ) ⊗Q(ζm ) =

e|d2

Q(ζmd1 er/m s ),

e|d2 h∈Gal(Q(ζms )/Q(ζm ))

where s is the greatest common divisor of r/m and d1 e. Since the ring Z[ζmr/m , 1r ][U ]/(U d − ζm ) is the integral closure of Z[ 1r ] in Q(ζmr/m )[U ]/(U d − ζm ), it equals 1 Z ζmrd1 e/m s , . r e|d2 h∈Gal(Q(ζms )/Q(ζm ))

Let

1/d ζmd12

1/d

be an md1 th root of unity satisfying (ζmd12 )d2 = ζmd1 . Then, 1/d

the (e, h)-component of the image of U is ζmd12 ζe . Thus, we have 1 Z[ζmr/m , ]((qr ))[T ]/(T a − ζm qrbr ) r 1 1/d Z ζmrd1 e/m s , ((qr ))[T ]/(T a −ζmd12 ζe qrbr ). = r e|d2 h∈Gal(Q(ζmd )/Q(ζm ))

(2) Take positive integers c, d satisfying bc − ad = 1, and deﬁne a morphism of Z[ζmn , 1r ]-algebras Z[ζmn , 1r ]((qr ))[T ]/(T a − ζm qrb ) → −c a −d b Z[ζmn , 1r ]((S)) by qr → ζm S , T → ζm S . Then, since the inverse is c −d deﬁned by S → T qr , this is an isomorphism. Since Z[ζmn , 1r ][[S]] is ﬁnitely generated as a Z[ζr , 1r ][[qr ]]-module, the integral closure of Z[ζr , 1r ][[qr ]] is Z[ζmn , 1r ][[S]]. The compactiﬁcations of Y0,∗ (N, r)Z[ r1 ] and Y1,∗ (N, r)Z[ r1 ] are deﬁned similarly to Deﬁnition 8.62. Definition 8.72. Let N ≥ 1 be an integer, and let r ≥ 3 be an integer relatively prime to N . (1) The scheme X0,∗ (N, r)Z[ r1 ] over Z[ 1r ] is deﬁned as the integral closure of X(1)Z[ r1 ] with respect to Y0,∗ (N, r)Z[ r1 ] → Y (1)Z[ r1 ] . (2) The scheme X1,∗ (N, r)Z[ r1 ] over Z[ 1r ] is deﬁned as the integral closure of X(1)Z[ r1 ] with respect to Y1,∗ (N, r)Z[ r1 ] → Y (1)Z[ r1 ] .

8.9. COMPACTIFICATIONS

55

If N = 1, X0,∗ (N, r)Z[ r1 ] = X1,∗ (N, r)Z[ r1 ] is denoted by X(r)Z[ r1 ] . Proposition 8.73. Let N ≥ 1 be an integer, and let r ≥ 3 be an integer relatively prime to N . (1) The scheme X0,∗ (N, r)Z[ r1 ] is a regular projective curve over Z[ 1r ]. X0,∗ (N, r)Z[ N1r ] is smooth over Z[ N1r ]. The ﬁeld of constants of the curve X0,∗ (N, r)Q = X0,∗ (N, r)Z[ r1 ] ⊗Z[ r1 ] Q over Q equals Q(ζr ). (2) Let p r be a prime number, and let N = M pe with (p, M ) = 1. For 0 ≤ a ≤ e, the closed immersion ja : Y0,∗ (M, r)Fp → Y0,∗ (N, r)Z[ r1 ] extends to a closed immersion ja : X0,∗ (M, r)Fp → X0,∗ (N, r)Z[ r1 ] . If C a is the image of ja , we have C a ∩ C a = Ca ∩ Ca ⊂ Y0,∗ (N, r)Z[ r1 ] for a = a. Moreover, if e = 1, X0,∗ (N, r)Z[ r1 ] is semistable at p and the closed ﬁber X0,∗ (N, r)Fp is the union of C 0 and C 1 . Proof. (1) By Corollary 8.56, Y0,∗ (N, r)Z[ r1 ] is regular, and Y0,∗ (N, r)Z[ N1r ] is smooth over Z[ N1r ]. Let X(r)Z[ r1 ] |∧ ∞ be the comple1 tion of X(r)Z[ r1 ] along the inverse image of ∞ = PZ A1Z . Then, by Lemma 8.68 and Corollary 8.70, we obtain an isomorphism 1 Spec Z ζr , [[qr ]] → X(r)Z[ r1 ] |∧ (8.60) ∞. r σ∈GL2 (Z/rZ)/V (Z/rZ)·{±1}

Let ir : Spec Z[ζr , 1r ] → X(r)Z[ r1 ] be the composition of the closed immersion deﬁned by qr → 0 and the extension e¯r : Spec Z[ζr , 1r ][[qr ]] → X(r)Z[ r1 ] . Then ir is a closed immersion. Let Dr ⊂ X(r)Z[ r1 ] be the image of ir , and let DN,r = Dr ×X(r)Z[ 1 ] X0,∗ (N, r)Z[ r1 ] . By r

Lemma 8.71, the scheme X0,∗ (N, r)Z[ r1 ] is regular on a neighborhood of DN,r , and smooth over Z[ N1r ]. Moreover, for σ ∈ GL2 (Z/rZ), X0,∗ (N, r)Z[ r1 ] is regular on a neighborhood of σ ∗ (DN,r ) and is smooth over Z[ N1r ]. By (8.60), we have X0,∗ (N, r)Z[ r1 ] Y0,∗ (N, r)Z[ r1 ] = σ ∗ (DN,r ). σ∈GL2 (Z/rZ)/V (Z/rZ)·{±1}

Thus, X0,∗ (N, r)Z[ r1 ] is regular everywhere and smooth over Z[ N1r ]. Since the ﬁeld of constants of Y0,∗ (N, r)Q is Q(ζr ), the ﬁeld of constants of X0,∗ (N, r)Q is also Q(ζr ).

56

8. MODULAR CURVES OVER Z

(2) By (1), the projective curve X0,∗ (M, r)Fp over Fp is a smooth compactiﬁcation of Y0,∗ (M, r)Fp . Deﬁne a reduced closed subscheme C a of X0,∗ (N, r)Z[ r1 ] as the closure of the image Ca of the closed immersion ja : Y0,∗ (M, r)Fp → Y0,∗ (N, r)Z[ r1 ] in X0,∗ (N, r)Z[ r1 ] . By the proof of (1), C a is smooth on a neighborhood of the intersection with g ∗ (DN,r ) over Fp for each g ∈ GL2 (Z/rZ). Thus, C a is also a smooth compactiﬁcation of Y0,∗ (M, r)Fp and is isomorphic to X0,∗ (M, r)Fp . Furthermore, by the proof of (1), the reduced part of X0,∗ (N, r)Fp is smooth on a neighborhood of the intersection with the inverse image of g ∗ (Dr ) for each g ∈ GL2 (Z/rZ). Thus, if a = a , C a and C a do not intersect each other on a neighborhood of the inverse image of each g ∗ (Dr ). Hence, the intersection C a ∩ C a is contained in Y0,∗ (N, r)Z[ r1 ] . The last assertion in the case of e = 1 follows easily from the above and Corollary 8.58. Proposition 8.74. Let N ≥ 1 be an integer, and let r ≥ 3 be an integer relatively prime to N . (1) The scheme X1,∗ (N, r)Z[ r1 ] is a regular projective curve over Z[ 1r ]. X1,∗ (N, r)Z[ N1r ] is smooth over Z[ N1r ]. The ﬁeld of constants of the curve X1,∗ (N, r)Q = X0,∗ (N, r)Z[ r1 ] ⊗Z[ r1 ] Q over Q equals Q(ζr ). (2) Let p r be a prime number, and let N = M pe with (p, M ) = 1. For 0 ≤ a ≤ e, let Ig(M pa , r)Fp be the smooth compactiﬁcation of the smooth aﬃne curve Ig(M pa , r)Fp over Fp . Then, the closed immersion ja : Ig(M pa , r)Fp → Y1,∗ (N, r)Z[ r1 ] extends to a closed immersion ja : Ig(M pa , r)Fp → X1,∗ (N, r)Z[ r1 ] . If a = a , the intersection of the image C a of ja and the image C a of ja is contained in Y1,∗ (N, r)Z[ r1 ] . The proof of Proposition 8.74 is similar to that of Proposition 8.73, and we omit it. Proof of Theorem 8.63. (1) It is clear from the deﬁnition that X0 (N )Z is a normal projective curve over Z. The geometric ﬁber X0 (N )Q at the generic point is connected by Theorem 2.10(3). Thus, we have Γ(X0 (N )Q , O) = Q and Γ(X0 (N )Z , O) = Z. Hence, by Theorem A.16, each geometric ﬁber of X0 (N )Z is connected. (2) X0 (N )Z is obtained by patching together the quotients of X0,∗(N, r)Z[ r1 ] by GL2 (Z/rZ). Thus by Propositions 8.73 and B.10(1),

8.9. COMPACTIFICATIONS

57

X0 (N )Z[ N1 ] is smooth over Z[ N1 ]. Furthermore if p N , X0 (N )Fp is a quotient of X0,∗ (N, r)Fp and it is a smooth compactiﬁcation of Y0 (N )Fp . (3) As in (2), if N = M p with p M , then X0 (N )Z is weakly semistable at p by Proposition 8.73 and Corollary B.11(2). Since the closures C 0 , C 1 of the images C0 , C1 of the closed immersions j0 , j1 : Y0 (M )Fp → Y0 (N )Z are regular at the cusps, both are isomorphic to X0 (M )Fp . Thus, the closed immersions j0 and j1 extend to closed immersions j0 , j1 : X0 (M )Fp → X0 (N )Fp . The facts X0 (N )Fp = C 0 ∪ C 1 and C 0 ∩ C 1 = Y0 (M )ss Fp follow easily from Proposition 8.73. Since the proof of Theorem 8.66 is similar to above, we omit it. Similarly to Theorem 8.63, we have Theorem 8.76 below. Definition 8.75. Let M ≥ 1 and N ≥ 1 be integers relatively prime to each other. Deﬁne a functor M1,0 (M, N ) over Z by associating to a scheme T the set ⎧ ⎫ isomorphism classes of triples (E, P, C),⎪ ⎪ ⎬ ⎨ where E is an elliptic curve over T , P is . M1,0 (M, N )(T ) = ⎪ ⎭ ⎩a section of E of exact order M , and C is a⎪ cyclic subgroup scheme of order N For a prime number p | M , we deﬁne the restrictions of functors j0 , j1 : M1 (M )Fp → M1,0 (M, p)Fp by [(E, P )] → [(E, P, Ker F )] and [(E, P )] → [(E (p) , P (p) , Ker V )]. Theorem 8.76. Let M ≥ 1 and N ≥ 1 be integers relatively prime to each other. (1) There exists a coarse moduli scheme Y1,0 (M, N )Z of the functor M1,0 (M, N ) over Z. Y1,0 (M, N )Z is a normal connected aﬃne curve over Z, and the morphism deﬁned by the j-invariant, j : Y1,0 (M, N )Z → A1Z , is ﬁnite ﬂat. (2) Let N = p be a prime number. The integral closure X1,0 (M, p)Z of P1Z with respect to the ﬁnite morphism j : Y1,0 (M, p)Z → A1Z is weakly semistable at p. The restrictions of functors j0 , j1 : M1,0 (M )Fp → M1,0 (M, p)Fp induce closed immersions j0 , j1 : Y1 (M )Fp → Y1,0 (M, p)Z , and they extend to closed immersions j0 , j1 : X1 (M )Fp → X1,0 (M, p)Z . X1,0 (M, p)Fp is the union of the image C 0 of j0 and the image C 1 of j1 . We omit the proof of this theorem, too.

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The integral closure X1,0 (M, N )Z of P1Z with respect to the ﬁnite morphism j : Y1,0 (M, N )Z → A1Z may also be denoted by X0,1 (N, M )Z in this work. X1,0 (M, N )Z is the quotient of X1 (M N )Z by (Z/N Z)× ⊂ (Z/M N Z)× . The quotient of X1,0 (M, N )Z by (Z/M Z)× identiﬁed with (Z/M N Z)× /(Z/N Z)× is X0 (M N )Z . For an integer r ≥ 3 relatively prime to M N , X1,0,∗ (M, N, r)Z[ r1 ] is also deﬁned similarly. Unlike Theorem 8.63(3), X1 (M p)Q , where p M , may not have semistable reduction at p. However, the extension of the base change X1 (M p)Q(ζp ) = X1 (M p)Q ⊗Q Q(ζp ) has semistable reduction at a prime ideal lying above p. Theorem 8.77. Let p be a prime number, let M ≥ 1 be an integer relatively prime to p, and let r ≥ 3 be an integer relatively prime be the normalization of the scheme to M p. Let X1,∗ (M p, r)bal Z[ 1 ,ζp ] r

X1,∗ (M p, r)Z[ r1 ] ⊗Z[ r1 ] Z[ 1r , ζp ]. Then, the curve X1,∗ (M p, r)bal Z[ 1 ,ζp ] r

over Z[ 1r , ζp ] is semistable at the prime ideal p = (ζp − 1). There exists a closed immersion (8.61)

j0 , j1 : Ig(M p, r)Fp −→ X1,∗ (M p, r)bal Z[ 1 ,ζp ] r

satisfying the following condition. Let C0 , C1 be the images of j0 , j1 . ⊗Z[ζp ] Fp = C0 ∪ C1 and C0 ∩ C1 = C0ss . We have X1,∗ (M p, r)bal Z[ r1 ,ζp ] The diagrams (8.62) Ig(M p, r)Fp −−→ X1,∗ (M, r)Fp X1,∗ (M, r)Fp ⏐ ⏐ ⏐ ⏐ ⏐ ⏐j j0 j0 0 X1,∗ (M p, r)bal −−→ X1,∗ (M p, r)Z[ r1 ] −−→ X1,0,∗ (M, p, r)Z[ r1 ] Z[ 1 ,ζp ] r

and Ig(M p, r)Fp ⏐ ⏐ j1

Ig(M p, r)Fp ⏐ ⏐ j1

−−→

X1,∗ (M, r)Z[ r1 ] ⏐ ⏐j 1

X1,∗ (M p, r)bal −−→ X1,∗ (M p, r)Z[ r1 ] −−→ X1,0,∗ (M, p, r)Z[ r1 ] Z[ 1 ,ζp ] r

are commutative. We will not prove this theorem. Let p be a prime number, and let M ≥ 1 be an integer relatively prime to p. Let r ≥ 3 be an integer relatively prime to M p, and let

8.9. COMPACTIFICATIONS

59

a ≥ 0 be an integer. The quotient of the Igusa curve Ig(M pa , r)Fp by GL2 (Z/rZ) is denoted by Ig(M pa )Fp . If a = 0, then we have Ig(M pa )Fp = X1 (M )Fp . Corollary 8.78. Let p be a prime number, and let M ≥ 1 be an integer relatively prime to p. Let X1 (M p)bal Z[ζp ] be the integral closure of X1 (M p)Z in X1 (M p)Q(ζp ) = X1 (M p)Q ⊗Q Q(ζp ). The curve X1 (M p)bal Z[ζp ] over Z[ζp ] is weakly semistable at the prime ideal p = (ζp −1). The closed immersion (8.61) induces a closed immersion (8.63)

j0 , j1 : Ig(M p)Fp −→ X1 (M p)bal Z[ζp ] .

Let C0 , C1 be the images of j0 , j1 . Then we have X1 (M p)bal Z[ζp ] ⊗Z[ζp ] Fp = C0 ∪ C1 and C0 ∩ C1 = C0ss . For i = 0, 1, the diagram

(8.64)

Ig(M p)Fp ⏐ ⏐ ji

−−−−→

X1 (M )Fp ⏐ ⏐j i

X1 (M p)bal Z[ζp ] −−−−→ X1,0 (M, p)Z is commutative. Proof. X1 (M p)bal is a quotient of X1,∗ (M p, r)bal by the Z[ r1 ,ζp ] Z[ r1 ,ζp ] action of GL2 (Z/rZ). By Lemma 8.41(2), the inertia group at the generic point of the ﬁber X1,∗ (M p, r)Q(ζp ) over Q(ζp ) is 1 if M p > 2, and {±1} if M p ≤ 2. Thus, the assertion follows from Theorem 8.77 and Corollary B.11(2). The morphisms sd : Y1 (N )Z → Y1 (M )Z and sd : Y0 (N )Z → Y0 (M )Z , which we deﬁned in Lemma 8.60, uniquely extend to the compactiﬁcation. Lemma 8.79. Let N ≥ 1 be an integer, and let dM | N . The morphisms of modular curves sd : Y1 (N )Z → Y1 (M )Z and sd : Y0 (N )Z → Y0 (M )Z extend uniquely to ﬁnite morphisms (8.65)

sd : X1 (N )Z −→ X1 (M )Z , sd : X0 (N )Z −→ X0 (M )Z .

Outline of proof. Let r ≥ 3 be an integer relatively prime to N . Since sd : Y1,∗ (N, r)Z[ r1 ] → Y1,∗ (M, r)Z[ r1 ] is a morphism of two-dimensional regular schemes, it uniquely extends to a morphism X → X1,∗ (N, r)Z[ r1 ] from the scheme X → X1,∗ (M, r)Z[ r1 ] obtained by blowing up ﬁnitely many times at ﬁnitely many closed points of

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8. MODULAR CURVES OVER Z

X1,∗ (N, r)Z[ r1 ] Y1,∗ (N, r)Z[ r1 ] . We then show that a morphism sd : X1,∗ (N, r)Z[ r1 ] → X1,∗ (M, r)Z[ r1 ] is obtained without taking blowups. Dividing these by the action of GL2 (Z/rZ) and patching them up, we obtain the morphism sd : X1 (N )Z → X1 (M )Z . A similar proof works for sd : X0 (N )Z → X0 (M )Z . Question. Complete the proof of Lemma 8.79.

10.1090/mmono/245/02

CHAPTER 9

Modular forms and Galois representations As we announced in Chapter 2, in this chapter we construct Galois representations associated with modular forms. We show that these satisfy the required conditions using Theorems 8.63 and 8.66, which are fundamental properties of modular curves over Z, shown in Chapter 8. In addition, we will prove Theorem 3.52 and a part of Theorem 3.55, which concern ramiﬁcations and levels of Galois representations associated with modular forms. In §9.1, we deﬁne some fundamental objects such as Hecke algebras with Z coeﬃcients, and then we study Galois representations associated with modular forms using properties of modular curves shown in Chapter 8. In §9.2, we show Theorem 9.16 about the construction of Galois representations associated with modular forms. The key fact here is the congruence relation (Lemma 9.18), which is a consequence of Theorem 8.63(3) concerning the semistable reduction of modular curves. In §9.3, we show the relation between the Hecke algebras with Z coeﬃcients and modular mod representations. In §9.4, we prove Theorem 3.52, which is about the ramiﬁcation of -adic representations associated with modular forms and the level of modular forms. In §9.5, we study the action of the Hecke algebras on the image of the space of modular forms of lower level. The proof of the statements in this section requires only the modular curves over C, and we do not need modular curves over Z. In §9.6, we study the reduction mod p of the Jacobian of X0 (M p), p M . The results here will play a crucial role in the proof of a part of Theorem 3.55 in §9.7. 9.1. Hecke algebras with Z coeﬃcients Let N ≥ 1 be an integer. Let J0 (N )Q be the Jacobian of the curve X0 (N )Q . J0 (N )Q is an abelian variety over Q. In Chapter 2, we deﬁned the space S(N ) of modular forms with Q coeﬃcients as Γ(X0 (N ), Ω1 ). From here on, we write it as S0 (N ) instead of S(N ) in 61

62

9. MODULAR FORMS AND GALOIS REPRESENTATIONS

order to distinguish with the spaces of modular forms S1 (N ) ⊂ S(N ), which we will deﬁne later. By the natural isomorphism (D.16), (9.1) Γ(J0 (N )Q , Ω1J0 (N )Q /Q ) −→ Γ(X0 (N )Q , Ω1X0 (N )Q /Q ) = S0 (N ), we identify as S0 (N ) = Γ(J0 (N )Q , Ω1 ). In Deﬁnition 2.31 we deﬁned the Hecke operator Tn : S0 (N ) → S0 (N ) for each integer n ≥ 1. There, we used the ﬁnite ﬂat morphisms s, t : X0 (N, n) → X0 (N )Q of curves over Q and deﬁned it as Tn = s∗ ◦ t∗ . From now on, we change the notation to denote s, t : X0 (N, n) → X0 (N )Q by sn , tn : I0 (N, n) → X0 (N )Q instead. The curve I0 (N, n) over Q is the compactiﬁcation of the coarse moduli scheme of the functor that associates to a scheme T over Q the set ⎧ ⎫ isomorphism classes of triples (E, C, Cn), where⎪ ⎪ ⎨ ⎬ E is an elliptic curve over T , C a cyclic sub. I0 (N, n)(T ) = ⎪ ⎩group scheme C of order N , and Cn a subgroup⎪ ⎭ scheme of order n such that C ∩ Cn = 0 The morphisms sn , tn : I0 (N, n) → X0 (N )Q are deﬁned by sending (E, C, Cn ) to (E, C) and (E/Cn , (C + Cn )/Cn ), respectively. Deﬁne the Hecke operator Tn : J0 (N )Q → J0 (N )Q as the endomorphism Tn = tn∗ ◦ s∗n of J(N )Q . If Ci (i ∈ I) are connected components of I0 (N, n), then Tn is the composition (sn |Ci )∗

J0 (N )Q = Jac X0 (N )Q −−−−−−−→ i

Jac Ci

i∈I

(tn |Ci )∗

−−−−−−→ Jac X0 (N )Q = J0 (N )Q . i

By the identiﬁcation of (9.1), S0 (N ) = Γ(J0 (N )Q , Ω1 ), the Hecke operator Tn : S0 (N ) → S0 (N ) coincides with the pullback by Tn : J0 (N )Q → J0 (N )Q . Definition 9.1. Let N ≥ 1 be an integer. Deﬁne the Hecke algebra T0 (N )Z as the subring of End(J0 (N )Q ) generated by the Hecke operators Tn , n = 1, 2, 3, . . . , i.e., (9.2)

T0 (N )Z = Z[Tn , n = 1, 2, 3, . . .] ⊂ End J0 (N )Q .

The Hecke algebra T0 (N )Z ⊂ End J0 (N )Q deﬁned above can be identiﬁed with a subalgebra of the Hecke algebra T (N ) ⊂ End S0 (N ) as follows.

9.1. HECKE ALGEBRAS WITH Z COEFFICIENTS

63

Lemma 9.2. T0 (N )Z is a commutative algebra and is ﬁnitely generated as a Z-module. To T ∈ T0 (N )Z ⊂ End J0 (N )Q , we associate T ∗ ∈ End Γ(J0 (N )Q , Ω1 ) = End S0 (N ). Then, we obtain an isomorphism of commutative algebras (9.3)

T0 (N )Z ⊗Z Q = T0 (N )Q −→ T (N ).

The isomorphism (9.3) sends the Hecke operator Tn : J0 (N ) → J0 (N ) to the Hecke operator Tn : S0 (N ) → S0 (N ). Proof. Since End J0 (N )Q is a ﬁnitely generated free Z-module by Proposition A.51(1), so is the submodule T0 (N )Z . Since the Hecke operator Tn : S0 (N ) → S0 (N ) is the pullback Tn∗ of the Hecke operator Tn : J0 (N )Q → J0 (N )Q , we obtain a surjective homomorphism T0 (N )Z ⊗Z Q → T (N ). By Proposition A.51(3), this is injective. By Proposition 2.32, T (N ) is commutative, so is T0 (N )Z . From now on, we identify T (N ) with T0 (N )Q through (9.3). Corollary 9.3. Let K be a ﬁeld of characteristic 0, and let f ∈ S0 (N )K be a primary form with K coeﬃcients. The subﬁeld Kf = Q(an (f ), n ≥ 1) of K is a ﬁnite extension of Q, and an (f ) is an algebraic integer for each integer n ≥ 0. Proof. Let ϕf : T (N ) = T0 (N )Q → K be the ring homomorphism deﬁned by the primary form f . Since Kf is the image of ϕf : T0 (N )Q → K, it is a ﬁnite extension of Q. Since an (f ) = ϕf (Tn ) is in the image of T0 (N )Z , it is an algebraic integer. ) = 0, then T0 (N )Z = 0. If g0 (N ) = 1, Example 9.4. If g0 (N ∞ then T0 (N )Z = Z. If f = n=1 an (f )q n is the unique primary form of level N , then Tn = an (f ). The Atkin–Lehner involution w = wN : X0 (N )Q → X0 (N )Q induces the involution of J0 (N )Q , w = w∗ : J0 (N )Q → J0 (N )Q . Lemma 9.5. Let N ≥ 1 and n ≥ 1 be integers. (1) The endomorphism Tn∗ = sn∗ ◦ t∗n of J0 (N )Q satisﬁes w ◦ Tn = Tn∗ ◦ w. (2) If n and N are relatively prime, then we have Tn∗ = Tn . Proof. (1) Let T be a scheme over Q. For a triple (E, C, Cn ) ∈ I0 (N, n)(T ), let E = E/(C + Cn ). If we let C be the kernel of the dual of E/Cn → E and Cn be the kernel of the dual of

64

9. MODULAR FORMS AND GALOIS REPRESENTATIONS

E/C → E , then we have (E , C , Cn ) ∈ I0 (N, n)(T ). Sending ˜ : I0 (N, n) → (E, C, Cn ) to (E , C , Cn ), we obtain a morphism w ˜ 2 = id, and the diagram I0 (N, n). We have w s

t

t

s

X0 (N )Q ←−−−− I0 (N, n) −−−−→ X0 (N )Q ⏐ ⏐ ⏐ ⏐w ⏐ ⏐ w w ˜ X0 (N )Q ←−−−− I0 (N, n) −−−−→ X0 (N )Q is commutative. The assertion follows easily from this. (2) Deﬁne a morphism v : I0 (N, n) → I0 (N, n) by sending (E, C, Cn ) to (E/Cn , C + Cn /Cn , E[n]/Cn ). Then, we have v 2 = id, and the diagram s

t

t

s

X0 (N )Q ←−−−− I0 (N, n) −−−−→ X0 (N )Q & & ⏐ & & ⏐ v & & X0 (N )Q ←−−−− I0 (N, n) −−−−→ X0 (N )Q is commutative. The assertion follows easily from this.

an

Let X0 (N ) be the compact Riemann surface associated with the curve X0 (N )Q , and let H1 (X0 (N )an , Z) be its singular homology group. The complex torus J0 (N )an deﬁned by the abelian variety J0 (N )Q can be identiﬁed with Hom(S0 (N ), C)/H1 (X0 (N )an , Z) through the isomorphism (D.9). Deﬁne the Hecke operator Tn on H1 (X0 (N )an , Z) by Tn = t∗ ◦ s∗ . Through the injective morphism of algebras End J0 (N )Q → End H1 (J0 (N )an , Z) = End H1 (X0 (N )an , Z), H1 (X0 (N )an , Z) is a T0 (N )Z -module. Proposition 9.6. H1 (X0 (N )an , Q) is a free T0 (N )Q -module of rank 2. Proof. By Proposition 2.55, Hom(S0 (N )Q , Q) is a free T0 (N )Q module of rank 1. Thus, Hom(S0 (N )C , C) is a free T0 (N )C -module of rank 1 and a free T0 (N )R -module of rank 2. The natural isomorphism (D.8) gives an isomorphism of T0 (N )R -modules H1 (X0 (N )an , Q) ⊗Q R → Hom(S0 (N )C , C). Thus, H1 (X0 (N )an , Q) ⊗Q R is a free T0 (N )R -module of rank 2. Since T0 (N )R is faithfully ﬂat over T0 (N )Q , H 1 (X0 (N )an , Q) is a free T0 (N )Q -module of rank 2.

9.1. HECKE ALGEBRAS WITH Z COEFFICIENTS

65

H1 (X0 (N )an , Z) is the dual of the singular cohomology group H (X0 (N )an , Z), and thus, by the Poincar´e duality, it is naturally isomorphic to H 1 (X0 (N )an , Z(1)). The cup product of cohomology H 1 (X0 (N )an , Z(1)) × H 1 (X0 (N )an , Z(1)) → Z(1) is nondegenerate, and it induces a nondegenerate alternating form 1

( , ) : H1 (X0 (N )an , Z) × H1 (X0 (N )an , Z) → Z(1). For a ∈ H1 (X0 (N )an , Z), deﬁne fa : H1 (X0 (N )an , Z) → Z(1) by fa (b) = (a, wb). By the Poincar´e duality, the mapping (9.4)

H1 (X0 (N )an , Z) −→ Hom(H1 (X0 (N )an , Z), Z(1))

that sends a to fa is an isomorphism of Z-modules. We consider Hom(H1 (X0 (N )an , Z), Z(1)) as a T0 (N )Z -module through T f (b) = f (T b). Lemma 9.7. The mapping (9.4) is an isomorphism of T0 (N )Z modules. Proof. It suﬃces to show that (9.4) is a morphism of T0 (N )Z modules. For a, b ∈ H1 (X0 (N )an , Z) and n ≥ 1, we have (Tn a, wb) = (a, Tn∗ wb). Thus, by Lemma 9.5(1), we have (Tn a, wb) = (a, wTn b), and thus (9.4) is a morphism of T0 (N )Z -modules. Corollary 9.8. (1) The T0 (N )Q -module Hom(T0 (N )Q , Q) is isomorphic to T0 (N )Q . (2) The T0 (N )Q -module S0 (N )Q is isomorphic T0 (N )Q . Proof. (1) Since we have HomQ (H1 (X0 (N )an , Q), Q(1)) = HomT0 (N )Q (H1 (X0 (N )an , Q), HomQ (T0 (N )Q , Q(1))), by Lemma 9.7 and Proposition 9.6, Hom(T0 (N )Q , Q(1))2 is a free T0 (N )Q -module of rank 2. The assertion follows immediately from this. (2) The T0 (N )Q -module HomQ (S0 (N )Q , Q) is isomorphic to T0 (N )Q . Thus, by (1), S0 (N )Q = HomQ (HomQ (S0 (N )Q , Q), Q) is isomorphic to T0 (N )Q . Definition 9.9. Let N ≥ 1 be an integer. Deﬁne a positive deﬁnite Hermitian form on the space of modular forms S0 (N )C by √ ' −1 f ∧ g¯ (9.5) (f, g) = 8π 2 X0 (N )(C) for f, g ∈ S0 (N )C = Γ(X0 (N )C , Ω1 ). (f, g) is called the Petersson product.

66

9. MODULAR FORMS AND GALOIS REPRESENTATIONS

Lemma 9.10. Let N ≥ 1 be an integer, and let n ≥ 1 be an integer relatively prime to N . Then, the Hecke operator Tn : S0 (N )C → S0 (N )C is self-adjoint with respect to the Petersson product. Proof. The adjoint operator of Tn is Tn∗ . Thus, the assertion follows immediately from Lemma 9.5(2). Corollary 9.11. Let N ≥ 1 be an integer. (1) The reduced Hecke algebra with R coeﬃcients T0 (N )R ⊂ T0 (N )R generated by Tn with (n, N ) = 1 is isomorphic to the product ring Rdim T0 (N )R . (2) The reduced Hecke algebra with Q coeﬃcients T0 (N )Q ⊂ T0 (N )Q is generated by Tn with (n, N ) = 1 is reduced. Let Φ0 (N ) = Spec T0 (N )Q , and Kf the residue ﬁeld for f ∈ Φ0 (N ). Then, we have T0 (N )Q = f ∈Φ0 (N ) Kf , and each Kf is a totally real ﬁeld. (3) Let f ∈ S0 (N )C be a primitive form with complex coeﬃcients. Then, an is a real number for all integers n ≥ 1. Proof. (1) Commutative Hermitian matrices can be diagonalized simultaneously. (2) Follows immediately from (1). (3) If f is a primitive form, then by Corollary 2.61, we have Q(an (f ); n ≥ 1) = Q(an (f ); (n, N ) = 1). Thus, the assertion follows immediately from (1). So far we have treated modular forms deﬁned as diﬀerential forms on X0 (N )Q , but modular forms deﬁned as diﬀerential forms on X1 (N ) also have similar properties. Let N ≥ 1 be an integer. Let S1 (N ) = Γ(X1 (N )Q , Ω1 ). Let J1 (N )Q be the Jacobian of the curve X1 (N )Q . It is an abelian variety over Q. Through the natural morphism (D.16) (9.6)

Γ(J1 (N )Q , Ω1 ) −→ Γ(X1 (N )Q , Ω1 ) = S1 (N ),

we identify as S1 (N ) = Γ(J1 (N )Q , Ω1 ). All the properties of S1 (N ) and J1 (N ) we describe below are proved similarly to the properties of S0 (N ) and J0 (N )Q . So we omit the proofs. For each integer n ≥ 1, the Hecke operator Tn : J1 (N )Q → J1 (N )Q is deﬁned as follows. Let I1 (N, n) be the compactiﬁcation of the coarse moduli scheme of the functor that associates to a scheme

9.1. HECKE ALGEBRAS WITH Z COEFFICIENTS

67

T over Q the set ⎧ ⎫ ⎨isomorphism classes of triples (E, P, C), where E is an⎬ elliptic curve over T , P is a point of E of exact order N , . ⎩ ⎭ and C is a subgroup of degree n such that P ∩ C = 0. I1 (N, n) is a proper smooth curve over Q. To (E, P, C), associating (E, P ) and (E/C, image of P ), respectively, we obtain ﬁnite ﬂat morphisms sn , tn : I1 (N, n) → X1 (N )Q of curves over Q. Deﬁne an endomorphism Tn by Tn = tn∗ ◦ s∗n . We call it the Hecke operator. We deﬁne the Hecke operator Tn : S1 (N ) → S1 (N ) as the pullback of Tn : J1 (N )Q → J1 (N )Q . For a ∈ (Z/N Z)× , the diamond operator a : X1 (N )Q → X1 (N )Q induces an automorphism a = a∗ on J1 (N )Q and an automorphism a = a∗ on S1 (N ). The Hecke algebra T1 (N )Z is deﬁned as the subalgebra (9.7)

T1 (N )Z = Z[Tn ; n ≥ 1, a; a ∈ (Z/N Z)× ]

of End J1 (N )Q . The Hecke algebra T1 (N )Z is commutative and we have (9.8)

∞ n=1

Tn n−s =

−1 1 − Tp p−s + pp · p−2s . p:prime

Here we deﬁned p = 0 for p | N . By (9.8), we have T1 (N )Z = Z[Tp ; p : prime, a; a ∈ (Z/N Z)× ]. T1 (N )Z is a ﬁnitely generated free Z-module. Deﬁne a morphism w : X1 (N )Q(ζN ) → X1 (N )Q(ζN ) as follows. For an elliptic curve E, let ( , )N : N : E[N ] × E[N ] → μN be the Weil pairing. Regard T as a scheme over Q(ζN ). Let E = E/P for the pair (E, P ) ∈ M1 (N )(T ), and let P be the image of Q ∈ E[N ] satisfying (P, Q)N = ζN . Then, we have (E , P ) ∈ M1 (N )(T ). Sending (E, P ) to (E , P ), we obtain a morphism w : X1 (N )Q(ζN ) → X1 (N )Q(ζN ) . We have w2 = −1. Let n ≥ 1 be an integer. We deﬁne an involution w ˜ : I1 (N, n)Q(ζN ) → I1 (N, n)Q(ζN ) as follows. Let T be a scheme over Q(ζN ). For a triple (E, P, C) ∈ I1 (N, n)(T ), let E = E/(P + C), let P ∈ E [N ] be the image of a point Q in E/C satisfying (image of P, Q) = ζN , and let C be the kernel of the dual of E/C → E . Then, we have (E , P , C ) ∈ I1 (N, n)(T ). Sending (E, P, C) to (E , P , C ), we ob˜ 2 = −1. The tain a morphism w ˜ : I1 (N, n) → I1 (N, n). We have w

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9. MODULAR FORMS AND GALOIS REPRESENTATIONS

diagram s

t

t

s

X1 (N )Q ←−−−− I1 (N, n) −−−−→ X1 (N )Q ⏐ ⏐ ⏐ ⏐ ⏐ ⏐w w w ˜ X1 (N )Q ←−−−− I1 (N, n) −−−−→ X1 (N )Q is commutative. From this we have w ◦ Tn = Tn∗ ◦ w. Let n ≥ 1 be an integer relatively prime to N . Let v : I1 (N, n) → I1 (N, n) be the morphism deﬁned by sending a triple (E, P, C) to (E/C, image of P, E/C). Then, v is an automorphism, and the diagram s

t

t

s

X1 (N )Q ←−−−− I1 (N, n) −−−−→ X1 (N )Q & ⏐ ⏐ & ⏐ ⏐ v n & X1 (N )Q ←−−−− I1 (N, n) −−−−→ X1 (N )Q is commutative. Thus, the endomorphism Tn∗ = s∗ ◦ t∗ of J1 (N )Q is equal to n ◦ Tn . We have w ◦ a = a ◦ w for a ∈ (Z/N Z)× . For an integer n ≥ 1, we deﬁne the Hecke operator Tn of the singular homology group H1 (X1 (N )an , Z) by Tn = t∗ ◦ s∗ , and for a ∈ (Z/N Z)× we deﬁne the diamond operator a by a = a∗ . H1 (X1 (N )an , Z) is a T1 (N )Z -module, and H1 (X1 (N )an , Q) is a free T1 (N )Q -module of rank 2. Deﬁne an isomorphism of T1 (N )Z -modules (9.9)

H1 (X1 (N )an , Z) −→ Hom(H1 (X1 (N )an , Z), Z(1))

by sending a ∈ H1 (X1 (N )an , Z) to the linear form H1 (X1 (N )an , Z) → Z(1) that send b to (a, wb). The T1 (N )Q -module Hom(T1 (N )Q , Q) is isomorphic to T1 (N )Q . The Petersson product on S1 (N )C is deﬁned by √ ' −1 (9.10) (f, g) = f ∧ g¯ 8π 2 X1 (N )(C) for f, g ∈ S1 (N )C = Γ(X1 (N )C , Ω1 ). For an integer n ≥ 1 relatively prime to N , the adjoint operator of the Hecke operator Tn is n−1 ◦ Tn , and the adjoint of n is n−1 . By the fact that commuting normal matrices can be diagonalized simultaneously, the reduced Hecke algebra with complex coeﬃcients T1 (N )C = C[Tn , n; (n, N ) = 1] ⊂ T1 (N )C is isomorphic to the product ring Cdim T1 (N )C . The reduced

9.1. HECKE ALGEBRAS WITH Z COEFFICIENTS

69

Hecke algebra with rational coeﬃcients T1 (N )Q = Q[Tn , n; (n, N ) = 1] ⊂ T1 (N )Q is isomorphic to the product ring of a ﬁnite number of number ﬁelds. The relation between S1 (N ) and S0 (N ) and that between T1 (N ) and T0 (N ) are as follows. The natural morphism X1 (N )Q → X0 (N )Q that sends (E, P ) to (E, P ) induces a surjective morphism of Jacobians J1 (N )Q → J0 (N )Q . The pullback by this morphism deﬁnes a natural injective morphism S0 (N ) → S1 (N ). Since X0 (N )Q is the quotient of X1 (N )Q by the action of the diamond operators × (Z/N Z)× , we identify S0 (N ) with the invariant part S1 (N )(Z/N Z) through the natural injection S0 (N ) → S1 (N ). For an integer n ≥ 1, the diagram s

t

s

t

n X1 (N )Q ←−− −− I1 (N, n) −−−n−→ X1 (N )Q ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ n X0 (N )Q ←−− −− I0 (N, n) −−−n−→ X0 (N )Q is commutative. Moreover, the two squares induce isomorphisms I1 (N, n) → (the normalization of I0 (N, n) ×X0 (N )Q X1 (N )Q )), the natural surjection J1 (N )Q → J0 (N )Q is compatible with the Hecke operator Tn . Thus, we deﬁne a ring homomorphism T1 (N )Z → T0 (N )Z by sending Tn to Tn for any integer n ≥ 1 and a to 1. The natural surjection J1 (N )Q → J0 (N )Q and the natural injection S0 (N ) → S1 (N ) are compatible with T1 (N )Z → T0 (N )Z . By Proposition 8.69, the pair (Eq , P ) of the Tate curve Eq and its point P = (0, ζN ) of order N deﬁnes a morphism e : Spec Q(ζN )[[q]] → X1 (N )Q . For an extension K of Q(ζN ), e∗ : S1 (N )K → K[[q]]dq is n dq injective. If e∗ f = ∞ n (f )q q for a modular form f ∈ S1 (N )K , n=1 a ∞ we call the power series n=1 an (f )q n the q-expansion of f . Since the diagram Spec Q(ζN )[[q]] −−−−→ X1 (N )Q ⏐ ⏐ ⏐ ⏐

Spec Q[[q]] −−−−→ X0 (N )Q is commutative, so is the diagram S0 (N )K −−−−→ K[[q]] ⏐ & ⏐ & ∩ & S1 (N )K −−−−→ K[[q]].

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Let K be a ﬁeld of characteristic 0. A modular form f ∈ S1 (N )K = S1 (N ) ⊗Q K is called a primary form if f is an eigenvector of all T ∈ T1 (N )Z and a1 (f ) = 1 ∈ K(ζN ). Deﬁne a ring homomorphism ϕf : T1 (N )Z → K by T f = ϕf (T )f , and we obtain a one-to-one correspondence between primary forms with K coeﬃcients and the ring homomorphisms ϕf : T1 (N )Z → K. For a primary form f with K coeﬃcients, each coeﬃcient an (f ) = ϕf (Tn ) of the q-expansion is an algebraic integer. The subﬁeld Q(an (f ), n ≥ 1) of K is a ﬁnite extension of Q. For a primary form f ∈ S1 (N )K with K coeﬃcients, the character ε = εf : (Z/N Z)× → K × deﬁned by af = ε(a)f is called the character of f . εf is the composition of : (Z/N Z)× → T1 (N )× Z × and ϕf : T1 (N )× Z → K . For an integer n ≥ 1, we have ϕf (Tn ) = an (f ), and for a ∈ (Z/N Z)× , we have ϕf (a) = εf (a). For relatively prime integers N, M ≥ 1, the quotient of X1 (N M ) by the subgroup (Z/N Z)× ⊂ (Z/N M Z)× is denoted by X0,1 (N, M ). Let J0,1 (N, M ) be the Jacobian of X0,1 (N, M ), and deﬁne the Hecke algebra T0,1 (N, M )Z as the subring Z[Tn ; n ≥ 1, a; a ∈ (Z/M Z)× ] of End J0,1 (N, M ) generated by the Hecke operators Tn , n ≥ 1, and the diamond operators a, a ∈ (Z/M Z)× . 9.2. Congruence relations In this section we construct Galois representations associated with modular forms. Let N ≥ 1 be an integer, and let be a ¯ be the Tate modprime number. Let T J0 (N ) = lim J0 (N )[n ](Q) ←n

ule of the Jacobian J0 (N ), and let V J0 (N ) = T J0 (N ) ⊗Z Q . These naturally possess the action of the absolute Galois group GQ = ¯ Gal(Q/Q). Deﬁne the action of T ∈ T0 (N )Z ⊂ End J0 (N ) as T∗ . Then, T J0 (N ) and V J0 (N ) become the T0 (N )Z ⊗Z Z = T0 (N )Z module and T0 (N )Z ⊗Z Q = T0 (N )Q -module, respectively. The absolute Galois group GQ acts on T J0 (N ) and V J0 (N ) as an automorphism of T0 (N )Z -module and T0 (N )Q -module, respectively, and these deﬁne -adic representations of GQ . An isomorphism of T0 (N )Z -modules H1 (X0 (N )an , Z) ⊗ Z/n Z → J0 (N )[n ] is induced by the natural isomorphism of complex tori S0 (N )C /H1 (X0 (N )an , Z) → J0 (N )an . It induces an isomorphism of T0 (N )Z -modules H1 (X0 (N )an , Z) ⊗ Z → T J0 (N ) and an isomorphism of T0 (N )Q -modules H1 (X0 (N )an , Z) ⊗ Q → V J0 (N ). Similar facts hold for T J1 (N ) and V J1 (N ).

9.2. CONGRUENCE RELATIONS

71

Lemma 9.12. Let N ≥ 1 be an integer, and let be a prime number. (1) V J0 (N ) is a free T0 (N )Q -module of rank 2. (2) V J1 (N ) = T J1 (N ) ⊗Z Q is a free T1 (N )Q -module of rank 2. (3) The natural morphism X1 (N )Q → X0 (N )Q induces an isomorphism of T0 (N )Q -modules (9.11)

V J1 (N ) ⊗T1 (N )Q T0 (N )Q −→ V J0 (N ).

Proof. (1) We have an isomorphism of T0 (N )Q -modules H1 (X0 (N )an , Z) ⊗ Q → V J0 (N ), and the assertion follows immediately from Proposition 9.6. (2) Similar to (1). (3) The mapping (9.11) is a surjective morphism of free modules of rank 2, which ought to be an isomorphism. Taking a basis of V J0 (N ) over T0 (N )Q , we obtain a continuous representation GQ → GL2 (T0 (N )Q ). The same is true for V J1 (N ). In this section we prove the following theorem. Theorem 9.13. Let N ≥ 1 be an integer, and let be a prime number. The -adic representation V J0 (N ) of GQ is unramiﬁed at primes p N . The characteristic polynomial det(1−ϕp t : V J0 (N )) ∈ T0 (N )Q [t] of the Frobenius substitution ϕp is given by det(1 − ϕp t : V J0 (N )) = 1 − Tp t + pt2 . Corollary 9.14. Let be a prime number, and let K be a ﬁnite extension of Q . Let f ∈ S0 (N )K be a primary form of level N with K coeﬃcients. Let Vf be the tensor product V J0 (N ) ⊗T0 (N )Q K with respect to the ring homomorphism T0 (N )Q → K deﬁned by f . The -adic representation Vf of GQ is unramiﬁed at primes p not dividing N , and we have det(1 − ϕp t : Vf ) = 1 − ap (f )t + pt2 . Theorem 3.18(1) follows immediately from Corollary 9.14 by taking a lattice in Vf stable under the action of GQ . n Example 9.15. Let f = ∞ n=1 an (f )q be the unique primitive form of level 11, and let E = X0 (11). If is a prime number, T E is the -adic representation associated with f , and we have det(1 − ϕp t : T E) = 1 − ap (f )t + pt2

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9. MODULAR FORMS AND GALOIS REPRESENTATIONS

for all primes p = , 11. The solution (5, 5) to the equation y 2 + y = x3 − x2 − 10x − 20 is a point of order 5, and if p = 11, we have 5|E(Fp ). Thus, we have ap (f ) ≡ p + 1 mod 5. By Lemma 9.12(3), Theorem 9.13 is reduced to the following theorem. Theorem 9.16. Let N ≥ 1 be an integer, and let be a prime number. The -adic representation V J1 (N ) of GQ is unramiﬁed at primes p not dividing N , and we have (9.12)

det(1 − ϕp t : V J1 (N )) = 1 − Tp t + ppt2 .

Corollary 9.17. Let be a prime number, and let K be a ﬁnite extension of Q . Let f ∈ S1 (N )K be a primary form of level N of character ε with K coeﬃcients. Deﬁne the tensor product Vf = V J1 (N ) ⊗T0 (N )Q K by the ring homomorphism T1 (N )Q → K corresponding to f . The -adic representation Vf of GQ is unramiﬁed at primes p not dividing N , and we have det(1 − ϕp t : Vf ) = 1 − ap (f )t + ε(p)pt2 . The character det Vf of GQ is the product of the -adic cyclotomic ε character and the composition GQ → Gal(Q(ζN )/Q) = (Z/N Z)× → K ×. Proof of Theorem 9.16. Let p be a prime number not dividing N . By Theorem 8.63(2), X1 (N )Z has good reduction at p. Thus, by Lemma D.18, J1 (N ) has good reduction at p and V J1 (N ) is an -adic representation good at p. We show (9.12). We regard V J1 (N )Fp as a T1 (N )Q -module through the injection End J1 (N ) → End J1 (N )Fp in Lemma D.11(2). The natural isomorphism V J1 (N )Q → V J1 (N )Fp in Lemma D.18 is an isomorphism of T1 (N )Q -modules. Thus, we have det(1 − ϕp t : V J1 (N )Q ) = det(1 − ϕp t : V J1 (N )Fp ). If F : J1 (N )Fp → J1 (N )Fp is the Frobenius endomorphism, then as in the proof of Proposition 3.15, the action of Frobenius substitution ϕp on V J1 (N )Fp is the same as the action of F on V J1 (N )Fp . Thus, it suﬃces to show det(1 − F t : V J1 (N )Fp ) = 1 − Tp t + ppt2 . Let V : J1 (N )Fp → J1 (N )Fp be the dual of F . V is the pullback F ∗ by the morphism F : X1 (N )Fp → X1 (N )Fp , and we have F V = V F = p.

9.2. CONGRUENCE RELATIONS

73

Lemma 9.18. Let N ≥ 1 be an integer, and let p be a prime number not dividing N . (1) The following relation of endomorphisms of J1 (N )Fp holds (9.13)

Tp = F + pV.

(2) Let ζN be an N th root of unity, and let w = wN be the Atkin– Lehner involution deﬁned by ζN . Then the following relation of endomorphisms of J1 (N )Fp (ζN ) holds V w = wpV. The relation (9.13) is called the congruence relation. Proof. (1) Since p is a prime, we have I1 (N, p) = X1,0 (N, p). If s, t : X1,0 (N, p) → X1 (N ) are natural morphisms, we have Tp = t∗ ◦ s∗ : J1 (N ) → J1,0 (N, p) → J1 (N ) by the deﬁnition of Hecke operators. By Theorem 8.76, X1,0 (N, p) is semistable at p. Let J1,0 (N, p)aFp be the abelian part of J1,0 (N, p)0Fp . By Theorem A.49(1), t∗ : J1 (N, p) → J1 (N ) induces J1 (N, p)aFp → J1 (N )Fp , and the Hecke operator Tp : J1 (N )Fp → J1 (N )Fp is the composition J1 (N )Fp → J1,0 (N, p)aFp → J1 (N )Fp . By Theorem 8.76 and Corollary D.21, the morphism (j0∗ , j1∗ ) : J1 (N, p)aFp −→ J1 (N )Fp × J1 (N )Fp induced by j0 , j1 : X1 (N )Fp → X1 (N, p)Fp is an isomorphism. Since the composition t∗ ◦ (j0∗ , j1∗ ) : J1 (N )Fp × J1 (N )Fp → J1 (N )Fp equals (t∗ ◦ j0∗ , t∗ ◦ j1∗ ), Tp = t∗ ◦ s∗ : J1 (N )Fp → J1 (N )Fp equals (t ◦ j0 )∗ ◦ (s ◦ j0 )∗ + (t ◦ j1 )∗ ◦ (s ◦ j1 )∗ . Hence, it suﬃces show s ◦ j0 = id,

s ◦ j1 = F,

t ◦ j0 = F,

t ◦ j1 = p.

Since j0 : X1 (N )Fp → X1 (N, p)Fp maps [(E, P )] to [(E, P, Ker F )], we have s ◦ j0 = id and t ◦ j0 = F . Since j1 : X1 (N )Fp → X1 (N, p)Fp maps [(E, P )] to [(E (p) , P (p) , Ker V )], we have s ◦ j1 = F , and since F V = p, we have t ◦ j1 = p. (2) Each side is the pullback of an endomorphism of X1 (N )Fp , w−1 ◦ F and F ◦ p−1 ◦ w−1 , respectively. Thus, it suﬃces to show w ◦F = F ◦w ◦p. For an elliptic curve E over Fp (ζN ) and its section P of order N , let Q be a section of order N satisfying (P, Q)N = ζN . p Then, we have (P (p) , Q(p) )N = ζN . Hence, we have w ◦ F (E, P ) = (p) (p) (p) (p) −1 (p) w(E , P ) = (E /P , p Q ), and thus w◦F = p−1 ◦F ◦w.

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On the other hand, since we have F ◦ w ◦ p = F ◦ p−1 ◦ w = p−1 ◦ F ◦ w, we conclude that w ◦ F = F ◦ w ◦ p. We now come back to the proof of Theorem 9.16. Since F ◦V = p, we have (1 − F t)(1 − pV t) = 1 − Tp t + ppt2 by Lemma 9.18. Taking the determinant of each side, we obtain det(1−F t : V J1 (N )Fp ) det(1−V t : V J1 (N )Fp ) = (1−Tp t+ppt2 )2 . As in (9.9), the composition V J1 (N )Fp → Hom(V J1 (N )Fp , Q ) of the limit of the isomorphism (D.15) and the transpose of the Atkin– ∗ is an isomorphism of T1 (N )Q -modules. By Lehner involution wN Lemma 9.18(2), this isomorphism maps the action of F to the action of the transpose of pV , we have det(1 − F t : V J1 (N )Fp ) = det(1 − pV t : V J1 (N )Fp ). Thus, we have det(1 − F t : V J1 (N )Fp )2 = (1 − Tp t + ppt2 )2 . If we regard det(1−F t), 1−Tp t+ppt2 ∈ 1+T1 (N )[[t]] ⊂ T1 (N )[[t]]× , we obtain det(1 − F t) = 1 − Tp t + ppt2 by taking the square root of each side. Corollary 9.19. Let be a prime number, and let K be a ﬁnite extension of Q . For an -adic representation ρ : GQ → GL2 (K) and an integer N ≥ 1, the following conditions (1) and (2) are equivalent. (1) ρ is modular of level N . (2) ρ is isomorphic to a subrepresentation of V J0 (N ) ⊗Q K. Proof. The reduced Hecke algebra T0 (N )K is isomorphic to the product ring f ∈Φ(N )K Kf . Thus, V J0 (N )⊗Q K is decomposed into ( the direct sum f ∈Φ(N )K V J0 (N ) ⊗T0 (N )Q Kf . By Theorems 9.13 and 3.18(2), the semisimpliﬁcation of V J0 (N )⊗T0 (N )Q Kf is the sum of -adic representations associated with f . The assertion follows from this immediately. As a matter of fact, we can prove that V J0 (N )⊗Q K is semisimple using Proposition 9.27 for example. However we do not prove it in this book. If we let p = in Corollary 9.17, the following holds. If V is a good p-adic representation of GQp , a ﬁltered Qp [F, V ]-module D(V ) is deﬁned as in §C.2. If A is an abelian variety over Qp having good reduction at p and V = Vp A, then we have D(V ) = D(AFp ).

9.2. CONGRUENCE RELATIONS

75

Corollary 9.20. Let p be a prime number, and let K be a ﬁnite extension of Qp . Let N ≥ 1 be an integer relatively prime to p, f ∈ S1 (N )K a primary form of lever N , character ε, with K coeﬃcients. Let Vf be the tensor product Vp J1 (N ) ⊗T1 (N )Qp K with respect to the ring homomorphism T1 (N )Q → K deﬁned by f . (1) The restriction of Vf to GQp is a good p-adic representation, and we have det(1 − F t : D(Vf )) = 1 − ap (f )t + ε(p)pt2 . The subspace D(Vf ) ⊂ D(Vf ) is one dimensional over K. (2) The restriction Vf |GQp of Vf to GQp is ordinary if and only if ap (f ) is a p-adic unit. Suppose ap (f ) is a p-adic unit, and we write 1 − ap (f )t + ε(p)pt2 = (1 − αt)(1 − pβt), where α and β are also p-adic units. We denote simply by α and β the unramiﬁed characters of GQp whose value at ϕp are α and β, respectively. Let χ be the p-adic cyclotomic character. Then, Vf |GQp is an extension of α by β · χ. Proof. By Theorem C.6(3), the submodule D(Vp J1 (N )) of the T1 (N )Qp -module D(Vp J1 (N )) equals S1 (N )Qp as a T1 (N )Qp -module. Also by Theorem C.6(3), the quotient D(Vp J1 (N ))/D(Vp J1 (N )) is Hom(S1 (N )Qp , Qp ) as a T1 (N )Qp -module. Thus, both D(Vp J1 (N )) and D(Vp J1 (N ))/D(Vp J1 (N )) are free T1 (N )Qp -modules of rank 1. Hence, D(Vp J1 (N )) = D(J1 (N )) is a free T1 (N )Qp -module of rank 2. By Lemma 9.18(1), we have det 1 − F t : D(Vp J1 (N )) det 1 − pV t : D(Vp J1 (N )) = (1 − Tp t + ppt2 )2 . By Lemma 9.18(2), using Theorem C.2 for k = Fp (ζN ), we see det 1 − F t : D(Vp J1 (N )) = det 1 − pV t : D(Vp J1 (N )) in the same way as the proof of Theorem 9.16. Hence, we have det(1 − F t : D(Vp J1 (N ))) = 1 − Tp t + ppt2 . Since D(Vf ) = D(Vp J1 (N ))⊗T1 (N )Qp K, this is a two-dimensional K-vector space, and det(1 − F t : D(Vf )) = 1 − ap (f )t + ε(p)pt2 . D(Vf ) = S1 (N )Qp ⊗T1 (N )Qp K is one dimensional. (2) Clear from (1) and Corollary C.8.

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9. MODULAR FORMS AND GALOIS REPRESENTATIONS

Theorem 9.21. Let K be a ﬁeld of characteristic 0, let f ∈ S1 (N )K be a primary form with K coeﬃcients, let ε be the character of f , and let p N be a prime number. If we write 1 − ap (f )t + ε(p)pt2 = (1 − αt)(1 − βt), then both α and β are algebraic integers √ and the absolute value of any of their conjugates is p. Proof. Similarly to Corollary 9.3, ap (f ) is an algebraic integer, and so are α and β. Replacing K by its subﬁeld Kf = Q(an (f ), n ≥ 1), we may assume K is a number ﬁeld. Take a prime number diﬀerent from p. Replacing K by its completion at the place dividing , we may assume K is a ﬁnite extension of Q . By Theorem 9.16, α and β are the eigenvalues of the action of the Frobenius substitution ϕp on V J1 (N )Fp . The eigenvalues of the action of the Frobenius substitution ϕp on V J1 (N )Fp are the eigenvalues of the action of the Frobenius morphism F : X1 (N )Fp → X1 (N )Fp on the ´etale cohomology group H 1 (X1 (N )Fp , Q ). Thus, by the Weil conjecture, any of the complex conjugates of α and β has √ the absolute value p. Theorem 2.47 follows immediately from Theorem 9.21. 9.3. Modular mod representations and non-Eisenstein ideals In this section, we give relations between modular mod representations and maximal ideals of Hecke algebras with Z coeﬃcients or the torsion points of the Jacobian of modular curves. We ﬁrst give the correspondence between modular mod representations and maximal ideals of Hecke algebras with Z coeﬃcients. Lemma 9.22. Let be an odd prime number, and let F be a ﬁnite extension of F . Let ρ¯ : GQ → GL2 (F) be an absolutely irreducible continuous representation, and let N ≥ 1. Then, the following conditions (i) and (ii) are equivalent. (i) ρ¯ is modular of level N . (ii) There exists a ring homomorphism ϕ¯ : T0 (N )Z → F to a ﬁnite extension F of F such that det(1 − ρ¯(ϕp )t) = 1 − ϕ(T ¯ p )t + pt2 for almost all prime numbers p N . Proof. Let K be a ﬁnite extension of Q , let f ∈ S0 (N )K be a primary form, and let ϕf : T0 (N )Q → K be the ring homomorphism

9.3. MOD REPRESENTATIONS AND NON-EISENSTEIN IDEALS

77

deﬁned by f . If FK is the residue ﬁeld of K, ϕf : T0 (N )Z → K induces a homomorphism ϕ¯f : T0 (N )Q → FK and ϕ¯f (Tp ) ≡ ap (f ) for all prime numbers p. Conversely, let F be a ﬁnite extension of F, and let ϕ¯ : T0 (N )Z → F be a ring homomorphism. Let m be the maximal ideal Ker ϕ¯ of T0 (N )Z . Take a maximal ideal m of the integral closure A of T0 (N )Z lying above m. Let O be the completion of A at m , and let K be its ﬁeld of fractions. K is a ﬁnite extension of Q , and O is its ring of integers. Let f be a primary form corresponding to the ring homomorphism T0 (N )Z → K, and replace F by the compositum with the residue ﬁeld FK of K. We have ϕ¯f (Tp ) ≡ ap (f ) for all prime numbers p. The assertion now follows immediately from this. Corollary 9.23. Let be an odd prime number, let F be a ﬁnite extension of F , and let ρ¯ : GQ → GL2 (F) be an absolutely irreducible continuous representation. Let N ≥ 1 be an integer, and let ϕ¯ : T0 (N )Z → F be a ring homomorphism such that ρ¯ is unramiﬁed at almost all prime numbers p N and satisﬁes det(1 − ρ¯(ϕp )t) = 1 − ϕ(T ¯ p )t + pt2 . Then, the following hold. (1) For all prime numbers p N , ρ¯ is unramiﬁed at p, and we have det(1 − ρ¯(ϕp )t) = 1 − ϕ(T ¯ p )t + pt2 . (2) If N , ρ¯ is good at . If D(ρ) is the F -module obtained by applying Theorem C.6 to the restriction of ρ¯ to GQ , we have det(1 − F t : D(¯ ρ)) = 1 − ϕ(T ¯ )t + t2 . Moreover, if ρ¯ is good and ordinary at , then Tr(F : D(¯ ρ)) = ϕ(T ¯ ) equals the value ρ¯I (ϕ ) ∈ F× at the Frobenius substitution ϕ of the unramiﬁed character ρ¯I deﬁned by the coinvariant quotient of ρ by the inertia group I . Proof. (1) Clear from the proofs of Lemmas 9.22 and 3.26. (2) Clear from Corollary 9.20 and the proof of (1). Definition 9.24. Let N ≥ 1 be an integer, and let m be a maximal ideal of the Hecke algebra T0 (N )Z . Let Fm = T0 (N )Z /m. We say that m is an Eisenstein ideal if there exists an integer M ≥ 1 and characters α, β : (Z/M Z)× → F× m such that Tp ≡ α(p) + β(p) mod m for almost all prime numbers p N M .

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If m is not an Eisenstein ideal, we say that m is a non-Eisenstein ideal . Example 9.25. The maximal ideal (5) of T0 (11)Z = Z is an Eisenstein ideal by Example 9.15. If ≥ 3 and = 5, then by Proposition 4.3, the mod representation deﬁned by the -torsion points of E = X0 (11) is irreducible. Thus, by Proposition 9.26 below, the maximal ideal () of T0 (11)Z is non-Eisenstein. A maximal ideal of a Hecke algebra with Z coeﬃcients corresponds to a modular irreducible mod representation if and only if it is a non-Eisenstein ideal. Proposition 9.26. Let N ≥ 1 be an integer, and let be an odd prime number. Let m be a maximal ideal of the Hecke algebra T0 (N )Z containing , and let Fm = T0 (N )Z /m be the residue ﬁeld. Then, the following conditions (i) and (ii) are equivalent. (i) There exist a ﬁnite extension F of the residue ﬁeld Fm and a modular irreducible representation ρ¯ : GQ → GL2 (F) of level N satisfying (9.14)

det(1 − ρ¯(ϕp )t) ≡ 1 − Tp t + pt2 mod m

for almost all prime numbers p N . (ii) m is a non-Eisenstein ideal. Proof. (i) ⇒ (ii). Let ρ¯ : GQ → GL2 (F) be a modular irreducible representation satisfying condition (i). Suppose m is an Eisenstein ideal and we derive a contradiction. If m is an Eisenstein ideal and the characters α, β : (Z/M Z)× → F× m satisfy Tp ≡ α(p) + β(p) mod m for almost all prime ideals p N M , then, by Proposition 3.4(3), ρ¯ is isomorphic to the direct sum of the characters α ⊕ β, which is a contradiction. (ii) ⇒ (i). As in the proof of Lemma 9.22, take a homomorphism T0 (N )Z → O to the ring of integers of a ﬁnite extension K of Q such that m is the inverse image of a maximal ideal of O. Let f ∈ S0 (N )K be a primary form corresponding to T0 (N )Z → O ⊂ K, and let V be the -adic representation of GQ associated with f . Take an O-lattice L of V stable under the action of GQ , and let F be the residue ﬁeld of O. V¯ = L ⊗O F deﬁnes a mod representation ρ¯. ρ¯ satisﬁes (9.14) for almost all prime numbers p N . It suﬃces to show that if m is a non-Eisenstein ideal, ρ¯ is reducible.

9.3. MOD REPRESENTATIONS AND NON-EISENSTEIN IDEALS

79

Taking the contrapositive, we assume ρ¯ is reducible and show m is an Eisenstein ideal. If ρ¯ is reducible, the semisimpliﬁcation of ρ¯ is the direct sum of characters α, β : GQ → F× . It suﬃces to show the image of α and β are in F× m . If σ∞ is the complex conjugate, we have αβ(σ∞ ) = −1, α(σ∞ ) = ±1, β(σ∞ ) = ±1, and thus we may assume α(σ∞ ) = 1, β(σ∞ ) = −1. By (9.14) and Theorem 3.1, we have det(1 − ρ¯(σ)t) ∈ Fm [t] for any σ∞ . Thus, for any τ ∈ Gal(F/Fm ), we have {τ ◦ α, τ ◦ β} = {α, β}. Since τ ◦ α(σ∞ ) = 1, we have τ ◦ α = α, and τ ◦ β = β. Thus, the image of α and β are in F× m , and m is an Eisenstein ideal. In general, for a two-dimensional irreducible representation, we have the following. Proposition 9.27. Let F be a ﬁeld, let G be a group, and let ρ : G → GL2 (F ) be an absolutely irreducible representation. Let V = F 2 be the representation space of ρ. Let W be a ﬁnite-dimensional representation of G over F , and suppose the action of each g ∈ G on W satisﬁes g 2 − Tr(ρ(g)) · g + det ρ(g) = 0. Then, W is isomorphic to the product of copies of V as a representation of G. Proof. The ring homomorphism from the group algebra ρ : F [G] → M2 [F ] induced by ρ : G → GL2 (F ) is denoted also by ρ. Since ρ : G → GL2 (F ) is absolutely irreducible, ρ : F [G] → M2 (F ) is surjective. Let J be the two-sided ideal (g 2 − Tr ρ(g) · g + det ρ(g); g ∈ G). The surjective homomorphism F [G]/J → M2 (F ) induced by ρ : F [G] → M2 (F ) is also denoted by ρ. Since the ring M2 (F ) is semisimple, and a simple M2 (F )-module is isomorphic to F 2 , it sufﬁces to show that the surjective homomorphism ρ : F [G]/J → M2 (F ) is an isomorphism. Deﬁne an anti-involution ∗ of F [G] by g ∗ = det ρ(g) · g −1 . We have ∗2 = 1. Since we have (g 2 − Tr ρ(g) · g + det ρ(g))∗ = det ρ(g)2 · g −2 − Tr ρ(g) · det ρ(g) · g −1 + det ρ(g) = det ρ(g)(g 2 − Tr ρ(g) · g + det ρ(g))g −2 , the anti-involution ∗ preserves the ideal J. The anti-involution of F [G]/J induced by ∗ is also denoted by ∗. Deﬁne an anti-involution ∗ of M2 (F ) by A∗ = Tr(A) − A. Since we have g ∗ − (Tr ρ(g) − g) = (g 2 − Tr ρ(g) · g + det ρ(g))g −1 ∈ J,

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the ring homomorphism ρ : F [G]/J → M2 (F ) is compatible with the anti-involution ∗. For any x ∈ F [G]/J, we have x + x∗ = Tr ρ(x). We show xx∗ = det ρ(x) for x ∈ F [G]/J. Since ρ(xx∗ ) = ρ(x)ρ(x)∗ = det ρ(x), xx∗ = det ρ(x) and xx∗ ∈ F are equivalent. Since (x + y)(x + y)∗ = xx∗ + yy ∗ + xy ∗ + (xy ∗ )∗ = xx∗ + yy ∗ + Tr ρ(xy ∗ ), {x ∈ F [G]/J|xx∗ ∈ F } is a linear subspace of F [G]/J. This subspace contains the image of G, which implies that it is the entire space. It follows immediately from this that the multiplicative group (F [G]/J)× equals {x ∈ F [G]/J| det ρ(x) = 0}. We show ρ : F [G]/J → M2 (F ) is injective. It suﬃces to show for x ∈ Ker ρ, that the annihilator Ann x = {y ∈ F [G]/J|yx = 0} equals the entire F [G]/J. If x ∈ Ker ρ, then x∗ = −x. If y ∈ F [G]/J, we have yx ∈ Ker ρ and thus yx = −(yx)∗ = −x∗ y ∗ = xy ∗ . Hence for y, z ∈ F [G]/J, we have yzx = x(yz)∗ = xz ∗ y ∗ = zxy ∗ = zyx. This implies Ann x ⊃ {yz − zy | y, z ∈ F [G]/J}, and thus Ann x is a two-sided ideal of F [G]/J. Since ρ(Ann x) is a two-sided ideal of M2 (F ) containing {AB − BA|A, B ∈ M2 (F )}, it is the entire M2 (F ). Thus, Ann x contains an element y satisfying ρ(y) = 1. Since this y is an invertible element of F [G]/J, we have Ann x = F [G]/J. This proves x = 0. We now give the relation between modular mod representations and the torsion points of the Jacobian of modular curves. Lemma 9.28. Let N ≥ 1 be an integer, let be an odd prime number, and let m be a non-Eisenstein maximal ideal of the Hecke algebra T0 (N )Z containing . Let F be a ﬁnite extension of Fm = T0 (N )Z /m, and let ρ¯ : GQ → GL2 (F) be a modular irreducible representation of level N satisfying det(1 − ρ¯(ϕp )t) ≡ 1 − Tp t + pt2 mod m for almost all prime numbers p N . Let V = F2 be the representation space. (1) J0 (N )[m] = {x ∈ J0 (N )(Q) | ax = 0 for all a ∈ m} is not 0. (2) J0 (N )[m] ⊗T0 (N )Z F is isomorphic to the direct sum of copies of V as a mod representation of GQ .

9.4. LEVEL AND RAMIFICATION OF -ADIC REPRESENTATIONS

81

Proof. (1) We show J0 (N )[m] = 0. If we let H1 (X0 (N )an , F ) = H1 (X0 (N )an , Z) ⊗Z F , then J0 (N )[m] is isomorphic to H1 (X0 (N )an , F )[m] = {x ∈ H1 (X0 (N )an , F ) | ax = 0 for all a ∈ m} as T0 (N )Z -modules. Since H1 (X0 (N )an , Z) is a ﬁnitely generated T0 (N )Z -module and T0 (N )Z → End H1 (X0 (N )an , Z) is injective, the localization H1 (X0 (N )an , F )m is not 0. Since the localization (T0 (N )Z /())m is a ﬁnite local ring, we have H1 (X0 (N )an , F )[m] = 0. (2) Let q N be a prime number. By Lemma 9.18(1), we have the relation Tq = Fq + Vq of the endomorphisms of J0 (N )Fq . Thus, as ¯ q ), we have ϕ2q − Tq ϕq + q = Fq2 − Tq Fq + endomorphisms of J0 (N )(F Fq Vq = 0. Hence, for each prime number q N , the action of the Frobenius substitution ϕq on J0 (N )Fq [m] satisﬁes ϕ2q −Tq ϕq +q = 0. If we let W = J0 (N )[m]⊗T0 (N )Z F, then by Theorem 3.1, the assumption of Proposition 9.27 is satisﬁed. Thus, W is the direct sum of copies of V . 9.4. Level of modular forms and ramiﬁcation of -adic representations In §3 of Chapter 3, we state the following theorem on the level of modular forms and ramiﬁcations of -adic representations. 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. 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) (i) p N . (ii) ρf is good at p. (2) (i) p2 N . (ii) ρf is semistable at p. Proof of (i) ⇒ (ii). By Theorem 8.63, X0 (N ) has good reduction at p if p N , has semistable reduction at p if p2 N . Thus, the assertion follows from Corollary 9.17, Lemmas D.18, D.16, and Corollary D.22, and in the case = p, together with the fact that det ρf is the -adic cyclotomic character. The rest of this section is devoted to the proof of (ii) ⇒ (i). It uses the detailed structure of X0,∗ (N, r) studied in Proposition 8.57.

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Let N ≥ 1 be an integer, and let p be a prime number. Write N = M pe , where M ≥ 1 is an integer relatively prime to p. If e = 0, we have nothing to prove. So, we assume e ≥ 1. For an integer 0 ≤ k ≤ e, the ﬁnite morphism of modular curves spk : X0 (M pe ) → X0 (M ) (8.65) is denoted by sk : X0 (M pe ) → X0 (M ) in this section. If e = 1, s0 , s1 : X0 (N ) → X0 (M ) are the same as s, t : I0 (M, p) → X0 (M ), which appeared in the deﬁnition of Hecke operators. Similarly, for an integer 0 ≤ k < e, the morphism spk : X0 (M pe ) → X0 (M p) is denoted by tk : X0 (M pe ) → X0 (M p). Let r ≥ 3 be an integer relatively prime to N = M pe . For an integer k ∈ Z, let σp∗k be the automorphism of Spec Z[ 1r , ζr ] deﬁned by pk ∈ (Z/rZ)× = Gal(Q(ζr )/Q). For a scheme X over Z[ 1r , ζr ], the k ﬁbered product X ×Spec Z[ r1 ,ζr ]σ∗k Spec Z[ 1r , ζr ] is denoted by X (p ) . p

If X is a scheme over Z[ 1r , ζr ]/(p) and if k ≥ 0, then this notation coincides with the one in §8.1. Similarly to sk : X0 (M pe ) → X0 (M ), we deﬁne −→ X0,∗ (M, r)Z[ r1 ]

sk : X0,∗ (M pe , r)Z[ r1 ]

−→

(E, (C, Cpe ), α)

(E/Cpk , C, α)

for 0 ≤ k ≤ e. Here, for a cyclic group Cpe of order pe , Cpk is its unique cyclic subgroup of order pk , and for a cyclic subgroup C of order M and a basis α of E[r], their images in E/Cpk are also denoted by C and α. For P, Q ∈ E[r], the Weil pairing satisﬁes k (image of P , image of Q)(E/Cpk )[r] = (P, Q)pE[r] . Thus, the diagram s

(9.15)

X0,∗ (M pe , r)Z[ r1 ] −−−k−→ X0,∗ (M, r)Z[ r1 ] ⏐ ⏐ ⏐ ⏐ Spec Z[ r1 , ζr ]

σp∗k

−−−−→ Spec Z[ 1r , ζr ]

is commutative. By the commutative diagram (9.15), the horizontal (pk )

arrow sk deﬁnes a morphism X0,∗ (M pe , r)Z[ r1 ] → X0,∗ (M, r)Z[ 1 ] of r

schemes over Spec Z[ 1r , ζr ]. Similarly, for 0 ≤ k < e, deﬁne tk : X0,∗ (M pe , r)Z[ r1 ] (E, (C, Cpe ), α)

−→

X0,∗ (M p, r)Z[ r1 ]

−→ (E/Cpk , (C, Cpk+1 /Cpk ), α).

9.4. LEVEL AND RAMIFICATION OF -ADIC REPRESENTATIONS

83

(pk )

tk deﬁnes a morphism X0,∗ (M pe , r)Z[ r1 ] → X0,∗ (M p, r)Z[ 1 ] of schemes r

over Spec Z[ 1r , ζr ]. Deﬁne a morphism p : X0,∗ (M pe , r)Z[ r1 ] → X0,∗ (M pe , r)Z[ r1 ] by sending a triple (E, C, α) to the triple (E, C, α ◦ p). p deﬁnes a morphism p : X0,∗ (M pe , r)Z[ r1 ] → (p2 )

X0,∗ (M pe , r)Z[ 1 ] of schemes over Spec Z[ 1r , ζr ]. r In what follows, we choose a maximal ideal p of Z[ζr ] lying above (p), and we denote by Fp (ζr ) its residue ﬁeld Z[ζr ]/p. Proposition 9.29. Let p be a prime number, let M ≥ 1 be an integer relatively prime to p, and let r ≥ 3 be an integer relatively prime to N = M pe . (1) To the abelian part J0,∗ (M pe , r)aFp (ζr ) , the morphism of abelian varieties s∗k : J0,∗ (M, r)(p ) → J0,∗ (M pe , r) over Q(ζr ) induces an isogeny e e ) (pk ) (9.16) s∗k : J0,∗ (M, r)Fp (ζr ) −→ J0,∗ (M pe , r)aFp (ζr ) . k

k=0

k=0

(2) To the torus part J0,∗ (M pe , r)tFp (ζr ) , the morphism of abelian varieties s∗k : J0,∗ (M, r)(p an isogeny

k

(9.17)

e−1 ) k=0

t∗k :

e−1

)

→ J0,∗ (M pe , r) over Q(ζr ) induces (pk ) t

J0,∗ (M p, r)Fp (ζr ) −→ J0,∗ (M pe , r)tFp (ζr ) .

k=0

Proof. (1) Suppose a ≤ e = a + b. The morphism ja : X0,∗ (M, r)Fp → X0,∗ (N, r)Fp is deﬁned by (E, (C, Ker V a F b ), α) a ≤ b, (E, C, α) −→ (pa−b ) (pa−b ) a b (pa−b ) , (C , Ker V F ), α ) b ≤ a. (E Thus, if a ≤ b, then ja is a morphism over Fp (ζr ). If b ≤ a, the diagram ja

X0,∗ (M, r)Fp −−−−→ X0,∗ (M pe , r)Fp ⏐ ⏐ ⏐ ⏐ ϕa−b p

Spec Fp (ζr ) −−−−→ Spec Fp (ζr ) is commutative, and the closed immersion ja deﬁnes a morphism (pe−2a ) X0,∗ (M, r)Fp → X0,∗ (N, r)Fp over Fp (ζr ).

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By Proposition 8.73 and Corollary D.20, we obtain an isomorphism from the abelian part e ) (pmin(0,e−2a) ) ja∗ : J0,∗ (M pe , r)aFp (ζr ) −→ J0,∗ (M, r)Fp (ζr ) . a

a=0

Thus, it suﬃces to show the morphism of abelian varieties e e ) (pk ) (pmin(0,e−2a) ) (sk ◦ ja )∗ : J0,∗ (M, r)Fp (ζr ) → J0,∗ (M, r)Fp (ζr ) a,e

k=0

a=0

is an isogeny. As in the proof of Lemma 9.18(1), sk ◦ ja is ⎧ Fk if a ≤ b, k ≤ b, i.e., 2a ≤ e, a + k ≤ e, ⎪ ⎪ ⎪ ⎪ k−b b−(k−b) ⎪ p F ⎪ ⎪ ⎪ ⎨ = pa+k−e F 2e−2a−k if a ≤ b ≤ k, i.e., 2a ≤ e, a + k ≥ e, ⎪ if k ≤ b ≤ a, i.e., 2a ≥ e, a + k ≤ e, F a−b+k = F 2a+k−e ⎪ ⎪ ⎪ k−b a−b+b−(k−b) ⎪ ⎪ p F ⎪ ⎪ ⎩ = pa+k−e F e−k if b ≤ a, b ≤ k, i.e., 2a ≥ e, a + k ≥ e. As consequence, the determinant det(sk ◦ ja ) of the matrix (sk ◦ ja ) ∈ Me+1 (Z[F, p]) equals e(e−1) (−1) 2 (p − F 2 )F e−a−1 × (p − F 2 )F a−1 . 1≤a≤e/2

e/2e Σ×[0,e−1] ( k 2a≤e ¯ ¯ of Z ⊕ ZΣ×[0,e−1] are detomorphism kF ⊕ kF ﬁned. The linear mappingZ[0,e] → Z[0,1]×[0,e−1] that sends the stan dard basis ea ∈ Z[0,e] to a+k 2 and n is odd, we have H q (GS , M ) = 0. (2) If n is odd, we have the exact sequence (11.14) ( 0 0 −→ H 0 (GS , M ) −→ H (Qp , M ) −→ H 2 (GS , M ∨ (1))∨ p∈S ( 1 −→ H 1 (GS , M ) −→ H (Qp , M ) −→ H 1 (GS , M ∨ (1))∨ p∈S ( 2 −→ H 2 (GS , M ) −→ H (Qp , M ) −→ H 0 (GS , M ∨ (1))∨ −→ 0. p∈S

Here, the mapping H (Qp , M ) → H 2−q (GS , M ∨ (1))∨ is the composition of the natural isomorphism H q (Qp , M ) → H 2−q (Qp , M ∨ (1))∨ (11.10) and the dual of the restriction mapping H 2−q (GS , M ∨ (1)) → H 2−q (Qp , M ∨ (1)). q

We do not give a proof of this proposition. A similar proposition holds for even n, but in that case we need to consider the inﬁnite places, and we omit it here. Example 11.26. Let the notation be as in Example 11.24. By the isomorphism in Lemma 11.4(1), we identify H 1 (GS , Z/nZ)∨ with

11.3. SELMER GROUPS

157

ab n ab Gab S /(GS ) . The maximal abelian quotient GS of GS is equal to the Galois group Gal(Q(ζpn ; p ∈ S, n ≥ 1)/Q) = p∈S Z× p . If n is odd, by the isomorphism (11.11), the second line of the exact sequence (11.14) for M = μn gives the isomorphism of class ﬁeld theory

( × × n × n ab n Coker Z× /(Z ) → Q /(Q ) −→ Gab p p S /(GS ) . S S p∈S

The third line of (11.14) is the bottom line of (11.13). About the order of the cohomology group H q (GS , M ), the following is known. Proposition 11.27. Let M be a ﬁnite GQ -module. Let S be a ﬁnite set of primes such that M is unramiﬁed outside S. Then, we have M H 1 (GS , M ) = . 0 2 H (GS , M ) · H (GS , M ) M GR We do not prove this either. Example 11.28. Let the notation be as in Example 11.24. From the exact sequence (11.13), we have H 2 (GS , μn ) = n#S−1 · gcd(n, 2). By the isomorphism (11.11), we obtain an exact sequence nth power

× 1 0 −→ H 0 (GS , μn ) −→ Z× S −−−−−−→ ZS −→ H (GS , μn ) −→ 0. S Since Z× S is isomorphic to Z ⊕ Z/2Z, we have

n S n H 1 (GS , μn ) = = . H 0 (GS , μn ) · H 2 (GS , μn ) n S−1 gcd(n, 2) gcd(n, 2) R This equals μn /μG n .

11.3. Selmer groups Definition 11.29. Let n ≥ 1 be an integer, and let M be a ﬁnite Z/nZ-GQ -module. Let S be a ﬁnite set of prime numbers such that M is unramiﬁed at all the primes outside S, and all the divisors of n are contained in S. (1) A family L = (Lp )p∈S of subgroups Lp ⊂ H 1 (Qp , M ) is called a local condition. SelL (M ) (2) Let L = (Lp )p∈S be a local condition. The Selmer group ( of M with respect to L is deﬁned as the inverse image of p∈S Lp ( by the restriction mapping H 1 (GS , M ) → p∈S H 1 (Qp , M ).

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11. SELMER GROUPS

(3) For a local condition L = (Lp )p∈S , the dual local condition ∨ 1 ∨ L∨ = (L∨ p )p∈S is deﬁned as the family of Lp ⊂ H (Qp , M (1)) ∨ for p ∈ S, where Lp is the annihilator of Lp with respect to the bilinear mapping H 1 (Qp , M ) × H 1 (Qp , M ∨ (1)) → n1 Z/Z in Proposition 11.18(2). By Proposition 11.25(1), the Selmer group SelL (M ) is a ﬁnite group. By defnition,

( 1 (H (Qp , M )/Lp ) . SelL (M ) = Ker H 1 (GS , M ) → p∈S

By Lemma 11.22, we have SelL (M )

H 1 (Q, M ) −→ . = Ker ( 1 (H (Qp , M )/Lp ) ⊕ (H 1 (Qp , M )/Hf1 (Qp , M )) p∈S

p∈S

For a ﬁnite set S ⊃ S, deﬁne a local condition L = (Lp )p∈S by Lp = Lp for p ∈ S and Lp = Hf1 (Qp , M ) for p ∈ S S. Then, we have SelL (M ) = SelL (M ). Example 11.30. Let the notation be as in Example 11.24. Let n ≥ 1 be an integer, and let S be a ﬁnite set of prime numbers that contains all the prime divisors of n. Deﬁne the Selmer group Sel(μn ) × ×n of μn by deﬁning the local condition Lp ⊂ H 1 (Qp , μn ) = Q× p /(Qp ) 1 × × ×n for p ∈ S to be the unramiﬁed part Hf (Qp , μn ) = Zp /(Zp ) . By the isomorphism (11.11), we obtain Sel(μn ) = Z× /(Z× )n = {±1}/{(±1)n }. Example 11.31. Let E be an elliptic curve over Q, and let n ≥ 1 be an integer. Let E[n] be the ﬁnite GQ -module of n-torsion points of E. Let S be a ﬁnite set of prime numbers that contains all the primes at which E does not have good reduction and all the prime divisors of n. Deﬁne the local condition Lp ⊂ H 1 (Qp , E[n]) for p ∈ S as the image of the natural injection E(Qp )/nE(Qp ) → H 1 (Qp , E[n]). The Selmer group SelL (E[n]) deﬁned by the local condition L = (E(Qp )/nE(Qp ))p∈S is called the Selmer group of E and is denoted by Sel(E, n). From the ﬁniteness of Sel(E, n) and the natural injection E(Q)/nE(Q) → Sel(E, n), we obtain the weak Mordel–Weil theorem (§1.3(b) in Number Theory 1 ), which says E(Q)/nE(Q) is a ﬁnite group.

11.3. SELMER GROUPS

159

Let E be the elliptic curve y 2 = x3 −x over Q. E has good reduction at p = 2. If we let S = {2}, then GS -module E[2] is isomorphic to (Z/2Z)⊕2 and H 1 (GS , E[2]) is isomorphic to (Z[ 12 ]× /(Z[ 21 ]×2 ))⊕2 . Sel(E, 2) is isomorphic to (Z× /(Z×2 ))⊕2 = {±1}⊕2 , and E[2](Q) → Sel(E, 2) is an isomorphism. Proposition 11.32. Let n ≥ 1 be an odd integer, and let M be a ﬁnite Z/nZ-GQ -module. Let S be a ﬁnite set of prime numbers such that M is unramiﬁed outside S and S contains all the prime divisors of n. Let L = (Lp )p∈S be a local condition, and let L = (Lp )p∈S be a family of subgroups Lp ⊂ Lp . Then, we have the exact sequence ( −−→ SelL (M ) −−→ 0 −−→ SelL (M ) p∈S Lp /Lp −−→ SelL∨ (M ∨ (1))∨ −−→ SelL∨ (M ∨ (1))∨ −−→

0.

Proof. The ﬁrst line is exact by the deﬁnition of Selmer groups. Similarly, we have an exact sequence ( ∨ ∨ 0 → SelL∨ (M ∨ (1)) → SelL∨ (M ∨ (1)) → Lp /Lp . p∈S ∨ By the deﬁnition of dual local condition, L∨ p /Lp is the dual of Lp /Lp , ( ∨ ∨ we obtain the exact sequence → p∈S Lp /Lp → SelL∨ (M (1)) SelL∨ (M ∨ (1))∨ → 0 by taking( the dual. We show the exactness at p∈S Lp /Lp . Deﬁne

( 1 H (Qp , M ) , A = Im H 1 (GS , M ) → p∈S

( 1 B = Im H 1 (GS , M ∨ (1)) → H (Qp , M ∨ (1)) . p∈S

By Proposition 11.25(2), to ( of B1 with respect ( A is 1the annihilator ∨ H (Q , M ) × H (Q , M (1)) → the bilinear mapping p p p∈S p∈S ( 1 Z/Z. The image of SelL (M ) → p∈S Lp /Lp is the image of A ∩ n ( ( ∨ the image of SelL∨ (M ∨ (1)) → p∈S L∨ p /Lp is p∈S Lp . Similarly, ( ( ∨ the image of B ∩ p∈S Lp . Thus, the kernel of Lp /Lp → p∈S ( SelL∨ (M ∨ (1))∨ is the annihilator of the image of B ∩ p∈S L∨ p ( ( ∨ with respect to the bilinear mapping p∈S Lp /Lp × p∈S L∨ p /Lp → ( 1 to show the image of A ∩ p∈S Lp is the n Z/Z. Thus, it suﬃces ( annihilator of B ∩ p∈S L∨ p with respect to the bilinear mapping ( ( 1 ∨ ∨ L /L × L /L p p → n Z/Z. p∈S p p∈S p

160

11. SELMER GROUPS

The image of A ∩ (A ∩

)

( p∈S

Lp ) +

p∈S

Lp is the image of

)

Lp =

p∈S

)

Lp ∩ (A +

p∈S

)

Lp ).

p∈S

( ( ( ∨ L ∩ (A + p∈S Lp ) is the annihilator of Since p∈S Lp + ( p∈S ∨p ( (B ∩ p∈S Lp ), the image of A ∩ p∈S Lp is the annihilator of B ∩ ( ( ∨ p∈S Lp . This shows the exactness at p∈S Lp /Lp . Proposition 11.33. Let n ≥ 1 be an odd integer, and let M be a ﬁnite Z/nZ-GQ -module. Let S be a ﬁnite set of prime numbers such that M is unramiﬁed outside S and S contains all the prime divisors of n. Let L = (Lp )p∈S be a local condition, and let L = (Lp )p∈S be a family of subgroups Lp ⊂ Lp . Then, we have Lp SelL (M ) M GS 1 = · · . G ∨ ∨ G Q S SelL∨ (M (1)) M (1) M p M GR p∈S Proof. SelL (M ) is the kernel of the composition ( of 1the mapping in the second line of (11.14) H 1 (GS , M ) → p∈S H (Qp , M ) ( ( and the surjection p∈S H 1 (Qp , M ) → p∈S H 1 (Qp , M )/Lp by the deﬁnition of Selmer group. Moreover, by the deﬁnitions of the dual lois identiﬁed cal condition and Selmer group, the dual SelL∨ (M ∨ (1))∨( with the cokernel of the composition of the inclusion p∈S Lp → ( 1 H (Q , M ) and the mapping of the second line of (11.14) (p∈S 1 p 1 ∨ ∨ p∈S H (Qp , M ) → H (GS , M (1)) . Thus, we obtain the exact sequence by Proposition 11.25(2) 0 −→ H 1 (GS , M ) −→

−→

SelL (M )

H 1 (Qp , M )/Lp −→ SelL∨ (M ∨ (1))∨

p∈S

−→ H 2 (GS , M ) −→

H 2 (Qp , M )

−→ H 0 (GS , M ∨ (1))∨ −→ 0.

p∈S

From this we obtain SelL (M ) H 1 (GS , M ) = SelL∨ (M ∨ (1)) H 0 (GS , M ∨ (1)) · H 2 (GS , M ) H 2 (Qp , M ) · Lp . × H 1 (Qp , M ) p∈S

11.4. SELMER GROUPS AND DEFORMATION RINGS

161

By Proposition 11.27, the ﬁrst factor of the right-hand side equals . By Proposition 11.20, the contribution of each p ∈ S

M GS M M ∨ (1)GS M GR Lp

is

M

GQ p

· (M ⊗Zp )

. The equality in question follows immediately from

this.

11.4. Selmer groups and deformation rings In §5.2, we deﬁned the deformation ring RΣ . In this section we relate deformation rings and Selmer groups, and we reduce Theorems 5.32(1) and 5.34 to properties concerning Selmer groups, Theorem 11.37 and Proposition 11.38. As in §5.2, let be an odd prime number, let F be a ﬁnite extension of F , and let ρ¯ : GQ → GL2 (F) be a modular semistable irreducible mod -representation. Let K be a ﬁnite extension of Q whose residue ﬁeld is F, and let f ∈ Φ(N∅ )K,¯ρ (K) be a primitive form with K coeﬃcients. Let O be the ring of integers of K. The -adic representation ρ = ρf : GQ → GL2 (O) associated with f is a lifting of ρ¯ to O of type D∅ . ρ : GQ → GL2 (O) is unramiﬁed outside Sρ¯ ∪ {}. As in Deﬁnition 5.5, let Σ be a ﬁnite set of prime numbers such that Σ ∩ Sρ¯ = ∅ and that if ∈ Σ, then ρ¯ is good and ordinary at . Let SΣ = Sρ¯ ∪ Σ ∪ {}. ρ : GQ → GL2 (O) induces a representation of GSΣ on O2 . We regard O2 as a GSΣ -module through ρ and denote it by V . Let W = End0 (V ) = {f ∈ End(V )| Tr f = 0}. W is a free O-module of rank 3 that possesses a natural action of GSΣ . Let π be a primitive element of O. For an integer n ≥ 1, let Vn , Wn be O/π n O-modules V /π n V, W/π n W = End0 (Vn ), respectively. Let V∞ = limn Vn = V ⊗ K/O, W∞ = limn Wn = W ⊗ K/O. For n = 1, −→ −→ let V1 = V¯ , W1 = W . We now give the local condition that deﬁnes the Selmer group. Since ρ : GQ → GL2 (O) is a lifting of ρ¯ of type D∅ , if ∈ Sρ¯, ρ¯ is good, and by Theorem 9.13, ρ is good at . If ∈ Sρ¯ ∪ Σ, then ρ¯ is ordinary at , and by Theorem 3.52(2) and Proposition 7.11, ρ is ordinary at . If ρ is good at , deﬁne the subgroup Hf1 (Q , Wn ) of H 1 (Q , Wn ) by Hf1 (Q , Wn ) = H 1 (Q , Wn ) ∩ Hf1 (Q , End(Vn )). If ρ is ordinary at , deﬁne the subgroup Hs1 (Q , Wn ) of H 1 (Q , Wn ). Let Vn0 ⊂ Vn be the subgroup on which I acts as the cyclotomic character. Vn0 , Vn /Vn0 are free O/π n O-modules of rank 1.

162

11. SELMER GROUPS

Let Wn0 ⊂ Wn be {f ∈ End0 (Vn )|f (Vn0 ) = 0, f (Vn ) ⊂ Vn0 }. Wn0 is also a free O/π n O-module. Deﬁne Hs1 (Q , Wn ) by Hs1 (Q , Wn ) = Ker(H 1 (Q , Wn ) → H 1 (I , Wn /Wn0 )). Definition 11.34. Deﬁne the Selmer group SelΣ (Wn ) ⊂ H 1 (GSΣ , Wn ) by the local condition LΣ = (LΣ,p )p∈SΣ deﬁned by ⎧ 1 H (Qp , Wn ) if p = , p ∈ Sρ¯, ⎪ ⎪ ⎪ f1 ⎨ H (Qp , Wn ) if p = , p ∈ Σ, LΣ,p = 1 ⎪ if p = ∈ / Sρ¯ ∪ Σ, H ⎪ f (Q , Wn ) ⎪ ⎩ 1 Hs (Q , Wn ) if p = ∈ Sρ¯ ∪ Σ. Deﬁne SelΣ (W∞ ) = limn SelΣ (Wn ). −→ We translate the local condition in terms of inﬁnitesimal deformations. Lemma 11.35. Through the bijection (11.2), identify Z 1 (Qp , Wn ) with Lift0O/πn O-GQp (Vn ). Let c : GQp → Wn be a 1-cocycle, and let 5 be the corresponding inﬁnitesimal lifting. M (1) Suppose p = , p ∈ Sρ¯. Then, [c] belongs to Hf1 (Qp , Wn ) if and 5 is ordinary. only if M (2) Suppose p = ∈ Sρ¯ ∪ Σ. In this case, ρ is ordinary at . Then, 5 is ordinary. [c] belongs to Hs1 (Qp , Wn ) if and only if M Proof. (1) Suppose p = , p ∈ Sρ¯. If σ ∈ Ip is a lifting of the generator of Ip /Ip [Ip , Ip ] Z/Z, then ρ¯(σ) = 1. If we take a suitable basis of Vn , the matrix representation of the action of σ is given by 5 is generated by σ, the ( 10 11 ). Since the image of Ip → AutO/πn O M 5 is ordinary means that the matrix representation of σ is fact that M 5. This implies that given by ( 10 11 ) if we choose a suitable basis of M 5 is isomorphic to Vn ⊗O/πn O O/π n O[ε] as the inﬁnitesimal lifting M n O/π O[ε]-Ip -modules, which is in turn equivalent to the condition that the restriction of [c] to Ip is 0. (2) Suppose p = ∈ Sρ¯ ∪ Σ. In this case, as we remarked above, the restriction of ρ to Ip is ordinary by Proposition 7.11. The part Vn0 on which the inertia group Ip acts as the cyclotomic character is a free O/π n -module of rank 1. If c|Ip ∈ Z 1 (Ip , Wn0 ), then Ip acts on Vn0 ⊗O/πn O O/π n O[ε] as the cyclotomic character, and it acts trivially

11.4. SELMER GROUPS AND DEFORMATION RINGS

163

5 is ordinary. Conversely, on (Vn /Vn0 ) ⊗O/πn O O/π n O[ε]. Thus, M 5 is ordinary. Then, Ip acts on the submodule M 50 of rank 1 suppose M 5 of M as the cyclotomic character, and it acts trivially on the quotient 5/M 50 . Thus if we choose a suitable basis of M 5, we have c|I ∈ M p 1 0 Z (Ip , Wn ). We give a relation between deformation rings and Selmer groups. Let RΣ be the deformation ring deﬁned in §5.2, and πΣ : RΣ → O the homomorphism deﬁned by ρ. Proposition 11.36. Let mRΣ be the maximal ideal of the deformation ring RΣ . Then, there is a natural isomorphism of F-linear spaces (11.15)

HomF (mRΣ /(m2RΣ , π), F) −→ SelΣ (W ).

Let pRΣ be the kernel of the ring homomorphism RΣ → O deﬁned by ρ : GQ → GL2 (O). Then, there is a natural isomorphism of Omodules (11.16)

HomO (pRΣ /p2RΣ , K/O) −→ SelΣ (W∞ ).

Proof. For an integer n ≥ 1, deﬁne a natural injection (11.17)

HomO/(πn ) (pRΣ /(p2RΣ , π n ), O/(π n )) −→ H 1 (GSΣ , Wn ).

First, we identify HomO/(πn ) (pRΣ /(p2RΣ , π n ), O/(π n )) with a subset of n n n Def ρ,D ¯ Σ (O/(π )[ε]). Let pn : O/(π )[ε] → O/(π ) be the natural surn jection. If f : RΣ → O/(π )[ε] is a morphism of O-algebras satisfying pn◦f = πΣ mod π n , then the restriction of f to pRΣ = Ker(πΣ : RΣ → O) induces an O/(π n )-linear mapping pRΣ /(p2RΣ , π n ) → O/(π n )ε. By this correspondence, HomO/(πn ) (pRΣ /(p2RΣ , π n ), O/(π n )) is identiﬁed with the set {f : RΣ → O/(π n )[ε] | f is a morphism of O-algebras with pn ◦ f = πΣ mod π n } n This set is identiﬁed with the inverse image Def ρ,D ¯ Σ (O/(π )[ε])[Vn ] n n of the class of Vn by Def ρ,D ¯ Σ (O/(π )[ε]) → Def ρ,D ¯ Σ (O/(π )) by the deﬁnition of the deformation ring RΣ . n Deﬁne Liftρ,D ¯ Σ (O/(π )[ε])Vn to be the inverse image of Vn by the n n mapping Liftρ,D ¯ Σ (O/(π )[ε]) → Liftρ,D ¯ Σ (O/(π )). Since an element n of Liftρ,D ¯ Σ (O/(π )[ε])Vn deﬁnes an inﬁnitesimal lifting that preserves the determinant of Vn , we obtain a natural injection

(11.18)

0 n Liftρ,D ¯ Σ (O/(π )[ε])Vn −→ LiftO/(π n )-GS (Vn ). Σ

164

11. SELMER GROUPS

By Proposition 11.8(2), we identify H 1 (GSΣ , Wn ) with the set of inﬁnitesimal deformations preserving the determinant Def 0O/(πn )-GS (Vn ). Σ We show that (11.18) induces an injection (11.19)

0 n Def ρ,D ¯ Σ (O/(π )[ε])[Vn ] −→ Def O/(π n )-GS (Vn ). Σ

The set Def 0O/(πn )-GS (Vn ) is the quotient of Lift0O/(πn )-GS (Vn ) Σ Σ by the group 1 + ε End(Vn ) = Ker(GL2 (O/(π n )[ε]) → GL2 (O/(π n )) n by deﬁnition. We show Def ρ,D ¯ Σ (O/(π )[ε])[Vn ] is also the quotient of n n Liftρ,D ¯ Σ (O/(π )[ε])Vn by 1+ε End(Vn ). For ρ ∈ Liftρ,D ¯ Σ (O/(π )[ε])Vn 1 n n and P ∈ U GL2 (O/(π )[ε]) = Ker(GL2 (O/(π )[ε]) → GL2 (F)), we ﬁrst show P ∈ (1 + mO/(πn )[ε] ) · (1 + ε End(Vn )) if ad(P )(ρ) ∈ n Liftρ,D In this case, the image P = P mod ε ∈ ¯ Σ (O/(π )[ε])Vn . 1 n U GL2 (O/(π )) = Ker(GL2 (O/(π n )) → GL2 (F)) satisﬁes the relation ad(P )(ρΣ mod π n ) = ρΣ mod π n . By the ﬁrst part of the proof of Proposition 7.15, M2 (O/(π n )) is generated by the image of ρΣ mod π n over O/(π n ). Thus, we have P ∈ 1 + mO/(πn ) , and P ∈ n (1 + mO/(πn )[ε] ) · (1 + ε End(Vn )). Therefore, Def ρ,D ¯ Σ (O/(π )[ε])[Vn ] n is the quotient of Liftρ,D ¯ Σ (O/(π )[ε])Vn by (1 + mO/(π n )[ε] )· (1+ε End(Vn )). Since the action of the scalar matrices 1+mO/(πn )[ε] is n n trivial, Def ρ,D ¯ Σ(O/(π ))[ε])[Vn ] is the quotient of Liftρ,D ¯ Σ(O/(π )[ε])Vn 0 n by 1+ε End(Vn ). Hence, Liftρ,D ¯ Σ (O/(π )[ε])Vn → LiftO/(π n )-GS (Vn ) Σ

0 n induces an injection Def ρ,D ¯ Σ (O/(π )[ε])[Vn ] → Def O/(π n )-GS (Vn ). Σ

By the natural isomorphism H 1 (GSΣ , Wn ) → Def 0O/(πn )-GS (Vn ) Σ (11.4), the injection (11.19) deﬁnes the injection (11.17). We show the image of the injection (11.17) is SelΣ (Wn ). It suﬃces to verify that the ramiﬁcation condition that a lifting is of type DΣ corresponds to the local condition that deﬁnes the Selmer group for each prime number p ∈ SΣ . First, suppose p = . If p ∈ Sρ¯, the ramiﬁcation condition is that the lifting is ordinary. Thus, by Lemma 11.35(1), this corresponds to the local condition Hf1 (Qp , Wn ). If p ∈ Σ, there is no ramiﬁcation condition, which corresponds to the local condition H 1 (Qp , Wn ). Suppose p = . If ∈ / Sρ¯ ∪Σ, the ramiﬁcation condition is that the lifting is good, which corresponds to the local condition Hf1 (Qp , Wn ). If ∈ Sρ¯ ∪ Σ, then by Proposition 7.11, the ramiﬁcation condition is that the lifting is ordinary. Thus, by Lemma 11.35(2), it corresponds to the local condition Hs1 (Qp , Wn ).

11.5. PROOF OF PROPOSITION 11.38

165

Therefore, the injection (11.17) deﬁnes a natural isomorphism HomO/(πn ) (pRΣ /(p2RΣ , π n ), O/(π n )) → SelΣ (Wn ). Letting n = 1, we obtain the isomorphism (11.15). Taking limn , we obtain the isomor−→ phism (11.16). By Proposition 11.36, Theorems 5.32(1) and 5.34 are reduced to Theorem 11.37 and Proposition 11.38, respectively. Theorem 11.37. Let r = dimF Sel∅ (W ). For any integer n ≥ 1, there exists a set Q consisting of r prime numbers q1 , . . . , qr satisfying the condition (5.19)n

qi ∈ / Sρ¯,

qi ≡ 1 mod n ,

and

Tr ρ¯(ϕqi ) ≡ ±2

such that (11.20)

dimF SelQ (W ) = r.

Proposition 11.38. Suppose p ∈ Σ, and let Σ = Σ {p}. Then, we have (p + 1)2 − ap (f )2 = 0, and (11.21) lengthO SelΣ (W∞ )/ SelΣ (W∞ ) ≤ ordO (p − 1)((p + 1)2 − ap (f )2 ) holds. We will prove Theorem 11.37 and Proposition 11.38 in §11.6 and §11.5, respectively. 11.5. Calculation of local conditions and proof of Proposition 11.38 In this section we prove Proposition 11.38 by calculating the order of local conditions Hf1 (Q , Wn ) and Hs1 (Q , Wn ). The proof of Theorem 11.37, which uses Proposition 11.39 below, will be postponed until the next section. We keep the notation in the previous section. Let Mn be the ﬁltered ϕ-O-module D(Vn ), and deﬁne End0O (Mn ) = Ker(Tr : HomO (Mn , Mn ) → O/(π n )) and its subgroup End0O (Mn ) = End0O (Mn ) ∩ HomO (Mn , Mn → O/(π n )) . M1 is also written M . Proposition 11.39. (1) If p = , then dim Hf1 (Qp , W ) = dim H 0 (Qp , W ). (2) If ∈ / Sρ¯, then dim Hf1 (Ql , W ) = dim H 0 (Ql , W ) + 1.

166

11. SELMER GROUPS

(3) If ∈ Sρ¯, then dim Hs1 (Ql , W ) = dim H 0 (Ql , W ) + 1. Proof. (1) Clear from Corollary 11.13. (2) As in the proof of Corollary C.10, we obtain an exact sequence 0 −→ H 0 (Q , W ) −→ End0F (D(W )) −→ End0F (D(W )) −→ Hf1 (Q , W ) −→ 0 by Proposition C.9. It now follows from dimF End0F (D(W )) = 3 and End0F (D(W )) = 2. (3) First, we show (11.22)

0

Hs1 (Q , W ) = Ker(H 1 (Q , W ) → H 1 (Q , W /W )).

The inclusion ⊃ is clear. We show ⊂. Suppose a class M of inﬁnitesimal liftings that preserve the determinant of V is contained in Hs1 (Q , W ). By Lemma 11.35(2), M is ordinary. If χ is the cyclotomic character, there exists an unramiﬁed characters α, β : GQ → F[]× such that M is the extension of α by βχ. Since V = M ⊗F[] F is not good by assumption, we have α = β by Corollary 11.17. Since M is an inﬁnitesimal lifting preserving the determinant, we have α2 = 1 and the image of α is contained in {±1} ⊂ F× . Thus, by choosing a basis suitably, we may assume that the restriction c|GQ of 0

the 1-cocycle c that gives M satisﬁes c ∈ Z 1 (GQ , W ). Thus, we 0 have [M ] ∈ Ker(H 1 (Q , W ) → H 1 (Q , W /W )), and Hs1 (Q , W ) = 0 Ker(H 1 (Q , W ) → H 1 (Q , W /W )) is proved. From the exact sequence 0

0

0

0

0 −−→ H 0 (Q , W ) −−→ H 0 (Q , W ) −−→ H 0 (Q , W /W ) −−→ H 1 (Q , W ) −−→ H 1 (Q , W ) −−→ H 1 (Q , W /W ) and (11.22), we obtain (11.23)

dim Hs1 (Q , W ) − dim H 0 (Q , W ) 0

0

0

= dim H 1 (Q , W ) − dim H 0 (Q , W ) − dim H 0 (Q , W /W ). By Propositions 11.20, and 11.18, the right-hand side of (11.23) is equal to 0

0

0

dim W + dim H 0 (Q , (W )∨ (1)) − dim H 0 (Q , W /W ).

11.5. PROOF OF PROPOSITION 11.38 0

167

0

We have dim W = dim H 0 (Q , W /W ) = 1. Since the action of GQ 0 0 on (W )∨ (1) is trivial, dim H 0 (Q , (W )∨ (1)) is also 1. Hence, the right-hand side of (11.23) equals 1. Proposition 11.40. (1) Suppose p = and p ∈ / Sρ¯. det(1 − pϕp : W ) = 0, then we have

If

lengthO H 1 (Qp , W∞ )/Hf1 (Qp , W∞ ) = ordO det(1 − pϕp : W ). (2) Suppose ρ¯ is good and ordinary at . If det(1−ϕp : (W 0 )∨ (1)) = 0, then we have lengthO Hs1 (Q , W∞ )/Hf1 (Q , W∞ ) ≤ ordO det(1 − ϕp : (W 0 )∨ (1)). Proof. (1) By Proposition 11.18(2), H 1 (Qp , W∞)/Hf1 (Qp , W∞) is the dual of Hf1 (Qp , W (1)). Thus, by Lemma 11.12(2), the length of O-module Hf1 (Qp , W (1)) = H 1 (Fp , W (1)) = Coker(1 − ϕp : W (1)) equals ordO det(1 − pϕp : W ). (2) It suﬃces to show the following: (11.24)

lengthO Hf1 (Q , Wn ) − lengthO H 0 (Q , Wn ) = n,

(11.25)

lengthO Hs1 (Q , Wn ) − lengthO H 0 (Q , Wn ) ≤ n + ordO det(1 − ϕp : (W 0 )∨ (1)).

As in the proof of Proposition 11.39(2), we obtain an exact sequence 0 −→ H 0 (Q , Wn ) −→ End0O (Mn ) −→ End0O (Mn ) −→ Hf1 (Q , W n ) −→ 0. The equality (11.24) follows from the facts lengthO End0O (Mn ) = 3n and lengthO End0O (Mn ) = 2n. We show (11.25). Hs1 (Q , Wn ) is the kernel of the composition res ◦u of u

res

H 1 (Q , Wn ) −→ H 1 (Q , Wn /Wn0 ) −→ H 1 (I , Wn /Wn0 ). Since Ker res = H 1 (F , (Wn /Wn0 )I ) by Proposition 11.5, we have lengthO Hs1 (Q , Wn ) ≤ lengthO Ker u+lengthO H 1 (F , (Wn /Wn0 )I ). As in the proof of Proposition 11.39(3), from the exact sequence 0 −−→ H 0 (Q , Wn0 ) −−→ H 0 (Q , Wn ) −−→ H 0 (Q , Wn /Wn0 ) u

−−→ H 1 (Q , Wn0 ) −−→ H 1 (Q , Wn ) −−→ H 1 (Q , Wn /Wn0 ),

168

11. SELMER GROUPS

we obtain lengthO Hs1 (Q , Wn ) − lengthO H 0 (Q , Wn ) ≤ lengthO H 1 (Q , Wn0 ) − lengthO H 0 (Q , Wn0 ) − lengthO H 0 (Q , Wn /Wn0 ) + lengthO H 1 (F , (Wn /Wn0 )I ). The right-hand side equals lengthO Wn0 + lengthO H 0 (Q , (Wn0 )∨ (1)) by Lemma 11.12(1) and Proposition 11.20. We have lengthO Wn0 = n, and elementary linear algebra on O-modules shows the inequality lengthO H 0 (Q , (Wn0 )∨ (1)) ≤ ordO det(1 − ϕp : (W 0 )∨ (1)). Thus, the inequality (11.25) is proved. Proof of Proposition 11.38. We show the inequality (11.21). First, consider the case p = . From the exact sequence 0 −→ SelΣ (W∞ ) −→ SelΣ (W∞ ) −→ H 1 (Qp , W∞ )/Hf1 (Qp , W∞ ), we obtain lengthO SelΣ (W∞ )/ SelΣ (W∞ ) ≤ lengthO H 1 (Qp , W∞ )/Hf1 (Qp , W∞ ). Thus, by Proposition 11.40(1), it suﬃces to show (11.26)

det(1 − pϕp : W ) = (1 − p)((p + 1)2 − ap (f )2 ) = 0.

Write 1 − ap (f )t + pt2 = (1 − αt)(1 − βt), and we have (11.27) (p + 1)2 − ap (f )2 = (1 + αβ)2 − (α + β)2 = (1 − α2 )(1 − β 2 ). By Theorem 9.13, the eigenvalues of ρ(ϕp ) are α and β. Since α2 , β 2 = 1 by Theorem 9.21, the right-hand side is not 0. Since we have det(1 − pϕp : W ) = (1 − p)(1 − pα/β)(1 − pβ/α) = (1 − p)(1 − α2 )(1 − β 2 ), the equality in (11.26) is also proved. This completes the proof in the case p = . Suppose p = ∈ Σ. As in the case p = , we have lengthO SelΣ (W∞ )/ SelΣ (W∞ ) ≤ lengthO Hs1 (Qp , W∞ )/Hf1 (Qp , W∞ ), and (1 − p)((p + 1)2 − ap (f )2 ) = 0. Thus, by Proposition 11.40(2), it suﬃces to show (11.28) ordO det(1−ϕp : (W 0 )∨ (1)) = ordO (1−p)((p+1)2 −ap (f )2 ).

11.6. PROOF OF THEOREM 11.37

169

Since ρ is ordinary at p = , we can write 1 − ap (f )t + pt2 = (1 − αt)(1 − p/α · t), where α is a p-adic unit by Corollary 9.20(2). Since both 1 − p and 1 − (p/α)2 are p-adic units, the right-hand side of (11.28) equals ordO (1 − α2 ) by (11.27). If we denote also by α the unramiﬁed character of GQp deﬁned by the property that the image of ϕp is α, the restriction of V to GQp is an extension of α by α−1 (1) by Corollary 9.20(2). Thus, we have W 0 = Hom(α, α−1 (1)). Hence, det(1 − ϕp : (W 0 )∨ (1)) = 1 − α2 . 11.6. Proof of Theorem 11.37 We ﬁrst give a summary of group theoretic facts about the absolutely irreducible mod -representation ρ¯ : G → GL2 (F ) that will be needed to prove Theorem 11.37. Proposition 11.41. Let be an odd prime number, and let 2 G ⊂ GL2 (F ) be a ﬁnite subgroup. Suppose V = F is absolutely irreducible as a representation of G. Let W = End0 (V ) = Ker(Tr(End(V ) → F)). Then, one of the following (i) and (ii) holds. (i) W is also absolutely irreducible as a representation of G. (ii) There exist a subgroup H of G of index 2 and a character χ : × H → F such that V IndG H χ. Let δ : G → G/H → {±1} be the character of order 2, and let χ be the conjugate of χ by g ∈ G H. Then, we have χ = χ, and W δ ⊗ IndG H (χ /χ). Proof. It suﬃces to show (ii), assuming W is absolutely reducible. If W is absolutely reducible, W possesses a one- or twodimensional subspace that is stable under the action of G. Since W is self-dual, we may assume that W has a G-stable one-dimensional subspace T . Let f be a basis of T . Since Ker f is a G-stable subspace of V and V is absolutely irreducible, f deﬁnes an isomorphism × V ⊗ T → V . Let δ : G → F be the character that the action of G on T deﬁnes. By the isomorphism V ⊗T → V , we have det V ·δ 2 = det V , and thus the order of δ is at most 2. If δ = 1, then the basis f of T deﬁnes an endomorphism of V that is not a scalar multiple. Thus, by Schur’s lemma, the order of δ is exactly 2. Let H = Ker δ. Again by Schur’s lemma, V is reducible × as a representation of H. Thus, there exists a character χ : H → F

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such that V IndG H χ. Since V is irreducible, we have χ = χ. The assertion on W follows immediately from this.

Corollary 11.42. Let the notation be as in Proposition 11.41. Then, the following hold. (1) W G = 0. × (2) If δ : G → F is a character, then either (i) or (ii) below holds. (i) (W ⊗ δ)G = 0. (ii) Let H = Ker δ. Then, we have [G : H] = 2, and there × exists a character χ : H → F such that V IndG H χ. Proof. (1) Clear from Schur’s lemma. (2) Suppose (W ⊗ δ)G = 0. Then, there exists a one-dimensional G-stable subspace T of W such that the action of G on T is given by δ −1 . As in the proof of Proposition 11.41, we have δ −1 = δ, and the assertion follows from this. Corollary 11.43. Let the notation be as in Proposition 11.41. If T = 0 is a G-stable subspace of W , then there exists a g ∈ G ⊂ GL2 (F ) that has mutually distinct eigenvalues and that satisﬁes T + (g − 1)W = W . Proof. We ﬁrst show the case where W is absolutely irreducible. In this case we have T = W , and it suﬃces to show that there exists an element g in G ⊂ GL2 (F ) that has distinct eigenvalues. Assuming such a g does not exist, we derive a contradiction. Since V is absolutely irreducible, the center of G is Z = {g ∈ G | g is a scalar matrix}. By the assumption, the order of any element of G/Z is either or 1. If H is the -Sylow subgroup of G, we have G = Z × H. Since an irreducible representation of H is one dimensional, this contradicts the irreducibility. Next, we show the case where W satisﬁes the condition (ii) of Proposition 11.41. In this case, the order of G is relatively prime to , the representation W of G is semisimple. By the proof of Proposi tion 11.41, we may prove it by assuming T = δ or T = IndG H (χ /χ) G when we decompose W = δ ⊗ IndH (χ /χ). If T = δ, we may take a g in H that is not a scalar matrix. If T = IndG H (χ /χ), we may take a g ∈ G H. We give a suﬃcient condition for H 1 (G, W ) = 0.

11.6. PROOF OF THEOREM 11.37

171

Lemma 11.44. Let be an odd prime number, and let F be a ﬁnite extension of F . If V = F2 and W = End0 (V ), then we have H 1 (SL2 (F), W ) = 0 except in the case F = F5 . Outline of proof. Let V0 = F ⊂ V be a one-dimensional subspace, and deﬁne subgroups B and U of SL2 (F) by B = {g ∈ SL2 (F) | g(V0 ) ⊂ V0 } U = {g ∈ B | g|V0 = 1}. The action on B/U → F× . Explicitly, we have 6Va0 0deﬁnes 1 uan$ isomorphism 7 × B = , ( 0 1 )$ a ∈ F , u ∈ F and U = { ( 10 u1 )| u ∈ F}, and 0 a−1 0 → a. Since the the isomorphism B/U → F× is given by a0 a−1 indices [SL2 (F) : B] and [B : U ] are relatively prime to , the restriction H 1 (SL2 (F), W ) → H 1 (B, W ) is injective and H 1 (B, W ) → H 1 (U, W )B/U is an isomorphism. Deﬁne subspaces of W by W1 = {f ∈ W |f (V0 ) ⊂ V0 } ⊃ W0 = {f ∈ W |f (V0 ) = 0}. Since the actions of U on W/W1 , W1 /W0 , and W0 are trivial, H 1 (U, W/W1 ), H 1 (U, W1 /W0 ), and H 1 (U, W0 ) are identiﬁed with Hom(U, W/W1 ), Hom(U, W1 /W0 ), and Hom(U, W0 ). Moreover, the invariant parts H 1 (U, W/W1 )B/U , H 1 (U, W1 /W0 )B/U , and H 1 (U, W0 )B/U are naturally identiﬁed with HomB/U (U, W/W1 ), HomB/U (U, W1 /W0 ), and HomB/U (U, W0 ), respectively. We show the following (i) If F = 3, 5, 9, then HomB/U (U, W/W1 ) = 0. (ii) If F = 3, then HomB/U (U, W1 /W0 ) = 0. (iii) HomB/U (U, W0 ) = HomF (U, W0 ). The isomorphisms of F-vector spaces W/W1 → HomF (V0 , V /V0 ), W1 /W0 → HomF (V0 , V0 ), W0 → HomF (V /V0 , V0 ) are isomorphisms of B/U -modules. Moreover, associating to g ∈ U the mapping induced by g − 1, we obtain an isomorphism of B/U -modules U → HomF (V /V0 , V0 ). Thus, the actions of B/U = F× on W/W1 , W1 /W0 , W0 , U are inverse square, trivial, square, square. Suppose the order of F is f . We show (i) by contradiction. Suppose HomB/U (U, W/W1 ) = 0. There exists a conjugate of the inverse square character of F× that is equal to the square character. Thus, there exists an integer 0 ≤ d < f satisfying −2d ≡ 2 mod (f −1). Since (f −1) | 2(d +1), we have f −1 ≤ 2(d +1), and thus (f −d −2)d ≤ 3. From this we have either f −d = 3 and d = 1, 3, or f −d = 5 and d = 1. Hence, F has to be one of 3, 5, 9, and (i) is proved. If F = 3, the square mapping is not trivial. (ii) is clear from

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this. Since F = F [F×2 ], we have HomB/U (U, W0 ) = HomF (U, W0 ), which shows (iii). Suppose F = 3, 5, 9, and we show H 1 (U, W )B/U that contains 1 H (SL2 (F), W ) itself is 0. By (i), (ii), (iii) above, HomF (U, W0 ) = H 1 (U, W0 )B/U → H 1 (U, W )B/U is surjective. Consider the long exact sequence H 0 (U, W ) −→ H 0 (U, W/W0 ) −→ H 1 (U, W0 ) −→ H 1 (U, W ). The ﬁrst morphism H 0 (U, W ) = W0 → H 0 (U, W/W0 ) is 0. Therefore, the second morphism induces an isomorphism H 0 (U, W/W0 ) = W1 /W0 → HomF (U, W0 ) = H 1 (U, W0 )B/U. Thus, the morphism H 1 (U, W0 )B/U → H 1 (U, W )B/U is 0. Hence, if F = 3, 5, 9, we have H 1 (SL2 (F), W ) ⊂ H 1 (U, W )B/U = 0. Suppose F = 9. Similarly to the case F = 3, 5, 9, we see that H 1 (U, W )B/U → H 1 (U, W/W1 )B/U = HomB/U (U, W/W1 ) is injective. Moreover, by the proof of (i), H 1 (U, W/W1 )B/U is an F-vector space of dimension 1. Suppose there exists an element of H 1 (U, W )B/U whose image in H 1 (U, W/W1 )B/U is nonzero. On the extension corresponding to this of the trivial representation element 0 (a ∈ F× ) and ( 10 u1 ) (u ∈ F) are given F by W , the actions of a0 a−1 by ⎞ ⎛ 2 ⎞ ⎛ 2 0 0 a 0 1 u u2 c(u) ⎜0 1 ⎜0 1 u d(u)⎟ 0 0⎟ ⎟ ⎜ ⎟ ⎜ ⎝ 0 0 a−2 0⎠ , ⎝0 0 1 u3 ⎠ 0 0 0 1 0 0 0 1 for some functions c, d : F → F. There exists no action satisfying such conditions, and thus H 1 (SL2 (F), W ) ⊂ H 1 (U, W )B/U = 0 in the case F = 9, also. Consider the case F = 3. It suﬃces to show H 1 (U, W ) = 0. Since U Z/3Z, take its generator σ and we have H 1 (U, W ) = Ker(1 + σ + σ 2 : W )/ Im(σ − 1 : W ). Since W is isomorphic to F[σ]/(σ − 1)3 as an F[σ]-module, we have H 1 (U, W ) = 0. Question. Fill in the details of the proof of Lemma 11.44. Proposition 11.45. Let be an odd prime number, and let G 2 be a ﬁnite subgroup of GL2 (F ) such that V = F is an irreducible × representation of G. Let δ : G → F be a character, and let W = × End0 (V ). Let Z = G ∩ F be the subgroup of all the scalar matrices

11.6. PROOF OF THEOREM 11.37

173

contained in G, and let G = G/Z be the image of G in P GL2 (F ) = × GL2 (F )/F . Then, either (i), or (ii) holds. (i) H 1 (G, W ⊗ δ) = 0. (ii) We have = 3 or 5. If = 3, then G is isomorphic to the alternating group A5 , and δ = 1. If = 5, then G is conjugate to either P SL2 (F5 ) or P GL2 (F5 ), and δ equals either 1 or the composition ( det 5 ) = ( 5 ) ◦ det : G → G → P GL2 (F5 ) → × 2 /(F ) → {±1}. F× 5 5 We deduce Proposition 11.45 from Theorem 10.28. We ﬁrst show that for the condition (ii) in Theorem 10.28, the following holds. Lemma 11.46. Let be a prime number, and let G be a ﬁnite subgroup of P GL2 (F ). If G is isomorphic to P GL2 (F ), it is conjugate to P GL2 (F ), and if G is isomorphic to P SL2 (F ), it is conjugate to P SL2 (F ). Proof. Consider the natural action of P GL2 (F ) on P1 (F ). Take an element g ∈ G of order . The action of g on P1 (F ) has only one ﬁxed point, and the orders of all other orbits are all equal to . Let X ⊂ P1 (F ) be a G-orbit containing the g ﬁxed point. The order d of X divides (G/)|(2 − 1), and d ≡ 1 mod . Since (2 − 1)/d ≡ −1 mod , we have (2 − 1)/d ≥ − 1, and thus d ≤ + 1. Hence, we have d = 1 or +1. If the order of X is 1, G is contained in a conjugate of the image of the group of upper triangular matrices, and thus, it has a normal subgroup of order , which is a contradiction. This shows the order of X is + 1. Choose the coordinates of P1 such that the ﬁxed point of g is the point at inﬁnity, another point diﬀerent from this is 0, and g(0) = 1. Then, X = P1 (F ). Since the action of G sends 0, 1, ∞ to points in P1 (F ), G is a subgroup of P GL2 (F ) of index less than 2. Since the abelianization of P GL2 (F ) is of order 2, the only subgroup of index 2 is P SL2 (F ), if = 2. Proof of Proposition 11.45. We have an exact sequence 0 → ¯ (W ⊗ δ)Z ) → H 1 (G, W ⊗ δ) → H 1 (Z, W ⊗ δ) by ProposiH 1 (G, tion 11.5. Since the order of Z is relatively prime to , we have H 1 (Z, W ⊗ δ) = 0 by Lemma 11.3. Thus, the natural mapping ¯ (W ⊗ δ)Z ) → H 1 (G, W ⊗ δ) is an isomorphism. Since the H 1 (G, action of Z on W is trivial, we have (W ⊗ δ)Z = 0 if δ|Z = 1. From now on, suppose δ|Z = 1, and we identify δ with a character of G. Furthermore, we identify H 1 (G, W ⊗ δ) with H 1 (G, W ⊗ δ). It suﬃces to show the assertion in each case where the conditions (i),

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(ii) or (iii) in Theorem 10.28 holds. Suppose ﬁrst condition (iii) holds. In this case, the order of G is relatively prime to by the assumption = 2. Thus, by Lemma 11.3, we have H 1 (G, W ⊗ δ) = 0. Next, suppose condition (i) holds. Suppose G is conjugate to P SL2 (F). We may assume G = P SL2 (F) by replacing a basis. Then, we have δ = 1. By Proposition 11.5, H 1 (P SL2 (F), W ) → ¯ W ⊗ δ) = H 1 (G, ¯ W) = 0 H 1 (SL2 (F), W ) is injective. Thus, H 1 (G, by Lemma 11.44 except for the case F = F5 . Suppose G is conjugate to P GL2 (F). If H ⊂ G is a subgroup conjugate to P SL2 (F), then ¯ W ⊗ δ) → H 1 (H, ¯ W ⊗ δ) is injective by Proposition 11.5 and H 1 (G, ¯ W ⊗ δ) = 0 except for F = F5 . Lemma 11.3 as above, and thus H 1 (G, × 2 det If F = F5 , the composition ( 5 ) : P GL2 (F5 )ab → F× 5 /(F5 ) → det {±1} is injective, and thus δ is either 1 or ( 5 ). Finally, suppose condition (ii) holds. If > 5, then the order of G is relatively prime to , and thus H 1 (G, W ⊗ δ) = 0 by Lemma 11.3. If = 3, the assertion is reduced to the case where condition (i) holds by Lemma 11.46, except for the case in which G is isomorphic to ab A5 . If G A5 , we have G = 1, and thus δ = 1. If = 5, except for the case G A5 , the order of G is relatively prime to , and thus H 1 (G, W ⊗ δ) = 0. If G is isomorphic to A5 , G is conjugate to P SL2 (F5 ) by Lemma 11.46. Corollary 11.47. Let the assumption be as in Proposition 11.45. If = 3, 5 or det G = 1, then we have H 1 (G, W ⊗ det) = 0. Proof. It suﬃces to show that condition (ii) in Proposition 11.45 does not hold assuming = 3, 5 and δ = det = 1. Since det = 1, if we assume condition (ii) holds, then we have = 5, det = −1 ( det 5 ) and det(G) = {±1}. However, we have −1 = ( 5 ) which is contradiction. In what follows let the notation be as in the previous section. Proof of Theorem 11.37. Since r = dim Sel∅ (W ), the equality r = dim SelQ (W ) is equivalent to the condition that the inclusion Sel∅ (W ) → SelQ (W ) is an isomorphism. Since W is self-dual, SelQ∨ (W (1))∨ is the Selmer group deﬁned by the dual local condition.

11.6. PROOF OF THEOREM 11.37

175

By Proposition 11.32, there is an exact sequence ( 1 H (Qp , W )/Hf1 (Qp , W ) 0 −→ Sel∅ (W ) −→ SelQ (W ) −→ p∈Q

−→ Sel∅∨ (W (1))∨ −→ SelQ∨ (W (1))∨ −→ 0. Thus, the equality r = dim SelQ (W ) is equivalent to the condition ( that p∈Q H 1 (Qp , W )/Hf1 (Qp , W ) → Sel∅∨ (W (1))∨ is injective. It is also equivalent to the condition that ) Sel∅∨ (W (1)) → H 1 (Fp , W (1)) p∈Q

is surjective. We now show the following lemma. Lemma 11.48. (1) dimF Sel∅∨ (W (1)) = dimF Sel∅ (W ) = r. (2) If a prime number q satisﬁes the condition (11.29)

q∈ / Sρ¯,

q ≡ 1 mod

and

Tr ρ¯(ϕq ) ≡ ±2,

1

then we have dimF H (Fq , W (1)) = 1. Proof. (1) Apply Proposition 11.33 to M = W . Since W is self-dual, it suﬃces to verify the following. (i) dimF H 1 (GFp , W Ip ) = dimF W GQp if p ∈ Sρ¯, = , / Sρ¯, (ii) dimF Hf1 (GQ , W ) − dimF W GQ = 1 if ∈ GQ 1 (iii) dimF Hs (GQ , W ) − dimF W = 1 if ∈ Sρ¯, (iv) dimF W GR = 1, (v) W GS = W (1)GS = 0. (i), (ii), and (iii) are Proposition 11.39(1), (2), and (3), respectively. (iv) follows from the fact that for a complex conjugate c, ρ¯(c) has eigenvalues 1 and −1 with multiplicity one. We show (v). Condition (ii) in Corollary 11.42(2) does not hold by Lemma 9.51. Thus, (v) follows from Corollary 11.42. (2) By the condition (10.22), ρ¯(ϕq ) has two distinct eigenvalues. The assertion follows immediately from this. ( 1 By Lemma 11.48, Sel∅∨ (W (1)) → p∈Q H (Fp , W (1)) is surjective if and only if it is injective. We show there exists a Q that makes ) ) H 1 (Fp , W (1)) = W (1)/(ϕp − 1)W (1) Sel∅∨ (W (1)) −→ p∈Q

p∈Q

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injective. By Theorem 3.1, it suﬃces to show there exist σ1 , . . . , σr ∈ GQ(ζn ) such that each ρ¯(σi ) has two distinct eigenvalues, and the direct sum of the restriction mapping r ) W (1)/(σi − 1)W (1) Sel∅∨ (W (1)) → i=1

is injective. For any nonzero element of Sel∅∨ (W (1)), let c : GQ → W (1) be a 1-cocycle that represents it. Then, it suﬃces to show there exists σ ∈ GQ(ζn ) such that ρ¯(σ) has two distinct eigenvalues, and that c(σ) ∈ W (1) is not contained in (σ − 1)W (1). Let GFn = Ker(¯ ρ : GQ(ζn ) → GL2 (F)). We show the restriction mapping (11.30)

H 1 (GQ , W (1)) → H 1 (GFn , W (1)) = Hom(GFn , W (1))

is injective. The kernel equals H 1 (Gal(Fn /Q), W (1)) by Proposition 11.5. Thus, it suﬃces to show H 1 (Gal(Fn /Q), W (1)) = 0. Moreover, by Proposition 11.5, we obtain an exact sequence (11.31) 0 −→ H 1 (Gal(F0 /Q), W (1)Gal(Fn /F0 ) ) −→ H 1 (Gal(Fn /Q), W (1)) −→ H 1 (Gal(Fn /F0 ), W (1))Gal(F0 /Q) . Since det ρ¯ is the mod cyclotomic character, we have Q(ζ ) ⊂ F0 . Thus, the action of Gal(Fn /F0 ) on W (1) is trivial. Therefore, it suﬃces to show the following: H 1 (Gal(F0 /Q), W (1)) = 0, H 1 (Gal(Fn /F0 ), W (1))Gal(F0 /Q) = HomGal(F0 /Q) (Gal(Fn /F0 ), W (1)) = 0. The determinant det ρ¯ is the mod cyclotomic character, and is nontrivial. Since ρ¯ is an absolutely irreducible faithful representation of Gal(F0 /Q), by Corollary 11.47, we have H 1 (Gal(F0 /Q), W (1)) = 0. We show HomGal(F0 /Q) (Gal(Fn /F0 ), W (1)) = 0. Since Fn = F0 · Q(ζn ), Gal(Fn /F0 ) → Gal(Q(ζn )/Q) is injective, and the conjugate action of Gal(F0 /Q) on Gal(Fn /F0 ) is trivial. Therefore, if f : Gal(Fn /F0 ) → W (1) is a morphism of Gal(F0 /Q)-modules, the image of f is contained in the invariant part W (1)Gal(F0 /Q) . Hence, we have HomGal(F0 /Q) (Gal(Fn /F0 ), W (1)) = 0. This completes the proof of H 1 (Gal(Fn /Q), W (1)) = 0, and H 1 (GQ , W (1)) → Hom(GFn , W (1)) is injective.

11.6. PROOF OF THEOREM 11.37

177

Take any nonzero element of Sel∅∨ (W (1)), and let c : GQ → W (1) be a 1-cocycle that represents it. The restriction of c to GFn deﬁnes a homomorphism c|GF n : GFn → W (1). Since H 1 (GQ , W (1)) → Hom(GFn , W (1)) is injective, c(GFn ) ⊂ W (1) is not 0. This is a subspace of W (1) stable under the action of GQ(ζn ) . The restriction ρ¯|GQ(ζn ) is absolutely irreducible by Corollary 9.52. Thus, by Corollary 11.43, there exists σ ∈ GQ(ζn ) such that ρ¯(σ) has two distinct eigenvalues, and satisﬁes c(GFn )+(σ −1)W = W . If c(σ) ∈ / (σ −1)W , this σ ∈ GQ(ζn ) satisﬁes the condition. If c(σ) ∈ (σ − 1)W , take τ ∈ GFn such that c(τ ) ∈ / (σ − 1)W . Since ρ¯(σ) = ρ¯(στ ) and c(στ ) = c(σ)+σc(τ ) ≡ c(τ ) ≡ 0 mod (σ −1)W , στ ∈ GQ(ζn ) satisﬁes the condition.

10.1090/mmono/245/05

APPENDIX B

Curves over discrete valuation rings B.1. Curves Definition B.1. (1) Let k be a ﬁeld. A separated scheme X of ﬁnite type over k such that each connected component is one dimensional is called a curve over k. If X is a proper smooth curve over k whose geometric ﬁber is connected, then g = dimk H 1 (X, O) is called the genus of X. (2) A ﬂat scheme X of ﬁnite presentation over a scheme S such that the geometric ﬁber Xs¯ for each geometric point s¯ → S is a curve over κ(¯ s) is called a curve over S. If X is a proper smooth curve over S such that each geometric ﬁber is connected and of genus g, we say that the genus of X is g. Lemma B.2. Let S be a scheme, and let X be a curve over S. (1) Suppose X is smooth over S. Then, for a closed subscheme D of X ﬁnite of ﬁnite presentation over S, the following conditions (i) and (ii) are equivalent. (i) D is ﬂat over S. (ii) D is a Cartier divisor of X. (2) If a closed subscheme D of X is a Cartier divisor of S and D is ´etale over S, then X is smooth over S on a neighborhood of D. Proof. (1) (i) ⇒ (ii). Since the assertion is local on S, we may assume that D is of degree N ≥ 1 over S. We prove by induction on N . First, we show the case N = 1. Since the assertion is local on X, we may assume S = Spec A, and there is an ´etale morphism X → A1S = Spec A[T ]. Thus, we may assume X = A1S . But in this case, the assertion is clear. We show the case in which N is general. Since D is a ﬂat covering of S and the assertion is ﬂat local on S, we may assume that D has a section P : S → D. Since the case N = 1 is already shown, P 179

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deﬁnes a Cartier divisor of X. Thus, there exists a closed subscheme D of D such that ID ⊂ OX and IP ⊂ OX satisfy ID = ID IP . Since we have an exact sequence 0 → OD → OD → OS → 0 of OD -modules locally on S, D is ﬂat over S of degree N − 1. It follows from the induction hypothesis that D is a Cartier divisor, and thus so is D = D + P . (ii) ⇒ (i). The deﬁning ideal ID of D is an invertible OX -module. Since Ds is a ﬁnite subscheme of Xs for any point s in S, the ideal ID,x OXs ,x ⊂ OXs ,x is generated by nonzero divisors for any point x in the smooth curve Xs . In other words, tensoring ⊗κ(s) to the exact sequence 0 → ID → OX → OD → 0 for each s ∈ S, we obtain an exact sequence 0 → IDs → OXs → ODs → 0. Thus, we have S TorO 1 (OD , κ(s)) = 0, and OD is a ﬂat OS -module. (2) By Proposition A.4(1), we may assume S = Spec k with k an algebraically closed ﬁeld. If x ∈ D, then the local ring OX,x is regular, and the assertion follows from Proposition A.4(2). We deﬁne an ordinary double point of a curve over a ﬁeld. Definition B.3. Let X be a curve over a ﬁeld k, and let x be a closed point in X. We call x a node of X if there exist ´etale morphisms u : U → X, f : U → Spec k[S, T ]/(ST ) and a point v ∈ U satisfying u(v) = x and f (v) = (S, T ). Lemma B.4. Let X be a curve over a ﬁeld k, and let x be a closed point of X. Then, the following conditions (i)–(iii) are equivalent. (i) x is a node. (ii) The residue ﬁeld κ(x) is a ﬁnite separable extension of k. X is reduced on a neighborhood of x, and the normalization X is smooth over k on a neighborhood of the inverse image of x. The length of the OX,x -module (OX /OX )x is 1, and X ×X x is ﬁnite ´etale over x of degree 2. X¯ ,¯x of the (iii) If k¯ is an algebraic closure of k, the completion O k ¯ ¯ local ring at each point x ¯ ∈ x ×k k ⊂ Xk¯ = X ×k k of the inverse ¯ ¯ image of x is isomorphic to k[[S, T ]]/(ST ) over k. Proof. (i) ⇒ (ii), (iii). The assertion is ´etale local. Thus, we may assume X = Spec[S, T ]/(ST ) and x = (S, T ). But in this case, the assertion is clear. (ii) ⇒ (i). Let X → X be the normalization of X. Replacing k by some ﬁnite separable extension if necessary, we may assume x and the points of its inverse image in X are k-rational points. Replacing

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X by a neighborhood of x, we may assume X = Spec A is reduced and aﬃne, and its normalization X = Spec B is smooth over k. Let x1 , x2 be the inverse images of x. We ﬁrst show the casewhere the normalization X is decomposed into a disjoint union V1 V2 = Spec B1 × B2 (xi ∈ Vi ). In this case A is the inverse image of the diagonal subring k ⊂ k × k by the surjection B1 × B2 → k × k deﬁned by x1 and x2 . Thus, if we let S ∈ B1 and T ∈ B2 be prime elements at x1 and x2 , respectively, we obtain a ring homomorphism k[S, T ]/(ST ) → A. The morphism X → Spec k[S, T ]/(ST ) deﬁned by it is ´etale on a neighborhood of x. We deduce the general case from the previous case. Let m be the maximal ideal of A corresponding to x. Then, we have mB = m, and B/mB is isomorphic to k × k. Take an element b in B such that its image in k × k is (1, 0), and let a = b2 − b ∈ mB = m ⊂ A. Furthermore, let g(Y ) = Y 2 − Y − a ∈ A[Y ], let A˜ = A[Y ]/g(Y ), and ˜ Since u : U → X is ﬂat, and the ﬁber at x is ´etale, we let U = Spec A. may assume U → X is ´etale by replacing X by a neighborhood of x if necessary. By Proposition A.13(3), U ×X X is a normalization of U . Since the ´etale covering U ×X X → X of degree 2 has a section deﬁned by thehomomorphism A˜ = A[Y ]/g(Y ) → B : Y → b, it is isomorphic to X X. Thus, the general case is reduced to the case where the normalization X is decomposed into the disjoint union V1 V2 . (iii) ⇒ (ii). We prove it assuming k is a perfect ﬁeld. In this ¯ we may assume k is algebraically closed. Since case, replacing k by k, X,x , it is reduced. the local ring OX,x is a subring of the completion O Thus, replacing X by a neighborhood of x, we may assume X is reduced. The normalization X is smooth over k. If we identify the completion of OX,x with k[[S, T ]]/(ST ), the completion of OX at the inverse image of x in X is k[[S]]×k[[T ]]. Thus, the remaining assertion follows easily. We deﬁne the dual chain complex for a proper curve over a perfect ﬁeld. Definition B.5. Let k be a perfect ﬁeld, and let X be a proper curve over k. Let k¯ be an algebraic closure of k. First, suppose X is reduced. Let X be the normalization of X. We call P = Spec Γ(X, O) the ﬁnite scheme consisting of irreducible components of X. Let Σ be the reduced closed subscheme of X consisting of all the singular points, and Σ = X ×X Σ. Let Γ0 = ¯ ZP (k) , and let Γ1 be the kernel of the surjective homomorphism

182

B. CURVES OVER DISCRETE VALUATION RINGS ¯

¯

ZΣ(k) → ZΣ(k) deﬁned by the natural morphism Σ → Σ. Let d : Γ1 → Γ0 be the homomorphism deﬁned by the natural morphism Σ → X. Then, we call the chain complex Γ = [Γ1 → Γ2 ] of length 1 the dual chain complex of X. For a general X, we deﬁne the dual chain complex of X as the dual chain complex of the reduced part of X. The condition that H0 (Γ) = Z is equivalent to the condition that the geometric ﬁber Xk¯ is connected. B.2. Semistable curve over a discrete valuation ring In what follows, let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. Definition B.6. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. (1) Let X be a ﬂat curve over O. If X is smooth over O except for a ﬁnite number of nodes in XF , then we say that X is weakly semistable. If X is regular and weakly semistable, we say X is semistable. (2) Let X be a weakly semistable curve over O, and let x ∈ XF ⊂ X be a node. We call the length of the OX,x -module Ω2X/O,x , the index of x. Definition B.7. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let XK be a proper smooth curve over K. (1) If there exist a proper smooth curve XO over O and an isomorphism XK → XO ⊗O K over K, we say that XK has good reduction. (2) If there exist a proper weakly semistable curve XO and an isomorphism XK → XO ⊗O K over K, we say that XK has semistable reduction. Let XQ be a proper smooth curve over Q. We say that XQ has good reduction at a prime p, or semistable reduction at p if, letting O = Z(p) , XQ has good reduction, or semistable reduction, respectively. Lemma B.8. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. A regular curve X over O is semistable if all of the following conditions hold: The ﬁber XK → X at the generic point is smooth. The closed ﬁber XF is reduced, and as

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183

a Cartier divisor, it is the sum XF = C1 + C2 of C1 , C2 ⊂ X, where C1 , C2 are smooth curve over F and the intersection C1 ∩ C2 = C1 ×X C2 is ´etale over F . Proof. Let π be a prime element of O. By Proposition A.4, X C1 ∩ C2 is smooth over O. We show that x ∈ C1 ∩ C2 is a node of XF . Since this assertion is ´etale local on X, we may assume the residue ﬁeld of x is F . Replacing X by a neighborhood of x, we suppose C1 is deﬁned by an element s. Then t = π/s deﬁnes C2 . Deﬁne a morphism of O-schemes X → Spec O[S, T ]/(ST − π) by S → s and T → t. We show this is ´etale at x. It suﬃces to show that the homomorphism of the completion X,x is an isomorphism. Since A0 = O[[S, T ]]/(ST − π) → A = O 2 2 mA0 /mA0 → mA /mA is an isomorphism, A0 → A is surjective. Since both A0 and A are two-dimensional regular local rings, A0 → A must be an isomorphism. Lemma B.9. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, let F be its residue ﬁeld, and let π be a prime element. Let X be a weakly semistable curve over O, let x be a node of XF , and let e be its index. If O is complete and F is algebraically closed, then the completion of the local ring OX,x is isomorphic over O to O[[S, T ]]/(ST − π e ). Proof. Let A be the completion of OX,x . We ﬁrst show there exist an integer m ≥ 1 and an isomorphism O[[S, T ]]/(ST − π m ) → A. Through the isomorphism F [[S, T ]]/(ST ) → A/(π), we identify F [[S, T ]]/(ST ) = A/(π). One of the following (i) and (ii) holds. (i) There exist liftings s, t ∈ A of the images of S and T , an integer m ≥ 1, and u ∈ A× such that st = uπ m . (ii) If liftings s, t ∈ A of the images of S and T , an integer m ≥ 1, and v ∈ A satisfy st = vπ m , then v is contained in the maximal ideal (π, s, t) of A. Suppose (i) holds. Let A0 = O[[S, T ]]/(ST − π m ), and deﬁne a morphism of O-algebras A0 → A by S → s = su−1 and T → t. Since the morphism of F -algebras A0 /(π) → A/(π) is an isomorphism, and A and A0 are both O-ﬂat, A0 /(π n ) → A/(π n ) is an isomorphism by induction on n. Taking the limit, A0 → A is an isomorphism. Since Ω2X/O,x is isomorphic to limn Ω2(A0 /(πn ))/(O/(πn )) A0 /(S, T ) = ←− O/(π m ), we have e = m.

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Next, we show that (ii) cannot hold. We construct a sequence (sm , tm )m=1,2,... of liftings of the images of S and T to A such that sm+1 ≡ sm mod π m , tm+1 ≡ tm mod π m , sm tm ∈ (π m ) inductively. We take s1 and t1 arbitrarily. Suppose we already have up to sm , tm . We can write sm tm = π m vm , vm = asm + btm + cπ (a, b, c ∈ A). If we deﬁne sm+1 = sm − π m b and tm+1 = tm − π m a, then we have sm+1 tm+1 = π m+1 (c + abπ m−1 ), and the required conditions hold. If we deﬁne s = limm→∞ sm and t = limm→∞ tm , then s and t are the liftings of the images of S and T , respectively, and st = 0. If we deﬁne a morphism of O-algebras O[[S, T ]]/(ST ) → A by S → s and T → t, this is an isomorphism as before. However, since XK is smooth, AK must be regular, which is a contradiction. Proposition B.10. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. Let X and Y be normal curves over O, and let f : X → Y be ﬁnite surjective morphism over O. (1) If X is smooth over O, then X → Y is ﬂat, and Y is also smooth over O. (2) Suppose X is semistable over O, and let C1 , C2 be Cartier divisors of X smooth over F satisfying the condition (B.1)

C1 + C2 = XF , C1 ∩ C2 = {nodes of XF }, and C1 = f −1 (f (C1 )), C2 = f −1 (f (C2 )). Then, Y is weakly semistable over O. Let D1 = f (C1 ) and D2 = f (C2 ) be reduced closed subschemes of Y . Then, D1 and D2 are smooth curves over F , and we have D1 ∪ D2 = YF and D1 ∩ D2 = {nodes of YF }. Furthermore, if x is a node of XF , then y = f (x) is a node of YF . Let ey be the index of y, let Fx and Fy be the residue ﬁelds of x and y, let A and B be the completions of the local rings OY,y and OX,x , and let Lx and Ly be the ﬁelds of fractions of A and B. Then, we have

(B.2)

[Fx : Fy ] · ey = [Lx : Ly ].

Proof. (1) Let O ⊃ O be a complete discrete valuation ring which has the same prime element, and whose residue ﬁeld F is an algebraic closure of F . We show that YO = Y ×O O is normal. The morphism X → Y is ﬂat except for a ﬁnite number of closed points of the closed ﬁber of Y . Thus, by Corollary A.14, Y is smooth except

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185

for a ﬁnite number of closed points of the closed ﬁber. Hence, YO is also smooth except for a ﬁnite number of closed points of the closed ﬁber. Since the closed ﬁber YF is reduced, so is the closed ﬁber YF . Thus, by Lemma A.41, YO is normal. Replacing O by O , we may assume O is complete and F is algebraically closed. We show X → Y is ﬂat. Let x ∈ XF be a closed point, and let y = f (x). Let A and B be the completions of the local rings OY,y and OX,x , respectively. It suﬃces to show B is ﬂat over A. Let Ly and Lx be the ﬁelds of fractions of A and B, respectively, and let d = [Lx : Ly ] be the degree of the extension. Then, A → B is ﬁnite ﬂat of degree d except at the maximal ideal. Choose an isomorphism O[[t]] → B and identify as O[[t]] = B. Deﬁne a morphism of O-algebras A0 = O[[t ]] → A by letting t = NB/A t. Since the valuation of t in B/(π) = F [[t]] is d, B/(π, t ) = B ⊗A0 F equals F [[t]]/(td ). Thus, X → Y is ﬂat by Lemma A.43. Since X is smooth over O, it is regular by Proposition A.13(3). Thus, Y is also smooth by Corollary A.14. (2) As in (1), we may assume O is complete and F is algebraically Y,y → B = closed. Let x be a node of XF , and let y = f (x). Let A = O X,x . Let Ly and Lx be ﬁelds of fractions of A and B, respectively, O and let d = [Lx : Ly ]. Choose an isomorphism O[[s, t]]/(st − π) → B, 1 and identify O[[s, t]]/(st − π) = B. Suppose the inverse image C of C1 by Spec B → X is V (s), and the inverse image C2 of C2 is V (t). Let s = NB/A s, and let t = NB/A t. Since s and t belong to the maximal ideal of A and satisfy s t = NB/A π = π d , we obtain a homomorphism A0 = O[[s , t ]]/(s t − π d ) → A. We show this is an isomorphism. 1 and we have f −1 (f (C1 )) = C1 by assumpSince s equals 0 on C 1 . Thus, there exists u ∈ B × satistion, s is invertible on Spec B C d fying s = us . Similarly, there exists v ∈ B × satisfying t = vtd , and thus B ⊗A0 A0 /mA0 = B/(sd , td , π) = F [[s, t]]/(sd , td , st) is a ﬁnitedimensional F = A0 /mA0 -vector space. Since B and A0 are complete, B is ﬁnitely generated as an A0 -module by Nakayama’s lemma. Since A0 is a two-dimensional integrally closed domain, A0 → B is injective. Thus, the only inverse image of (t ) by Spec B → Spec A0 is (t), and the degree of the extension F ((s))/F ((s )) of residue ﬁelds is d since s = usd . If L0 is the ﬁeld of fractions of A0 , then Lx is an extension of L0 of degree d. Therefore, we have Ly = L0 , and A = A0 . This

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B. CURVES OVER DISCRETE VALUATION RINGS

shows Y is weakly semistable over O, and the remaining assertions have also been shown. Corollary B.11. Let O be a discrete valuation ring, and let X be a curve over O. Suppose an action of a ﬁnite group G on X over O is given, and suppose the quotient Y = X/G exists. Furthermore, if x is the generic point of an irreducible component of XF and ηx is the generic point of the irreducible component of X containing x, then suppose the inertia groups Ix and Iηx are the same. (1) If X is smooth over O, then so is Y , and we have YF = XF /G. (2) Suppose X is semistable over O, and let C1 and C2 be smooth Cartier divisors of X over F stable under the G-action satisfying condition (B.1). Then Y is also weakly semistable over O, and if D1 = f (C1 ) and D2 = f (C2 ) are reduced closed subschemes, then we have D1 = C1 /G and D2 = C2 /G. Moreover, if x is a node of XF and ηx is the generic point of an irreducible component of x, then the index ey of the node y = f (x) equals the index [Ix : Iηx ]. Proof. (1) By Proposition B.10(1), Y is smooth over O. We show XF /G → YF is an isomorphism. Since both curves are normal over F , it suﬃces to show that the residue ﬁelds at the generic points of irreducible components are isomorphic. Let x be the generic point of an irreducible component of XF , and let y = f (x) ∈ YF be its image. Let Lx be the ﬁeld of fractions of the completion of OX,x , and let Ly be the ﬁeld of fractions of the completion of OY,y . Then, Lx is a Galois extension of Ly , and its inertia group is Ix /Iηx . Thus, by assumption, Lx is an unramiﬁed extension of Ly . Hence, the residue ﬁeld κ(x ) of the image x in XF /G of x is equal to κ(y), and XF /G → YF is an isomorphism. (2) By Proposition B.10(2), Y is weakly semistable over O, and by Corollary B.11(1), the closed immersions C1 → Y and C2 → Y induce closed immersions C1 /G → Y and C2 /G → Y except at nodes. Moreover, by Proposition B.10(2), they deﬁne closed immersions at nodes. Let x be a node, and let y = f (x) be its image. Let Lx be the ﬁeld of fractions of the completion of OX,x and let Ly be the ﬁeld of fractions of the completion of OY,y . Then Lx is a Galois extension of Ly , and its inertia group is Ix /Iηx . Thus, by Proposition B.10(2), we have ex = [Ix : Iηx ].

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B.3. Dual chain complex of curves over a discrete valuation ring Let O be a discrete valuation ring, and let F be its residue ﬁeld. Let X be a proper curve over O, and let Γ = [Γ1 → Γ0 ] be the dual chain complex of XF . If X is regular, we deﬁne a symmetric bilinear form ( , )0 : Γ0 × Γ0 → Z, and if X is semistable, we deﬁne a symmetric bilinear form ( , )1 : Γ1 × Γ1 → Z. Let X be a proper ﬂat regular curve over O. For irreducible components C1 and C2 of XF , we deﬁne the intersection product (C1 , C2 )X by (C1 , C2 )X = deg OX (C1 )|C2 . If C1 = C2 , it is equal to dimF Γ(X, OC1 ⊗OX OC2 ) and satisﬁes (C1 , C2 )X = (C2 , C1 )X . Let Z1 (XF ) be the free Z-module generated by the irreducible components of XF . Then, the intersection pairing deﬁnes a symmetric bilinear form ( , )X : Z 1 (XF ) × Z 1 (XF ) → Z. Since the divisor XF is a principal divisor, (XF , ) is the 0-mapping. Lemma B.12. If XF is connected, then the kernel of the symmetric bilinear form ( , )X is generated over Q by XF . Let F be a perfect ﬁeld, and let Γ = [Γ → Γ0 ] be the dual chain complex of XF . Let O be the completion of the maximal unramiﬁed extension of the completion of O. O is a complete discrete valuation ring, and its residue ﬁeld is an algebraic closure F of F . Then, XO = X ×O O is a proper regular curve over O , and we have XO ×O F = XF and Γ0 = Z1 (XF ). Deﬁne a symmetric bilinear form (B.3)

( , )0 : Γ0 × Γ0 −→ Z

by the intersection pairing of XO . If XF is connected, then, by Lemma B.12, the kernel of the bilinear form ( , )0 is generated by XF over Q. Deﬁne a linear mapping (B.4)

α0 : Γ0 −→ Γ∨ 0 = Hom(Γ0 , Z)

by α0 ([C])([C ]) = (C, C )0 , and deﬁne a linear mapping (B.5)

β : Γ∨ 0 −→ Z

∨ ∨ as the dual of the linear mapping β : Z → Γ0 deﬁned by β (1) = [XF¯ ] = C eC [C].

Corollary B.13. Let O be a discrete valuation ring, and suppose its residue ﬁeld F is perfect. Let X be a proper regular curve over

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O, and let Γ = [Γ1 → Γ0 ] be the dual chain complex of XF . Then, the composition β ◦ α0 : Γ0 → Γ∨ 0 → Z is the 0-mapping. Moreover, if the geometric ﬁber XF is connected, Ker β/ Im α0 has ﬁnite order. Proof. Since (XF , )0 is the 0-mapping, we have β ◦ α0 = 0. By Lemma B.12, if XF is connected, the kernel of the bilinear form ( , )0 is generated by XF over Q. Thus, Ker β/ Im α0 is a ﬁnite abelian group. Let O be a discrete valuation ring, and let XO be a proper weakly semistable curve over O. Let Γ = [Γ1 → Γ0 ] be the dual chain complex of the closed ﬁber XF . We use the notation of Deﬁnition B.5. For each x ∈ Σ(F ), let x1 , x2 be the inverse images in Σ(F ), and deﬁne fx = [x1 ] − [x2 ]. Then, fx , x ∈ Σ(F ) is a basis of the free Z-module Γ1 . Deﬁne a symmetric bilinear form (B.6)

( , )1 : Γ1 × Γ1 −→ Z

by deﬁning (fx , fx ) to be the index ex and letting (fx , fx ) = 0 if x = x . This does not depend on the numbering of x1 and x2 . Deﬁne the linear mapping (B.7)

α1 : Γ1 −→ Γ∨ 1 = Hom(Γ1 , Z)

∨ by α1 (fx )(fx ) = (fx , fx )1 . If Γ∨ = [Γ∨ 0 → Γ1 ] is the dual complex ∨ of Γ, then α : Γ1 → Γ1 induces ∨ (B.8) α ¯ 1 : H1 (Γ) = Ker(Γ1 → Γ0 ) −→ H 1 (Γ∨ ) = Coker(Γ∨ 0 → Γ1 ).

If X is weakly semistable, a minimal resolution of singularities X is constructed as follows. Let x be a node of XF , and let e be its index. Suppose e ≥ 2, and let X1 be the blow-up of X at x. If e = 2, the exceptional divisor E of X1 is a smooth conic over x, and X1 is semistable on a neighborhood of E. If e ≥ 3, the exceptional curve E is a singular conic and it is smooth over x except at its unique node x1 , and the residue ﬁeld of x1 equals that of x. X1 is weakly semistable, and it is semistable on a neighborhood of E except possibly at x1 , where the index equals e − 2. For each node x in X, repeat this procedure [ e2x ] times, and we obtain the minimal resolution of singularities X of X. Let Γ = [Γ1 → Γ0 ] be the dual complex of the closed ﬁber XF of the minimal resolution of singularities X . We deﬁne a natural morphism Γ → Γ of complexes. We use the notation in Deﬁnition B.5. For the minimal resolution of singularities X , let X F be the normal ization of XF , and deﬁne P = Spec Γ(X F , O), Σ = {nodes of XF },

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189

etc. Since the natural morphism X → X is an isomorphism outside the nodes of XF , it induces an open immersion P → P . Deﬁne Γ0 → Γ0 to be P → P . The natural morphism X → X induces Σ → Σ, and for each x ∈ Σ(F ), the number of elements of its inverse image in Σ(F ) is the index ex . For x ∈ Σ(F ), let x1 and x2 be its inverse images in Σ(F ). Let x1 , . . . , xex be the inverse image of x in Σ (F ), and we choose xi,1 , xi,2 to be the inverse images of

xi , i = 1, . . . , ex , in Σ (F ) such that x1 and x1,1 , x2 and xex ,2 , and xi,2 and x(i+1),1 for 1 ≤ i < ex are contained pairwise in the same connected component of the normalization X F . Then, we deﬁne homomorphism Γ1 → Γ1 by letting the image of ([x1 ] − [x2 ]) ∈ Γ1 be ex i=1 ([xi,1 ]−[xi,2 ]) ∈ Γ1 . The homomorphisms Γ0 → Γ0 and Γ1 → Γ1 deﬁne a morphism of chain complexes (B.9)

Γ −→ Γ .

The morphism Γ → Γ induces homomorphisms of homology groups H0 (Γ) → H0 (Γ ), H1 (Γ) → H1 (Γ ). The symmetric bilinear form ( , ) : Γ1 × Γ1 → Z induces a symmetric bilinear form ( , )1 : Γ1 × Γ1 → Z through the linear mapping Γ1 → Γ1 . Thus, α1 : Γ1 → Γ∨ 1 is obtained as the composition of α1 : Γ1 → Γ∨ 1 with Γ1 → Γ1 and its dual. Proposition B.14. Let O be a discrete valuation ring, and let X be a proper weakly semistable curve over O. Let Γ = [Γ1 → Γ0 ] be the dual chain complex of XF . (1) If X is the minimal resolution of singularities of X, then the homomorphisms H0 (Γ) → H0 (Γ ) and H1 (Γ) → H1 (Γ ) induced by the morphism of chain complexes Γ → Γ in (B.9) are isomorphisms. (2) Suppose X is semistable, and let α ¯ 1 : H1 (Γ) → H 1 (Γ∨ ) be the linear mapping (B.8). Then, we have a natural homomorphism (B.10)

Coker(α ¯ 1 : H1 (Γ) → H 1 (Γ∨ )) ∨ −→ Ker(β : Γ∨ 0 → Z)/ Im(α0 : Γ0 → Γ0 ).

Proof. (1) This is easily veriﬁed. (2) Under the notation in Deﬁnition B.5, Γ0 is a free Z-module generated by P (F ). Sending a basis P (F ) of Γ0 to its dual basis, we deﬁne an isomorphism γ0 : Γ0 → Γ∨ 0 . Since XF is reduced, the composition β = β ◦ γ0 : Γ0 → Z sends each element of P (F ) to 1. The

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composition β ◦ d : Γ1 → Γ0 → Z equals 0. Since XF is connected, β induces an isomorphism Coker d = H0 (Γ) → Z. Since X is semistable, α1 : Γ1 → Γ∨ 1 is an isomorphism. Deﬁne δ0 : Γ∨ 0 → Γ0 to be the composition α−1

d∨

∨ 1 Γ∨ 0 −→ Γ1 −→ Γ1 −→ Γ0 . d

∨ Identify Γ1 and Γ∨ 1 through the isomorphism α1 : Γ1 → Γ1 , and ∨ ∨ ∨ regard δ0 : Γ0 → Ker β as the composition of d : Γ0 → Γ∨ 1 and d : Γ1 → Ker β . Then, we obtain an exact sequence

(B.11)

d◦α−1

1 1 Ker d −→ Coker d∨ −→ Ker β / Im δ0 −→ Ker β / Im d.

α ¯

Since we have Ker d = H1 (Γ), Coker d∨ = H 1 (Γ∨ ), and Ker β / Im d = 0, (B.11) gives an isomorphism (B.12)

Coker(α ¯ 1 : H1 (Γ) → H 1 (Γ∨ )) −→ Ker(β : Γ0 → Z)/ Im(δ0 : Γ∗0 → Γ0 ).

By the isomorphism (B.12), it suﬃces to show that the diagram α

β

δ

β

0 → Γ∨ Γ0 −−−− 0 −−−−→ ⏐ ⏐ ⏐ ⏐ −γ0 γ0−1

Z & & &

0 Γ∨ 0 −−−−→ Γ0 −−−−→ Z

commutes. The right square commutes by the deﬁnition of β . We show that the left square commutes. Let D be an irreducible component of XF . It suﬃces to show that −δ0 ◦γ0 (D) = −d◦α1−1 ◦d∨ ◦γ0 (D) ◦ α0 (D) = D (D, D )0 · D . is equal to γ0−1 Let ΣD = D =D (D ∩ D ) ⊂ Σ(F¯ ) be the union of the intersections of D and the other irreducible components than D. For each x ∈ ΣD , number the inverse images x1 , x2 in Σ(F ) so that x1 ∈ D. Then, we have ([x1 ] − [x2 ]). α1−1 ◦ d∨ ◦ γ0 (D) = x∈ΣD

Thus, we have = −(ΣD )·D+ D =D (D, D )0 ·D . Since (XF¯ , D)0 = 0, we have (D, D)0 = −ΣD . This shows the left square is commutative. −d◦α1−1 ◦d∨ ◦γ0 (D)

10.1090/mmono/245/06

APPENDIX C

Finite commutative group scheme over Zp C.1. Finite ﬂat commutative group scheme over Fp First, we give a description of the category of ﬁnite ﬂat commutative group schemes. Theorem C.1. Let p be a prime, and let n ≥ 1 be an integer. Let G be a ﬁnite Z/pn Z-module scheme over Fp . (1) There is an equivalence of abelian categories D : (ﬁnite Z/pn Z-module schemes over Fp ) → (ﬁnite Z/pn Z[F, V ]/(F V − p)-module). (2) G is ´etale over Fp if and only if F : G → G is an isomorphism and if and only if F : D(G) → D(G) is an isomorphism. If GFp ¯ p ) ⊗ Zur G is ´etale, D(G) is the invariant subgroup (G(F p ) with respect to the diagonal action of the absolute Galois group ¯ p ) ⊗ Zur )GFp is the GFp , and the action of F on D(G) = (G(F p restriction of ϕp ⊗ 1 (3) If G∨ is the Cartier dual of G, D(G∨ ) = Hom(D(G), Qp /Zp ), and F = V ∨ , V = F ∨ . (4) Let A be an abelian variety over Fp of dimension g. Then, the Z/pn Z[F, V ]-module D(A[pn ]) is a free Z/pn Z-module of rank 2g. D(A) = limn D(A[pn ]) ⊗Zp Qp is a Qp -vector space ←− of dimension 2g. For an endomorphism f : A → A, we have deg f = det(f : D(A)). We omit the proof. D(G) is what is usually called the contravariant Dieudonn´e module D∨ (G∨ ) of the Cartier dual G∨ . The Frobenius endomorphism FG of G induces its transpose VG∨ on G∨ , and thus it acts on D(G) = D∨ (G∨ ) as D∨ (VG∨ ). In this book we agree on writing F instead of (FG )∗ = D∨ (VG∨ ). If R is a commutative ring and G has a structure of R-module, so does D(G). 191

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C. FINITE COMMUTATIVE GROUP SCHEME OVER Zp

If G = Z/pn Z, we have D(G) = Z/pn Z and F = 1, V = p. If G = μpn , we have D(G) = Z/pn Z and F = p, V = 1. Theorem C.1 is generalized to a general perfect ﬁeld k of characteristic p > 0. Let Wn (k) be the ring of Witt vectors of length n with k coeﬃcients, and let F : Wn (k) → Wn (k) be the Frobenius endomorphism. Deﬁne a noncommutative Wn (k)-algebra Wn (k)F, V generated by F and V by the relations F V = V F = p, F a = F (a)F , aV = V F (a) (a ∈ Wn (k)). With this notation we have the following. Theorem C.2. Let p be a prime, and let n ≥ 1 be an integer. If k is a perfect ﬁeld of characteristic p, we have an equivalence of abelian categories " #

n Wn (k)F, V -modules of ﬁnite ﬁnite Z/p Z-module D: → length as Wn (k)-module schemes over k . If k is a ﬁnite extension of k, the following diagram commutes: " #

D Wn (k)F, V -modules of ﬁnite ﬁnite Z/pn Z-module −−− −→ length as Wn (k)-module schemes over k ⏐ ⏐ ⏐ ⏐⊗ ⊗k k Wn (k) Wn (k ) " #

D Wn (k )F, V -modules of ﬁnite ﬁnite Z/pn Z-module −−− − → length as Wn (k )-module schemes over k . C.2. Finite ﬂat commutative group scheme over Zp Finite ﬂat commutative group schemes over Zp can be described by the following linear algebraic objects. Definition C.3. Let p be a prime, and let n ≥ 1 be an integer. (1) A Z/pn Z-module M is a ﬁltered ϕ-module if it is endowed with a submodule M and linear mappings ϕ : M → M and ϕ : M → M satisfying ϕ|M = pϕ . (2) A ﬁnite ﬁltered ϕ-module M is strongly divisible if M = ϕ(M )+ ϕ (M ). (3) Let M, N be ﬁltered ϕ-R-modules. An R-linear mapping f : M → N is a morphism of ﬁltered ϕ-R-modules if f (M ) ⊂ N , ϕ ◦ f = f ◦ ϕ and ϕ ◦ f |M = f ◦ ϕ . (4) A strongly divisible ﬁnite ﬁltered ϕ-module (M, M ) is ´etale if M = M . If M = 0, the strongly divisible ﬁltered ϕ-module (M, M ) is said to be multiplicative.

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193

Lemma C.4. A ﬁnite ﬁltered ϕ-module M is strongly divisible if and only if (C.1)

(p,−can)

(ϕ ,ϕ)

0 −→ M −−−−−→ M ⊕ M −−−−→ M −→ 0

is exact. Proof. The composition M → M ⊕M → M is the 0-mapping. By Deﬁnition C.3(2), M is strongly divisible if and only if M ⊕ M → M is surjective. The assertion is now clear by counting the number of elements. Corollary C.5. (1) The category of ﬁnite strongly divisible ﬁltered ϕ-modules is an abelian category. (2) If M is a ﬁnite strongly divisible ﬁltered ϕ-module, there exists a linear mapping F : M → M satisfying F ◦ ϕ = p and F ◦ ϕ = idM . (3) Let O be the ring of integers of a ﬁnite extension of Qp . Let n ≥ 1 be an integer, and let R = O/mnO . If (M, M ) is a ﬁnite strongly divisible ﬁltered ϕ-R-module, M is a direct summand of M as an R-module. (4) A strongly divisible ﬁltered ϕ-module (M, M ) is ´etale if and only if F : M → M is an isomorphism. Proof. (1) Clear from Lemma C.4. (2) We have M = Coker (p, −can) : M → M ⊕ M from the exact sequence (C.1). The assertion follows immediately from this. (3) Let F be the residue ﬁeld of O. The exact sequence (C.1) remains exact after tensoring ⊗O F . Thus, M → M is injective after tensoring ⊗O F . (4) If F is an isomorphism, we have ϕ = p ◦ F −1 , and thus M = ϕ(M ) + ϕ (M ) ⊂ pM + ϕ (M ). Thus, by Nakayama’s lemma, ϕ : M → M is surjective and we have M = M . The converse is clear. By Corollary C.5(2), an additive functor (ﬁnite strongly divisible ﬁltered ϕ-Z/pn Z-module) −→ (ﬁnite Z/pn Z[F, V ]/(F V − p)-module) is deﬁned by letting V = ϕ.

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C. FINITE COMMUTATIVE GROUP SCHEME OVER Zp

Theorem C.6. Let p be an odd prime, and let n ≥ 1 be an integer. (1) There is an equivalence of abelian categories (C.2) " # " # ﬁnite strongly divisible ﬁnite ﬂat Z/pn Z-module → D: ﬁltered ϕ-Z/pn Z-module . schemes over Zp The diagram " " # # ﬁnite strongly divisible D ﬁnite ﬂat Z/pn Z-module −−→ ﬁltered ϕ-Z/pn Z-module schemes over Zp ⏐ ⏐ ⏐ ⏐ ⊗Zp Fp " " # # n n D ﬁnite ﬂat Z/p Z-module ﬁnite Z/p Z[F, V ]/(F V − p)−−→ schemes over Fp module is commutative. (2) For a ﬁnite ﬂat Z/pn Z-module scheme G over Zp , G is ´etale if and only if the corresponding strongly divisible ﬁltered ϕ-module D(G) is ´etale. G is multiplicative if and only if D(G) is multiplicative. (3) Let A be an abelian scheme over Zp of relative dimension g, and let D(A) = limn D(A[pn ]) ⊗Zp Qp . Then, the subspace D(A) = ←− limn D(A[pn ]) ⊗Zp Qp ⊂ D(A) is a Qp -vector space of dimen←− sion g, and there exist natural isomorphisms of (End A) ⊗Q Qp -modules D(A) → Γ(AQp , Ω1AQ /Qp ) and D(A)/D(A) → p

Hom(Γ(AQp , Ω1AQ

), Qp ). p /Qp

We do not prove this theorem either. If G = Z/pn Z, we have D(G) = D(G) = Z/pn Z and ϕ = p, ϕ = 1. If G = μpn , we have D(G) = Z/pn Z, D(G) = 0 and ϕ = 1. If R is a commutative ring and G has a structure of R-module, so is D(G). Let V be a good p-adic representation of GQp . A ﬁltered Qp [F ]module D(V ) is deﬁned as follows. Let T be a GQp -stable Zp -lattice in V . For an integer n ≥ 1, the ﬁnite ﬂat commutative group scheme Gn deﬁned by the Galois representation T /pn T may be extended uniquely to a ﬁnite ﬂat commutative group scheme Gn,Zp over Zp . Thus, by Theorem C.6, strongly divisible ﬁltered modules D(Gn,Zp ) are deﬁned, and these form a inverse system. D(V ) = limn D(Gn,Zp )⊗Zp ←− Qp is a ﬁltered Qp -ϕ-module, and it does not depend on the choice of the Zp -lattice T . If we let F = ϕ−1 ◦ p, then D(V ) becomes a Qp [F ]-module. This coincides with limn D(Gn,Fp ) ⊗Zp Qp . Let A be ←−

C.2. FINITE FLAT COMMUTATIVE GROUP SCHEME OVER Zp

195

an abelian scheme over Zp , and deﬁne a p-adic representation Vp AQp by Vp AQp = Tp AQp ⊗Zp Qp , where Tp AQp is the Tate module. Then, we have D(Vp AQp ) = D(A). Similarly, for a good mod p-representation V of GQp , the strongly divisible ﬁltered Fp -ϕ-module D(V ) and also the Fp [F, V ]/(F V − p)-module D(V ) are deﬁned. We give a condition for a good two-dimensional representation of GQp to be ordinary. Proposition C.7. Let p be an odd prime, and let G be a ﬁnite ﬂat commutative group scheme over Zp . Let G = G/pG, and let D(G) be the corresponding strongly divisible ﬁltered ϕ-module, let n be the multiplicity of the eigenvalue 0 of ϕ : D(G) → D(G), and let h = dim D(G) . (1) The following conditions are equivalent. (i) There exists a multiplicative closed subgroup scheme H of G such that G/H is ´etale. (ii) n = h. (2) Suppose the conditions in (1) hold. Then, D(H) is the maximal submodule of D(G) such that the restriction of ϕ is an isomorphism, and D(G)/D(H) is the maximal quotient module such that ϕ induces the 0 homomorphism. Proof. (1). (i) ⇒ (ii) It suﬃces to show it assuming G = G and G is either ´etale or multiplicative. If G is ´etale, we have n = h = dim D(G) by Theorem C.6(2) and Corollary C.5(4). Similarly, if G is multiplicative, we have n = h = 0. (ii) ⇒ (i) The subring Zp [ϕ] ⊂ End D(G) is the direct product of the part ϕ is invertible and the part ϕ is nilpotent. Thus, D(G) also decomposes to the direct sum of the part D(G)o in which ϕ is invertible and the part ϕ is nilpotent. We show D(G)o ∩ D(G) = 0. Assume 0 = x ∈ D(G)o ∩ D(G) , and we deduce a contradiction. We may assume px = 0. Since 0 = x ∈ D(G)o , we have ϕ(x) = 0. But, since x ∈ D(G) and px = 0, we have ϕ(x) = pϕ (x) = ϕ (px) = 0, which is a contradiction. Hence, the natural mapping D(G)o ⊕ D(G) → D(G) is injective. ¯ = h, the in¯ = dim D(G) ¯ o + n and dim D(G) Since dim D(G) o ¯ ⊕ D(G) ¯ → D(G) ¯ is an isomorphism if n = h. Thus, jection D(G) by Nakayama’s lemma, D(G)o ⊕ D(G) → D(G) is surjective, and an isomorphism. From this we obtain an exact sequence of strongly

196

C. FINITE COMMUTATIVE GROUP SCHEME OVER Zp

divisible ﬁltered ϕ-modules 0 −→ (D(G)o , 0) −→ (D(G), D(G) ) −→ (D(G) , D(G) ) −→ 0. The assertion now follows from Theorem C.6(2). (2) Clear from the proof of (1) (ii) ⇒ (i).

Corollary C.8. Let p be an odd prime, let K be a ﬁnite extension of Qp , and let O be its ring of integers. Let V be a good representation of GQp on a two-dimensional K-vector space, and let D = D(V ) be the corresponding ﬁltered K[F ]-module. Suppose dim D(V ) = 1. Then, for V to be ordinary, it is necessary and suﬃcient that there exist p-adic units satisfying det(1 − F t : D) = (1 − αt)(1 − pβt). Suppose this condition holds, and let α, β denote the unramiﬁed characters of GQp such that the value of ϕ is α, β, respectively, and χ the p-adic cyclotomic character. Then, V is an extension of α by β · χ. Proof. It is clear that the condition is necessary. We show it is suﬃcient. Suppose det(1 − F t : D) = (1 − αt)(1 − pβt) and α, β are p-adic units. Let Do ⊂ D be the eigenspace belonging to the eigenvalue pβ of F , and T ⊂ V a GQp -stable O-lattice. D(T )o = D(T ) ∩ Do is a free O-module of rank 1, and the action of ϕ on D(T )o is the multiplication by 1/β, and its action on D(T )/D(T )o is the multiplication by p/α. For an integer m ≥ 1, let Gm be the ﬁnite ﬂat commutative group ¯ p ) = T /pm T . Then, n and h in scheme over Zp deﬁned by Gm (Q Proposition C.7 are both equal to [K : Qp ]. Thus, by Proposition C.7, there exists a subrepresentation T o of T such that D(T o ) = D(T )o and that both T /T o and T o (−1) are unramiﬁed. Since the action of ϕ on D(T /T o ) = D(T )/D(T )o is p/α, the action of F is α, and by Theorem C.1(2), the action of ϕp on T /T o is also α. Similarly, the action of ϕp on T o (−1) is multiplication by β. We give a description of an extension of strongly divisible ﬁltered ϕ-R-modules. Let R be a ﬁnite commutative algebra over Zp , and let M, N be strongly divisible ﬁltered ϕ-R-modules. As in §11.1, an exact sequence of strongly divisible ﬁltered ϕ-R-modules (E) : 0 → N → E → M → 0 is called an extension of M by N .

C.2. FINITE FLAT COMMUTATIVE GROUP SCHEME OVER Zp

197

If there is a commutative diagram (E) : 0 −−−−→ N −−−−→ & & &

E −−−−→ ⏐ ⏐

M −−−−→ 0 & & &

(E ) : 0 −−−−→ N −−−−→ E −−−−→ M −−−−→ 0, we say that the extensions (E) and (E ) are isomorphic. The set of isomorphism classes Extϕ R (M, N ) has a structure of R-module. By the equivalence of categories (C.2), for ﬁnite ﬂat R-module schemes G, H over Zp , the group ExtR (G, H) of isomorphism classes of extensions of G by H is naturally identiﬁed with Extϕ R (D(G), D(H)). For ﬁltered ϕ-R-modules M, N , deﬁne HomR (M, N ) = {(f, g) ∈ HomR (M, N ) × HomR (M , N ) | f |M = pg}, HomR (M, N ) = {f ∈ HomR (M, N ) | f (M ) ⊂ N }, and deﬁne a homomorphism δ : HomR (M, N ) → HomR (M, N ) by δ(f ) = (ϕ ◦ f − f ◦ ϕ, ϕ ◦ f |M − f ◦ ϕ ). We denote by Homϕ R (M, N ) the set of homomorphisms M → N of ﬁltered ϕ-R-modules. For (f, g) ∈ HomR (M, N ), deﬁne an extension E = Ef,g of M by N by E = M ⊕ N , E = M ⊕ N , ϕ(x, y) = (ϕ(x), ϕ(y) + f (x)), ϕ (x, y) = (ϕ (x), ϕ (y) + g(x)). A homomorphism HomR (M, N ) → Extϕ R (M, N ) is deﬁned by assigning to (f, g) the isomorphism class Ef,g . Proposition C.9. Let p be an odd prime. Let R be a Zp -algebra. Let G, H be ﬁnite ﬂat R-module schemes over Zp , and let M = D(G), N = D(H) be corresponding strongly divisible ﬁltered ϕ-R-modules. Then, there exists an exact sequence of R-modules (C.3) 0 → HomR (G, H) −→ HomR (M, N ) −→ HomR (M, N ) δ

−→ ExtR (G, H). Furthermore, if M is a free R-module, then there exists an exact sequence (C.4) 0 → HomR (G, H) −→ HomR (M, N ) −→ HomR (M, N ) δ

−→ ExtR (G, H) → 0. Proof. First we show the exact sequence (C.3). We identify ϕ HomR (G, H) = Homϕ R (M, N ) and ExtR (G, H) = ExtR (M, N ) by

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C. FINITE COMMUTATIVE GROUP SCHEME OVER Zp

Theorem C.6(1). It is clear from the deﬁnition that Homϕ R (M, N ) is the kernel of δ : HomR (M, N ) → HomR (M, N ). We show the exactness at HomR (M, N ). If for (f, g) ∈ HomR (M, N ), the class of Ef,g is 0, we have an isomorphism Ef,g → E0,0 . Its component h : M → N gives an element of HomR (M, N ) satisfying (f, g) = δ(h). Thus, the sequence is exact at HomR (M, N ). We show HomR (M, N ) → Extϕ R (M, N ) is surjective assuming M is a free R-module. Let E be an extension of M by N . Since M is the direct sum of M , if M is a free R-module, M is a projective R-module. Thus, there exists a direct sum decomposition E = M ⊕N of R-modules that is an extension of the direct sum decomposition E = M ⊕ N as R-modules. The surjectivity of HomR (M, N ) → Extϕ R (M, N ) follows from this and the deﬁnition. Corollary C.10. Let p be an odd prime, and let O be the ring of integers of a ﬁnite extension of Qp . Let n ≥ 1 be an integer, and let R = O/mnO . Let G, H be ﬁnite ﬂat R-module schemes over Zp , and suppose G(Qp ) is a free R-module. Let M and N be strongly divisible ﬁltered ϕ-R-module corresponding G and H, respectively. Then, we have ExtR (G, H) = HomR (M , N/N ). HomR (G, H) Proof. We show M is a free R-modules. By Nakayama’s lemma, for M to be a free R-module, it is necessary and suﬃcient that M = (M/mO M )n . Thus, for M to be free, it is necessary and suﬃcient that G(Qp ) is a free R-module. R (M,N ) By Proposition C.9, the left-hand side equals Hom HomR (M,N ) . By Corollary C.5(3), M is a direct summand of M . Thus, HomR (M, N ) and HomR (M, N ) are the kernels of the surjections HomR (M, N ) ⊕ HomR (M , N ) → HomR (M , N ) : (f, g) → f |M − pg, HomR (M, N ) ⊕ HomR (M , N ) → HomR (M , N ) : (f, g) → f |M − g. Therefore, the left-hand side is equal to the right-hand side.

HomR (M ,N ) HomR (M ,N ) ,

and equal to

10.1090/mmono/245/07

APPENDIX D

Jacobian of a curve and its N´ eron model The group of divisors of degree 0 of Riemann surfaces has the structure of a compact complex torus. For a curve over any ﬁeld, or more generally, over a scheme, the group of divisors of degree 0 has a natural algebraic geometric structure. We call it the Jacobian of a curve. In Chapter 9, we constructed a Galois representation associated with modular forms using this algebraic structure for the modular curves. For a prime number at which a curve has bad reduction, its Jacobian may not have good reduction. Even for such a prime number, we can still study its properties using the N´eron model. D.1. The divisor class group of a curve Let X be a proper normal connected curve over a ﬁeld k. Let K be the function ﬁeld of X. A formal linear combination of closed points with Z coeﬃcients is called the divisor of X. The free abelian group generated by the closed points of X ) Z · [x] (D.1) Div(X) = x:closed point of X

is called the divisor group. For a rational function f ∈ K × on X, the divisor of f , noted div f , is deﬁned by x ordx f · [x] ∈ Div(X). By associating to f ∈ K × the divisor div f ∈ Div(X), we obtain a homomorphism of abelian groups div : K × → Div(X). An element of the image of this homomorphism is called a principal divisor , and the cokernel (D.2)

Pic(X) = Div(X)/ div K ×

is called the divisor class group. The divisor class group Pic(X) has the following cohomological expression. Denote by K × the constant sheaf on X, and for a closed point x, let Zx be the extension to X of the constant sheaf on x. 199

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´ D. JACOBIAN OF A CURVE AND ITS NERON MODEL

Since the local ring at each closed point of X is a discrete valuation ring, we obtain an exact sequence of sheaves on X (D.3)

⊕ ord

x 0 −→ Gm −→ K × −−x−−−→

)

Zx −→ 0.

x:closed point of X

The long exact sequence induced by this yields an isomorphism (D.4)

Pic(X) −→ H 1 (X, Gm ).

In what follows we identify Pic(X) = H 1 (X, Gm ) through this isomorphism. For D = x nx [x] in X, we call deg D = x nx [κ(x) : k] the degree of the divisor D. Associating to the divisor D ∈ Div(X) to its degree deg D ∈ Z, we obtain a homomorphism of abelian groups deg : Div(X) → Z. Since the degree of a principal divisor is 0, this induces a homomorphism deg : Pic(X) → Z. Its kernel Ker(deg : Pic(X) → Z) is denoted by Pic0 (X), and is called the divisor class group of degree 0. Let f : X → Y be a ﬁnite ﬂat morphism of proper normal connected curves. Deﬁne f ∗ : Div(Y ) → Div(X) by f ∗ ([y]) = [X ×Y y] = e x→y x/y [x] for a closed point y in Y . Here, ex/y indicates the ramiﬁcation index at x. f ∗ : Div(Y ) → Div(X) induces f ∗ : Pic(Y ) → Pic(X) and f ∗ : Pic0 (Y ) → Pic0 (X). f∗ : Div(X) → Div(Y ) is deﬁned by f∗ ([x]) = fx/f (x) [f (x)] for a closed point x in X. Here, fx/f (x) indicates the degree of extension [κ(x) : κ(f (x))] of the residue ﬁeld. f∗ : Div(X) → Div(Y ) induces f∗ : Pic(X) → Pic(Y ) and f∗ : Pic0 (X) → Pic0 (Y ). The Jacobian of a curve X is deﬁned as the divisor class group Pic0 (X) of degree 0 equipped with a geometric structure. In the next section we deﬁne the Jacobian of X as the moduli space of the Picard functor. In this section we give an analytic expression of the Jacobian when k is the complex number ﬁeld. Let X be a smooth connected curve over C of genus g, and let X an be the Riemann surface associated with X. Consider the singular chain complex (Cq (X an , Z), dq )q∈Z of X an . The fact that H0 (X an , Z) = Z implies Div0 (X) = Ker(C0 (X an , Z) → H0 (X an , Z)) = Im(C1 (X an , Z) → C0 (X an , Z)),

D.2. THE JACOBIAN OF A CURVE

201

and we obtain a surjection C1 (X an , Z) → Div0 (X). Deﬁne a homomorphism C1 (X an , Z) −→ H 0 (X, Ω1X )∨ = Hom(H 0 (X, Ω1X ), C) 8 by associating to 1-chain γ the linear form ω → γ ω. (D.5) induces a homomorphism

(D.5)

(D.6)

Div0 (X) −→ H 0 (X, Ω1X )∨ / Im H1 (X an , Z).

The mapping induced by (D.5) (D.7)

H1 (X an , Z) −→ H 0 (X, Ω1X )∨

induces an isomorphism of R-vector spaces (D.8)

H1 (X an , Z) ⊗Z R −→ H 0 (X, Ω1X )∨ .

In other words, the free abelian group H1 (X an , Z) of rank 2g is a lattice in the C-vector space H 0 (X, Ω1X )∨ of dimension g. Thus, H 0 (X, Ω1X )∨ / Im H1 (X an , Z) is a complex torus of dimension g. By Abel’s theorem, (D.6) induces an isomorphism (D.9)

Pic0 (X) → H 0 (X, Ω1X )∨ / Im H1 (X an , Z).

In this way Pic0 (X) has a structure of compact complex torus of dimension g. √ Let Z(1) be the constant sheaf 2π −1Z on X an . The trace mapping H 2 (X an , Z(1)) → Z is an isomorphism. By the Poincar´e duality, H 1 (X an , Z(1)) is identiﬁed with H1 (X an , Z), the dual of H 1 (X an , Z). D.2. The Jacobian of a curve We deﬁne the Picard functor, and give an algebraic geometric structure to the divisor class group of degree 0 of a curve. Let X be a scheme. Denote by Pic(X) the set of isomorphism classes of invertible sheaves on X. Deﬁne the product of the classes of invertible sheaves L and L by [L] · [L ] = [L ⊗OX L ]. Then, Pic(X) is a commutative group, called the Picard group of X. If X is a normal connected curve over a ﬁeld k, Pic(X) coincides with the divisor class group of X. If L is an invertible sheaf on X, Isom OX (OX , L) deﬁnes a Gm torsor over X. Thus, we obtain a natural homomorphism (D.10)

Pic(X) −→ H 1 (X, Gm ).

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Conversely, if a Gm -torsor over X is given, we obtain an invertible sheaf by patching, and thus (D.10) is an isomorphism. In what follows we identify Pic(X) = H 1 (X, Gm ) via (D.10). Let S be a scheme, and let X be a scheme over S. Deﬁne a functor PX/S over S by associating to a scheme T over S the commutative group Pic(X ×S T ). The deﬁnition of the functor PX/S is too naive, and we cannot expect in general that such a functor is representable. So, we give the following deﬁnition. Definition D.1. Let S be a scheme, and let X be a scheme a of the functor PX/S over S deﬁned over S. The ﬂat sheaﬁﬁcation PX/S by (D.11)

PX/S (T ) = Pic(X ×S T )

is called the Picard functor and is denoted by PicX/S . If k is a ﬁeld and S = Spec k, then PicX/S is also written as PicX/k . s) = Pic(Xs¯) → For a geometric point s¯, the natural map PX/S (¯ PicX/S (¯ s) is an isomorphism. If X is a smooth conic over k, the degree mapping deﬁnes an isomorphism PicX/k (k) → Z, and the natural map Pic(X) → PicX/k (k) is injective. If X has a rational point, this mapping is an isomorphism; if not, its image is 2Z. Let S be a scheme, and let f : X → Y be a morphism of schemes over S. Then, the pullback of an invertible sheaf by f deﬁnes a morphism of functors f ∗ : PicY /S → PicX/S . If f : X → Y is ﬁnite ﬂat of ﬁnite presentation, the norm of an invertible sheaf deﬁnes a morphism of functors f∗ : PicX/S → PicY /S . If L is an invertible sheaf on X, the norm Nf L is deﬁned as an invertible sheaf on Y as follows. For a point y in Y , there exists an open neighborhood V and a basis V of an Of −1 (V ) -module L|f −1 (V ) . Nf L is an invertible sheaf that has a basis N (V ) over V , and for a change of bases = a we have N ( ) = NX/Y a · N (). Definition D.2. Let S be a scheme, and let X be a proper curve over S. (1) Let k be a ﬁeld, and let S = Spec k. Let X be a normalization of X, and let X1 , . . . Xn be its connected components. Deﬁne (D.12)

Pic0 (X) =

n 9 i=1

deg Ker Pic(X) → Pic(Xi ) → Z .

D.2. THE JACOBIAN OF A CURVE

203

A subfunctor Pic0X/S of PicX/S is deﬁned by 9 " inverse image of PicX/S (T ) → # 0 (D.13) PicX/S (T ) = PicX/S (t¯) = Pic(Xt¯) by Pic0 (Xt¯) t¯:geometric

point of T

for a scheme T over S. The following theorem is fundamental. Theorem D.3. Let S be a scheme, and let f : X → S be a proper smooth curve over S of genus g such that each geometric ﬁber is connected. (1) Pic0X/S is representable by an abelian scheme j : J → S over S of relative dimension g. (2) There is a natural isomorphism (D.14)

j∗ ΩJ/S → f∗ ΩX/S .

(3) Let n ≥ 1 be an integer. The Weil pairing deﬁnes a bilinear form J[n]×J[n] → μn and deﬁnes an isomorphism to the Cartier dual (D.15)

J[n] −→ J[n]∨ .

The moduli space J of the functor Pic0X/S is call the Jacobian of X. Example D.4. Let S be a scheme, and let E be an elliptic curve over S. The morphism of functors E → Pic0E/S , deﬁned by associating to a scheme T over S and a section P : T → E over T the invertible sheaf OET ([P ] − [O]), is an isomorphism. By this isomorphism, the Jacobian of E is identiﬁed with E itself. Corollary D.5. Let k be a ﬁeld, and let X be a proper smooth curve over k of genus g such that the geometric ﬁber Xk¯ is connected. (1) The Jacobian J = Pic0X/k is an abelian variety over k of dimension g. (2) There is a natural isomorphism (D.16)

Γ(J, ΩJ/k ) −→ Γ(X, ΩX/k ).

(3) Let n ≥ 1 be an integer. The Weil pairing deﬁnes a bilinear form J[n] × J[n] → μn and the isomorphism J[n] → J[n]∨ to its Cartier dual.

204

´ D. JACOBIAN OF A CURVE AND ITS NERON MODEL

The Jacobian of X is sometimes denoted by Jac(X). Let k = C. Identify J[n] = H1 (X an , Z/nZ) = H 1 (X an , Z/nZ(1)). The Weil pairing J[n] × J[n] → μn may be identiﬁed with the pairing H1 (X an , Z/nZ) × H1 (X an , Z/nZ) → Z/nZ(1) that is induced by the composition of the cup product H 1 (X an , Z(1)) × H 1 (X an , Z(1)) → H 2 (X an , Z(2)) and the trace mapping H 2 (X an , Z(2)) → Z(1). Let f : X → Y be a ﬁnite ﬂat morphism of proper smooth curves over k. The morphisms of functors f∗ : PicX/k → PicY /k , f ∗ : PicY /k → PicX/k induce morphisms of Jacobians f∗ : JX → JY , f ∗ : JY → JX . Lemma D.6. Let f : X → Y be a ﬁnite ﬂat morphism of proper smooth curves over k. (1) The kernel of f ∗ : JY → JX is ﬁnite over k. (2) Let X be a Galois covering of Y , and let G be its Galois group. If is a prime number invertible in k, f ∗ : V JY → V JX deﬁnes an isomorphism f ∗ : V JY → (V JX )G to the G-invariant part (V JX )G . Proof. (1) Since f∗ ◦ f ∗ : JY → JY is the multiplication-by[X : Y ] mapping, Ker f ∗ is ﬁnite. ∗ (2) ∗ It is clear from the fact that f ◦ f∗ : JX → JX equals g∈G g . Theorem D.7. Let k be a ﬁeld, and let X be a proper curve over k. (1) The functor Pic0X/k is represented by a smooth connected commutative group scheme J over k. (2) Suppose X is smooth. Let X = ni=1 Xi be the decomposition into connected components, and let ki = Γ(Xi , O) be the ﬁeld of deﬁnition of Xi for i = 1, . . . , n. Then, J is an abelian variety that is isomorphic to the product ni=1 Reski /k Ji of the Weil restrictions to k of the Jacobian Ji = Pic0Xi /ki of Xi over ki . (3) Suppose X is smooth except for a ﬁnite number of ordinary double points. Let X be its normalization. Then J is an extension of the Jacobian J of X by a torus. (4) Assume k is perfect. Let X be the normalization of X, and let Γ be the dual chain complex of X. Then, the morphism J → J induced by the natural map X → X gives an isomorphism from the abelian part J a of J (D.17)

J a −→ J.

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205

The character group of the torus part J t of J is naturally isomorphic to H1 (Γ). D.3. The N´ eron model of an abelian variety Theorem D.8. Let O be a discrete valuation ring, and let K be its ﬁeld of fractions. Let AK be an abelian variety over K. Then, there exists a smooth commutative group scheme A over O having the following property: for any smooth scheme X over O, the restriction mapping {morphism X → A of schemes over O} −→ {morphism XK → AK of schemes over K} is an isomorphism. The smooth commutative group scheme over O satisfying the condition in Theorem D.8 is unique up to natural isomorphisms. Definition D.9. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. Let AK be an abelian variety over K. The smooth commutative group scheme A over O that satisﬁes the property in Theorem D.8 is called the N´eron model of AK . The open subgroup scheme A0 of A, which is deﬁned by the conditions A0 ⊗O K = A ⊗O K and that A0 ⊗O F is the connected component A0F of A ⊗O F containing the identity element, is called the connected component of the N´eron model A. Definition D.10. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. Let AK be an abelian variety over K, and let A be the N´eron model of AK . (1) If A is an abelian scheme over O, AK is said to have good reduction. (2) If the connected component A0F of the closed ﬁber AF of A is an extension of an abelian variety by a torus, AK is said to have semistable reduction. Lemma D.11. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. Let AK be an abelian variety over K that has good reduction, let A be the N´eron model of AK , and let AF = A ⊗O F be its reduction. Let be a prime number diﬀerent from the characteristic of F .

206

´ D. JACOBIAN OF A CURVE AND ITS NERON MODEL

(1) The Tate module T AK is an unramiﬁed representation of GK , and the natural isomorphism T AK → T AF is compatible with the natural surjection GK → GF . (2) The natural isomorphism End AK → End AF is injective, and it is compatible with the natural isomorphism T AK → T AF . Proof. (1) It follows easily from Lemma A.47(1). (2) It follows easily from Proposition A.51(2).

Proposition D.12. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. Let AK be an abelian variety over K, and let G be a ﬁnite ﬂat commutative group scheme over O. Let A be the N´eron model of AK , and let GK → AK be a morphism of commutative group schemes over K. Then, the following hold. (1) If one of the conditions (i) or (ii) holds, there exists a morphism G → A of commutative group schemes over O that extends GK → AK . (i) G is ´etale over OK . (ii) If p is the characteristic of the residue ﬁeld F of K, then the valuation e = ordK p of p in K is less than p − 1. Moreover, the connected component AF = A ⊗O F is an extension of an abelian variety by a torus. (2) Suppose that GK → AK is a closed immersion and that either condition (ii) in (1) holds or e = p − 1 and the degree of G is relatively prime to p. Then, the morphism G → A of commutative group schemes over K extending GK → AK is also a closed immersion. The case (i) in (1) is clear from the deﬁnition of N´eron models. We omit the proof of the other cases. Corollary D.13. Let AK be an abelian variety over K, and let A be its N´eron model. Let I ⊂ GK be the inertia group. (1) Let N be an positive integer relatively prime to p. Then, there is a natural isomorphism of the ﬁnite abelian group AF [N ](F ) → AK [N ](K)I . (2) Let be a prime number diﬀerent from the characteristic of the residue ﬁeld. Then, there is a natural isomorphism of the ﬁnitedimensional Q -vector space V AF → (V AK )I compatible with the action of the natural morphism GK → GK /I = GF . Suppose F is a perfect ﬁeld, and let AaF be the abelian part of AF , and let

´ D.3. THE NERON MODEL OF AN ABELIAN VARIETY

207

AtF be the torus part of AF . Then, we obtain an exact sequence 0 → V AtF → V AF → V AaF → 0. (3) Let L be a ﬁnite Galois extension of K, let IL/K ⊂ Gal(L/K) be the inertia group, and let E be the residue ﬁeld of L. Suppose F is a perfect ﬁeld, and let AaE be the abelian part of the closed ﬁber AE of the N´eron model of AL , and let AtE be the torus part. Then, the natural morphisms V AaF → (V AaE )IL/K and V AtF → (V AtE )IL/K are isomorphisms. Proof. (1) Since the multiplication-by-N morphism [N ] : A → I A is ´etale, A[N ] is ´etale over OK . Therefore, if K ur = K is the maxur )→ imal unramiﬁed extension of K, the natural morphism A[N ](OK AF [N ](F ) is an isomorphism. By the deﬁnition of N´eron model, nr ¯ I is also an isomorphism. A[N ](OK ) → AK [N ](K nr ) = AK [N ](K) I (2) By (1), V AF → (V AK ) is an isomorphism. By Corollary A.50, we obtain the exact sequence 0 → V AtF → V AF → V AaF → 0. (3) By (2), the natural morphism AF ⊗F E → AE induces the isomorphism V AF → (V AE )IL/K . By taking the IL/K -invariant part of the exact sequence 0 → V AtE → V AE → V AaE → 0, we obtain 0 → V AtF → V AF → V AaF → 0 by Corollary A.50(2). Corollary D.14. Let K be a discrete valuation ﬁeld, and suppose its residue ﬁeld F is perfect. Let be a prime number diﬀerent from the characteristic of F . (1) Let AK → BK be a morphism of abelian varieties over K, and let A → B be the morphism induced on their N´eron models. Let AtF ⊂ A0F and BFt ⊂ BF0 be the torus parts of the closed ﬁbers. Suppose the kernel of AK → BK is ﬁnite. Then, V AK → V BK is injective. If we identify V AK and V BF as the subspaces of V BK , we have V AF = V AK ∩V BF and V AtF = V AK ∩V BFt . (2) Let XK → YK be a Galois covering of proper smooth curves over K, and let G be its Galois group. Let AK and BK be the Jacobians of XK and YK , respectively. Let AF and BF be the closed ﬁbers of the N´eron models of AK and BK , and let AaF , BFa , AtF , BFt be their abelian parts and torus parts. We denote by G the G-invariant part. Then, the natural mappings V AaF → (V BFa )G and V AtF → (V BFt )G are isomorphisms. Proof. (1) It is clear that V AK → V BK is injective. By Corollary D.13(2), V AF and V BF are invariant subspaces by the

208

´ D. JACOBIAN OF A CURVE AND ITS NERON MODEL

inertia group I. Thus, V AF = V AK ∩ V BF follows from (V AK )I = V AK ∩ (V BK )I . By Corollary A.50(2), V AaF = V AF /V AtF → V BFa = V BF /V BFt is injective. Thus, V AtF = V AK ∩ V BFt . (2) By Lemma D.6(2), V AK is identiﬁed with (V BK )G . Thus, by (1), we have V AF = (V BF )G and V AtF = (V BFt )G . Moreover, taking the G-invariant part of the exact sequence 0 → V BFt → V BF → V BFa → 0, we obtain the isomorphism V AaF → (V BFa )G . Whether an abelian variety AK over a discrete valuation ﬁeld K has good reduction or semistable reduction can be determined by the -adic representation V A of GK . Definition D.15. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let p be the characteristic of F . Let be a prime number that is invertible in K, and let V be an -adic representation of the absolute Galois group GK . (1) A projective system G = (Gn )n∈N of surjections of ﬁnite ﬂat commutative group schemes over O is called -divisible group if the following conditions are satisﬁed: n : Gn → Gn is the 0 morphism. [] : Gn → Gn is decomposed into the surjection Gn → Gn−1 and the closed immersion in−1 : Gn−1 → Gn . The kernel of [] : Gn → Gn is in−1 ◦ · · · ◦ i1 : G1 → Gn . (2) V is said to be a good -adic representation if there is an divisible group such that V = limn Gn (K) ⊗Z Q . ←− (3) V is said to be a semistable -adic representation if there exist a good -adic representation V0 ⊂ V such that V /V0 is unramiﬁed. If = p, then V is good if V is unramiﬁed. If = p, the deﬁnition of good or semistable -adic representation is limited to this book, and they are much stronger conditions than usual ones. Lemma D.16. Let p and be prime numbers, and let K be a ﬁnite extension of Q . Let ρ be an -adic representation of GQp to a two-dimensional K-vector space V . If the action of GQp on det V is the -adic cyclotomic character, the following are equivalent. (1) V is semistable in the sense of Deﬁnition D.15. (2) V is semistable in the sense of Deﬁnition 3.35. Proof. (2) ⇒ (1) is clear. We show (1) ⇒ (2). It suﬃces to show it assuming V has a good one-dimensional subrepresenstation V0 such that V /V0 is unramiﬁed. By the assumption that the determinant

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209

det V is the -adic cyclotomic character, the action of the inertia group Ip on V0 is also the -adic cyclotomic character, and thus V is ordinary. The following is a generalization of Proposition 3.46. Theorem D.17. Let K be a local ﬁeld, and let AK be an abelian variety over K. Let be a prime number diﬀerent from the characteristic of K. In each of (1) and (2), the conditions (i) and (ii) are equivalent. (1) (i) AK has good reduction. (ii) The -adic representation V of GK is good. (2) (i) AK has semistable reduction. (ii) The -adic representation V of GK is semistable. (i) ⇒ (ii) in (1) is clear from Lemma D.11. We omit the proof of the others. D.4. The N´ eron model of the Jacobian of a curve Let O be a discrete valuation ring, let XK be a curve over its ﬁeld of fractions. By Theorem D.3, if XK has good reduction, so does its Jacobian JK = Pic0 XK . More precisely, we have the following. Lemma D.18. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. Let X be a proper smooth curve over O such that each geometric ﬁber is connected. Then, the functor Pic0X/O is represented by an abelian scheme J over O. JK = J ⊗O K is the Jacobian of XK ⊗O K, and J is the N´eron model of JK . The closed ﬁber JF = J ⊗O F is the Jacobian of XF = X ⊗O F . For a prime number diﬀerent from the characteristic of K, the -adic representation V JK of Gal(K/K) is a good -adic representation. If is diﬀerent from the characteristic of F , the natural mapping V JK → V JF is an isomorphism of ﬁnite-dimensional Q -vector spaces that is compatible with Gal(K/K) → Gal(F /F ). Proof. It follows easily from Theorem D.3 and Lemma D.11. Even if a curve does not have good reduction, we have the following theorem on regular model. Theorem D.19. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. Let X be a proper regular connected curve over O such that each geometric ﬁber is connected.

´ D. JACOBIAN OF A CURVE AND ITS NERON MODEL

210

Suppose XK = X ⊗O K is smooth, F is perfect, and the greatest common divisor of the multiplicities of components of XF = X ⊗O K in XF is 1. Then, (1) The functor Pic0X/O is represented by the connected component J 0 of the N´eron model J of the Jacobian JK of XK . (2) Let Γ be the dual chain complex of XF , and let α0 : Γ0 → Γ∗0 and β : Γ∗0 → Z be the linear mappings in (B.4) and (B.5), respectively. Then the group of connected components of the closed ﬁber JF = J ⊗ F of the N´eron model J is naturally isomorphic to Ker β/ Im α0 . Corollary D.20. Let the notation be as is in Theorem D.19. Let J F be the Jacobian of the normalization X F of the reduced part of XF . Then, the morphism JF → J F induced by the natural morphism X F → XF gives an isomorphism from the abelian variety part of the connected component JF0 (D.18)

JFa → J F .

Proof. It is clear from Theorem D.19(1) and Theorem D.7(4). If a curve over a discrete valuation ﬁeld has semistable reduction, so does its Jacobian. More precisely, we have the following. Corollary D.21. Let O be a discrete valuation ring, let K be its ﬁeld of fractions, and let F be its residue ﬁeld. Let X be a proper weakly semistable curve over O such that each geometric ﬁber is connected. Let J be the N´eron model of the Jacobian JK of XK = X ⊗O K. (1) Let X F be the normalization of XF , and let Γ = [Γ1 → Γ2 ] be the dual chain complex of the closed ﬁber XF . The connected component JF0 of the closed ﬁber JF = J ⊗O F is an extension of the Jacobian of X F by a torus Hom(H1 (Γ), Gm ). Let C1 , . . . , Cm be the components of X F , let Σ be the reduced closed subscheme of XF consisting of singular points. Let g be the genus of XF , Fi the ﬁeld of deﬁnition of Ci , and let gi be the genus of the curve Ci over Fi . Then we have (D.19)

g = 1 + deg Σ +

m i=1

[Fi : F ](gi − 1).

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211

(2) The group of connected components of JF is naturally isomorphic to the cokernel of the linear mapping α ¯ 1 : H1 (Γ) → H 1 (Γ∨ ) of (B.8). Proof. (1) Let X be the minimal resolution of singularities of . Since all the exceptional divisors are of X. We have XK = XK genus 0, the Jacobian of the normalization of XF equals that of X F . If Γ is the dual chain complex of XF , then by Proposition B.14(1), we have a natural isomorphism H1 (Γ) → H1 (Γ ). Thus, by replacing X by X , we may assume X is semistable. By Theorem D.19(1) and Theorem D.7(3), JF0 is an extension of the Jacobian by a torus. Then, by Corollary D.21, the character group of the torus part of JF0 equals H1 (Γ). Let a be the dimension of the abelian variety part of JF , and let t be that of the torus part. Then,we have g = a + t. Since m m a = i=1 [Fi : F ]gi and t − 1 = deg Σ − i=1 [Fi : F ], we have (D.19). (2) As in (1), we may replace X by X by Proposition B.14(1). The assertion follows from Theorem D.19(2) and Proposition B.14(2). Corollary D.22. Let K be a discrete valuation ﬁeld, let XK be a proper smooth curve over K, and let JK be the Jacobian XK . If XK has semistable reduction, so does the Jacobian JK . For a prime number diﬀerent from the characteristic of K, the -adic representation V JK of Gal(K/K) is a semistable -adic representation. Proof. It follows from Corollary D.21(1) and Theorem D.17(2) (i) ⇒ (ii).

Bibliography

References for theorems and propositions that were not proved in the text Chapter 8 [1] J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math., Springer, 106, 1986. [2] P. Deligne, M. Rapoport, Les sch´emas de modules de courbes elliptiques, in Modular Functions of One Variable II, Lecture Notes in Math., Springer, 349, 1973, 143–316. [3] N. Katz, B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Math. Studies, Princeton Univ. Press, 151, 1994. [4] H. Hida, Geometric modular forms and elliptic curves, World Scientiﬁc, 2000. [5] B. J. Birch, W. Kuyk (eds.), Modular Functions of One Variable IV, Lecture Notes in Math., Springer, 476, 1973.

Lemma 8.37: [2] III Corollaire 2.9, p.211, [3] Corollary 4.7.2. Lemma 8.41: p = 2, 3: [1] Appendix A, Proposition 1.2 (c). Example 8.65: [5] Table 6, p.143. Proposition 8.69: [2] VII Costruction 1.15, p.297. Theorem 8.77: [2] V Th´eor`eme 2.12, [3] Theorem 13.11.4. Chapter 9

[6] H. Carayol, Sur les repr´esentations galoisiennes modulo attach´ees aux formes modulaires, Duke Math. J. 59 (1989), 785– 801. [7] T. Miyake, Modular forms. Translated from the Japanese by Yoshitaka Maeda. Springer-Verlag, Berlin, 1989. x+335 pp. 213

214

BIBLIOGRAPHY

[8] K. Ribet, On modular representations of Gal(Q/Q) arising from modular forms, Inventiones Math., 100 (1990), 431–476. Theorem 3.55(2) (ii) ⇒ (i): [6]. Theorem 9.40: [7] Corollary 4.6.20. Theorem 3.55(1) (ii) ⇒ (i) the case p ≡ 1 mod : [8].

Chapter 10 [9] J.-P. Serre, Arbres, amalgames, SL2 , Ast´erisque 46, Soci´et´e Math´ematique de France, Paris, 1977. , Le probl`eme des groupes de congruence pour SL2 , Ann. [10] of Math. 92 (1970), 489–527. [11] L. E. Dickson, Linear groups with an exposition of the Galois ﬁeld theory, Teubner, Leipzig, 1901. Theorem 10.15(1): [9] Chapitre II 1.4 Th´eor`eme 3 p.110. Theorem 10.15(2): [10] 2.6 Corollaire 3 p.449 (If we let K = Q, ˜ ) and Eq = S = {p, ∞}, and q = (N ), then we have Γq = Γ(N ˜ E(N ).) Theorem 10.28: [11] sections 255, 260.

Chapter 11 [12] J.-P. Serre, Corps Locaux, 3e ed., Hermann, Paris, 1980. [13] , Cohomologie galoisienne, 5e ed., Lecture Notes in Math., Springer-Verlag, Berlin, 5, 1994. [14] J. S. Milne, Arithmetic duality theorems, Perspectives in Math. 1, Academic Press, Boston, 1986. General theory of Galois cohomology and duality theorem: [4]. Proposition 11.11(1): [12] Chapitre X §3 b), (2): ibid., Proposition 9. Proposition 11.18: [14] Corollary 2.3. Proposition 11.20: [14] Theorem 2.8. Proposition 11.25(1): [14] Corollary 4.15. Proposition 11.25(2): [14] Theorem 4.10. Proposition 11.27: [14] Theorem 5.1.

BIBLIOGRAPHY

215

Appendix B [65] P. Deligne, N. Katz, Groupes de Monodromie en G´eom´etrie Alg´ebrique (SGA 7) II, Lecture Notes in Math., Springer, 340, (1973). Lemma B.4 (iii) ⇒ (i) The case where k is general: [65] Exp. XV, Th´eor`eme 1.2.6. Lemma B.12: [65] Exp. X, Corollaire 1.8.

Appendix C [15] J.-M. Fontaine, Groupes p-divisible sur les corps locaux, Ast´erisque 47–48, Soc. Math. de France, 1977. [16] J.-M. Fontaine, G. Laﬀaille, Construction de repr´esentations p´ adiques, Ann. Sci. Ecole Norm. Sup. (4) 15 (1982), 547–608 (1983). [17] N. Wach, Repr´esentations cristallines de torsion, Compositio Math. 108 (1997), 185–240. Theorem C.1:[15] Chapitre III. Theorem C.6:[16], [17].

Appendix D [18] S. Mukai, An introduction to invariants and moduli, Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003. xx+503 pp. [19] M. Artin, N´eron models, in G. Cornell, J. Silverman, (eds.), Arithmetic Geometry, Springer, 1986 pp.213–230. [20] J. S. Milne, Jacobian varieties, in G. Cornell, J. Silverman, (eds.), Arithmetic Geometry, Springer, 1986 pp.167–212. [21] S. Bosch, W. L¨ utkebohmert, M. Raynaud, N´eron models, Springer, 1990. [22] M. Raynaud, Jacobienne des courbes modulaires et op´erateurs de Hecke, in “Courbes modulaires et courbes de Shimura”, Ast´erisque 196–197, Soc. Math. de France, 1991, 9–25.

216

BIBLIOGRAPHY

[23] A. Grothendieck, Mod`eles de N´eron et monodromie, in Groupes de Monodromie en G´eom´etrie Alg`ebrique, SGA 7I, Lecture Notes in Math., Springer, 288, 1972, 313–523. Proposition 5.13. Jacobians and N´eron models: [19], [20], [21] Curves and their Jacobians: [18] Chapter 8. Abel’s theorem: [18] Theorem 9.8.6. Theorem D.3: [21] Theorems 8.4/3 and 9.3/1. Theorem D.7: [21] Theorem 8.2/3 and Propositions 9.2/5 and 10. Theorem D.8: [21] Corollary 1.3/1. Proposition D.12: [22], Proposition 6. Theorem D.17(1) = p: [21] Theorem 7.4/5 (b)⇔(d). Theorem D.17(2) = p: [21] Theorem 7.4/6. Theorem D.17(2) = p: [23] Proposition 5.13. Theorem D.19(1): [21] Theorem 9.5/4, p.267. Theorem D.19(2): [21] Theorem 9.6/1, p.274.

Symbol Index ˜ ), 118 E(N e E (p ) , 2 Eqrr , 51 End0R (M ), 147 ExtR-G (M, N ), 148

∪, 149 1-cocycle, 144 αq , 125 a, 22 a−1/2 , 132 Af , 111, 131 AΣ , 113

fa , 65 fQ , 132 F , 2, 191, 193 Fe , 2 FS , 1 FΣ ,Σ , 116

Br(F ), 150 150

n Br(F ),

C q (G, M ), 149 C(X, F), 137 ΔQ , 126 Δq , 125 Δp , 117 Def R-G (M ), 146 Def 0R-G , 146 n Def ρ,D ¯ Σ (O/(π )[ε])[Vn ] , 163 div f , 199 D(G), 191, 194 D(¯ ρ), 108 D(X), 199

Γ0,∗ (p, N ), 118 Γ(N ), 118 Γ1 (N ), 38 ˜ ), 118 Γ(N Γ(r), 29 G-coinvariant, 144 G× , 10 G(a,b) , 33 GF , 143 GS , 155 Grass(OE[N ] , N ), 41

ε, 70 εf , 70 e¯, 51 ex , 24 E(N ), 118

H 0 (G, M ), 144 H 1 (G, M ), 144 Hf1 (Q , Wn ), 161 Hf1 (Qp , M ), 151 Hf1 (Qp , HomR (M, N )), 152 217

218

Hf1 (Qp , N ), 152 Hs1 (Q , Wn ), 162 H q (F, M ), 150 H q (G, M ), 149 iΣ , 110 I0 (N, n), 62 I1 (N, n), 66 I0 (N, n), 62 Ig(M pa , r)Fp , 33 =0 Ig(M pa , r)P Fp , 34 Ig(M pa , r)Fp , 56 j, 21, 29 ja , 35, 42, 55, 56 j0 , j1 , 49, 58, 59, 118 J0,1 (N, M ), 70 J0 (N )Q , 61 J1 (N )Q , 66 Kf , 63 λ, 31 -divisible group, 208 (Lp )p∈S , 157 (L∨ p )p∈S , 158 LΣ , 162 Lift0R-G , 146 LiftR-G (M ), 146 n Liftρ,D ¯ Σ (O/(π )[ε])Vn , 163 μ, 25 μ× N , 12 mQ , 130 m∅,Q , 135 m∗Σ ,Σ , 114 mΣ ,Σ , 115 mQ , 131 mRΣ , 163 mΣ , 110 mΣ , 110

SYMBOL INDEX

˜ , 146 M 5 Mρ˜, 146 M0 (N )E , 41 M1 (N )E , 15 M (1), 150 M G , 144 MG , 144 MQ , 134 MΣ , 114 M ∨ , 150 M, 21 M0,∗ (N, r)Z[ r1 ] , 25 M0 (N ), 20 M0 (N )E , 13 M0 (N )Fp , 23 M1,∗ (N, r)Z[ r1 ] , 25 M1,0 (M, N ), 57 M1 (N ), 20 M1 (N )E , 13 M(r)Z[ r1 ] , 25 Nf L, 202 N -IsogE , 41 ord, 2 ϕΣ , 108 Φ, 97 pRΣ , 163 P , 13 Pf,p (U ), 111, 131 Pp (U ), 94, 109 Pq (U ), 130 PX/S (T ), 202 Pic(X), 199, 201 Pic0 (X), 202 Pic0X/S , 203 PicX/k , 202

SYMBOL INDEX

Q, 126 Q, 126 4 125 Q, QS , 155 RQ , 135 RΣ , 114 R[ε], 146 s0 , s1 , 118 sd , 45, 59 sk , 82 sk (q), 50 S0 (N ), 61 S1 (N ), 66 S ss , 2 Sel(E, n), 158 SelL (M ), 157 SelΣ (Wn ), 162 tk , 82 t˜∅,Q , 129 tΣ ,Σ , 113 t∅,Q , 130 T (N )Z , 108 T (NΣ )O , 110 T [N ], 16 T (N ) , 16 T J0 (N ), 70 T0 (N )Z , 62 T1 (N )Z , 67 TN,E , 41 TΣ , 110 TQ , 130 T , 113 Tq , 129 T , 97 T , 103 Tor, 137

U , 115 Up , 92, 115 V , 2, 72, 161, 191 Vn0 , 161 Ve , 2 Vf , 71 V J0 (N ), 70 2 wN , 22 W , 161 Wn0 , 162

X0 (N )Z , 48 X0,∗ (N, r)Z[ r1 ] , 54 X1 (M p)bal Z[ζp ] , 59 X1 (N )Z , 48 , 58 X1,∗ (M p, r)bal Z[ r1 ,ζp ] X1,∗ (N, r)Z[ r1 ] , 54 X1,0 (M, N )Z , 58 X1,0 (N, M )Z , 58 X (p) , 1 X(1)Z |∧ ∞ , 51 Y0,∗ (N, r)Z[ r1 ] , 42 Y0 (N )Z , 23 Y1 (4), 20, 47 Y1,∗ (N, r)Z[ r1 ] , 31 Y1,0 (M, N )Z , 57 Y1 (N )an , 38 Y1 (N )Z , 24 Y (1)Z , 29 Y (2), 31 Y (3), 25 Y (r)an , 29 Y (r)Z[ r1 ] , 26 Z 1 (G, M ), 144 144 Z, Z((q)), 51

219

Subject Index

absolute Frobenius morphism, 1 annihilator, 153 Atkin–Lehner involution, 23

ﬁnite G-module, 143 ﬁnite R-G-module, 143 Frobenius morphism, 23 full Hecke algebra, 110, 130 full set of sections, 6

Brauer group, 150

G-coinvariant, 144 generator, 9 genus, 179 good, 208 good reduction, 182, 205

character of f , 70 congruence relation, 73 connected component, 205 cup product, 149 curve, 179 cyclic group scheme, 9

Hecke algebra, 62, 67 Hecke module, 114, 134 Hecke operator, 67

diamond operator, 22, 67 Dieudonn´e module, 191 divisor, 199 divisor class group, 199 divisor group, 199 Drinfeld level structure, 13 dual chain complex, 182 dual local condition, 158

Igusa curve, 33 index, 182 inﬁnitesimal deformation, 146 inﬁnitesimal lifting, 146 Jacobian, 203 -divisible group, 208 local condition, 157

Eisenstein ideal, 77 ´etale ﬁltered ϕ-module, 192 exact order N , 13 extension, 147, 196

minimal resolution of singularities, 188 multiplicative ﬁltered ϕ-module, 192

ﬁltered ϕ-module, 192 221

222

N´eron model, 205 node, 180 non-Eisenstein, 123 non-Eisenstein ideal, 78 norm of an invertible sheaf, 202 old part, 91 1-cocycle, 144 ordinary, 2 perfect complex, 136 Petersson product, 65, 68 Picard functor, 202 Picard group, 201 preserve the determinant, 146 primary form, 70 principal divisor, 199 proﬁnite group, 143

INDEX

relative Frobenius morphism, 2 right bounded, 136 scheme of generators, 10 Selmer group, 157 of an elliptic curve, 158 semistable, 182, 208 semistable reduction, 182, 205 singular chain complex, 137 strongly divisible, 192 supersingular, 2 Tate curve, 51 Tate twist, 150 unramiﬁed part, 151 weakly semistable, 182