Fermat's Last Theorem The Proof
Translations of
MATHEMATICAL MONOGRAPHS
Volume 245
Fermat's Last Theorem The Proof Takeshi Saito Translated from the Japanese by Masato Kuwata
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American Mathematical Society Providence. Rhode Island
FERUMA YOSO (Fermat Conjecture)
7
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JL/x-.:P�
by Takeshi Saito
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© 2009 by Takeshi Saito First published 2009 by lwanami Shoten, Publishers, Tokyo. T his 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 llFll, 11F80, 11G05, 11Gl8.
Library of Congress Cataloging-in-Publication Data ISBN 978-0-8218-9849-9
( Translations
Fermat's last theorem: the proof
of mathematical monographs ; volume 245)
The first volume was catalogued as follows: Saito, Takeshi, 1961Fermat'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.
1. Fermat's last theorem.
paper )
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 © 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: I /www.ams.org/
10 9 8 7 6 5 4 3 2 1
19 18 17 16 15 14
Contents Basic Tools xi
Preface Preface to the English Edition Chapter
0.1. 0.2. 0.3. 0.4.
0.5.
xvii
0.
Synopsis Simple paraphrase Elliptic curves Elliptic curves and modular forms Conductor of an elliptic curve and level of a modular form .e-torsion points of elliptic curves and modular forms
1.
Chapter
Elliptic curves Elliptic curves over a field Reduction mod p Morphisms and the Tate modules Elliptic curves over an arbitrary scheme Generalized elliptic curves
Chapter
Modular forms The j-invariant Moduli spaces Modular curves and modular forms Construction of modular curves The genus formula The Hecke operators The q-expansions Primary forms, primitive forms Elliptic curves and modular forms
1.1. 1.2. 1.3. 1.4. 1.5.
2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2. 7. 2.8. 2.9.
2.
v
1 1 3 5 7 9 13 13 15 22 26 29 35 35 37 40 44 52 55 58 62 65
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
3.
Chapter
Galois representations Frobenius substitutions Galois representations and finite group schemes The Tate module of an elliptic curve Modular t'-adic representations Ramification conditions Finite fiat group schemes Ramification of the Tate module of an elliptic curve Level of modular forms and ramification
Chapter
The trick Proof of Theorem Summary of the Proof of Theorem
3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8.
4.1. 4.2.
Chapter
5.1. 5.2. 5.3. 5.4. 5.5. 5.6.
4.
3-5
2.54
5. R = T
What is R = T? Deformation rings Hecke algebras Some commutative algebra Hecke modules Outline o f the Proof o f Theorem
6.
5.22
Chapter
Commutative algebra Proof of Theorem Proof of Theorem
Chapter
Deformation rings Functors and their representations The existence theorem Proof of Theorem Proof of Theorem
6.1. 6.2.
7.1. 7.2. 7.3. 7.4.
0.1
5.25 5.27
7.
5.8 7.7
Appendix A. Supplements to scheme theory A.l. Various properties of schemes Group schemes Quotient by a finite group Flat covering
A.2. A.3. A.4.
66 70 74 77 81 82 86 89 91 96 100 103 108 111 111 116 119 119 122 126 131 135 137 143 143 149 159 159 161 162 166 171 171 175 177 178
vii
CONTENTS
A.5. A.6. A.7. A.8.
G-torsor Closed condition Cartier divisor Smooth commutative group scheme
Bibliography Symbol Index Subject Index
179 182 183 185 189 197 199
The Proof Preface
ix
Preface to the English Edition
xv
8.
Chapter
Modular curves over Z Elliptic curves in characteristic Cyclic group schemes Drinfeld level structure Modular curves over Z Modular curve Y(r)z [ � ] Igusa curves Modular curve Y1 (N) z Modular curve Yo (N)z Compactifications
Chapter
Modular forms and Galois representations Hecke algebras with Z coefficients Congruence relations Modular mod £ representations and non-Eisenstein ideals Level of modular forms and ramification of £-adic representations Old part Neron model of the Jacobian Level of modular forms and ramification of mod £ representations
8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9.
9.1. 9.2. 9.3.
9.4. 9.5. 9.6. 9.7.
p>0
9.
J0 (Mp)
1 1 6 12 20 25 32 37 41 48 61 61 70 76 81 90 97 102
viii
CONTENTS
Chapter 10. Hecke modules 10.1. Full Hecke algebras 10.2. Hecke modules 10.3. Proof of Proposition 10.11 10.4. Deformation rings and group rings 10.5. Family of liftings 10.6. Proof of Proposition 10.37 10.7. Proof of Theorem 5.22 Chapter 11. Selmer groups 11.1. Cohomology of groups 11.2. Galois cohomology 11.3. Selmer groups 11.4. Selmer groups and deformation rings 11.5. Calculation of local conditions and proof of Proposition 11.38 11.6. Proof of Theorem 11.37 Appendix B. Curves over discrete valuation rings B. l. Curves B.2. Semistable curve over a discrete valuation ring B.3. Dual chain complex of curves over a discrete valuation ri� Appendix C. Finite commutative group scheme over Zp C.l. Finite fl.at commutative group scheme over FP C.2. Finite fl.at commutative group scheme over Zp Appendix D. Jacobian of a curve and its Neron model D.l. The divisor class group of a curve D.2. The Jacobian of a curve D.3. The Neron model of an abelian variety D.4. The Neron model of the Jacobian of a curve Bibliography Symbol Index Subject Index
107 108 113 118 125 129 136 140 143 143 149 157 161 165 169 179 179 182 1� 191 191 192 199 199 201 205 209 213 217 221
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 pow ers) , 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 num ber 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.
ing Co., New York, 1966.
II: ix
Diophantine analysis,
Chelsea Publish
x
PREFACE
curves, modular forms, and Galois representations. Using these rela tions, we reduce Fermat's Last Theorem to the modularity of certain R-adic representations ( Theorem 3.36) and a theorem on the level of mod e representations ( Theorem 3.55) . Next, we introduce the no tions 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 Fer mat's Last Theorem is derived from Theorem 0.13, which is about the connection between elliptic curves and modular forms, and Theo rem 0.15, which is about the ramification and level of e-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 Chap ter 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 alge bras. 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
Basic Tools,
a translation of the Japanese original.
Fermat's Last Theorem:
xi
PREFACE
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 mod ular 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 divid ing 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 cen tury 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 cele brated proof of Theorem 3.55 in the case of p 1 mod £ by K. Ri bet because it requires heavy preparations, such as the p-adic uni formization 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 Proposi tion 5.33. In Chapter 1 1 , 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 1 1 up to § 1 1 .3 may be read inde pendently 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 valua tion 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 fl.at 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 Neron 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 Chap ters 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 el liptic 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, in cluding the lecture course at the University of Tokyo in the first se mester 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 Universite Paris-Nord, Max Planck-Institut fiir Mathematik, and Universitii.t 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 The orem 1first 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 de velopments, 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 Math
ematics. 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 vol ume, 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 twen tieth 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
CHAPTER 8 Modular curves over Z
In Chapter 2, we used modular curves over Q to define modu lar 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 de fine modular curves over Z using the Drinfeld level structure, which will be introduced in §8.3 after some preparations in §8.2. The Drin feld level structure plays an important role when we study the struc ture 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 x s S by Fs : S-+ S. The morphism X -+ X (P) defined by the commutative diagram x�x
1
1
s�s
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 + a 1 xy+ a3 y = x3 +a2 x 2 +a4x + a5 , then E ( P) is defined by the equation y 2 + af xy + a� y = x 3 + a� x2 + a� x + a� , and F : E --+ E (P) is defined by ( x, y ) H ( xP , yP ) . For a nonnegative integer e, we1 denote by pe : E --+ E (p•) the composition of F : E (p ') --+ E (P '+ ) , i = 0, . . . , e - 1, and let V e : • p E ( ) --+ E be the dual of pe . 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 F2 , we say E is supersingular. A supersingular elliptic curve over a field is smooth and thus non singular. 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 Yo(Mp)F p (p f M) corresponding to supersingular elliptic curves coincide with the sin gular 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 3ord the maximal open subscheme U of S such that the restriction Eu is ordinary. Meanwhile, if we apply Corollary A.37(2) to Ker F2 and E[p] , the condition P that ET = E x s T is supersingular for an S-scheme T is a closed condition on S. We denote by 855 the closed subscheme of S defined by the closed condition P. We show that 3ord is the complementary open subscheme of sss. More precisely, we have the following. o
o
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. (1) The order of the abelian group E[p] (k) is either p or 1 . (2) The following conditions (i)-(iv) are e quivalent. (i) The order of the abelian group E[p] (k) is p. (ii) E is ordinary. (iii) For any integer e 2::: 1 , Ker y e is etale and the abelian group Ker v e (k) is isomorphic to Z/pe z. (iv) For any integer e ;:::: 1 , the group scheme E[pe ],;; is isomor phic to Z/pez x µp•. (3) The following conditions (i)-(iii) are e quivalent. (i) The order of the abelian group E[p] (k) is 1 . (ii) E is supersingular. (iii) For any integer e ;:::: 1 , the only closed subgroup scheme of E of order pe is Ker Fe . PROOF. It suffices to prove the case where k = k. (1) As in Proposition 3.45, let E[p ] 0 the connected component of E[p] containing 0, and let E[p ]et be the maximal etale quotient. Consider the exact sequence (8. 1) 0 --+ E[p] 0 --+ E[p] --+ E[p ] e t --+ O. Since we assumed k = k, the exact sequence (8. 1) gives an isomor phism of finite groups E[p] (k) --+ E[p]e t (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 ]e t is connected. Thus, the Cartier dual (E[p]e t )v is a closed subgroup scheme of E[p] 0 . Hence, ( lt E[p] (k)) 2 = (deg E[p]e t ) (deg(E[p ]e t )v) divides (deg E[p]e t ) (deg E[p ] 0 ) = deg E[p ] = p2 . (2) (i) =?- (ii) . Since [p] = V o F, we obtain the exact sequence (8.2)
0 --+ Ker(F : E --+ E (P) ) --+ E[p ] � Ker(V : E (P) --+ E) --+ 0. 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. (ii) =?- (iii) . Since y e = V o v
4
8. MODULAR CURVES OVER
Z
(iii) :::::} (iv) . In the exact sequence (8.3) 0 --+ Ker(F e : E --+ E (p• ) ) ---+ E[pe] � Ker(Ve : E (p• ) --+ E) --+ 0, Ker Fe is connected and Ker ve is etale. Let E[pe]red be the reduced part of E[pe]. Then, the composition E[pe]rect --+ E[pe] --+ Ker ve is an isomorphism since k is algebraically closed. Since Ker ve is isomorphic to Z/pe z, Ker Fe is isomorphic to its Cartier dual µp• . (iv):::::} (i) . This is clear from µp(k) = {l}. (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 of the local ring OE , o, we have G = Spec OE,o/mf = Ker F e . (iii) :::::} (ii) . Since E[p] is a closed subgroup scheme of degree p2 , we have E[p] = Ker F2 . D (ii) :::::} (i) . We have E[p] (k) = Ker F2 (k) = 0. 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) PROOF. It is clear from Proposition 8.2.
D
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 2:: 1 be an integer. (1) If E is ordinary, then ve : E (p• ) --+ E is etale, and Ker ve 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' (p• ) satisfying f = ve g . :
0
PROOF. (1) Similar to the proof of Proposition 8.2(2) (ii):::::} (iii) . (2) Let t f : E' --+ E be the dual of f. Since f t f = [pe], Kert f 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, Kertf is a closed subgroup scheme. Hence, we have Kert f = Ker F e , and an isomorphism g : E --+ E' (p• ) satisfying Fe = g ot f is induced. Clearly, g satisfies the required condition and g is unique. Since ve : E' (p• ) --+ E' is etale, E' is ordinary. Moreover, since D E and E' (p• ) are isomorphic, E is also ordinary. o
8 . 1 . ELLIPTIC CURVES IN CHARACTERISTIC p >
5
0
PROPOSITION 8.5. Let p be a prime number, and let E be an elliptic curve over Fp. Let a = 1 + p ltE(Fp)· Then, the following conditions (i)-(iii) are equivalent. (i) E is ordinary. -
(ii) pf a .
(iii) If p = 2, a = ±1 . If p = 3, a = ±1, ±2. If p;:::: 5, a -:f 0. PROOF. As in Proposition 1.21, we have 1 at + pt 2 det(l-Ft : 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 eigenval ues 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 J a l < 2...JP and Theorem 1.15. 0 -
EXAMPLE 8.6. Let p be an odd prime number, and let E be the elliptic curve over FP given by y2 = 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, O)} is a subgroup of E(Fp), we have ltE(Fp) = p + 1 - a = 0 mod 4. Thus, if p = 1 mod 4, we have a = 2 mod 4, which means a -:f 0. If p = -1 mod 4, then - 1 is not a quadratic residue mod p. Hence, if x -:f 0, ±1, either x3 - x or (-x)3 (-x) = -(x3 - x) is a quadratic residue, and not both. Thus, we have ltE(Fp) = p + 1, which means a = 0. Similarly, for a prime number p ;:::: 5, let E be the elliptic curve over FP defined by y2 = 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 oo,
-
-
equivalent. (i) The p-adic representation VpE of GQ p is ordinary. (ii) Either E has good reduction and EF,, is ordinary or E has mul tiplicative reduction.
PROOF. First, we assume E has good reduction, and we show
VpE is ordinary if and only if EF,, is ordinary. By Theorem C.6(3) , the
subspace D'(E) c D(E) is one dimensional. Thus, by Corollary C.8, VpE is ordinary if and only if there exist p-adic units a and f3 such that 1 - at+ pt 2 = deg(l - Ft : D(E)) decomposes into (1 - at ) ( l pf3t) This is in turn equivalent to that EF,, is ordinary by Proposition 8.5. -
.
6
8. MODULAR CURVES OVER
Z
Furthermore, by Proposition 3.46(2) , E has stable reduction if VpE is ordinary. This shows (i)::::} ( ii) . Suppose E has multiplicative reduction. In this case, we have al ready proved that VpE is ordinary in the proof of Proposition 3.46(2) D (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 fiat 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 N (8.5) NxR/ R(f) = IT f (Pi ) i= l for any commutative ring R, any morphism Spec R --+ S, and any element f E r(X X s Spec R, 0). LEMMA 8.9. Let S be a scheme, let N � 1 be an integer, and let X be a finite fiat 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 )
is surjective.
N
II Pi : SU · · ·U S -+ X
i= l
PROOF. It suffices to show it when S = Spec k, where k is an D algebraically closed field, but it is clear in this case. Even if the morphism ( 8.6 ) is surjective, Pi, . . . , PN may not be a full set of sections of X. For example, let k be a field, let S = Spec k[t: ]/(t: 2 ) , and let X = Spec k[t: , t:'] (t: 2 , t:'2 ) . Define sec tions P1 , P2 : S --+ X by t-+ 0 and t:' t-+ t:, respectively. Then, P1 1l P2 : S 1l 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. €
€.
1 € ,
8.2. CYCLIC
G ROUP 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 . ( ii ) IJ!1 Pi : S IJ · · · IJ S ---+ X is an isomorphism. IJ S ---+ X is surjective. (iii) IJ!1 Pi : S IJ ·
·
·
PROOF. ( ii ) ::::} ( i ) is clear. ( i ) ::::} ( iii ) holds by Lemma 8.9. Since S IJ · · · IJ S and X are both finite etale of degree N, ( ii ) and ( iii ) are 0 equivalent. PROPOSITION 8. 11. Let S be a scheme, let N ;:::: 1 be an integer, and let X be a finite fiat sche me of finite presentation over S of degree N. Let P1, ... , PN S ---+ X be a family of sections of X . The :
condition P that Pi, ... , 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 gi, ..., 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[Ti , ..., TN] and f = L::f=1 g1T1 E B[T1 , ..., TN]· For such R and f, ( 8.5 ) becomes ( 8.7 )
N N N = g T NB[T1,...,TN]/A[T1, ...,TN]c�= j j) rrc�=gj(Pi)T1)· i=l j=l j=l
If I C 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 Ti, ... , TN with A coefficients, I is finitely 0 generated. If X is a closed subscheme of a smooth curve, we have the propo sition below. Note that if E is a smooth curve over S and X is a closed subscheme of E that is finite fl.at 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 fiat 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:
N X = L[Pi]· i=l PROOF . (ii) => (i) . Let Spec R -+ S be a morphism of schemes. We show NxR / R(f) = TI!1 f(Pi) for f E r(XR, 0). Replacing s by Spec R, we may assume S = Spec R. For i = 1, . . . , N, let I;, be the defining ideal sheaf of the Cartier divisor [Pi] of E. By the equality of divisors X = E!i [Pi], the finitely generated free Os-module Ox is a successive extension of the invertible Os-modules n�:i Ij/ TI� = l Lj· Since the multiplication-by-f map of 0x induces the multiplicationby- f (Pi) map of n�:i Lj/ TI� = l Lj, we have Nx; sU ) = TI!1 f(Pi) · (i) => (ii) . Both X and E!1 [Pi] are finite fl.at of finite presen tation over S of degree N. Thus, it suffices to show X is a closed subscheme of E!1 [Pi] · Let s E S. We may replace S by Spec Os, s · By Lemma 8.9, we have X = LJ!1 Pi(S). Since A = Os , s is a local ring, Pi(s) -:j:. Pi(s) implies Pi(S) n Pj(S) = 0 for i, j = 1, . . . , N. Thus X = Ux>-+s LJP;(s)=x Pi(S). Hence, for an inverse image x of s, we have Spec O x x = LJP;(s)=x Pi(S) and X = Ux>-+s Spec O x x· Therefore, it suffices to show the assertion assuming X = Spec O x 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 E B. For i = 2, . . . , N, t - t(Pi) E 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) E 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) E B for i = l, . . . , N. Let
,
,
9
8.2. CYCLIC GROUP SCHEMES
in the defining ideal Ker(B ---+ B) of X. This shows that X is a closed D subscheme of L::�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 fl.at 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 - l)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 fl.at 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 fiat
commutative group scheme of finite presentation of degree M. Then, the multiplication-by-M mapping of G is the 0 mapping.
It suffices to show gM = 1 for any scheme T over S and any section g E 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 av = Spec AV , where Av = HomR(A, R) , be the Cartier dual of G. The multiplica tion of Av is the dual of the ring morphism A ---+ A ©R A that defines the group operation µ : G x s G ---+ G. Let g E G(S) , let µ9 : G ---+ G be the translation by g, and let µ9 : A ---+ A be the corresponding isomorphism of rings. We identify g E G(S) = HomR- aig(A, R) C HomR(A, R) with an ele ment of Av = HomR(A, R) . Furthermore, we identify µ9 E G(G) = HomR- alg(A, A) C EndR A with an element of EndR A = A ©R Av . Since G(G) is a subgroup of (A ©R Av) x , µ9 E A ©R Av is invertible. Moreover, since µ9 : G ---+ G is the product of id = µ 1 : G ---+ G and G ---+ S � G, we have µ9 = µ 1 (1 © g) E A ©R Av . Thus, taking the norm N = NA®RAv /Av of both sides, we obtain N(µ9) = N(µ 1 ) gM E Av x . On the other hand, since µ9 (µ 1 ) = µ 1 µ9 = µ9 , the homomorphism of A v-algebras (µ9 © 1) : A © R A v ---+ A © R Av maps µ 1 to µ9 • Thus, we have N (µ9) = N (µ 1 ) E Av x . This shows M D g = 1 E G(S) c Av . PROOF .
·
·
o
10
8. MODULAR CURVES OVER
Z
LEMMA 8.15. Let S be a scheme, let N � 1 be an integer in vertible on S, and let G be a finite fiat 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. ( ii ) For any geometric point s in S, P8 E G(s) is a generator of G(s) . ( iii ) There exists an isomorphism of commutative group schemes Z/NZ -+ 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/NZ -+ G etale locally on S. PROOF. Since G is etale over S by Proposition 8.14 and Defini0 tion A. 17, the assertions are clear from Corollary 8 . 1 0 . LEMMA 8.16. Let S be a scheme, let N � 1 be an integer, and let G be a finite fiat commutative group scheme over S of finite pre sentation 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 i s locally of finite type. (2) The functor that associates the set {P E G(T) IP is a generator of G T} ( 8.8 ) to a scheme T over S is represented by a closed subgroup scheme ex of G. PROOF. ( 1 ) It suffices to apply Proposition 8.11 to the sections
0, P, . . . , (N - l)P of G.
(2) It suffices to apply ( 1 ) to the pullback section of G to G 0 defined by the diagonal map G -+ G x s G. The closed subgroup scheme ex of G is called the scheme of gen erator s of G. If ex is faithfully fiat over S, then ex is a fiat 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 fiat commutative group schemes of finite
8.2. CYCLIC
G ROUP 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 MP is a generator of G' .
PROOF. (i) ::::::} (ii) . Let P be a generator of G, and let N be the degree of G. By Lemma 8.9, 11!0 1 iP : S 11 · · · 11 S -t G -t G" is surjective. Thus, by Proposition 8.14, 11!0 1 iP" : Bil · · · 11 S -t G" is also surjective. Therefore, by Corollary 8.10, P" defines an isomorphism Z/MZ -t G" , and P" is a generator of G". Moreover, since 11�01 ( +iP) : 11� 01 G' -t G is an isomorphism of schemes, P is a generator of G if and only if MP is a generator of G". (ii) ::::::} (i) . P" defines an isomorphism Z/MZ -t G" , by Corol lary 8.10. The remaining part is similar to the proof of (i) ::::::} (ii) . D 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 -t 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. (ii) The Cartier divisor E !0 1 [iP] of E is a closed subgroup scheme of E. ( 2 ) If one of the equivalent conditions of ( 1 ) holds, then NP = 0, and G = E !o 1 [iP] . ( 3) Let G be a closed subgroup scheme of E finite fiat of finite pre sentation of degree N over S. Let P be a section of E over G defined by the diagonal map G -t E x s G. The scheme ax 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 subschemes of Er, Gr and E !0 1 [iP]r are equal.
PROOF. ( 1 ) Clear from Proposition 8.12. ( 2 ) Since the cyclic subgroup scheme 2: !0 1 [iP] is a finite fl.at commutative group scheme of finite presentation of degree N, the image of the multiplication-by-N map of E !o 1 [iP] is 0 by Proposition 8.14. Thus, we have NP = 0. G = E !0 1 [iP] is clear from Proposition 8.12. D ( 3 ) Clear from ( 2 ) .
12
8. MODULAR CURVES OVER
Z
LEMMA 8.19. Let N � 1 be an integer. The group scheme µ N = Spec Z [X]/(X N - 1) consisting of Nth roots of unity is a cyclic group scheme. If
N (X) E Z[X] is the Nth cyclotomic polynomial, the scheme µ;, of generators of µ N is Spec Z[X]/( N (X)) . PROOF. Spec Z[X]/( N (X)) is a fl.at covering of Spec Z. Thus, if we show µ;, = Spec Z[X]/( N (X)), then it follows that µ N is a cyclic group scheme. Clearly, µ�,SpecZ[1f] = Spec Z[k] [X]/(N (X)) over Spec Z[kJ by Lemma 8.15. Thus, applying Corollary A.44(1) to S = Spec Z, X = Gm,s , A= µ;,, 8 , and B = Spec Z[X] /(N (X)), it suffices to show the inequality deg µ;, k:::; cp(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 e = µ N = Spec k[X] /(X N ) . Let P: e-+ Gm x k e be the diagonal section. By Lemma 8.18(3) , the closed subscheme ex of e is defined by the closed condition that the two closed subschemes of Gm x ke = Spec k[X, (l + X) - 1 , T] /(TN ) , e xk e = Spec k[X, T]/(X N , TN ) and the pullback of
� [iP]
=
Spec k[X, ( 1 + x) - 1 , T]/
Cfi1 (1 + x - (1 + T) i) , TN)
are equal. Thus, if we let N-1 N-1 IJ ( 1 + x - ( 1 + T) i) = x N - L aj (T)XJ , j= O i=O we have ex = Spec k[T]/(TN , ao(T) , . . . , aN - 1 (T)). Since we have ( 1 + T) i - 1 i T mod T2 , the T-adic valuation ord((l + T) i - 1 ) is 1 if p f i, and at least 2 if p I i . Thus, we have ord a N/p (T) = U{i I p f i, O :::; i < N} = N - N/p = cp(N) , and we have deg ex = D min(N, ord ao (T) , . . . , ord aN - 1 (T)) :::; cp(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 supersingu lar 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
13
8.3. DRINFELD LEVEL STRUCTURE
scheme. The level structure defined in such a way is called the Drin
feld level structure .
DEFINITION 8.20. Let S be a scheme, and let Ebe 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 I: �� 1 [iP] is a closed subgroup scheme of E. If P has exact order N, we call (8.9) (P) = L [iP] :
iEZ/NZ
the cyclic subgroup scheme of order N generated by P. (2) The functor M 0 (N)e over S is defined by associating to a scheme T over S the set (8.10) Mo(N)e(T) = { cyclic subgroup scheme of Er of order N}. (3) The functor M 1 (N)e over S is defined by associating to a scheme T over S the set (8.11) M 1 (N)e(T) = { section of Er 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 E M 1 (N)e(T) of exact order N, associate the cyclic subgroup scheme (P) E M 0 (N)e(T) , and we obtain a natural morphism of functors M 1 (N)e --+ Mo(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 equiv :
alent. ( i) P has exact order N. ( ii) There exists a closed immersion Z/NZ --+ E of commutative group schemes over S such that P : S --+ E is defined by 1 E
Z/NZ.
( iii) N P = 0, and for any geometric point s in S, the element P 8 of the abelian group E(s) has exact order N.
PROOF .
Clear from Lemmas 8. 18(2) and 8.15.
0
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 2". 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[OJ of E is a cyclic subgroup scheme
of order pe. If one and hence all of these conditions hold, we have G
Ker pe. PROOF. (i):::::} (ii) , (iii) . If S is a scheme over Fp, pe[OJ = Ker pe 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 [OJ of Eis defined by a section T of OE on a neighborhood of [OJ. Then, we have G = SpecA[T]/(TP0). Let F(T, S) be the image of T by the ring homomorphism A[T]/(TP0) -+ A[TJ/(TP0) ©A A[T]/(TP") = A[T, SJ/(TP", SP") corresponding to the group operation G x G -+ G. Here, we identify T = T © 1, and let S = 1 © T. F(T, S)P" equals 0 as an element of A[T, S]/(TP", SP"). Since F(T, 0) = F(O, T) = T, there exists f(S, T) E A[T, S]/(TP", SP") such that F(T, S) = T + S + ST f (S, T). Looking at the homoge neous degree pe part of F(T, S)P", all the coefficients of (T + S)P" (TP" + SP") = I::f:�1 (P")TP"-isi are 0 as elements of A. Since i the greatest common divisor of (P;) and (P�:1) is p, A is an Fp D 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 2". 1 be an integer, and let P : S -+ E be a section of E. For a scheme T over S, the condition P that Pr is a section of exact order N of E r is a =
closed condition on S.
PROOF. Let G = L��1[iP J. By Definition 8.20(1) , the condi tion Pis that Gr is a closed subgroup scheme of Er. This condition is in turn equivalent to the following. The closed subscheme Gr x r Gr
8.3. DRINFELD LEVEL STRUCTURE
is
of Er x r Er is a closed subscheme of the inverse image of Gr by the addition + Er x r Er, and Gr is equal to the inverse image of Gr by the multiplication-by- ( - ! ) morphism Er --+ Er. Thus, by Corollary A.37, the condition Pis a closed condition. D COROLLARY 8.24. Let S be a sche me, and let E be an elliptic curve over S. Let N :2'.: 1 be an integer. The functor Mi ( N) E over S is represented by a scheme Mi (N)E finite of finite presentation over S. If N is invertible on S, Mi (N)E is etale over S. PROOF. Applying Lemma 8.23 to the diagonal section E[N] --+ E x s E[N] over E[N] , Mi (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 Corol lary 1.27. Since the assertion is etale local on S, we may assume that E[N] is isomorphic to the constant group scheme (Z/NZ)2 . But, in this case Mi (N)E is isomorphic to ilaE(Z/NZ)",ord{a)=N S . D LEMMA 8.25. Let k be a field of characteristic p, and let E be an elliptic curve over k. Let N :2'.: 1 be an integer. (1) If p > 0 and G is a closed subgroup sche me 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 deg G x :5 r.p (N) . (8 . 1 2) The equality holds unless p I N and E is supersingular. ( 3 ) We have an equality deg Mi (N)E = r.p (N)'l/J(N) ( 8 . 13 ) unless p I 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 I N and E is supersingular, can we prove directly the equality ( 8 . 1 2 ) 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 pe by Proposition 8 . 2 ( 3 ) , and 0 is a generator of this. If E is ordinary, E[p e] is isomorphic to Z/pez x µp• by Propo sition 8.2(2) . Let G be a closed subgroup scheme of E[p e ] of de gree pe . Since G n µp• is a closed subgroup scheme of µp•, we have G n µp• = µPb for some b :2'.: e. Since k is algebraically closed, G is :
16
8. MODULAR CURVES OVER
Z
isomorphic to Z/p a z x µPb' a + b = e. By Lemma 8.17, (1, 1) is a generator of Z/pa z x 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 G = Ker F e . Choosing isomorphism k[[X]] ---+ 8E , o , we identify k[[X]] = 8E,O· Then, we have G = Spec k[[X]]/(X N ) . For an integer i, we denote by [i] * the ring homomorphism k[[X]] ---+ k[[X]] induced by the multiplication-by-i mapping [i] : E ---+ E. ex is a closed subscheme of G = Spec k[T]/(TN ) defined by the condition that the ideal (IJ �(/ (X - [i] * T) ) is equal to the ideal (X N ) , and we have [i] * (T) iT mod T2 • 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/p a x 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/p a z x µPb is a generator if and only if the projec tion of P to Z/p a 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, ex is equal to (Z/pa z) x x 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 iden tify E[N] with G = Z/NZ x µN . M1 (N)E is the closed subgroup scheme consisting of all the sections of G of exact order N. Decompose G = lli EZ/NZ Gi = lli EZ/NZ µ N , and M1 (N)E = lli EZ/NZ M1 (N) k. If i E Z/ NZ has order pa , and a :::; e = a + b, then by Lemma 8. 17, M1 (N) kis the inverse image of µ;b by the multiplication-by-pa mapping Gi = µp• ---+ µpb· Thus, M1 (N) k= µ;. if b > 0, and M1 (N) k= Gi if b = 0. The equality (8. 13) is clear. 0 an
=
a
As 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 2:: 1 be an integer. Define a morphism Z ---+ Z x 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/NZ by Gm, and the kernel T[N] of the multiplication-by-N map T (N) ---+ T (N) is an extension of Z/NZ by µN . For i E Z/NZ, let T (N) i and T[N] i be the inverse
17
8.3. DRINFELD LEVEL STRUCTURE
images of the natural morphisms to Z/NZ. We have r < N) and (8. 14)
N-1
N-1 =
T[N] =
r < N) i II Spec Z[q, q - 1 ] [T, T - 1 ] II i=O i=O =
N-1
II i=O
T[N] i =
N- 1
II Spec Z[q, q - 1 ] [T]/(TN - qi ) .
i=O
N 2: 1 be an integer. (1) The functor M0 (Nhc NJ over Z[q, q- 1 ] is represented by Spec Z[(d" ] [q, q - 1 ] [T]/(Td1 - (d" Qd� ). dd' =N Here, for d and d' satisfying dd' N, d" is the greatest common divisor of d and d' , and d 1 = d/ d" , d� d' / d" . (2) The functor M 1 (NhcNJ over Z[q, q- 1 J is represented by N-1 Spec Z[(d' ] [q, q - 1 ] [T]/(Td - (d' q' ) . PROPOSITION 8.26. Let
II
=
=
II i=O
·I
Here, for 0 :::; i < N, d' is the greatest common divisor of N and i , and d = N/d' , i' i/d' . =
Let S be a scheme Z [q, q - 1 ] . For a finite fl.at closed subgroup scheme G C T�N ) over S and i E Z, let Qi = G n T�N) i_ G0 is the kernel of G -t T�N ) -t Z/NZ and is a closed subgroup scheme of T�N ) o = Gm, s that is finite fl.at over S. Let d be a divisor of N, and let d' N/d. Define the subfunctor Mo (N)�fNJ of Mo (N) T
=
=
=
dd' 2: 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
(1) Let G be a closed subgroup scheme of T�N ) that is finite fiat of finite presentation over S of degree N and such that G0 = µd' ,s . Then, there exists a unique section s : S -t T ( N ) d" such that for each 0 � i < N, the diagram Gd' i � T ( N ) d' i [ d' ] (8.15) 1 s � T ( N ) d' 2 i is Cartesian. Moreover, s satisfies s d = 1 . (2) Conversely, suppose s : S -t T ( N ) d' 2 is a section that satisfies s d = 1 . Define a closed subgroup scheme G = u:,:� Gd' i ofT�N ) by the condition that diagram (8. 15) is Cartesian for each i . Then, G is a finite fiat scheme of finite presentation over S of degree N, and T�N ) is a closed subgroup scheme. (3) Let G and s be as above. Then, G is a cyclic subgroup scheme if and only if s d 1 is a generator of µd" ,s . PROOF. (1) Since [d'] : Ts( N ) d' -t Ts( N ) d"2 is faithfully fl.at, a section s : S -t T ( N ) d"2 that satisfies the condition is unique once it exists. Thus, the assertion is fl.at local on S. Since Gd' is a fl.at covering of S, we may assume Gd' has a section t : S -t Gd' . For each i, the section t : S -t Gd' defines the vertical isomorphisms in the diagram µd' ------+ G m
l
x ti
1
Gd' i ------+ Ts( N ) d' i Thus, by letting s = t d' , diagram (8.15) is Cartesian for each i. We have s d = t N = 1 by Proposition 8.14. (2) Since [d'] : G m -t G m is finite fl.at of degree d', G is a finite fl.at scheme of finite presentation of degree N. If s d = 1, it is easy to see that G is a closed subgroup scheme of T�N ) . (3) By Lemma 8.17, G is a cyclic subgroup scheme if and only if there exists a section P of Gd' such that pd is a generator of µd' fl.at locally on S, or equivalently, Gd' [ ] x µ;, -t S has a section fl.at d �Gm locally on S.
19
8.3. DRINFELD LEVEL STRUCTURE
T ( N ) d' x µ;, by We have Gd' x µ;, = S s ( N )xd'2 [d] '\.Gm [d] '\.Gm '-,. T / [d'] (8.15) . By Lemma 8.19, all the vertical morphisms in the diagram T ( N ) d' � G m f--- µ;,
l ( '
[d' ]
T N) d 2
--------"-------,
[d 1 ]
are faithfully fl.at. Thus, Gd'
l
[d� ]
Gm
x
[d] '-,. G,,,
L___
..,-----
µ;, = Ss
l
x
µd'' x
T ( N ) d' x [d) '-,.G,,, ' ( T N) d 2 x
'-,. T ( N ) d12 / [d']
µ;, is faithfully fl.at over the closed subscheme Ss x( N ) d12 [d 1 ) '-,.G,,, '-,. T µ;,, of S. Therefore, the condition that Gd' x µ;, -+ S has a [d] '-,.G,,, section fl.at locally on S is equivalent to the equality S = S x '-,. T ( N ) d'2 d ' 1 ( ) 2 d N x µ;,, , and it is also equivalent to s being a generator T [d 1 ] '\.Gm D of µd" , S · Back to the proof of Proposition 8.26. By Lemma 8.27( 1 ) , we define injection of functor Mo(N)c;!{N , -+ T ( N ) d" 2 by associating d s to G. Since d = d" di , if s 1 is a generator of µ " , s , then s d = 1 . d Furthermore, by Lemma 8.27(2) and (3) , the functor Mo(N)c;!{N , is represented by the closed subscheme T ( N ) d" 2 x µ;,, of T ( N ) d" 2 • [d 1 ] '-,. G,,, By Lemma 8.19, T ( N ) d" 2 x µ;,, is the spectrum of 8
an
[d 1 ] '\.Gm
Z[q, q - 1 ] [T, r - 1 i ® z [q,q- 1 ] [T, T - 1 ] Z[(d11] [q, q - 1 ] 1 = Z [ ( " ] [q, q- ] [T]/(T d 1 - ( " Q d� ) . d d Here, the tensor product is taken with respect to the homomorphism sending T to Td 1 / q d 1 d" 2 IN = Td 1 / qd� and the homomorphism send ing T to (d" · (2) Suppose 0 � i < N. Let d I N be the order of i E Z/NZ, 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 pd is a generator of µd' . Thus, we have N- 1 M l ( N ) T ( N ) = 11 T ( N ) i x µ;, . (8. 16) i =O
[d] '-,.G,,,
20
8. MODULAR CURVES OVER
Z
X µ;, is the spectrum of T(N)i [d]�G,.,. Z[q , q - 1 ] [T, r - 1 i © z [q , q - 1 ] [T,T - 1 ] Z[(d1 l [q , q - 1 ]
Z[(d' ] [q , q - 1 ][T]/(Td - (d' l ) . Here, the tensor product is taken with respect to the homomorphism sending T to Td / qdi /N and the homomorphism sending T to (d" . 8.4. Modular curves over Z DEFINITION 8.28. Let N ;::: 1 be an integer. (1) Define a functor Mo (N) over Z by associating to a scheme T the set isomorphism classes of pairs ( E, C) , (8. 17) M 0 (N) ( T ) = where E is an elliptic curve over T, . and C is its cyclic subgroup scheme of order N (2) Define a functor M 1 (N) over Z by associating to a scheme T the set isomorphism classes of pairs (E, P) , (8. 18) M i (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)
M i (N) ----+ Mo (N) . LEMMA 8.29. Let N = 4. The restriction of the functor M 1 ( 4 ) to Z l-:l] is represented by (8.20)
Y1 (4)z [ i l = Spec Z
[�, d, d(d � 4) ] ·
The universal elliptic curve E and the universal section P of order 4 are given by E : dy 2 = x3 + (d 2) x 2 + x, P = (1, 1 ) . (8.21) -
PROOF. Let S be a scheme over Z[!J , let f : E --+ S be an elliptic curve over S, and let P be a section of exact order 4. Let 0 be the 0section, and let Q = 2P and R = 3P. Let x be the inverse image of 1 by the isomorphism f* 0(2[0] - 2[Q]) --+ Op, and let y be the inverse image of 1 by the isomorphism f* 0(3[0] - (2[P] + [Q])) --+ Op . Define the immersion E --+ P� by the basis x, y, 1 of f* 0(3[0]). Then x and
8. 4 . MODULAR CURVES OVER
Z
21
y satisfy an equation of the form y 2 + aixy + a3 y = aox 3 + a 2 x 2 + a4x + a5 (ai E Os ), which defines E. Since the three points 0, P, R are collinear, the coordinates of 0, 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 a5 = a3 = 0, aox 3 + a 2 x 2 + a4x + a5 = aox(x - 1) 2 , and a i = - 1 . Hence, the elliptic curve E is defined by the equation y 2 - xy = a0x(x - 1) 2 , where a0 E O� . Let a0 = d/4, and substitute x - 2y by y. Then, the equation 4d(y2 - xy) = x(x - 1) 2 becomes dy2 = dx 2 + x(x - 1) 2 . The condition that the right-hand side has a triple root is d = 0, 4. D
If N = 1 , we have Mo(l) = Mi (l). 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) Mo (N) --+ M and Mi (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 -+ Ab over Q. This is extended naturally to a morphism of functors j : M -+ A� over Z. LEMMA 8.30. (1) There exists a unique morphism of functors j : M -+ A� over Z that extends the morphism of functors j : Mq -+ Ab over Q. ( 2 ) For any algebraically closed field k, the mapping j : M (k) -+ AH k) = k is a bijection. QUESTION . Prove Lemma 8.30. (Hint: To show the uniqueness, it suffices to consider the elliptic curve E over A = Z [a 1 , a2 , a3 , a4 , a5]
[�]
defined by the equation y 2 + aixy + a 3 y = x3 + a 2 x 2 + a4x + a5 . To show the existence, verify j (E) 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 y2 (x - a ) (x - f3) (x 'Y ) over Z[ � , a, /3 , ')'] [�- i ] , where � ((a - /3 ) ( /3 - 'Y) ('Y - a ) ) 2 , is given by -
(8.24)
Thus the j-invariant of the elliptic curve y2 Z[ � , a, b] [b- i , (a2 - 4b)- i ] is given by
=
x(x 2 + ax + b) over
(a2 - 3b) 3 s 2 . 2 2 . b ( a - 4b)
(8.25)
-'----'--
The j-invariant of the universal elliptic curve dy2 = x(x 2 + (d- 2)x+ l) over Yi (4)z [ ! J is given by 28 .
(8.26)
(d2 - 4d + 1) 3 . d(d - 4)
Suppose a E ( Z /NZ) x . 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) : Mi (N) --+ Mi (N) . This is called the diamond operator. The diamond operator (a) Mi (N) -+ Mi (N) is an automorphism over Mo ( 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 Mo(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") , E is an elliptic curve over T, P' is . M i (N) (T) = where 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 iso morphism class of triples (E/C', E[N']/C', (C" + C')/C') , we obtain a morphism of functors (8.28) WN' : Mo (N) --+ Mo(N),
{
}
{
}
8.4. MODULAR CURVES OVER
23
Z
called the Atkin-Lehner involution. It is easy to see Wh, is the iden tity. Let N � 1 be an integer, and let p be a prime number. We denote by Mo (N)Fp the restriction of the functor Mo (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 : Mo (N)Fp --+ Mo (N)Fp · We call it the Probenius morphism. Define a subfunctor Mo (N)�p of Mo (N)Fp by associating to a scheme T over F P the set isomorphism classes of pairs ( E, C) , where E is a super�i�gular �lliptic . ( ss ) (8.29) Mo (N) F T = curve over T, and C is its cychc subgroup scheme of order N Define the restriction of functor M 1 (N)Fp and the Frobenius mor phism 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 = Mp. Define a morphism of functors over FP io : Mo (M)Fp --+ Mo (N)Fp by associating to the isomorphism class of a pair (E, C) the isomorphism class of (E, C, Ker F) . Also, define ii : Mo(M)Fp --+ Mo (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 Mo (N) over Z. Yo ( N) z is a normal connected affine curve over Z. (2) The morphism Yo (N)z ---+ Y(l)z (8.30) induced by the morphism of functors Mo (N) --+ M (8.23) is
{
}
P
finite fiat of degree t/J(N) .
(3) Yo (N)z is smooth over Z[kJ . For a prime number p f N,
Yo (N)Fp = Yo (N)z ©z Fp is the coarse moduli scheme of the restriction Mo (N)Fp · (4) Let p be a prime number, and assume N = Mp for an inte ger M prime to p. Then, Yo (N)z is weakly semistable at p . The morphisms jo , j1 : Mo (M)Fp --+ Mo (N)Fp over Fp in duce closed immersions jo , j1 : Yo (M)Fp --+ Yo (N)z. The fiber Yo (N)Fp is the union of the image Co of jo and the image Ci of i i - The intersection of Co = Yo (M)Fp and Ci is the
24
8. MODULAR CURVES OVER
Z
coarse moduli Yo (M)� of Mo(M)�p . The index ex of the ordinary double point x = [(E, C)] E Y0 (M)�p is the order of Aut(.Ep.p , Gp;p )/{±1}. The fiber Yo (N)Q = Yo (N)z © z Q of Yo (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) y2 = 4x 3 - 4x2 - 40x - 79 of Xo(ll)Q becomes (8.31) y 2 + y = x3 - x 2 - lOx - 20. The homogeneous equation of (8.31) defines an elliptic curve over Z[ AJ, and the fiber of F 11 is the Neron 1-gon. If we let s = �=�� ' (8.31) becomes s2 (x - 16) + 121s = x2 + 15x + 230. The morphism Spec Z[x, s]/(s 2 (x - 1 6 ) + 121s - (x 2 + 15x + 230)) -+ Spec Z[x, y]/(y 2 + y - (x 3 - x2 - lOx - 20)) is an open immersion over Z[AJ. Through this immersion, we identify Y0 (11)z = Spec Z[x, s]/(s2 (x - 16) + 121s - (x 2 + 15x + 230)) . We have Yo (ll)F 11 = Spec F11 [x, s]/((s2 - ( x - 2)) (x - 5)) , and the curve Yo (ll)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 Mi (N) over Z . If N � 4, Y1 (N)z l -k l 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(l)z induced by the morphism of functors Mi (N) -+ M (8.23) is finite flat of degree 'l/J(N)(N)/2 if N � 3, and of degree 3 if N = 2. (3) Y1 (N)z is smooth over Z[:kl · For a prime number p f N, Y1 (N)F p = Y1 (N)z © z Fp is a coarse moduli scheme of the restriction M 1 ( N) F p P
•
8.5. MODULAR CURVE Y ( r )z[ � J
25
REMARK. The modular curves Yo (N) z and Y1 (N) z are the in tegral closures of Yo (N)z [ -kJ and Y1 (N)z [ -k l ' respectively, by Theo rems 8.32 and 8.34. However, if we defined Yo (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 [ � J
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 Mo, * (N, r)z [ � J over Z [ � ] by associating to a scheme T over Z [ � ] the set (8.33)
isomorphism classes of triples (E, C, a ) , whe�e is an elliptic curve over T, C is its = cychc E subgroup scheme of order N, and . a (Z/rZ) 2 --7 E [r] is an isomorphism (2) Define the functor M 1 ,* (N, r)z [ � J over Z [ � ] by associating to a scheme T over Z [ � ] the set
1
M o, * (N' r) Z [ ;: ] ( T )
(8.34)
{
}
:
isomorphism classes of triples (E, P, a ) , wh is. an elliptic curve over T, . = P �isrea Esection of exact order N, and a (Z/rZ) 2 --7 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 Mo,* ( 1 , r)z [ � J and on M 1 ,* (l, r)z [ � J · If N = 1 , we have Mo, * ( 1 , r) z [ �J = M 1 ,* ( 1 , r) z [ � ] ' which we denote simply by
1
M 1• * (N' r) Z [ ;: ] ( T )
{
}
:
M (r)z [ �J ·
by
EXAMPLE 8.36. Let r = 3. The functor M (3) z [ t J is represented
(8.35)
Y(3)z [ t J = Spec Z
[ �, (3 , µ , µ3 � 1 ] ·
8. MODULAR CURVES OVER Z 26 The universal curve (E, 0) and the universal base a are given respec tively by E : X 3 + Y3 + Z3 - 3µX Y Z = 0, 0 : (0, 1 , - 1 ) , (8.36)
a : ( 1 , 0)
t-+
(0, (3 , - l ) , (0, 1)
t-+
( 1 , 0, - 1 ) .
QUESTION . Verify Example 8.36. (Hint: Use the proof of Theo rem 2.21 for the case N = 3.) Let r, s :?: 1 be integers. Let H be the kernel Ker(GL 2 (Z/rsZ) -+ GL2 (Z/rZ)) . Define a morphism of functors over Z [ ..1s ] (8.37) M (r)z [ -f. J --+ [M (rs)z [ -!; J f H] as follows. Let S be a scheme over Z[..18 ] , let E be an elliptic curve over S, and let a (Z/rZ)2 -+ E[r] be an isomorphism. Define a functor Isoma ((Z/rsZ)2, E[rs]) over S by associating to a scheme T over S the set Isoma ( (Z/rsZ) 2 , E[rs]) (T) = {isomorphism (3 : (Z/rsZ) 2 -+ ET [rs] I (3 induces a } . The functor Isoma ((Z/rsZ)2, E[rs]) is represented by an H-torsor P over S. The element of M (rs)z [ -f. J (P) defined by the pair (Ep, (3) of the pullback Ep of E to P, and the universal isomorphism (3 over P is compatible with the action of H. Thus, it defines ele ment of [M(rs) z[ -!; J f H] ( S ) . We thus have a morphism M(r) z[ -f. J -+ [M(rs)z [ -f.J / H] . Similarly, for relatively prime integers N, r :?: 1 , we define mor phisms of functors over Z [;] Mo(N) z [ � l --+ [Mo A N, r) z[ � 1 /GL2 (Z/rZ)] , (8.38) Mi (N) z[ � J --+ [M1 ,* (N, r) z[ � 1 /GL2 (Z/rZ)] . LEMMA 8.37. Let r :?: 3 be an integer. The functor M (r) z [ � ] over z[;J is represented by a smooth affine connected curve Y(r)z [ � ] over z [;J . The field of constants of Y(r)q = Y(r) z[ � ] ©z[ � ] Q is Q((r ) Y(r)z[ � ] ©z [ � l Q is the modular curve Y(r) constructed in The orem 2.21. Lemma 8.37 is proved in the same way as in §2.4. :
an
8.5. MODULAR CURVE Y ( r ) z [ � l
27
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 w and P is the universal element of order 4. Let Q be the diagonal section of E over E[4] . Let Y(4)z [ !J 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 [ !J is represented by Y(4)z [ !J · If r is a multiple of 3, M (r)z [ �J is represented by the finite etale scheme Y(r)z [ �J over Y(3)z [ �J · 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 2:: 3 be an integer, and let S be a scheme over Z [�] . If the restriction 9IE [r] is the identity morphism, then so is g .
(2) Let N 2:: 4 be an integer, and let S be a scheme over Z [ k l · Let C be a cyclic subgroup scheme of E of order N. If the restriction g l c is the identity morphism, then so is g .
PROOF. The invertible sheaf £ = 0(3[0] ) on E defines a closed immersion E -+ P ( f* .C) . (1) Let D 2:: 0 be the Cartier divisor defined by D = [E[r]] - [OJ . Since r is invertible on S, we have D n 0 = 0 , and .CID = O \ D · Moreover, since r 2:: 3, we have deg D = r 2 - 1 > 3, and f*.C -+ £ \ D is injective. Thus, if g \ E [r] is the identity, the action of g on f* .C c £\ D = OD is trivial, and so is the action on P ( f* .C) . Hence, the action of g on E is also trivial. (2) Let D = C- [O] . If N 2:: 5, we have deg D = N - 1 > 3. Then, the proof goes similarly to (1). If N = 4, let P E C be the section of exact order 2. Then, by Example D.4, we have £ ( -D ) � O( [P] - [O] ) locally on S. Thus f* .C -+ £ \ D is injective, and the rest is similar to D the proof of (1). COROLLARY 8.39. (1) Let s 2:: 1 be an integer, and let H = Ker {GL 2 (Z/rsZ) -+ GL 2 (Z/rZ)) . If r 2:: 3, the morphism of functors M (r)z [,_l; J -+ [M (rs)z [,_l; J /H] (8.37) over z [ ;8 ] is an isomorphism.
(2) Let N 2:: 4 be an integer relatively prime to r. The morphism of functors M i (N)z [ Jr l -+ [M 1 ,* (N, r)z [ Jr l /GL 2 (Z/rZ)] (8.38) over Z [ �r l is an isomorphism.
28
8.
MODULAR CURVES OVER
Z
PROOF. (1) We construct the inverse morphism. Let S be a scheme over Z[ r18 ] . Let P be an H-torsor over S, and let (E, /3) E M (rs) z [ � J (P) be an H-invariant pair of an elliptic curve over P and an isomorphism f3 : (Z/rsZ) 2 -+ E[rs] . Let a : (Z/rZ) 2 -+ E[r] be the isomorphism induced by /3. Let g E H. Then, by the assump tion r ;::: 3 and Lemma 8.38(1), there exists a unique isomorphism g * (E, /3) = P x (E, /3) -+ (E, /3 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 x s P -+ E is an isomorphism. Moreover, the isomorphism a : (Z/rZ) 2 -+ E[r] is the pullback of an isomorphism as : (Z/rZ) 2 -+ Es [r] . Sending (E, /3) to (Es , as), we obtain the inverse morphism M (r)z [ �] -+ o
[M (rs)z [ � J /H] .
(2) We construct the inverse morphism. Let S be a scheme over Z[ Jr ] , and let Q be a GL 2 (Z/rZ)-torsor over S . Let (E, P, a) E Mi, * (N, r)z [ Jr l (Q)
be a GL2 (Z/rZ)-invariant triple of an elliptic curve E over Q, a sec tion P of exact order N, and an isomorphism a : (Z/rZ) 2 -+ E[r] . The section P defines an isomorphism Z/NZ -+ (P) c E. Suppose g E GL 2 (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 D rest is similar to the proof of ( 1). We show Lemma 8.37 when r is general. Suppose s = 3 or 4. The functor M (rs)z [ �] is represented by Y(rs) z [ � ] · By Lemma 8.38(1), the natural action of H = Ker (GL 2 (Z/rsZ) -+ GL 2 (Z/rZ)) on Y(rs)z [ �] is free. Thus, by Lemmas A.31 and A.33, the natural morphism Y(rs) z [ � ] -+ Y(rs)z [ �J /H is finite and etale, and the quotient Y (rs)z [ �J /H represents the functor [M (rs)z [ �J /H] over Z[ r18 ] . By Corollary 8.39(1), the quotient Y (r) z [ �] = Y(rs)z [ �] /H represents the functor M (r) z [ � ] · Moreover, Y(r) z [ � ] is a smooth affine curve over z [...!.. ] . Y (r)z [ 1.r ] is obtained by gluing Y(r)z [ ..L ] and Y(r) z [ fr-J on Y(r)z [ f,:J · Let ( , )E [r] : E[r] x E [r] -+ µr be the Weil pairing. Associating to the pair (E, a) the root of unity (a( l , O) , a(O, l))E [r] > we obtain Y(r)z [�] -+ Z[� , (r] · To show that the field of constants of Y(r)Q = Y(r)z [ �] ©z [ � ] Q is Q((r), it suffices to show that the Riemann surface �
�
8.5. MODULAR CURVE Y( r )z[ � J
29
Y(r)an defined by Y(r)c = Y(r)Q ®Q ( (r ) C is connected. Let r(r) be the subgroup of SL 2 (Z) defined by (8.39) I'(r) = Ker (S L 2 (Z) --+ SL 2 (Z/rZ)) , and consider the natural action of I'(r) on the upper half-plane H = { E C I Im > O}. As in Corollary 2.66, we obtain an isomorphism of Riemann surfaces T
T
I'(r)\H --+ Y(r) an . Thus Y(r) an is connected and Y(r)Q ( (r ) is a smooth connected affine curve over Q ( (r ) · 0 (8.40)
COROLLARY 8.40. (1) There exists a coarse moduli scheme Y(l) z of the functor M . (2) The morphism of functors M --+ A� defined by the j -invariant
induces an isomorphism j : Y(l) z --+ A� . (8.41) (3) Let r � 3 be an integer. The restriction of the natural morphism j : Y(r)z [ �J --+ Y(l) z = A� to U = Spec Z [j, iU _!1 2 a ) ] c A� Y (r)z [ �J x A� U --+ U is a GL2 (Z/rZ) /{±l}-torsor.
PROOF. (1) As in the proof of Lemmas 2.27 and 8.37, the coarse moduli scheme Y(l) z of M is obtained by gluing the quotient of Y(3)z [ t l by GL 2 (Z/3Z) and the quotient of Y(4)z [ ! l by GL 2 (Z/4Z ) . (2) By the construction in (1), Y(l) z is a normal affine curve over z. Since j : Y(l) z --+ A� is an isomorphism over Q, it is a birational morphism. Moreover, by Lemma 8.30(2) , the morphism of normal schemes j : Y(l) z --+ A� induces a bijection on each geomet ric fiber, and thus it is an isomorphism. (3) The natural action of GL 2 (Z/rZ) on Y(r)z [ �] is an action as an automorphism over Y(l) z . Since the multiplication-by-(-1) morphism is an automorphism of the universal elliptic curve, the ac tion of -1 E GL 2 (Z/rZ) on Y (r)z [ � ] is trivial. Since Y(r) z [ � ] --+ Y(l)z [ � ] is a finite morphism of regular schemes, it suffices to show, by Lemma A.34, that each geometric fiber over U = Spec Z[�] (j, j(j _!1 2 a ) ] is a GL 2 (Z/rZ)/ { ±l } -torsor. Let k be an algebraically closed field with r E kx , and let E be an elliptic curve over k with j(E) =/:- 0, 12 3 . Since Y(r)z [�] is a fine moduli scheme, the fiber of the morphism
30
8. MODULAR CURVES OVER
Z
Y(r)z [ i l -+ Y(l) at j(E) E A 1 (k) = Y(l)z [ iJ (k) is identified with Isom ((Z/ r Z) 2 , E[r])/{±1}. This is a GL 2 (Z/rZ)/{±1}-torsor since
0 we have Aut(E) = {±1} by Lemma 8.41 below. LEMMA 8.41 . Let k be an algebraically closed field of character istic p � 0, and let E be an elliptic curve over k . (1) The automorphism group Aut(E) is finite, and the order of g E Aut(E) is either a divisor of 4 or a divisor of 6. (2) If j (E) =/= 0, 12 3 , then Aut(E) = {±1}. (3) If p =/= 2, 3 and j(E) = 0 , then Aut(E) = µ5 . (4) If p =I= 2, 3 and j (E) = 12 3 , then Aut(E) = µ4 • (5) If p = 3 and j (E) = 0 = 123 , then U Aut(E) = 12, and 1 -+ {±1} -+ Aut(E) -+ Aut(E[2]) -+ 1 is an exact sequence. (6) If p = 2 and j(E) = 0 = 12 3 , then U Aut(E) = 24, and the natural mapping Aut(E) -+ Aut(E[3] , ( , ) 3 ) SL 2 (F 3 ) 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 E Aut(E) , then the order of g is finite. The charac teristic polynomial det(T - g) E 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 y2 = x3 + ax + b, a, b E k. An automorphism of E is then given by (x, y) t-t (u2 x, u3 y) with u E P satisfying u4 a = a, u6 b = b. If j =/= 0, 1728, then we have a =I= 0, b =I= 0, and thus u = ±1. If j = 0, then we have a = 0, b =I= 0, and thus u is a 6th root of unity. If j = 1728, then we have a =I= 0, b = 0, and thus u is a 4th root of unity. 0 We omit the proof of (5) and (6) . EXAMPLE 8.42. Define an elliptic curve E over the open scheme Uz [ ! J = Spec Z[j l ( i , j(j !1 23 ) J c Y(l) z = A� = Spec Z[j] by 123 j - 243 j . (2.24) y 2 = 4x 3 j - 12 3 x j - 12 3 As we showed in Proposition 2.15(1), the j-invariant of E equals j. Let � 3 be an integer. The functor associating to a scheme T over Uz [ t,: J the set {isomorphisms (Z/rZ) 2 -+ E[r]r of group schemes over T} is represented by a GL 2 (Z/rZ)-torsor M(r)E,Uz[ -1... J . The r
6r
8.5. MODULAR CURVE Y ( r ) zl tJ
31
..L
morphism M(r) E ' uZ [ 6r ] ---+ Y(r) z [ .!.r ] X Y ( l ) z Uz [ ...L6r ] over Uz [ ...L6r ] defined by the universal isomorphism (Z/rZ) 2 ---+ E[r] is compatible with the action of GL 2 (Z/rZ). This induces an isomorphism M(r) E ' uZ l i / {± 1 } ----+ Y(r)z [.!. ] x Y ( l )z Uz [ ...L 6r ] sr l r
of GL 2 (Z/rZ)-torsors over Uz [ j,, ] · Y(r)z [ � l is isomorphic to the in tegral closure of Y(l)z [�] in M(r)E.Uz ii;: / {± 1 } .
L
Let E be the elliptic curve over Spec Z [ ! , >., >. ( >. ) ] 2 defined by y = x(x - l ) (x - >.) , and let a (Z/2Z) 2 ---+ E[2] be the isomorphism defined by the basis (0, 0) , ( 1 , 0) . Show that defines an isomor the pair (E, a ) E M (2)z [ ! J ( Spec Z [! , >., phism Spec Z[! , >., .>. ( l� .>. ) ] ---+ Y(2)z [ ! l ' where Y(2)z [ ! J is the coarse moduli scheme of M (2)z [ !J · QUESTION .
:
.x cL.xi J )
COROLLARY 8.43. Let N ;::: 1 be an integer, and let r ;::: 3 be an integer relatively prime to N. The functor M 1 ,* (N, r)z [ �] over Z[�] is represented by a finite scheme Y1 ,* (N, r)z [ �J over Y (r)z r n Y1 , * (N, r )z [ rN1 l is smooth over Z [ r1N ] . The field of constants of Y1 ,. (N, r)Q = Y1 ,. (N, r)z [ �J ®z m Q is Q( (r ) · PROOF. Let E be the universal elliptic curve over Y(r)z [ � ] · By Corollary 8.24, the functor M i ,. (N, r)z [ � ] is represented by the finite scheme Mi (N)E = Y1 ,. (N, r)z [�] over Y(r)z [�] · Since Y1 ' . (N, r) z [ ..L ---+ Y(r)z [ ..L J is etale and Y(r)z [ ..L J is a Nr J Nr Nr smooth affine curve over Z [ Jr ] , Y1 ,. (N, r)z [ Jr l is also a smooth affine curve over Z[ Jr l · The proof of the fact that Q ( ( ) is the field of constants of Y1 ,. (N, r)Q = Y1 ,. (N, r)z [ � ] ©z [ � ] Q is similar to that of Lemma 8.37, D and we omit it. r
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 [�] using Igusa curves and prove Theorem 8.34. Finally in §8.8, we study the modular curve Yo,. (N, r)z [�] and prove Theorem 8.32.
32
8. MODULAR CURVES OVER
Z
8.6. lgusa 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)F p = Y(r)z [ � ] @ Fp , and let Y(r)Fp be the closed subscheme of Y(r)F p 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)5F5p is a Cartier divisor of Y(r)F p and is finite etale over Fp . PROOF. We show S = Y(r)Fp is etale over Fp . It suffices to show that the absolute Frobenius morphism F : S-+ S is an automorphism of S of finite order. F : S-+ S is the endomorphism defined by send ing the isomorphism class [(E, a ) ] of a pair of a supersingular elliptic curve E and a basis a of E[r] to the isomorphism class [(E (P) , a
Since E is supersingular, we have Ker[p] = Ker F2 • Thus, we have an isomorphism fE E -+ ECP2 ) satisfying F2 = fE [p] . This implies that F2 maps [(E, a)] to [(ECP2 ) , aCP2 ) )] = [( E, a p) ] . Hence, if n is the order of p E (Z/rz) x , F2n is the identity morphism of S. Y ( )Fp is a closed subscheme that is etale over FP of the smooth curve Y (r)F p over Fp . Thus, Y(r)Fp is a Cartier divisor of Y(r)F p and is finite over Fp · 0 COROLLARY 8.45. Let p be a prime number, let S be a scheme over F 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 F2 a . 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 a : ( Z/ r Z ) 2 -+ E [r] , where r � 3 is an integer relatively prime to p. Let S -+ Y(r)F p be the morphism defined by (E, a ) . Since S-+ Y(r)F p factors through the closed subscheme of Y(r)F p defined by the closed condition Ker[pa ] = Ker F2 a , it suffices to show the assertion assuming S to be the closed subscheme of Y(r)F P defined by the condition Ker[pa ] = Ker F2 a . Then, similarly to the proof of Lemma 8.44, the absolute Frobenius morphism of S is automorphism of S of finite order, and thus S is etale over Fp · The assertion now follows from Proposition 0 8.2(3) (i) ::=? (ii) . o
:
o
r
P,
an
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. !GUSA CURVES
33
an integer, and define G (a , b) by
(8.42)
G (a , b )
=
{
Ker ( V a pb : E --+ E (Pb -''> ) Ker ( V a pb : E (P"- b) --+ E)
if a :S b, if a � b.
(1) G (a , b) is a finite fiat scheme of finite presentation over S of degree t.p(pe) .
(2) Gc a, b) is a cyclic subgroup scheme of order pe . (3) If P is a generator of Gc a ,o) = Ker v a , P has exact order pe as a section of E (P" ) .
PROOF. (1) Let r � 3 be an integer relatively prime to p. Since the assertion is fl.at local on S, we may assume there is a basis a of E [r] over S. Since a 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) , ex has degree t.p(pe) at each point of 5ord , and degree :::; t.p(pe) at each point of sss . By Lemma 8.44, ex is a finite fl.at scheme of degree t.p(pe) over a dense open subscheme U C 5ord of S. Thus, by Lemma A.43, ex is a finite fl.at scheme of degree t.p(pe) at every point of S. (2) By (1), G (a , b) is a fl.at covering of S and G (a , b) has a universal generator on G (a , b) " (3) As in (1), the assertion is fl.at local on S. Thus, it suffices to prove, assuming that r � 3 is an integer relatively prime to p, S is G (a ,o) over Y(r)Fp and P is the universal generator of Gc a,O ) · Since 0 is a generator of Ker p b , the assertion on 5ord follows by applying Lemma 8.17 to the exact sequence 0 --+ Ker pb --+ Gc a , b) --+ Ker v a --+ 0. Since S is fl.at over Y(r)Fp , there is no closed subscheme of S other than S itself that contains 5ord as an open subscheme. Thus, the assertion follows from Lemma 8.23. D 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)F The finite fl.at scheme x ) ( Ker v a · E (P" ) --+ E) x G (a,O of degree t.p(pa ) over Y1 , * (M, r)Fp is called the Igusa curve and is denoted by Ig(Mpa , r)Fp · P
-
•
.
8. MODULAR CURVES OVER
34
Z
If a = 0, we have Ig(M, r)F,, = Y1 , * (M, r)F,, · The Igusa curve Ig(Mpa , r)F,, represents the functor that associates to a scheme T over FP the set isomorphism classes of quadruples (E, P, P', a ) , where is an elliptic curve over T, P is a generator of Gc a ,o ) = . (8 . 43) E Ker v a c ECP°" ) , P' is a section of E of exact order M, and a (Z/rZ) 2 --+ E[r] is an isomorphism. If P is the universal generator of the cyclic subgroup scheme G c a , O ) over the Igusa curve Ig(Mpa , r)F,, , we denote by Ig(Mpa , r):;= 0 the closed subscheme of lg(Mpa , r)F,, 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 lgusa curve Ig(Mpa , r)F,, is a smooth affine curve over Fp. (2) The natural morphism lg(Mpa , r)F,, --+ Y1 , * (M, r)F,, is etale over Y1 , * (M, r)'F: . (3) Suppose a :'.'.'. 1 . The closed subscheme Ig(Mpa , r):;= 0 is the reduced part of lg(Mpa , r)� . The morphism Ig(Mpa , r): = o --+ Y1 , * (M, r)�,, is an isomorphism. PROOF. The morphism Ig(Mpa , r)F,, --+ Y1 , * (M, r)F,, is the base change of the morphism Ig(pa , r)F,, --+ Y(r)F,, by the etale morphism Y1 , * (M, r)F,, --+ Y(r)F,, · Thus, it suffices to show them assuming M = 1. (2 ) Since Ker va is et ale over y ( r w:' the morphism lg (pa ' r) F " --+ Y(r)F,, is etale over Y(r) �.r: . (3) We show Ig(pa , r ): = ° C Ig(pa , r )� . To do so, it suffices to show that the universal elliptic curve E over S = Ig(pa , r):"= 0 is supersingular. Since P = 0 is a generator of Ker V a C E (P°" ) over S, we have Ker va = Ker pa , and thus Ker[pa ] = Ker F2 a c E. Hence, by Corollary 8.45, E is supersingular. Since we have Ig(pa , r):"= 0 c Ig(pa , r)�" , we obtain a morphism Ig(pa , r):"= o --+ Y(r)�" . This is an isomorphism since the inverse morphism Y(r)�" --+ Ig(pa , r):"= 0 is defined by sending (E, a ) to (E, 0, a ) . For a supersingular elliptic curve over a field k of characteristic p, the only section P E E(k) that generates Ker Va is P = 0. Thus,
{
}
:
p
"
p
p
8.6. !GUSA CURVES
35
Ig(pa , r) :; o --+ Ig(pa , r)�P is surjective. Since Ig(pa , r) :; 0 = Y (r)�P is reduced, this is the reduced part of Ig(pa , r )�p . (1) It suffices to show it assuming a ;::: 1. Let E be the universal elliptic curve over Ig(pa , r)F p · By Lemma B.2(1) , the 0-section of E is a Cartier divisor of E. Thus, the closed subscheme Ig(pa , r ):p= 0 c Ig(pa , r )F p is defined locally by a principal ideal. Since Ig(pa , r )F p is fl.at over Y ( r)F P ' Ig(pa , r) :; o is a Cartier divisor of lg(pa , r)Fp by Lemma A.40. Ig(pa , r) :p= o is etale over Fp · Thus, by Lemma B.2(2) , Ig(pa , r )F p is smooth over FP on a neighborhood of Ig(pa , r ):; 0 . Since the complementary open subscheme Ig(pa , r )F"pd of Ig(pa , r ):p= 0 is etale over Y ( r w:, it is smooth over FP . Thus, Ig(pa , r )F P is smooth everywhere over Fp · D Let 0 � a � e be integers, and let N = Mpe . 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' of exact order M and a basis a of E[r] , P' (P" ) and aCP" ) define a section of E (P" ) of exact order M and a basis of E (P" ) [r] , respec tively. Thus, by Lemma 8.46(3) , a morphism ia : Ig(Mpa , r)F p --+ Y1 , * (N, r)F p c Y1 , * (N, r)z [ �] is defined by sending the isomorphism class [(E, P, P', a)] to the isomorphism class [(ECP" ), (P, pi (p" ) ) , aCP" l ) ] . 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 = Mpa . (1) For 0 � a � e, the morphism (8.44) ia : Ig(Mpa , r)F p -+ Y1 , * (N, r)F p
is a closed immersion. � a � e, if we denote by Ca the image of the closed immersion ia : Ig(Mpa , r )F p --+ Y1 , * (N, r)F p , then we have Y1 , * (N, r)F p = LJ: =o Ca . For each 0 � a � e, the inclusion c�s --+ Y1 , * (N, r)�p is bijective. (3 ) For 0 � a < a' � e, the intersection Ca X y1 ( N ,r )F p Ca ' is c�s .
(2) For 0
.•
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 F • E is an elliptic curve over S, and P is a section of E over S, then byP 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 condition, and let (P) pa = L::f: � 1 [iP] over T. The condition that the kernel of the dual of Er ---+ E' = Er/ (P) pa equals is a closed condition on T. Define a closed subscheme Ca of Y1 , * (N, r)F p by the closed con dition: The universal generator P has exact order pa , and (8.45) the kernel of the dual of E ---+ E / (P)pa equals Ker p a . The morphism ja : Ig(pa , r)F p ---+ Y1 , * (N, r)F p defines a morphism Ig(pa , r)F p ---+ Ca . Define a morphism Ca ---+ Ig(pa , r )F p by sending ( E, P, a) to the isomorphism class of ( E', P, the image of a p - a ) . Since E = E' (Pa ) over Ca , and y a pa = [pa ] , this is the inverse morphism of lg(pa , r)Fp ---+ Ca . Thus, the morphism Ig(pa , r)Fp ---+ Ca is an isomorphism, and ja : Ig(pa , r)F p ---+ Y1 , * (N, r)F p is a closed immersion. (2) For a rational point [ (E, P, a) ] E Y1 , * (N, r) (k) over an al gebraically closed field of characteristic p, let pa be the order of P E E(k) . Then we have [ (E, P, a) ] E Ca (k) . Thus, we have Y1 , * (N, r)F p = LJ� =O Ca If E is supersingular, then P = 0. In this case, for any 0 ::; a ::; e, we have [(E, P, a) ] E Ca (k) , and thus the mapping Ig(pa , r)Fp ---+ Y1 , * (N, r)Fp is bijective. (3) Let (E, P, a) be an elliptic curve with the universal level structure over the intersection Ca X y1 _ . (N ,r ) Fp Ca ' Consider E ---+ E' = Ej(P)pa ---+ E" = Ej(P) pa ' · Since pa p = 0, we have Ker(E' ---+ E") = Ker pa ' - a . Since the kernel of its dual is also Ker pa ' - a , we have E' [pa' - a ] = Ker F 2( a' - a) . Thus, by Corollary 8.45, E' is supersingular. Hence, E '.:::'. E' (Pa ) is also supersingular. The in tersection Ca n Ca ' is a closed subscheme of c�s . We now show c�s c Ca n Ca ' · Let (E, P, a) be the universal elliptic curve with level structure over c�s. Since Ker[pa ] = Ker F2 a , the kernel of E ---+ E' = E/ (P) pa equals Ker Fa . Since pa p = 0, (P) pa ' = l::f�0- 1 [iP] is the inverse image of Ker pa ' - a by E ---+ E'. Since (P) pa ' = Ker pa ' and Ker[pa ' ] = Ker F2 a ' , we have c�s c o
o
·
·
�-
0
8.7. MODULAR CURVE Y1 (N)z
37
8.7. Modular curve Y1 (N) z PROPOSITION 8.50. Let N 2:'.: 1 be an integer, and let r 2:'.: 3 be an integer relatively prime to N. The fine moduli scheme Y1 ,* (N, r)z [ �] The natural morphism is a regular affine curve over Z[�]. Y1 , * (N, r)z [ � ] ---+ Y1 (r)z [�] is finite fiat of degree cp(N)'lf;(N) . PROOF. Let p be a prime number. We first show Y1 ,* (pe , r)z [ �J is regular in the case where N = pe > 1. Since Y1 * (pe , r )z [ ...L ] is smooth over z[;r ], it suffices to examine a neighborhood of Y1 ,* (pe , r) F · Let S = Y1 ,* (pe , r)z [�] · For 0 ::; a ::; e, define a closed subscheme Da of S = Y1 ,* (pe , r)z [ �J as follows. Let P be a universal element over S of exact order p e . If a < e, let Da be the closed subscheme defined by the condition pa p = 0, and define De = Y1 ,* (pe , r) F · 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 ia : lg(pa , r) ---+ S (8.44) . Let D!t be the open subscheme of Da defined by the condition that the divisor Lf: � 1 [iP] is etale over Da . LEMMA 8.51. (1) Co = Do . (2) If 0 < a ::; e , we have c�rd = D!t . PROOF . (1) Co is the closed subscheme of Yi ,* (pe , r)z [ �J defined by the condition p = 0 and P = 0, and Do is the closed subscheme defined by . the condition P = 0. Since Do is a scheme over F by Lemma 8.22, we have Co = Do . (2) On c�r
pr
p
p
P
is etale. Thus, we have c�r
38
8. MODULAR CURVES OVER
Z
We show S - S:!p is regular. Sz [ i l --+ Y(r) z [t;J is finite etale of degree r.p(N)'ljJ(N) . Thus, by Lemmas 8.25 and A.43, we see that S - SFp --+ Y(r) z [ �J - Y (r)�P is finite fiat. Since Y(r) z [ � J is smooth over Z[ � ] , S - S:!P is also finite fiat over Z[ � ] . Since Da is defined by a principal ideal locally on S, c�rd is a Cartier divisor of S - SWp by Lemmas 8.51 and A.40. By Lemma 8.48, Ca is a regular subscheme of s. Since SF p = u:= O Ca by Proposition 8.49(2) , s SFp is regular. Applying Corollary A.42 to the closed immersion S:! c Do = C0 --+ S, we see that S is regular on a neighborhood of SWp . S --+ Y(r) z [�J is finite fiat of degree r.p(N)'ljJ(N) except for S:!p · Since both S and Y(r)z [ �] are regular, S --+ Y(r) z [�] is finite fiat everywhere and of degree r.p(N)'ljJ(N) by Proposition A.13(2) . This proves that Y1, * (N, r) z [ �J is a regular affine curve over Z[ � ] in the case N = pe . We now show it for general N. If N = Ilp l N pe P , Y1, * (N, r) z [ �J has an open covering up lN Y1 , * (N, r) z [� ] " Since the natural mor phism Y1 , * (N, r) z [�] --+ Y1 , * (pe P , r) z [�] is finite etale of degree r.p(N/pe P )'ljJ(N/pe P ) , the proof is reduced to the case N = pe . D
-
p
PROOF OF THEOREM 8.34. (1) Let r � 3 be an integer rela tively prime to N. Consider the action of the group GL 2 (Z/rZ) on Y1 , * (N, r)z [ �J · As in Proposition 2.23, the quotient Y1 (N) z [ �J of Y1 , * (N, r )z [ �J by the action of the group GL 2 (Z/rZ) is the coarse moduli scheme of the restriction of the functor M 1 (N) over Z to Z[ � ] . By Proposition 8.50, Y1 (N)z [�] is a normal affine curve over Z[ � ] . Since the coarse moduli scheme Y1 (N)z is obtained by patching together Y1 (N) z [ �J with (N, r) = 1, it is a normal affine curve over Z. If N � 4, then by Lemma 8.38(2), the action of GL 2 (Z/rZ) on Y1 , * (N, r)z [ �J is free. Similarly to the proof of Lemma 8.37, Y1 (N)z [ Jr l represents the functor [M 1 , * (N, r) z [ k rJ /GL 2 (Z/rZ)] by Lemmas A.31, and A.33. Thus, by Corollary 8.39(2) , Y1 (N)z [ Jr l is a fine moduli scheme of M i (N)z [ Jr l · 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 r 1 (N) be the subgroup of SL2 (Z) defined by
(8.46) r 1 (N) =
{( � � ) E SL2 (Z) l a = l mod N, c = O mod N } ,
8.7. MODULAR CURVE Y1 (N)z
39
and consider the action of I'1 (N) on the upper half-plane H { E C I Im > O}. As in Corollary 2.66, we obtain an isomor phism of Riemann surfaces (8.47) 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 [ �] -+ Y1 (r) z [�] is a finite surjective morphism of two-dimensional normal schemes by Proposition 8.50. Thus, the induced morphism Y1 (N)z -+ Y(l)z is also a finite surjective mor phism of two-dimensional normal schemes. Since Y(l)z is isomorphic to A� , it is thus regular. Hence, Y1 (N)z -+ Y(l)z is finite flat by Proposition A.13(2) . Y1, * (N, r)z [�] -+ Y1 (N) z [ �] is a Galois covering whose Galois group equals GL 2 (Z/rZ) for N 2:: 3 and GL 2 (Z/rZ)/{±l} for N = 1, 2 by Lemma 8.41 (2) . Since the degree of Y1, * (N, r) z [ �] -+ Y1 (r) z [�] is 1/J(N)cp(N) by Proposition 8.50, the degree of Y1 (N) z [�] -+ Y(l) z [ �] equals 1/J(N)cp(N)/2 for N 2:: 3, and 1/J(N)cp(N) for N = 1, 2. (3) Since Y1, * (N, r) z [ �] is a smooth affine curve over Z[ � ] , its quotient Y1 (N) z [ �] is a smooth affine curve by Proposition B.10(1). Suppose p f N. If N 2:: 4, Y1 (N) z [Jl,-J is a fine moduli scheme, and thus Y1 (N)z ©z Fp is a fine moduli scheme of the restriction M1 (N)F p · As for the action of GL 2 (Z/rZ) on Y0 , * (N, r) z [ �J · 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 =F 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 GL 2 (Z/rZ) , and it is a coarse moduli scheme of the restriction of the functor M1 (N)F D T
T
P
.
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 2:: 1 be an integer. (1) The finite scheme Mi (N)E over S is fiat of finite presentation over S of degree cp(N)'l/J(N) . The scheme M1 (N)E is a Cartier divisor of E. If N = N'N" with (N', N") = 1, then we have Mi (N)E = M1 (N')E Xs M1 (N")E · If N = pe with e 2:: 1, then we have an
40
8. MODULAR CURVES OVER
Z
equality of Cartier divisors
(8.48) Mi (N)E = [Ker[pe ] ] - [Ker[pe -l l]
E[pe ] p 1 x E p (E[p] - [OJ ) . [ • - ]',, [ ] (2) Let P be a section of E of exact order N. Let G = (P) = L a E Z /N z [aP] . Then, we have ex = L a e ( Z /N Z ) X [aP] . PROOF. (1) It suffices to show the following case: N = pe , r 2: 3 an integer relatively prime to N, S = Y(r) z [ �] ' and E the universal elliptic curve over S. In this case, we have Mi (N)E = Y1 AN, r) z [ �] · By Proposition 8.50, M1 (N)E --+ S is finite fl.at of degree cp(N)'l/J(N) . We show (8.48) . Since S is fl.at over Z, it suffices to apply Corol lary A.44 to U = S[l/p] , A = M1 (N)E, B = [Ker[pe l] - [Ker[pe - 1 ] ] , and E[pe ] p 1 x E p (E[p] - [OJ ) . [ • - ]',, [ ] (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 fl.at over Z, it suffices to apply Corollary A.44 to the closed subscheme A = ex of Er and B = L a e ( Z /N z ) x [aP] . D 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 fiat 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 ex of generators of G is a finite fiat scheme over S of degree cp(N) . (2) The condition for a scheme T over S that Gr is a cyclic subgroup scheme of Er 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 fl.at 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 Gr is a cyclic subgroup scheme of Er is equivalent to the condition that c;_ is a finite fl.at scheme on T of degree cp(N) . Moreover, by Proposition 8.52(2) and Nakayama ' s lemma, the quasi-coherent sheaf Oax over S is generated by cp(N) sections locally on S. Hence, by Corollary A.38, this condition is a closed condition. D =
41
8.8. MODULAR CURVE Y0 (N)z
8.8. Modular curve Yo (N)z
PROPOSITION 8.54. Let S be a scheme, and let E be an elliptic curve over S. Let N 2: 1 an integer. The functor Mo ( N) E over S is represented by a finite flat scheme of finite presentation Mo (N)E over S of degree 'l/J(N) . The natural morphism M1 (N)E -+ Mo (N)E is finite flat of finite presentation of degree cp(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 2: 1 be an integer. The functor N -Isog E defined by associating to a scheme T over S the set
{
closed subgroup scheme of Er that is a finite
}
N- Isog E (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 Or-module of rank N of the locally free Or-module OE [ N] x r} is represented by the Grassmannian scheme denoted by Grass(OE [N] • N) , which is a proper scheme over S. If G C Er is a closed subgroup scheme flat finite of finite presentation of degree N as a scheme over T, then Oc is a locally free quotient Or modules of rank N of OE [N] x s r· Thus, as in the proof of Lemma 8.23, the functor N-Isog E 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 pe by Proposition 8.2. If E is ordinary, we have E[N] Z/pez x µp• , and G is of the form Z/pa z x µP b ' a + b = e, D and there are e + 1 of such G . PROOF OF PROPOSITION 8.54. We show Mo (N)E is represen table. Let GMo (N)E C EMo (N)E be the universal closed subgroup scheme. Applying Corollary 8.53(2) to GrN , E c ErN ,E , we see that Mo(N)E is represented by a closed subscheme Mo (N)E of TN , E· We now show M1 (N)E -+ Mo(N)E is finite flat of degree cp(N) . Let GMo (N)E C EMo (N)E be the universal cyclic subgroup scheme of �
42
8. MODULAR CURVES OVER Z
degree N. Then, since M1 (N)e = G �o (N)E ' M1 (N)e -t Mo (N)e is finite fl.at of degree 1..p ( N) by Corollary 8.53(1). Since M1 (N)e is finite flat over S of degree 1..p ( N)'l/!(N) by Proposition 8.52(1), Mo (N)e is also finite flat over S of degree '1/J(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 isomor phic to ( Z /N Z ) 2 . In this case, we have, by Corollary 8.10, M0 (N)e = ilc:cyclic subgroup of(Z/NZ)2 of order N S, and the assertion is now clear. 0
COROLLARY 8.56. Let N ::::: 1 be an integer, and let r ::::: 3 be an integer relatively prime to N. The functor Mo , * (N, r)z [ � ] over Z[�] is represented by a regular finite flat scheme Yo, * (N, r)z [ � ] over Y(r)z [ � ] of degree '1/!(N) . Yo, * (N, r)z [ r'.vl is smooth over Z [ r�l The field of constants of Yo, * (N, r)q = Yo, * (N, r)z [ � ] ®z [ � ] Q is Q((r ) PROOF. Let E be the universal elliptic curve over Y (r)z [ � ] · The functor Mo, * (N, r)z [ � ] is represented by Mo (N)e = Yo, * (N, r) z [ � ] · This is a finite flat scheme over Y(r)z [ � ] of degree '1/J(N) . By Propo sition 8.54, Y1 , * (N, r)z [ � J = M1 (N)e -t Yo, * (N, r)z [ � ] = Mo (N)e is faithfully fl.at. Since Y1 , * (N, r)z [ � ] is regular, so is Yo, * (N, r)z [ � ] by Proposition A.13( 1 ) . Since Yo * (N, r)z [ ..L J is etale over Y(r)z [ ..L J , it is smooth over Z [ r� ] . We omit the proof that the field of constants of Y0 , * (N, r)q = Yo, * (N, r) z [ �] ®z [ � ] Q equals Q((r ) because it is similar to that of Lemma 8.37. 0 Let p be a prime number, N = Mpe 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 F and let E be an elliptic curve over S. Then, by Lemma 8.46(2) , for e = a + b ::::: a ::::: 0, ( G(a,b) = Ker ( V a F b : E -t E b ) ) is a cyclic subgroup scheme of E of order pe if a ::; b, and ( - b) -t E ) G(a, b) = Ker ( V a F b : E is a cyclic subgroup scheme of E( - b) of order pe if a ::::: b. Define a morphism of schemes Ja : Yo, * (M, r)Fv -t Yo, * (N, r)z [ � ] by '
rN
rN
P>
P
P"
P"
"
8.8. MODULAR CURVE Y0 (N)z
[(E, C, a) ] H [(E, (G ( a , b ) > C) , a)] if a $ b, and by [(E, C, a) ] [(E (Pa- b) , ( G ( a,b) , C (Pa- b) ) , a<Pa- b > )] if a 2: b.
43 H
PROPOSITION 8.57. Let p be a prime number, let N = Mpe with (p, M) = 1 an integer, and let r 2: 3 be an integer relatively prime to N. (1) For 0 $ a $ e, the morphism (8.49) is a closed immersion.
(2) For 0 $ a $ e, let Ca be the image of the closed immersion ia : Yo , * (M, r)F p --+ Yo, * (N, r)F p · Then, we have Yo, * (N, r)F p = LJ: = o Ca , and the inclusion C�s --+ Yo , * (N, r )�p is a bijection. The multiplicities of Co and of Ce in Yo, * (N, r)F p are 1 . (3) If 0 $ a $ e and 0 $ a' $ e with a f:. a' , then the intersection Ca n Ca ' = Ca X yo (N ,r ) F p Ca' equals c�s = c�� PROOF. As in the proof of Proposition 8.49, it suffices to show the assertions when M = 1 and N = pe 2: 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 Yo, * (N, r)F p · For an integer 0 $ a $ b = e - a, define a closed subscheme of Yo, * (N, r)F p by the closed condition (8.50) The morphism ia : Y(r)Fp --+ Yo, * (N, r) z [ �] defines an isomorphism Y(r)F p --+ Ca by definition. Thus, ia : Y(r)F p --+ Yo, * (N, r) z [ � ] is a closed immersion. If b = e - a $ a, a similar proof works if we define Ca by the closed condition the kernel Ker(E' --+ E) of the dual of E --+ E' = E/G (8.51 ) equals Ker(V b pa : E' --+ E'(Pa- b > ) . (2) Let k be an algebraically closed field of characteristic p, let [(E, G, a)] E Yo, * (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, a)] E Co(k) . If E is ordinary, we also have [ (E, G, a) ] E Ca (k) . Thus, we have Yo , * (N, r)Fp = LJ: = o Ca - If E is supersingular, we have G = Ker Fe and [(E, G, a)] E Ca (k) for all 0 $ a $ e. Thus, the mapping ia : Y(r)�P --+ YoAN, r)�P is a bijection. .•
44
8. MODULAR CURVES OVER
Z
By Lemma 8.4, C8rd equals the open subscheme of Yo, * (N, r )F P defined by the condition that the dual of E -7 EI G is etale. Thus, the multiplicity of Co equals 1. Similarly, c�rd equals the open subscheme of YoAN, r)F p defined by the condition that G is etale, and thus its multiplicity equals 1. (3) Let (E, G, a ) be an elliptic curve with universal level structure over the intersection Ca n Ca ' . Let b = e a and b' = e a' . If a < a' � e /2 then since we have G = Kerpa Fb' - a = Kerpa 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' -7 E be the dual of E -7 E' = E/G. The composition E -7 E' -7 E equals pb' p a' - b' pa pb - 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 n Ca ' is a closed subscheme of c�s. We now show c�s c Ca n Ca' · Let G be the universal cyclic subgroup scheme over c�s . If a, a' � e /2, then G = Ker pa pb - a = Ker pa F b' - a , and thus c�s c Ca' . Similarly, if a, a' 2:: e /2, we also have c�s c Ca ' . If a < e /2 < a' , then the kernel of the dual of E -7 E' = E / G equals Ker pa y b - a = Ker pa' pb' - a' , and again we D have c�s c Ca' · The case a' < e /2 < a is also similar. COROLLARY 8.58. If e = 1, the regular curve YoAN, r) z [ �] over Z [ � ] is semistable at p. The fiber Yo, * (N, r)F p is the union of Co -
-
I
I
,
o
I
I
and C1 .
PROOF. It follows easily from Corollary 8.56, Proposition 8.57 D 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 2:: 3 be an integer relatively prime to p. As for the action of GL 2 (Z/rZ) on Yo, * (Mp, r) z [ �] ' 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 (Mp) is weakly semistable, and jo , j1 : Yo (M)F p -7 Yo (Mp)F p are closed immersions. The in tersection of the image Co of io and that of C1 of j1 is Yo (M)�p = Yo, * (M, r)�p /GL 2 (Z/rZ). Let x = [(E, C)] E Yo (M)�p be an ordinary double point of Yo (Mp)Fp ' let x' = [(E, C, a)] E Yo, * (M, r)�P be a point in the inverse image of x, and let 17 = [(Eo, Co , ao )] be the generic point of
8.8. MODULAR CURVE Y0 (N)z
45
Yo, * (N, r) z . Then, the inertia group Ix' is the image of the injection Aut(EF , CF ) -? GL 2 (Z/rZ) , and by Lemma 8.41, the inertia group I,, is {±1 } C GL2 (Z/rZ) . Thus, by Corollary B.11(2) , the index of x equals [Ix' : I71 ] = lt Aut( Bjj< , CF )/{±1}. D I'
p
I'
p
We define morphisms between modular curves. P ROPOSITION 8.59. Let S be a scheme, and let E be an elliptic curve over S. Let N = M dM' 2: 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 = I:: �,:� [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 2: 3 is integer relatively prime to N, S = Y1 , * (N, r) z [ f: l ' E is the universal elliptic curve over S, and P is the universal section of exact order N. The assertion is clear on S[k l · Thus, it suffices to apply Corol lary 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 = e x , S' = Mo (d) E , and let H be the universal cyclic subgroup scheme of Es' of order d. By (1 ) , we obtain a morphism f : X -? Y defined by P >--+ (M'P) c Ex / (:L:�,:� [i l,f Pl ) . 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 2: 3 be an integer relatively prime to N. Since the assertion is flat local on S, we may assume there exists a basis a for E[r] . Then, (E, C, a ) defines a morphism S -? Yo, * (N, r) z [ f: ] · Thus, it suffices to show the case S = Yo , * (N, r) z [ f:J · On Sz [kl the assertion is clear. It now suffices to apply Lemma A.45. D an
We define a morphism of functors (8.52 )
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) s d : Mo (N) --t Mo (M) by sending the isomorphism class of ( E, C) to the isomorphism class (E', C' ) .
LEMMA 8.60. Let M d I N � 1 be integers. The morphisms of modular curves defined by the morphism of functors s d s d : Yi (N) z --t Yi (M) z , (8.54) s d : Yo (N) z --t Yo (M) z are finite. PROOF. We show the morphism s d : Yi (N) z --+ Yi (M) z is fi nite. Let r � 3 be an integer relatively prime to N. Define s d : Yi, * (N, r)z [�] --+ Yi, * (M, r)z [ �] in the same way as s d : Yi (N) z --+ Yi ( M) z . We show this morphism is finite. Let ( E, P, a) be the uni versal elliptic curve with level structure over S = Yi, * (M, r)z [ �] · Let A = Mo (d) E , and let G C 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 = Mi (N)E' , and let P' : B --+ Ek be the universal section of exact order N. The condition that g( jJd P') = P and If P' is a generator of the kernel of g : Ek --+ 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. Let a0 : (Z/rZ) 2 --+ E(;. [r] be the composition of a and the inverse of the isomorphism E(;. [r] --+ Ec [r] . Then, the triple (E(;. , P(;, a' (X)) defines a morphism C --+ Yi, * (N, r) z [n We show that the morphism C --+ Yi, * (N, r )z [ �l is an isomor
phism and that the composition of the inverse of this and the natural morphism C --+ Yi, * (M, r) z [ � ] is s d : Yi, * (N, r)z [ �J --+ YiAM, r) z [ �] · Let (E' , P', a') be the universal elliptic curve over Yi, * (N, r)z [ �] with level structure. The dual E --+ E' of g : E' --+ E = E' / (If P') defines a morphism Yi, * ( N, r )z [ � 1 --+ A that extends B d - The universal section P' defines Yi, * (N, r)z [�] --+ C C B. It is easy to see that this is the inverse. Thus, s d : Yi, * (N, r )z [ �] --+ Yi, * (M, r )z [ � ] is finite. Letting r � 3 run integers relatively prime to N, we obtain a finite morphism s d : Yi (N) z --+ Yi (M) z by taking the quotients and patching them. Similarly, s d : Yo (N) z --+ Yo (M) z is finite. D
8.8.
47 EXAMPLE 8.6 1 . Since Yo (4)z is the quotient of Y1 (4)z by the diamond operator (Z/4Z) x = { (±1 ) } , we have Yo (4)z = Y1 (4)z. We show Y1 (4)z = Yo (4)z = Spec Z[s, t, u]/(st-2 8 , u(s+24 ) -24 t, u(t+24 ) -t 2 ) . By the Remark after Theorem 8.34, Y1 (4)z is the integral closure of Y(l)z = Spec Z[j] in Y1 (4) z [ �J · Let A = Z[s, t, u]/(st - 2 8 , u(s + 24 ) - 24 t, u(t + 24 ) - t 2 ) . We have A[!l = za] [s, ( ! ) ] This is isomorphic to z a , d, d(L4 ) ] by s H- 4(d - 4) . Through this isomorphism we identify Y1 (4) z [ !] = Spec A[!J We show A is an integrally closed domain. A[!J = Z[!] [s, ( ! ) ] is an integrally closed domain. A/2A = F 2 [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] x F 2 [u] x F 2 [s] given by { (!, g, h) E F 2 [t] x F 2 [u] x F 2 [s] \ f(O) = g ( O ) = h ( O ) } by the mapping s H- (0, 0, s), t H- (t, 0, 0) , u H (t, u, O) . 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) = 24d(d - 4) , we have ku = 2 12 . By (8.26) , the morphism j : Y1 (4) z [ !J -+ Y(l)z defined by the j-invariant is defined by Z[j] -+ A A[!J 2 1 k d ) k 2 + 3 · 24k + 3 · 2 8 + u. j H- 2 8 . ( d d-( d4-+4 ) 3 - ( +24)3 k We show that the integral closure of Z[j] in A[!J 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 - (k 2 +3 · 24k+3 - 28 ) and t 2 = 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 - 2 12 ) . The surjective morphisms of rings A/2A -+ F 2 [t] , A/2A -+ F 2 [u] and A/2A -+ F 2 [s] define closed subschemes Co, C1 and C2 C Yo (4)F 2 , respectively. We define isomorphisms Ji : Y(l)F 2 -+ Ci (i = 0, 1 , 2) by t H- j, u H- j and s H- j. The Atkin-Lehner involution w : Yo (4)z -+ Yo (4)z is defined by s H- t, t H- s, u H- v = s + t -4 u - 24. It suffices show that MODULAR CURVE Yo (N)z
s s
24
.
s s
--
24
8 . MODULAR CURVES OVER
48
Z
w4 : Yo (4) z[ ! J ---+ Yo (4) z[ ! ] is defined by d i--+ d4_!4 . Extending the coefficients to Z [ J=l, !J, we compute W4 Y1 (4) z[ v'=T , ! l ---+ Y1 (4) z[ v'=T, ! J · Let P = ( 1 , 1) E E be the universal section of or der 4. The quotient E' = E / (2P) of the universal elliptic curve E : dy2 = x(x2 + (d - 2)x + 1) is given by dy'2 = x ' ( x' + d ) ( x ' + 4 ) , and E ---+ E' is given by x' = x + � - 2, y' = � (x - � ) . Moreover, E" = E/ (P) is given by dy"2 = x" (x"2 - 2(d + 4)x" + (d - 4) 2 ) . E' ---+ E" is given by x" = x' + d + 4 + ;� , y" = � (x' - !�) . If we let x" = -(d - 4)x 1 and y" = (d - 4) 2 2J=ly1 , E" is given by d4!4 y� = X 1 (x� + 2 ��;) x 1 + 1 ) . Since the universal section of E" is given by ( 1 , 1 ) , we have w4 ( d ) = d4!4 . If we let l = t ( t + 24 ) and v = s+ t - u - 24 , 84 = so ow4 : Yo (4)z ---+ Y(l)z is defined by j 1---t l 2 + 3 2 4 l + 3 · 2 8 + v. Since the j-invariant of E' is j (E') = 24 { d:2(!����) 3 = { k��8 ) 3 = k + 3 · 2 8 + 3 · 24 u + u2 , s 2 : Yo (4)z ---+ Y(l)z is defined by j i--+ k + 3 2 8 + 3 24 u + u2 . The image of j = sij, s2j, s 4,j in F 2 [t] x F 2 [u] x F 2 [s] is given by (t, u, s4 ) , ( t 2 , u2 , s2) , ( t 4 , u , s) , respectively. :
·
·
·
8.9. Compactifications
In this book we define the compactification Xo (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 8.6 2 . Let N ;::: 1 be an integer. (1) Define Xo (N)z as the integral closure of P� with respect to j : Yo (N)z ---+ A� . (2) Define X1 (N)z as the integral closure of P� with respect to j : Y1 (N)z ---+ A� . In this section we prove the following fundamental properties of Xo(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 geo
metric fiber is connected. (2) Let p f N be a prime number. Then, Xo (N)z is smooth at p . The fiber Xo (N) F,, = Xo (N)z ®z F is a smooth compactification of
Yo (N) F,, ·
P
8.9. COMPACTIFICATIONS
49
(3) Let N
= Mp with p f M. Then, Xo (N)z is weakly semistable at p. The closed immersions io : Yo (M)F,, -t Yo (N)F,, and j1 : Yo (M)F,, -t Yo (N)F,, extend to closed immersions io : Xo (M)F,, -t Xo (N)F,, and j 1 : Xo (M)F,, -t Xo (N)F,, · The fiber Xo (N)F,, is the union of the image Co of io and the im age C1 of j1 , and the intersection of Co and C1 is the coarse moduli scheme Yo (M)p" of Mo (M)pp . The index of the ordinary double point x = [(E, C)] E Y0 (M)pp is the order of
Aut(E:F ,, , C:F ,, )/{±1}. Using Theorem 8.63, we compute the number of isomorphism classes of supersingular elliptic curves, which is equal to deg Y (1 )p . COROLLARY 8.64. Let p be a prime number. Then, we have Y(l)p # 0 . The number of isomorphism classes of supersingular elliptic curves over FP equals deg Y (1 )p , which equals "
p
"
-a 1 + 9o(P) = 1 + p� ( if p = a = 2, 3, 5, 7, -1, 13 mod 12) .
PROOF. By Theorem 8.63, Xo (p)F,, = Co U C1 is connected. Thus, C0 n C 1 = Y(l)p is nonempty. Since the coarse moduli scheme Y(l)p is reduced, we have deg Y(l)p = HY(l)p (Fp ) · We have {isomorphism classes of supersingular elliptic curves over Fp } = M (l)p (Fp ) = Y(l)p (Fp ) · By Corollary D.21(1) and the fact that go(l) = 0, we have go (p) = deg Y(l)p - 1. By Proposition 2.15 and Lemma 2.14, we have 1 1 g0 (p) = l + (p + l) - 2 - 2 12 (p 2 mod 3) (p 3 mod 4) 1 - -41 1 (p = 2) - -3 1 (p = 3) 2 (p 1 mod 4) . 0 2 (p 1 mod 3) p
p
p
p
p
p
{O
"
=
{O
=
=
=
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 supersin gular if p - 1 mod 3. Thus, by Lemma 8.41 and Corollary 8.64, we obtain 1 p-1 --- = 24 H Aut E 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. 2
p
3 5
7
11
13
17
19
...
... 2 U of s.s. curves a 1 1 1 1 2 1 2 s.s. j-invariant0 0 0 0 - 1 0, 1 6 0, 8 - 1 , 7 . . . indexc 1 2 6 3 2 3, 2 1 3, 1 2, 1 . . . anumber of isomorphism classes of supersingular elliptic curves b j-invariant of supersingular elliptic curve cindex at each point
=
� U Aut (E)
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 f N. Then, X1 (N) z is smooth at p. The fiber X1 (N)F p = X1 (N) z ® z Fp is a smooth compactification of Y1 (N)F p ·
In order to describe the compactification, we define the Tate curves. In Chapter 2, (2.34 ) , we defined the power series Ek (q)
=
1+
2 ((l - k )
�O"k - 1 (n)qn 00
E
Q [[q]] ,
where O"k - 1 ( n) = L dl n d k - l . We then defined an elliptic curve over the field of power series Q((q)) by ( 2 . 3 5)
By the change of coordinates x = x' + l2 , y = 2y' + x', the equation (2.35) becomes 1 (8.55) y'2 + x ' y' = x '3 - - (E4 (q) - l)x ' 48 1 1 (E5 (q) - 1) . (E4(q) - 1) + 4 . - 4 . 12 2 216 Let s k (q ) = L:: := l O"k - 1 (n)qn = ( ( l; k ) (Ek (q) - 1) E Z[[q]] . Since ( 3) = 1 �0 and ((-5) = - � , the coefficient of the degree 1 term ( -the right-hand side of (8.55)2 2equals -5s4 (q) , and the constant term of is - 112 (5s4 (q) + 7s5 (q) ) .
8.9. COMPACTIFICATIONS
51
QUESTION. Verify that 112 (5s4 (q) + 7s5 (q)) E Z [[q]] . Let Z ((q)) = Z[[q]] [q- 1 ] . Show that the equation y 2 + xy = x 3 - 5s4 (q)x -
(8.56)
1 (5s4 (q) + 7s5 (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(l) z = A� defined by the Tate curve Eq extends uniquely to e : Spec Z [[q]] --+ X(l) z = P� . Let X(l) z l � be the completion of X ( l ) z = P� along oo . Then e induces an isomorphism e : Spec Z[[q]] --+ X(l) z l � PROOF. As we have seen Example 2.37 in Chapter 2, the j invariant of the Tate curve is E ( )3 1 = q + 744 + 196884q + 21493760q 2 + . . . . j (q) =
� (�)
q f1�= 1 ( 1 - qn ) 24 E qZ [[q]] x and E4 (q) E Z[[q]] x , we have j (q) E i · Z [ [q]) X . The assertion now
Since we have D.(q) 1 + Z[[q]]
C
=
follows easily. D The following is a proposition concerning torsion points of the Tate curve, for which we omit the proof. PROPOSITION 8.69. Let N 2:: 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 J --+ Z ((q)) of rings. From now on, we identify Eq [N] and T[N] through the isomor phism in Proposition 8.69. Let r 2:: 1 be an integer. Define a ring homomorphism Z (( q )) --+ Z[ � , (r ] ((Qr)) by q ...+ q; . Let Eq-;_ be the pullback of the Tate curve Eq over Z [ � , (r] ((qr)) by this ring homomorphism. We define an isomor phism ar : (Z/rZ) 2 --+ Eq; [r] = T[r] ® z [q,q-1 ] Z [ � , (r] ((qr)) of group schemes over Z[� , (r] ((qr )) by ar ( ( l , O) ) = (O, (r ) and ar ( (0, 1)) = (1, qr ) · The morphism er : Spec Z [ � , (r] ((qr)) --+ Y (r)z [ f: J defined by the pair ( Eq-;_ , ar) is called the morphism defined by the Tate curve. We have el = e. For a E (Z/rz) x and b E Z/rZ, let aa , b = ( 0 n E GL 2 (Z/rZ) . Define a subgroup V(Z/rZ) = { aa , b I a E (Z/rz) x , b E Z/rZ} C GL 2 (Z/rZ), and define an action of V(Z/rZ) on Z [ � , (r] ((qr )) by
52
8. MODULAR CURVES OVER Z
O"a , b ((r) = C: and O"a , b (qr) = (� qr. We define an action of -1 E GL 2 (Z/rZ) as the trivial action. COROLLARY 8. 70. Let r ;::: 1 be an integer. The morphism defined by the Tate curve
[ � ] ((qr)) --+ Y(r) z [ �]
er : Spec Z (r i
is compatible with the action of V(Z/rZ) · { ±1} commutative diagram induced by er lJ
aEGL 2 (Z/rZ)/V(Z/rZ) · {±l}
C
Spec Z[(r i :] ((qr))
(8.57)
1
Spec z[:J ((q))
GL2 (Z/rZ) . The
------+
�
Y(r)z[ �]
1
Y(l) z[ �J
is Cartesian. The morphism e r : Spec Z[(r, :J ((qr)) --+ Y(r) z[ �] uniquely extends to er : Spec Z[(r, :] [[qr]] -+ X(r) z[ �J ·
PROOF. It is easy to see that er : Spec Z[(r, :]((qr)) --+ Y(r) z[ �] is compatible with the action of the subgroup V(Z/rZ) · {±1} C GL 2 (Z/rZ). The morphism defined by the diagram (8.57)
II
aEGL2 ( Z/rZ)/V(Z/rZ)·{±l}
� ] ((qr)) --+ Spec Z [� ] [[q]] x Y(l)
Spec Z [(r,
Y(r)z[ .!Jr is, by Corollary 8.40(3) , a morphism of GL 2 (Z/rZ)/{±1}-torsors over Spec Z [:] (( q )) , and thus it is an isomorphism. The integral closure of z[:] [[q]] in Z[(r, :] ((qr)) is Z[(r, :] [[qrlJ · Thus, the morphism er : Spec Z[(r, :]((qr)) --+ Y(r) z[ �] uniquely ex D tends to er : Spec Z[(r, :] [[qr]] -+ X(r)z[ n x Yo, * (N, r) z[ �] is The fibered product Spec Z[(r i :] ((qr)) T
e r '\,.Y( r )z [ � )
1
Z[ r )
isomorphic to the spectrum of the ring Z (r, ;:1 ((qr)) ©z (( q )) Z[(d" ] ((q)) [T]/(Td 1 - (d" qd'1 ) (8.58 ) II dd' =N by Proposition 8.26(1). The integral closure of Z[(r, :] [[qr]] in this ring is calculated as follows.
[ ]
8.9. COMPACTIFICATIONS
LEMMA 8.71 . Let m ;::: 1 and r be relatively prime integers.
;:::
53
1 be integers, and let a, b
(1) If we define a ring homomorphism q i-+ q�, the tensor product
2:
Z[q, q -1] ---+ Z[(r i :J (( qr))
1 by
is isomorphic to
Z [(mdr/m1d1 , :;:-l] (( qr)) [Tl /(T - (md qr ) . IT gEGal ( Q ( (m1 ) / Q) �
b�
eld2 h EGal ( Q { (,.,, 41 ) / Q ( (,.,, ) )
Here, m' = (m, r) , d = (a, r) , d' = (d, r/m') , and we let a = a'd, r = r'd. Moreover, d = d 1 d 2 , where the prime factors of d 1 are prime factors of m and (m, d 2 ) = 1 . For a divisor e of d 2 , s is the greatest common divisor of r/m' and die . (;.,(f12 is an md 1 th root of unity satisfying ( (;ff: ) d2 = (md1 • (2) Let n 2: 1 , and let r I mn. The integral closure of Z[(r , :][[qr lJ in Z[(mn, :J (( qr )) [T] /(Ta - (m q� ) is isomorphic to Z[(mn, :rns]] . The homomorphism Z[(r i :] [[qr ]] ---+ Z[(mn, :rnsn is given by qr i-+ c;;,c sa for some positive integer c relatively prime to a.
Since Z[(r , :J ® z Z[(m ] is the integral closure of in Q ((r) ®Q Q ((m) = n gEGal ( Q ( (,.,, i ) / Q) Q ((mr/m1 ) , it equals n gEGal ( Q { (,.,, 1 ) / Q) Z[(mr /m1 , n Thus, the ring (8.59) equals
z( :J
PROOF. (1)
Z [(mr /m1 , � ] (( qr )) [T] / (Ta - (m q�r ) IT gEGal ( Q { (,.,, 1 ) / Q)
·
Moreover, we have
Z [(mr/m' ' � ] (( qr)) [T] /(Ta - (m q�r ) ( z [(mr/m' > �] (( qr )) [U] /(Ud - (m)) [T] /(Ta' - Uq�r ' ) . =
54
8. MODULAR CURVES OVER
Z
Factorize d = did2 , where prime factors of di are prime factors of m and ( m , d2 ) = 1. Then we have Q ( (mr/m' ) [U]/(Ud - (m ) = Q ( (mr/m ' ) ® Q((m ) Q ( (m ) [U]/(U d - (m ) = Q ( (mr/m ' ) ® Q((,,, ) Q ( (m dJ [U]/(Ud2 - (m d J = Q ( (mr/m ' ) ® Q((,,, ) II Q ( (m d 1 e )
Q ( (md 1 er/m 1s ) , II II e l d2 hEGal(Q((,,, . )/Q((,,, )) where is the greatest common divisor of r / m and die. Since the ring Z[(mr/m ' ' �] [U]/(U d - (m ) is the integral closure of Z[�] in Q ( (mr/m ' ) [U]/(U d - (m ) , it equals z cmrd 1 e /m ' s · · II II le d2 hEGal(Q((,,, . )/Q((,,, )) Let (!(112 be an mdi th root of unity satisfying ( (;,(f12 ) d2 = (md i . Then, the ( e, h)-component of the image of U is (;,(f12 (e · Thus, we have Z[(mr/m' ' !r ] ((qr )) [T]/(Ta - (m q�r ) = II II )/Q((,,, Z)) (mrd1 e /m' s ' ((qr )) [T]/(Ta'- (!(f12 (e q�r ' ) . e l d2 hEGal(Q((,,, d' (2) Take positive integers c, d satisfying be - ad = 1, and define a morphism of Z[(mn, � ]-algebras Z[(mn, �] ((qr )) [T]/(Ta - (m q� ) -+ Z[(mn, �] ((S)) by qr t-+ C;;,c sa , T t-+ C;;,,d Sb . Then, since the inverse is defined by S t-+ Tcq; d , this is an isomorphism. Since Z[(mn, �] [[SJ] is finitely generated as a Z[(r i � ] [[qr]]-module, the integral closure of 0 Z[(r , �] [[qr ]] is Z[(mn, �] [[SJ] . The compactifications of Yo, * (N, r)z [ � l and YiAN, r)z [ � l are de fined similarly to Definition 8.62. DEFINITION 8.72. Let N 2':: 1 be an integer, and let r ;,::: 3 be an integer relatively prime to N. (1) The scheme Xo, * (N, r)z [ �] over Z[�] is defined as the integral closure of X(l)z [ � ] with respect to Yo, * (N, r)z [ �] -+ Y(l)z [ �] · (2) The scheme Xi,* (N, r)z [ �] over Z[�] is defined as the integral closure of X(l)z [ � l with respect to Yi, * (N, r)z [ �J -+ Y(l)z [ �] · =
s
'
[
[
�J
�]
8.9. COMPACTIFICATIONS
55
If N = 1, Xo, * (N, r)z [ � ] = X1 , * (N, r)z [ �] is denoted by X(r)z [ �] · PROPOSITION 8.73. Let N 2".: 1 be an integer, and let r 2".: 3 be an
integer relatively prime to N. (1) The scheme X0, * (N, r)z [ � ] is a regular projective curve over Z[·:J Xo, * (N, r)z [ Jr l is smooth over Z [ Jr l · The field of con stants of the curve Xo, * (N, r)q = Xo, * (N, r)z [ � ] ®z [ �] Q over Q equals Q((r ) · (2) Let p f r be a prime number, and let N = Mpe with (p, M ) = 1 . For 0 :::; a :::; e, the closed immersion Ja : Yo, * (M, r)F p --+ Yo, * (N, r)z [ � ] extends to a closed immersion j a : Xo, * (M, r)F p --+ XoAN, r)z l �J · If C a is the image of Ja , we have Ca n C a ' = Ca n Ca ' C YoAN, r)z [ � ] for a =f a . Moreover, if e = 1 , Xo, * (N, r)z [ � ] is semistable at p and the closed fiber Xo, * (N, r)F p is the union of Co and C 1 .
PROOF. ( 1 ) By Corollary 8.56, Y0, * (N, r) z [� ] is regular, and Yo, * (N, r)z [ Jr l is smooth over Z[ Jr l · Let X (r)z [ �J I � be the comple tion of X (r)z [ � ] along the inverse image of oo = P� - A� . Then, by Lemma 8.68 and Corollary 8.70, we obtain an isomorphism (8.60 )
II
uEGL2 (Z/ r Z)/V(Z/ r Z)·{± 1 }
[
Spec z (r, �] [[qr]] -+ X (r)z [ �J l � -
Let ir : Spec Z[(r , � ] --+ X (r)z [ �] be the composition of the closed im mersion defined by Qr t-+ 0 and the extension er : Spec Z[(r �] [[qr]] --+ X(r)z [n Then ir is a closed immersion. Let Dr c X (r)z [ � ] be the image of ir , and let DN,r = Dr X x (r ) z [ l. J Xo, * (N, r)z [n By Lemma 8.71 , the scheme Xo, * (N, r)z [ � ] is regular on a neighborhood of DN,r , and smooth over Z[ Jr l · Moreover, for E G L2 ( Z /r Z ) , Xo, * (N, r)z [ � ] is regular on a neighborhood of * ( DN , r ) and is smooth over Z[ Jr l · By ( 8.60) , we have
i
a
a
a*(DN,r ) · Thus, Xo, * (N, r)z [ �J is regular everywhere and smooth over Z[ Jr l · Xo, * (N, r)z [ � ] - Yo, * (N, r)z [�]
=
II
uEGL2 (Z/ r Z)/V(Z/ r Z)·{± l }
Since the field of constants of Yo, * (N, r)q is Q ( (r ) , the field of con stants of Xo, * (N, r)q is also Q ( (r ) ·
56
8. MODULAR CURVES OVER Z
(2) B y ( 1 ) , the projective curve X0 , * (M, r)F,, over Fp is a smooth compactification of Yo, * (M, r )F,, . Define a reduced closed subscheme C a of XoAN, r) z [*] as the closure of the image Ca of the closed im mersion ia : Yo, * (M, r)F,, ---+ YoAN, r) z [*] in XoAN, r) z [*] · By the proof of ( 1 ) , Ca is smooth on a neighborhood of the intersection with g * (DN ,r) over Fp for each g E GL 2 (Z/rZ) . Thus, Ca is also a smooth compactification of Yo, * (M, r)F,, and is isomorphic to X0 , * (M, r)F,, · Furthermore, by the proof of ( 1 ) , the reduced part of Xo, * (N, r)F,, is smooth on a neighborhood of the intersection with the inverse im age of g * (Dr) for each g E GL 2 (Z/rZ) . Thus, if a =f. a', Ca and C a ' do not intersect each other on a neighborhood of the inverse im age of each g * (Dr) · Hence, the intersection Ca n Ca' is contained in Yo, * (N, r) z [*] · The last assertion in the case of e = 1 follows easily from the D 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 [*] is a regular projective curve over Z[�] . X1, * (N, r) z [Jr l is smooth over Z[Jr l · The field of con stants of the curve X1, * (N, r)q = Xo, * (N, r) z [*] © z [*] Q over Q equals Q((r) · (2) Let p f r be a prime number, and let N = Mp e with (p, M) = 1 . For 0 ::::; a ::::; e, let Ig(Mpa , r)F,, be the smooth compactification of the smooth affine curve Ig(Mpa , r)F,, over Fp . Then, the closed immersion ia : Ig(Mpa , r)F,, ---+ Y1, * (N, r) z [*] extends to a closed immersion ia : Ig(Mpa , r)F,, ---+ X1, * (N, r) z [*] · If a =f. a', the intersection of the image C a of ia and the image Ca ' of ia' is contained in Y1AN, r) z [*] ·
The proof of Proposition 8.74 is similar to that of Proposition
8. 73, and we omit it.
PRO OF 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 Xo (N) Q at the generic point is connected by Theorem 2. 10(3) . Thus, we have r(Xo(N)q, 0) = Q and r(Xo(N)z, 0) = Z. Hence, by Theorem A. 16, each geometric fiber of Xo (N)z is connected. ( 2 ) Xo (N)z is obtained by patching together the quotients of XoAN, r)z [*] by GL 2 (Z/rZ). Thus by Propositions 8.73 and B.10(1),
8.9. COMPACTIFICATIONS
57
Xo (N) z [Jl;-J is smooth over Z [ -:kr l · Furthermore if p f N, Xo (N)F p is a quotient of Xo, * (N, r)F p and it is a smooth compactification of Yo (N)Fp · (3) As in (2) , if N = Mp with p f M, then Xo (N)z is weakly semistable at p by Proposition 8.73 and Corollary B. 1 1 (2) . Since the closures Co , C1 of the images Co, C1 of the closed immersions jo, j1 : Yo (M)F p -+ Yo (N)z are regular at the cusps, both are isomorphic to Xo (M)F p · Thus, the closed immersions io and j1 extend to closed immersions jo , j1 : Xo (M)F p -+ Xo(N)F.,, · The facts Xo (N)F.,, = CoU C1 and Co n C1 = Yo (M)�.,, follow easily from Proposition 8.73. D 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 M 1,o (M, N) over Z by associating to a scheme T the set isomorphism classes of triples (E, P, C) , E is an elliptic curve over T, P is M i,o (M ' N) ( T ) = awhere section of E of exact order M, and C is a · cyclic subgroup scheme of order N For a prime number p I M, we define the restrictions of functors jo, j1 : Mi (M)F p -+ Mi,o (M, p)F.,, by [(E, P)] H [(E, P, Ker F)] and [(E, P)] r-+ [(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,o (M, N)z of the functor Mi,o (M, N) over Z. Y1,o (M, N)z is a normal connected affine curve over Z, and the morphism defined by the j -invariant, j : Y1,o (M, N)z -+ A�, is finite flat. (2) Let N = p be a prime number. The integral closure X1,o (M, p)z of P� with respect to the finite morphism j : Y1,o (M, p)z -+ A� is weakly semistable at p. The restrictions of functors jo , j1 : Mi,o (M)F.,, -+ Mi,o (M, p)F.,, induce closed immersions io, i1 : Y1 (M)F p -+ Y1,o (M, p)z, and they extend to closed immersions jo, j1 : X1 (M)F .,, -+ X1,o (M, p)z . X1,o(M, p)F.,, is the union of the image Co of io and the image C1 of i1 . We omit the proof of this theorem, too.
8. MODULAR CURVES OVER Z
58
The integral closure X1 , 0 (M, N)z of P� with respect to the fi nite morphism J : Y1 ,o (M, N)z --+ A� may also be denoted by Xo, 1 (N, M)z in this work. X1,o(M, N)z is the quotient of X1 (MN) z by (Z/Nz) x c (Z/MNz) x . The quotient of X1 ,o (M, N)z by (Z/Mz) x identified with (Z/MNZ) X /(Z/Nz) x is X0 (MN)z. For an integer r ;::::: 3 relatively prime to MN, X1 ,o, * (M, N, r) z[ � ] is also defined similarly. Unlike Theorem 8.63(3) , X1 (Mp)q, where p f M, may not have semistable reduction at p. However, the extension of the base change X1 (Mp)Q ( (p ) = X1 (Mp)q © Q Q((p) 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 to Mp. Let X1 , * (Mp, r)�f� , (p ] be the normalization of the scheme X1 , * (Mp, r) z[ � ] ©z[ � ] Z[�, (p] · Then, the curve X1 , * (Mp, r)�f� , (p ] over Z[� , (p] is semistable at the prime ideal p = ((p - 1) . There exists a closed immersion (8.61) Jo, J 1 : Ig (Mp, r)Fp ---+ X1 , * (Mp, r)�f� . C:,, J satisfying the following condition. Let Co, C1 be the images of Jo, J i . We have X1 , * (Mp, r)�[tc:,, J ©z[c:,, J Fp = Co U C1 and Co n C1 = CQ5 • The diagrams (8.62)
lg ( Mp, r)F,,
�1
X1, * (Mp, r)�(�.c:,, J
----+
X1 , * (M, r)F,,
----+
X1 , * (Mp, r) z[ � J
�1
X1 , * (M, r)F,,
l�
----+
X1 ,o, * (M, p, r) z[ � J
----+
X1 A M, r) z r � 1
----+
X1,o, * (M, p, r) z[ � J
and
X1 , * (Mp, r)�(� .c:,, J
----+
X1, * (Mp, r) z[ � J
lj1
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 Mp, and let
8.9. COMPACTIFICATIONS
59
a 2:: 0 be an integer. The quotient of the Igusa curve Ig(Mpa , r)F " by GL 2 (Z/rZ) is denoted by Ig(Mpa )F " · If a = 0, then we have
Ig(Mpa )F " = X1 (M)F " · COROLLARY 8.78. Let p be a prime number, and let M ?: 1 be an integer relatively prime to p. Let X1 (Mp) �[t i be the integral closure of X1 (Mp)z in X1 (Mp) Q ( (p ) = X1 (Mp)Q ®Q Q((p) · The curve X1 (Mp) �(t i over Z [(p] is weakly semistable at the prime ideal p = ((p - 1 ) . The closed immersion (8.61) induces a closed immersion io, j1 : Ig(Mp)F " ---+ X1 (Mp) �(t 1 · ( 8.63 ) Let Co, C1 be the images of jo, j1 . Then we have X1 (Mp) �(�"l ®z[("] Fp = Co u C1 and Co n C1 = c�s . For i = 0, 1 , the diagram (8.64) is commutative.
PROOF. X1 (Mp) �[� , ("] is a quotient of X1, * (Mp, r) �[� , ("] by the action of GL 2 (Z/rZ) . By Lemma 8.41 (2) , the inertia group at the generic point of the fiber X1, * (Mp, r) Q ( (") over Q((p) is 1 if Mp > 2, and {±1} if Mp :5 2. Thus, the assertion follows from Theorem 8.77 0 and Corollary B . 1 1 (2). The morphisms s d : Y1 (N)z --t Y1 (M)z and s d : Yo (N)z --t Yo (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 I N. The mor phisms of modular curves s d : Y1 (N)z --t Y1 (M)z and s d : Yo (N)z --t Yo ( M) z extend uniquely to finite morphisms s d : X1 (N)z ---+ X1 (M)z , (8.65) s d : Xo (N)z ---+ Xo (M) z . OUTLINE OF PROOF. Let r ?: 3 be an integer relatively prime to N. Since s d : Y1, * (N, r) z [ �] --t Y1, * (M, r) z [ �] is a morphism of two-dimensional regular schemes, it uniquely extends to a morphism X' --t X1, * (N, r) z [ �J from the scheme X' --t X1, * (M, r) z [ �J obtained by blowing up finitely many times at finitely many closed points of
60
8. MODULAR CURVES OVER Z
X1 , * (N, r) z [�] - Y1 , * (N, r) z [�] · We then show that a morphism s d : X1 , * (N, r) z [�] --+ X1 , * (M, r) z [�] is obtained without taking blowups. Dividing these by the action of GL 2 (Z/rZ) and patching them up, we obtain the morphism s d : X1 (N)z --+ X1 (M)z . A similar proof D works for Sd Xo (N)z --+ Xo (M) z . QUESTION. Complete the proof of Lemma 8.79. :
CHAPTER 9 Modular forms and G alois representations
As we announced in Chapter 2, in this chapter we construct Ga lois 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 al gebras 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 con struction 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 mod ular 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 Xo (Mp) , p f 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 Jo (N)Q be the Jacobian of the curve Xo (N) Q · Jo (N)Q is an abelian variety over Q. In Chapter 2, we defined the space S(N) of modular forms with Q coefficients as r(X0 (N) , 0 1 ) 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 81 (N) C S(N) , which we will define later. By the natural isomorphism (D.16) , (9.1) r(Jo (N)q , n}o (N ) Q / Q ) -----+ r(Xo(N)q, nt(N) Q / Q ) = So (N) , we identify as So (N) = I'(J0 (N)q, 0 1 ) . In Definition 2.31 we defined the Hecke operator Tn : So (N) ---+ S0 (N) for each integer n � 1. There, we used the finite fl.at morphisms s, t : X0 (N, n) ---+ X0 (N)q of curves over Q and defined it as Tn = s * o t * . From now on, we change the notation to denote s, t : Xo (N, n ) ---+ Xo (N)q by Sn , t n : Io (N, n ) ---+ Xo (N)q instead. The curve Io (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 ( is an elliptic curve over T, C a cyclic subIo (N' n ) T) = E group scheme C of order N, and Cn a subgroup . scheme of order n such that C n Cn = 0 The morphisms Sn , t n : I0 (N, n ) ---+ Xo (N)q are defined by sending (E, C, Cn ) to (E, C) and (E/Cn , (C + Cn )/Cn ) , respectively. Define the Hecke operator Tn : Jo (N)q ---+ Jo (N)q as the endomorphism Tn = t n * o s� of J(N)q. If Ci (i E I) are connected components of Io (N, n ) , then Tn is the composition
}
{
Jo (N)q = Jac Xo (N)q
fl (sn l ci ) * ;
IJ Jac Ci iEJ
fl
_ ; _--+
Jac Xo (N)q = Jo (N)q. By the identification of (9.1), So(N) = r(Jo(N)q, 0 1 ) , the Hecke operator Tn : So(N) ---+ So (N) coincides with the pullback by Tn : Jo (N)q ---+ Jo (N)q. DEFINITION 9.1. Let N � 1 be an integer. Define the Hecke al gebra To ( N) z as the subring of End(J0 (N)q) generated by the Hecke operators Tn , n = 1, 2, 3, . . . , i.e., (9.2) To(N) z = Z [Tn , n = 1, 2, 3, . . . ] C End Jo (N)q. The Hecke algebra To(N) z C End Jo (N)q defined above can be identified with a subalgebra of the Hecke algebra T (N) C End S0 (N) as follows.
9 . 1 . HECKE ALGEBRAS WITH
Z
COEFFICIENTS
63
LEMMA 9.2. To (N)z is a commutative algebra and is finitely gen erated as a Z-module. To T E To (N)z C End Jo (N)q, we associate T * E End r(J0 (N)q , 0 1 ) = End S0 (N) . Then, we obtain an isomor phism of commutative algebras (9.3) To (N) z © z Q = To (N)q --+ T(N) . The isomorphism (9.3) sends the Hecke operator Tn : J0 (N) --+ J0 (N) to the Hecke operator Tn : So (N) --+ So (N) . PROOF. Since End Jo (N)q is a finitely generated free Z-module by Proposition A.51(1), so is the submodule To (N)z . Since the Hecke operator Tn : S0 (N) --+ S0 (N) is the pullback T,'.;'. of the Hecke operator Tn : Jo (N)q --+ Jo (N)q, we obtain a surjective homomorphism To (N)z © z Q --+ T(N) . By Proposition A.51 (3) , this is injective. By Proposition 2.32, T(N) is commutative, so is 0 To (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 E So (N)K be a primary form with K coefficients. The subfield K1 = Q(an (f) , n 2:'.: 1) of K is a finite extension of Q, and a n (!) is an algebraic integer for each integer n 2:'.: 0. PROOF. Let 'P J T(N) = To (N)q --+ K be the ring homomor phism defined by the primary form f . Since Kf is the image of 'P J : To (N)q --+ K, it is a finite extension of Q. Since a n (!) = 'P J (Tn ) 0 is in the image of To (N)z , it is an algebraic integer. EXAMPLE 9.4. If 9o (N) = 0, then To (N)z = 0. If 9o (N) = 1, then To (N)z = Z. If f = L::= l an (f)qn is the unique primary form of level N, then Tn = an (!) . The Atkin-Lehner involution w = WN : X0 (N)q --+ X0 (N)q induces the involution of Jo (N)q, w = w * Jo (N)q --+ Jo (N)q. LEMMA 9.5. Let N 2:'.: 1 and n 2:'.: 1 b e integers. (1) The endomorphism T,'.;'. = Sn * o t� of Jo (N)q satisfies w o Tn = T,'.;'. o w . (2) If n and N are relatively prime, then we have T,'.;'. = Tn . PROOF. (1) Let T be a scheme over Q. For a triple (E, C, Cn ) E Io (N, n) (T) , let E' = E/(C + Cn ) · If we let C' be the kernel of the dual of E/Cn --+ E' and C� be the kernel of the dual of :
:
64
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
E/C --+ E' , then we have (E' , C', C�) E I0 (N, n) (T) . Sending (E, C, Cn) to (E' , C' , C� ) , we obtain a morphism w : Io (N, n) --+ I0 (N, n) . We have w 2 = id, and the diagram
Xo (N)Q
wl
�
Io(N, n)
wl
�
Xo (N)Q
lw
Xo (N)Q � Io(N, n) � Xo (N)Q is commutative. The assertion follows easily from this. (2) Define a morphism v Io (N, n) --+ Io(N, n) by sending (E, C, Cn) to (E/Cn, C + Cn/Cn, E[nl /Cn ) · Then, we have v 2 = id, and the diagram Xo (N)Q � Io (N, n) � Xo (N)Q :
I
v
I
1
Xo(N)Q � Io (N, n) � Xo (N)Q D is commutative. The assertion follows easily from this. Let X0 (N)an be the compact Riemann surface associated with the curve X0 (N)Q, and let H1 (X0 (N)an , Z) be its singular homol ogy group. The complex torus J0 (N)an defined by the abelian vari ety Jo (N)Q can be identified with Hom(So(N) , C ) / H1 (Xo(N)an , z) through the isomorphism (D.9) . Define the Hecke operator Tn on H1 (Xo(N)an , Z) by Tn = t * s * . Through the injective morphism of algebras End Jo (N)Q --+ End H1 (Jo (N) an , Z) = End H1 (Xo(N) an , Z) , H1 (Xo (N)an , Z) is a To (N)z -module. PROPOSITION 9.6. H1 (Xo(N)an, Q) is free To (N)Q -module of o
rank 2.
a
PROOF. By Proposition 2.55, Hom(So (N)Q, Q) is a free To (N)Q module of rank 1. Thus, Hom(So(N)c , C) is a free To (N)c-module of rank 1 and a free To (N)R-module of rank 2. The natural isomorphism ( D.8 ) gives an isomorphism of To (N)R-modules H1 (Xo (N) an , Q) ©Q R --+ Hom(So(N)c, C ) . Thus, H1 (Xo (N)an, Q) ©Q R is a free To (N)R-module of rank 2 . Since To(N)R is faithfully flat over To (N)Q, H 1 (Xo (N)an, Q) is a free To (N)Q-module of rank 2. D
9. 1 .
HECKE ALGEBRAS WITH Z COEFFICIENTS
65
H1 (X0 (N)an, Z) is the dual of the singular cohomology group H 1 (X0 (N)=, z) , and thus, by the Poincare duality, it is naturally isomorphic to H 1 (X0 (N)=, Z(l)). The cup product of cohomology H 1 (X0 (N)an, Z(l)) x H 1 (X0 (N)=, Z(l)) -+ Z(l) is nondegenerate, and it induces a nondegenerate alternating form ( , ) : H1 (Xo(N) an , Z) x H1 (Xo(N) an , Z) -+ Z(l). For a E H1 (Xo(N)an, Z) , define fa : H1 (Xo(N)an , Z) -+ Z(l) by fa (b) = (a, wb) . By the Poincare duality, the mapping (9.4) H1 (Xo (N) an , Z) ----+ Hom(H1 (Xo(Nrn , Z) , Z(l)) that sends a to fa is an isomorphism of Z-modules. We consider Hom(H1 (Xo (N)an, Z) , Z(l)) as a To (N)z-module through Tf (b) = f (Tb) . LEMMA 9.7. The mapping (9.4) is an isomorphism of To (N)z modules.
PROOF. It suffices to show that ( 9.4 ) is a morphism of T0 (N)z modules. For a, b E H1 (X0 (N)an, z) and n ;::: 1 , we have (Tn a, wb) = (a, T;:: w b) . Thus, by Lemma 9.5(1) , we have (Tn a, wb) = (a, wTn b) , D 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 To (N)Q · (2) The To (N) Q -module So (N)Q is isomorphic To (N)Q · PROOF. ( 1 ) Since we have HomQ (H1 (X0 (N)an , Q) , Q(l)) HomTo(N) Q (H1 (Xo(N)=, Q), HomQ (To (N)Q , Q(l))), by Lemma 9.7 and Proposition 9.6, Hom(To(N)Q, Q(1)) 2 is a free T0 (N)Q-module of rank 2. The assertion follows immediately from this. (2) The To (N)Q-module HomQ (So (N)Q , Q) is isomorphic to To (N)Q · Thus, by ( 1 ) , So (N)Q = HomQ (HomQ (So(N)Q, Q) , Q) is D isomorphic to To (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 ( 9.5 )
(f, g)
for f, g E S0 (N)c product.
=
=
Rj
87r 2
f /\ g
Xo(N)(C) r(X0 (N)c , n 1 ) . (f, g) is called the Petersson
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 : So (N)c ---+ So(N)c is self-adjoint with respect to the Petersson product.
PROOF. The adjoint operator of Tn is T_;: . Thus, the assertion follows immediately from Lemma 9.5(2) . D COROLLARY 9.11. Let N � 1 be an integer. (1) The reduced Hecke algebra with R coefficients T0(N)R c T0 (N)R generated by Tn with ( n, N) = 1 is isomorphic to the product ring Rdim To (N) � .
(2) The reduced Hecke algebra with Q coefficients T0 (N)Q c T0 (N)Q is generated by Tn with ( n, N) = 1 is reduced. Let
PROOF. (1) Commutative Hermitian matrices can be diagonal ized 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). D So far we have treated modular forms defined as differential forms on Xo ( 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) = r(X1 (N)Q , 0 1 ) . 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) we identify as S1 (N) = r(J1 (N)Q , n 1 ) . All the properties of S1 (N) and J1 (N) we describe below are proved similarly to the properties of So (N) and Jo (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 Ii (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) n C = 0. Ii (N, n) is a proper smooth curve over Q. To (E, P, C) , associat ing (E, P) and (E/C, image of P) , respectively, we obtain finite fl.at morphisms sn , tn : I1 (N, n) ---+ X1 (N)Q of curves over Q. Define an endomorphism Tn by Tn = tn* s� . We call it the Hecke operator. We define the Hecke operator Tn : 81 (N) ---+ 81 (N) as the pullback of Tn : J1 (N)Q ---+ J1 (N)Q · For a E (Z/Nz) x , 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 81 (N) . The Hecke algebra T1 (N)z is defined as the subalgebra (9.7) T1 (N)z = Z[Tn ; n :;::: 1, (a) ; a E (Z/Nz) x ] of End J1 (N)Q· The Hecke algebra T1 (N)z is commutative and we have (9.8) L Tn n -s = II (1 - Tpp -s + (p)p . p - 2 s ) - 1 . p :prime n= l Here we defined (p) = 0 for p I N. By (9.8) , we have T1 (N)z = Z [Tp; p : prime, (a) ; a E (Z/NZ) x ] . T1 (N)z is a finitely generated free Z-module. Define a morphism : X1 (N) Q ( (N ) ---+ X1 (N)Q ( (N ) as follows. For an elliptic curve E, let ( , ) N : N : E(N] x E[N] ---+ µ N be the Weil pairing. Regard T as a scheme over Q((N ) · Let E' = E/(P) for the pair (E, P) E M1 (N) (T) , and let P' be the image of Q E E(N] satisfying (P, Q) N = (N · Then, we have (E', P') E M l (N) (T) . Sending (E, P) to (E', P') , we obtain a morphism X1 (N) Q ( (N ) ---+ X1 (N) Q ( (N ) · We have 2 = (-1) . Let n :;::: 1 be an integer. We define an involution w : J1 (N, n) Q ( (N ) ---+ li (N, n) Q ( (N ) as follows. Let T be a scheme over Q((N ) · For a triple (E, P, C) E I1 (N, n) (T) , let E' = E/( (P) + C) , let P' E E' (N] be the image of a point Q in E / C satisfying (image of P, Q) = (N , and let C' be the kernel of the dual of E/C ---+ E'. Then, we have (E', P', C') E I1 (N, n) (T) . Sending (E, P, C) to (E', P', C') , we ob tain a morphism w : J1 (N, n) ---+ Ii (N, n) . We have w 2 = (-1). The
{
}
o
00
w
w
w :
68
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
diagram X1 (N)Q w
l
�
li (N, n)
wl
�
X1 (N)Q
lw
X1 (N)Q � li (N, n) � X1 (N)Q is commutative. From this we have w o Tn = T� o w. Let n ;::: 1 be an integer relatively prime to N. Let v Ii ( N, n) ---+ Ii (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 dia gram X1 (N)Q � li (N, n) � X1 (N)Q (n) X1 (N)Q � li (N, n) � X1 (N)Q is commutative. Thus, the endomorphism T� = s * o t * of J1 (N)Q is equal to (n) o Tn . We have w o (a) = (a) o w for a E (Z/Nz) x . For an integer n ;::: 1, we define the Hecke operator Tn of the singular homology group H1 (X1 (N)an, z) by Tn = t * o s * , and for a E (Z/Nz) x 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(l)) by sending a E H1 (X1 (N)an , Z) to the linear form H1 (X1 (N)an , Z) -+ Z(l) 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 :
l
(9.10)
v
( f, g)
=
l
I
87r2 l A
-
X 1 ( N ) (C)
-
f /\ g
for f, g E S1 (N)c = r(Xi (N)c , 0 1 ) . For an integer n ;::: 1 relatively prime to N, the adjoint operator 1of the Hecke operator Tn is (n) - 1 Tn , and the adjoint of (n) is (n) - . By the fact that commuting nor mal matrices can be diagonalized simultaneously, the reduced Hecke algebra with complex coefficients T{ (N)c = C[Tn , (n) ; (n, N) = 1] C T1 ( N)c is isomorphic to the product ring C dim T{ ( N ) c . The reduced o
9 . 1 . HECKE ALGEBRAS WITH Z COEFFICIENTS
69
Hecke algebra with rational coefficients T{ (N)Q = Q[Tn , (n) ; (n, N) = 1] c T1 (N)Q is isomorphic to the product ring of a finite number of number fields. The relation between S1 (N) and So(N) and that between T1 (N) and To(N) are as follows. The natural morphism X1 (N)Q --+ Xo (N)Q that sends (E, P) to (E, (P) ) induces a surjective morphism of Ja cobians J1 (N)Q --+ Jo (N)Q · The pullback by this morphism de fines a natural injective morphism So (N) --+ S1 (N) . Since Xo(N)Q is the quotient of X1 (N)Q by the action of the diamond operators (Z/Nz) x , we identify S0 (N) with the invariant part S1 (N) ( Z /N Z ) x through the natural injection S0 (N) --+ S1 (N) . For an integer n � 1, the diagram
l
l
l
Xo(N)Q � Io (N, n) � Xo(N)Q is commutative. Moreover, the two squares induce isomorphisms Ii (N, n) --+ (the normalization of Io (N, n) x xo(N) Q X1 (N)Q)), the natural surjection J1 (N)Q --+ Jo (N)Q is compatible with the Hecke operator Tn. Thus, we define a ring homomorphism T1 (N)z --+ To(N)z by sending Tn to Tn for any integer n � 1 and ( a) to 1. The natural surjection J1 (N)Q --+ Jo (N)Q and the natural injection So(N) --+ S1 (N) are compatible with T1 (N)z --+ To (N)z. By Proposition 8.69, the pair (Eq , P ) of the Tate curve Eq and its point P = (0, (N ) of order N defines a morphism e : Spec Q( (N ) [[q]] --+ X1 (N)Q · For an extension K of Q((N ) , e* : S1 (N)K --+ K[[q]]dq is injective. If e* f = :L::'= i an ( f) qn � for a modular form f E S1 (N)K, we call the power series :L::'= l an ( f) qn the q-expansion of f. Since the diagram Spec Q((N ) [[q]] --+ X1 (N)Q
l
l
Spec Q[[q]] --+ X0 (N)Q is commutative, so is the diagram So(N)K --+ K[[q]]
nl
S1 (N)K
--+
I
K[[q]].
70
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
Let K be a field of characteristic 0. A modular form f E 81 (N ) K 81 (N) ©Q K is called a primary form if f is an eigenvector of all T E T1 (N)z and a 1 ( f ) = 1 E K((N ) · Define a ring homomorphism 'Pf : T1 (N)z -+ K by Tf = r.pf (T)f, and we obtain a one-to-one correspondence between primary forms with K coefficients and the ring homomorphisms 'Pf : T1 (N)z -+ K. For a primary form f with K coefficients, each coefficient an (!) = r.p f (Tn) of the q-expansion is an algebraic integer. The subfield Q (an ( f ) , n 2:: 1) of K is a finite extension of Q . For a primary form f E 81 ( N) K with K coefficients, the character c = cf : (Z/Nz) x -+ Kx defined by (a)f = c(a)f is called the character of f . cf is the composition of ( ) : (Z/Nz) x -+ T1 (N)� and 'Pf : T1 (N)� -+ Kx . For an integer n 2:: 1, we have r.pf (Tn ) = an ( ! ) , and for a E (Z/Nz) x , we have r.pf ((a) ) = cf (a) . For relatively prime integers N, M 2:: 1, the quotient of X1 (NM) by the subgroup (Z/Nz) x C (Z/NMz) x is denoted by X0 , 1 (N, M) . Let Jo,1 (N, M) be the Jacobian of Xo, 1 (N, M) , and define the Hecke algebra To, 1 (N, M)z as the subring Z [Tn ; n 2:: 1, (a) ; a E (Z/Mz) x ] of End Jo, 1 (N, M) generated by the Hecke operators Tn, n 2:: 1, and the diamond operators (a) , a E (Z/Mz) x . =
9.2. Congruence relations
In this section we construct Galois representations associated with modular forms. Let N 2:: 1 be an ninteger, and let i be a prime number. Let T£Jo(N) = lim Jo(N) [i ] (Q) be the Tate mod-f-- n ule of the Jacobian Jo(N), and let Ve Jo(N) = T£Jo (N) ©z t Q£. These naturally possess the action of the absolute Galois group GQ = Gal( Q/ Q ) . Define the action of T E T0 (N)z c End Jo(N) as T* . Then, T£Jo(N) and Ve Jo(N) become the To (N)z ©z Z£ = To (N)z t module and To (N)z ©z Q£ = To (N)Q t -module, respectively. The absolute Galois group G Q acts on T£Jo(N) and Ve Jo(N) as an au tomorphism of To (N)z t -module and To(N)Q t -module, respectively, and these define £-adic representations of GQ . An isomorphism of T0 (N)z-modules H1 (X0 (N ) an , Z) © Z/ en z -+ n J0 (N) [t ] is induced by the natural isomorphism of complex tori 80 (N)c/ H1 (X0 (N ) an , Z) -+ J0 (N) an. It induces an isomorphism of To (N)z t -modules H1 (Xo( N ) an , Z) © Z£ -+ T£Jo(N) and an isomor phism of To (N)Q t -modules H1 (Xo (N ) an , Z) © Q£ -+ Ve Jo(N) . Similar facts hold for T£J1 (N) and VeJ1 (N) .
9. 2 . CONGRUENCE RELATIONS
71
LEMMA 9.12. Let N ;:::: 1 be an integer, and let i be a prime number. (1) VeJo (N) is a free To (N)Qt -module of rank 2 . (2) VeJ1 (N) = T1.J1 (N) ® zt Q 1. is a free T1 (N)Qt -module of rank 2 . (3) The natural morphism X1 (N)Q -+ Xo (N)Q induces an isomorphism of To (N)Qt -modules (9. 11) VeJ1 (N) ®T1 (N) Qt To (N)Qt � VeJo (N) .
PROOF. (1) We have an isomorphism of To (N)Qt -modules H1 (Xo (N)a:n , Z) ® Q1. -+ VeJo (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 0 of rank 2, which ought to be an isomorphism. Taking a basis of VeJo (N) over T0 (N)Qt , we obtain a continuous representation GQ -+ GL 2 (To (N)Qt ) . The same is true for VeJ1 (N) . In this section we prove the following theorem. THEOREM 9.13. Let N ;:::: 1 be an integer, and let i be a prime
number. The i-adic representation VeJ0 (N) of GQ is unramified at primes p f Ni. The characteristic polynomial det(l -
Theorem 3.18(1) follows immediately from Corollary 9.14 by tak ing a lattice in V1 stable under the action of GQ . EXAMPLE 9.15. Let f = E:= l an (f)qn be the unique primitive form of level 1 1 , and let E = X0 ( 1 1 ) . If i is a prime number, T1.E is the i-adic representation associated with f, and we have det(l
72
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
for all primes p -:/:- i, 11. The solution (5, 5) to the equation y2 + y = x3 - x 2 - lOx - 20 is a point of order 5, and if p -:/:- 11, we have 5 IU 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 i be a prime number. The i-adic representation VeJi (N) of GQ is unramified at =
primes p not dividing Ni, and we have det(l -
COROLLARY 9.17. Let i be a prime number, and let K be a fi nite extension of Qf. . Let f E Si (N)K be a primary form of level N of character c with K coefficients. Define the tensor product VJ = VeJi (N) ®To (N)Q t K by the ring homomorphism Ti (N)Q t ---7 K cor responding to f . The i-adic representation VJ of GQ is unramified at primes p not dividing Ni, and we have det(l -
PROOF OF THEOREM 9.16. Let p be a prime number not divid ing Ni. By Theorem 8.63(2) , Xi (N)z has good reduction at p. Thus, by Lemma D.18, Ji (N) has good reduction at p and VeJi (N) is an i-adic representation good at p. We show (9. 12) . We regard VeJi (N)F p as a Ti (N)Q t -module through the injection End Ji (N) ---7 End Ji (N)F p in Lemma D.11(2). The natural isomorphism VeJi (N)Q ---7 VeJi (N)F p in Lemma D.18 is an isomorphism of Ti (N)Q t -modules. Thus, we have det(l -
9.2. CONGRUENCE RELATIONS
73
LEMMA 9.18. Let N 2:'. 1 be an integer, and let p be a prime number not dividing N. (1) The following relation of endomorphisms of Ji (N)F p holds (9.13)
Tp = F + (.p) V.
(2) Let (N be an Nth root of unity, and let w = W N be the Atkin Lehner involution defined by (N . Then the following relation of endomorphisms of Ji (N)F p ( t;N ) holds
Vw = w(p) V. The relation (9.13) is called the congruence relation. PROOF. (1) Since p is a prime, we have Ii (N, p) = Xi,o (N, p) . If s, t : Xi,o(N, p) � Xi (N) are natural morphisms, we have Tp = t * o s * : Ji (N) � Ji,o (N, p) � Ji (N) by the definition of Hecke operators. By Theorem 8.76, Xi,o (N, p) is semistable at p. Let Ji,o (N, p) F p be the abelian part of Ji,o (N, p)i p . By Theorem A.49(1), t * : Ji (N, p) � Ji (N) induces Ji (N, p) F p � Ji (N)F p , and the Hecke operator Tp : Ji (N)F p � Ji (N)F p is the composition Ji (N)Fp � Ji,o (N, p) F p � Ji (N)F p · By Theorem 8.76 and Corollary D.21, the morphism (j(j , j i ) : Ji (N, p) F p -----+ Ji (N)F p x Ji (N)Fp induced by jo, ji : Xi (N)F p � Xi (N, p)Fp is an isomorphism. Since the composition t * o (j0 , ji) : Ji (N)F p x Ji (N)F p � Ji (N)F p equals (t * o io * ' t * o j 1 * ) , Tp = t * o s * : Ji (N)Fp � Ji (N)F p equals (t o jo) * o (s o io) * + (t o ii ) * o (s o ii ) * . Hence, it suffices show s o j0 = id, s o ji = F, t o j0 = F, t o ji = (p) . Since jo : Xi (N)F p � Xi (N, p)F p maps [(E, P)] to [(E, P, Ker F)] , we have s o io = id and t o io = F. Since ii : Xi (N)F p � Xi (N, p)F p maps [(E, P)] to [(E (P) , p (P) , Ker V)] , we have s o ii = F, and since FV = p, we have t o ji = (p) . (2) Each side is the pullback of an endomorphism of Xi (N)F p , w - i o F and F o (.p) - i o w - 1 , respectively. Thus, it suffices to show w o F = F ow o (.p) . For an elliptic curve E over FP ( (N ) and its section P of order N, let Q be a section of order N satisfying ( P, Q) N = (N . Then, we have (P (P) , Q (P ) ) N = (Kr. Hence, we have w o F(E, P) = w(E (P ) , p (P ) ) = (E (P ) /(P (P ) ) , p - i Q (P ) ) , and thus woF = (p) - i oFow.
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
74
On the other hand, since we have F (p) = F (p) 1 = 1 0 F we conclude that F = F (p) . (p) We now come back to the proof of Theorem 9. 16. Since Fo V = p, we have ( 1 - Ft )(l - (p) Vt ) = 1 - Tp t + p (p)t 2 by Lemma 9.18. Taking the determinant of each side, we obtain det(l - Ft : VeJ1 (N)F p ) det(l - Vt : VeJ1 (N)F p ) = (1 - Tp t +p(p)t 2 ) 2 . As in (9.9) , the composition VeJ1 (N)F p --+ Hom(VeJ1 (N)F p , Qe) of the limit of the isomorphism (D.15) and the transpose of the Atkin Lehner involution jV is an isomorphism of Tl (N)ql-modules. By Lemma 9.18(2) , this isomorphism maps the action of F to the ac tion of the transpose of (p) V, we have det(l - Ft : VeJ1 (N)Fp ) = det(l - {p) Vt : VeJ1 (N)Fp ) . Thus, we have det(l - Ft : lfe J1 (N)F p ) 2 = (1 - Tp t + p(p)t 2 ) 2 . If we regard det(l -Ft) , 1 -Tp t+p {p)t 2 E l +T1 (N) [[t]] c Tl (N) [[tl] X , we obtain det(l - Ft ) = 1 - Tp t + p(p)t 2 by taking the square root of 0 each side. COROLLARY 9.19. Let f be a prime number, and let K be a finite extension of Qe . For an f-adic representation p : Gq --+ GL 2 (K) and an integer N ;:::: 1, the following conditions (1) and (2) are equivalent. (1) p is modular of level N. ( 2 ) p is isomorphic to a subrepresentation of VeJo(N) @ql K. PROOF. The reduced Hecke algebra T� (N)K is isomorphic to the product ring rrf E � (N)K K1 . Thus, VeJo(N) @ql K is decomposed into the direct sum ffi f E � (N)K VeJo (N) ®T6 (N) Ql K1 . By Theorems 9.13 and 3.18 ( 2 ) , the semisimplification of VeJo (N) ®T6 (N) Q t Kf is the sum of f.-adic representations associated with f. The assertion follows from this immediately. 0 As a matter of fact, we can prove that VeJo(N) @q t K is semisim ple using Proposition 9.27 for example. However we do not prove it in this book. If we let p = f in Corollary 9.17, the following holds. If V is a good p-adic representation of Gq p , a filtered Qp [F, VJ-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(AF p ) . -
o
o w,
w o
w
o w o
o w o
o
-
o w
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 E S1 (N)K a primary form of lever N, character E:, with K coefficients. Let Vt be the tensor product VpJ1 (N) ©r1 (N)Q ,, K with respect to the ring homomorphism T1 (N)Q t --+ K defined by f. (1) The restriction of Vt t o GQ,, is a good p-adic representation, and we have det(l - Ft : D(Vt)) = 1 - ap (f) t + c(p)pt2 • The subspace D(Vt )' c D(Vt ) is one dimensional over K. (2) The restriction Vt l a Q ,, of Vt to GQ,, is ordinary if and only if ap(f) is a p-adic unit. Suppose ap (f) is a p-adic unit, and we write l - ap (f) t + c (p ) pt 2 = (l - t )(l - p/3t ) , where and /3 are a
a
also p-adic units. We denote simply by a and /3 the unramified characters of GQ,, whose value at <.pp are a and /3, respectively. Let x be the p-adic cyclotomic character. Then, Vt l aQ ,, is an extension of a by /3 X . ·
PROOF. By Theorem C.6(3) , the submodule D(V,, J1 (N))' of the T1 (N)Q,,-module D(VpJ1 (N)) equals S1 (N)Q,, as a T1 (N)Q,,-module. Also by Theorem C.6(3) , the quotient D(Vp J1 (N))/D(VpJ1 (N))' is Hom(S1 (N)Q,, , Qp) as a T1 (N)Q,,-module. Thus, both D(V,, J1 (N))' and D(VpJ1 (N))/D(VpJ1 (N))' are free T1 (N)Q,,-modules of rank 1. Hence, D(Vp J1 (N)) = D(J1 (N)) is a free T1 (N)Q,,-module of rank 2. By Lemma 9.18(1) , we have
det ( l - Ft : D(VpJ1 (N)) ) det ( l (p) Vt : D(V,, J1 (N)) ) = (1 - Tpt + p(p)t 2 ) 2 • By Lemma 9.18(2) , using Theorem C.2 for k = Fp((N ) , we see det ( l - Ft : D(VpJ1 (N)) ) = det ( l - (p) Vt : D(Vp J1 (N)) ) in the same way as the proof of Theorem 9. 16. Hence, we have det(l - Ft : D(V,, J1 (N))) = l - Tp t + p(p)t 2 . Since D(Vt) = D(Vp J1 (N) ) ©r1 (N)Q ,, K, this is a two-dimensional K-vector space, and det(l - Ft : D(Vt)) = 1 - ap (f)t + c(p)pt2 . D(Vt )' = S1 (N)Q,, ©r1 (N) Q ,, K is one dimensional. D (2) Clear from (1) and Corollary C.8. -
76
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
THEOREM 9.21. Let K be a field of characteristic 0, let f E 81 (N)K be a primary form with K coefficients, let c be the char acter of f , and let p f N be a prime number. If we write 1 - ap( f ) t + c(p)pt 2 = (1 - at) (l - /3t) , then both a and f3 are algebraic integers and the absolute value of any of their conjugates is VP·
PROOF. Similarly to Corollary 9.3, ap( f ) is an algebraic integer, and so are a and /3. Replacing K by its subfield Kt = 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 Qe. By Theorem 9.16, a and f3 are the eigenvalues of the action of the Frobenius substitution
In this section, we give relations between modular mod f repre sentations 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 f representations and maximal ideals of Hecke algebras with Z coefficients. LEMMA 9.22. Let f, be an odd prime number, and let F be a finite extension of Fe . Let 15 : GQ -+ GL 2 (F) be an absolutely irreducible continuous representation, and let N � 1 . Then, the following condi tions (i) and (ii) are equivalent. (i) j5 is modular of level N. (ii) There exists a ring homomorphism cp : To (N)z -+ F' to a finite extension F' of F such that det(l 15(
E So (N)K be a primary form, and let
9.3. MOD
e REPRESENTATIONS AND NON-EISENSTEIN IDEALS
77
defined by f. If FK is the residue field of K, 'Pf To (N)z --+ K induces a homomorphism 'PJ : To (N)q --+ FK and '1J1 (Tp) ap ( f ) for all prime numbers p. Conversely, let F' be a finite extension of F, and let rp : To (N)z --+ F' be a ring homomorphism. Let m be the maximal ideal Ker rp of To (N)z . Take a maximal ideal m' of the integral closure A of To (N)z lying above m. Let 0 be the completion of A at m' , and let K be its field of fractions. K is a finite extension of Qe , and 0 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 '1J1 (Tp) ap ( f ) for all prime numbers p. The assertion now follows immediately from this. D COROLLARY 9.23. Let R. be an odd prime number, let F be a finite extension of Fe, and let 15 : Gq --+ GL 2 ( F) be an absolutely irreducible continuous representation. Let N ;:::: 1 be an integer, and let rp : To ( N) z --+ F be a ring homomorphism such that {> is unramified at almost all prime numbers p f N and satisfies det ( l p ( is unramified at p, and we have det(l p (
=
=
-
-
-
-
-
-
Moreover, if 15 is good and ordinary at R., then Tr(F : D (15) ) = rp(Te) equals the value Prt (
D (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 max imal ideal of the Hecke algebra To (N) z . Let Fm = To (N)z/m. We say that m is Eisenstein ideal if there exists an integer M ;:::: 1 and characters a, {3 : (Z/Mz) x --+ F� such that Tp a(p) + f3(p) mod m for almost all prime numbers p f NM. an
=
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
78
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 i � 3 and i -:/:- 5, then by Proposition 4.3, the mod i representation defined by the £-torsion points of E = Xo (ll) 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 corre sponds to a modular irreducible mod i representation if and only if it is a non-Eisenstein ideal. PROPOSITION 9.26. Let N � 1 be an integer, and let i be an odd prime number. Let m be a maximal ideal of the Hecke algebra To (N)z containing l, and let Fm = To (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 p : GQ --+ GL 2 (F) of level N satisfying
1 - Tp t + pt 2 mod m for almost all prime numbers p f N. (ii) m is a non-Eisenstein ideal. PROOF. (i) ==? (ii) . Let p : GQ --+ GL 2 (F) be a modular ir reducible representation satisfying condition (i) . Suppose m is an Eisenstein ideal and we derive a contradiction. If m is an Eisenstein ideal and the characters a, f3 : (Z/ MZ) x --+ F� satisfy Tp a(p) + f3(p) mod m for almost all prime ideals p f NM, then, by Proposition 3.4(3) , p is isomorphic to the direct sum of the characters a E9 /3, which is a contradiction. (ii) =} (i) . As in the proof of Lemma 9.22, take a homomorphism To (N)z --+ 0 to the ring of integers of a finite extension K of Q£ such that m is the inverse image of a maximal ideal of 0. Let f E So ( N) K be a primary form corresponding to To (N)z --+ 0 c K, and let V be the £-adic representation of GQ associated with f. Take an 0-lattice L of V stable under the action of GQ, and let F be the residue field of 0. V = L ©o F defines a mod i representation p. p satisfies (9. 14) for almost all prime numbers p f N. It suffices to show that if m is a non-Eisenstein ideal, p is reducible. (9. 14)
det(l - p( r.pp ) t )
=
=
9.3. MOD
R.
REPRESENTATIONS AND NON-EISENSTEIN IDEALS
79
Taking the contrapositive, we assume p is reducible and show m is an Eisenstein ideal. If p is reducible, the semisimplification of p is the direct sum of characters a, f3 : GQ --+ F x . It suffices to show the image of a and f3 are in F,; . If a00 is the complex conjugate, we have af3(a00) = -1, a(a00) = ±1, f3(a00) = ±1, and thus we may assume a(a00) = 1, /3(a00) = - 1. By (9. 14) and Theorem 3. 1, we have det(l - p (a) t ) E F m [t] for any a00 • Thus, for any E Gal(F /Fm), we have {r o a, r o /3} = {a, /3}. Since r o a(a00) = 1, we have r o a = a, and o f3 = /3. Thus, the image of a and f3 are in F,; , and m is an 0 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 p : G --+ GL 2 (F) be an absolutely irreducible representation. Let V = F2 be the representation space of p. Let W be a finite-dimensional representation of G over F, and suppose the action of each g E G on W satisfies g2 - Tr(p(g)) · g + det p(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 p : F[G] --+ M2 [F] induced by p : G --+ GL 2 (F) is denoted also by p. Since p : G --+ GL 2 (F) is absolutely irreducible, p : F[G] --+ M2 (F) is surjective. Let J be the two-sided ideal (g2 - Tr p(g) · g + det p(g) ; g E G) . The surjective homomorphism F[G]/ J --+ M2 (F) induced by p : F[G] --+ M2 (F) is also denoted by p. Since the ring M2 (F) is semisimple, and a simple M2 (F)-module is isomorphic to F2 , it suf fices to show that the surjective homomorphism p : F[G]/ J --+ M2 (F) is an isomorphism. Define an anti-involution * of F[G] by g * = det p(g) · g - 1 . We have * 2 = 1. Since we have (g 2 - Tr p(g) · g + det p(g)) * = det p(g) 2 · g - 2 - Tr p(g) · det p(g) · g - 1 + det p(g) = det p(g) (g 2 - Tr p(g) · g + det p(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 p(g) - g) = (g2 - Tr p(g) · g + det p(g) )g - 1 E J, T
T
80
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
the ring homomorphism p : F[G] / J � M2 ( F ) is compatible with the anti-involution For any x E F[G] / J, we have x + x * = Tr p(x) . We show xx * = det p(x) for x E F[G] /J. Since p(xx * ) = p(x)p(x) * = det p(x) , xx * = det p(x) and xx * E F are equivalent . Since (x + y) (x + y) * = xx * + yy * + xy * + (xy * ) * = xx * + yy * + Tr p(xy * ) , *·
{ x E F [G] / Jlxx * E F } is a linear subspace of F[G]/ J . This sub space 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) x equals {x E F[GJ /JI det p(x) # O}. We show p : F[G] / J � M2 ( F ) is injective. It suffices to show for x E Ker p, that the annihilator Ann x = {y E F[G] / Jlyx = O} equals the entire F[G]/ J. If x E Ker p, then x * = -x. If y E F[G]/ J, we have yx E Ker p and thus yx = - (yx) * = -x * y * = xy * . Hence for y, z E F[G] /J, we have yzx = x(yz) * = xz * y * = zxy * = zyx. This implies Ann x � {yz - zy I y, z E F[G] / J} , and thus Ann x is a two-sided ideal of F[G]/J. Since p(Ann x) is a two-sided ideal of M2 ( F ) containing {AB - BAIA, B E M2 ( F ) }, it is the entire M2 ( F ) . Thus, Ann x contains an element y satisfying p(y) = 1 . Since this y is an invertible element of F[G] / J, we have Ann x = F[G] / J. This 0 proves x = 0. We now give the relation between modular mod i representations and the torsion points of the Jacobian of modular curves. LEMMA 9.28. Let N 2: 1 be an integer, let i be an odd prime num ber, and let m be a non-Eisenstein maximal ideal of the Hecke algebra To (N)z containing £. Let F be a finite extension of F m = To (N)z/m, and let p : GQ � GL 2 (F) be a modular irreducible representation of level N satisfying det(l - p (cpp ) t )
=
1 - Tpt + pt 2 mod m
for almost all prime numbers p f N. Let V = F 2 be the representation space. (1) Jo (N) [m] = {x E Jo (N) (Q) I ax = 0 for all a E m} is not 0. (2) Jo (N) [m] ®To ( N )z F is isomorphic to the direct sum of copies of V as a mod i representation of GQ .
9.4. LEVEL AND RAMIFICATION OF £-ADIC REPRESENTATIONS
81
PROOF. (1) We show Jo (N) [m] f:. 0. If we let H1 (Xo (N)an, Fe) = H1 (Xo (N)an , Z) ©z Fe, then Jo (N) [m] is isomorphic to H1 (Xo (N) an , Fe) [m] = {x E H1 (Xo (N) an , Fe) I ax = 0 for all a E m} as T0 (N) z modules . Since H1 (X0 (N)an , Z) is a finitely generated To (N)z -module and To (N)z --+ End H1 (X0 (N)an, Z) is injective, the localization H1 (Xo (N)an , Fe)m is not 0. Since the localization (To (N)z/ (f) )m is a finite local ring, we have H1 (X0 (N)an , F e) [m] f:. 0. (2) Let q f Nf be a prime number. By Lemma 9.18(1), we have the relation Tq = Fq + Vq of the endomorphisms of J0 (N)F q · Thus, as endomorphisms of Jo (N) ( F q ), we have cp � - Tq cpq + q = Fi - Tq Fq + Fq Vq = 0. Hence, for each prime number q f Nf, the action of the Frobenius substitution cp q on Jo (N)F q [m] satisfies cp� -Tq cp q +q = 0. If we let W = Jo (N) [m] ©To ( N ) z F, then by Theorem 3.1, the assumption of Proposition 9.27 is satisfied. Thus, W is the direct sum of copies -
D
� V.
9.4. Level of modular forms and ramification of f-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 0 be the ring of integers of a finite extension K of Qe, and let f be a primitive form of level N with K coefficients.
Let PJ : GQ --+ GL 2 (0) be the f-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 f N (ii) PJ is good at p. (2) (i) p 2 f N. (ii) PJ is semistable at p. PROOF OF (i) =? (ii) . By Theorem 8.63, Xo (N) has good reduc tion at p if p f N, has semistable reduction at p if p2 f N . Thus, .
the assertion follows from Corollary 9. 17, Lemmas D.18, D.16, and Corollary D.22, and in the case f = p, together with the fact that D
82
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
Let N 2: 1 be an integer, and let p be a prime number. Write N = Mpe , where M 2: 1 is an integer relatively prime to p . If e = 0, we have nothing to prove. So, we assume e 2: 1. For an integer 0 ::::; k ::::; e, the finite morphism of modular curves Spk : X0 (Mpe ) --+ X0 (M) (8.65) is denoted by Bk : X0 (Mpe ) --+ Xo (M) in this section. If e = 1, so, s 1 : Xo (N) --+ Xo (M) are the same as s, t : Io (M, p) --+ Xo(M) , which appeared in the definition of Hecke operators. Similarly, for an integer 0 ::::; k < e, the morphism Spk : Xo (Mpe ) --+ Xo(Mp) is denoted by tk : Xo (Mpe ) --+ Xo (Mp) . Let r 2: 3 be an integer relatively prime to N = Mpe . For an integer k E Z, let a;k be the automorphism of Spec Z[� , (r] defined by p k E (Z/rz) x = G al( Q ((r) / Q ) . For a scheme X over Z[� , (r] , the fibered product x X spec Z [ � , (rl /O"*p k Spec Z[� , (r] is denoted by x
(llJ, (<7, <7p• ) , a)
t---7
(llJ/<7pk , <7, a)
for 0 ::::; k ::::; e. Here, for a cyclic group <7p• of order pe , <7pk is its unique cyclic subgroup of order pk , and for a cyclic subgroup <7 of order M and a basis a of JlJ[r] , their images in llJ/<7pk are also denoted by <7 and a. For P, Q E JlJ[r] , the Weil pairing satisfies k ( image of P, image of Q ) ( E/Cp k ) [r] = (P, Q)� [r] " Thus, the diagram Xo, * (Mpe , r) z [ �] � Xo , * (M, r) z [ �J (9.15)
1
Spec Z[� , (r]
1
is commutative. By the commutative diagram (9.15) , the horizontal k arrow Bk defines a morphism Xo, * (Mpe , r) z [ �] --+ Xo, * (M, r) z(p[ �]) of schemes over Spec Z [ � , (r] . Similarly, for 0 ::::; k < e, define Xo, * (Mp, r) z [ �J tk : Xo, * (Mpe , r) z [ �] ----+ (llJ, (<7, <7p• ) , a)
i---+ ( llJ I <7pk , ( <7, <7pk +l / <7pk ) , a) .
9.4. LEVEL AND RAMIFICATION OF l-ADIC REPRESENTATIONS
83
�i
t k defines a morphism Xo,. (Mpe , r) z[ � ] � Xo,. (Mp, r) l of schemes over Spec Z[� , (r] . Define a morphism (p) : X0,* (Mpe , r) z[ � ] � to the triple X0 ,. (Mpe , r) z[ � ] by sending a triple o p) . (p) defines a morphism (p) : Xo, * (Mpe , r)z [ � ] � X0 ,. (Mpe , r)�[ ] of schemes over Spec Z[� , (r] . In what follows, we choose a maximal ideal p of Z[(r] lying above (p) , and we denote by Fp( (r ) its residue field Z [(rJ/lJ. PROPOSITION 9.29. Let p be a prime number, let M 2".: 1 be an integer relatively prime to p, and let r 2".: 3 be an integer relatively prime to N = Mpe . (1) To the abelian part Jo, * (Mpe , r) F p ( (r ) ' the morphism of abelian varieties s j. : Jo, * (M, r ) CP k ) � Jo, * (Mpe , r) over Q ( (r) induces
(E,C,a)
(E,C,a
i
an isogeny
e e sj. : II Jo,. (M, r) �:(i;r ) ffi k =O k =O
(9.16)
-7
Jo, * (Mpe , r) F p ( (r ) ·
(2) To the torus part Jo, * (Mpe , r) � p ( (r ) ' the morphism of abelian varieties sj. : J0,* (M, r) CP k) � J0,* (Mpe , r) over Q((r) induces an isogeny
e-1 e-1 E9 tj. : II Jo, * (Mp, r) ( ) -7 Jo, * (Mpe , r) �p ( (r ) · k = O k =O PROOF. (1) Suppose a :::::; e = a + b. The morphism Xo, * (M, r)F p � Xo, * (N, r)F p is defined by
�:{ :
(9.17)
a Fb),a) (E C ) { ((EE(P,("C-,KerV b ) , (C (P" - b ) , v a Fb), a CP" - b l ) Ja FP ( ,
, Q
Thus, if a diagram
i--+
:::::;
Ker is a morphism over
b, then
1
a-b
Spec Fp ( (r) �
Ja
a :5 b, b :::::; a.
(r) . If b :::::; a, the
1
Spec Fp ( (r)
is commutative, •- and the closed immersion defines a morphism Xo, * (M, r) F(pP 2 " ) � Xo,. (N, r)F p over Fp( (r ) ·
Ja
84
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
By Proposition 8.73 and Corollary D.20, we obtain an isomor phism from the abelian part Thus, it suffices to show the morphism of abelian varieties
is an isogeny. As in the proof of Lemma 9.18(1), B k ia is pk if a � b, k � b, i.e., 2a � e, a + k � e, (p) k-b pb-(k-b ) · , 2 a <_ e, a + k > e, = (p ) a+k-ep2 e- 2 a-k " f a _< b < k , i.e. pa-b+k = p2 a+k-e if k � b � a, i.e., 2a � e, a + k � e , (p) k-b pa-b+b-(k-b ) = (p ) a+k-e pe-k if b � a, b � k, i.e., 2a � e, a + k � e. As consequence, the determinant det ( B k ia ) of the matrix ( B k ia ) E Me+ i (Z[F, (p) ]) equals II ( (p) - F2 ) pa- 1 . (- 1) •Ce; l l II ( (p) - F2 ) pe-a- 1 1 :5a:5e/ 2 Since (p) - F2 : Jo,* (M, r) F,, ( (r ) -+ Jo,* (M, r) �:{(r ) is etale, the de terminant det ( s k ia ) is an isogeny. Thus, EB a , e (s k ia ) * is also an isogeny. (2) Let Xe be the character group of the torus Jo,* (Mpe , r)� ,, ( (r ) · Similarly, define X1 for Jo,* (Mp, r)�,, ( (r ) · It suffices to show that the homomorphism of Z-modules EB k t k * : Xe -+ x? e induced by EB k t k : n�:,� Jo,* (Mp, r)
i
_
_
o
o
X
o
o
9.4. LEVEL AND RAMIFICATION OF e-ADIC REPRESENTATIONS
85
is identified with [O , e] = {a E N I 0 ::; a ::; e }, and the set of the singular points of the reduced part Xo, * (Mpe , r)Fp ( (r) , red is identified with E. Furthermore, the inverse image of E in the normalization of Xo, * (Mpe , r)Fp ( (r ) ,red is identified with E x [O , e] . Thus, if we let z � + i = Ker(z n +l s� Z) , the dual chain complex of Xo, * (Mpe , r) Fp ( (r) is identified with [(zg+ i )t -+ z e + i ] . Thus, by Corollary D.20, the character group Xe is identified with Ker((zg+ i )t -+ z e +l ) . Similarly, the character group X1 can be identified with Ker((Z � )t -+ Z 2 ) . Since the rank of Xe and the rank of Xf are equal, it now suffices to show that ffi k th : Xe -+ XfB e is injective. Similarly to B k o ia , t k o ia is calculated as follows: if a ::; b, k < b, i.e., 2 a ::; e, a + k < e, io o F k if a ::; b ::; k, i.e., 2 a ::; e, a + k ?: e, i 1 (p) k - b pb - ( k - b) - 1 = j l (p) a + k - e p 2 e - 2 a - k - 1 jo o pa - b+ k = jo o p2 a + k - e if k < b ::; a, i.e., 2 a ?: e, a + k < e, j l (p) k - b p a - b+ b - ( k - b) - 1 if b ::; a, k, i.e. , 2a ?: e, a + k ?: e. = j l (p) a + k - e pe - k - 1 For a supersingular elliptic curve, we have [p] = F 2 . Thus, (p) : E -+ :E (P2 ) equals F2 . Hence, the mapping th : E x [O , e] -+ E x [O , l] induced by t k is given by (F k (x) , 0) if 2 a ::; e, a + k < e, k - 1 (x) , 1) (F if 2 a ::; e, a + k ?: e, t k (x, a) = k 2 (F a + - e (x) , 0) if 2 a ?: e, a + k < e, (F 2 a + k - e -l (x) , 1) if 2 a ?: e, a + k ?: e. 0
0
0
0
{
Since the Frobenius morphism F : E -+ :E (P ) is a bijection, the automorphism ffi 2 a ::; e id EB ffi 2 a > e p2 a - e of zt x [O , e] and the au tomorphism ffi k p k EB ffi k p k -l of ztx [O , e -l ] EB ztx [O , e -l ] are de fined. The linear mapping z [o, e] -+ z [O , l ] x [O , e -l ] that sends the stan , , dard basis ea E z [O , e] to "'"°"' L...- a + k < e eo , k + "'"°"' L...- a + k >- e e l k E z [O l ] x o,[O e -l ] , is injective. The mapping ffi k th : ztx [O e] -+ ffi�= � ztx [ 1] = ztx [O , e -l ] EB ztx [O , e- l ] is obtained by composing the above automor phisms to the direct sum of injections. Thus, this is also injective. D •
86
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
COROLLARY 9.30. Let p be a prime number, let M ?: 1 be an integer relatively prime to p, let e ?: 0 be an integer, and let N = Mpe . Consider the morphisms of abelian varieties over Q
e EB s 'k EB Jo (M) --+ Jo (Mpe ) , k =O k e- 1 t : EB 'k EB Jo (Mp) --+ Jo (Mpe ) . k=O k
(9.18)
:
(9.19)
(1) Let i =f. p be a prime number. The linear mapping ffi k s 'k Ve Jo(M) ffi e + l --+ Ve J0 (N) is injective and induces an isomor phism
Ve Jo(M)ffiFpe + l --+ Ve Jo(N) � p ·
(9.20)
The mapping of the torus parts defined by the morphism ( 9.19) induces an isomorphism
Ve Jo(Mp) tFffip e --+ Ve Jo(N) �p . The invariant part VeJ0 (N) 1P by the inertia group is equal to the
( 9.21)
image of
(9.22)
(EB s 'k , EB tk) : VeJo(M )ffi e+ l EB VeJo (Mp) 1P ffi e --+ VeJo (N) , k
k
and it is the direct sum of the abelian part Ve Jo(N) a and the torus part Ve Jo (N r . ( 2 ) The kernel of the morphism ( 9.18 ) is finite. The morphism ( 9.18 ) induces an isogeny ffi k s k : E9 � =0 Jo(M)F p --+ Jo (Mpe )� p · (3) The morphism ( 9.19 ) induces an isogeny of the torus parts ffi k t 'k : ffi�:� Jo(Mp)� p --+ Jo(Mpe ) �p . PROOF. ( 1 ) Let r ?: 3 be an integer relatively prime to N = Mpe. Since Q((r) is unramified at a prime ideal p lying above p, it suffices to show the assertion after extending the base field to Q((r) · Xo(Mpe )Q ( (r ) = Xo(Mpe )Q © Q Q((r) is the quotient of Xo, * (Mpe , r)Q by SL2 (Z/rZ). Thus, by Corollary D.14(2) , the nat ural mappings V £ JO (Mp e ) aFp ( (r ) --+ ( V £ JO,* (Mp e , r ) aF p ( (r ) ) SL2 ( Z/rZ) ' Ve Jo (Mpe ) �p ( (r ) --+ (VeJo, * (Mpe , r ) � p ( (r ) ) SL2 ( Z/ r Z ) TT
T
TT
T
9.4. LEVEL AND RAMIFICATION OF £-ADIC REPRESENTATIONS
87
are isomorphisms. By the isogenies (9.16 ) and (9.17) , we obtain the isomorphism (9.10) and (9.21) by taking the SL 2 (Z /r Z) invariant part. The remaining assertions follow immediately from the exact sequence (A.5) 0 -+ VlJo (Mpe ) t -+ VeJo(Mpe) lp -+ VeJo (Mpe) a -+ 0. (2) By the isomorphism (9.20) , VeJ0 (M)ffi e +l -+ VeJo(N) is in jective, and the kernel of the morphism ( 9.18 ) is finite. By the iso morphism (9.20) , ffi k s j. : Jo (M)�+p i -+ Jo (Mpe )�p is an isogeny. D ( 3 ) Clear from the isomorphism (9.21). PROOF OF (ii) ::::} (i) . We show the case p f:. f.. The case p = f. is similar, and we omit the proof. ( 1 ) Suppose a modular f.-adic representation VJ of level N is good at p, and we show VJ is modular of level M. By Corollary 9.19, we may assume VJ is a subrepresentation of VeJo(N) © K. Let W be the image of ffi k s j. : VeJo (M)ffie +l -+ VeJo (N) . Then, we have vfp = VJ , and thus VJ c VeJo (N)1,, © K. We show VJ c W © K. By Theorem 9.21, the eigenvalues of the action of
an integer. Let f. be a prime number, and let K be a finite extension of Qe . Let f E S1 ,o (Mpe , N)K be a primary form with K coefficients, and let VJ be the f.-adic representation associated with f.
88
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
(1) The following conditions (i) and (ii) are equivalent. (i) Vf is good at p. (ii) Vf is modular of level I' 1 ,o(M, N) . More precisely, there exists a primary form g E 81 ,o (M, N)K such that Vf is isomorphic to the R.-adic representation associated with g .
(2) The following conditions (i) and (ii) are equivalent. (i) Vf is semistable at p. (ii) Vf is modular of level I' 1 ,o(M, Np ) . More precisely, there exists a primary form g E 81 ,o (M, Np ) K such that Vf is
isomorphic to the R.-adic representation associated with g . PROOF. If f E 81 ,o (M, Npe )K, the proof is similar to that of
Theorem 3.52, and we omit it here. We reduce it to the case f E 81 ,o(M, Np e )K · It suffices to show f E 81 ,o(M, Np e ) if Vf is semistable at p. We show in the case f, =f p. We omit the case R. = p. Suppose Vf is semistable at p. Then, deg pf is unramified at p. On the other hand, by Theorem 9.16,
relatively prime to p. Let f, =f p be a prime number, and let K be a finite extension of Qf . Let f E 81 (Mp)K be a primary form with K coefficient, let c : (Z/Mp z) x -+ K x be the character of f, and let Vf be the R.-adic representation associated with f. Assume the restriction cl ( Z /pZ ) x is nontrivial. Then, we have ap (f) =f 0. Let a : GQ,, -+ K x be the unramified character defined by a(cpp ) = ap(f) , and let x : GQ -+ Qj be the R. adic cyclotomic character. Then, the restriction of Vf to GQ,, is the direct sum of the unramified character a and the ramified character ( x · c ) l o Q ,, · a - 1 .
We first show the consequence of Theorem 8. 77
9.4. LEVEL AND RAMIFICATION OF £-ADIC REPRESENTATIONS
89
PROPOSITION 9.33. Let p be a prime number, and let M ;:::: 1 be an integer relatively prime to p. Define an abelian variety J over Q to be the cokernel of the pullback J = Coker( Ji, o(M, p ) --* J1 (Mp) ) of the natural map X1 (Mp) --* X1 ,o (M, p ) . (1) The base change JQ ( (p ) = J ®Q Q((p) has good reduction at the prime ideal p = ((p - 1) . If Jp is the reduction mod p of JQ ( (p ) ' the closed immersions io, i 1 : Ig(Mp )Fp --* X 1 (Mp ) �[t i (8.61) induce an isomorphism (2) Let R. be a prime number different from p, and let VeJ:P be the invariant part by the action of the inertia group Ip G al( Qp ( (p ) /Qp ) · Then, ii induces an isomorphism
(3) The isomorphism (jo , j 1 ) * is compatible with the Hecke operators
Tq (q =/: p) and the diamond operators (a) (a E (Z/Mz) x ) . The isomorphism ii sends the action of the Hecke operator Tp to the action of the Frobenius homomorphism F.
PROOF. (1) By Corollaries 8.78 and D.21, J1 (Mp ) Q ( (p ) has semistable reduction at p . Moreover, the character group of the torus part J1 (Mp )� P of the mod p reduction J1 (Mp )Fp is naturally isomorphic to the character group of the torus part J1 ,o (M, p)� 'P of the mod p reduction of J1 ,o (M, p )F p · Thus, the natural mor phism J1 ,o (M, p )�'P --* J1 (Mp)� P is an isomorphism, and JQ ( (p ) = Coker(J1 ,o (M, p ) Q ( (p ) --* J1 (Mp ) Q ( (p ) ) has good reduction at p . By the diagram (8.64) , the isomorphisms from the abelian part j0 EB ii : J1 (Mp) f. p --* Jac lg(Mp) � p and io EB ii : J1 ' o(M, p ) f.p --* J1 (M) �'P induce the isomorphism (9.23) . (2) By Theorem 8.77 and Corollary D.21, we obtain a commu tative diagram J1,o,.(M, p , r)f.P ---+ ---+ J1,. (Mp , r)f.P J1,. (Mp , r)f. 'P (jo ,i1 ) "
l
Ci o ,ii ) *
l
l Uo ,i1 ) *
J1,. (M, r)} P ---+ J1,.(M, r)F p Jac lg(Mp , r)F p ---+ Jac lg(Mp , r)}P , x
90
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
and the vertical arrows are isomorphisms. Taking the SL 2 (Z/rZ) invariant part of Ve of these abelian varieties, we obtain the commu tative diagram VlJ1 (Mp)F p VlJ1 (Mp)Fp VlJ1,o (M, p)Fp (i o .il ) •
l
�
(i o .il ) •
2
l
�
1 (io .il ) •
-
2
Vl Jac lg(Mp)Fp > VlJ1 (M) Fp Vl Jac lg(Mp)F p VlJ1 (M)Fp and the vertical arrows are isomorphisms. By Corollary D.13(3) , the natural morphism VeJ1 (Mp ) Fp = (VeJ1 (Mp ) F p )1P is an isomorphism. Hence, we obtain the isomorphism ve J:P -+V£ J1 (Mp ) F)°Ve J1 ,o(M, p ) F/; -+V£ Jac lg(Mp )F p /VeJ1 (M)F p · (3) We omit the proof. D PROOF OF THEOREM 9.32. By Theorem 9.16, under the nota tion of Proposition 9.33, we may assume VJ is a subrepresentation of VeJ ©Q e K. Thus, by Proposition 9.33(2) , we have vfp = VJ n ( V£ (Jac lg(Mp)F p /J1 (M)F p ) ©Q e K) , and VJ = V/I E9 VJ/VJIP · If we write T1 (Mp)Qp = T1 ,o (M, p)Qp x A, then by Proposition 9.33 and Lemma 9.12(2), Ve(Jac lg(Mp )F p /J1 (M)F p ) is a free A-module of rank 1. Thus, both vfP and VJ ;vfP are one dimensional. The action of the Frobenius substitution cpp on Ve Jac lg(Mp )F p ©Qe K is the same as the action of the Frobenius homomorphism F, and in turn, the same as the action of the Hecke operator Tp by Proposition 9.33 ( 3 ) . Thus, the action of cpp on the invariant part VfP = VJ n (Ve(Jac lg(Mp)F p /J1 (M)F p ) ©Q e K) is the multiplication by ap(f) . Hence, ap(f) =j:. 0. Since det pJ = x · c , the representation VJ I G Q p is the sum of the characters GQ p -+ K x and (x c ) l a Q p · - 1 . D x
�
a :
�
·
a
9.5. Old part
Let p be a prime number, and let M � 1 be an integer relatively prime to p . In the proof of Theorem 3.52, the natural morphisms e E9 k s k EB Jo (M) --+ Jo (Mpe ) , k=O (9.25) e- 1 E9 k tk EB Jo ( Mp ) --+ Jo (Mpe ) :
:
k=O
91
9.5. OLD PART
played an important role. In this section we study these morphisms in more detail. In general, if M is a proper factor of N, the image of the natural morphism ffi d s;t : ffi d l N/M Jo (M) -+ Jo (N) is called the old part of Jo (N) . We first study the action of s;t on the q-expansion. LEMMA 9.34. Let d, M, N ;::: 1 be integers, and suppose dM I N . The image of f = I:: :'= i an qn E So (M) c by s;t : So (M) c -+ So (N) c is given by (9.26)
00
s;tf = 2:: dan q dn . n= l
PROOF. Let e : .6. -+ X0 (N)an be the morphism in Proposi tion 2.68(1) . The morphism e is defined by the family of elliptic curves Eq = e x /q z over .6. * = .6. - {O} and its cyclic subgroup µN . We denote the dth power mapping .6. -+ .6. also by s d . By the dth power mapping e x jqZ -+ e x jq dZ , the quotient Eq/µd is isomorphic to the pullback of Eq by s d : .6. -+ .6., and µM d / µd is mapped to µM . Thus, the diagram
8d 1
18d
.6. � Xo (M)an is commutative. Hence, we obtain s;t(f � ) = I:: :'= l an q dn
·
�-
D
From now on, as in the previous section, let p be a prime number, and for an integer M ;::: 1, we denote simply by B k the morphism of modular curves Sp k : Xo (Mpk ) -+ Xo (M) . LEMMA 9.35. Let p be a prime number, let M ;::: 1 be an integer relatively prime to p, and let e ;::: 0 be an integer. Let N = Mpe . (1) There is an isomorphism of curves a : X0 (Mpe+l ) -+ I0 (Mpe , p) that makes the diagram Xo (Mpe + l ) � Xo (Mpe )
(9.27)
commutative.
92
9.
MODULAR FORMS AND GALOIS REPRESENTATIONS
(2) Suppose e ;::: 2. Then, the diagram Xo (Mpe+l ) �
Xo (Mpe )
(9.28) Xo (Mpe )
� Xo (Mpe - l )
is commutative, and i t induces an isomorphism X0 (Mp e+l ) ---+ (the normalization of Xo (Mpe ) X xo (Mp • - 1 ) Xo (Mpe ) ) . ( 3 ) Suppose e ;::: 1, and let w = Wp : Xo (Mp) ---+ Xo (Mp) be the Atkin-Lehner involution. Then the diagram ( s o ,id )
Xo (Mpe +l ) II Xo (Mpe )
(9.29)
( s. , w os e - 1 )
l
Xo (Mpe )
� Xo (M)
Xo (Mp)
is commutative, and it induces an isomorphism X0 (Mpe+l ) II Xo ( Mpe ) ---+ (the normalization of Xo (Mpe ) x xo (M) Xo (Mp) ) .
PROOF. ( 1 ) Define a morphism a Xo (Mpe+l ) ---+ J0 (Mpe , p) by sending (E, CM , Cp•+i ) to (E/Cp , CM , Cp•+i /Cp , Efp] /Cp) · The inverse morphism I0 (Mpe , p) ---+ X0 (Mpe+l ) is obtained by sending (E, CM , Gp • , C) to (E/C, CM , fp] - 1 Cpe /C) . The lower left triangle of (9.27) is clearly commutative. Since the image of Cp•+1 /Cp in E = (E/Cp ) / (E fp] /Cp) is Gp • , the upper right triangle is also commutative. (2) The commutativity is easy to prove. Since X0 (Mpe+l ) ---+ ( normalization of Xo (Mpe ) X xo (Mp • - 1 ) Xo (Mpe ) ) is a morphism of coverings of degree p of X0 (Mpe ) , it is an isomorphism. ( 3 ) Since s0 ow : X0 (Mp) ---+ X0 (M) equals Sq , the diagram (9.29) is commutative. The proof of the isomorphism is similar to (2) . D PROPOSITION 9.36. Let p be a prime number, and let M ;::: 1, e ;::: 1 be integers. Let N = Mpe . Define Up E Me+ 1 (End Jo (M)) by 0 0 0 0 :
p
(9.30 )
Up =
0 0
0
0 0 0 -1
p
Tp
9.5. OLD PART
93
if p f M, and
0
0 0
p (9.31)
Up = 0
0
0 0 0 p Tp
if p \ M . Then, the diagram e Jo (M) k=O
EB
Up x 1
(9.32)
e EB Jo (M) k=O
ffik sZ
Jo (Mpe )
---..:.:..+
l Tp
ffik s Z
Jo (Mp e )
---..:.:..+
is commutative.
PROOF. It suffices to show (9.33)
By Lemma 9.35(1), we have Tp Xo (Mpe ) .
= so*
o
if 0 :::; k < e, if k = e and p f M, if k = e and p \ M. si for so , s1 : Xo (Mpe+l )
First, we show the case k < e. Since equals s k +l so , we have o
Xo (M)
Sk
o
s 1 : Xo ( Mp e+l )
-+ -+
Next, we show the case k = e. If p f M, then by Lemma 9.35(3) and so = s1 we have o w,
* ,..,., * * * * * * ( B O .J. p = B O S o* O 8 1 = So* O S + S e - 1 O W ) O 8 1 e e e = so* o s ; + s; _ 1 = Tp o s; + s; _ 1 . +l
If p \
M,
then by Lemma 9.35(2) , we have
COROLLARY 9.37. Let p be a prime number, let M ;::: 1 be an integer relatively prime to p, and let e ;::: 1 . Let N = Mpe . Let
94
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
Pp (U) E To (M) z [U] be the characteristic polynomial of the matrix Up Pp (U) = det(U - Up) = u e - 1 (U2 - TpU + p) . (9.34) (1) Sending Tq E To (N) z to Tq if q =f. p, and to U if q = p, we obtain a ring homomorphism (9.35) To (N) z --+ To (M) z [U] /(Pp (U) ) . (2) Regard ffi� =O Jo (M) as a To (M) z [U]/(Pp(U)) -module by defin ing the action of U as the multiplication of the matrix Up in (9.30) . Then, ffi k s k : ffi� = O Jo (M) -+ Jo (Mpe ) is compatible with the ring homomorphism To (N) z -+ To (M) z [U] / (Pp(U))
(9.35) . PROOF. Let T be the polynomial ring Z [Tq , q : prime] of infinite variables. Define a surjective homomorphism T -+ To (N) z by Tq H Tq , and T -+ To (M) z [U] / (Pp (U)) by Tq H Tq (q =f. p) and Tp H Up · By Proposition 9.36, ffi k sk : EB� =O Jo (M) -+ Jo (N) is T-linear. By Corollary 9.30(2) , the kernel of ffi k s k : EB� =O Jo (M) -+ Jo (Mpe) is finite. Thus, the kernel of T -+ To (N) z c End Jo (N) is contained in the kernel of T -+ To (M) z [U]/(Pp (U) ) c End(ffi� = O Jo (M) ) . The assertion follows from this. D Let N = M Tip e S pe p , where p f M if p E S. Applying Corol lary 9.37 repeatedly, we obtain the ring homomorphism To (N) z --+ To (M) z [Up ; p E S] /(Pp (Up) ; p E S) . (9.36 ) PROPOSITION 9.38. Let p be a prime number, and let M 2".: 1 be
an integer relatively prime to p. (1) The action of the Hecke operator Tp on the cokernel Coker(s0 EB si : Jo (M) 2 -+ Jo (Mp)) equals -1 times the action of the Atkin-Lehner involution Wp and satisfies r; = 1 . ( 2 ) Let e 2".: 2 be an integer. The action of r; - 1 o n the cokernel Coker(ffi k s k : Jo (Mp)e -+ Jo (Mpe)) equals 0.
PROOF. ( 1) By Lemma 9.35 ( 1 ) and ( 3 ) , we have Tp + wp = s0 * o si+wp = s i o s o * - Thus, the image of Tp +wp is contained in the image of s0 EB si : Jo (M) 2 -+ Jo (Mp) , and Tp + Wp induces the 0 morphism on the cokernel J = Coker(s0 EB si : J0 (M) 2 -+ J0 (Mp) ) . Therefore, we have Tp = -wp as an endomorphism of J. Since w� = 1, we have r; = i .
9.5. OLD PART
95 o
(2) By Lemma 9.35(1), we have Tp = sh s0 . Thus, we have Tp = s0 sh by Lemma 9.35(2) . Repeating this, we obtain = Hence, as in (1), induces the 0 morphism on the s() D cokernel Coker(ffi k s;;, Jo (Mp)e ---+ Jo (Mpe) ) . COROLLARY 9.39. Let p be a prime number, and let M � 1 be an integer relatively prime to p. Let K be a field of characteristic 0. (1) If f E So (Mp) K is a primitive form of level Mp, then we have ap (f) = ±1. (2) Let e � 2 be an integer. If f E So (Mpe)K is a primitive form of level Mpe , then we have ap (f) = 0. PROOF. Clear from Proposition 9.38. D THEOREM 9.40. Let N � 1 be an integer. For a primitive form f E �(N) (C), we denote its level by NJ I N. Then, we have So ( N ) c = E9 E9 C s d_ f. 0
o
Se - 1 * ·
r;-1
r;-1
:
·
fE if.>{N)(C) d lN/Nt
PROOF. For a primitive form f E �(N) (C) of level N1 IN, we only show that s d_ f (dlN/N1 ) are linearly independent, and the other part of the proof is omitted. Let f = an qn E �(N) (C) be a primitive form of level N1 IN. Suppose there is a nontrivial relation L d l N/Nt cdsd_f = 0, and let d be the smallest integer d I N/N1 such that Cd "I- 0. By ( 9.26 ) , the coefficient of qd is dcda 1 = 0. Since f is primitive, a = 1, which is a contradiction. D Let N � 1 be an integer, let K be a field of characteristic 0, and let f be a primitive form of level NJ I N over K. For a prime number p I N/Nf , let ep = ordp N/Nf , and define a polynomial PJ,p (Up) E K1 [Up] of degree ep + 1 by - ap (f) Up + p) if p f N1 , ( 9.37 ) PJ,p (Up) = if ordp N1 = 1, (Up - ap (f)) + e " f p2 1Nf · u.p By Corollary 9.39(2) , PJ,p (Up) is the characteristic polynomial of the matrix Up in Proposition 9.36 with Tp replaced by ap (f) . COROLLARY 9.41 . Let N � 1 be an integer, and let �(N) be the finite etale scheme over Q consisting of primitive forms of level dividing N, defined as Spec T6 (N)Q · For f E �(N) , let K1 be its
L.:;:'=1
1
{u;Pu;p-1 (U'£ p1
I
96
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
residue field, and let Nt be the level of f . Then, we have T0 (N)Q = I1 t e if> (N) Kt, and we obtain a ring homomorphism
(9.38)
To (N)Q ---+
II
t E if> (N)
Kt [Up, p lN/Ntl/(Pt,p(Up ) , p lN/Nt)
by defining the f component of the image of Tp equal to ap( f ) if P f N/Nt , and Up if p I N/Nt . PROOF. We have TO(N)Q = f1t e if> (N) Kt since T0 (N)Q is re duced. Therefore, we have To (N)Q = I1 t e if> (N) To (N)Q ©r0 (N) Q K1 We define a ring isomorphism (9.39) K1 [Up, p lN/N1]/(PJ ,p (Up) , pl N/N1 ) ---+ To (N)K ®ro (N) f< K1 . For subrings T0 (N) [Tp Pf N/N1] c To (N) and T0 (NJ ) [Tp : Pf N/N1] C To (NJ ) the natural surjection T0 (N) --+ T0 (NJ) induces an iso morphism T0 (N1) ©rc) (N) [Tp : p f N/N1] --+ T0 (N1) [Tp : p f N/N1] Thus, by Corollary 2.61, we obtain an isomorphism K1 ©rc) (N) K TMTP : P f N/N1] --+ K1 Thus, we have To (N)K © ro (N) f< K1 = K1 [Tp, plN/N1] C End(So(N)Q ©rc) (N) Q K1) Consider the polynomial ring K1 [Up, plN/N1] , and define a surjection K1 [Up, plN/N1] --+ To (N)K ®ro (N) f< K1 by Up H- Tp. We prove that its kernel is generated by PJ,p(Up) (plN/N1) - It suffices to show it after tensoring C over Kf . We have So (N)Q ©rc) (N) Q /''P t C = ffi d lN/Nt C sd,f by Theo rem 9.40. By Proposition 9.36, ffi d l N/Ni C sd,f is generated by sif as a C[Tp , plN/N1 ]-module. Moreover, ffi d l N/Ni C si f is a basis of the free C [Up , p lN/N1l/(PJ,p(Up) , p lN/N1)-module ffi d l N/Ni C s d_ f . Thus, we obtain the isomorphism (9.39) . It is clear that the product of the inverse of (9.39) gives the isomorphism (9.38) . D COROLLARY 9.42. Since g0 (11) = 1 and g0 (22) = 2, we have To (ll)z = Z and the surjection To (22)z --+ To (ll)z [U2 l/(Ui-T2 U2 +2) is an isomorphism. If f = L:: := l an (f)qn is the unique primitive form of level 11, we have a2 (f) = -2, and thus To (22)z = Z[U2 ]/(U? + 2U2 + 2) is isomorphic to Z[v'-IJ by letting U2 = -1 + :
,
·
·
·
·
A.
9.6. NERON MODEL OF THE JACOBIAN Jo (Mp)
97
J0 (Mp) In this section, p is a prime number, M � 1 is an integer relatively prime to p, and N = Mp. As a preparation of the proof of Theo rem 3.55 in the next section, we study the mod p reduction Jo (Mp)Fp of the Neron model of J0 (Mp). By Theorem 8.63(3) , X0 (Mp) has semistable reduction at p, so does J0 (Mp) by Corollary D.22. Thus, Jo(Mp)F p is a successive extension of the group
98
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
is induced by a1 : r 1 --+ q . These are also homomorphisms of T-modules. We identify r l with a submodule of ri through a1 : r 1 --+ ri . We denote H1 (r) = Ker(r 1 --+ ro) by r�. By Corol lary D.21(2) , the group
Consider the diagram of T-modules o ------+ r 1/ r� ------+ r *1 /r01 (9.40) 1 ro rif (r� + r0) =
9.6. NERON MODEL OF THE JACOBIAN
Jo ( Mp )
99
runs all the subgroups of E of order r. Since L e; ( (Tr - (r + 1)) · ex ) = 0, E y E ( Fp ) it suffices to show that e ; ( (Tr - ( r + 1)) · ex) = e ; (Tr · ex) is divisible by " Gx for x =J y. If x = [(E, C)] =J y = [(E', C')] , then e ; (Tr · ex ) is the number of elements of the set H is a subgroup of E of order such that Ix, y = H C E (E / H, C + H/ H) is isomorphic to (E', C')
{
I
}
r
·
Thus, it suffices to show the natural action of Gx = Aut(E, C)/{±1} on Ix , y is a free action. We show that the action of Gx on Ix , y is a free action. Suppose a: =J ±1 E Aut(E, C) and H E Ix , y satisfy a:(H) = H. Then, by Lemma 8.41(1), a: E End(E, C) satisfies one of the relations a: 2 + 1 = 0, a:2 + a: + 1 = O and a: 2 - a: + 1 = 0, and the subring Z[a:] C End(E, C) is isomorphic to either Z[H] or z[ - i+/=3 ] . Moreover, the group E[r] of r-torsion points of E is a free Z[a:]/(r)-module of rank 1 and H is a submodule of order r. Thus, if such an H exists, r is decomposed to the product r = of prime elements in the principal ideal domain Z[a:] , and we have H = Ker(rr : E --+ E) . In this case, the multiplication by rr induces the isomorphism (E / H, C + H/ H) --+ (E, C) , and it contradicts the assumption. Hence, we proved the action of Gx on Ix,y is a free action. Thus, we proved J fi/f 1 = 0. We now prove J . q;r� = 0. Since J fi/f 1 = 0, J . q;r� is a sub group of fi/f�. By Lemma 8.41, the order of Gx = Aut(E, C)/{±1} is a divisor of 12 for each x E E. Thus, q/r 1 is a finite abelian group, and 12 q/r 1 = 0. Thus, we have 12rifr� c ri1r�, and 12J q/r� c Jrifr� = 0. Thus, J · fi/f� is a subgroup of ri1r� c ro = z{O , l } that becomes 0 after multiplying by 12, which implies J · q;r� = 0. Since the vertical arrow q ;r� --+ � in (9.40) 0 is surjective, we conclude J · � = 0. PROPOSITION 9.45. The action of the Frobenius substitution 'Pp E GF p on the character group X = Hom( Jo ( Mp ) tF_ p , Gm) of Jo(Mp)� p equals the action of the Hecke operators Tp , and cp� = 1 holds. PROOF. Since the Atkin-Lehner involution Wp satisfies w� = 1, it suffices to show 'Pp = - wp by Proposition 9.38. The action of the Frobenius substitution 'Pp on X equals the action of the Frobenius endomorphism F. By Corollary D.20, X is identified with H1 (r) c 1rn'
·
·
·
·
100
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
ri = Ker ( z E (F p ) x { O , l } -t z E (F p ) ) . Thus, it suffices to compare the actions of the Atkin-Lehner involution wp and the Frobenius homo morphism on E(Fp) x {O, 1}. Let (E, C, C') be a supersingular point of X0 (Mp ) Fp Since C' = Ker F, the action of the Atkin-Lehner involution wp is given by wp(E, C, Ker F) = (E (P ) , C (P) , Ker F) . Thus, the action of Wp on the set E(Fp) of supersingular points equals the action of F . Moreover, Wp permutes irreducible components of Xo(Mp ) F p and F preserves irreducible components. Hence, the action of Wp on x c r 1 coincides D with that of -F. COROLLARY 9.46. Let p be a prime number, let M 2:: 1 be an in teger relatively prime to p, let e 2:: 1 be an integer, and let N = Mpe . If i =f. p is a prime number, then the action of the Probenius substitu tion 'Pp on the cokernel Coker(ffi k si:, : ffi�:t VeJo(M) -t Ve Jo(N) lp ) ·
'
is the multiplication of pTp, and its eigenvalue is ±p.
PROOF. The morphism ffi t i:, induces a surjection Ve Jo(Mp)�p -t Coker(ffik si:, : ffi�:t VeJo(M) -t Ve Jo(N) Ip ) by Corollary 9.30(1) . If X is the character group of the torus Jo (Mp)� p , then Ve Jo(Mp)�p is isomorphic to Hom(X, Q£(1)). Thus, by Proposition 9.45, the action of the Frobenius substitution r.pp on VeJ0 (Mp)�p is the multiplication D by pTP , and its eigenvalue is ±p. From Proposition 9.45, we have the following proposition on the properties of £-adic representations associated with modular forms. PROPOSITION 9.47. Let p be a prime number, let M 2:: 1 be an integer relatively prime to p, and let N = Mp. Let i be a prime number, and let ('.) be the ring of integer of a finite extension K of Q£ . Let f E So (N)K be a primary form of level N with K coefficients. Suppose the i-adic representation VJ of GQ associated with f is bad at p. Then, V1 is ordinary at p. The coinvariant quotient Vf , Ip of the restriction of V1 to the decomposition group G Q p by the inertia Ip defines a one-dimensional representation of GF and the action of the Probenius substitution r.pp E GF on Vf ,Ip is the multiplication by ap (f) = ±1. PROOF. Since p2 f N, the £-adic representation V1 is semistable at p by Theorem 3.52(2) (ii) =? (i) . Since V1 is bad at p, VJ is ordinary at p. By Corollary 9.19, We may assume VJ is a subrepresentation of VeJ1 (N) ® Q t K. Moreover, since V1 is not good at p, it is not P
P
,
9.6. NERON MODEL OF THE JACOBIAN
Jo ( Mp )
101
modular of level M and is a subrepresentation of Coker(s * ffi t * : Ve Jo(M)EEl 2 --+ Ve Jo(N)) ®Q e K by Theorem 3.52(1) (ii) =} (i) . We deal with the case p =f f. The case p = f is similar, and we omit it. Since VJ is ordinary and it is not unramified, the Ip invariant part vfp c V1 is one dimensional, and the coinvariant quotient V1 , 1,, equals V1 /V1IP . We have VfP = V1 n Coker(s * ffi t * : Ve Jo (M) EEl 2 --+ Ve Jo(N) 1P ) ®Q e K. By Corollary 9.46, the action of
PIP ( r.pp) = rp(Tp) = ±1 for the Frobenius substitution
(2) Let K be a finite extension of Qe, and let f E So(N)K be a primitive form of level N. Let F be a finite extension of the residue field of K, and suppose Tr ( r.pq) aq (f) for almost all =
q f Nf. Then, 15 is ordinary at p. Furthermore, suppose p = f and 15 is good at p = f. Let D(15) be the F-module obtained by applying Theorem C.6 to the restriction of 15 to GQ t . Then, we have Tr ( F : D(15) ) = PIP ( r.pp) = ap( f ) .
PROOF. (1) Let K be a finite extension of Qe, and let f E So(N)K be a primary form of level N satisfying PJ 15. If 15 is not good at p, then neither is Pi · By Proposition 9.47, Pi is ordinary, and thus so is 15. Since the coinvariant quotient PIP is not the entire space;' =
1 02
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
it is one dimensional and Pf,Ip Prp · The assertion now follows from Proposition 9.47. (2) Since the .e-adic representation V1 is semistable but not good at p, it is ordinary by Theorem 3.52. Thus, p is also ordinary. Suppose p = .e and p is good at p = .e. Since det p is the mod .e cyclotomic character x, the restriction of p to GQ p is the extension of the unramified character PIP by x p41 . Thus, the F[F, VJ-module Dp is an extension of an F [F, VJ-module on which F acts as the multiplication by Prp (cpp) by an F [F, VJ-module on which F acts as the multiplication by p · PIP ( cpp) - 1 = 0. This shows the equality. The D congruence relation is clear from Proposition 9.47. =
·
9.7. Level of modular forms and ramification of mod .e representations
In §3.8 in Chapter 3, we stated the following theorem of the level of modular forms and the ramification of mod .e representations. THEOREM 3.55. Let .e be an odd prime, let F be a finite extension of Fe and let p : GQ :-+ GL2 (F) be an irreducible continuous repre sentation. Suppose p is modular of level N, and M is the prime to p part of N. Then, the following conditions (i) and (ii) in each of (I) and (2) are equivalent. (1) (i) p is modular of level M. (ii) p is good at p. (2) (i) p is modular of level pM. (ii) p is semistable at p.
PROOF OF (i) ::::} (ii) . Similar to the proof of Theorem 3.52 D (ii) . In this book we do not prove Theorem 3.55(2) (ii) ::::} (i) . We prove (I) (ii) ::::} (i) in the case p ¢. 1 mod .e, assuming (2) (ii) ::::} (i) . In other words, we prove the following theorem in the case p ¢. 1 mod .e. THEOREM 9.49. Let .e be an odd prime number, and let F be a (i)
::::}
finite extension of Fe . Let p be a prime number, and let M 2::: 1 be an integer relatively prime to p. Suppose p : GQ -+ GL2 (F) is a modular irreducible continuous representation of level N = Mp. Then, if p is good at p, p is modular of level M .
In this book we give a proof of Theorem 9.49 only in the case
p ¢. 1 . In this case the proof uses results of §9.6, which are derived
9.7. LEVEL AND RAMIFICATION OF MOD
R.
REPRESENTATIONS
103
from Theorem 8.63(3) on the structure of the modular curve Xo(Mp) . We regret we are unable to include the proof of the case p 1 in this book because it requires more preparation. For the proof of Theorem 9.49, we introduce notation for Hecke algebras. Define (9.41) T = To(Mp)z, T' = T0 (M)z [Uv J l (U; - Tv Uv + p) . The natural surjection T --+ T' (9.35) which was defined in Corol lary 9.37 is compatible with (9.42) (s * , t * ) : J0 (M) x J0 (M) --+ Jo (N) . We give a sufficient condition for a mod f, representation to be modular of level M in terms of the ring homomorphism T --+ T' . LEMMA 9.50. Let p : GQ --+ GL 2 (F) be a modular absolutely irreducible continuous representation of lever Mp, and let rp : T --+ F be a ring homomorphism satisfying det(l - p(cpq)t) = 1 - rp(Tq)t + qt 2 for almost all prime numbers q . If there exist a finite extension F' of F and a ring homomorphism rp' : T' --+ F' that makes the diagram T � F =
1
1
-..L.+ F' commutative, then p is modular of level M. D PROOF. Clear from Lemma 9.22 (ii) ::::} (i) . PROOF OF THEOREM 9.49 IN THE CASE p "¢. 1. Let p : GQ --+ GL2 (F) be a mod f, representation that satisfies the assumption of Theorem 9.49. Replacing F by its finite extension if necessary, take a ring homomorphism rp : T = To ( N) z --+ F corresponding to p by Lemma 9.22. Let m be the maximal ideal of T that is the kernel of rp : T --+ F. Let Jo(Mp) [m] = {x E Jo(Mp) (Q) I ax = 0 for all a E m}. J0 (Mp) [m] defines a mod R. representation of GQ · The finite etale group scheme over Q corresponding to the mod f, representation Jo (Mp) [m] is also denoted by Jo (Mp) [m] . By Lemma 9.28 , Jo(Mp) [m] is nonzero, and Jo(Mp) [m] ®T F is isomorphic to the direct sum of copies of p as a mod R. represen tation of GQ . By assumption, p is good at p, and the finite etale T'
104
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
group scheme Jo (Mp) [m]q p over Qp extends to a finite etale group scheme Jo (Mp) [m]z p over Zp. If p # £, then Jo(Mp) [m]z p is finite etale, the inclusion J0 (Mp) [m] ---+ J0 (Mp) extends to a morphism Jo (Mp) [m]z p ---+ Jo(Mp)zp over Zp by the properties of Neron mod els. By Proposition D.12(2) , this is a closed immersion. In the case p = £ also, the inclusion Jo (Mp) [m] ---+ Jo(Mp) extends to a closed immersion Jo(Mp) [m]z p ---+ Jo(Mp)z p . Taking the reduction mod p, we obtain a closed immersion Jo(Mp) [m]F" ---+ Jo(Mp)F " over Fp. Here, Jo(Mp)F " denotes the reduction mod p of the Neron model of Jo(Mp) . Let
-
:::::
:
:
:
R. REPRESENTATIONS
9.7. LEVEL AND RAMIFICATION OF MOD
(1),
1 05
By a morphism Jo (Mp) [m] Fp --+ Jo (Mp)F- P is induced. The m-torsion part Jo (Mp) [m] F p is a T/m-module, and Jo (Mp)F p is a T' module. Thus, if the morphism above is not the 0-morphism, T/m is a quotient of T' since m is a maximal ideal. In this case p is modular of level M. (3) If Jo (Mp) [m] Fp --+ Jo (Mp)F p is the 0 morphism, a morphism Jo (Mp) [m] F p --+ Jo (Mp)i- P is induced and we have Jo (Mp) [m] F p = Jo (Mp)i- p [m] . If X is the character group of the torus Jo (Mp)i- p , we have Jo (Mp)i- p [m] = Hom(X/mX, µR.) · Thus, we obtain an isomorphism Jo (Mp)i- p [m] = Hom(X/mX, µR.) of a finite flat commutative group scheme over Zp· We first show the case p "I i. By Proposition 9.45 the action of the Frobenius substitution
1
=
1
V(-1(21
�
1.
106
9. MODULAR FORMS AND GALOIS REPRESENTATIONS
However, there is no modular form of level 1 and weight 2 other D than 0. This is a contradiction. COROLLARY 9.52. Let j5 be a semistable absolutely irreducible rep resentation. If n 2: 1 is an integer, then the restriction f51o Q C
CHAPTER 10 Hecke modules
As we announced in Chapter 5, we will define Hecke modules as the completion of singular homology groups of modular curves. Then, we show that the families of liftings and RTM-triples satisfy the conditions of two theorems we proved in Chapter 6. In Chapter 5, we stated Proposition 5.30, which is about the surjectivity of morphisms of Hecke modules, Theorem 5.32(2) , which says Hecke algebras are free modules over group rings, and Proposition 5.33, which is about the multiplier of the surjection of RTM-triples. These propositions will be proved as Propositions 10.11, 10.37, and 10.14. The main result of Chapter 5 that says the deformation ring R coincides with the Hecke algebra T (Theorem 5.22) will thus become a consequence of the computations of Selmer groups in the following chapter. We will first define full Hecke algebras and Hecke modules in §10.1 and §10.2, respectively, and we will construct RTM-triples. We then compute the multiplier of a surjection of RTM-triples constructed out of Hecke modules in Proposition 10.14 in §10.2. The proof of Proposi tion 10.11 on the surjectivity of the morphism of RTM-triples defined in §10.2 will be given in §10.3. In §10.4, we will study the relation between the deformation ring and the group ring that is defined from a set of prime numbers satisfying certain conditions. In §10.5, we will define another kind of full Hecke algebras and Hecke modules, and construct a family of liftings. The fact that these Hecke modules de fine a family of liftings follows from Proposition 10.37. The proof of Proposition 10.37 will be given in §10.6. In §10.7, we put these results together, and reduce the proof of Theorem 5.22 to Theorems 5.32(1) and 5.34, which concern Selmer groups and will be proved in Chap ter 11. The properties of Hecke algebras presented in this chapter mainly follow from the properties of Hecke algebras with Q coefficients, and are proved using the ramification of Galois representations associated 107
108
10 .
HECKE MODULES
with modular forms and properties of old forms. On the other hand, the properties of Hecke modules depend essentially on the fact that the coefficients are in Z, and these are proved by topological prop erties of modular curves and the fact that some maximal ideals are non-Eisenstein. The proof of the main result is given by combining these elements in an appropriate way.
10. 1 . Full Hecke algebras
We review briefly the notation we used in Chapter 5. Throughout this chapter i is an odd prime number, and K is a finite extension of Q£ . We denote by 0 its ring of integers and by F the residue field. 15 GQ -+ GL2 (F) is an irreducible modular semistable £-adic representation, and Sp is the set of prime numbers p at which p is not good. By Corollary 3.56, 15 is modular with level Np = N0 = Tip E S;; p. Let :E be a finite set of prime numbers such that Sp n :E = 0 . In Chapter 5 we defined the deformation ring Rr; , a positive inte ger Nr; = N0 · Tip E :E ,p,e e p2 Tip E E ,p=l p, the set of primitive forms
•
:
=
I1 t E 4> (NE ) K , p Kt ·
Let V = F 2 be the representation space of 15. If 15 is good at p = i, the restriction of V to GQp defines an F [F, VJ-module D(V) by Theorem C.l. Regard the action of F on D(V) as a representation of F, and denote it by D(15) . By Corollary 9.23(2) , we have Tr ( F : D(15) ) = cp (Te ) . In this section, we denote the Hecke algebra T0 (N)z simply by T(N)z. We define the full Hecke algebra Tf as the completion of the Hecke algebra T(Nr;) o = T(Nr;)z © z 0 at some maximal ideal. We first find the maximal ideal at which we take the completion. PROPOSITION 10.1 . There exists a unique 0-algebra homomor phism cpr; : T(Nr;)o = T(Nr;)z ©z 0 -+ F satisfying the following
10 . 1 . FULL HECKE ALGEBRAS
109
condition: if p =/. £, ¢. S,o u E, if p = f ¢. S,o, (10.1) if p E S,o, if p =/. £, E E. PROOF. Since T(NE)o is generated by Tp (p: prime) , the unique ness is clear. We show the existence. First, we show it when E = 0 . By Corollary 3.56, p is modular of level N,o = N0 . Thus, by Lemma 9.22 and Corollary 9.23(1), there exists a ring homomorphism VJ : T0 (N0)z --+ F' to a finite extension F' of F that satisfies det(l - p( ipp)t) = 1 - VJ(Tp)t + pt2 for all prime numbers p f N0 £ . If £ ¢. S,o, we have £ f N0 , and by Corollary 9.23(2), p is good at £ and det(l - Ft : D(p)) = 1 VJ(T£)t + ft2 . If p E S,o, then p is ordinary at p and bad at p. Thus, by Corollary 9.48(1 ) , we have hp (ipp) = VJ(Tp) · Hence, VJ : T(N0)o --+ F' satisfies the condition (10.1), and its image is contained in F. This proves the case E = 0 . Next, we show the general case. For p E E, define a polynomial Pp(U) E T(N0) o [U] as in (9.34 ) by - TpU + p) if p =/. £, (10.2) Pp(U) = U(U2 if p = £. U2 - T£U + £ Define a ring homomorphism T(Nr. )o --+ T(N0)o [Up, P E E]/(Pp(Up), p E E) using Corollary 9.37 repeatedly, and sending Tp to Tp if p ¢ E, and Tp to Up if p E E. We denote by Pp(U) E F[U] the image of Pp(U) by the ring homomorphism VJ0 : T(N0 ) o --+ F. If £ E E, we have P£(U) = U(U - VJ0 (T£)). Define a ring homomorphism F[Up, P E E]/(Pp(Up), p E E) --+ F by sending the image of Up to 0 if p =f. £, and to VJ0 (T£) if p = £ E E. It is clear from the definition that the composition VJr. : T(Nr. )o --+ T(N0)o [Up, p E E]/(Pp(Up), p E E) �0 F[Up, P E E] j (Pp(Up), p E E) --+ F --+ D satisfies condition (10.1).
{
1 10
1 0 . HECKE MODULES
DEFINITION 10.2. Let mi:: be the kernel of the surjective ring ho momorphism T(Ni:: ) o --+ F. The completion T(Ni:: ) o,mr. of T(Ni:: ) o at the maximal ideal mi:: is called the full Hecke algebra, and we denote it by Tf . Since T(Ni:: ) o is finitely generated as an 0-module, it is a com plete semilocal ring and decomposes into the product of local rings. Thus, the completion T(Ni:: ) o , mr. equals the localization of T(Ni:: )o at the maximal ideal mi:: . LEMMA 10.3. The full Hecke algebra Tf is a finitely generated free 0-module. PROOF. The 0-module T(Ni:: ) o is finitely generated and free since the Z-module T(Ni:: ) z is finitely generated and free. Since the 0-module Tf is a direct summand of T(Ni:: ) o, it is also finitely gen erated and free. D We now study the relation between the full Hecke algebra Tf and the reduced Hecke algebra Ti:: defined in Definition 5. 16. Let T(Ni:: ) 'o be the subring of T(Ni:: ) o generated by Tp , p f Ni:: f , T(Ni:: ) 'o = O[Tp, p f Ni:: £] c T(Ni:: ) o, and define a maximal ideal mE of T(Ni:: ) 'o as the inverse image of mi:: by the inclusion T(Ni:: ) 'o --+ T(Ni:: ) o. By Corollary 2.60, T'(Ni:: ) K = T' (Ni:: ) o ©o K coincides with the reduced Hecke algebra T' (Ni:: ) K in Definition 2.57. PROPOSITION 10.4. ( 1 ) As a subset of the set 'P (Ni:: ) K of prim itive forms over K defined as Spec T(Ni:: ) 1c, the set 'P (Ni:: ) K ,p is equal to Spec T(Ni:: ) 'o , m� ©o K. The reduced Hecke algebra Ti:: coincides with the localization T(Ni:: ) 'o m' of T(Ni:: ) 'o at mE : T(Ni:: ) 'o m' = Ti:: . ( 10.3 ) ( 2 ) The morphism of 0-algebras induced by the inclusion T(Ni:: ) 'o --+ T(Ni:: ) o ii:: : Ti:: = T(Ni:: ) o , m � --+ Tf = T(Ni:: )o,mr. ( 10.4 ) '
'
E
E
is injective and induces an isomorphism Ti:: , K --+ Tf . K · In Theorem 10.46, we will show ii:: : Ti:: --+ Tf is an isomorphism.
111
1 0 . L FULL HECKE ALGEBRAS
PROOF. (1) We show 'P(NE ) K,p = Spec T(NE )o , m E ® o K . Let f E 'P(NE ) K be a primitive form. The image of the homomorphism T(NE ) o ---+ T(NE ) K ---+ K1 is in the ring of integers Of , and the inverse image mf of the maximal ideal 0f is the unique maximal ideal of T(NE )o containing Ker( T (N:r: ) 0 ---+ K1 ) . Thus, we have Spec T(NE ) o , m E ® o K = { f E 'P(NE ) K I m1 = mD . On the other hand, by Definition 5.13, we have Tp = Tr p ( cpp) mod mt . E 'P(NE ) K 'P(NE ) K, p- = for all prime p f NE£ The restriction T(N:r: ) 0 ---+ F of VJ:r: is determined by VJdTp) = Tr p ( cpp) for p f N:r:i, and m� is its kernel. Thus, the right-hand side of the above two formulas are equal. Since T(N:r: ) K is reduced by Proposition 2.58, its subring T(NE ) o is reduced and so is its localization T(NE )o ' m 'E . Thus, T(NE ) o ' m' ---+ T(NE ) o , m E ® o K = rrf E q/ (NE ) K . p Kf is injective. T(NE ) o , m E is a subring of T(NE ) o , m E ® o K = rrf E g? (NE ) K . p K1 generated by Tp (P f NE£) over 0. Thus, by Definition 5. 16, TE = T(NE ) o ' m E' . (2) Since TE is a finitely generated free 0-module, it suffices to show that iE : TE ---+ Tf induces an isomorphism TE,K ---+ Tf , K · Since TE,K = rrf E g? (NE )K .;; Kf ' it suffices to show that Kf ---+ Tf , K ® TE.K Kf is an isomorphism for each primitive form f E 'P(N:r: ) K,p· Let f be a primitive form belonging to 'P(NE ) K,p, and N1 I NE the level of f. By definition, N0 is the product of p at which p is bad. The mod £ reduction of PJ is isomorphic to p, and P t is bad at the prime p where p is bad. By Theorem 3.52(1), a prime p at which PJ is bad divides Nf , and we have N0 I NJ . Thus, if p I NE/NJ , then p E :E. For a prime p I NE /NJ , define a polynomial Pf,p (U) E 01 [U] as in ( 9.37 ) by U(U2 - ap (f) U + p) if ordp NE /NJ = 2, PJ,p (U) = U(U - ap (f) ) ordp NE /NJ = 1 and p I NJ , ordp NE/NJ = 1 and p f NJ . U2 - ap (f) U + p By Corollary 9.41, we have
{1
I
}
I::
{
T (NE ) Q ®r' (NE ) Q K1 = K1 [Up , p l NE /N1 l /(PJ,p (Up ) , p l NE/N1 ) . Thus, if we let A J = 01 [Up , p l NE /N1]/(PJ,p (Up) , p l NE /N1 ) , we ob tain a morphism of 0-algebras T(NE ) o ---+ A 1 by sending the image
112
1 0 . HECKE MODULES
of Tp to ap (f) if p f Nr./NJ , and Up if p \ Nr./NJ . Moreover, we have Tf , K ©rE , K KJ = Tf ,K ©r(NE )K KJ [Up , p \ Nr./NJ]/(PJ,p (Up) , p\ Nr. /NJ ) = ( Tf © r(NE ) o AJ) © o K.
Thus, it suffices to show that OJ -+ Tf © r(NE ) o AJ is an isomorphism. Since AJ is finitely generated as an 0-module, it is a complete semilocal ring, and it decomposes to the product of local rings. Tf is by definition the localization of T(Nr. ) o at mr. . Thus, it suffices to show that there is a unique maximal ideal m of AJ lying above mr., and the local ring AJ, m is isomorphic to 0J . Let F J be the residue field of OJ . By the definition (10.1) of mr. , a maximal ideal m lies above mr. if and only if for each prime number p E E D(p)) mod m if p = f ¢ S-p, a (!) (10.5) 0 mod m if p # f, E E, if p f NE/NJ , and D(p)) mod m if p = f ¢ S-p, (l0.6) U. 0 mod m if p # f, E E, if p\Nr. /NJ . Thus, if m exists, it is unique. If p = f E E and f f Nr./NJ , then f ¢ Sp and £\ NJ . In this case, by Corollary 9.48(2), we have ap (f) D(p)). If p # £, p E E and p f Nr./NJ , then ordp NJ = 2. Thus, by Corollary 9.39(2) , we have ap (f) = 0. If p = f.\Nr./NJ , then p f NJ . Thus, by Corollary 9.23(2) , we have Pu (U) U(U D(p))). In this case, p is ordinary at £, and again by Corollary 9.23(2) , we have D(p)) # 0. Thus, in this case, the multiplicity of the root D(p)) of Pu (U) U(U D(p))) E F J [U] is l. Suppose p\Nr. /NJ and p # £. If p\NJ , then ap (f) = ±1 by Proposition 9.47 and the multiplicity of the root 0 of Pp,J (U) U(U - ap (f)) E F J [U] is one. If p f NJ , the multiplicity of the root 0 of Pp, J ( U) U(U2 - ap ( f ) U + p) E F J [U] is l. Thus, the only maximal ideal m of A J, F i = F J [Up , p \ Nr. /NJ J /(PJ,p (Up) , p\ Nr. /NJ ) , whose pullback equals mr. , is m = (Up (p # f, p\Nr. /NJ ) , U1. D(p)) (if f\Nr. /NJ ) ) . Moreover, the local ring A J, F 1 ,m is P
{Tr(F : {Tr(F :
=
P =
=
=
Tr(F :
Tr(F :
Tr(Tr(F :F :
Tr(F :
=
Tr(F :
=
=
1 0 .2. HECKE MODULES
1 13
isomorphic to Ff . Hence, there exists a unique maximal ideal m of A 1 lying above mE , and the local ring Af , m is isomorphic to 01. D
The full Hecke algebra Tf is also reduced. PROOF. By Lemma 10.3 and Proposition 10.4(2) , we have Tf C D Tf ,K = TE , K = TI 1E � (NE ) K , p K1 . COROLLARY 10.5.
Let E' be a finite set of prime numbers that does not intersect with S-p. If i ¢ E' and p is good and ordinary at p = i, T£ is an invertible element of T� by Corollary 9.48(2) . By Hensel ' s lemma, there exists a unique root of U2 - T£U + i that is an invertible element of Tf, . In what follows, we denote it by Tl. PROPOSITION 10.6. Let E = E' II {p} . There exists a unique morphism of 0-algebras t� ' ,E : Tf -+ Tf, satisfying the condition Tq if q =/= p, t�, ,E (Tq) = 0 (10.7) if q = p =/= i, Tl if q = p = i. PROOF. The uniqueness is clear. Let Pp(U) E T(N0) [U] be the polynomial we defined by (10.2) in the proof of Proposition 10.1. Let AE ' = T(NE ' )o [U]/(Pp(U)). Define a ring homomorphism T(NE)o -+ AE ' by sending Tp to U as in Corollary 9.37. Define a ring homomor phism Tf, © T( NE' ) o AE ' -+ Tf, by letting the image of U equal to 0 if p =/= i, and Tl if p = i. The inverse image of the maximal ideal of Tf, by the composition (10.8) is mE , and a ring homomorphism Tf -+ Tf, is induced. It is clear D that this homomorphism satisfies the condition. The restriction of t� ' , E : Tf -+ Tf, to TE is nothing but tE ' ,E : TE -+ TE ' defined in §5.5 in Chapter 5.
{
10.2. Hecke modules
In this section we define the Hecke module ME as the completion of the singular cohomology of a modular curve, and we construct RTM-triples.
10 .
11 4
HECKE MODULES
DEFINITION 10.7. We regard the Tf -module ME = H1 ( Xo ( N ) an , Z ) o © T ( NE ) o Tf as a TE-module through the ring homomorphism iE : TE ---+ Tf , and we call it Hecke module. In this chapter we may omit the superscript an to indicate a com plex manifold. For example, we sometimes write H1 ( Xo ( N ) , Z ) o in stead of H1 ( Xo ( N ) an , Z ) o . PROPOSITION 10.8. Let f E �(N0 ) K,p( K ) be a primitive form with K coefficients, and let 7rE : TE ---+ 0 be the ring homomorphism defined by f. Then, the RTM-triple RE = ( RE , TE , ME , fE : RE ---+ TE , 7rE : TE ---+ 0 ) is an RTM-triple of rank 2. PROOF. We verify the conditions of Definition 5.26. For the first three obj ects, RE is a profinitely generated complete local 0-algebra, TE is a local 0-algebra, and ME is a TE-module. As we remarked after Definition 5.16, TE is finitely generated as an 0-module. Since H1 ( Xo ( N ) , Z ) is a finitely generated free Z-module, ME is a finitely generated free 0-module. Since TE ,K = Tf ,K , ME , K is a free TE,K module of rank 2 by Proposition 9.6. If we let P TE = Ker 7rE, we have P TE /p}E © o K = 0 since TE is reduced. Thus, the finitely generated 0-module P TE / p}E has finite length. D We will show in Theorem 10.46 that ME is a free TE-module of rank 2. Let E = E' II {p}, and we define morphisms of Hecke mod ules mE ' , E : ME ---+ ME ' and m i; ' E : ME ' ---+ ME . First we de fine m i; ' E : ME ' ---+ ME as follows. Let si : H1 ( Xo ( NE1 ) , Z ) o ---+ H1 (X0 ( NE ) , Z ) o be the pullback of the morphism of modular curves Spi as in §§9.4-9.5. LEMMA 10.9. Let E = E' II {p} . Define an 0-linear mapping m * : ME ' ---+ H1 ( Xo ( NE ) , Z ) o by s o l ME ' - s i l ME ' o Tp · +p · s 2 1 ME ' ifp =f £, (10.9) if p = £. s0 I ME ' - s i l ME ' o Tf . The 0-linear mapping m * : ME ' ---+ H1 ( Xo ( NE ) , Z ) o induces an 0linear mapping mi; , ' E : ME ' ---+ ME compatible with the ring homomorphism t� , E ' : Tf ---+ Tf , . ,
,
{
10 . 2 .
HECKE MODULES
11 5
PROOF. As in the proof of Proposition 10.6, define a polynomial Pp(U) E T(N0) [U] by (10.2) , and let AE ' = T(NE ' )o [U]/(Pp (U) ) . We first show the case p =f. e . As in (9.30) , define a matrix Up E 00 0 M3 (T(NE ' )o) by Up = P0 po -Tp1 . Considering the action of U as the multiplication-by-Up, we regard H1 (X0 (NE ' ) , Z) b as an AE1-module. By Proposition 9.36, the morphism ( s0 , si , s2 ) : H1 (Xo(NE1 ) , Z) b -+ H1 (Xo (NE), Z)o is compatible with the ring homomorphism T(NE)o -+ AE ' · Since up - p = 0, the 0-linear mapping ( 1, -Tp , p) : ME ' -+ H1 (Xo(NE ' ) , Z) b is compatible with the ring homomorphism AE ' -+ Tf , ® T ( N'I:. ' ) o AE ' -+ Tf , defined by letting the image of Up be equal to 0. The linear mapping m * : ME ' -+ H1 (Xo(NE), Z)o is the composition of these linear mappings, and thus it is compatible with the ring homomorphism T(NE)o -+ AE ' -+ Tf , . t� ' ,E : Tf -+ Tf , is induced by this composition of ring homomorphisms. This shows the case p =f. e. In the case of p = i, define the action of U on H1 (X0 (NE ' ) , Z) b by the multiplication of the matrix Ut = ( � "T; ) E M2 (To(NE ' )o). Since Ut ( -�" ) = Tf ( -�" ) , the rest of the proof is similar to the case D p =/. i. COROLLARY 10.10. Let E = E' II {p} . We denote by Bi* the composition of natural surjection H1 (Xo(NE ' ) , Z)o -+ ME ' and si* : H1 (Xo (NE), Z)o -+ H1 (Xo(NE ' ) , Z)o, and define a morphism of 0modules m * : H1 (Xo(NE), Z)o -+ ME ' by if P =f. e , (10. 10) if P = e . Then, the morphism m * : H1 (Xo (NE ) , Z)o -+ ME ' induces a mor phism of 0-modules mE ' , E : ME -+ ME ' that is compatible with the ring homomorphism t�, , E : Tf -+ Tf , . PROOF. The isomorphism of T(NE)z-modules ( 9.4 ) H1 (Xo(NE), Z) -+ Hom(H1 (Xo(NE), Z) , Z) ; x ...+ ( y ...+ ( x, wy)) induces an isomorphism of Tf -modules ME -+ Homo (ME , 0), and similarly, an isomorphism of Tf ,-modules ME ' -+ Homo (ME ' , 0) . Let mi;Y,E : Homo (ME , O) -+ Homo (ME1 , 0) be the dual of mi; ' , E :
( )
( �)
e
e
10.
116
HECKE MODULES
M"E' -+ M"E . It suffices to show that the mapping (10.10) induces the composition (10.11) We show the case p "I £. By the definition of mE' , "E • the compo sition (10.11) is induced by (so* r; · s1 * + p · s2*) Since p f N"E' , we have r; = Tp by Lemma 9.5, and Tp = Tv More over, since Si = s 2 i : Xo (N"E) -+ Xo (N"E' ) , (10.11) is induced by s2• - Tp · sh + p s0 * . This shows the case p "I £. In the case of p = £, we prove it similarly using the fact that - TR.u S-1 * ) W = S-1 * TR.u So* o ( So* W w o
w o
-
-
w o
o w
o w.
o w.
·
o
-
•
o
-
•
-
PROPOSITION 10.11. If :E = :E' II {p} , then m"E',"E : M"E -+ M"E'
is surjective.
The proof will be given in the next section. Proposition 5.30 in Chapter 5 follows from Proposition 10.11 by the induction on the number of elements in :E - :E' . COROLLARY 10.12. If :E = :E' II {p} , then mE ' ' "E : M"E' -+ M"E is injective, and its image is a direct summand as an O-module.
PROOF. By the proof of Corollary 10. 10, mi; ' "E : M"E' -+ M"E is the dual of m"E',"E : M"E -+ M"E' · Thus, the assertion, follows immedi ately from Proposition 10.11. 0 PROPOSITION 10.13. Let f E
F"E',"E = (r"E',"E : R"E -+ R"E' , t"E',"E : T"E -+ T"E' , m"E',"E : M"E -+ M"E' , mE ',"E : M"E' -+ M"E) is a surjection of RTM-triples 'R,"E -+ 'R,"E' . PROOF. r"E',"E : R"E -+ R"E' and t"E',"E : T"E -+ T"E' are surjective local morphisms of local 0-algebras, and the diagram R"E � T"E � 0
rE', E 1
R"E'
fE1
-)-
ltE', E
T"E'
11"E'
-)-
I
0
10 . 2 .
11 7
HECKE MODULES
is commutative. mr:' , E : Mr; -+ Mr; , and m E ' , E : Mr: ' -+ Mr; are 0linear mappings compatible with t r: ' , E · By Proposition 10. 11, mr: ' , E : Mr; -+ Mr; 1 is surjective. By Corollary 10.12, m E ' E : Mr: ' -+ Mr; is injective, and its image is a direct summand as an ' O-module. D PROPOSITION 10.14. Let 1: = l:' ll {p}. Define an element .6.p in Tf, by (p - l)((p + 1)2 - Ti) if p =I £, .6.P if p £. (TJ2 - 1)(2TJ - Tp) Suppose also .6.p E Tr: ' when p = £. Then .6.p is the multiplier of the surjection Fr: ' , E of RTM-triples. Proposition 5.33 in Chapter 5 follows from Proposition 10.14.
{
=
=
PROOF. It suffices to show that the composition mr; 1 , E o m E , , E : Mr: ' -+ Mr: ' is the multiplication-by-.6.P mapping. We show the case p =I £. By the definition of mr: ' , E and mE ' , E • the composition mr: ' , E o m E ' E is induced by multiplication by ,
We show if i = j, if i = j ± 1, if i j ± 2. =
If j = j, then Si * o si = deg Si = p(p + 1). Let t o , ti : Xo(Nr:) -+ Xo(Nr;1p) and ro, r1 : Xo(Nr;1p) -+ Xo(Nr; 1 ) be morphisms s 1 and Sp in (8.65) . By Lemma 9.35(1), we have so * o si = ro * o to * o t0 o ri deg t o ro * o r i pTp. By Lemma 9.5(2) , we have s1 * os0 = pT; = pTp. Similar statements hold for s 2 * o si and s 1 * o s2 . By the proof of Proposition 9.38(1), we have to * o ti + w = ri o ro * · Thus, we have =
·
=
so * o s2
r0* o ri o ro * o ri - ro * o w * o ri = T; - deg ro * = T; - (p + 1). =
1 18
10. HECKE MODULES o
The same is true for s 2 * s0 . Thus, in the case p =f. C, the assertion follows from the equality p(p + 1) pTP r; (p + 1) (p -Tp 1 ) pTp p(p + 1) pTp - p r; - (p + 1) pTp p(p + 1 )
- ) ( :)
(
=
(p -TP 1 )
�
(
(p2 - l) (p + 1 ) - (p l ) Ti (p - 1 ) ( (p + 1 ) 2 - r;) . Similarly, in the case p = C, since we have p = TJ Tp - TJ 2 , the assertion follows from 1 + 1 Tp tt tt2 ( -TP" 1 ) Tp p + l -TJ - TP Tp + Tp - 2 (p + 1 ) TP = ( TJ 2 - 1) ( 2TJ - Tp) · D
- )
=
(P
)( )
_
For general E' C E, we define the morphism of Hecke algebras tb , E : Tf --+ Tf, and the morphisms of Hecke modules mr: ' , E : Mr; --+ Mr;, and m i; , ,E : ME ' --+ Mr; inductively, but we omit the details here. 10.3. Proof of Proposition 10. 1 1
In this section we prove Proposition 10.11 while admitting the following Theorem 10.15. Let p be a prime number, and let N ::::: 1 be an integer relatively prime to p. Define I'(N) Ker (SL2 (Z) --+ SL 2 (Z/NZ)) , =
I'0 , * (p, N) =
{ (� �) E I'(N) l c = O mod p } ,
f' (N) Ker (SL 2 (Z[ � ]) --+ SL 2 (Z/NZ)) . In addition, define =
E(N) = E (N) =
( G �) 1 ( G �) I A
A- 1 A
A
A- 1 A
) r(N) , E sL 2 (z[ � D ) c r (N) .
E SL2 (Z)
c
Let so : ro , * (p, N) --+ r(N) be the inclusion, and let S 1 : ro, * (p, N) --+ I'(N) be the conjugate A H o ; n - l A ( g n . Let Jo : I'(N) --+ f' (N) be the inclusion, and let j1 : I'(N) --+ f' (N) be the conjugate
1 0 .3.
AH
PROOF OF PROPOSITION
10 . 1 1
119
( {; � ) A ( {; � ) - 1 . The amalgamated sum of so , s 1 : ro,* (p, N)
�
r(N) is denoted by r(N) *r o ,. (p, N ) r(N) . For a group r, rab r / [r, rJ is its abelianization. THEOREM 10.15. Let p be a prime number, and let N ;::: 1 be an integer relatively prime to p. (1) The homomorphism r(N) *ro ,. (p, N ) r(N) � f'(N) defined by the commutative diagram r(N) So r
�
=
f'(N)
r Ji
ro,* (p, N) � r(N) is an isomorphism. (2) f'(N) = E(N) .
We do not give a proof of Theorem 10.15 in this book. COROLLARY 10.16. (1) E(N)ab � f'(N)ab is surjective. (2) so EB S 1 : ro,* (p, N)ab � (r(N)ab I E(N) ) 2 is surjective. PROOF. (1) Let A be any element of SL2 (Z[ � ] ) , and M A ( fi 1f ) A 1 . By Theorem 10.15(2) , it suffices to show that M is con jugate to an element E(N) by an element of f'(N) . If pn 1 mod N, then ( fi 1f ) is conjugate to ( fi 1f l" by ( P; p �"' ) E f'(N) . Thus, M is conjugate to MP2"' by f'(N) . Therefore, it suffices to show that 2" MP E E(N) for sufficiently large n > 0. Since (M - 1) 2 0 and M - 1 E M2 (Z[ � ] ) , we have MP2" 1 + p2n ( M - 1 ) E SL2 (Z) for sufficiently large n > 0. Since (MP2 "' - 1) 2 = 0, there is an integer b E Z such that MP2 "" is conjugate to ( fi � ) by an element of SL 2 (Z) . Since MP2" 1 mod N, we have 2" Nib and thus MP E E(N) . ( 2 ) By Theorem 10.15(1), we obtain an exact sequence =
-
=
=
=
=
soEEls1 io -ii ro,* (p, Nrb --+ r(Nrb 2 --+ r(Nrb -+ 0.
D The assertion is clear from this and ( 1 ) . Let p be a prime number, and let N ;::: 1 be an integer relatively prime to p. Modular curves X (N) and XoAP, N) over Q are curves over the cyclotomic field Q((N ) . so , s 1 : Xo,* (p, N) � X (N) are
1 20
1 0 . HECKE MODULES
defined just as so , s 1 : Xo (Np) ---+ Xo (N) . Embed Q((N ) into C by (N = exp 2,,.r ) , and let X (N)an and X0 ,* (p, N)an be the compact Riemann surfaces defined by X (N) ©Q ( (N)
(
an integer relatively prime to p. Then, so* ffi sh : H1 (Xo,* (p, N)an, Z) ---+ H1 (X (N)an, Z) 2 is surjective.
PROOF. If N = 1, 2, then the genus of X (N) is 0, and thus we may assume N � 3. If N � 3, then, by Lemma 8.37, r(N) acts freely on the upper half-plane H = {T E C I Im T > O}, and we have I'(N)\H = Y(N)an and I'o , * (p, N)\H = Yo,* (p, N)an. Thus, we obtain isomorphisms of fundamental groups r(N) ---+ 7r1 (Y(N)an) and I'o,* (p, N) ---+ 7r1 (Yo,* (p, N)an). Moreover, the inertia groups at the cusps of Y(N) are conjugate to ( ( fi lf. ) ) = r(N) n ( ( fi i ) ) , these isomorphisms define a commutative diagram
1
(r(N)ab I E(N) ) 2
1
The left vertical arrow is surjective, and the right vertical arrow is an isomorphism. Thus, the assertion follows immediately from Corol D lary 10.16 ( 2 ) . COROLLARY 10.18. Let p be a prime number, and let N � 1 be an integer relatively prime to p. Then, s o * ffi s 1 * : H1 (X1 ,o (N, p) an, Z) ---+ H1 (X1 (N)an , Z) 2 is surjective.
This is a more precise version of an analog of Corollary 9.30 ( 2 ) . PROOF. Consider the subgroups I'(N) C I' 1 (N) of SL 2 (Z) . By the isomorphisms of Riemann surfaces ( 8.47 ) and ( 8.40 ) , the ho momorphism H1 (Y(N)an, z) ---+ H1 (Y1 (N)an, z) is identified with that of abelianizations r(N)ab ---+ I' 1 (N)ab . H1 (X (N)an, Z) and H1 (X1 (N)an, Z) are the quotients by the images of inertia groups of H1 (Y(N)an, Z) and H1 (Y1 (N)an, Z) , respectively. The cokernel
10 . 3 .
PROOF OF PROPOSITION
10 . 11
121
of r(N) -+ r 1 (N) is generated by the images of the inertia group ( ( 6 U ) at the cusp, and H1 (X(N)an, z) -+ H1 (X1 (N)an , z) is sur jective. Hence, the assertion follows from Proposition 10.17 and the commutative diagram H1 (X* , o(N, p)an, Z) ---+ H1 (X(N)an, Z)
l
H1 (X1 ,o (N, p)an, Z)
---+
l
H1 (X1 (N)an, Z) .
D
PROPOSITION 10.19. Let p be a prime number, and let N 2: 4 be an integer relatively prime to p. Then, H1 (Y1 , o (N, p)an, Z) 2 H1 (Y1 (N)an , Z) --+ 0
(10.12) is an exact sequence of Z-modules.
PROOF. By the assumption N 2: 4, Y1 ( N) is a fine moduli scheme. Similarly to Lemma 9.35(3) , the commutative diagram of finite etale morphisms of curves Y1 , o (N, p2 ) II Y1 ,o (N, p) ( t i , id) Y1 , 0 (N, p) (10.13) ( to , woto)
l
Y1 ,o (N, p) is Cartesian. Therefore, the commutative diagram rr 1 (Y1 , o (N, p2 )) � rr1 (Y1 , o (N, p)) (10.14) so . to•
l
l
rr 1 (Y1 ,o (N, p)) � rr 1 (Y1 (N)) induces an isomorphism rr1 (Y1 , o (N, p)) *1ri ( Yi, o (N,p2 )) rr 1 (Y1 , o (N, p)) -+ D rr1 (Y1 (N)). We obtain the exact sequence (10.12) from this. LEMMA 10.20. Let N 2: 5 be an integer, and let p be a prime number that does not divide N. The action of the Hecke operator Tp on the kernel of H1 (Y1 (N)an , Z) -+ H1 (X1 (N)an, Z) is multiplication by (p) + p.
10 .
122
HECKE MODULES
PROOF. Let Z1 (N) = X1 (N)an - Y1 (N)an be the set of cusps. Then, we have an exact sequence of T1 (N)z-modules
Thus, it suffices to study the action of the Hecke operator Tp on the free Z-module z Zi ( N ) generated by the cusps. By the assumption N ;:::: 5, X1 (N) is a fine moduli scheme of gen eralized elliptic curves with level structure. Thus, Z1 (N) is identified with the set
{
{
isomorphism classes of pairs (Pd , P) , where Pd (d I N) is the Neron d-gon over C , and P is a point of exact order N of the smooth part PJ.m � Gm x Z/dZ such that (P) intersects with all the connected component of PJ.m
Similarly, Z1 , o (N, p) the set
=
} ·
}
X1 , o(N, p)an - Y1 ,o (N, p)an is identified with
isomorphism classes of triples (Pd , P, C) , where Pd (d I Np) is the Neron d-gon over C , P is a point of ex act order N of the smooth part PJ.m � Gm x Z/dZ, and . C C PJ.m is a cyclic subgroup of order p such that (P) + C intersects with all the connected component of PJ.m
The mappings s, t : Z1 , o(N, p) ---+ Z1 (N) are described as follows. For a triple (Pd , P, C) , let Q be the Neron polygon obtained by contracting the irreducible components of Pd that do not intersect with (P) . Then we have s(Pd , P, C) = (Q, image of P) . Also we have t(Pd , P, C) = (Pd /C, image of P) . The inverse image of the point (Pd , P) by s Z1 , 0 (N, p) ---+ Z1 (N) consists of two points (Pd , P, µP x 1) and (Pdp , P, 1 x dZ/dpZ) , and their ramification indices are 1 and p , respectively. Here, we identified PJ.m = Gm x Z/dZ with the subgroup of PJ.;' = Gm x Z/dpZ through the multiplication-by-p mapping Z/dZ ---+ Z/dpZ. Furthermore, the images of (Pd , P, µp x 1) and (Pdp , P, l x dZ/dpZ) by t : Z1 , o(N, p) ---+ Z1 (N) are (Pd , pP) and (Pd , P) , respectively. Hence, the image of [(Pd , P)] E z Zi ( N ) by Tp = t * s * is [(Pd , pP)]+ D p[(Pd , P) ] = ((p) + p) [(Pd , P)] . :
o
10 . 3 .
PROOF OF PROPOSITION
10 . 11
123
LEMMA 10.21. Let N 2: 1 be an integer, and let p be a prime number that does not divide N. The action of the Hecke operator Tp on Coker(H1 (X1 (N)an, Z) -+ H1 (X0 (N)an, Z)) is the multiplication by p + 1 . PROOF. X0 (N) is the quotient of X1 (N) by (Z/Nz) x acting as the diamond operators. Since H1 (X1 (N)an, Z) and H1 (X0 (N)an, Z) are the abelianizations of the fundamental groups 7r 1 (X1 (N)an) and 7r1 (Xo(N)an), Coker(H1 (X1 (N)an, Z) -+ H1 (Xo(N)an, Z)) is identi fied with the quotient of (Z/Nz) x corresponding to the maximal un ramified intermediate coverings of X1 (N) -+ Xo (N) . Since the mor phisms s, t : X1 , 0 (N, p) -+ X1 (N) are compatible with the action of (Z/Nz) x , the induced mappings s* , t* : Coker(H1 (X1 ,o(N, p)an, Z) -+ H1 (Xo(Np)an, Z)) -+ Coker(H1 (X1 (N)an, Z) -+ H1 (Xo (N)an, Z)) are 0 equal. Thus, we have Tp = t * s * = s* s * = deg s = p + 1. o
o
Let 0 be the ring of integers of a finite extension K of Qe, let M 2: 1 be an integer, and let O[Tp, p f M] be the ring of polyno mials with infinite number of variables. Let m be a maximal ideal of O[Tp, P f M] such that the residue field F = O[Tp , P f M]/m is a finite field. If there do not exist characters a., {3 : (Z/Mz) x -+ F x such that Tp a(p) + {3(p) mod m for all p f M, then m is said to be non-Eisenstein. For a factor M' of M, H1 (X0 (M')an, 0) and H1 (X1 (M')an, O) are O[Tp, p f M]-modules, and for a maximal ideal m, the localizations H1 (Xo(M')an, O)m and H1 (X 1 (M')an, O)m at m are defined. =
PROPOSITION 10.22. Let p be a prime number, and let N 2: 1 be an integer relatively prime to p. Let m be a non-Eisenstein maximal ideal of the polynomial ring O[Tq , q f Np2 ] . Then,
is surjective.
124
10 . HECKE MODULES
PROOF. If N :::;; 4, then the genus of X1 (N) is 0, and thus we may assume N ;:::: 5. Consider the following diagram. 0 H1 (X1 ,o (N, p2 )an, O) m ti. -to.
(10.16)
l
H1 (X1 , o (N, p)an, 0)� so. +si.
1
so. E!lsi . E!ls 2 .
( so. E!lsi. ) 2
.!.
l 1 (�l n
H1 (X1 (N)an, O)�
0 0 -1
H1 (X1 (N)an, O)�
1 (1
0
0
H1 (X1 (N)an, O) m
H1 (X1 (N)an, O) m
.!.
.!.
0
o.
1)
The right column is clearly exact. Since m is non-Eisenstein, we may replace X by Y in the left vertical column by Lemma 10.20. Thus, by Proposition 10.19, the left column is exact. Turthermore, by Corollary 10.18, the middle horizontal arrow is surjective. Thus, by the snake lemma, the upper horizontal arrow is also surjective. Consider the commutative diagram (10.17)
1
1
Since m is non-Eisenstein, the right vertical arrow is surjective by Lemma 10.21. Thus, the lower horizontal arrow is also surjective. D PROOF OF PROPOSITION 10.1 1 . By Proposition 9.26, the maxi mal ideal m:r; is non-Eisenstein. Thus, the inverse image of m:r; by O [T , p f M] --+ T(N:r;)o is also non-Eisenstein. In the case of p # i, the assertion follows immediately from Proposition 10.22. The case D p = i is similar, but we omit the details. p
10 .4. DEFORMATION RINGS AND GROUP RINGS
12 5
10.4. Deformation rings and group rings
As a preliminary for the construction of the family of liftings of (R, M) = (R0 , M0) in the next section, we study the relationship between deformation rings and group rings. Let Q be the set of all the prime numbers satisfying the conditions (10.18) q � Sp q =J f. and Tr p(cp q ) � ±(q + 1). The condition Tr p('P q ) � ±( q + 1) can also be stated as (Tr p('P q ) ) 2 � (q + 1) 2 . For q E Q, let a q and i3q be the eigenvalues of p(cp q ) · Since q a q i3q , the condition a q i3q ± (a q +i3q ) +l =J 0 in (10.18) is equivalent to the condition aq =J ±1 and i3q =J ±1, or aq / i3q =J q± 1 . Thus, if q 1 mod £, we have a q =J i3q- In what follows, we assume that aq and i3q belong to F for each q E Q after replacing K by its unramified quadratic extension if necessary. For q E Q, let � q be the maximal f.th power quotient of (Z/qz) x identified with G al(Q q ( (q ) / Q q ) If q � 1 mod £, then we have � q = 1 . We identify � q with a quotient of the inertia group lq through the surjection Iq -+ G al(Q q ( (q ) / Q q ) = (Z/qz) x . � q is identified with the maximal f.th power quotient of the image in the abelianization Im(Iq -+ G�t ) . The condition Tr p(cpq ) � ±(q + 1) is made for the following proposition. PROPOSITION 10.23. Let q E Q, and let R be a profinitely gener ated complete local 0-algebra. Let FR be the residue field of R. Let pq : GQ. -+ GL2 (R) be a lifting of the restriction of p to GQ0 • (1) Suppose aq =J i3q · There is a unique pair of characters aq , /3q : GQ. -+ Rx such that pq is isomorphic to the direct sum a q (f) /3q , and the composition GQ. -+ Rx -+ F� are unramified charac ters whose values at
=
·
( �).
=
1 26
10 . HECKE MODULES
if A ( � � ) (a, b, c, d E mR) satisfies PAP- 1 (1 + A ) q - 1, it suffices to show that b = c 0. Since PAP- 1 (1 + A ) q - 1, there exist polynomials f and g of four variables with Z coefficients whose constant terms are 0 that satisfy &j3- 1 b (q + f (a, b, c, d) )b and j3&- 1 c (q + g ( a , b , c, d) )c. By assumption, we have &j3- 1 and j3&- 1 ¢. q and f(a, b, c, d) g(a, b, c, d) 0 mod mR· Thus, we have b c 0. As above, we identify !:lq with the maximal eth power quotient of the image in the abelianization Im(Iq --+ G�) · Since l +mRM2 (R) c GL2 (R) is a pro-e group, the restriction of a q , /3q to Iq passes through !:lq. (2) We first show it in the case iiq =f. f3q· From (1) we see that there exist characters aq , /3q : G�q --+ Rx satisfying pq a q EB /3q · By the assumption q ¢. 1 mod e , the image !:l q of the inertia group is trivial, and aq and /3q are unramified. We show it in the case iiq f3q· We show that the image pq (lq) of the inertia group Iq is trivial. Replacing R by R/mR. if necessary, we may assume R is finite. Suppose the order of the image pq (Iq) c 1 + mRM2 (R) is em . We want to show m 0. Since e =f. q, pq (Iq) is a cyclic group of order em . Let A be a generator of the cyclic group pq (lq)· Suppose F E GQ q is a lifting of <.pq E GQ q · Let a E R be a root of unity of order prime to e that satisfy a := iiq mod ffiR · If we let p(F) aP, we have PAP - 1 Aq . If the order of P is N, then we have Aq N = A, and thus q N 1 mod em . Since the only eigenvalue of p mod ffiR is 1, N is a power of e, and we have q N q mod e. Since q ¢. 1 mod e, we conclude that m 0. D As we noted above, if q 1 mod e, we have iiq =f. j3q, and thus the condition of Proposition 10.23(1) is satisfied in this case. COROLLARY 10.24. The notation is as in Proposition 10.23. As sume furthermore that q 1 mod e and that the restriction of
=
=
=
=
=
=
=
=
=
�
=
=
=
=
=
=
=
=
=
=
q
=
=
=
=
1 0 .4. DEFORMATION RINGS AND GROUP RINGS
127
Ti q E Q (Z/qZ) x , we identify AQ with a quotient group of (Z/Qz) x . For each q E Q, choose one of two eigenvalues of p(cpq) and denote it by iiq E Fx . We define a ring homomorphism (10. 19) Let PQ : GQ -+ GL2 (RQ) be the universal representation. For each q E Q, apply Proposition 10.23(1) to the restriction pq : GQ . -+ GL2 (RQ) of PQ to the decomposition group GQ0 , and we define a character aq : G�. -+ Ra · By Proposition 10.23(1) , the character aq to the inertia group Iq induces a character Aq -+ Ra · Define O[AQ] -+ RQ as the ring homomorphism induced by the product of these characters, AQ -+ Ra . LEMMA 10.25. Let Q be a finite set of prime numbers in Q = {q E Q I q 1 mod t'}, and let O[AQ] -+ 0 be the augmentation =
morphism. Then, the diagram
O [AQ ]
-------+
0
-------+
1
RQ
1
R0 is commutative, and the induced morphism RQ © o [ � Q ] 0 -+ R0 is an isomorphism.
PROOF. Since the universal representation p 0 : GQ -+ GL2 (R0) is unramified at Q, the commutativity of the diagram is clear from the definition of O[AQ ] -+ RQ · If p is a lifting of p of type 'DQ , then, by Corollary 10.24, p is a lifting of type 'D if and only if aq : GQ0 -+ Rx is unramified for any q E Q. Thus, RQ ©o [� Q ] 0 represents the functor Def,o , v.0' over 0. Hence, RQ ©o [� Q ] 0 -+ R0 is an isomorphism. D QUESTION. Let Q be a finite subset Q, and let Q = Q n Q. Show that the natural morphism RQ -+ RQ is an isomorphism. (Hint: You may prove this similarly to the proof of Lemma 10.25.) PROPOSITION 10.26. There exist infinitely many odd prime num bers q E Q such that q ¢. 1 mod t'. PROOF. det p is the mod t' cyclotomic character and is of even order. Thus, by Theorem 3.1 and Lemma 9.51, the assertion is now D reduced to the following lemma of group theory.
1 28
10 . HECKE MODULES
LEMMA 10.27. Let F be a finite field of characteristic £ ;::: 3, let p : G ---+ GL 2 (F) be an absolutely irreducible representation of a finite group G, and let x : G ---+ y x be a character. Suppose the order of x is an even integer 2d. In case 2d = 2, we assume further that the restriction of p to H = Ker x is absolutely irreducible. Then, there exists an element a of G satisfying the condition p( a )) 2 =I- (l + x ( a )) 2 (10.20) x(a) =I- 1 and ( Tr det p(a) x(a) We deduce the above lemma from the following theorem. THEOREM 10.28. Let £ be a prime number, let F,, be an algebraic closure of F,,, and let G C GL 2 (F,,) be a finite subgroup. Suppose V = F; is absolutely irreducible as a representation of G. Then, the image G of G in PGL 2 (F,,) = GL 2 (F,,)/F; satisfies one of the following conditions (i) , (ii) , and (iii) . (i) There exists a finite extension F of Fi' such that G is conjugate to either PGL 2 (F) or PSL 2 (F) . (ii) G is isomorphic to the symmetric group 64 , the alternating group m4 ' or ms .
(iii) G is isomorphic to the dihedral group D2n of order 2n with n
and £ relatively prime to each other. We omit the proof of Theorem 10.28. PROOF OF LEMMA 10.27. We prove it by contradiction. Let
a(a) , (3(a) be eigenvalues of p(a) . Then the condition (a:C��g(;;i2 =f. ( l +C�J ) ) 2 is equivalent to �(:\ + �(:\ =I- x( a) + 1<7 , and in turn equiv x x( ) alent to �i:\ =I- x(a)± 1 . Suppose the negation of condition (10.20) , that is 1 (10.21) if a E G, then x(a) = 1 or = x(a)± holds, and we derive a contradiction to the assumption of Lemma
�i:�
10.27.
Let Z = {a E G I p(a) is scalar matrix}. By (10.21) , we have Z C Ker x = H. Thus, we have Ker p C Z C Ker x, and replacing G by p(G) , we may assume G C GL 2 (F) . Take an element a E G such that x(a) is a generator of x(G) . Let - : G ---+ G = G/Z be the natural surjection, and let H = H/Z. We show that G = (a) H and H is an abelian group. By (10.21) , �
1 29
10.5. FAMILY OF LIFTINGS
replacing F by its quadratic extension and changing bases if necessary, we may assume u ( 6 xfa) ) mod Fx . From this G = ( u ) H follows immediately. Suppose T E H. Since we have x ( udT ) = x ( u ) d = and ( udT ) 2 E Z. -1 , the eigenvalues of udT is of the form 1 Hence, we have TudTu-d mod Z. Thus, T t--t T- 1 defines an automorphism of H = H/Z, H is an abelian group. Since G = (a) H, the order of a is even, and H is abelian, it follows from Theorem 10.28 that G c PGL 2 (F) satisfies condition (iii) of Theorem 10.28. Suppose G is a dihedral group of order 2n (£ f n ) . Since (Gab ) 2 = 1, the order of x is 2. Since the order of any element of G - H is 2, H is a cyclic group of order n. Thus, H is an abelian group of order prime to f, and the restriction of p to H is 0 absolutely reducible, which is a contradiction. =
�
a, - a ,
=
�
10.5. Family of liftings
Q = { q E Q \ q 1 mod £} is the set of all the prime numbers q satisfying the condition (10.22) q r:j. S-p, q 1 mod f and Tr p(cpq) ¢. ±2. Condition (10.22) is the same as condition (5.19 ) . Choose a prime number q' E Q - Q satisfying the condition of Proposition 10.26, and fix it once and for all. For a finite set Q of prime numbers in Q, we define variants T� , M� of the full Hecke algebra T� and the Hecke module MQ and a homomorphism f� : RQ --+ T� from the deformation ring to the Hecke algebra, and we will show that these form a family of liftings. As in the previous section, for each q E Q, choose an eigenvalue of p(cp9) and denote it by i.i.9. By Hensel's lemma, for each q E Q, there exists a unique root in T� of U 2 - T9U + q = 0 whose image in F is i.i.9. We write it r: . We identify a finite set Q c Q with the integer Il q E Q q. In what follows, the Hecke algebra To, 1 (N0 , Q q'2 )z = Z[Tn ( n :2'.: 1 ) , (a) (a E (Z/Qq'2Z) x )] C End Jo , 1 (N0 , Q q 2 ) will be written by T(N0 , Qq'2)z. Let T(N0 , Q q'2 ) o = T(N0 , Qq'2)z © z 0. Just as Proposition 10.6, we show the following. PROPOSITION 10.29. Let Q be a finite subset of Q, and let q' be =
=
'
a prime number satisfying the condition in Proposition 10.26. Then there exists a unique ring homomorphism t0,Q : T(N0 , Q q'2 ) o --+ T�
1 30
10 . HECKE MODULES
satisfying the conditions
ifp ¢ Q U {q'}, if P = q E Q, if p = q', (10.24 ) a E (Z/Qq'2 Z) x . i0,Q ( (a) ) = 1 PROOF. The uniqueness is clear. We show the existence. The natural mapping Xo, 1 (N0 , Qq'2) --+ Xo (N0Qq'2) induces a ring ho momorphism T(N0 , Qq'2) o --+ T(N0Qq'2) o. For a prime number p, the image of Tp is Tp, and for a E (Z/Qq'2Z) x , the image of (a) is 1 . Thus, it suffices to give a ring homomorphism T(N0Qq'2)o --+ T� satisfying condition (10.23 ) . For q E Q U {q'}, we define a polynomial Pq(U) E T(N0 ) [U] as in the proof of Proposition 10.1 by if q E Q, p +q q (U) = U2 - TqU U(U2 - Tq ' U + q') if q = q'. Let A' = T(N0)o [Uq, q E Q U {q'}]/(Pq(Uq) , q E Q U {q'}). Define a ring homomorphism T(N0Qq'2)o --+ A' as in Corollary 9.37 by letting the image of Tp be equal to Tp if p ¢ Q U { q'} and equal to Up if p = q E Qu{ q'}. Define a ring homomorphism A' ©T ( NllJ ) o T� --+ T� by letting the image of Up equal to TJ if q E Q, and equal to 0 if q = q'. Then, the composition (10.25 ) T(N0Qq12 )o ---t A' ---t A' © T ( NllJ ) o T� ---t T� satisfies condition (10.23 ) D DEFINITION 10.30. Define a maximal ideal m� of T(N0 , Qq'2) o = T(N0 , Qq'2)z ©z 0 to be the inverse image of the maximal ideal m0 by the ring homomorphism t0,Q : T(N0 , Qq'2)o --+ T� . We call the completion T( N0 , Qq'2) 0 ' Q of T( N0 , Qq'2) o at the maximal ideal m � the full Hecke algebra, and denote it by T�. The ring homomorphism induced by t0,Q is denoted by t� ,Q : T� ---t T� . (10.26 ) We can define an isomorphism TQ --+ T�; Tp � (p) - 1 12Tp, p f Nr:,Qq' f, but we do not use it in this book. PROPOSITION 10.31. T� is reduced. (10.23 )
t0,Q (Tp) =
Tp J
{�
{
m�
1 0 .5. FAMILY OF LIFTINGS
131
PROOF. The proof is similar to that of Proposition 10.4. Define a subring T' (N0 , Qq'2 )o of T(N0 , Qq'2 )0 as the subring generated over 0 by Tp (p f N0Qq'2 ) and (a) (a E (Z/Qq' 2 z) x ) . Let 'P(N0 , Qq'2 )K = Spec T' (N0 , Qq'2 )K · T(N0 , Qq'2 )0 is reduced and T(N0 , Qq'2 ) K = I1 J E
m' Q
1
, m'Q
01
{
1
1
1,
mb Q
_
,
_
,
Q
1 32
1 0.
HECKE MODULES
By (10.24) , the image of ( ) : (Z / Qq'2 z) x -t T�x is contained in 1 + m� . Thus, the order M of this image is a power of £. Since £ -:f. 2, M is odd. For a E (Z/Qq'2z) x , define (a) -1 1 2 = (a) <M - l ) / 2 . The character ( ) - 1 / 2 : (Z / Q q 2 z) x -t 1 + m� c T�x satisfies 1 . L et PQ : GQ -t GL (Rq) be the universal ( ) . ( ( ) - 1 12 ) 2 2 representation. '
=
PROPOSITION 10.32. (1) There exists a unique ring homomor
phism (10.27 ) f� : Rq --+ T� that satisfies ( 10.28 ) for any prime number p f N0 Qq' £. ( 2 ) The image of a E (Z / Q Z) x by the mapping (Z/Qz) x -t D.q -t T�x induced by the composition O[D.q] -t Rq -t T� is (a) -1 1 2 E T� .
PROOF. ( 1 ) We define a ring homomorphism f� : Rq -t T� . By the proof of Proposition 10.31 , Tqb ®o K -t Ti t E 4> (N Qq'2 ) K , p , o. Q Kf is an isomorphism. We identify Tqb ®o K with Ti t E 4> (N Qq'2 ) K , p , a Q Kt through this isomorphism. Let pf : GQ -t GL ( 0 f) be the £-adic representation associated with f E (f!(N0 , Qq'2)K2, p , a q . By Proposition 10.23(2) , P! is unrami fied at p = q'. Let Cf : (Z / Qq'2 Z) X -t a; be the character of f. Cf is the composition of ( ) : (Z / Qq'2 z) x -t T�x and T�x -t Oj . Let 81 be the composition of ( ) - 1 1 2 : (Z / Qq'2z) x -t T�x and T�x -t O j . Then, we have c18J = 1 . Since PJ is unramified at q' and det pf is the product of c t and the cyclotomic character, the conductors of cf and 8f are divisors of Q. From now on, we regard cf and 8f as char acters of (Z / Q z) x = Gal(Q((q)/Q). Regarding 8f as a character GQ -t Oj , we define an £-adic representation Pt : GQ -t GL 2 (01) by Pt = P f ® 8f . By Corollary 9. 17, the £-adic representation Pt = Pf ® 8f : GQ -t GL 2 (0f ) is unramified at p f N0 Qq' £ , and we have det ( l - pt (cpp )t) = 1 - 8f (p )a p ( f ) t + 8f (p)2cf (p ) pt 2 = 1 - 8f (p )ap ( f)t + pt 2 . Thus, Pt : GQ -t GL 2 ( 0f) is a lifting of p, and
_
0,
_
_
_
1 33 character. Furthermore, by Theorem 9.31, pj is semistable at p E Sp U {t'}, and if f, fl. Sp, pj is good at p = £. By Proposition 10.23(2) , pj is unramified at p = q'. Thus, pj : Gq -+ GL 2 (01) is a lifting of j5 of type 'DQ The lifting pj : Gq -+ GL2 (01) of p of type 'DQ defines a ring homomorphism RQ -+ 01 . For p f N0Qq'£, we have Tr(pj ( ( N Q q' 2 ) K , p , o. Q 01 is injective, and the image of (p) -112TP is (cj 1 /2 (p )ap(f)) . Thus, for p f N0Qq'£, the image of Tr(pQ ( ( N Q q ' 2 ) K , p , a Q 01 coincides with the image of (p) - 112Tp. By Theorem 5 . 8 , the subring O[Tr(pQ ( ( N Q q ' 2 ) K , p , o. Q 0 I is a subring of T� , and the ring homomorphism fQ : RQ -+ T� has been constructed. The equation (10.28) has already been shown. The uniqueness is clear. (2) Let f E
0,
_
_
0,
_
_
0,
_
_
=
=
1 34
10 . HECKE MODULES
COROLLARY 10.33. t � , Q : T� --+ T� induces an isomorphism of 0-algebras T� ,K ©o[a q ] 0 --+ T0,K = T�,K · The diagram RQ
_!L
T�
R
---+
T00
TflJ, Q 1
f
1t�, Q
is commutative.
PROOF. We identify as TQQ K = I1 J e � (N Q q'2 ) K ,p,etQ Kf , T0,K = I11e� (Nfl1 ) K ,;; K1 . By Theorem 9.31 (1) and Proposition 10.32(2) , we have
fl/ ,
_
_
10 .5. FAMILY OF LIFTIN G S
13 5
PROPOSITION 10.35 . The 0-linear mapping (10.29) L ad sd* : H1 (Xo , 1 (N0 , Qq'2 ) , Z)o ----+ M0 d ! Q q'2 is compatible with the ring homomorphism t0,Q : T(N0 , Qq'2 ) ---+ T� in Proposition 10.29.
The proof is similar to that of Corollary 10.10, and we omit it. The 0-linear mapping induced by homomorphism (10.29) is de noted by (10.30) m � ,Q : M� ----+ M0 . LEMMA 10.36. (1) T� and M� are finitely generated free 0-
modules. (2) M� , K = M� ©o K is free T� ,K - module of rank 2 . (3) m � . Q : M� ---+ M0 is surjective. The proof of (1) is similar to that of Lemma 10.3, the proof of (2) is similar to that of Proposition 10.8, and the proof of (3) is similar to that of Proposition 10. 1 1 . Thus, we omit all these proofs. PROPOSITION 10.37. Let Q C Q be a finite subset. Through the homomorphism O[AQ] ---+ RQ ---+ T� , we consider M� as an O[AQ] module. Then, M� is a free O[AQ] -module.
This is Theorem 5.32(2) in Chapter 5. We will prove Proposi tion 10.37 in the next section. PROPOSITION 10.38. The RTM-triple 'R.� = (RQ , M� , O [AQ] ---+ RQ , r0 ,Q : RQ ---+ R0 , m0,Q : M� ---+ M0) is a lifting of (R, M) = (R0 , M0) along (O[AQ] , O[AQ] ---+ 0) . PROOF. We verify the conditions of Definition 5.24. R, RQ are profinitely generated complete local 0-algebra, and the residue field of R is the same as that of 0. M is an R-module, and M� is an RQ module. O[AQ] is a profinitely generated complete local 0-algebra, and O [A Q ] ---+ 0, O [AQ] ---+ RQ and r0,Q : RQ ---+ R are morphisms of local 0-algebras. By Lemma 10.25, the diagram O[A Q ] -----+ RQ
1
0
-----+
l r0. Q
R
136
1 0 . HECKE MODULES
is commutative, and RQ ©o [6. Q } CJ ---+ R is an isomorphism. m � ,Q : M� ---+ M = M0 is an CJ-linear mapping compatible with r0,Q : RQ ---+ R by Proposition 10.35 and Corollary 10.33. We show M� ©o [6. Q } CJ ---+ M is an isomorphism. By Proposi tion 10.37, the CJ-linear mapping M� is a free CJ[� Q ] -module. By Lemma 10.36(3) , M� ©o [6. Q } CJ ---+ M0 is a surjection of free CJ modules. Thus, it suffices to show ranko M� ©o [6. Q } CJ = ranko M0 . Through the isomorphism T� ,K ©o [6. Q } CJ ---+ T0,K in Corollary 10.33, we consider M� ,K ©0 [6. Q } CJ as a T0,K-module. It suffices to show that both M� ,K ©o [6. Q } CJ and M0,K are free T0,K-modules of rank 2. By Proposition 9.6 and Lemma 10.36(2) , M0 , K is a free T0,K-module of rank 2, and M� ,K is a free T� ,K -module of rank 2. By Corol lary 10.33, we have M� ,K ©o [6. Q } CJ = M� ,K © T� T0,K, and this is .K 0 also a free T0,K-module of rank 2. 10.6. Proof of Proposition 10.37
We give a brief summary of perfect complex. Let A be a commu tative ring. DEFINITION 10.39. A complex of A-modules c = (Cq, dq : Cq ---+ Cq - l ) q E Z is said to be right bounded if there exists a such that Cq = 0 for q :::; a. A complex of A-modules bounded below is called a perfect com plex if there exist a complex P = (Pq, dq)q of finitely generated projective A-modules such that Pq = 0 except for finitely many q and a morphism f : P ---+ C of complexes of A-modules such that f* : Hq (P) ---+ Hq (C) are isomorphisms for all q. In this section we assume A to be a commutative noetherian ring. LEMMA 10.40. Let C be a right-bounded complex of A-modules, and let a :::; b be integers. The following are equivalent. (i) There exist a complex P = (Pq , dq)q of finitely generated pro jective A-modules such that Pq = 0 except for a :::; q :::; b and a morphism f : P ---+ C of complexes of A-modules such that f* : Hq (P) ---+ Hq (C) are isomorphisms for all q.
1 0 .6.
PROOF OF PROPOSITION
1 0 . 37
13 7
(ii) For any integer q, Hq (C) is a finitely generated A-module, and except for a ::::; q ::::; b, Hq (C) = 0 and Tor : (c, A/m) = 0 for any maximal ideal m of A. PROOF. (i) ::::} (ii) is clear. We show (ii) ::::} (i) . Since A is noetherian, there exist a complex L of finitely generated free A modules and a morphism of complexes of A-modules f L ---+ C such that Lq = 0 for q < a and f* : Hq ( L ) ---+ Hq ( C ) are isomor phisms for all q. Let Pb = Coker(db+ l : L b+ l ---+ L b ) , and let P be the complex obtained by replacing L b by Pb and Lq (q > b) by 0. Then, for all q, Hq(P) ---+ Hq (C) are isomorphisms. Pb is a finitely generated A-module, and for any maximal ideal m of A, we have Tor 1 (Pb , A/m) = Tor�+ 1 (C, A/m) = 0. Thus, Pb is a finitely gener ated projective A-modules, and P ---+ C satisfies the condition. 0 COROLLARY 10.41. Let B be a commutative A-algebra, and let C be a right-bounded complex of B-modules. Let m be a maximal ideal of A, let E be a fiat B-module, and let q 1 be an integer. If Hq ( C ) © B E is a finitely generated A-module for any integer q and for q "I- q1 , Hq ( C ) m = 0, and Tor : (c, A/m) = 0, then Hq1 ( C ) m © B E is a finitely generated free Am -module. PROOF. Since E is a fiat B-module, we have Hq ( C © B E)m = Hq (C)m © B E and Tor : (C © B E, Ajm) © B E for any integer q. Thus, applying Lemma 10.40(ii) ::::} (i) to the complex of Am-modules Am © A C © B E and a = b = Q 1 , we see that Hq1 ( C ) m © B E = Pq1 is a finitely generated projective Am-module. Since Am is a local ring, this is a 0 free module. For a locally constant sheaf F on a topological space X, the singular chain complex C ( X, F) = ( Cq ( X, F) , dq) is defined. For any integer q 2:'.: 0, we have Cq ( X , F) = ffif:6.q-+x r(�q , f * F) . Here, �q is a standard q-simplex, and f : � q ---+ X runs all continuous mappings. If q < 0, we define Cq(X, F) = 0. dq : Cq (X, F) ---+ Cq_ 1 (X, F) is the alternating sum of the pullbacks by face mapping �q- l ---+ �q . For any integer q, we have Hq ( X, F) = Hq ( C ( X, F)) . LEMMA 10.42. Let G be a finite abelian group, let Y be a complex manifold, and let X ---+ Y be a G-torsor. Let R be a commutative :
7f :
noetherian ring, and let R[G] be the group algebra. (1) With the action of a E G defined as a* , the singular chain com plex C ( X, R) is a complex of R[G] -modules. For an ideal I in
1 0 . HECKE MODULES
1 38
R[G] , let F be the sheaf 7r* R ®R[G] R[G]/I on Y. F is a locally constant sheaf, and Hq (Y, F) = Tor : [Gl (C(X, R) , R[G]/I) for
any integer q. (2) Furthermore, let sA, h : YA -+ Y, .A E A be families of finite etale coverings, let XA -+ YA , .A E A be families of G-torsors, and let SA , tA : XA -+ x, A E A be finite etale coverings of G-torsors. For .A, µ E A, let iA,µ : XA,µ = XA '-.,,.t>, Xx s ,. / Xµ -+ Xµ,A be a homeomorphism that makes the diagram x
I
S ). Opr1 �
s ,. opr1
xA,µ
i )..µ
1
t,. opr 2
t ). opr 2
X
I
x x +-'---- Xµ , A commutative. Let T be the polynomial ring R[G] [TA, .A E A] . Then the singular chain complex C(X, R) is a complex of T-modules, the multiplication by TA defined as h * o st for .A E A. PROOF. (1) It is clear that C(X, R) is a complex of R[G]-mod
ules. We show it is a perfect complex. Since X -+ Y is a G-torsor, 7r* R is a locally constant sheaf of invertible R[G]-modules on Y. Thus, Fr is a locally constant sheaf on Y. Since C(Y, Fr) = C(Y, 7r * R) ®R[GJ R[G]/I and C(Y, 7r* R) = C(X, R) , we have Hq (Y, Fr) = Tor : [Gl (C(X, R) , R[G]/I) . (2) For .A, µ E A, we have TA o Tµ = h * o st o tµ * o s � = (h opr2 ) * o (sµ opr 1 ) * . Similarly, we have Tµ o TA = (tµ o pr2 ) * o (sA o pr 1 ) * . Thus, by the assumption, we have TA o Tµ = Tµ o TA . Similarly, we have a* o TA = TA o a* , and thus C(X, R) is a complex of T-modules. D COROLLARY 10.43. Let the notation be as in Lemma 10.42. Let E be a fiat T -module, and let q 1 be an integer. Let m be a maximal ideal of R[G] , and let Fm be the locally constant sheaf 7r * R ®R[G] R[G]/m on Y. Suppose Hq (X, R) ®T M is a finitely generated R[G] -module for any q and for any integer q =f. Q1 , Hq (X, R) ® R[G] R[G]m ®T M = 0 and Hq (Y, Fm) ®T M = 0. Then, Hq1 (X, R) ®R[G] R[G] m ®T M is a 7r :
finitely generated free R[G]m-module. PROOF . Let A = R[G] , B = T, and C = C(X, R) . The singu
lar chain complex C(X, R) is a right-bounded complex of T-modules by Lemma 10.42. Since Tor : [Gl (C(X, R) , R[G]/m) = Hq (Y, Fm), it D suffices to apply Corollary 10.41 .
10 . 6.
PROOF OF PROPOSITION
10 . 37
139
LEMMA 10.44. Let N, M ;::: 1 be relatively prime integers, and let p be a prime number not dividing NM. The action of the Hecke operator Tp = s * ot * on Ho (Yo, 1 (N, M) , Z) is the multiplication by p + 1. PROOF. It follows immediately from deg t = p + 1 . 0 PROPOSITION 10.45. Let N, M ;::: 1 be relatively prime inte gers, and let r ;::: 4 be an integer relatively prime to NM. Let 0 be the ring of integers of a finite extension of Qe , let m be a non Eisenstein maximal ideal of To , 1 (N, Mr) o , and let mo be the inverse image of m by O[(Z/Mz) x ] --+ T0 , 1 (N, M r ) o . Then, the localization H1 (Xo , 1 (N, Mr ) , Z)o ,m is a finitely generated free O[(Z/Mz) x ]m0 module. PROOF. Let G = (Z/Mz) x and R = 0, and let T be the polyno mial ring O[G] [Tp , P : prime] of infinite number of variables over O[G] . Let T be the image of the ring homomorphism T --+ To, 1 (N, Mr)o that sends Tp to Tp , and let mr c T, mr C T be the inverse images of m C To, 1 (N, Mr)o. Since To , 1 (N, Mr)o and T are both complete semilocal rings, the localization H1 (X0 , 1 (N, Mr) , Z)o , m is a direct summand of H1 (Xo, 1 (N, Mr) , Z)o,m'f' = H1 (Xo, 1 (N, Mr) , Z)o,m T · As in Lemma 10.20, the O[GJm0-linear mapping H1 (Yo , 1 (N, Mr) , Z)o,m T --+ H1 (Xo, 1 (N, Mr) , Z)o,m T is an isomorphism. Thus, it suffices to show H1 (Yo, 1 (N, Mr) , Z)o , m T is a finitely generated free O[G]m0-module. Since we assume r ;::: 4, X = Yo , 1 (N, Mr ) and Y = Yo , 1 (NM, r) are fine moduli schemes, and : X --+ Y is a G-torsor. Similarly to Sn, tn : Io (N, n ) --+ Xo (N) in §9. 1 , we define finite etale cover ings that induce Hecke operators. This family satisfies the condi tion of Lemma 10.42(2) . We show H1 (X, Z)o , m T = H1 (X, 0) ©o [G] O[G]m0 ©rTm T is a finitely generated free O[G]m0 module by applying Corollary 10.43. The localization M = TmT is a fl.at T-module. Since Hq (X, Z)o is a finitely generated 0-module, the image of T in Endo (Hq (X, Z)o) is finitely generated as 0-module, and the product of a finite number of complete local rings. Thus, Hq (X, Z)o , m T is a direct summand of Hq (X, Z)o, and a finitely generated O[G]-module. Since X is an affine curve, we have Hq (X, Z)o , m T = 0 and Hq (Y, Fm0 ) = 0 for q # 0, 1 . By Lemma 10.44, we have Ho (X, Z)o,m T = 0. Moreover, since the natural morphism Ho (X, 0) = Ho (Y, * O) --+ Ho (Y, Fm0 ) 7r
7r
1 40
1 0. HECKE MODULES
is surjective, we have Ho (Y, Fm0 )m T = 0. Thus, by Corollary 10.43, D H1 (X, Z) o mT is a finitely generated free O[G]m0-module. PROOF OF PROPOSITION 10.37. We apply Proposition 10.45 to N = N0 , M = Q, and r = q'2 . Let G = (Z/Qz) x , and let mo [G] C O[G] be the inverse image of the maximal ideal m� of To,1 (N, Mr)o by the ring homomorphism O[G] -+ To ,1 (N, Mr) o . Since the image of G by O[G] -+ To ,1 (N , Mr) o/m� equals 1 and D.. Q is the maximal .eth power quotient of the finite abelian group G, the localization of O[G] at mo (G] is O[D.. Q ] · Thus, by Proposition 10.45, M� = H1 (Xo,1 (N0 , Qq'2 ) , Z) o , m � is a finitely generated free O[D.. Q ]-module. D ,
10.7. Proof of Theorem 5.22
In this section, assuming Theorems 5.32(1) and 5.34, we will prove Theorem 5.22 along the idea explained in §5.6. Theorems 5.32(1) and 5.34 are related to Selmer groups, and will be proved in the next section. We prove the following stronger theorem. THEOREM 10.46. Let E be a finite set of primes such that EnS.o = If .e E "E , we assume p to be ordinary at .e. Then, fa : Rr:; -+ Tr:; is an isomorphism, Rr:; is a complete intersection, Mr:; is a free Tr:; module, and ir:; : Tr:; -+ Tf is an isomorphism.
0.
PROOF. For the ring of integers O' of a finite extension K' of K, the natural mappings Rr:;,o ©o O' -+ Rr:;,01 and Tr:;,o ©o O' -+ Tr:;,01 are isomorphisms by Corollaries 5 . 10 and 5. 19, and the natural map pings Mr:;,o ©o O' -+ Mr:;,01 and Tf ,0 ©o O' -+ Tf , o 1 are isomor phisms by definition. Thus, we may replace K by a finite extension. By Proposition 5. 14(2) , 'P(N0 ) K,.o is nonempty. Thus, replacing K by its finite extension if necessary, we may assume there exists a primitive form f E 'P(N0 ) K,.o(K) with K coefficients. Replacing further by an unramified quadratic extension if necessary, we may assume the eigenvalues of p(cpp) are elements of F for each prime p f Nr:;.e.
Let 7r0 : T0 -+ 0 be a ring homomorphism defined by a primitive form f E 'P(N0 ) K,.o (K) with K coefficients, and let 7rr:; : Tr:; -+ 0 be the composition 7r0 t0,r:; . By Proposition 10.8, 'Rr:; = (Rr:; , Tr:; , Mr:; , fa : Rr:; -+ Tr:; , 7rr:; : Tr:; -+ 0) is an RTM-triple of rank 2. By Proposition 5.29, if the RTM-triple 'Rr:; is complete, i-r; : T-r; -+ Tf o
1 0.7. PROOF OF THEOREM 5.22
141
is an isomorphism. Thus, we only have to show that the RTM-triple Ry;, is complete, in other words, Theorem 5.31 . We first show the RTM-triple R0 is complete. Let
By Proposition 10.26, there exists an odd prime q' E Q such that q' ¢. 1 mod .e, and so choose such a q' . By Theorem 5.32 ( 1 ) , which will be proved in the next section, there exists for any positive integer n, a set Qn = {qi , . . , qr } of r prime numbers such that .
Choose such Qn for each n. By Proposition 10.38, the quintuple � (RQ " , Mt , 0[.6.Q J --+ RQ " , r0,Q " : RQ " --+ R0 , m0,Q " : Mt --+ M0 ) is a lifting of (R, M) = (R0 , M0 ) along O[AQ " ] --+ 0. Let On be the group algebra O[(Z/.en z rJ . For each n, take a sur n jection AQ " --+ (Z/.e zt, and let 0[.6. QJ --+ On be the induced ring homomorphism. Define a lifting Rn = (Rn, Mn, On --+ Rn, Tn : Rn --+ R, mn : Mn --+ M) of (R, M) along the augmentation morphism On --+ 0 by R� " © o [a q" J On = (RQ " © o [a q "J On , Mt © o [a q" J On, On --+ RQ " © o [a q"J On, r0,Q " © 1 , m0,Q " © 1 ) . By Proposition 10.37, Mt is a free O[AQ " ]-module, and thus Mn = Mt © o (D. q" ] On is also a free On-module. Applying Theo rem 5.25, we conclude that Ry;, is a finitely generated 0-module, the ring R0 is a complete intersection, and M0 is a finitely generated free R0-module. Thus, by Proposition 5.28, the RTM-triple R0 is com plete. Moreover, by Proposition 10.4(2) , the assumption of Proposi tion 5.29 is satisfied, and thus i0 : T0 --+ T� is an isomorphism. This completes the proof of the case E = 0 . We show the general case by induction on the number of elements of E. We have already shown the case E = 0 , and so we take p E E and let E' = E {p} . We show Ry;, is complete assuming Ry;,1 is complete. By Proposition 10.13, the quadruple Fy;,1 ,E = (ry;,1,E : Ry;, --+ Ry;,1 , ty;,1 E : Ty;, --+ Ty;,1 , my;,1 ' E : My;, --+ My;,1 , mi; 1 ' E : My;,1 --+ My;,) is a surje�tion Ry;, --+ Ry;,1 of RTM-triples. By the assumption (hypothesis) of induction, iy;,1 : Ty;,1 --+ Tf 1 is an isomorphism. Thus, even when p = .e, Tp E Ty;,1 , and Ap E Ty;,1 . R " =
-
1 0 . HECKE MODULES
1 42
If p = £, let TJ E TE' be the root of T2 - TpT + p = 0 that is invertible, and define �P E TE' by �P - (p - l)((p + 1)2 - Ti) if p # £, if p = £. (TJ2 - 1) (2TJ - Tp) By Proposition 10.14, �P is the multiplier of the surjective morphism FE',E : RE � RE' · If p # £, we have 7l"E 1 (�p) = (p - l)((p + 1) 2 - ap( / ) 2 ) . If p = £ , then 2TJ - Tp, p - 1 and 1 -p2 /TJ 2 are all invertible elements of TE ' , and thus we have ((p + 1) 2 - T;) = (1 + Tp + p) (l - Tp + p) = (1 - TJ 2 ) (1 - p2 /TJ 2 ) . Hence, in this case also, we have ordo 7l"E1 (�p) = ordo (p - l)((p + 1) 2 - ap(/) 2 ) . By Theorem 9.21, we have (p + 1)2 - ap( / )2 # 0 and length0 Ker ( P RE / p �E � P RE ' / p �E ' ) :::; ordo (p - 1) ( (p + 1) 2 - ap( / ) 2 ) = ordo 7l"E' (�p) · Thus, applying Theorem 5.27, the RTM-triple RE is also complete. This completes the proof of Theorem 5.31, and thus Theorems 10.46 and 5.22. D _
{
CHAPTER 1 1 Selmer groups
In this chapter we study Selmer groups and describe the rela tion between the Selmer groups and the deformation rings. Then, by studying the order of Selmer groups, we prove Theorems 5.32(1) and 5.34, which completes the proof of Theorem 5.22. In § 1 1 . 1 and § 1 1 .2, we will introduce cohomology of groups and Galois co homology and show their fundamental properties. In § 1 1 .3, we will introduce Selmer groups and describe its rela tion with deformation rings. We translate Theorems 5.32(1) and 5.34 into the properties of Selmer groups in Theorem 1 1 .37 and Proposi tion 1 1 .38. In § 1 1 .5, we will calculate the order of local cohomology groups and prove Proposition 1 1 .38. In § 1 1 .6, we will prove Theo rem 1 1 .37 using group theoretic properties of subgroups of GL 2 (F) . 1 1 . 1 . Cohomology of groups
DEFINITION 1 1 . 1 . (1) The projective limit G = � AE A GA of a projective system ( G>. ) AE A of surjections of finite groups is called a profinite group. (2) Let G be a profinite group, and let R be a commutative ring. A finite R-module M is called a finite R-G-module if a contin uous homomorphism G -+ AutR(M) of a finite discrete group AutR(M) is given. If R = Z, a finite Z-G-module is simply called a finite G-module . If F is a field, its absolute Galois group GF is a profinite group. For a profinite group G and a finite R-G-module M, the cohomology group Hq (G, M) , q = 0, 1 , 2, . . . , is defined as R-modules. The cases where q = 0, 1 are particularly important, and we first give a definition in these cases. DEFINITION 1 1 .2. Let G be a profinite group, let R be a commu tative ring, and let M be a finite R-G-module. 143
1 44
11 .
SELMER GROUPS
( 1) H0 ( G, M) is defined as the G-invariant part of M M0 = { x E M I gx = x for any g E G}. (2) A continuous mapping c : G -* M is called a 1 -cocycle if it satisfies c( g h) = c(g) + g c(h) for any g, h E G. For x E M, let Cx : G -* M be the 1-cocycle defined by Cx (g) = gx - x. Let Z 1 (G, M) be the R-module of all 1-cocycles, and define H 1 (G, M) = Z 1 (G, M)/{cx I x E M} . H0 ( G, M) and H 1 ( G, M) naturally have a structure of R-module. LEMMA 1 1 .3. Let G be a finite group of order n, and let R be a commutative ring. For any R-G-module M, we have n · H 1 (G, M) = 0. In particular, if n is invertible in R, we have H 1 ( G, M) = 0. PROOF. Let c E Z 1 (G, M) be a 1-cocycle, and let b = Lh EG c(h) . Since b - gb = LhE G (c(gh) - gc(h) ) = nc(g), we have n[c] = 0 E 0 H 1 (G, M) . For a finite G-module M, the G-coinvariant quotient is defined by M/(gx - x : g E G, x E M) and is denoted by Mc . LEMMA 1 1 .4. Let G be a profinite group, let R be a commutative ring, and let M be a finite R-G-module. ( 1) If the action of G on M is trivial, H 1 ( G, M) equals the R-module Hom(G, M) of all continuous homomorphisms G -* M. (2) Let G = Z = �n Z /n Z be the profinite completion of Z . Then, the mapping that sends the class of a 1 -cocycle c : G -* M to the class of c(l) (11.1) is an isomorphism.
PROOF. (1) Clear from Definition 1 1 .2. (2) The mapping Z 1 (G, M) -* M : c r-+ c(l) is a bijection. This induces the isomorphism ( 1 1 . 1 ) . 0 If G is a profinite group, a homomorphism of finite R-G-modules M -* N induces an R-linear mapping Hq (G, M) -* Hq (G, N) . Also, if G -* H is a continuous homomorphism of profinite groups and M is a finite R-H-module, then an R-linear mapping Hq (H, M) -* Hq (G, M) is defined by regarding M as an R-G-module through
11 . 1 . COHOMOLOGY OF GROUPS
1 45
G ---+ H. If a homomorphism G ---+ H is injective, the induced ho momorphism H q (H, M) ---+ H q (G, M) is called the restriction map ping. If (N>.. h . EA is a fundamental neighborhood system of the iden tity element consisting of open normal subgroups of G, the morphism H 1 (G, M) ---+ � >.. H 1 (G/N>.. , M N" ) is an isomorphism. PROPOSITION 11.5. Let G be a profinite group, and let N be its closed normal subgroup. If M is a finite G-module, the natural mappings form an exact sequence PROOF. It is easy to see that the image of the restriction map ping H 1 (G,a M) ---+ H 1 (N, M) is contained in the G-invariant part H 1 (N, M) . Also, the composition H 1 (G/N, M N ) ---+ H 1 (G, M) ---+ H 1 (N, M) G is the 0 mapping. We show H 1 (G/N, M N ) ---+ Ker(H 1 (G, M) ---+ H 1 (N, M) G ) is surjective. Let c : G ---+ M be a 1-cocycle. Assuming the class [c] belongs to Ker(H 1 (G, M) ---+ H 1 (N, M) G ) , we show [c] is in the image of H 1 (G/N, M N ) ---+ H 1 (G, M) . There exists an x E M such that c( h) = hx - x for any h E N. Replacing c by the 1-cocycle c' : G ---+ M defined by c' (g) = c(g) - (g(x) - x) , we may assume c l N = 0. Then, for any g E G and h E N, we have c(gh) = c(g) + gc(h) = c(g) , hc(g) = c(hg) - c(h) = c(g g - 1 hg) = c(g) , and thus c : G ---+ M induces c : G/N ---+ M N . Since the class [c] E H 1 (G, M) is the image of [c] E H 1 (G/N, M N ) , we conclude that H 1 (G/N, M N ) ---+ Ker(H 1 (G, M) ---+ H 1 (N, M) G ) is surjective. We show H 1 (G/N, M N ) ---+ H 1 (G, M) is injective. Let c : G/N ---+ N M be a 1-cocycle. Suppose the image of [c] E H 1 (G/N, M N ) in H 1 (G, M) is 0. Then, there exists an x E M such that c(g) = gx - x for any g E G. Since hx = x + c(ii) = x for any h E N, we see 0 x E M N . This proves [c] = 0. ·
COROLLARY 11.6. Let p be a prime number. Let G be a profinite group, and let N c G be a closed normal subgroup. Suppose N is the projective limit of a projective system of finite groups of order relatively prime to p, and suppose G/N is isomorphic to Zp - If M is a finite Zp-G-module, then there is a natural isomorphism H 1 (G, M) ---+ Hom(G/N, MN ) ·
11 .
1 46
SELMER GROUPS
PROOF. By Lemma 1 1 .3, we have H 1 (N, M) = 0. Thus, we obtain an isomorphism H 1 (G/N, M N ) --+ H 1 (G, M) , by Proposi tion 11.5. The invariant part M N is naturally identified with coin variant quotient MN . The rest of the proof goes similarly with Lemma 1 1 .4(2) . D For a commutative ring, define R[c] = R[X]/(X 2 ) . e represents the image of X. R[c] = R + Re is a free R-module of rank 2 with c2 = 0. For an R-module M, define an R[e]-module M by R[c]©RM = M + cM. We have a natural isomorphism M / cM --+ M. DEFINITION 1 1 .7. Let G be a profinite group, let R be a commu tative ring, and let M be a finite R-G-module. (1) For a continuous homomorphism p G --+ AutR [eJ (M) , con sider M as a !_nite R[c]-G-module through p, and denote it by Mp. We call Mp an infinitesimal lifting if the natural homomor phism Mp/cM --+ M of R-modules is an isomorphism of R-G modules. The set of all infinitesimal liftings of M is denoted by Lift R- G ( M) . (2 ) An isomorphism of infinitesimal liftings Mp and Mp' is an iso morphism of R[c]-G-modules Mp --+ Mp' such that the induced map M --+ M on the quotients is the identity. An isomorphism class of infinitesimal liftings of M is called an infinitesimal de formation of M, and is denoted by DefR-a (M) . (3) Let M be a finitely generated free R-module, and let Mp be an infinitesimal lifting. We say Mp preserves the determinant if det p G --+ R[c] x and det p G --+ Rx C R[c] x are equal. The set of all infinitesimal liftings that preserve the determinant is denoted by Lift�_ 0 , and the set of all infinitesimal deformations that preserve the determinant is denoted by Def�- G . Let G be a profinite group, let R be a commutative ring, and let M be a finite R-G-module. Define an action of G on EndR(M) by g(f) = g f g - 1 for f E EndR(M) , g E G. EndR(M) becomes a finite R-G-module. For f E EndR(M) , define an automorphism l +cf of M by (l+cf) (x+cy) = x+c(f(x)+y) . Define a homomorphism of groups EndR(M) --+ AutR [eJ (M) by associating l + cf to f E EndR(M) , and we obtain an isomorphism of groups EndR(M) --+ Ker(AutR [eJ (M) --+ AutR(M)). :
:
o
:
o
11 . 1 . COHOMOLOGY OF GROUPS
1 47
For a 1-cocycle c : G ---+ EndR(M) , define an action Pc of G on M by (1 + c ( g )) (1 © g) . Then, M is an infinitesimal lifting of M. Thus, we obtain a natural mapping (11.2) PROPOSITION 11.8. Let G be a profinite group, let R be a com mutative ring, and let M be a finite R-G-module. (1) The natural mapping (11.2) is bijective, and it induces a bijec c:
o
tion
H 1 (G, EndR(M)) ----* DefR-a (M) . (11.3) (2) Suppose M is finitely generated and free as an R-module, and let End� (M) be the kernel of Tr : EndR(M) ---+ R. Then, the mapping Z 1 (G, End� (M)) ---+ Lift � _0 (M) induced by the bijec tion (11.2) is also a bijection. Furthermore, if the rank of M is invertible in R, it induces a bijection
(11.4) H 1 (G, End� (M)) ----* Def�_ 0 (M) . PROOF. (1) Clear from the definition. ( 2) The determinant of the action Pc (g) is det g ( 1 + Tr c(g) ) . Thus, the bijection Z 1 (G , EndR(M)) ---+ LiftR-a (M) induces the bi jection Z 1 (G, End� (M)) ---+ Lift�_ 0 (M) . We show the mapping (11.4) is bijective if the rank r of M is invertible in R. By definition, Def�_ 0 (M) is the image of Lift �_ 0 (M) ---+ DefR-a (M) . Thus, it suf fices to show that H 1 ( G, End� ( M)) ---+ H 1 ( G, EndR ( M)) is injective. If the rank r is invertible in R, we have EndR(M) = End� (M) E9 R, D from which the assertion follows immediately. Let G be a profinite group, let R be a commutative ring, and let M, N be finite R-G-modules. A finite R-G-module E together with an exact sequence of finite R-G-modules 0 ---+ N ---+ E ---+ M ---+ 0 is called an extension of M by N. Two extensions E and E' of M by N are said to be isomorphic if there exists a commutative diagram 0 ----+ N ----+ E ----+ M ----+ 0 ·
I
1
I
c:
N ----+ E' ----+ M ----+ 0 . For extensions E and E', the sum is defined to be Ker(E E9 E' ..::+ M)/ Im(N 4 E E9 E') . 0
----+
·
1 48
11 .
SELMER GROUPS
Here, - : E EB E' --+ M is the difference between E --+ M and E' --+ M, and + : N --+ E EB E' is the sum of N --+ E and N --+ E' . For an extension E and a E R, define the multiplication by a to be Coker((l, -a) : N --+ E EB N) . The set of isomorphism classes of extensions of M by N, denoted by ExtR-a (M, N), has a structure of an R-module by this operation. If G = 1, we write ExtR-a (M, N) simply by ExtR(M, N) . An infinitesimal deformation M of M defines an extension of M by M. This defines a natural injection DefR-a (M) --+ ExtR-a (M, M) , and its image equals the kernel of the mapping ExtR-a (M, M) --+ ExtR(M, M). Through this injection, we consider DefR-a(M) as a submodule of ExtR-a (M, M) . PROPOSITION 11.9. Let G be a profinite group, and let R be a commutative ring. For R-G-modules M and N, we have a natural injection
H 1 (G, HomR(M, N)) ---+ ExtR-a (M, N) . The image of (11 .5) is the kernel of ExtR-a (M, N) --+ ExtR(M, N) . If M = N, the image of (11.5) is DefR-a (M) C ExtR-a (M, N) , and the natural morphism (11.5) coincides with (11.3) . If M is a projective R-module, the morphism (11.5) is an isomorphism. In par ticular, if M = R, H 1 (G, N) --+ ExtR-a(R, N) is an isomorphism. PROOF. If we define an action of G on M EB N by g(x, y) = (gx, c ( g ) g x + gy) for 1-cocycle c : G --+ HomR(M, N) , then M EB N becomes an R-G-module. We denote by Ee this extension of M by N. Sending a 1-cocycle c : G --+ HomR(M, N) to the class of the extension Ee, we obtain an R-linear mapping Z 1 (G, HomR(M, N)) --+ ExtR-a (M, N) . It is easy to see that the extensions Ee and Eo are isomor phic if and only if there exists an R-linear mapping f : M --+ N satisfying c (g ) = gfg - 1 - f. Thus, we obtain a natural injection H 1 (G, HomR(M, N)) --+ ExtR-a (M, N) . By the definition of (11.5) , it is clear that the image coincides with the kernel of ExtR-a (M, N) --+ ExtR(M, N) . It is clear from the definition that (11.5) and (11.3) coincide if M = N. If M is a projective R-module, we have ExtR(M, N) = 0, 0 and thus (11.5) is an isomorphism. q For a general integer q � 0, the cohomology of group H (G, M) is defined as follows. Let G be a profinite group, let R be a commutative (11.5)
11.2. GALOIS COHOMOLOGY
1 49
ring, and let M be a finite R-G-module. For an integer q � 0, define C q ( G , M) = {continuous mapping G q -+ M} . For f E Cq ( G , M) , define dq f E Cq +l ( G, M) by dq f ( 90 , . . · 9q ) = 9o f ( 9i , . . 9q ) - f ( go 9i , 92 , . . . , 9q ) + ! ( 90 , 9192 , . . 9q ) ,
·
,
·
,
+ (- l) q f ( 90 , . . . , 9q - 2 , 9q - 19q ) + (- l) q +l ! ( 90 , . . . , 9q - 2 , 9q - 1 ) . For q = - 1 , define c - 1 (G , M) = 0 and d - 1 = 0. For q � 0, define Ker (dq : Cq (G, M) -+ Cq +l (G, M)) . Hq (G ' M) = Im (dq - l : Cq - 1 (G , M) -+ Cq (G, M) ) This definition coincides with Definition 1 1 . 2 for q = 0, 1 . For an exact sequence of finite R-G-modules 0 -+ M' -+ M -+ M" -+ 0, we - · · ·
have a long exact sequence
0 --+ H0 (G , M') --+ H0 (G , M) --+ H0 (G, M") --+ H 1 (G, M') --+ H 1 (G , M) --+ H 1 (G, M") --+ H2 (G , M') --+ H 2 (G , M) --+ H 2 (G , M") --+ · · · . Let M, N , L be finite R-G-modules. An R-bilinear mapping M x N -+ L is a bilinear mapping of finite R-G-modules if ( 9x , 9y) = 9 (x , y) for any x E M, y E N, 9 E G. A bilinear mapping of R-G-modules M x N -+ L induces a bilinear mapping of R-modules U : HP (G , M) x H q (G , N) -+ HP+ q (G , L ) . This is called the cup product .
Let £ be a prime number, and let 0 be the ring of integers of a finite extension of Qe. Let R be a profinitely generated complete local 0-algebra, and let m be its maximal ideal. A finitely generated R-module M is a G-module if a continuous homomorphism G -+ AutR(M) = � AutR;mn (M/mn M) is given. For an integer q � 0, n define Hq (G , M) = � Hq (G, M/mn M) . n
Hq (G , M) is an R-module.
1 1 . 2 . Galois cohomology
The group cohomology of the absolute Galois group G F of a field is called Galois cohomology. In this section we present fundamen tal properties of Galois cohomology of p-adic fields and the rational
1 50
11 .
SELMER GROUPS
number fields, Propositions 11.18, 11.20, 11.25, and 11.27. We do not give proofs to them, but instead we give typical examples in Exam ples 11.15, 11.19, 11.21, and 11.26. DEFINITION 11.10. Let F be a field, and let Gp = Gal(F/F) be the absolute Galois group of F. For a finite Gp-module M, define Hq (F, M) = H q (Gp, M) . For an integer n ;:::: 1, we consider Z/nZ as a trivial Gp-module. By Lemma 11.4(1), we have (11.6) H 1 (F, Z/nZ) = Hom(G}b , Z/nZ) . Let n ;:::: 0 be an integer invertible in F. Let µn be the finite Gp module of nth roots of unity {x E px lxn = l}. For a finite Z/nZ Gp-module M, let Mv be the dual module Hom(M, Z/nZ), and M(l) the Tate twist M © µn . PROPOSITION 11.11. Let F be a field, and let n ;:::: 1 be an integer invertible in F. (1) There is a natural isomorphism (11.7) p x /(F x ) n --+ H l (F, µn ) · (2) Let Br(F) be the Brauer group of F (§8.2(c) in Number Theory 2 ) . Let n Br(F) = {x E Br(F) I nx = O} be its n-torsion subgroup. Then, we have a natural isomorphism
(11.8) We do not give a proof in this book. The natural isomorphism F x / (F x ) n --+ H 1 ( F, µn ) is defined as follows. For a E F x , take its nth root E F, and let Ca Gp --+ µn be the 1-cocycle defined by ca (<7) = <7(a)/a. The natural isomorphism is defined by sending a to the class of Ca . Let E be an elliptic curve over F, and let n ;:::: 1 be an integer invertible in F. Let E[n] be the finite Gp-module of n-torsion points of E. Similarly to the natural isomorphism px /(F x ) n --+ H 1 (F, µn ) , we define a natural injection E(F)/nE(F) --+ H 1 (F, E[n]) as follows. For a E E(F) , take E E(F) satisfying a = na, and let Ca Gp --+ E[n] be the 1-cocycle defined by ca ( u ) = u ( ) The natural injection is defined by sending a to the class of Ca · Let F be a finite field Fp . We have an isomorphism Z --+ GFp from the profinite completion of Z by sending 1 to the Frobenius a
:
a
a
- a.
:
11 . 2 . GALOIS COHOMOLOGY
151
substitution 'Pp E GF ,, · Thus, by Lemma 11.4(2) , we obtain a nat ural isomorphism H 1 (F M) --+ MaF p : c ....+ c (
P'
module with a continuous action of GFp . If the action of the Frobenius substitution
By Lemma 11.4(2) , we obtain an isomorphism H 1 (Fp, M) --+ Ma F ,, --+ Coker(! -
15 2
11. SELMER GROUPS
ExtR- G Qp (M, N) . Define the unramified part H} (Qp, HomR(M, N)) of H 1 (Qp, HomR(M, N) ) as the subgroup consisting of the isomor phism classes of extensions good as GQ p representations. Consider ing R as the trivial representation of GQ p and identifying N with HomR(R, N) , we define a subgroup Hj (Qp, N) of H 1 (Qp, N) . LEMMA 11.14. Let p be an odd prime, and let 0 be the ring of integers of a finite extension of Qp . Let n � 1 be an integer, and let R = 0 /m0 . Let M and N be finite R-GQ p -modules. Suppose M is a free R-module, and suppose M and N are good as representations of GQ p . Let D( M ) and D( N) be the strongly divisible filtered cp-R modules corresponding to M and N, respectively. Then, we have ttH} (Qp , HomR ( M , N) ) = tt HomR (D( M ) ' , D(N) / D(N) ' ) . tt HomR-G Qp (M, N) In particular, if M = R, we have ttH} (Qp, N) = llD(N) / D(N) ' . ttH 0 (Qp, N) II PROOF . Clear from Corollary C. 10.
D
EXAMPLE 11.15. Let p be a prime number, and let n � 1 be an integer. (1) Suppose M = Z/nZ. The unramified part H} (Qp , Z/nZ) of H 1 (Qp, Z/nZ) = Hom(GQ p , Z/nZ) is { x E Hom(GQ p , Z/nZ) \ x ( Ip) = O} = Hom (GQ p / Ip, Z/nZ) = Hom(GF p , Z/nZ) . (2) If M = µn , we identify H 1 (Qp, µn) = Q; /(Q; ) n , Then, we have H} (Qp, µn ) = z; /( z; ) n . If n and p are relatively prime, the orders of H0 (Qp, µn ) and H} (Qp , µn ) are both equal to the greatest common divisor ( n, p - 1). If p � 3 and n is a power of p, then we have H0 (Qp, µn) = 0, and the order of H} (Qp, µn) is n. LEMMA 11.16. Let p be a prime number, and let n be a power of p. Let R be a finite commutative Z/nZ-algebra, let GQ p -+ R x be an a :
unramified character, and let N be an R-GQ p -module obtained from the R-module R by defining the action of GQ p by a . Let Q�r be the maximal unramified extension of Qp . Then, the natural isomorphism (11.7) induces an isomorphism (11.9) (Q �r x / (Q�r x ) n ©z / n Z N) G al( Q �r /Q p ) ---+ H l (Qp, N( l ) ) .
1 53
11 . 2 . GALOIS COHOMOLO GY
By the isomorphism ( 1 1 .9), the unramified part H} (Qp, N(l)) is the image of the subgroup (z�rx /(z�rx ) n © z; n z N) G al (Q �r /Qp) of the left hand side. PROOF. The invariant subgroup by the inertia group N(1)1P is 0. Thus, similarly to Proposition 1 1 .5, the restriction H 1 (Qp , N(l)) -+ H 1 (Q�r , N(l)) G ai ( Q�r/Qp ) is an isomorphism. H 1 (Q�r , N(l)) is iden tified with Q�rx /(Q�rx ) n © z / nZ N by Proposition 1 1 . 1 1 ( 1 ) , and we obtain the isomorphism ( 1 1 .9) . Define the unramified part H} (Q�r , N(l)) C H 1 (Q�r , N(l)) just as H} (Qp , N(l) ) . Then, the subgroup H} (Q�r , µn) C H 1 (Q�r , µn) = Q�r X / (Q�r X ) n equals z�r x ;(z�rx ) n . Since H} (Qp , N(l)) is the in verse image of H} (Q�r , N(l)) G al(Q�r/Qp) , the assertion follows. D COROLLARY 1 1 . 17. Let p be an odd prime number, and let R be a finite commutative Zp-algebra. Let a, (3 : Gq p -+ R x be unramified characters, and let M and N be R-Gq p -modules obtained from the R-module R by defining the action of Gq p by a and (3, respectively. Let F be the residue field of R, and let M = M © R F , N = N ©R F . If a # (3, the image of the natural map H 1 (Qp, HomR(M, N ( l ) ) ) -+ H 1 (Qp, HomF (M, N(l))) is contained in H} (Qp, HomF (M, N(l)) ) . PROOF. Replacing a, (3 by l , a -1 , we may assume M = R. By Lemma 1 1 . 16, we have a commutative diagram of exact sequences 0 ----t H} (Qp, N(l)) ----t H 1 (Qp, N(l)) ----t H0 (Qp, N) ----t 0
1
1
1
0 ----t H} (Qp, N(l)) ----t H 1 (Qp, N(l)) ----t H0 (Qp, N) ----t 0. If (3 # 1, we have H0_(Qp, J'!) � N, and by Nakayama' s lemma, D H0 (Qp, N) -+ H0 (Qp, N) c N is the 0 mapping. Let n 2:: 1 be an integer, and M = µn . By the natural isomor phism Br(Qp ) -+ Q/Z (Theorem 8.25 in Number Theory 2 ) , we have a natural isomorphism 1 - Z/Z ----t n Br (Qp) ----t H2 (Qp, µn) ·
n Let f : A x B -+ � Z/Z be a bilinear mapping of finite Z/nZ-modules. For a submodule A' of A, the submodule of B defined as { y E B I f ( x, y ) = 0 for all x E A'} is called the annihilator of A'.
1 54
11 .
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PROPOSITION 11.18. Let p be a prime number, let n � 1 be an integer, and let M be a finite Z/nZ-GQ p -module. (1) H0 (Qp, M) , H1 (Qp , M) , and H2 (Qp, M) are finite abelian groups. If q > 2, we have Hq (Qp, M) = 0. (2) For an integer q � 0, the linear mapping (11.10)
Hq (Qp, M) ----+ H2 - q (Qp, M v (l)) v
defined by the cup product
Hq (Qp , M) x H2 - q (Qp, M v (l)) ----+ H2 (Qp, µn ) ----+ I.n z ; z
is an isomorphism. n is relatively prime to p. Then, the annihilator of the subgroup H} (Qp , M) of H1 (Qp , M) with respect to the bi linear mapping H1 (Qp, M) x H1 (Qp, Mv (l)) -+ �Z/Z equals
(3) Suppose
H} (Qp , MV (l)) . We do not prove this proposition. EXAMPLE 11.19. Let n � 1 be an integer. The isomorphism for q = 1 and M = µn , H1 (Qp, µn ) -+ H1 (Qp, Z/nz)v (11.10) gives the isomorphism of local class field theory Q; /CQ; ) n ----+ G�p /(G�,, ) n by the isomorphisms (11.6) and (11.7) . If p f n, the annihilator of the unramified part H} ( Qp, µn ) = z; I ( z; ) n is the unramified part H} (Qp, Z / nZ) = {X E Hom(GQ,, , Z/nZ) \ x(Ip) = O}. We have the following for the order of the p-adic cohomology group H q ( GQp , M) . PROPOSITION 11.20. Let M be a finite GQ P -module. Then we have U H1 (Qp, M) = (M © Zp) · 0 U H (Qp, M) . UH2 (Qp, M) U We do not prove this proposition, either. EXAMPLE 11.21. By the exact sequence 0 ----+ H o ( Qp, µn ) -+ Qp nt h Qp -+ H 1 ( Qp, µn ) ----+ 0 and the natural isomorphism � Z/Z -+ H2 (Qp, µn ) , we have n · U Zp/nZp = p power part of n. U H1 (Qp, µn ) = O 2 . n U H (Qp , µn ) UH (Qp, µn ) This equals the order of µn © Zp. x
power
x
1 55 11 . 2 . GALOIS COHOMOLO GY Finally, we consider the case where F is the rational number field Q. For a finite set of prime numbers S, let Qs be the compositum of all the subfields of Q that are unramified outside S. Let Gs = Gal(Qs/Q) . Gs is the quotient of the absolute Galois group GQ by the closed normal subgroup generated by the images of the inertia groups Ip , p (j. S. Let M be a finite GQ-module. Let S be a finite set of prime numbers such that M is unramified outside S. Then, the action of GQ on M induces an action of Gs, and M is regarded as a Gs-module naturally. LEMMA 11.22. Let M be a finite GQ -module, and let S be a finite set of prime numbers such that M is unramified outside S. Then, we have
(
)
n H 1 (Ip, M) . p :pnme p '/.S PROOF. Let N = Ker(GQ ---+ Gs) . The action of N on M is triv ial by assumption. Thus, by Proposition 11.5, we obtain an exact se quence 0 ---+ H1 (Gs , M) ---+ H1 (Q, M) ---+ H1 (N, M) = Hom(N, M) . Since N is the closed normal subgroup generated by the image of the inertia groups Ip for prime numbers p not in S, Hom(N, M) ---+ D IJp ll' S Hom( Ip , M) is injective, which proves the assertion. COROLLARY 11.23. Assume furthermore that the order of M is H 1 (Gs, M) = Ker H 1 (Q, M) ---+
invertible outside S. If S' :) S is a finite set of prime numbers, we have
(
)
H 1 (Gs, M) = Ker H 1 (Gs1 , M) -+ ffi H1 (Qp , M)/H} (Qp, M) . p ES'- S PROOF. It follows immediately from Lemma 11 .22 and the defiD nition of Hj (Qp , M) . EXAMPLE 11.24. Let n :2:: 1 be an integer, and let S be a finite set of prime numbers containing all the prime divisors of n. Let Zs = Z[ � , p E SJ . Then, the group H1 (Gs, µn ) is identified with the kernel of q x /(Q x r ---+ npll' S q; /(z; . (Q; ) n ) by Proposition 11.11 ( 1 ) and Example 11.15 ( 1 ) . From this we obtain a natural isomorphism ( 11.11 )
1 56
11 . SELMER GROUPS The natural mapping H2 (Gs, µn ) ---+ H2 (Q, µn ) induces an iso morphism (11.12) H2 (Gs, µn ) ---+ Ker H2 (Q, µn ) ---+ E£1 H2 (Qp, µn ) . p
(
0 --+ H 2 (Gs, µn ) --+
)
EB Z/Z EB gcd Cn , 2) Z/Z --+ � Z/Z --+ 0.
p ES �
PROPOSITION 11 .25. Let n 2: 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 n is invertible and M is unramified outside S. (1) For an integer q 2: 0, Hq (Gs, M) is a finite abelian group, and if q > 2 and n is odd, we have Hq (Gs, M) = 0. (2) If n is odd, we have the exact sequence ( 1 1 . 14) 0 --+ H0 (Gs , M) --+ EB H0 (Qp, M) --+ H2 (Gs, Mv (l))v p ES
--+ H1 (Gs, M) --+ EB H1 (Qp, M) --+ H1 (Gs, Mv (l))v
p ES 2 H --+ (Gs, M) --+ EB H2 (Qp, M) --+ H0 (Gs, Mv (l))v --+ 0. p ES q Here, the mapping H (Qp, M) ---+ H2 - q (Gs, Mv (l))v is the compo sition of the natural isomorphism H q (Qp , M) ---+ H 2 - q (Qp, Mv (l))v ( 1 1 . 10) and the dual of the restriction mapping H 2 - q ( Gs , Mv (1)) ---+ H2 - q (Qp, MV (1)) . 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 1 1 .26. Let the notation be as in Example 11 .24. By the isomorphism in Lemma 1 1 .4(1), we identify H1 (G 8 , Z/nz)v with
1 57 G8j;b / ( G8j;b ) n . The maximal abelian quotient G8j;b of Gs is equal to the Galois group Gal(Q( (pn ; p E s, n 2'. 1)/Q) = TipES z; . 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 Coker Z � /(Z � t ---+ EJ1 Q; /(Q; ) n __,, G8j;b /(G8j;b ) n . 1 1.3. SELMER GROUPS
(
)
pES
The third line of (11.14) is the bottom line of (11.13) . About the order of the cohomology group ltH q (Gs, M) , the fol lowing 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
lt H1 (Gs, M) ltM . ltH 0 (Gs, M) ltH2 (Gs, M) ltAf 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 ltH 2 (Gs, µn ) = n# S - l · gcd(n, 2) . By the isomorphism (11.11), we obtain an exact sequence Z 8 ---+ H 1 ( Gs, µn ) ---+ 0. 0 ---+ Ho ( Gs, µn ) ---+ Z s nth s Since Z� is isomorphic to z EB Z/2Z, we have n S 2 0 · gcd(n, 2) · ) nU gcd(n, 2) H (Gs, µ ) (Gs, µ H 1 lt n n lt This equals ltµn / ltµ�R . ·
x
power
x
1 1 .3. Selmer groups
DEFINITION 11.29. Let n 2'. 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)pES of subgroups Lp C H1 (Qp, M) is called a local condition.
(2) Let L = (Lp )pES be a local condition. The Selmer group SelL (M) of M with respect to L is defined as the inverse image of ffipES Lp by the restriction mapping H1 (Gs, M) ---+ ffipES H1 ( Qp, M) .
11 . SELMER GROUPS 1 58 (3) For a local condition L = (Lp)p ES , the dual local condition Lv = (L�)p ES is defined as the family of L� C H1 (Qp, Mv (l)) for p E S, where L� is the annihilator of Lp with respect to the bilinear mapping H1 (Qp, M) x H1 (Qp, Mv (l)) -t �Z/Z in Proposition 11.18(2) . By Proposition 11.25(1), the Selmer group SelL (M) is a finite group. By defnition, SelL (M) = Ker H 1 (Gs, M) -t pffi (H 1 (Qp, M)/Lp) . ES By Lemma 11.22, we have SelL (M)
(
= Ker
(Hffi1 (Q,(HM)1 (Q , M)/Lp) -t
p
)
E9
TI
(H 1 (Qp, M)/H} (Qp, M))
)
.
p ES p �S For a finite set S' :J S, define a local condition L' = (L�) p ES ' by L� = Lp for p E S and L� = H} (Qp,l\11) for p E 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 ) of µn by defining the local condition Lp c H1 (Qp, µn ) = Q; /(Q; ) x n for p E S to be the unramified part Hj (Qp, µn ) = z; /(z; ) x n . By the isomorphism (11.11), we obtain Sel(µn ) = z x /(z x ) n = {±1}/{(±l) 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 di visors of n. Define the local condition Lp C H1 (Qp, E[n]) for p E S as the image of the natural injection E(Qp )/nE(Qp) -t H1 (Qp , E[n]). The Selmer group SelL (E[n]) defined by the local condition L = (E(Qp)/nE(Qp)) p ES 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) -t 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.
159
11 . 3 . SELMER GROUPS
Let E be the elliptic curve y2 = x 3 - x over Q. E has good reduc tion at p =f 2. If we let S = {2} , then Gs-module E [2] is isomorphic to (Z/2Z)$2 and H 1 (G 8 , E[2]) is isomorphic to (Z[!J x /(Z[!J x2))$2. Sel ( E, 2) is isomorphic to (zx /(ZX2))$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 E S be a local condition, and let L' = (L� )p E S be a family of subgroups L � c Lp · Then, we have the exact sequence 0 --+
Selu (M)
--+ Seluv (Mv (l))v --+ SelLv (Mv (l))v --+
0.
PROOF. The first line is exact by the definition of Selmer groups. Similarly, we have an exact sequence 0 ---+ SelLv (M v ( l ) ) ---+ Seluv (M v (l)) ---+ ffi L�v /L� . ES p
By the definition of dual local condition, L�v / L : is the dual of Lp/ L � , we obtain the exact sequence ffip E S L P /L� ---+ SelLv (Mv (l))v ---+ Seluv (Mv (l))v ---+ 0 by taking the dual. We show the exactness at ffip E S Lp/ L � . Define
( (
)
= Im H 1 ( Gs, M) -+ ffi H 1 (Qp, M) , pE S H 1 (Qp , M v (l)) . B = Im H 1 (Gs , M v ( l ) ) ---+ ffi pE S A
)
By Proposition 11.25(2) , A is the annihilator of B with respect to the bilinear mapping E9P E8 H 1 (Qp, M) x E9P E8 H 1 (Qp, Mv (l)) ---+ *Z/Z. The image of SelL (M) ---+ ffip E S Lp/ L � is the image of A n ffip E S Lp. Similarly, the image of S el uv (Mv (1)) ---+ ffip E S L �v / L : is the image of B n ffip E S L�v . Thus, the kernel of ffip E S Lp/ L � ---+ SelL'v (Mv (l))v is the annihilator of the image of B n ffip E S L �v with respect to the bilinear mapping ffip E S Lp/ L � x ffi p E S L �v / L : ---+ *Z/Z. Thus, it suffices to show the image of A n ffip E S Lp is the annihilator of B n ffip ES L�v with respect to the bilinear mapping ffip E S Lp/ L � X ffip E S L �v / L : ---+ *Z/Z.
1 60
11 . SELMER GROUPS The image of A n EBp e s Lp is the image of p ES
p ES
p ES
p ES
Since EBp e S Lp n (A + ffip e s L�) is the annihilator of EBp e s L� + (B n EBp e S L�v ) , the image of A n EBp ES Lp is the annihilator of B n
ffip e S L�v . This shows the exactness at ffip ES Lp/ L�.
0
PROPOSITION 1 1 .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 e s be a local condition, and let L' = (L�)p ES be a family of subgroups L� c Lp · Then, we have
PROOF. SelL (M) is the kernel of the composition of the map ping in the second line of ( 1 1 . 14) H 1 (Gs, M) -+ ffip e s H1 (Qp, M) and the surjection EBp ES H1 (Qp, M) -+ ffip e s H 1 (Qp, M)/ Lp by the definition of Selmer group. Moreover, by the definitions of the dual lo cal condition and Selmer group, the dual SelLv (Mv (l))v is identified with the cokernel of the composition of the inclusion ffip ES Lp -+ ffip e s H1 (Qp, M) and the mapping of the second line of ( 1 1 . 14 ) ffip e s H1 (Qp, M) -+ H1 (Gs, Mv (l))v . Thus, we obtain the exact sequence by Proposition 1 1 .25 ( 2 ) SelL (M)
0 --+
H1 (Gs , M)
--+
--+
H 2 (Gs , M)
--+
From this we obtain
E9 H1 ( Qp , M)/Lp pES E9 H 2 ( Qp , M) pES
--+
SelLv (Mv (l))v
--+
H 0 (Gs , Mv ( l))v
--+
O.
161 11 .4. SELMER GROUPS AND DEFORMATION RINGS By Proposition 11.27, the first factor of the right-hand side equals u J�7;a 5 u l{'tR . By Proposition 11.20, the contribution of each p E S is UM U·LU (pM ® Z p ) The equality in question follows immediately from 6Q
�-
P
•
D
1 1 .4. Selmer groups and deformation rings
In §5.2, we defined the deformation ring R�;. In this section we relate deformation rings and Selmer groups, and we reduce Theo rems 5.32(1) and 5.34 to properties concerning Selmer groups, Theo rem 11.37 and Proposition 11.38. As in §5.2, let i be an odd prime number, let F be a finite ex tension of Ft, and let p : GQ ---+ GL 2 (F) be a modular semistable irreducible mod £-representation. Let K be a finite extension of Qi whose residue field is F, and let f E
=
=
11 . SELMER GROUPS 1 62 Let W� c Wn be {! E End0 (Vn ) lf(V,?) = 0, f (Vn ) c V,?}. W� is also a free 0/7rn 0-module. Define H! (Qe, Wn ) by H; (Qe, Wn ) = Ker(H 1 (Qe, Wn ) ---+ H 1 ( Ie , Wn /W�) ) . DEFINITION 1 1 .34. Define the Selmer group Selr; (Wn ) C H 1 ( GsE , Wn ) by the local condition L r; = (Lr: ,p)p E SE defined by H} (Qp, Wn ) if p f .e , p E Sp , H1 (Qp, Wn ) if p f .e, p E E, L E,p Hfl (Qe, W:n ) if p = .e � Sp u E, H! (Qe, Wn ) if p = .e E Sp U E. Define Selr; (W00) = �n Selr; (Wn ) · We translate the local condition in terms of infinitesimal defor mations. LEMMA 1 1 .35 . Through the bijection ( 1 1 .2) , identify Z1 (Qp, Wn ) with Lift� ; .,,. n o-G Q p (Vn ) · Let c GQ P ---+ Wn be a 1 -cocycle, and let M be the corresponding infinitesimal lifting. (1) Suppose_!. f .e, p E Sp . Then, [c] belongs to H} (Qp, Wn ) if and only if M is ordinary. _
{
:
(2) Suppose p = .e E Sp U E . In this case, p is ordinary at .e. Then, [c] belongs to H! (Qp, Wn ) if and only if M is ordinary. PROOF. (1) Suppose p f .e, p E Sp . If a E Ip is a lifting of the generator of Ip / I� [Ip , Ip ] � Z/.ez, then p(a) f 1 . If we take a suitable
basis of Vn , the matrix representation of the action of a is given by ( fi i ) . Since the image of Ip ---+ Auto ;.,,. n o M is generated by a, the fact that M is ordinary means that the matrix representation of a is given by ( fi i ) if we choose a suitable basis of M. This implies that the infinitesimal lifting M is isomorphic to Vn ®o ; .,,. n o 0/7rn 0[c] as 0 /7rn 0[c]-Ip-modules, which is in turn equivalent to the condition that the restriction of [c] to Ip is 0. (2) Suppose p = .e E Sp U E. In this case, as we remarked above, the restriction of p to Ip is ordinary by Proposition 7. 1 1 . The part V,? on which the inertia group Ip acts as the cyclotomic character is a free 0/ 7r n-module of rank 1 . If cl 1p E Z1 (Ip, W�) , then Ip acts on V,? ® 0 ; .,,. n o 0/1fn 0[c] as the cyclotomic character, and it acts trivially
11 .4.
SELMER GROUPS AND DEFORMATION RINGS
1 63
on (Vn /V�) ®o / rr "' O O/rrn O[c] . Thus, M is ordinary. Conversely, suppose M is ordinary. Then, Ip acts on the submodule M0 of rank 1 of M as the cyclotomic character, and it acts trivially on the quotient M/M0 . Thus if we choose a suitable basis of M, we have c l 1P E 0 Z 1 (Ip, W�) We give a relation between deformation rings and Selmer groups. Let RE be the deformation ring defined in §5.2, and 1l"E : RE -t 0 the homomorphism defined by p . PROPOSITION 11.36. Let mRE be the maximal ideal of the defor mation ring RE . Then, there is a natural isomorphism of F -linear spaces
(11.15) Let p RE be the kernel of the ring homomorphism RE -t 0 defined by p : GQ -t GL 2 (0) . Then, there is a natural isomorphism of 0modules
(11.16) PROOF. For an integer n 2:: 1, define a natural injection
(11. 17) Homo /( 7r "') ( P RE /(p�E ' rr n ) , 0 /(rrn )) ---7 H 1 (GsE , Wn ) First, we identify Homo /( 7r" ) ( P RE /( p�E , rrn ) , 0 /(rrn )) with a subset of Def,o,vE (O/(rrn ) [c]). Let pn : O/(rrn ) [c] -t O/(rrn ) be the natural sur jection. If f : RE -t 0 / ( rr n ) [c] is a morphism of 0-algebras satisfying Pn f = 1l"E mod rr n , then the restriction of f to p RE = Ker( 1l"E : RE -t 0) induces an O/(rrn )-linear mapping P RE /( p�E ' rrn ) -t O/(rrn ) € . By this correspondence, Homo /( 7r "') ( P RE /( p�E ' rrn ) , 0 /(rrn )) is identified with the set {! : RE -t O/(rrn ) [c] I f is a morphism of 0-algebras with Pn f = 1l"E mod rr n } This set is identified with the inverse image Def,o ,vE (O/(rrn ) [c] ) [ v,. ] of the class of Vn by Def,o ,vE (O/(rr n ) [c] ) -t Def,o,vE (O/(rr n )) by the definition of the deformation ring RE . Define Lift,o ,vE (O/(rr n ) [c] )v,. to be the inverse image of Vn by the mapping Lift,o ,vE (O/(rr n ) [c] ) -t Lift,o ,vE (O/(rrn )). Since an element of Lift,o ,vE (O/(rrn ) [c])v,. defines an infinitesimal lifting that preserves the determinant of Vn , we obtain a natural injection Lift,o ,VE ( 0 /( rrn ) [c] )v,. ---7 Lift� /( 7r "') - G s E (Vn ) (11.18 ) o
o
1 64 11 . SELMER GROUPS By Proposition 11.8(2) , we identify H 1 (GsE , Wn ) with the set of infin itesimal deformations preserving the determinant Def� /( 11"' ) - G s E (Vn ) · We show that (11. 18) induces an injection (11.19 ) The set Def� /( 1l " ) - G sE (Vn ) is the quotient of Lift � /( 11" ) - Gs E (Vn ) by the group 1 + e- End(Vn ) = Ker(GL2 (0/(1I"n ) [e]) --* GL2 (0/(1I"n )) by definition. We show Defp ,'.DE ( 0 / ( 'll"n ) [e-]) [Vn ] is also the quotient of Liftp , '.DE (0/(1I"n ) [e] )v" by l+e- End(Vn ) · For p E Liftp , '.DE (0/(1I"n ) [e])v" and P E U1GL2 (0/(1I"n ) [e-]) = Ker(GL2 (0/(1I"n ) [e-]) --* GL2 (F)), we first show P E ( 1 + mo /( 11" )[e:J ) · ( 1 + e- End(Vn ) ) if ad(P) (p) E Liftp , '.DE (0/(1I" n ) [e-])vn · In this case, the image P = P mod e E U1GL 2 (0/(1I"n )) = Ker(GL 2 (0/(1I"n )) --* GL 2 (F)) satisfies the re lation ad(P) (PE mod 'll"n ) = PE mod 'll"n . By the first part of the proof of Proposition 7. 15, M2 (0/(1I"n )) is generated by the image of PE mod 'll"n over 0/(1I"n ) . Thus, we have P E 1 + mo /( 11" ) • and P E (1 + mo /( 11" )[e:J ) · ( 1 + e End(Vn )). Therefore, Defp,'.DE (0 /(1I"n ) [e-]) [ Vn] is the quotient of Liftp,'.DE (0/(1I"n ) [e-])v" by (1 + mo /( 11" )[e:J ) · (l+e End(Vn )). Since the action of the scalar matrices l+mo ; ( 11" )[e:] is trivial, Defp,'.DE( 0/(1I"n )) [e-]) [ Vn ] is the quotient of Liftp,'.DE( 0/(1I"n ) [e-])v" by l+e End(Vn ) · Hence, Liftp,'.DE (0 /(1I"n ) [e-])v" --* Lift� /( 11" ) - G sE (Vn ) induces an injection Defp,'.DE (0/(1I"n ) [e]) [ Vn] --* Def� /( 11" ) - G s E (Vn ) · By the natural isomorphism H 1 (GsE , Wn ) --* Def� /( 11" ) - Gs E (Vn ) ( 11.4 ) , the injection ( 11.19 ) defines the injection ( 11.17 ) . We show the image of the injection ( 11.17) is SelE (Wn ) · It suffices to verify that the ramification condition that a lifting is of type VE corresponds to the local condition that defines the Selmer group for each prime number p E SE . First, suppose p =fa £. If p E Sp, the ramification condition is that the lifting is ordinary. Thus, by Lemma 11.35(1), this corresponds to the local condition H} (Qp, Wn ) · If p E E, there is no ramification condition, which corresponds to the local condition H1 (Qp, Wn ) · Suppose p = £. If £ fj: SpUE, the ramification condition is that the lifting is good, which corresponds to the local condition H} (Qp, Wn ) · If £ E Sp U E, 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 Hi (Qp, Wn ) ·
11 .5. PROOF OF PROPOSITION 11 .38
165
Therefore, the injection (11.17) defines a natural isomorphism Hom o/( 7rn) (PRE / (p �E ' 7r n ) , 0/(7r n )) --* SelE (Wn ) · Letting n = 1, we obtain the isomorphism (11.15). Taking �n ' we obtain the isomor phism (11.16) . 0 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 Sel0 (W) . For any integer n � 1, there exists a set Q consisting of r prime numbers Q 1 , . . . , Qr satisfying the condition
(5.19) n such that
(11.20)
dimF SelQ (W) = r. E - {p} . PROPOSITION 11.38. Suppose p E E, and let E' Then, we have (p + 1) 2 - ap( / ) 2 f:. 0, and (11.21) length0 SelE (Woo )/ SelE' (W00) :$ ord o (p - l ) ( (p + 1) 2 - ap(/) 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 1 1 .38
In this section we prove Proposition 11.38 by calculating the or der of local conditions H} (Qe, Wn ) and Hi (Qe, Wn ) · The proof of Theorem 11.37, which uses Proposition 11.39 below, will be post poned until the next section. We keep the notation in the previ ous section. Let Mn be the filtered cp-0-module D(Vn ) , and define End � ( Mn ) = Ker(Tr : Homo (Mn , Mn ) --* 0/(7r n )) and its sub group End� (Mn ) = E nd� ( Mn ) n Homo ( Mn , Mn --* 0/(7rn ))'. Mi is also written M. PROPOSITION 11.39. (1) If p f:. i, then d1. m H11 (Qp, W) = d1. m H0 (Qp, W) . '
-
(2) If i (j. Sp , then
-
1 66 11 . SELMER GROUPS (3) If i E Sp, then dim H51 (Q1 , W) = dim H0 (Q1 , 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 (Qt, W) -+ End� (D(W))' -+ End� (D(W)) -+ H 1 (Q1., W) -+ O by Proposition C.9. It now follows from dimF End� (D(W)) = 3 and End� (D(W))' = 2. ( 3) First, we show (11.22) The inclusion :::> is clear. We show c. Suppose a class M of in finitesimal liftings that preserve the determinant of V is contained in H1 (Q1., W). By Lemma 11.35(2) , M is ordinary. If x is the cyclotomic character, there exists an unramified characters (3 Gqe -+ F[i:JX such that M is the extension of by f3x. Since V = M ©F [•] F is not good by assumption, we have = (3 by Corollary 11.17. Since M is an infinitesimal lifting preserving the determinant, we have a. 2 = 1 and the image of is contained in {±1} c Fx . Thus, by choos ing a basis suitably, we may assume that the restriction cl aQ e of the 1-cocycle c that gives M satisfies c E Z 1 (Gqe , W° ) . Thus, we have [M] E Ker(H1 (Qt, W) -+ H1 (Q1., W/W )), and HI (Qt, W) = Ker(H1 (Qt, W) -+ H1 (Q1., W /W )) is proved. From the exact sequence 0 -+ H0 (Qt, W ) -+ H0 (Qt, W) -+ H0 (Q1., W /W ) -+ H1 (Qt, W° ) -+ H1 (Qt, W) -+ H1 (Qt, W/W° ) and (11.22) , we obtain (11.23) dim H51 (Q1., W) - dim H0 (Q1., W) 1 = dim H (Qt, W ) - dim H0 (Q1., W ) - dim H0 (Qt, W /W ) . By Propositions 11.20, and 11.18, the right-hand side of (11.23) is equal to dim W° + dim H0 (Q1., (W0 )v (l)) - dim H0 (Q1., W /W ) . a
a
a
a,
:
11 .5.
PROOF OF PROPOSITION
11 . 38
1 67
We have dim W° = dim H0 (Qe, W /W° ) = 1. Since the action of Gq t on (W° )v (l) is trivial, dim H0 (Qe, (W° )v (l)) is also 1. Hence, the 0 right-hand side of (11.23) equals 1. PROPOSITION 11.40. (1) Suppose p =f. f, and p i. Sp . Jf det(l - p
1 68
11 .
SELMER GROUPS
we obtain length0 H; (Qe, Wn ) - length0 H0 (Qe, Wn ) ::; length0 H 1 (Qe, W�) - length0 H0 (Qe, W�) - length0 H0 (Qe, Wn /W�) + length0 H 1 (Fe, (Wn /W�) 1t ) . The right-hand side equals length0 W� + length0 H0 (Qe, (W� )v (l)) by Lemma 11.12(1) and Proposition 11.20. We have length0 W� = n, and elementary linear algebra on CJ-modules shows the inequality length0 H 0 (Qe, (W�)v (l)) ::; ordo det(l - cpp : (W 0 )v (l)). Thus, the 0 inequality (11.25) is proved. PROOF OF PROPOSITION 11.38. We show the inequality (11.21) . First, consider the case p =f. f. From the exact sequence 0 -+ Selr;1 (W00) -+ Selr; (W00) -+ H 1 (Qp, W00)/H} (Qp, W00) , we obtain length0 Selr; (W00)/ Selr;1 (W00) ::; length0 H 1 (Qp, W00)/ Hj (Qp, Woo ) · Thus, by Proposition 11.40(1), it suffices to show det(l - pcpp : W) = (1 - p) ((p + 1) 2 - ap(/) 2 ) =f. 0. (11.26) Write 1 - ap(f)t + pt2 = (1 - at) (l - 13t) , and we have (11.27) (p + 1) 2 - ap(/) 2 = (1 + al3 ) 2 - (a + 13) 2 = (1 - a 2 ) (1 - 132 ) . By Theorem 9.13, the eigenvalues of p (cpp ) are a and 13 . Since a 2 , 13 2 =f. 1 by Theorem 9.21, the right-hand side is not 0. Since we have det(l - pcpp : W) = (1 - p) (l - pa/13) (1 - pl3 /a) = (1 - p) (l - a 2 ) ( 1 - 13 2 ) , the equality in (11.26) is also proved. This completes the proof in the case p =f. f. Suppose p = f E :E. As in the case p =f. f, we have length0 Selr; (W00)/ Selr;1 (W00) ::; lengtho H; (Qp, Woo )/ H} (Qp, Woo ) , and (1 - p) ((p + 1) 2 - ap(/) 2 ) =f. 0 . Thus, by Proposition 11.40(2), it suffices to show (11.28) ordo det(l - cpp : (W0 ) v (1)) = ordo ( l -p) ((p+ 1) 2 - ap( J ) 2 ) .
1 1 .6. PROOF OF THEOREM 1 1.37
169
p is ordinary at p = f, we can write 1 - ap (f)t + pt 2 = ( 1 - at ) (l - p / a t) , where a is a p-adic unit by Corollary 9.20(2) . Since both 1 - p and 1 - (p / a ) 2 are p-adic units, the right-hand side of (11.28) equals ordo (l - a 2 ) by (11.27) . If we denote also by a the
Since
·
unramified character of GQ P defined by the property that the image of 'Pp is a, the restriction of V to GQ p is an extension of a by a- 1 ( 1 ) by Corollary 9.20(2). Thus, we have w0 = Hom( a, a- 1 ( 1 ) ) . Hence, det(l - 'Pp : (W0 )v (l)) = 1 - a2 . D 1 1 .6. Proof of Theorem 1 1 .37
We first give a summary of group theoretic facts about the abso lutely irreducible mod £-representation p : G -+ GL 2 (Ft) that will be needed to prove Theorem 11.37. PROPOSITION 1 1 .41 . Let f be an odd prime number, and let G C GL 2 (Ft) be a finite subgroup. Suppose V = F t2 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 x : H -+ F; such that V Ind� x- Let 8 : G -+ G/H -+ { ± 1 } be the character of order 2, and let x' be the conjugate of x by g E G - H. Then, we have x' =I x, and W 8 ® Ind� (x' /x) . PROOF. It suffices to show (ii) , assuming W is absolutely re ducible. If W is absolutely reducible, W possesses a one- or two dimensional 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 sub space of V and V is absolutely irreducible, f defines an isomorphism V ® T -+ V. Let 8 : G -+ F; be the character that the action of G on T defines. By the isomorphism V@T -+ V, we have det V - 82 = det V, and thus the order of 8 is at most 2. If 8 = 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 8 is exactly 2. Let H = Ker 8. Again by Schur ' s lemma, V is reducible as a representation of H. Thus, there exists a character x : H -+ F; �
�
1 70
11 .
SELMER GROUPS
such that V � Ind� X · Since V is irreducible, we have x' =f. X · The D assertion on W follows immediately from this. COROLLARY 11.42. Let the notation be as in Proposition 11.41 . Then, the fallowing hold.
(1) w 0 = 0. (2) If 8 : G --+ F; is a character, then either ( i ) or ( ii ) below holds. ( i ) (W © 8 ) 0 = 0. ( ii ) Let H = Ker 8. Then, we have [ G HJ = 2, and there exists a character x : H --+ F; such that V � Ind� X · PROOF. (1) Clear from Schur ' s lemma. (2) Suppose (W © 8 ) 0 =f. 0. Then, there exists a one-dimensional G-stable subspace T of W such that the action of G on T is given by 8 - 1 . As in the proof of Proposition 11.41, we have 8 - 1 = 8, and the assertion follows from this. D :
COROLLARY 11.43. Let the notation be as in Proposition 11.41 . If T =f. 0 is a G-stable subspace of W, then there exists a g E G C GL2 (Ft) that has mutually distinct eigenvalues and that satisfies T + (g l ) 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 c GL 2 (Ft) that has distinct eigenvalues. Assuming such a g does not exist, we derive a contradiction. Since V is abso lutely irreducible, the center of G is Z = { g E G I g is a scalar matrix } . By the assumption, the order of any element of G IZ is either £ or 1. If H is the £-Sylow subgroup of G, we have G = Z x 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 = 8 or T = Ind� ( x' I x ) when we decompose W = 8 © Ind� ( x' I x ) . If T = 8, we may take a g in H that is not a scalar matrix. If T = Ind� ( x' I x ) , we may take D a g E G - H. We give a sufficient condition for H 1 ( G, W) = 0.
1 1 .6. PROOF OF THEOREM 1 1 .37
1 71
LEMMA 1 1 .44. Let f be an odd prime number, and let F be a finite extension of Fe. If V = F 2 and W = End0 (V) , then we have
H 1 (SL 2 (F) , W) = 0 except in the case F = F 5 . OUTLINE OF PROOF. Let V0 = F C V be a one-dimensional sub space, and define subgroups B and U of SL2 (F) by B = {g E SL 2 (F) I g(Vo ) C Vo } I> U = {g E B I g i ve = l } . The ac tion on Vo defines an isomorphism B/U -+ y x . Explicitly, we have B = ( ( � a� 1 ) , ( 6 Y ) l a E F x , u E F) and U = { ( li Y ) l u E F} , and the isomorphism B /U -+ y x is given by ( � a� 1 ) H- a. Since the indices [SL 2 (F) : BJ and [B : U] are relatively prime to f, the re striction H 1 (SL 2 (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 = {! E W l f(Vo) C Vo } :::> Wo = {! E W l f(Vo ) = O}. Since the actions of U on W/Wi , Wi fWo , and Wo are trivial, H 1 (U, W/W1 ) , H 1 (U, Wi fWo ) , and H 1 (U, Wo) are identified with Hom(U, W/W1 ) , Hom(U, Wi /Wo ) , and Hom(U, Wo ) . Moreover, the invariant parts H 1 (U, W/W1 ) 8fu , H 1 (U, Wi fW0 ) B f U , and H 1 (U, W0 )8fu are naturally identified with Hom8 ; u (U, W/W1 ) , HomB ; u ( U, Wi /Wo ) , and Hom B ; (U, Wo ) , respectively. We show u the following (i) If lt F =/:. 3, 5, 9, then HomB ; u (U, W/W1 ) = 0. (ii) If lt F =/:. 3, then HomB ; u (U, Wi fWo ) = 0. (iii) HomB ; u (U, Wo ) = HomF (U, Wo ) . The isomorphisms of F-vector spaces W/W1 -+ HomF (Vo , V/Vo ) , Wi /Wo -+ HomF ( Vo , Vo ) , Wo -+ HomF (V/Vo , Vo ) are isomorphisms of B/U-modules. Moreover, associating to g E U the mapping in duced by g - 1 , we obtain an isomorphism of B/U-modules U -+ HomF (V/Vo , Vo ) . Thus, the actions of B/U = y x on W/W1 , Wi /Wo , Wo , U are inverse square, trivial, square, square. Suppose the order of F is ff . We show (i) by contradiction. Suppose Hom8 ; u (U, W/W1 ) =/:. 0. There exists a conjugate of the inverse square character of y x that is equal to the square character. Thus, there exists an integer 0 :::; d < f satisfying -2fd 2 mod (f f - 1 ) . Since (f f - 1 ) I 2(fd + l ) , we have ff - 1 :::; 2(fd + l ) , and thus (ff - d _2)fd :::; 3. From this we have either ff -d = 3 and fd = 1 , 3, or ff - d = 5 and fd = 1 . Hence, ltF has to be one of 3, 5, 9, and (i) is proved. If lt F =/:. 3, the square mapping is not trivial. (ii) is clear from =
1 72
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SELMER GROUPS
this. Since F = Fl [Fx 2 ] , we have Hom8 ; u(U, Wo) = HomF (U, Wo) , which shows (iii) . Suppose U F =/= 3, 5, 9, and we show H 1 (U, W) B / U that contains H 1 (SL 2 (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 H0 (U, W) ----t H0 (U, W/W0 ) ----t H 1 (U, W0 ) ----t H 1 (U, W) . The first morphism H0 (U, W) = W0 � H0 (U, W/W0 ) is 0. There fore, the second morphism induces an isomorphism H0 (U, W/Wo) = Wi/Wo � HomF (U, Wo) = H 1 (U, Wo) B / U_ Thus, the morphism H 1 (U, W0 ) B / U � H 1 (U, W) B / U is 0. Hence, if U F =/= 3, 5, 9, we have H 1 (SL 2 (F) , W) c H 1 (U, W) B / U = 0. Suppose U F = 9. Similarly to the case U F =/= 3, 5, 9, we see that H 1 (U, W) B / U � H 1 (U, W/W1 ) B / U = Hom8 ; 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 IU is nonzero. On the extension corresponding to this element of the trivial representation F by W, the actions of ( � a� 1 ) (a E F x ) and ( fi Y ) ( u E F) are given by a2 O O 0 1 u u; ( ) 0 1 0 0 0 1 u d (u) 0 0 a- 2 0 ' 0 0 1 u3 0 0 0 1 0 0 0 1 for some functions d : F � F. There exists no action satisfying such conditions, and thus H 1 (SL2 (F) , W) c H 1 (U, W) B / U = O in the case U F = 9, also. Consider the case U F = 3. It suffices to show H 1 (U, W) = 0. Since U Z/3Z, take its generator a and we have H 1 (U, W) = Ker(l + a + a 2 : W)/ Im(a - 1 : W) . Since W is isomorphic to 0 F[a]/(a - 1) 3 as an F[a]-module, we have H 1 (U, W) = 0.
) (
(
c u
)
c,
�
QUESTION. Fill in the details of the proof of Lemma 11.44. PROPOSITION 11.45. Let f, be an odd prime number, and let G be a finite subgroup of GL2 (F£ ) such that V = F� is an irreducible representation of G. Let 8 : G � F lx be a character, and let W = End0 (V) . Let Z = G n F; be the subgroup of all the scalar matrices
11 .6.
PROOF OF THEOREM
11.3 7
1 73
contained in G, and let G = G/Z be the image of G in PGL 2 (Fe) = GL2 (Fe)/F; . Then, either (i) , or (ii) holds. (i) H 1 (G, W ® 8) = 0. (ii) We have f = 3 or 5 . If f = 3, then G is isomorphic to the alternating group 21s , and 8 = l . If f = 5, then G is conju gate to either PSL 2 (Fs) or PGL 2 (Fs), and 8 equals either 1 or the composition ( d;t ) = ( g ) o det : G -+ G -+ PGL 2 (F 5 ) -+ F� /(F� ) 2 -+ {±1 } . We deduce Proposition 1 1 .45 from Theorem 10.28. We first show that for the condition (ii) in Theorem 10.28, the following holds. LEMMA 1 1 .46. Let f be a prime number, and let G be a finite sub group of PGL 2 (Fe) . If G is isomorphic to PGL 2 (Fe), it is conjugate to PGL 2 (Fe), and if G is isomorphic to PSL 2 (Fe) , it is conjugate to
PSL 2 (Fe) .
PROOF. Consider the natural action of PGL 2 (Fe) on P 1 (Fe). Take an element g E G of order £. The action of g on P 1 (Fe) has only one fixed point, and the orders of all other orbits are all equal to £. Let X c P 1 (Fe) be a G-orbit containing the g fixed point. The order d of X divides (ltG/£) 1 (£2 - 1), and d 1 mod £. Since (£2 - 1)/d - 1 mod f, we have (£2 - 1)/d � £ - 1 , and thus d ::; f + 1 . Hence, we have d = 1 or f + 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 f, which is a contradiction. This shows the order of X is f + 1. Choose the coordinates of P 1 such that the fixed point of g is the point at infinity, another point different from this is 0, and g (O) = 1 . Then, X = P 1 (Fe). Since the action of G sends 0, 1 , to points in P 1 (Fe), G is a subgroup of PGL 2 (Fe) of index less than 2. Since the abelianization of PGL2 (Fe) is of order 2, the only subgroup of index 2 is PSL2 (Fe), if f =I- 2. D PROOF OF PROPOSITION 1 1 .45. We have an exact sequence 0 -+ H1 ( G , (W ® 8) z ) -+ H 1 (G, W ® 8) -+ H 1 (Z, W ® 8) by Proposi tion 1 1 .5. Since the order of Z is relatively prime to f, we have H1 (Z, W ® 8) = 0 by Lemma 1 1 .3. Thus, the natural mapping H 1 ( G , (W ® 8) z ) -+ H 1 (G, W ® 8) is an isomorphism. Since the action of Z on W is trivial, we have (W ® 8) Z = 0 if 8l z =I- 1 . From now on, suppose 8l z = 1 , and we identify 8 with a character of G. Furthermore, we identify H 1 ( G, W ® 8) with H 1 ( G, W ® 8) . It suffices to show the assertion in each case where the conditions (i) , =
=
oo
11 .
1 74
SELMER GROUPS
(ii) or (iii) in Theorem 10.28 holds. Suppose first condition (iii) holds. In this case, the order of G is relatively prime to f. by the assumption f. =f. 2. Thus, by Lemma 1 1 .3, we have H 1 (G, W © 8) = 0. Next, suppose condition (i) holds. Suppose G is conjugate to PSL2 (F) . We may assume G = PSL 2 (F) by replacing a basis. Then, we have 8 = 1 . By Proposition 1 1 .5, H 1 (PSL 2 (F) , W) -+ H 1 (SL 2 (F) , W) is injective. Thus, H 1 ( G , W © 8) = H 1 ( G , W) = 0 by Lemma 11 .44 except for the case F = F 5 . Suppose G is conjugate to PGL 2 (F) . If H C G is a subgroup conjugate to PSL2 (F) , then H 1 ( G , W © 8) -+ H 1 (fI, W © 8) is injective by Proposition 11.5 and Lemma 11.3 as above, and thus H 1 ( G , W © 8) = 0 except for F = Fs . If F = Fs , the composition ( d�t ) : PGL 2 (F 5 )ab -+ F; /(Fi ) 2 -+ {±1} is injective, and thus 8 is either 1 or ( d�t ) . Finally, suppose condition (ii) holds. If f. > 5, then the order of G is relatively prime to f., and thus H 1 (G, W © 8) = 0 by Lemma 1 1 .3. If f. = 3, the assertion is reduced to the case where condition (i) holds by Lemma 1 1 .46, except for the case in which G is isomorphic to 2l5 . If G 2ls , we have Gab = 1 , and thus 8 = 1 . If f. = 5, except for the case G 2l5 , the order of G is relatively prime to f., and thus H 1 (G, W © 8) = 0. If G is isomorphic to 2ls , G is conjugate to D PSL2 (Fs) by Lemma 1 1 .46. �
�
COROLLARY 1 1 .47. Let the assumption be as in Proposition l l .45 . If f. =f. 3, 5 or det G =f. 1 , then we have H 1 (G, W © det) = 0. PROOF. It suffices to show that condition (ii) in Proposition 1 1 .45 does not hold assuming f. = 3, 5 and 8 = det =f. 1. Since det =f. 1, if we assume condition (ii) holds, then we have f. = 5, det = ( d�t ) and det(G) = {±1}. However, we have -1 =f. Ce/) which is contradiction. D In what follows let the notation be as in the previous section. PROOF OF THEOREM 1 1 .37. Since r = dim Sel0 (W) , the equal ity r = dim SelQ (W) is equivalent to the condition that the inclu sion Sel0 (W) -+ SelQ (W) is an isomorphism. Since W is self-dual, SelQv (W(l))v is the Selmer group defined by the dual local condition.
11 .6.
PROOF OF THEOREM
11 . 3 7
1 75
By Proposition 1 1 .32, there is an exact sequence 0 � Sel0 (W) � SelQ (W) � E9 H 1 (Qp, W)/Hj (Qp, W) pEQ � Sel0v (W(l)) v � Sel Q v (W(l)) v � 0. Thus, the equality r = dim SelQ (W) is equivalent to the condition that E9p E Q H 1 (Qp , W)/Hj (Qp, W) --+ Sel0v (W(l))v is injective. It is also equivalent to the condition that Sel0v (W(l)) --+ EB H 1 (Fp, W(l)) pEQ is surjective. We now show the following lemma. LEMMA 1 1 .48. (1) dimF Sel0v (W(l)) = dimF Sel0 (W) = r. (2) If a prime number q satisfies the condition ( 1 1 .29 ) q � S-p , q = 1 mod f. and Tr p (cp q ) ¢. ±2, then we have dimF H 1 (F q , W(l)) = 1 .
PROOF. (1) Apply Proposition 1 1 .33 to M = W . Since W is self-dual, it suffices to verify the following. (i) dimF H 1 (GFp ' W1P ) = dimF w 0 Q p if p E S-p , f. f., (ii) dimF Hj (GQ e , W) - dimF W 0Q e = 1 if f. � S-p , (iii) dimF H°1 (GQ e , W) - dimF W 0 Qe = 1 if f. E S-p , (iv) dimF W GR = 1 , (v) w a s = W(l) G s = 0 . (i) , (ii) , and (iii) are Proposition 1 1 .39(1 ) , ( 2 ) , and ( 3 ) , respec tively. (iv) follows from the fact that for a complex conjugate c, p(c) has eigenvalues 1 and - 1 with multiplicity one. We show (v) . Con dition (ii) in Corollary 1 1 .42 ( 2 ) does not hold by Lemma 9.51. Thus, (v) follows from Corollary 1 1 .42. ( 2 ) By the condition ( 10.22 ) , p(cpq) has two distinct eigenvalues. D The assertion follows immediately from this. By Lemma 1 1 .48, Sel0v (W(l)) --+ E9p E Q H 1 (Fp , W(l)) is sur jective if and only if it is injective. We show there exists a Q that makes Sel0v (W(l)) � EB H 1 (Fp, W(l)) = EB W(l)/(cpp - l)W(l) pE Q pE Q
1 76
11 . SELMER G ROUPS
injective. By Theorem 3.1, it suffices to show there exist O"i , , O"r E GQ ( (en ) such that each p(O"i ) has two distinct eigenvalues, and the direct sum of the restriction mapping .
Sel0v (W(l)) -+
ffi= W(l)/(O"i - l)W(l)
•
.
r
i l is injective. For any nonzero element of Sel0v (W ( 1)), let : GQ -+ W(l) be a 1-cocycle that represents it. Then, it suffices to show there exists O" E GQ ( (en ) such that p( O") has two distinct eigenvalues, and that ( ) E W(l) is not contained in ( - l)W(l). Let GF,, = Ker(p : G Q ( (en ) -+ GL 2 (F)). We show the restriction mapping (11.30) H 1 (GQ , W(l)) -+ H 1 (GF,. , W(l)) = Hom(Gp,. , W(l)) is injective. The kernel equals H 1 (Gal(Fn /Q) , W(l)) by Proposi tion 11.5. Thus, it suffices to show H 1 (Gal(Fn /Q) , W(l)) = 0. More over, by Proposition 11.5, we obtain an exact sequence (11.31) 0 --+ H l (Gal(Fo/Q) , W(l) G al ( F,. / Fo) ) --+ H 1 (Gal(Fn /Q) , W(l)) --+ H 1 (Gal(Fn / Fo) , W(l)) G al ( Fo/ Q ) . Since det p is the mod R. cyclotomic character, we have Q((e) c F0 . Thus, the action of Gal(Fn / Fo) on W(l) is trivial. Therefore, it suffices to show the following: H 1 (Gal(Fo/Q) , W(l)) = 0, H l (Gal(Fn / Fo), W(l)) Gal ( Fo/ Q ) = Homa I ( Fo/ Q ) (Gal(Fn /Fo), W(l)) = 0. a The determinant det p is the mod R. cyclotomic character, and is nontrivial. Since p is an absolutely irreducible faithful representation of Gal(Fo/Q) , by Corollary 11.47, we have H 1 (Gal(Fo/Q) , W(l)) = 0. We show HomaaI ( Fo/ Q ) (Gal(Fn /Fo), W(l)) = 0. Since Fn = Fo · Q((e.. ) , Gal(Fn /Fo) -+ Gal(Q((e.. )/Q) is injective, and the con jugate action of Gal(Fo/Q) on Gal(Fn / Fo) is trivial. Therefore, if f : Gal(Fn / F0 ) -+ W(l) is a morphism of Gal(Fo/Q)-modules, the image of f is contained in the invariant part W(l) GaI ( Fo/ Q ) . Hence, we have Homa aI ( Fo/ Q J (Gal(Fn /Fo) , W(l)) = 0. This completes the proof of H 1 (Gal(Fn /Q) , W(l)) = 0, and H 1 (GQ, W(l)) -+ Hom(GF,. , W(l)) is injective. c
c CT
CT
11.6. PROOF OF THEOREM 11 .37
177
Take any nonzero element of Sel0v (W ( l ) ) , and let c : GQ -+ be a 1-cocycle that represents it. The restriction of c to GP,. de fines a homomorphism c l a ,. : Gp,. -+ W ( l ) . Since H 1 (GQ , W ( l ) ) -+ Hom(Gp,. , W ( l ) ) is injective, c(Gp,. ) c W ( l ) is not 0. This is a sub space of W ( l ) stable under the action of GQ((tn ) · The restriction .Pla Q «tn l is absolutely irreducible by Corollary 9.52. Thus, by Corol lary 11.43, there exists a E GQ((tn ) such that p(a) has two distinct eigenvalues, and satisfies c(Gp.. ) + (a - l ) W = W. If c(a) fj. (a - l ) W , this a E GQ((tn ) satisfies the condition. If c(a) E (a - l ) W , take E Gp,. such that c(r) fj. (a - l ) W . Since p(a) = p(ar) and c(ar) = c(a) + ac(r) c(r) ¢. 0 mod (a - l ) W , ar E GQ((tn ) satisfies D the condition. W(l)
F
T
=
APPENDIX B Curves over discrete valuation rings
B . 1 . Curves
DEFINITION B.l. (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. I f X is a proper smooth curve over k whose geometric fiber is connected, then g = dim k H 1 (X, 0) is called the genus of X .
(2) A fl.at scheme X of finite presentation over a scheme S such that the geometric fiber X8 for each geometric point s --+ S is a curve over K(s) is called a curve over S. If X is a proper smooth curve over S such that each geo metric 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 fiat 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 --+ A1 = Spec A[T] . Thus, we may assume X = A1 . But in this case, the assertion is clear. We show the case in which N is general. Since D is a fl.at covering of S and the assertion is fl.at local on S, we may assume that D has a section P S --+ D. Since the case N = 1 is already shown, P :
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B. CURVES OVER DISCRETE VALUATION RINGS
18 0
defines a Cartier divisor of X. Thus, there exists a closed subscheme D' of D such that Iv' c Ox and Ip C Ox satisfy Iv = Iv,Ip . Since we have an exact sequence 0 ---+ 0 V' ---+ 0 v ---+ 0 s ---+ 0 of Ov-modules locally on S, D' is fl.at over S of degree N l . It follows from the induction hypothesis that D' is a Cartier divisor, and thus so is D D' + P. (ii) => (i) . The defining ideal Iv of D is an invertible Ox-module. Since Ds is a finite subscheme of Xs for any point s in S, the ideal Iv,xOx. ,x C Ox. ,x is generated by nonzero divisors for any point x in the smooth curve X5 • In other words, tensoring ®11:(s) to the exact sequence 0 ---+ Iv ---+ Ox ---+ Ov ---+ 0 for each s E S, we obtain an exact sequence 0 ---+ Iv. ---+ Ox. ---+ Ov. ---+ 0. Thus, we have Tor f ( 0v , 11:(s)) = 0, and Ov is a fl.at Os-module. (2) By Proposition A.4(1 ) , we may assume S = Spec k with k an algebraically closed field. If x E D, then the local ring Ox,x is D 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 E 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 11:(x) is a finite separable extension of k. X -
=
5
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 x is finite etale over x of degree 2 . (iii) If k i s an algebraic closure of k, the completion Ox,. ,x of the local ring at each point x E x x k k C Xk = X 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 X 1 , X 2 be the inverse images of x. We first show the case where the normalization X is decomposed into a disjoint union Vi 11 Vi = Spec B 1 x B2 ( xi E Vi ) . In this case A is the inverse image of the diagonal subring k C k x k by the surjection B 1 x B2 --+ k x k defined by x 1 and xz . Thus, if we let S E B 1 and T E B2 be prime elements at x 1 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 x k. Take an element b in B such that its image in k x k is ( 1 , 0) , and let a = b2 - b E mB = m c A. Furthermore, let g(Y) = Y 2 - Y - a E A[Y] , let A = A[Y]/ g(Y) , and let u = Spec A. Since u --+ x is fl.at, and the fiber at x is etale, we may assume U --+ X is etale by replacing X by a neighborhood of x if necessary. By Proposition A. 13(3) , U x x X is a normalization of U. Since the etale covering U xx X --+ X of degree 2 has a section defined by the homomorphism A = A[Y]/g(Y) --+ B Y f-+ b, it is isomorphic to X 11 X. Thus, the general case is reduced to the case where the normalization X is decomposed into the disjoint union V1 lJ V2 . (iii) ::::} (ii) . We prove it assuming k is a perfect field. In this case, replacing k by k, we may assume k is algebraically closed. Since the local ring Ox,x is a subring of the completion Ox , x , it is reduced. 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]] x k[[T]]. Thus, the remaining assertion D 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 r(X, 0) the finite scheme consisting of irreducible components of X. Let E be the reduced closed subscheme of X consisting of all the singular points, and � = X x x E. Let ro = z P (k ) , and let r 1 be the kernel of the surjective homomorphism u :
:
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zE ( k ) ---+ zE( k ) defined by the natural morphism E ---+ I:. Let d r 1 ---+ r0 be the homomorphism defined by the natural morphism E ---+ X. Then, we call the chain complex r = [f 1 ---+ f 2 ] 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 (f) = Z is equivalent to the condition that the geometric fiber X;o is connected. B.2. Semistable curve over a discrete valuation ring
In what follows, let 0 be a discrete valuation ring, let K be its field of fractions, and let F be its residue field. DEFINITION B.6. Let 0 be a discrete valuation ring, let K be its field of fractions, and let F be its residue field. (1) Let X be a fl.at curve over 0. If X is smooth over 0 except for a finite number of nodes in Xp, 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 0, and let x E Xp C X
be a node. We call the length of the Ox , x-module n �/ O,x • the index of x . DEFINITION B . 7. Let 0 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 0 and an iso morphism 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 0 = Z (p) i XQ has good reduction, or semistable reduction, respectively. LEMMA B . 8 . Let 0 be a discrete valuation ring, let K be its field of fractions, and let F be its residue field. A regular curve X over 0 is semistable if all of the following conditions hold: The fiber XK ---+ X at the generic point is smooth. The closed fiber Xp is reduced, and as
B. 2 . SEMISTABLE CURVE OVER A DISCRETE VALUATION RIN G 1 83 a Cartier divisor, it is the sum Xp = C1 + C2 of C1 , C2 c X, where C1 , C2 are smooth curve over F and the intersection C1 n C2 = C1 x x C2 is etale over F. PROOF. Let 7r be a prime element of 0. By Proposition A.4, X - C1 n C2 is smooth over 0. We show that x E C1 n 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 = 7r / s defines C2. Define a morphism of 0-schemes X -+ Spec O[S, T]/(ST - 7r) by s H s and T H t. We show this is etale at x . It suffices to show that the homomorphism of the completion Ao = O[[S, T]]/(ST - 7r) -+ A = Ox,x is an isomorphism. Since mA 0 / m � 0 -+ mA / m � is an isomorphism, Ao -+ A is surjective. Since both Ao and A are two-dimensional regular local rings, Ao -+ A must be an isomorphism. D LEMMA B.9. Let 0 be a discrete valuation ring, let K be its field of fractions, let F be its residue field, and let 7r be a prime element. Let X be a weakly semistable curve over 0, let x be a node of Xp, and let e be its index. If 0 is complete and F is algebraically closed, then the completion of the local ring Ox,x is isomorphic over 0 to
O[[S, T]]/(ST - 7re) . PROOF. Let A be the completion of Ox,x · We first show there exist an integer m 2:'.: 1 and an isomorphism O[[S, T]]/(ST - 7rm) -+ A. Through the isomorphism F[[S, T]]/(ST) -+ A/(7r) , we identify F[[S, T]]/(ST) = A/(7r) . One of the following (i) and (ii) holds. (i) There exist liftings s, t E A of the images of S and T, an integer m 2:'.: 1 , and u E Ax such that st = U7rm. (ii) If liftings s, t E A of the images of S and T, an integer m 2:'.: 1 , and v E A satisfy st = V7rm, then is contained in the maximal ideal (7r, s, t) of A. Suppose (i) holds. Let Ao = O [ [S, T]]/(ST - 7rm) , and define a mor phism of 0-algebras Ao -+ A by S H s' = su- 1 and T H t. Since the morphism of F-algebras Ao/(7r) -+ A/(7r) is an isomorphism, and A and Ao are both 0-flat, Ao/(7rn) -+ A/(7rn) is an isomorphism by induction on n. Taking the limit, Ao -+ A is an isomorphism. Since 0 �/ 0, x is isomorphic to � O(Ao / ( ""'' )) / ( O / (-•r" )) Ao/(S, T) = n 0/(7rm), we have e = m . v
'.:::::'.
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Next, we show that ( ii ) cannot hold. We construct a sequence of liftings of the images of S and T to A such that Sm+l Sm mod 7rm , tm+l tm mod 7r m , Smtm E ( 7rm ) inductively. We take s 1 and t 1 arbitrarily. Suppose we already have up to Sm , tm . We can write Smtm = 11"m Vm , Vm = asm + btm + C11" ( a, b, E A ) . If we define Sm+l = Sm - rrm b and tm+l = tm - rrm a, then we have m+ Sm+itm+l = rr 1 ( c + abrrm - l ) , and the required conditions hold. If we define s = limm --+ oo Sm and t = limm --+ oo 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 0-algebras O[[S, T]] / (ST) -+ A by S H s and T H t, this is an isomorphism as before. However, since XK is smooth, AK D must be regular, which is a contradiction. PROPOSITION B. 10. Let 0 be a discrete valuation ring, let K ( sm , tm ) m=l, 2 , ... ::=
=
c
be its field of fractions, and let F be its residue field. Let X and Y be normal curves over 0, and let f : X -+ Y be finite surjective morphism over 0. (1) If X is smooth over 0, then X -+ Y is flat, and Y is also smooth over 0. (2) Suppose X is semistable over 0, and let C1 , C2 be Cartier divi sors of X smooth over F satisfying the condition C1 + C2 = Xp , C1 n C2 = { nodes of Xp }, (B.1) and C1 = f - 1 (f(C1 ) ) , C2 = f - 1 (f(C2 ) ) . Then, Y is weakly semistable over 0. Let D 1 = f(C1 ) and D2 = f (C2 ) be reduced closed subschemes of Y. Then, D 1 and D2 are smooth curves over F, and we have D 1 U D2 = YF and D 1 n D2 = { nodes of Yp } . Furthermore, if x is a node of Xp , then y = f (x) is a node of Yp . 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) PROOF. (1) Let O' :::> 0 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 x o O' is normal. The morphism X -+ Y is fl.at except for a finite number of closed points of the closed fiber of Y. Thus, by Corollary A. 14, Y is smooth except
B. 2 . SEMISTABLE CURVE OVER A DISCRETE VALUATION RIN G 1 85
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 Yp is reduced, so is the closed fiber YF' . Thus, by Lemma A.41 , Yo ' is normal. Replacing 0 by O', we may assume 0 is complete and F is algebraically closed. We show X --+ Y is fl.at. Let x E Xp 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 fl.at 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 fl.at of degree d except at the maximal ideal. Choose an isomorphism O [ [t]] --+ B and identify as O [ [t]] = B. Define a morphism of 0-algebras Ao = O [ [t']] --+ A by letting t' = NB /At. Since the valuation of t' in B/ (rr) = F [ [t]] is d, B/(rr, t') = B ®Ao F equals F[[t] ] / (td) . Thus, X --+ Y is fl.at by Lemma A.43. Since X is smooth over 0, it is regular by Proposition A. 13(3) . Thus, Y is also smooth by Corollary A. 14. (2) As in ( 1 ) , we may assume 0 is complete and F is algebraically closed. Let x be a node of Xp , and let y = f (x) . Let A = 8Y,y --+ B = Ox , x · Let Ly and Lx be fields of fractions of A and B, respectively, and let d = [Lx : Ly ] · Choose an isomorphism O [ [s , t]] / (st - rr) --+ B, and identify O [[s , t]] / (st - rr) = B. Suppose the inverse image 01 of C1 by Spec B --+ X is V ( s ) , and the inverse image C2 of C2 is V (t) . Let s' = NB / As, and let t' = NB /At. Since s' and t' belong to the maximal ideal of A and satisfy s't' = NB /A1f = rrd , we obtain a homomorphism Ao = O [[s' , t'] ] / (s't' - rrd) --+ A. We show this is an isomorphism. Since s' equals 0 on C1 and we have 1 -1 (J(C1 ) ) = C1 by assump tion, s' is invertible on Spec B 01 . Thus, there exists u E B x satis fying s' = usd . Similarly, there exists v E Bx satisfying t' = vtd , and thus B ®Ao Ao /mA 0 = B/(sd, td, rr) = F [ [s , t]] / ( sd , td, st) is a finite dimensional F = Ao/mA 0-vector space. Since B and Ao are complete, B is finitely generated as an Ao-module by Nakayama ' s lemma. Since Ao is a two-dimensional integrally closed domain, Ao --+ 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 Lo is the field of fractions of A0 , then Lx is an extension of Lo of degree d. Therefore, we have Ly = Lo, and A = Ao. This ,
-
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shows Y is weakly semistable over 0, and the remaining assertions 0 have also been shown. COROLLARY B . 1 1 . Let 0 be a discrete valuation ring, and let X be a curve over 0. Suppose an action of a finite group G on X over 0 is given, and suppose the quotient Y = X/G exists. Furthermore,
if x is the generic point of an irreducible component of Xp and TJx is the generic point of the irreducible component of X containing x, then suppose the inertia groups Ix and I.,,., are the same. (1) If X is smooth over 0, then so is Y, and we have YF = Xp/G. (2) Suppose X is semistable over 0, and let C1 and C2 be smooth Cartier divisors of X over F stable under the G-action satisfying condition (B. l) . Then Y is also weakly semistable over 0, and if D 1 = f(Ci ) and D 2 = f(C2 ) are reduced closed subschemes, then w e have D 1 = Ci f G and D2 = C2 /G. Moreover, if x is a node of Xp and T/x is the generic point of an irreducible component of x, then the index ey of the node y = f (x) equals the index [Ix : I.,,.,] . PROOF. (1) By Proposition B.10(1), Y is smooth over 0 . We show Xp/G ---+ Yp 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 Xp , and let y = f (x) E YF be its image. Let Lx be the field of fractions of the completion of Ox , x , and let Ly b e the field of fractions of the completion of Oy, y . Then, Lx is a Galois extension of L y , and its inertia group is Ix / I.,,., . Thus, by assumption, Lx is an unramified extension of L y . Hence, the residue field 11: ( x' ) of the image x' in Xp/G of x is equal to 11: ( y ) , and Xp/G ---+ Yp is an isomorphism. (2) By Proposition B. 10(2) , Y is weakly semistable over 0, and by Corollary B . 1 1 ( 1 ) , the closed immersions C1 ---+ Y and C2 ---+ Y induce closed immersions Ci /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 L y be the field of fractions of the completion of OY,y . Then Lx is a Galois extension of L y , and its inertia group is Ix /I.,,,, . Thus, by Proposition B. 10(2) , we have e x = [Ix : I77,, ] . 0
B.3.
DUAL CHAIN COMPLEX OF CURVES OVER A DVR
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B.3. Dual chain complex of curves over a discrete valuation ring
Let 0 be a discrete valuation ring, and let F be its residue field. Let X be a proper curve over 0, and let r = [r1 --+ ro] be the dual chain complex of XF . If X is regular, we define a symmetric bilinear form ( ' ) o : ro x r o --+ z, and if x is semistable, we define a symmetric bilinear form ( ' ) i : rl x ri --+ z. Let X be a proper fl.at regular curve over 0. For irreducible components C1 and C2 of XF , we define the intersection product (C1 , C2)x by (Ci , C2)x = deg Ox (C1 ) b . If C1 "I- C2 , it is equal to dimF r( X , Oc1 ®ox Oc2 ) and satisfies (C1 , C2)x = (C2 , C1 ) x . Let Z1 (XF) b e the free Z-module generated by the irreducible com ponents of XF. Then, 1the intersection pairing defines a symmetric bilinear form ( ' )x : Z ( XF ) x 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 symmet ric bilinear form ( , )x is generated over Q by XF . Let F be a perfect field, and let r = [r --+ r o ] be the dual chain complex of XF . Let O' be the completion of the maximal unramified extension of the completion of 0. O' is a complete discrete valuation ring, and its residue field is an algebraic closure F of F. Then, X0' = X x o O' is a proper regular curve over O' , and we have Xo' x o' F = Xp and r o = Z1 (Xp) · Define a symmetric bilinear form ( ' )o : ro x ro
(B.3)
�
z
by the intersection pairing of Xo' . If XF is connected, then, by Lemma B. 12, the kernel of the bilinear form ( , ) o is generated by XF over Q. Define a linear mapping (B.4) ao : ro � rti = Hom(ro, Z) by ao ( [C] ) ( [C'] ) = (C, C') o , and define a linear mapping (B.5) f3 : rti � Z as the dual of the linear mapping 13v : z --+ r o defined by 13 v ( 1 ) = [Xp ]
2:: 0 ec [C] . COROLLARY B.13. Let 0 be a discrete valuation ring, and sup pose its residue field F is perfect. Let X be a proper regular curve over =
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0, and let r = [r 1 ---+ r0] be the dual chain complex of Xp . Then, the composition /3 a o : fo ---+ f6 ---+ Z is the 0-mapping. Moreover, if the geometric fiber XF is connected, Ker /3 / Im ao has finite order. PROOF. Since (X-p, )0 is the 0-mapping, we have /3 a0 = 0. By Lemma B.12, if X-p is connected, the kernel of the bilinear form ( , ) o is generated by XF over Q. Thus, Ker /3 / Im ao is a finite D abelian group. Let 0 be a discrete valuation ring, and let Xo be a proper weakly semistable curve over 0. Let r = [f 1 ---+ fo] be the dual chain complex of the closed fiber Xp . We use the notation of Definition B.5. For each x E "E(F) , let xi , x 2 be the inverse images in �(F) , and define fx = [x 1 ] - [x 2 ] . Then, fx , x E "E(F) is a basis of the free Z-module r 1 . Define a symmetric bilinear form (B.6) ( , h : r 1 x r 1 ---+ z by defining (!x , fx ) to be the index ex and letting (!x , fx ' ) 0 if x =J. x' . This does not depend on the numbering of x 1 and x 2 . Define the linear mapping (B.7) a 1 : f 1 ---+ rj' Hom(f 1 , Z) by a 1 Ux ) Ux 1 ) = Ux , fx ' h · If rv = [r6 ---+ ry] is the dual complex of r ' then a : r 1 ---+ ry induces (B.8) a 1 : H1 (f) = Ker(f 1 ---+ fo) ---+ H 1 (f v ) = Coker(f6 ---+ ri ) . If X is weakly semistable, a minimal resolution of singularities X' is constructed as follows. Let x be a node of Xp , and let e be its index. Suppose e 2:'.: 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 2:'.: 3, the exceptional curve E is a singular conic and it is smooth over x except at its unique node x 1 , and the residue field of x 1 equals that of x. X1 is weakly semistable, and it is semistable on a neighborhood of E except possibly at x 1 , where the index equals e - 2. For each node x in X, repeat this procedure [ �] times, and we obtain the minimal resolution of singularities X' of X. Let r' [r� ---+ r�] be the dual complex of the closed fiber X_F of the minimal resolution of singularities X'. We define a natural mor phism r ---+ r' of complexes. We use the notation in Definition B.5. For the minimal resolution of singularities X', let X � be the normal ization of X_F, and define P' Spec f(X � , 0) , "E' = {nodes of X_F }, o
o
=
=
=
=
=
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DUAL CHAIN COMPLEX OF CURVES OVER A DVR
1 89
etc. Since the natural morphism X' ---+ X is an isomorphism out side the nodes of XF , it induces an open immersion P ---+ P'. Define r0 ---+ r0 to be P ---+ P'. The natural morphism X' ---+ X induces E' ---+ E, and for each x E E(F) , the number of elements of its in verse image in E(F) is the index ex . For x E E(F) , let x1 and x2 be its inverse images in � ( F) . Let x� , . . . , x� ., be the inverse image of x in E' ( F) , and we choose x� 1 , x� , 2 to be the inverse images of x� , i = 1, . . . , ex , in E (F) such that x1 and x1,i , x2 and Xe ., ,2 , and Xi,2 and X(i+l),l for 1 ::; i < ex are contained pairwise in the same connected component of the normalization X F . Then, we define homomorphism rl ---+ r� by letting the image of ( [x1] - [x2] ) E rl be :L:� 1 ( [xi,1] - [xi,2] ) E r� . The homomorphisms ro ---+ ro and r 1 ---+ r� define a morphism of chain complexes r ---t r' . (B.9) The morphism r ---+ r' induces homomorphisms of homology groups H0 (r) ---+ H0 (r' ) , H1 (r) ---+ H1 (r' ) . The symmetric bilinear form ( , ) : r� x r� ---+ z induces a symmetric bilinear form ( , h : r 1 x r1 ---+ Z through the linear mapping r1 ---+ r� . Thus, a1 : r1 ---+ rt is obtained as the composition of a1 : r� ---+ r�v with rl ---+ r� and its dual. PROPOSITION B.14. Let 0 be a discrete valuation ring, and let X be a proper weakly semistable curve over 0. Let r = [r1 ---+ r 0 ] be _, _
,
_,
the dual chain complex of XF . (1) If X' is the minimal resolution of singularities of X, then the homomorphisms H0 (r) ---+ H0 (r') and H1 (r) ---+ H1 (r') induced by the morphism of chain complexes r ---+ r' in (B.9) are iso morphisms. (2) Suppose X is semistable, and let a1 : H1 (r) ---+ H 1 (rv ) be the linear mapping (B.8) . Then, we have a natural homomorphism (B.10) Coker (a1 : H1 (r) ---+ H 1 (rv ) )
---t Ker(,6 : rti ---+ Z)/ Im(ao : ro ---+ rti ) . PROOF. (1) This is easily verified. (2) Under the notation in Definition B.5, r 0 is a free Z-module generated by P(F) . Sending a basis P(F) of r 0 to its dual basis, we define an isomorphism 'Yo : ro ---+ r6 . Since XF is reduced, the com position ,6' = ,6 'Yo : r 0 ---+ Z sends each element of P(F) to 1. The o
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composition (3' o d : r1 --+ r0 --+ Z equals 0. Since X-p is connected, /3' induces an isomorphism Coker d = Ho (r) --+ z. Since X is semistable, a 1 : r 1 --+ r¥ is an isomorphism. Define 80 : r6 --+ ro to be the composition V
1
V
d dv <>1 r l ---=ro ---+ -t r i ---+ ro. Identify r and rr through the isomorphism a 1 : r 1 --+ rr , and regard 80 : r6 --+ Ker /3' as the composition of dv : r6 --+ r¥ and d : r 1 --+ Ker (3'. Then, we obtain an exact sequence doa - 1 (B. 1 1 ) Ker d � Coker dv =.!t Ker /3' / Im 80 ---+ Ker /3' / Im d. Since we have Ker d = H1 (r) , Coker dv = H 1 (rv) , and Ker (3' / Im d = 0, (B.11) gives an isomorphism (B. 12) Coker(a 1 : H1 (r) --+ H 1 (rv ) ) ---+ Ker(/3 ' : ro --+ Z)/ Im(8o : r0 --+ ro) . By the isomorphism (B. 12) , it suffices to show that the diagram 1
ro � r6 � z
1
- -ro r6
-11
'Yo 0
�
ro
{3' --'------+
I
z
commutes. The right square commutes by the definition of (3'. We show that the left square commutes. Let D be an irreducible compo nent of X-p. It1suffices to show that -8o o'"Yo (D) = -doa1 1 odv o'"Yo (D) is equal to '"Yo o a0 (D) = "Ev, (D, D') o · D' . Let :Ev = LJ v , ,= v (D n D') C :E (F) be the union of the intersec tions of D and the other irreducible components than D. For each x E :Ev , number the inverse images x 1 , X 2 in E(F) so that x 1 E D . Then, we have Thus, we have -doa1 1 odv o'"Yo (D) = - rn:Ev) · D + "Ev ' ,e v (D, D')o · D' . Since (Xp , D)o = 0, we have (D, D)o = -":Ev . This shows the left D square is commutative.
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 fl.at commu tative group schemes. THEOREM C.l. Let p be a prime, and let n � 1 be an integer. Let a be a finite Z/p n 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] /(FV - p) -module) .
(2) a is etale over FP if and only if F : a -* a is an isomorphism and if and only if F : D( a) -* D( a) is an isomorphism. If a is etale, D(a) is the invariant subgroup (a(Fp) @ z�r ) a Fp with respect to the diagonal action of the absolute Galois group aF p , and the action of F on D(a) = (a(Fp) @ z�r ) a Fp is the restriction of 'Pp @ 1 (3) If av is the Cartier dual of a, D(av ) Hom (D(a) , Qp/Zp ) , and F = vv , V = Fv . (4) Let A be an abelian variety over Fp of dimension g . Then, the Z/pn Z [F, VJ -module D(A[pn ] ) is a free Z/pn Z-module of rank 2g. D(A) �n D(A[pn ] ) @ z p Qp is a Qp-vector space of dimension 2g. For an endomorphism f : A -* A, we have deg / = det (f : D(A) ) . =
=
We omit the proof. D( a) is what is usually called the contravari ant Dieudonne module D v (av ) of the Cartier dual av . The Frobe nius endomorphism Fa of a induces its transpose Vav on av , and thus it acts on D(a) = Dv (av) as Dv (Vav ) . In this book we agree on writing F instead of (Fa ) * = Dv (Vav ) . If R is a commutative ring and a has a structure of R-module, so does D(a) . 191
C. FINITE COMMUTATIVE G ROUP SCHEME OVER z ,,
192
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 = l . Theorem C . l is generalized to a general perfect field k of char acteristic 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 en domorphism. Define a noncommutative Wn (k) -alg b a Wn (k) (F, V) generated by F and V by the relations FV = VF = p, Fa = F(a)F, aV = VF(a) (a E Wn (k) ) . With this notation we have the following. THEOREM C.2. Let p be a prime, and let n � 1 be an integer. e
r
If k is a perfect field of characteristic p, we have an equivalence of abelian categories n n (k) (F, V) -modules of finite D .· finite Z/p z-module --+ W length as Wn (k) -module schemes over k . If k' is a finite extension of k, the following diagram commutes: finite Z/pn Z-module _E__,, Wn (k) (F, V) -modules of finite length as Wn (k) -module schemes over k
(
)
(
®kk'
1
)
(finite Z/pn Z-module) schemes over k'
_E__,,
(
)
(
)
1 ®wn (k) Wn ( k' )
(Wlength n (k') (F, V) -modules of finite) as Wn (k') -module
.
C.2. Finite flat commutative group scheme over Zp
Finite fl.at 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/p n Z-module M is a filtered cp-module if it is endowed with a submodule M' and linear mappings cp' : M' --+ M and cp : M --+ M satisfying 'PI M ' = pep'. (2) A finite filtered cp-module M is strongly divisible if M = cp(M) + cp' ( M ') . (3) Let M, N be filtered cp-R-modules. An R-linear mapping f : M --+ N is a morphism of filtered cp-R-modules if f (M') C N', cp o f = f o cp and cp' o J I M' = f o cp'. (4) A strongly divisible finite filtered cp-module (M, M') is etale if M = M'. If M' = 0, the strongly divisible filtered cp-module (M, M') is said to be multiplicative.
C. 2 . FINITE FLAT COMMUTATIVE G ROUP SCHEME OVER
z ,,
1 93
LEMMA C.4. A finite filtered cp-module M is strongly divisible if
and only if
(C.l)
0 ----+ M'
(p, -can )
M' EEl M
' ( cp ,cp)
M ----+ 0
is exact. PROOF . The composition M' -7 M' EEl M --+ M is the 0-mapping. By Definition C.3(2) , M is strongly divisible if and only if M' EEl M --+ M is surjective. The assertion is now clear by counting the number D of elements.
C.5. (1) The category of finite strongly divisible filtered cp-modules is an abelian category. (2) If M is a finite strongly divisible filtered cp-module, there exists a linear mapping F : M --+ M satisfying F o cp = p and F o cp' = COROLLARY
idM' · (3) Let 0 be the ring of integers of a finite extension of Qp . Let n ;::: 1 be an integer, and let R = 0 /m0 . If ( M, M') is a finite strongly divisible filtered cp-R-module, M' is a direct summand of M as an R-module. ( 4) A strongly divisible filtered cp-module (M, M') is etale if and only if F : M --+ M is an isomorphism. P ROOF . (1) Clear from Lemma C.4. (2) We have M = Coker ((p, -can) : M' --+ M' EEl M) from the exact sequence (C.1). The assertion follows immediately from this. (3) Let F be the residue field of 0. The exact sequence (C.l) remains exact after tensoring ®oF. Thus, M' --+ M is injective after tensoring ®oF. (4) If F is an isomorphism, we have cp = p p- 1 , and thus M = cp(M) + cp'(M') c pM + cp' (M'). Thus, by Nakayama ' s lemma, cp' : M' --+ M is surjective and we have M = M'. The converse is D �ar. o
By Corollary C.5(2) , an additive functor (finite strongly divisible filtered cp-Z/pn Z-module) ----+ (finite Z/pn Z[F, V]/(FV - p)-module) is defined by letting V = cp.
C. FINITE COMMUTATIVE G ROUP SCHEME OVER Z p
1 94
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)
(
strongly divisible ) ) (finite filtered
finite flat Z/pn z-module · schemes over Zp The diagram finite flat Z/pn Z-module � schemes over Zp
D
.
(
®zp F P
l
(finite flat Z/pn Z-module) schemes over
�
--r
1 (finite Z/pn z [F, module
.
)
V] / ( F V p ) FP 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
Hom (r(Aqp , n � Q p / Q p ) , Qp ) · We do not prove this theorem either. If G = Z/pn z, we have D (G) = D(G)' = Z/pn z and
C. 2 . FINITE FLAT COMMUTATIVE G ROUP SCHEME OVER Zp
195
an abelian scheme over Zp , and define a p-adic representation VpA Qp by Vp AQ p = TpAQp ©zp Qp, where TpAQ p is the Tate module. Then, we have D(VpAq P ) = D(A) . Similarly, for a good mod p-representation V of Gq p , the strongly divisible filtered Fp-cp-module D(V) and also the Fp[F, V]/(FV - p)-module D(V) are defined. We give a condition for a good two-dimensional representation of Gq p to be ordinary. PROPOSITION C.7. Let p be an odd prime, and let G be a finite flat commutative group scheme over Z p . Let G = G/pG, and let D( G) be the corresponding strongly divisible filtered cp-module, let n be the multiplicity of the eigenvalue 0 of cp : 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 GI 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 cp is an isomor phism, and D( G) / D( H) is the maximal quotient module such that cp 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 [cp] C End D(G) is the direct product of the part cp is invertible and the part cp is nilpotent. Thus, D( G) also decomposes to the direct sum of the part D(G)0 in which cp is invertible and the part cp is nilpotent. We show D(G)0 n D(G)' = 0. Assume 0 =F x E D(G)0 n D(G)', and we deduce a contradiction. We may assume px = 0. Since 0 =F x E D(G)0, we have cp(x) =F 0. But, since x E D(G)' and px = 0, we have cp(x) = pcp' (x) = cp' (px) = 0, which is a contradiction. Hence, the natural mapping D( G)0 EB D( G)' -+ D( G) is injective. Since dim D( G ) = dim D( G )0 + n and dim D( G )' = h, the in jection D( G )0 EB D( G )' -+ D( G ) is an isomorphism if n = h. Thus, by Nakayama ' s lemma, D(G)0 EB D(G)' -+ D(G) is surjective, and an isomorphism. From this we obtain an exact sequence of strongly
196
C. FINITE COMMUTATIVE G ROUP SCHEME OVER Zp
divisible filtered cp-modules 0 --+ ( n ( G ) 0 , 0 ) --+ (D( G ) , n ( G ) ' ) --+ (D( G ) ' , D ( G ) ' ) --+ 0 .
The assertion now follows from Theorem C.6(2) . (2) Clear from the proof of (1) (ii) => (i) .
0
COROLLARY C.8. Let p be an odd prime, let K be a finite ex tension of Qp, and let 0 be its ring of integers. Let V be a good representation of G Q P on a two-dimensional K -vector space, and let n = n (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(l - Ft : n) = (1 - at) (l - p(3t) . Suppose this condition holds, and let a, (3 denote the unramified characters of G Q ,, such that the value of cp is a, (3, respectively, and x the p-adic cyclotomic character. Then, V is an extension of a by (3 . X · PROOF. It is clear that the condition is necessary. We show it is sufficient. Suppose det(l - Ft : n) = ( 1 - at) (l - p(3t) and a, (3 are p-adic units. Let n° c D be the eigenspace belonging to the eigenvalue p(3 of F, and T c V a GQ p -stable 0-lattice. n (T)0 = D (T) n n° is a free 0-module of rank 1, and the action of cp on D (T)0 is the multiplication by 1/(3, and its action on n (T)/n(T)0 is the multiplication by pf a . For an integer m � 1, let Gm be the finite fl.at commutative group scheme over Zp defined by Gm (Qp ) = T/pm T. Then, n and h in Proposition C.7 are both equal to [K : Qp] · Thus, by Proposition C.7, there exists a subrepresentation T0 of T such that n (T0) = D (T)0 and that both T/T0 and T0 ( -1) are unramified. Since the action of cp on D (T/T0 ) = n (T)/D(T) 0 is p/a, the action of F is a, and by Theorem C.1(2) , the action of cpp on T/T0 is also a. Similarly, the 0 action of 'Pp on T0 ( - 1 ) is multiplication by (3.
We give a description of an extension of strongly divisible fil tered cp-R-modules. Let R be a finite commutative algebra over Zp , and let M, N be strongly divisible filtered cp-R-modules. As in §11.1, an exact sequence of strongly divisible filtered cp-R-modules ( E) : 0 -+ N -+ E -+ M -+ 0 is called an extension of M by N.
C. 2 . FINITE FLAT COMMUTATIVE G ROUP SCHEME OVER
If there is a commutative diagram (E) : 0 -------+ N -------+ (E') : 0
I
E
1
-------+
M
I
-------+
Zp
197
0
N -------+ E' -------+ M -------+ 0, we say that the extensions (E) and (E') are isomorphic. The set of isomorphism classes Extit(M, N) has a structure of R-module. By the equivalence of categories (C.2) , for finite fl.at 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 Extit(D(G) , D(H)). For filtered cp-R-modules M, N, define HomR(M, N) = { (f, g) E HomR(M, N) x HomR(M ', N) \ f \ M 1 = pg}, HomR(M, N)' = {f E HomR(M, N) \ f(M') c N'}, and define a homomorphism o : HomR(M, N)' --+ HomR(M, N) by o ( f ) = (cp o f - f o cp, cp' o f \ M ' - f o cp') . We denote by Homit(M, N) the set of homomorphisms M --+ N of filtered cp-R-modules. For (f, g) E HomR(M, N) , define an extension E = E1 , 9 of M by N by E = M ffi N, E' = M' ffi N', cp( x , y) = (cp(x) , cp(y) + f (x) ) , cp' ( x , y) = (cp'( x ), cp'(y) + g(x) ) . A homomorphism HomR(M, N) --+ Extit(M, N) is defined by assigning to (!, 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 cp-R-modules. Then, there exists an exact sequence of R-modules
(C.3) 0 -+ HomR(G, H) ----+ HomR(M, N)' � HomR(M, N) ----+ 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) ----+ ExtR(G, H) --+ 0. PROOF. First we show the exact sequence (C.3) . We identify HomR(G, H) = Homit(M, N) and ExtR(G, H) = Extit(M, N) by
198
C. FINITE COMMUTATIVE G ROUP SCHEME OVER Zp
Theorem C.6(1). It is clear from the definition that Hom�(M, N) is the kernel of 8 : HomR(M, N)' -+ HomR(M, N) . We show the exactness at HomR(M, N) . If for (f, g) E HomR(M, N) , the class of Ef,g is 0, we have an isomorphism E1,9 -+ Eo,o - Its component h : M -+ N gives an element of HomR ( M, N)' satisfying (!, g) = 8 ( h) . Thus, the sequence is exact at HomR(M, N) . We show HomR(M, N) -+ Ext�(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 ffiN of R-modules that is an extension of the direct sum decomposition E' = M' ffi N' as R-modules. The surjectivity of HomR(M, N) -+ Ext�(M, N) follows from this and the definition. D COROLLARY C.10. Let p be an odd prime, and let 0 be the ring of integers of a finite extension of Qp . Let n ;:::: 1 be an integer, and let R = O/m0 . Let G, H be finite fiat R-module schemes over Zp, and suppose G(Qp ) is a free R-module. Let M and N be strongly divisible filtered
-
APPENDIX D J acobian of a curve and its Neron 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 as sociated with modular forms using this algebraic structure for the modular curves. For a prime number at which a curve has bad re duction, its Jacobian may not have good reduction. Even for such a prime number, we can still study its properties using the Neron 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 (D.1) Div(X) = Z · [x] x :closed point of X
is called the divisor group . For a rational function f E K x on X, the divisor of f, noted div f, is defined by L x ordx f · [x] E Div(X) . By associating to f E K x the divisor div f E Div(X), we obtain a homomorphism of abelian groups div : K x -+ 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 x is called the divisor class group. The divisor class group Pic(X) has the following cohomological expression. Denote by K x the constant sheaf on X, and for a closed point x, let Z x be the extension to X of the constant sheaf on x. 199
200
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
EB
(D.3)
x :closed point of X
Zx ----+ 0.
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 iso morphism. For D L x nx [x] in X, we call deg D = L x nx [K(x) : k] the degree of the divisor D. Associating to the divisor D E Div(X) to its degree deg D E 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 fl.at morphism of proper normal con nected curves. Define f * : Div(Y) """""* Div(X) by J * ( [y] ) = [X x y y] = L x >-+ y ex;y [x] for a closed point y in Y. Here, ex/y indicates the ram ification 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 [K(x) : K(j(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 xan be the Riemann surface associated with X. Consider the singular chain complex (Cq (Xan , Z) , dq ) q EZ of xan . The fact that Ho (xan , Z) = Z implies Div0 (X) = Ker ( Co ( Xan , Z) ---* Ho (X an , Z)) = Im( C1 (X an , Z) """""* Co (Xan , Z)) , =
:
:
D. 2 .
THE JACOBIAN OF A CURVE
201
and we obtain a surjection C1 (Xan, Z) -t Div0 (X) . Define a homo morphism (D.5) C1 (X an , Z) ---+ H0 (X, 01:)v = Hom(H0 (X, 01 J , C) by associating to 1-chain 'Y the linear form w t-t J""Y w . (D.5) induces a homomorphism (D.6) Div0 (X) ---+ H0 (X, Oi ) v / Im H1 (X an , Z) . The mapping induced by (D.5) (D.7) H1 (x an , Z) ---+ H0 (X, Oi)v induces an isomorphism of R-vector spaces (D.8) In other words, the free abelian group H1 (xan, Z) of rank 2g is a lattice in the C-vector space H0 (X, Oi )v of dimension g. Thus, H0 (X, Oi )v / 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) -t H0 (X, Oi)v/ Im H1 (X an , z) . In this way Pic0 (X) has a structure of compact complex torus of dimension g . Let Z(l) be the constant sheaf 27rHZ on x an_ The trace map ping H2 (xan, Z ( l ) ) -t Z is an isomorphism. By the Poincare duality, H 1 (Xan, Z ( l ) ) is identified with H1 (Xan, Z) , the dual of H 1 (xan, Z ) . D . 2 . The Jacobian o f 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 .C and .C' by [.CJ · [.C'] = [.C ©o x .C'] . 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 .C is an invertible sheaf on X, Isomox (Ox , .C) defines a Gm torsor over X. Thus, we obtain a natural homomorphism (D.10)
202
D. JACOBIAN OF A CURVE AND ITS NERON MODEL
Conversely, if a G m -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, G m ) 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 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.l. Let S be a scheme, and let X be a scheme over S. The fiat sheafification PX.; s of the functor Px; s over S defined by (D.11) Px; s (T) = Pic(X 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 · For a geometric point s, the natural map Px; s (s) = Pic(X8) --+ 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 fiat of finite presentation, the norm of an invertible sheaf defines a morphism of functors f* Picx; s --+ Picy; s· If £ is an invertible sheaf on X, the norm N1£ 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 .ev of an 01- 1 ( V ) -module Cl 1- 1 ( v ) · N1£ is an invertible sheaf that has a basis N(.ev) over V, and for a change of bases .e' = a.e we have N(.e') = Nx; ya · N(.e) . :
:
:
:
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 n (D.12) Pic0 (X) = n Ker ( Pic(X) --+ Pic(Xi ) � Z ) .
i=l
D.2. THE JACOBIAN OF A CURVE
2 03
A subfunctor Pic�; s of Picx; s is defined by
. ox; s (T ) (D. 13) Pic
=
_
n
.
t : geometr1c point of T
( inverse image of Picx; s (To) ---+ ) Picx; s ( t) = Pic(Xf) by Pic (Xf)
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) Pic�; s is representable by an abelian scheme j : J ---+ S over S of relative dimension g . (2) There is a natural isomorphism
(D. 14) (3) Let n � 1 be an integer. The Weil pairing defines a bilinear form J[n] x J[n] ---+ µn and defines an isomorphism to the Cartier dual (D.15)
J[n] --+ J[n] v .
The moduli space J of the functor Pic�; 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 ---+ Pie� 1 8 , 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 = Pic�/ k is an abelian variety over k of dimen :
sion g .
(2) There is a natural isomorphism (D. 16) (3) Let n � 1 be an integer. The Weil pairing defines a bilinear form J[n] x J[n] ---+ µn and the isomorphism J[n] ---+ J[n]v to its Cartier dual.
2 04
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 (xan, Z/nZ) = H 1 (xan, Z/nZ(l)). The Weil pairing J[n] x J [n] ---+ µn may be identified with the pairing H1 (xan, Z/nZ) x H1 (xan, Z/nZ) ---+ Z/nZ (l) that is induced by the composition of the cup product H 1 (xan , Z(l)) x H 1 (xan, Z(l)) ---+ H2 (xan, Z (2)) and the trace mapping H2 (xan , Z(2) ) ---+ Z(l) . Let f : X ---+ Y be a finite fl.at 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 fiat 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 f, is a prime number invertible in k, f * : V[Jy ---+ V[Jx defines an isomorphism f * : V[Jy ---+ (VlJx ) 0 to the G-invariant part ( V£ Jx ) G . PROOF. (1) Since f* o f * : Jy ---+ Jy is the multiplication-by [X : Y] mapping, Ker f * is finite. (2) It is clear from the fact that f * o f* : Jx ---+ Jx equals D l: g E G g * . THEOREM D.7. Let k be a field, and let X be a proper curve over k . (1) The functor Pic�/ k is represented b y a smooth connected com mutative group scheme J over k . (2) Suppose X is smooth. Let X = IJ �= l Xi be the decomposition into connected components, and let ki = r(Xi , 0) be the field of definition of Xi for i = 1, . . . , n . Then, J is an abelian variety that is isomorphic to the product Il�= l Res k ; / k Ji of the Weil restrictions to k of the Jacobian Ji = Pic�; / k ; of Xi over ki . (3) Suppose X is smooth except for a finite number of ordinary dou ble points. Let X be its normalization. Then J is an extension of the Jacobian ] of X by a torus. (4) Assume k is perfect. Let X be the normalization of X, and let r be the dual chain complex of X . Then, the morphism J ---+ ] induced by the natural map X ---+ X gives an isomorphism from the abelian part Ja of J (D.17) Ja ----+ J.
D.3. THE NERON MODEL OF AN ABELIAN VARIETY
20 5
The character group of the torus part Jt of J is naturally iso morphic to H1 (r) . D.3. The Neron model of an abelian variety THEOREM D . 8 . Let 0 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 0 having the following property: for any smooth scheme X over 0, 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 0 satisfying the condition in Theorem D.8 is unique up to natural isomorphisms. DEFINITION D . 9 . Let 0 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 0 that satisfies the property in Theorem D.8 is called the Neron model of AK . The open subgroup scheme A 0 of A, which is defined by the conditions A0 ®o K = A ®o K and that A0 ®o F is the connected component A'j.. of A ®o F containing the identity element, is called the connected component of the Neron model A. DEFINITION D.10. Let 0 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 Neron model of AK . ( 1 ) If A is an abelian scheme over 0, AK is said to have good reduc tion.
(2) If the connected component A'j.. of the closed fiber Ap of A is an extension of an abelian variety by a torus, AK is said to have semistable reduction.
LEMMA D . 1 1 . Let 0 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 Neron model of AK , and let Ap = A ®o F be its reduction. Let f. be a prime number different from the characteristic of F.
206
D . JACOBIAN OF A CURVE AND ITS NERON MODEL
(1) The Tate module TtAK is an unramified representation of GK, and the natural isomorphism TtAK ---+ TtAF is compatible with the natural surjection GK ---+ G F . (2) The natural isomorphism End AK ---+ End AF is injective, and it is compatible with the natural isomorphism TtAK ---+ TtAF . PROOF. (1) It follows easily from Lemma A.47(1). (2) It follows easily from Proposition A.51 (2) . D PROPOSITION D.12. Let 0 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 fiat commutative group scheme over 0. Let A be the Neron 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 mor phism G ---+ A of commutative group schemes over 0 that ex tends 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 commuta tive group schemes over K extending GK ---+ AK is also a closed immersion. -
-
The case ( i ) in (1) is clear from the definition of Neron 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 Neron model. Let I c 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 ) 1 . (2) Let £ be a prime number different from the characteristic of the
residue field. Then, there is a natural isomorphism of the finite dimensional Qt-vector space VeAF ---+ (VeAK ) 1 compatible with the action of the natural morphism GK ---+ GK/ I = G F . Suppose F is a perfect field, and let A} be the abelian part of AF, and let
D.3. THE NERON MODEL OF AN ABELIAN VARIETY
207
A} be the torus part of Ap . Then, we obtain an exact sequence 0 --r VeA} --r VeAF --r VeA} --r 0. (3) Let L be a finite Galois extension of K, let h / K C Gal(L/K) be the inertia group, and let E be the residue field of L. Suppose F is a perfect field, and let Af; be the abelian part of the closed fiber AE of the Neron model of AL , and let Ak be the torus part. Then, the natural morphisms VeA} --r (VeAE;)htK and VeA} --r (VeAk)ht K are isomorphisms. PROOF . (1) Since the multiplication-by-N morphism [NJ : A --r A is etale, A[N] is etale over OK . Therefore, if Kur = K 1 is the max imal unramified extension of K, the natural morphism A[N] (O't() --r Ap [N] (F) is an isomorphism. By the definition of Neron model, A[N] (OW) --r AK [N] (Knr ) = AK [N] (K)1 is also an isomorphism. (2) By ( 1 ) , VeAF --r (VeAK )1 is an isomorphism. By Corol lary A.50, we obtain the exact sequence 0 --r VeA} --r VeAF --r VeA} --r 0. (3) By (2) , the natural morphism Ap @p E --r AE induces the isomorphism VeAF --r (VeAE)ht K . By taking the h / K-invariant part of the exact sequence 0 --r VeAk --r VeAE --r VeA'E --r 0, we obtain D 0 --r VeA} --r VeAF --r VeA} --r 0 by Corollary A.50(2) . COROLLARY D . 14. Let K be a discrete valuation field, and sup pose its residue field F is perfect. Let l be a prime number different from the characteristic of F. (1) Let AK --r BK be a morphism of abelian varieties over K, and let A --r B be the morphism induced on their Neron models. Let A} C A'f.. and B} c Bi be the torus parts of the closed fibers. Suppose the kernel of AK --r BK is finite. Then, VeAK --r VeBK is injective. If we identify VeAK and VeBF as the subspaces of VeBK, we have VeAF = VeAKn lfe Bp and VeA} = VeAK n VeB} . (2) Let XK --r 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 Ap and Bp be the closed fibers of the Neron models of AK and BK , and let A}, B}, A}, B} be their abelian parts and torus parts. We denote by G the G-invariant part. Then, the natural mappings VeA} --r (VeB}) 0 and VeA} --r (VeB};.) 0 are isomorphisms. PROOF . (1) It is clear that VeAK --r VeBK is injective. By Corollary D . 13(2) , VeAF and VeBF are invariant subspaces by the
208
D . JACOBIAN OF A CURVE AND ITS NERON MODEL
inertia group I. Thus, ViAF = VeAK n VeBF follows from (ViAK )1 = VeAK n (VeBK)1. By Corollary A.50(2) , VeA} = VeAF/VeA}.. --+ VeB} = VeBF/VeB} is injective. Thus, VeA}.. = VeAK n VeB}. (2) By Lemma D.6(2) , VeAK is identified with (VeBK) 0 . Thus, by (1) , we have VeAF = (VeBF) G and VeA}.. = (VeB}.. ) 0 . More over, taking the G-invariant part of the exact sequence 0 --+ VeB} --+ VeBF --+ VeBP, --+ 0, we obtain the isomorphism VeA'F --+ (VeBP,) 0 .
D
Whether an abelian variety AK over a discrete valuation field K has good reduction or semistable reduction can be determined by the £-adic representation VeA of GK. DEFINITION D.15. Let 0 be a discrete valuation ring, let K be its field of fractions, and let p be the characteristic of F. Let .e 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)nEN of surjections of finite fl.at commutative group schemes over 0 is called £-divisible group if the following conditions are satisfied: .en : 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 i n 1 i i G 1 --+ Gn . (2) V is said to be a good £-adic representation if there is an £ divisible group such that V = �n Gn (K) ® z e Qe . (3) V is said to be a semistable £-adic representation if there exist a good £-adic representation Vo C V such that V/Vo is unramified. If .e =f. p, then V is good if V is unramified. If .e = 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 .e be prime numbers, and let K be a -
o
·
·
·
o
:
finite extension of Qe . Let p be an l-adic representation of Gqp to a two-dimensional K -vector space V. If the action of Gqp on
D.4. THE NERON MODEL OF THE JACOBIAN OF A CURVE
209
Let 0 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 0 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 0 such that each geometric fiber is connected. Then, the functor Pic�; o is represented by an abelian scheme J over 0. JK J ©o K is the Jacobian of XK ©o K, and J is the Neron model of JK . The closed fiber Jp = J ©o F is the Jacobian of Xp = X ©o F. For a prime number .e different from the characteristic of K, the .e-adic representation VeJK of Gal(K / K) is a good .e-adic representa tion. If .e is different from the characteristic of F, the natural map ping VeJK -t VeJF is an isomorphism of finite-dimensional Qt-vector spaces that is compatible with Gal(K/K) -t Gal(F/F) . PROOF. It follows easily from Theorem D.3 and Lemma D.11. =
D
Even if a curve does not have good reduction, we have the fol lowing theorem on regular model. THEOREM D.19. Let 0 be a discrete valuation ring, let K be its field of fractions, and let F be its residue field. Let X be a proper regu lar connected curve over 0 such that each geometric fiber is connected.
21 0
D . JACOBIAN OF A CURVE AND ITS NERON MODEL
Suppose XK = X ®o K is smooth, F is perfect, and the greatest com mon divisor of the multiplicities of components of Xp = X ®o K in Xp is 1 . Then, (1) The functor Pic�/ O is represented by the connected component J0 of the Neron model J of the Jacobian JK of XK . (2) Let r be the dual chain complex of Xp , and let ao : ro � r0 and /3 : rQ � Z be the linear mappings in (B.4) and (B.5) , re spectively. Then the group of connected components of the closed fiber Jp = J ® F of the Neron model J is naturally isomorphic to Ker /3/ Im ao .
COROLLARY D.20. Let the notation be as is in Theorem D.19. Let ]F be the Jacobian of the normalization X F of the reduced part of Xp . Then, the morphism Jp � ]F induced by the natural morphism Xp � Xp gives an isomorphism from the abelian variety part of the connected component Ji (D.18) PROOF. It is clear from Theorem D.19(1) and Theorem D.7(4) .
D
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 0 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 0 such that each geometric fiber is connected. Let J be the Neron model of the Jacobian JK of XK = X ®o K. (1) Let Xp be the normalization of Xp , and let r = [r 1 � r 2 ] be the dual chain complex of the closed fiber Xp . The connected component Ji of the closed fiber JF = J ®o F is an extension of the Jacobian of Xp by a torus Hom(H1 (r) , Gm ) · Let C1 , . , Cm be the components of Xp, let E be the re duced closed subscheme of Xp consisting of singular points. Let g be the genus of Xp , Fi the field of definition of Ci, and let Yi be the genus of the curve Ci over Fi . Then we have .
.
m
(D.19)
g = 1 + deg E + L [Fi : F] ( gi - 1). i= l
D.4. THE NERON MODEL OF THE JACOBIAN OF A CURVE
211
( 2 ) The group of connected components of Jp is naturally isomor phic to the cokernel of the linear mapping ii. 1 : H1 (r) ---+ H 1 (rv ) of ( B.8 ) .
PROOF. ( 1 ) Let X' be the minimal resolution of singularities of X. We have XK Xk . Since all the exceptional divisors are of genus 0, the Jacobian of the normalization of XJ.. equals that of Xp. If r ' is the dual chain complex of XJ.. , then by Proposition B.14 ( 1 ) , we have a natural isomorphism H1 (r) ---+ H1 (r' ) . Thus, by replacing X by X', we may assume X is semistable. By Theorem D.19 ( 1 ) and Theorem D.7 ( 3 ) , Ji is an extension of the Jacobian by a torus. Then, by Corollary D.21, the character group of the torus part of Ji equals =
H1 (r) .
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 a = L::,i [Fi : F]gi and t - 1 = deg E - r::, 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 ) . =
0
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 VlJK of Gal ( K / K ) is a semistable £-adic rep resentation.
(i )
PROOF. It follows from Corollary D.21 ( 1 ) and Theorem D.17 ( 2 ) 0 ( 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 schemas de modules de courbes el liptiques, 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 Sci entific, 2000. [5] B. J. Birch, W. Kuyk (eds.), Modular Functions of One Vari able 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 Theoreme 2.12, [3] Theorem 13.11.4. =
CHAPTER 9 [6] H. Carayol, Sur les representations galoisiennes modulo f, at tachees aux formes modulaires, Duke Math. J. 59 (1989) , 785801. [7] T. Miyake, Modular forms. Translated from the Japanese by Yoshitaka Maeda. Springer-Verlag, Berlin, 1989. x+335 pp. 213
214
BIBLIO G RAPHY
[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 1 0 [9] J.-P. Serre, Arbres, amalgames, SL 2 , Asterisque 46, Societe Mathematique de France, Paris, 1977. [10] , Le probleme des groupes de congruence pour SL2 , Ann. 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 Theoreme 3 p.110. <> Theorem 10.15(2) : [10] 2.6 Corollaire 3 p.449 (If we let K Q, S {p, oo}, and q (N) , then we have r q r (N) and Eq E (N) .) <> Theorem 10.28: [11] sections 255, 260. ___
=
=
=
=
=
CHAPTER 1 1 [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., Propo sition 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 APPENDIX
215
B
[65] P. Deligne, N. Katz, Groupes de Monodromie en Geometrie Algebrique ( SGA 7 ) II, Lecture Notes in Math., Springer, 340, ( 1973 ) . Lemma B.4 ( iii ) ::::} ( i ) The case where k is general: [65] Exp. XV,
Theoreme 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,
Asterisque 47-48, Soc. Math. de France, 1977.
[ 16] J.-M. Fontaine, G. Laffaille, Construction de representations p adiques, Ann. Sci. Ecole Norm. Sup. ( 4 ) 15 ( 1982 ) , 547- 608 ( 1983 ) . [ 17] N. Wach, Representations cristallines de torsion, Compositio Math. 108 ( 1997) , 185- 240. Theorem C.1: [ 15] Chapitre III. Theorem C.6: [ 1 6] , [ 17] . o o
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, Neron 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. Liitkebohmert, M. Raynaud, Neron models, Springer, 1990. [22] M. Raynaud, Jacobienne des courbes modulaires et operateurs de Hecke, in "Courbes modulaires et courbes de Shimura" , Asterisque 196- 197, Soc. Math. de France, 1991, 9- 25.
BIBLIOGRAPHY
216
[23] A. Grothendieck, Modeles de Neron et monodromie, in Groupes de Monodromie en Geometrie Algebrique, SGA 71, Lecture Notes in Math., Springer, 288, 1972, 313-523. Proposition 5.13. Jacobians and Neron 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) f =f. p : [21] Theorem 7.4/5 ( b )tj-( d ) . Theorem D.17(2) f =f. p: [21] Theorem 7.4/6. Theorem D. 17(2) f = 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. o
o
o
o
o
o o
o
o
o
o
o
Symbol Index
U, 149 1-cocycle, 144 Gq , 125 (a) , 22 (a) - 1 ! 2 , 132 At , 111, 131 A E 1 , 113 Br(F) , 150 n Br(F) , 150 Cq (G, M) , 149 C(X, F) , 137 !}.. Q , 126 /}.. q , 125 !}..p , 117 DefR-a(M) , 146 Def�- G , 146 Def,o,vE (0 /(7rn ) [c] ) [ v,,. ] , 163 div f, 199 D(G), 191, 194 D(p) , 108 D(X), 199 70 f , E: 70 e, 51 ex , 24 E(N) , 118
E (N) , 118 · E (P l , 2 Eq; , 51 End�(M) , 147 ExtR-a (M, N) , 148 fa , 65 f� , 132 F, 2, 191, 193 Fe , 2 Fs, 1 FE 1 ,E , 116 I'o, * (p, N) , 118 I'(N) , 118 I' 1 (N) , 38 f' (N) , 118 r(r) , 29 G-coinvariant, 144 e x , 10 G(a , b ) , 33 CF , 143 Gs, 155 Grass(OE [NJ , N) , 41 H0 (G, M), 144 H 1 (G, M), 144 H} (Q£, Wn ) , 161 H} (Qp, M) , 151 H} (Qp, HomR(M, N)) , 152
E: ,
217
218
SYMBOL INDEX
H} (Qp, N ) , 152 H� (Q£, Wn ) , 162 Hq (F, M) , 150 H q (G, M) , 149 iE , 110 Io (N, n) , 62 li (N, n) , 66 Io (N, n) , 62 lg (Mpa , r) F P ' 33 lg (Mpa , r)�:0 , 34 lg (Mpa , r) F p , 56 j , 21, 29 ia , 35, 42, 55, 56 jo , j1 , 49, 58, 59, 118 Jo , 1 (N, M) , 70 Jo (N) Q , 61 J1 (N) Q , 66 K1 , 63 A, 31 £-divisible gro p , 208 (Lp) p ES , 157 (L�)pE S , 158 LE , 162 Lift�_ G , 146 LiftR-c(M), 146 Liftp ,VE (O/(nn ) [e])v,, , 163 µ, 25 µ � , 12 m � , 130 m � Q ' 135 , m i; ' , E ' 114 mE ' , E ' 115 mQ , 131 ffiRE ' 163 ffiE , 110 m� , 110 u
M, 146 Mp , 146 Mo (N) E , 41 Mi (N) E , 15 M ( l ) , 150 Mc , 144
Mc, 144 M� , 134 ME , 114 Mv , 150 M , 21
M o ,* (N, r ) z[�] ' 25 M o (N) , 20 Mo (N) E , 13 Mo (N)F p ' 23 M 1 ,* (N, r ) z[�] ' 25 M i , o (M, N) , 57 M i (N) , 20 M i (N) E , 13 M ( r ) z[ �l ' 25 N1£ , 202 N-l sogE , 41
ord, 2 108 97 PRE , 163 (P) , 13 PJ ,p(U) , 111, 131 Pp(U) , 94, 109 Pq (U) , 130 Px;s (T), 202 Pic(X) , 199, 201 Pic0 (X) , 202 Pic'1-18 , 203 Picx;k , 202 VJE , '
SYMBOL INDEX
Q, 126 Q, 126 Q, 125 Qs , 155
� 135
n , n'E, ,
114 R[c] , 146
S o , S 1 , 118 S d , 45, 59 Bk, 82 Sk ( q ) , 50 So(N), 61 S1 (N) , 66 sss, 2 Sel(E, n) , 158 SelL (M) , 157 SelE (Wn ) , 162 tk, 82 i0,Q , 129 tb ,E , 113 Q t 0,Q > 130 T(N)z , 108 T(NE)'c » 110 T[N] , 16 T (N) , 16 Te Jo (N), 70 To(N)z, 62 T1 (N)z, 67 TN,E, 41 Tf , 110 T� , 130 Tf , 113 TJ , 129 T, 97 T' , 103 Tor, 137
115 92, 115 v, 2, 72, 161, 191 vi , 161 Ve , 2 Vf , 71 VeJo(N), 70 w'Jv, , 22 w, 161 w� , 162 Xo(N)z, 48 Xo ,* (N, r) z[ � ] ' 54 X1 (Mp) �[t J ' 59 X1 (N)z, 48 X1 ,* (Mp, r) �[L: l ' 58 X1,* (N, r) z[ � ] ' 54v X1,o(M, N)z , 58 X1,o(N, M)z , 58 X (P) , 1 X(l)z I�, 51 Yo ,* (N, r) z[ � ] ' 42 Yo(N)z, 23 Y1 (4), 20, 47 Y1, * (N, r) z[ � ] ' 31 Y1,o (M, N)z, 57 Y1 (N) an, 38 Y1 (N)z, 24 Y(l)z, 29 Y(2), 31 Y(3), 25 Y(r) an, 29 Y(r) z[ � ] ' 26 � 1 (G, M), 144 z, 144 Z ((q)) , 51 Ue , Up ,
2 19
Subject Index
absolute Frobenius morphism, 1 annihilator, 153 Atkin-Lehner involution, 23 Brauer group, 150 character of f , 70 congruence relation, 73 connected component, 205 cup product, 149 curve, 179 cyclic group scheme, 9 diamond operator, 22, 67 Dieudonne module, 191 divisor, 199 divisor class group, 199 divisor group, 199 Drinfeld level structure, 13 dual chain complex, 182 dual local condition, 158 Eisenstein ideal, 77 et ale filtered
finite G-module, 143 finite R-G-module, 143 Frobenius morphism, 23 full Hecke algebra, 110, 130 full set of sections, 6 G-coinvariant, 144 generator, 9 genus, 179 good, 208 good reduction, 182, 205 Hecke algebra, 62, 67 Hecke module, 114, 134 Hecke operator, 67 Igusa curve, 33 index, 182 infinitesimal deformation, 146 infinitesimal lifting, 146 Jacobian, 203 £-divisible group, 208 local condition, 157 minimal resolution of singularities, 188 multiplicative filtered
222
Neron 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
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