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
I
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
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 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 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 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 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 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 ( 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
(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 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 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 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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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
=
=
=
=
=
B.3.
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
1 90
B.
CURVES OVER DISCRETE VALUATION RIN G S
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 -+ 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 -
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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 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