Introduction to the Arithmetic Theory of Automorphic Functions (Publications of the Mathematical Society of Japan 11)

PUBLICATIONS OF THE MATHEMATICAL SOCIETY OF JAPAN 11 INTRODUCTION TO THE ARITHMETIC THEORY OF AUTOMORPHIC FUNCTIONS B...

Author: Goro Shimura

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PUBLICATIONS OF THE MATHEMATICAL SOCIETY OF JAPAN

11

INTRODUCTION TO THE ARITHMETIC THEORY OF AUTOMORPHIC FUNCTIONS BY

GORO SHIMURA

KANÔ MEMORIAL LECTURES 1

Princeton University Press Princeton, New Jersey

Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex Copyright © 1971 by the Mathematical Society of Japan All Rights Reserved Library of Congress Cataloging-in-Publication Data Shimura, Gore), 1930— Introduction to the arithmetic theory of automorphic functions / by Gor6 Shimura. p. cm. —(Publications of the Mathematical Society of Japan ; 11. Kant) memorial lectures ; 1) Originally published: Tokyo : Iwanami Shoten ; Princeton, N.J. : Princeton University Press, 1971. Includes bibliographical references and index. ISBN 0-691-08092-5 (pbk. : acid-free) 1. Automorphic functions. I. Title. II. Series: Publications of the Mathematical Society of Japan ; 11. III. Series: Publications of the Mathematical Society of Japan. Kan6 memorial lectures ; 1. QA353.A9S55 1994 515.9—dc20 94-5898 Kanô Memorial Lectures In 1969, the Mathematical Society of Japan received an anonymous donation to encourage the publication of lectures in mathematics of distinguished quality in commemoration of the late K6kichi Kan6 (1865-1942). K. Kan6 was a remarkable scholar who lived through an era when Western mathematics and philosophy were first introduced in Japan. He began his career as a scholar by studying mathematics and remained a rationalist for his entire life, but enormously enlarged the domain of his interest to include philosophy and history. In appreciating the sincere intentions of the donor, our Society has decided to publish a series of "Kan6 Memorial Lectures" as a part of our Publications. This is the first volume in the series. Originally copublished in 1971 by Iwami Shoten, Publishers, and Princeton University Press; reprinted in paperback by arrangement with the Mathematical Society of Japan First Princeton Paperback printing, 1994 Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources 10 9 8 7 6 5 4 3 2 1 Printed in the United States of America

PREFACE There are two major topics treated in this volume : I. Complex multiplication of elliptic or elliptic modular functions. II. Applications of the theory of Hecke operators to the zeta-functions of algebraic curves and abelian varieties. Although these will form the "raison d'être" of the book, I have also attempted, in the first few chapters, to present an introductory account of the theory of automorphic functions of one complex variable, along with the fundamentals of Hecke operators. Our discussion is mainly concerned with elliptic modular functions of arbitrary level and the geometric objects directly related to them, except that we consider automorphic functions of a more general type in the first two and the last two chapters, and abelian varieties of higher dimension with complex multiplication in a few places. As to the first topic, we shall give two formulations, both in terms of adeles. One is concerned with the behavior of an elliptic curve and its points of finite order under automorphisms of the number field in question. The other is closely connected with the structure of the field of all modular functions of all levels whose Fourier coefficients belong to cyclotomic fields. It will be shown that the group of all automorphisms of is isomorphic to the adelization of GL2(Q) modulo rational scalar matrices and the archimedean part. Then the reciprocity-law in the maximal abelian extension of an imaginary quadratic field is given as a certain commutativity of the action of the adeles with the specialization of the functions of . The second topic is a development of the result of Eichler in his paper appeared in the Archly der Mathematik vol. 5, 1954. The conjecture of Hasse and Weil will be verified for the algebraic curves uniformized by modular functions. Further we shall show that if a cusp form of weight 2 is a common eigen-function of the Hecke operators, then the product of several Dirichlet series associated with it coincides, up to finitely many Euler factors, with the zeta-function of a certain abelian variety which is specifically given. As an application of this result, it will be shown that the arithmetic of a real quadratic field—its units, abelian extensions, etc.—is closely connected with the modular forms of " Neben "-type in Hecke's sense. My excuse for including this rather immature subject is that I think it gives a positive, if not complete, answer to the question "Can one construct abelian extensions of a real quadratic field by an analytic means? ", which arises naturally after the detailed discussion of the corresponding problem for an imaginary quad-

PREFACE

vi

ratic field in Chapters 5 and 6. The present book has grown out of my lectures at Princeton University and the University of Tokyo on various occasions during 1963-69. The notes taken by Larry Goldstein (Fall Term 1965) and by Alain Robert (Spring Term 1969) were most helpful in preparing the first draft. Here I gratefully acknowledge my indebtedness to them. I wish to express my hearty thanks to K. Doi, H. Naganuma, and H. Trotter who made the table of eigen-values of Heck e operators in § 7.7; and to W. Casselman, S. Lang, T. Miyake, A. Robert, and A. Weil, who read the manuscript as a whole or in part. Many of their suggestions have been incorporated in the present volume. My thanks are also due to S. Iyanaga and Y. Kawada, who took an interest in this work, and invited me to publish it in Publications of the Mathematical Society of Japan. Finally I would like to extend thanks to the audience of my lectures, whose enthusiasm was very encouraging. Princeton, May 1970

Goro Shimura

CONTENTS Preface Notation and terminology List of symbols Suggestions to the reader Chapter 1.1. 1.2. 1.3. 1.4. 1.5. 1.6.

1. Fuchsian groups of the first kind Transformation groups and quotient spaces Classification of linear fractional transformations The topological space r\tyr The modular group SL,(Z) The quotient r\o* as a Riemann surface Congruence subgroups of SL,(Z)

Chapter 2.1. 22. 2.3. 2.4. 2.5. 2.6.

2. Automorphic forms and functions Definition of automorphic forms and functions Examples of modular forms and functions The Riemann-Roch theorem The divisor of an automorphic form The measure of r\ei The dimension of the space of cusp forms

I/ xi xii xiv

1 1 5 10 14 17 20 28 28 32 34 37 40 45

Chapter 3. Hecke operators and the zeta-functions associated with modular forms 51 51 3.1. Definition of the Hecke ring 3.2. A formal Dirichlet series with an Euler product 55 3.3. The Hecke ring for a congruence subgroup 65 3.4. Action of double cosets on automorphic forms 73 3.5. Hecke operators and their connection with Fourier coefficients 77 3.6. The functional equations of the zeta-functions associated with modular forms 89 Chapter 4. Elliptic curves 4.1. Elliptic curves over an arbitrary field 4.2. Elliptic curves over C 4.3. Points of finite order on an elliptic curve and the roots of unity 4.4. Isogenies and endomorph isms of elliptic curves over C

96 96 98 100 102

viii

CONTENTS

4.5. Automorphisms of an elliptic curve 4.6. Integrality properties of the invariant j

106 107

Chapter 5.

5.1. 5.2. 5.3. 5.4. 5.5.

Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves 111 111 Preliminary considerations 115 Class field theory in the adelic language Main theorem of complex multiplication of elliptic curves 117 Construction of class fields over an imaginary quadratic field 121 Complex multiplication of abelian varieties of higher dimension 124

Chapter 6. Modular functions of higher level 133 6.1. Modular functions of level N obtained by division of elliptic 133 curves 6.2. The field of modular functions of level N rational over Q(e2x") 136 6.3. A generalization of Galois theory 141 6.4. The adelization of GL, 143 6.5. The action of U on 146 6.6. The structure of Aut() 149 6.7. The canonical system of models of r\o* for all congruence subgroups r of G I. 2(Q) 152 6.8. An explicit reciprocity-law at the fixed points of G Q.,. on 0 157 6.9. The action of an element of GQ with negative determinant 163 Chapter 7. Zeta-functions of algebraic curves and abelian varieties 167 7.1. Definition of the zeta-functions of algebraic curves and abelian 167 varieties ; the aim of this chapter 7.2. Algebraic correspondences on algebraic curves 168 7.3. Modular correspondences on the curves V s 172 7.4. Congruence relations for modular correspondences 176 7.5. Zeta-functions of V s and the factors of the jacobian variety of V s 179 7.6. 1-adic representations 189 . 7.7. Construction of class fields over real quadratic fields 197 7.8. The zeta-function of an abelian variety of CM-type 211 7.9. Supplementary remarks 220 Chapter 8. The cohomology group associated with cusp forms 223 223 8.1. Cohomology groups of Fuchsian groups 8.2. The correspondence between cusp forms and cohomology classes 230 236 8.3. Action of double cosets on the cohomology group 8.4. The complex torus associated with the space of cusp forms 239

CONTENTS

Chapter 9. Arithmetic Fuchsian groups 9.1. Unit groups of simple algebras 9.2. Fuchsian groups obtained from quaternion algebras Appendix References Index Errata

ix

241 241 243 253 260 - 265 269

NOTATION AND TERMINOLOGY 0.1. The symbols Z, Q, R, C, and H denote respectively the ring of rational integers, the rational number field, the real number field, the complex number field, and the division ring of Hamilton quaternions. For a rational prime p, zp and Qp denote the ring of p-adic integers and the field of p-adic numbers, respectively. For z E C, we denote by 2, Re (z), and Im (z) the complex conjugate, the real part, and the imaginary part of z, respectively. The symbol 0 denotes the upper half complex plane : 0= (z E C I IM (Z) > 0) .

If we discuss a Fuchsian group of the first kind r on 0, then 0* denotes the union of 0 and the cusps of r, see §§1.2, 1.3. (Therefore depends on F.) For an associative ring T with an identity element, we denote by TX the group of all invertible elements of T, and by M(T) the ring of all square matrices of size n with coefficients in T. Then we put GL(T) = M(T)". The identity element of M(T) is denoted by 1„, and often simply by 1. The transpose of X E M u (T) is denoted by X. If T is commutative, we denote by det (X) and tr (X) the determinant and trace of X E M(T), and put 0.2.

SL„(T) = {X E GL„(T)Idet(X)=1}. If there is no risk of confusion, we write T n for the product of n copies of T, and often consider the elements of T as row-vectors or column-vectors with components in T. This applies especially to the cases T =Z,Q, R, C, or H. If V is a T-module, End (V, T) denotes the ring of all T-linear endomorphisms of V. 0.3. For an arbitrary field K, we denote by Aut (K) the group of all automorphisms of K. If F is a subfield of K, Aut (K/F) denotes the subgroup of Aut (K) consisting of the elements which are trivial on F. When K is a finite or an infinite Galois extension of F, we put Aut (KIF). Gal (K/F). If x„ -• • , Xn are elements of K, F(x,, , x r,) stands for the subfield of K generated over F by x1 , , F„, of K, , x,„ (See also Appendix 1.) For subfields F„ we denote by F, F„, the composite of F1 , , e., the smallest subfield of K containing F„ ,Fm . If a is an isomorphism of K to another field, we denote by xe the image of X E K under a, so that (x7 = x'r. 0.4. The symbol

CI denotes the algebraic closure of Q in C. By an

algebraic number field, we understand a subfield of O. A prime divisor, or

xii

LIST OF SYMBOLS

simply a prime, of an algebraic number field F means an equivalence class of non-trivial valuations of F. The maximal order of F is the ring of all algebraic integers in F. If F is of finite degree over Q, a non-archimedean prime divisor of F corresponds uniquely to a prime ideal of the maximal order of F, which we simply call a prime ideal in F. If g is a fractional ideal in F, NW denotes its absolute norm, i. e., the positive rational number which generates the fractional ideal NF,e(g) in Q. Occasionally, the complex conjugate of an element x of 0 is denoted by xP.

0.5. If a and b are rational integers, we denote by (a, b) the positive integer d such that dZ = aZ±bZ (unless a= b = 0). Especially (a, b) = 1 if and only if a and b have no common divisors other than +1. 0.6. The notation [X: Y ] means the index of a subgroup Y of a group X, or the dimension of a vector space X over a field Y, especially the degree of an algebraic extension X of a field Y. The distinction will be clear from the context. If f is a homomorphism of a group to a group, the kernel of f is denoted by Ker (f). Occasionally, an isomorphism means an injective homomorphism. For example, we speak of an isomorphism of a quadratic extension of Q into WO, instead of an isomorphism of K onto a subfield of M2(Q). 0.7. The symbol id. stands for the identity map for which the set in question is clear from the context. If a map f defined on a set X is the identity map on a subset Y of X, we write f= id. on Y. 0.8. As for the terminology and notation concerning algebraic geometry, see Appendix at the end of the book.

LIST OF SYMBOLS (in alphabetical order, except for a few at the end)

30 Aut ( ) xi, 106, 141 xi C 4 (semi-group) 54, 55 4, 66 67 'Ni .4', 68,175 4' 78 4, zl(z) (discriminant, cusp form) 33,97 Ak (I ')

deg (A) (A: a divisor) 35 deg (x) (x=Ekck • (II 2« kr p)) 51 deg ()) (): a rational map) 112 det ( ) xi, 144 div ( ) 35, 36, 38 e, e,, e„ e, 107 xi End ( , ) End ( ), End Q ( ) 96,258 fa,

a

f 1 (, fa', J.:

133 146,248

LIST OF SYMBOLS

oN

137

154, 248 as 155-156, 247 Vs GA GA+, GQ, GQ,, G., G.+, Go 143-144, 242 xi Gal ( ) Gk(r) 30 GL( ) xi 6 GU(R) r, 67, 175 r[,, ro 78 ro w) 24 TN , r(N) 20, 55 rs 154, 247 73 ar23k 79 Maria,0 33 go(z), g(z) 99 g2(10 1, (02), gs(wi , cos) xi, 242 H 6 0 10, 153 0* 4 107 id xii il( ) 115 Im ( ) xi e (main involution) 72, 243 6, 28 j(a, z) 97 3E, i(E) j(z) 99 J(z) 33, 99 156, 247 frs( ) k 1,, 139, 140 144, 247 ks Ker ( ) xii 93 L(s, f, X) 66 AN xi AU ) ,

cri

N( )

xiii

xii

I„

xi P(u, co„ co,), P'(u, co1 , co0 98-99 xi Q, Q, Q, xi R Re( ) xi 54 R (1 4) p (complex conjugation) ....xii, 124 Sk(r) 30 (r1 79 xi SL„( ) 144, 247 a(x) (x E GA) 72, 176 aq 57 T(a„ — , aft)

sk ;, 0)

T(m) r(m), T'(a, d) r(m)k ,,, r(a, d)k,sb

60

70 79 tr ( ) xi, 243 86, 176 r (matrix) r(x) (x E GA+) 149,248 144 U, U N U' 174 155-156, 247 Vs W(X) 91 172 X75( ) xi Z, Zp C(s • V/ k) 167-168 192 as ; Alk, F) (T: Tx an associative ring) xi Ka, K; (K: a number field) ....115 115 Es, K3 (s E K1) 28 1 Ea3k (a E GUM) Y: groups or fields)..x ii (X, Y3 EX: k(V) (V: a variety defined over a field k) 111,255 XX) Op : a prime ; X: algebrogeometric object) ..114-115, 176

SUGGESTIONS TO THE READER This book is not homogeneously written ; it is intended for readers with various mathematical backgrounds. The reader who is familiar with elementary properties of topological groups and Riemann surfaces will have no difficulty in Chapters 1, 2, 3. In § 2.3, the Riemann-Roch theorem for a compact Riemann surface is needed. Also, in the proof of Prop. 2.15, one needs the divisibility property of the jacobian variety. Further, in § 3.5, a theorem of Wedderburn about an algebra with radical is employed. If the reader is not acquainted with any of these theorems, he is advised simply to accept the statements, since the rest of the chapters does not require them again. After the first three chapters, the reader may go directly to Chapter 8, which demands only a very elementary knowledge of homology and cohomology of groups and simplicial complexes. Chapters 4, 5, 6 presuppose the knowledge of elliptic curves and class field theory. The reader is advised to go through the Appendix before reading these chapters, to make sure of the terminology of algebraic geometry, even if he is an expert on the subject. The last section of Chapter 5 and a large part of Chapters 7, 9 are intended for the most advanced reader. The style is therefore somewhat different from the rest of the book, although the author believes that the degree of sophistication is still tolerable for inexperienced readers. There are a few exercises at the end of each section. Some of them are routine applications of the material of the text. But they are often statements of secondary importance which could be given as theorems or examples with detailed proofs in a more extensive book. At any rate, there should be no great difficulty in working them out by the methods developed in the text. Theorems, propositions, lemmas, remarks, and exercises are numbered in one sequence throughout each chapter. Displayed formulas, statements, and assumptions are cross-referred to in parentheses such as (3.5.7), which means the seventh of those in Section 3.5.

CHAPTER 1 FUCHSIAN GROUPS OF THE FIRST KIND 1.1.

Transformation groups and quotient spaces

In this section we shall discuss some elementary properties of a group of transformations acting o n a topological space. All topological groups are assumed Hausdorff. Let G be a topological group, and S a topological space. We say that G acts continuously on S, or G is a transformation group on S, if a continuous map GxS (g, s).---)gs E S is given and satisfies the following conditions : (i) (ab)s = a(bs) for a e G, b e G, s e S; (ii) es= s for all s e S, where e denotes the identity element of G. We see that, for every g e G, the map s.--.gs is a homeomorphism of S onto itself. We shall write also g(s) for gs. For every s E S, we put Gs= igsigeG), and call it the orbit of s under G, or simply the G orbit of s. Two points with the same G orbit are often called Gequivalent, or equivalent under G. We say that G acts transitively on S if there is only one G-orbit, S itself. Let us denote by G\S the set of all G-orbits of points on S. Let Tr : S .G\S denote the natural projection defined by 7r(s)=Gs. Call a subset X of G\S open if 7r -1(X) is open in S. It can easily be verified that this defines a topology on G\S, which we call the quotient topology. Then 7r is clearly continuous. Moreover, 7r is open, since if Y is an open subset of S, then 7r -1 (7r(Y))-= U gE G g(Y), and this is obviously open. It should be noted that G\S is not necessarily Hausdorff, even if S is Hausdorff. Let K be a closed subgroup of G. Consider the action of K on G by right multiplication. Then the K-orbit of an element g of G is just a left coset gK. Introduce the quotient topology in GIK as above. The closedness of K implies that G/K is Hausdorff. To show this, let aKbK. Define a continuous map f: GxG .G by f(x, y). x - 51. Then (a, b)EE f -,(K). Since 1.- '(K) is closed, there exist open sets U resp. V containing a resp. b, such that (Ux V) n f-1(K) =0. If h: C .GI K is the natural projection, this means h(U)nh(V). 0, q. e. d. Now let G act on G/K as usual by the rule g.(xK)=gxK for ge G, xe G. The map (g, xK). gxK of Gx(G/K) to G/K is obviously continuous. Furthermore, this action is transitive. Let S be an arbitrary Hausdorff space on which G acts continuously and transitively. Fix any point t of S, and put K= {ge Glgt=t). Then K is a closed subgroup of G, and called the isotropy subgroup of G at t, or the stability -

-

—

—

—

—

2

FUCHSIAN GROUPS OF THE FIRST KIND

There is a natural one-to-one map 2:G/K .S defined by 2(gK) = gt. For any subset X of S, one has 2-1 (X). h({ge GI gt e X }), where h is the projection map : G .G/K. This equality shows that 2 - (X) is open if X is open. Hence A is continuous. But 2 is not necessarily a homeomorphism. One can at least prove the following criterion :

group of t.

—

—

THEOREM 1.1. The map 2:G/K—..3 is a homeomorphism if both G and S are locally compact, and G has a countable base of open sets. Let U be an open set in G, and let ge U. It is sufficient to show that gt is an interior point of Ut. Take a compact neighborhood V of the identity element of G so that V=V' and gV 2 c U.. If Vt contains an interior point vt with v e V, then gt= gtr'vt is obviously an interior point of Ut. By our assumption, G is a union U.g,,V with countably many (g.) c G. Then S= U. g„V t, and Vt must contain an interior point, on account of the following Lemma, so that our theorem is proved. PROOF.

LEMMA 1.2. Let S be a (non-empty) locally compact Hausdorff V„ , V., ••• be countably many closed subsets of S such that S=U4Z- 1 V. Then at least one of the V. has an interior point. Assuming that no V„ has interior points, let us derive a contradiction. Take a non-empty open subset W1 of S whose closure W, is compact. Define W2, W„ -•• successively so that W. is non-empty and open, and W. + , C V„. Then the Win form a decreasing sequence of non-empty compact sets, hence (1. W„*0 . But this is a contradiction, since the intersection is disjoint with any V., q. e. d. PROOF.

PROPOSITION 1.3. Let G be a topological group acting continuously on a locally compact Hausdorff exists a compact subset C of S such that GC=S. Let ir denote the natural map of S to G\S. If GC=S, we have x(C)=G\S, so that the 'if '-part is obvious. Conversely, cover S by open sets with compact closures, and map them by 7r. If G\S is compact, we have G\S = U t t ) with finitely many open sets th whose closures C/i are compact. Then S = G • (Ut CI t ), q. e. d. PROOF.

Let G be a topological group. In general, a subset M of G may have limit points in G even if the induced topology of M is discrete. But, for a subgroup of G, we have : PROPOSITION 1.4. Let r be a subgroup of G. Suppose that the induced topology of r is locally compact. Then r is closed in G. Especially, if r is discrete, then r is closed, and has no limit point in G.

1.1

3

TRANSFORMATION GROUPS AND QUOTIENT SPACES

We call discrete.

r a discrete subgroup

of G, if the induced topology of

r

is

r

PROOF. Suppose that has a compact neighborhood C of the identity element e. Take an open neighborhood U of e in G so that Unr c C. Let x be an element of the closure of We can find a neighborhood V of x so that V -1 V c U. Then (v n r)i(v c C. Note that Vnr and take an element y of Vn E. Then Vnr c yC. Now for every neighborhood W of x, we have Wr)Vnr #0, hence x belongs to the closure of V" r. Since is closed. The last assertion is obvious. yC is compact, xe yCcr, hence

r.

0,

r

PROPOSITION 1.5. Let G be a locally compact group, and K a compact subgroup of G. Put S=G/K, and let h:G—S be the natural map. If A is a compact subset of S, h -1 (A) is compact. Take an open covering of G whose members have compact closures, and consider their images on S by h. Then we see that A c Ui h(Vi) with finitely many open sets lit whose closures are compact. Hence h -1 (A) cUt 17,K. Observe that I7,K is compact. Therefore, h-1 (A), being a closed subset of a compact set, is compact. PROOF.

r

PROPOSITION 1.6. Let G, K, S, and h be as in Prop. 1.5, and a subgroup of G. Then the following two statements are equivalent: is q discrete subgroup of G. (2) For any two compact subsets A and B of S, fger Ig(A)nB# 0) is a finite set.

(1)r

Let A and B be compact subsets of S, and let C=I1 - '(A), D= h -1(B), ge If g(A)nB# 0, one has gCr\D# 0, hence g rn(DC'). is discrete, By Prop. 1.5, C and D are compact, hence DC-1 is compact. If r n 3c-1) is both compact and discrete, hence must be finite. This shows (1) (2). To prove the converse, let V be a compact neighborhood of e in G, and let t= h(e). Then I' nV c { g g t h(V)}. Viewing t and h(V) as A and B of (2), we find that rn V is a finite set. Therefore is discrete. PROOF.

r.

r

(

r

Hereafter till the end of this section, G, K, S, h will be the same as in Prop. 1.5, and a discrete subgroup of G. By (2) of Prop. 1.6, {ge 11 g(z)=z} is a finite set for every z e S.

r

1.7. For every z

S, there exists a neighborhood U of z such that (gerig(U)r\U#01={ge rig(z)=4. PROPOSITION

Let V be a compact neighborhood of z. By Prop. 1.6, {geri o} is a finite set, say (g„ , g, ) . Suppose that gi (z)=z or * 2

PROOF.

g(v)nv

E

4

FUCHSIAN GROUPS OF THE FIRST KIND

according as 1 s or s r. For every i> s, take a neighborhood Il i of z and a neighborhood Wi of g(z) so that. Vi W=ø, and put U= V n {n i>,(V e(W i))}. Then U has the required property.

PROPOSITION 1.8. If two points z and w of S are not I-equivalent, then there exist neighborhoods U of z and V of w such that g(U)nV =0 for every

g e r. Let X and Y be compact neighborhoods of z and w respectively. By Prop. 1.6, { g Er I g(X)(-)Y0} is a finite set, say {g„••• ,g„.}. Since z and w are not 1-equivalent, we have g,(z)ow for every i. Therefore we find neighborhoods Ui of g(z) and V i of w such that Ur V=0. Put U= X n gi-1(U1) ng,Two, v =Y nv ,- V t.. Then U and V have the desired property. PROOF.

Let I\S denote the set of all 1-orbits of the points of S. Prop. 1.8 implies that T\S, with the quotient topology, is a Hausdorff space. Now we have an obvious commutative diagram : G

T\G

*-S=G/K

r\S=I\GIK

We see easily that all maps in this diagram are open and continuous.

PROPOSITION 1.9.

r\G is

compact if and only if I\S is compact.

By Prop. 1.3, if I\S is compact, we have S. IC with a compact subset C of S, so that G= • h - '(C). By Prop. 1.5, h - J(C) is compact, hence, by Prop. 1.3, r\G is compact. The converse part is obvious. PROOF.

PROPOSITION 1.10. Let G, and G2 be locally compact groups, I a closed subgroup of G 1 xG2, and r, compact.

(1) (2) (3)

Then the following assertions hold: r, is closed in G,. /VG, x G 2) is compact if and only if is discrete in G 1 XG 2, then ri If

r,

Let V be a compact neighborhood of the identity in G 1. Then (VX / is compact, and Vnri G 1 XG 2 to G,. Therefore v n r, in G,. If further I is discrete, then (V X GOn / is finite, so that V nr1 is finite, hence (3). The assertion (2) follows easily from Prop. 1.3. PROOF.

1.2

CLASSIFICATION OF LINEAR FRACTIONAL TRANSFORMATIONS

r

rt r

5

In general, two subgroups and of a group G are said to be commensurable if is of finite index in and in The following proposition can easily be verified, and may therefore be left to the reader as an exercise.

r nr/

PROPOSITION

1.11. (1)

r.,

mensurable with

r

If

r,.

r,

r, 1.'1 r,

(2) Let and be commensurable subgroups of a topological group G. If is discrete, then is discrete. (3) Let and be commensurable closed subgroups of a locally compact group G. If is compact, then is compact.

r

r r\G

r,\G

1.2. Classification of linear fractional transformations Although our main interest is in the transformations on the upper half plane, let us first consider more generally a linear fractional transformation on CU {co } .

For a =La db ] E

GL,(C) and z e CV./{co), put a(z) , (az+b)1(cz+d).

Suppose that this is not the identity transformation, i. e., a is not a scalar matrix. From the theory of the Jordan canonical form, we see that the matrix a is conjugate to one of the following two forms :

(ii)

(i)

20p.

a] ,

[02

Therefore, our transformation is essentially of the following types : ;

(i)

(ii)

z

cz ,

c

1.

In the first case, we call a parabolic. In the second case, we call a elliptic if c I =1, hyperbolic if c is real and positive, and loxodromic otherwise. This definition applies to both matrices and transformations. The identity transformation is excluded from this classification. We see that the number of fixed points of a is one or two, according as a is parabolic or not. If we impose the condition det (a)=1, then the classification can be done by means of tr (a): PROPOSITION

1.12.

Let a

E

a is parabolic

PROOF.

SL,(C),

a

0 +12. Then

t=5 tr (a) , ±2,

elliptic

tr (a) is real and I tr (a)I< 2 ,

hyperbolic

tr (a) is real and I tr (a)I> 2,

loxodromic

tr (a) is not real.

Since det (a)=1, the Jordan canonical form for a is either

6

FUCHSIAN GROUPS OF THE FIRST KIND ±i 1 0 0 4.1 ] or [ 0 2 _,], 2* ±1.

Therefore the first three = and the first

[

and tr (a). is can easily be checked. Now suppose that a =[ 2 ° 0 2-' real. If A is real, a must' be hyperbolic. If A is imaginary, A and ,1 are the roots of the equation x 2 —tr (a)x Fl= 0, hence Ai= 1. Therefore a is elliptic. Thus a cannot be loxodromic if tr (a) is real. This proves the last =. Since the conditions on the right hand sides are mutually exclusive, this completes the proof. -

Let us now restrict ourselves to the transformations with real matrices. For z

E

C and a=[

qs ]

GL,(R), put

(1.2.1)

j(a, z)=rz-Fs.

If w= a(z), we have

a _ z+b, , rzir Lcz±dJ=L 1- 1

`4

(a, z).

Further if w' = (1.2.2)

=[

a z

L1

[ j (a6 z)

(a0 zo ]

Substituting 2 and u for z' and w', and taking the determinant, we obtain (1.2.3)

det (a) - lm (z)=Im (a(z)) • I j (a, 2)1 2 .

Let 0 denote the complex upper half plane, i. e., 0= {z E C I Im (2) > 01 . Further, put GLgR)= {a e GL 2(R) I det (a) > O). If a e GL:(R), a maps 0 onto itself. It is also well known that every holomorphic automorphism of 0 is obtained from an element of GLt(R). a induces the identity map if and only if it is a scalar matrix. Therefore the group of all holomorphic automorphisms of 0 is isomorphic to GL.1 - (R)/CRx • 121 and to SL2(R)/{±1 2 }From (1.2.2) we obtain easily (1.2.4)

j(ap, z)= j(a, p(z))j(13, z).

Furthermore, substituting z Fdz (formally) for z' in (1.2.2), and taking the determinant, we obtain -

(1.2.5)

d

dz

a(z)= det (a) • j(a, z) -2 .

CLASSIFICATION OF LINEAR FRACTIONAL TRANSFORMATIONS

1.2

If a = [Pr gs ]E SL,(R) and i=s/L we have a(i)=i if and only if p=s,

q= r, p2 +q2 = 1. —

Therefore, the special orthogonal group

SO(2)= {a E SL 2(R) I 'era = 12} is the isotropy subgroup of SL 2(R) at i. The action of SL 2(R) tive, since, for a >0, a - "2 •

[g in sends i to ai+b.

Therefore, by Th. 1.1, œfp

is homeomorphic to SL 2(R)/S0(2), through the map a'. cr(i). We shall now study more closely the transformations obtained from the elements of SL 2(R). By Prop. 1.12, SL(R) contains no loxodromic transformations. For every z E t), we can find an element r of SL,(R) so that r(i).z. Then r • SO(2)• {a E SL2(R) I cr(z)= z} • -

Since every element of SO(2) has characteristic roots of absolute value 1, this shows that an element of SL 2(R) with at least one fixed point in must be either +12 or elliptic. For every s e RU {co}, put

F(s)= {a E SL2(R) a(s)= sl , P(s)= la E F (s) I a parabolic or = ±12 }. Since SL,(R) acts transitively on RU ico 1, we can find an element (l of SL 2(R) so that o(co)=s. Then F(s)= F(00)a-1, P(s)= aP(oo)a- '. Now we see easily that F(00) = {[ oa ab _d I a E R", b

R},

p (co) = { ± [ 01 hi ] I h E 14=:..Rx{±-.11. This shows that if an element a of SL,(R), # +1 2, has at least one fixed point on RL) fool, then a is either parabolic or hyperbolic. From these considerations, we obtain PROPOSITION

a is parabolic

elliptic

1.13.

Let a e SL 2(R), a # ±12 .

Then

a has only one fixed point on RU {co},

4=:+ a has one fixed point z in 0, and the other fixed point 2,

hyperbolic ,:=) a has two fixed points on RU (co). PROPOSITION 1.14. Let a E SL,(R), a # +12 , and let mEZ, a# +12. Then a is parabolic (resp. elliptic, hyperbolic) if and only if am is parabolic (resp. elliptic, hyperbolic).

8

FUCHSIAN GROUPS OF THE FIRST KIND

The 'only if '-part follows immediately from Prop. 1.13 or the Jordan form of a. Then the 'if '-part is obvious. PROOF.

EXERCISE 1.15. Let a and p be elements of SL,(R), # +1 2, such that ap = pa. Prove : (1) If a is parabolic (resp. elliptic, hyperbolic), then is parabolic (resp. elliptic, hyperbolic). (2) If a(z)=. z for some z E CV {co}, then j3(z)=z.

p

Let us now fix a discrete subgroup r of SL,(R). A point z of 0 is called an elliptic point of r if there exists in elliptic element a of r such that cr(z)= z. Similarly, a point s of Rk...) {co} is called a cusp of r if there exists a parabolic element r of r such that z-(s)= s. If w is a cusp (resp. an elliptic point) of r and r e r, then we see easily that r(w) is also a cusp (resp. an elliptic point) of r. PROPOSITION

1.16. If z is an elliptic point of

r,

then {a E

r Ia(z)= z}

is

a finite cyclic group.

If reSLXR) and r(i)= z, we have { aerki(z)=z}=rS0(2)1-- 'nr. Since r is discrete and SO(2) is compact, this intersection must be a finite group. Now SO(2) is isomorphic to RIZ, and its finite subgroups are all cyclic, q. e. d. PROOF.

r,

r, =

r 1 c(s) = s). Then r, is either -I- 12

PROPOSITION 1.17. Let s be a cusp of and {a E 1-',/(1-1 n{-E1 2 }) is isomorphic to Z. Moreover, an element of nP(s). or parabolic, i. e.,

r, =r

We have seen that P(s) is isomorphic to Rx {±1}. Therefore, (P(s)nr)/(r n {±1}) is isomorphic to a non-trivial discrete subgroup of R, hence isomorphic to Z. Now, without losing generality, we may assume that + 01 _i_ hi ] s= co. Take a generator a =[-(modulo ±1) of this group. Assume ra b 1 contains a hyperbolic element 7= that l a l # 1. Taking 7 --1 L 0 a - 'J' instead of r, if necessary, we may assume that lai <1. Then raz- -1 PROOF.

r,

±1 a

=[

fore

0 :_t. 1 :11 e P(s)r'['. But this is a contradiction, since l a2h l
-

r,=P(s)nr.

1.18. The elements of elements together with +1 2. PROPOSITION

PROOF.

r of finite order consist

of the elliptic

If an element a of SL,(R) is of finite order, a is conjugate in

SL 2(C) to a matrix [; 9 ] with a root of unity C. By our definition, such

0(

CLASSIFICATION OF LINEAR FRACTIONAL TRANSFORMATIONS

1.2

9

a a is elliptic if C +1. The converse part follows immediately from Prop. 1.16.

PROPOSITION 1.19.

The set of all elliptic points of

r

has no limit point

in 0. PROOF. Assume that there is a sequence of distinct elliptic points {z„} of r converging to w e 0. By Prop. 1.7, we can find a neighborhood U of w such that, for 1 r, r(u) nu #o if and only if T(w)=w. For sufficiently large n, we have z U and z„* w. One has r(z n)-= z„ for some elliptic element r of r. Then r(U)r'\ U# 0, hence r(w) , w. Thus r has two fixed points on 0, a contradiction. Each matrix of SL 2(R) (or of GLAR)) should not be confused with the transformation on 0 represented by it. Especially one should be careful about the order of an elliptic element : PROPOSITION 1.20. Let a be an elliptic element of r. If a, as a matrix, is of an even order 2h, then r contains —1 2 , and the transformation z.--40.(z) is of order h. PROOF. One can find an element r of GL,(C) so that rar-i =[05 9.] with a primitive (2h)-th root of unity C. Then C' = —1, hence ai —1„ q.e. d. COROLLARY 1.21. is of an odd order.

If

r

does' not contain —1 2 , every elliptic element of

r

This is an immediate consequence of Prop. 1.20. To distinguish the transformation group from the matrix group, we shall denote by r the image of r by the natural map

SL 2(R) For an elliptic point z of

r,

SL2(R)/{±-1 2} -

the order of the group {a e

r 10
is called the order of the elliptic point z (relative to

r).

1.22. Neither elliptic nor parabolic element a of SL,(R) conjugate in SL 2(R) to a'. PROPOSITION

is

PROOF. Assume that Tar'. a - i for some r E SL 2(R). If a is elliptic, as is observed above, there exists an element r of SI4R) such that rcur-1 e SO(2). 4 ] and rrr -1 Lca dbi Then we have q *0 since a is Put rar '=[ — Pq P elliptic, and -

10

FUCHSIAN GROUPS OF THE FIRST KIND

ía b11 p qi_rp Lc (In —q 15J — Lc/

—nra pnc di'

d, b=c. Then 1= det (7-)= —(a 2±b2), which is impossible since ri a and b are real. If a is parabolic, we can take z- so that ray' 1=±L o a bir 1 hi Then [ d][0 1 i = [0 ca db 1 so that c= 0, a = d, hence 1= det (r) = —a', which is again impossible. so that a=

—

—

Note that a hyperbolic element a is conjugate in SL,(R) to a - '. 1.3.

The topological space

r\o*

Hereafter till the end of this section, we denote by r any discrete subgroup of SL2(R), and by 0* the union of 0 and the cusps of r. The set 0* depends on f ; of course 0*= 0 if r has no cusps. We observe that r acts on 0*, hence the quotient space r\o* is meaningful. We shall consider a structure of Riemann surface on r\tyi, in the next section. For that purpose we first define a topology of 0*. For every z e 0, as a fundamental system of open neighborhoods of z, we take the usual one. For a fundamental system of open neighborhoods of a cusp s* co, we take all sets of the form : s} {the interior of a circle in 0 tangent to the real axis at s . If co is a cusp, we take the sets }

(1.3.0)

{co } U {z e

I 1m (z)> cl

for all positive numbers c, as a fundamental system of open neighborhoods of 00. We shall write (1.3.0) also as {z 0*1Im (z)> c } . It can easily be seen that this defines a Hausdorff topology on 0*, and every element of r acts on 0* as a homeomorphism. However, 0* is not locally compact, unless 0'0 =0. For a cusp sœ of r, put P(s)= faeSL,(R)Ia(s)=s, a parabolic or = ±.1 21 -

T.= P(s)nr = {rE

I T(s)= s}

(see Prop. 1.17).

The neighborhoods of s of the above type are obviously stable under P(s). To study the structure of r\o*, let us assume that co is a cusp of r. We need the formula (1.3.1)

Im (a(z))= det (a) 1m (z)/1 cz+dI 2

for a =[ca db ] e GL,(R),

which was proved in § 1.2. For every a e r, we let c, denote the lower left entry of the matrix a. Then roo= {a e r J c,,= 0 } . By Prop. 1.17, we can find 1 h a generator ±[ o 1 ] of r. modulo ±12.

11

THE TOPOLOGICAL SPACE r\t*

1.3

I I depends only on the double coset

LEMMA 1.23.

This can be verified by a simple matrix computation. LEMMA 1.24. Given M> 0, there are only finitely many double cosets such that a E and Icu l M.

r

PROOF. Since

r,,,,={0ErIcu=o}, it is sufficient to consider only those

a for which c, O. Take a generator z= ±[ 01 hi ] of roe modulo ±1 2. Let [ da bl E I 0*Icl M. We are going to find an element a* in r,J1r.. a= c c/J such that aff(i) is contained in a compact set K which depends only on M and h. First we can find an integer n so that 1..-Fnhc_1-Flhcl. Put n = fa' 6' 1 1 Then I c' I = ICI, d' I = d + nhc. By (1.3.1), lm (a'(1)) Lc' d,J. a=" = 11(c" - F d'2). We have 1 d' 1 1+ I hc I, and l c IM, hence 1 c"-Fd" < M 2 +(l+ IhiMY . Therefore a'(i) belongs to the domain

1 Im (z):1/[1t1 2-F(1+Ih1M)2].

(1.3.2)

Now the transformation z.-4rni(z). z-Fmh does not change Im (z). take m so that el a'(i) satisfies (1.3.2) and

0

(1.3.3)

Re (z)

We can

hi.

The conditions (1.3.2) and (1.3.3) define a compact set K in 0. We have thus found an element aff = rinarn finitely many such a" in r. This proves the lemma.

r,

There exists a positive number r, depending only on for all a E Moreover, for such an r, one has Im (z)-Im(a(z))11r 2 for all z E and all a E

LEMMA 1.25. such that

r —roo.

PROOF. The existence of r follows immediately from Lemma 1.24. If Cr=[

cci t ] (is

e

r and

c * 0, we have

Im (a (z)) = Im (z) • I cz-F dl -2 Im (z) (c lm (z)) -2 r- Im (z) -1,

q. e. d.

r,

LEMMA 1.26. For every cusp s of there exists a neighborhood U of s in 0* such that a(U) r\U * 0}. E

r,={0. rI

PROOF. We may assume that s=œ. Let U= E I IM (Z) > 1/r}, with a number r of Lemma 1.25. If a e r—r , and ze U, we have, by Lemma 1.25, Im (a(z)) < 1/r. Thus U has the required property.

r

Observe that two points of the set U are equivalent under only if they are so under r,, and hence Au may be identified with a subset of moreover U" contains no elliptic point of

r

r.

r

12

FUCHSIAN GROUPS OF THE FIRST KIND

LEMMA 1.27. For every cusp s of r and for every compact subset K of D, there exists a neighborhood U of s such that Un1(K)=0 for every r Er. Assume again s=00. We can find two positive numbers A and B so that A < Im(z) Max (B, 1/A r2)} . PROOF.

Let ZE K. By Lemma 1.25, if crer—r., Im (a(z)) < 1/Ar2. If a Er., im (cr(z)) = Im (z)< B. Thus U has the required property. Let us now consider the quotient topology of r\* as defined in § 1.1. Namely we take {Xcr\D* Jr -1 (X) is open in D*) to be the class of all open sets in r\D*, where r is the natural projection of D* to r\D*. If U is as in Lemma 1.26 (and its proof), then r(U) can be identified with T A U, and is a neighborhood of r(s). THEOREM

1.28. The quotient space r\D*, with the above topology, is a

Hausdorff space.

By Prop. 1.8, r\D is a Hausdorff space. Since r\D* is the union of r \D and the equivalence classes of cusps, it remains to show that an equivalence class of cusps can be separated from an equivalence class of points in D, and also from another equivalence class of cusps. Lemma 1.27 takes care of the former case. Therefore let us consider two cusps s and t which are not r-equivalent. Without losing generality, we may assume t= co. PROOF.

Let

r„ and ±[

h7 1J

as before. Define three\sets L, K, and V as follows:

L=(z e C 1 Im (z)= u} , K= fzELIO5_Re(z) u} , where u is a positive number. Since K is compact, we can find, by Lemma 1.27, a neighborhood U of s so that Kn PU-= 0. We may assume that the boundary of U is a circle tangent to the real line R. Let us show that VnTU=0. Assume, on the contrary, that r(U)r) V # 0 for some r e r. Since r(s) # co, the boundary of r(U) is a circle tangent to R. Therefore, if r(U)r) V # 0, then r(U)nL #0, hence r(U) intersects some translation of K by an element of r, i. e., there exists an element 3 of r. such that r(U)n6(K)* 0. Then 3- IT(U)r\ K# 0, a contradiction. This completes the proof. PROPOSITION 1.29.

The quotient space

r\o* is

locally compact.

THE TOPOLOGICAL SPACE

1.3

r\o.

13

Our task is to show that if s is a cusp of T and if 7 denotes the natural map of ef,* to r\el*, then r(s) has a compact neighborhood. We may assume that s = 00. By Lemma 1.26 and the remark after it, there is a neighborhood V = {z E I Im c} with a positive constant c such that (modulo ±1), VIT., is identified with r(V). If [ 0 1 h ] is a generator of 1 we see that r(V) coincides with the image of { z e V I z= 00 or 0 7.i: Re (z) hi } by r. The latter set is obviously compact, hence r(V) is compact, q. e. d. (See also § 1.5, where we shall show that r\o* has a structure of a Riemann surface.) PROOF.

PROPOSITION 1.30. Let r and lit be mutually commensurable discrete subgroups of SL2(R) (see p. 5). Then r and rl have the same set of cusps. It suffices to consider the case in which ri z:r and Er :ri]
PROPOSITION 1.31. Let r and r, be as in Prof,. 1.30. compact if and only if ri\o* is compact.

Then

r\D* is

Again we may assume that ri cr , [r: r']
PROPOSITION 1.32. If r\tyt is compact, then the number of rinequivalent cusps (resp. elliptic points) is finite. Let C (resp. E) denote the set of all cusps (resp. all elliptic points) so that U,nE is of T. For each take a neighborhood U, of z in either empty, or possibly (z). This is possible in view of Prop. 1.7. By Lemma 1.26, for each se C, we can find a neighborhood U, of s containing no elliptic points. Let r denote the projection map of t)* to r\f,r. If r\t* is compact, we can select a finite number of sets of the form 7r(U1) or r(U,) which cover r\to.. Then the number of points in r(C) (resp. :r(E)) is at most the number of r(Us) (resp. raa, which are necessary to cover r\o*, q. e. d. PROOF.

FUCHSIAN GROUPS OF THE FIRST KIND

14 PROPOSITION

1.33. If

r\D is

compact, then

r has

no parabolic element.

rw.

Let r denote the projection map of 0 to Suppose that co is a cusp of T. Take an infinite sequence { z. } of points of 0 such that In(zi,)--- co. By Lemma 1.26, there exists a neighborhood PROOF.

U= {z E cti* I IM (z) > cl of cc such that r.= fr e T i r(U)n U #0). Then z„ e U for sufficiently large n. Since no element of changes Im (z),. if two points of { zi,} have distinct and sufficiently large imaginary parts, then they are not r-equivalent. Therefore {r(z7,)} contains a sequence of infinitely many distinct points of is compact, there exists a point w of 0 such that r(w) is a limit point If of {r(z,)). Let K be a compact neighborhood of w. By Lemma 1.27, there exists a neighborhood V of co such that Kr) r V =O. This is a contradiction, since r(z„)er(K)nr(V) for sufficiently large n.

r.

r\o.

r\o

1.4. The modular group SL 2(Z) In this section we shall illustrate the preceding discussion by studying the modular group SL 2(Z). It is clear that SL 2(Z) is a discrete subgroup of SL,(R). Let us determine its cusps and elliptic points. are exactly the points in First let us show that the cusps of Qu {co } . It is clear that oo is a fixed point under the parabolic element

r=SL2(Z)

1] ri L01 point s.

r.

If [a c d ] is a parabolic element of If s is finite, it satisfies

of

cs2+(d a)s b =0 , —

—

r, it has only

one fixed

c #0O.

Since the discriminant of this equation vanishes, s must be contained in Q. Conversely, for P/qe Q with pE Z, q E z, (p, q)= 1, take integers t and u so that pt—qu =1. Then a .[P q t ]e l', and a(oo)=p/q.

Since the image of a

r

is a cusp, this shows that all points of QU {co) cusp under any element of Moreover we have shown that all cusps are equivalent to are cusps of the cusp at co. Thus /1 \0* = (r\t) u {c0}. Next let us determine the elliptic points of SL 2(Z). If a is an elliptic element of SL 2(Z), I tr MI is an integer and <2 by Prop. 1.12. Therefore the characteristic polynomial of a is either x 2 +1 or x 2 +x+1, so that cri =1 or as= 1, and cr2 # 1". If 1 8 = 1, we have a' = +1. In the case a' = 1, we have

r.

—

1) One can reason also as follows: If C is a characteristic root of a, satisfies a quadratic equation with rational coefficients, so that [Q(C) : Q]2. Therefore en=1 with m=2, 4,3, or 6. This reasoning is applicable to the case with an algebraic number field of higher degree in place of Q. :

1.4

THE MODULAR GROUP SL2(Z)

15

(-6 )3 = 1. Thus, for the determination of elliptic elements (or points), it is sufficient to consider the cases a'= 1 and 63 =1. = 1. Let Z 2 denote the module of all column vectors [ a ] with b a and b in Z. Let the elements Z[a] act on Z2 by left multiplication. Since Z[a] is isomorphic to Z[i], Z[a] is a principal ideal domain. The module Z2 over Z[cr] is torsion-free, since (a+ba)x=0 implies (a2+Y)x= 0, hence x=0, if a+ba O. Therefore Z 2 must be a free Z[a]-module of rank 1, i. e., Z2 = Z[a]u for some u Œ Z2 . Put v= au. Then u and y form a basis of Z2 over Z. We have CASE 1: a

a • Di

=

ro —i -' Li

0

det En y] = +1.

If det Eu y] = 1, this shows that a is conjugate to

—01 ] in SL.,(Z).

If

0 1 1 , det Eu y] = —1, then a with r=[v u]. Thus every elliptic =71 1 o element a in SL,(Z) of order 4 is conjugate to ±[? —01 ] in SL,(Z). -

Therefore every elliptic point of order 2 is equivalent to the fixed point of 1 ro —10 10 ] on account of 0 j, that is j. ([ °1 — 01 is not conjugate to [ Li Prop. 1.22. )

2: a' =1. We see that Z[a] is isomorphic to Z[e 2'"], which is a principal ideal domain. Therefore we have again Z2 = Z[a]u for some u. Put u = au. Then ,r0 —11 a •[u v]=[u vJLi det[u y] = -±1 . ' CASE

Or r 2 = Therefore a is conjugate to either r [21.i 1 in SL 2(Z). = [1 — 1 -1 Thus every elliptic point of order 3 is equivalent to the point e2". (r is not conjugate to r2 in SL 2(R), see Prop. 1.22.) ]

For any discrete subgroup T of SL 2(R), we call F a fundamental domain for r\cti (or simply for r), if (i) F is a connected open subset of (D, (ii) no two points of F are equivalent under V ; (iii) every point of t• is equivalent to some point of the closure of F under T. It can be shown that every T has a fundamental domain. An explicit construction of a fundamental domain for a given r and its exact shape have been the object of much research. Here we shall not go into the details of this topic, but just find the standard fundamental domain for r.sL2(z).

16

FUCHSIAN GROUPS OF THE FIRST KIND

Let z e .f),

and a =[ a db ]

E

SL 2(Z).

Then Im (a(z))=Im(z)/ f ez±c1 1 2.

Now {cz Fdic EZ, de Z} is a lattice in C. Therefore Min I cz+dl, for (c, d) * (0, 0) with c EZ, de Z, exists. Thus, for a given z, Max, er 1m (o(z)) exists. 01 If a is such that Im (a(z)) is maximum, and w = a(z)= x±iy, r =[_ 1 0 ], then -

Im(ra(z))=Im( - 1/w)=Yilw1 2 -5Y, hence I wl .. 1.

rl li we have Im (rha(4)=Im (a(z)) for every h E Z, L 0 1 J' 1. Choosing a suitable h, we see that z is equivalent to a

If r =

hence I rha(z)I point of the region

{wECI —1/2 _Re(w).. 1/2, lwl >1} . Let us show that the interior F of this set is a fundamental domain for SL,(Z). Let z and z' be distinct points of F. Assume that z' = a(z) with

CI =[ ca d1 ] E r. We may assume that

Im (z)

Im (z')= Im (z)/Icz+d1 2. Then

1c1.1m(z)_5_1cz+d1_1.

(*)

If c= 0, then a = d= +1, hence z' = z+b, which is impossible. Therefore c * 0. Looking at the shape of F, we observe that 1m (z)> -Oa, hence by (*), I cl = 1. Then from (*) we obtain lz+c/I.. 1. But if z E F and I cil 1, we have I z±dl> 1. Therefore we must have d = 0, so that IzI .. 1. This contradicts that z E F. Thus we have proved that F is a fundamental domain for r. It can easily be verified that the set FI =FU{zeC11z1 > 1, Re(z)= —1/2}U {zECI Izi= 1, —1/2 Re(z)-_ 0} -.

•

rvv* =(rvt)u }

is a set of representatives for modulo r. It follows that {co is compact. By Prop. 1.31, T'Vt* is compact if is a discrete subgroup of SL,(R) commensurable with SL,(Z).

r,

1.34.

Give another proof for the results about the elliptic points of SL 2 (Z) by determining such points belonging to F'. 11 The modular group SL,(Z) is generated by two elements a =[0 1 ] and EXERCISE

r.: [01 —1] 0 . To show this, let T be the subgroup of SL,(Z) generated by a and r. Then —1 = r2 E T. Observe that every element of SL 2(Z) of the form r a bi [0* *4 ] is contained in T, and if b 1 E T, then r - c -d = 1- • L a b] U dJ L c di E T. Suppose T # SL 2(Z), and take an element [ca db ] of SL 2(Z) T so that Min (1a1,Ic1) is the smallest among such elements. We may assume —

1.5

THE QUOTIENT TAD* AS A RIEMANN SURFACE

17

lallcl > O. Take integers q and r so that a= cq l r and 0..r < I cl. Then a _ gr- a bi . r r *ITT, and r=Min(r, IcI)
EXERCISE 1.35. Let P denote the closure of F, and A the subgroup of SL,(Z) generated by the elements a such that a(P)n.P# 0. Let U be the union of r(P) for all r e A. Using the connectedness of 0, show that U=(71, A = SL,(Z), and SL,(Z) is generated by a and r. Observe that this method is applicable to any r for which a fundamental domain is (explicitly) given.

1.5.

The quotient

r\e,*

as a Riemann surface

Throughout this section, r will denote a discrete subgroup of SL,(R), and .0* the union of .0 and the cusps of r. Recall the main result of § 1.3 which asserts that r\,* is a Hausdorff space. By a Riemann surface, we shall mean, as usual, a one-dimensional connected complex analytic manifold. More specifically, a Riemann surface is a connected Hausdorff space TB on which there is defined a "complex structure" S with the following properties: (1) S is a collection of pairs (11„, pa) with a in a set A of indices, where {U„}crEA is an open covering of Ti), and pa is a homeomorphism of 1.1„ onto an open subset of C. (2) If U anUp#0, the map As 0 Ail : Pa(U. n (J)- P,3(u.('Up)

is holomorphic. (3) S is maximal under the conditions (1) and (2). The map pa is often called a local parameter at a point contained in U. Requirement (3) is not essential, since given any S satisfying (1) and (2), there exists a unique complex structure S' containing S. In fact, S' is given as the set of all pairs (V, q) formed by an open subset V of S and a homeomorphism q of V onto an open subset of C such that per oe and qopV are holomorphic whenever Vnu a.* 0. Let us now define a complex structure on r\el*. Denote by 9 the natural projection map of 0* to rw. For each veto*, put

I-, = fr E r i go. 0 . By Prop. 1.7 and Lemma 1.26, there exists an open neighborhood U of y such that

r0={rErigu)nu*o} .

18

FUCHSIAN GROUPS OF THE FIRST KIND

row

Then we have a natural injection rvw----.r\o*, and is an open neighborhood of 92(y) in r\O*. If y is neither an elliptic point nor a cusp, r. contains only 1 and possibly —1, so that the map go : u ----.r,,\LI is a homeomorphism. We take (ro p-i) as a member of the complex structure of r\b*. Next assume that y is an elliptic point, and denote by Pi, the transformation group (T',,. I+ 1})/{± 1 } . Let A be a holomorphic isomorphism of to onto the unit disc D such that 2(y) =O. If ii„ is of order n, then 21%2 -1 consists of the transformations c = e 2rVn . W . .) W, k= 0, 1, •-• , n— 1,

w, g

-

Then we can define a map p : rvw----.0 by p(v(z)) = 2(z)n. We see that p is a homeomorphism onto an open subset of C. Thus we include (T„\U, p) in our complex structure. Let s be a cusp of r, and let p be an element of SUR) such that p(s)= 00. Then

pr3 p-1.

{±1} = {±[ 01 hl ] 1 m e Z}

with a positive number h. Then we can define a homeomorphism p of rsw into an open subset of C by p(99(z))= exp [27rip(z)/h], and include (rs\u, p) in our complex structure. It is now easy to check the condition (2) for our complex structure. Thus we have been able to make TVD* a Riemann surface. By abuse of language, we sometimes call a point of r\o* an elliptic point or a cusp, if it corresponds to an elliptic point or a cusp on el* with respect to r. EXERCISE 1.36. Let lf" be a subgroup of r of finite index. the natural map of rAvic to TW is holomorphic.

Prove that

Let us now recall some elementary properties of the homology groups of a compact Riemann surface V. If Ii1(0, Z) denotes the i-dimensional homology group of V with coefficients in Z, we have :

A(0, Z) -'' Z, H, (03, Z ) ''H

zu ,

(YJB, Z) = 0

for p> 2 .

The non-negative integer g is called the genus of V. The Euler characteristic X of 23 is defined by

X =E:,- 0 (-1)P dim Hp(VB, Z)= 2-2g .

THE QUOTIENT

1.5

r\o* AS A RIEMANN SURFACE

19

If we take a triangulation of 8 and let c i, denote the number of p-simplexes, then X =c 3 c1 i c2. Let 8 and al3' be two compact Riemann surfaces, and j: 13'—'1 a holomorphic mapping. Then f is either constant or surjective. Suppose that f is surjective. Then (8', f) is called a covering of B. If z o e 1Y, wo and if u and t are local parameters at zo and wo, respectively, which map z o and wo to the origin, then we can express f in the form —

- -

a, # 0

t(f (z)). a,u(z)e + ae„,u(z)e+1 + • • ,

in a neighborhood of z„ with a positive integer e. The integer e is independent of the choice of u and t, and called the ramification index of the covering (23',f) at z o. There are only finitely many, say h, inverse images of w o by f. If e„.-. , eh are their respective ramification indices, the number

depends only on V, f, and is independent of wo. We call n the degree of the covering. It is known that the number of ramified points (i. e., those zo for which e> 1) is finite. If g and g' are the genera of B and 8', respectively, then these integers are connected by the Hurwitz formula (1.5.1)

2g ' —2 = n(2g —2) + E,, EP (es

-

1) ,

where es is the ramification index at z. This can be proved as follows. Triangulate TB so that among the 0-simplexes are included all points any of whose inverse images under f is ramified, and so that each 1-simplex lies within a single parametric disc. Taking the inverse image of this triangulation under f, we get a triangulation of 8'. If co, c„ c, and 4, c;, c; denote the number of 0-, 1-, 2-simplexes in these triangulations, then one has 2-2g= co — c i c, ,

2-2g' = c—

.

Observe that c= nc„ c0'=nc0— L.72a , (es 1). The formula now follows immediately. By a Fuchsian group of the first kind, we shall mean a discrete subgroup r of SL,(R) (or of SL 2(R)I{±1}) such that r\ti* is compact. Endowed with the complex structure defined above, r\o* becomes a compact Riemann surface. If 1/ is a subgroup of r of finite index, the natural map 1 ,\0* denote the images defines a covering in the above sense. Let and of r and I"' by the natural map -

r

SL 2(R)

—

SL 2(R)/{±1} .

Then the degree of the covering is exactly For every z E 0*, put

Er :

r,

20

FUCHSIAN GROUPS OF THE FIRST KIND

P,= {r e I' 1 r(z). z} ,

1%=-- P,

nri .

Consider a commutative diagram

identity

D*

D* W

f

- T\0*

where each map is a natural projection. Let ze D*, p = q)(z), and f-'(P) = { q1, ••• , qh ). Choose points W k of 0* so that qk =V(w k). PROPOSITION 1.37. The ramification index ek of f at qk is [11

:110.

U

Moreover, if w k =ak(z) with ak EP, then ek=Crz: cr";ir'crknri, and P=U P'Grkr. (disjoint).

Especially if

[r:p3=eih.

r/

is a normal subgroup of P, then e1 = •-• = eh, and

PROOF. The first assertion follows immediately from the definition of ramification index. Since = (jkl is ail and P' k = r' nak rz aii, we obtain the second assertion. Let r e F. Since f(V(r(z)))= v)(7(z))= ço(z)=p, we have =q k for some k, hence V(r(z))= stAcr k(z)). Therefore, r(z) = Jak(z) for some 3 e P`. Then we have T -13ak e P„, so that r E Picr k r,. This shows that r=uz..,r/akrz. If e e fig krz, we have io'(e(z))= çoi(w k)=qk . Therefore the union is disjoint. The remaining part of our proposition is obvious.

rick

1.6. Congruence subgroups of SI,a) The shape of the fundamental domain for SL 2(Z)\0*, given in § 1.4, tells us that the Riemann surface SL. 2(Z)\0* is isomorphic to the Riemann sphere. Let us now study T\D* for certain subgroups I of SL,(Z). In this section D* means DUQU{03}. For every positive integer N, put (1.6.1)

TN= T(N)= fr e SL,(Z)I r

1 2 mod (N)}

={[° db ]eSL 2 (Z)la-adm1, b-ac-a0 mod NZ}. Then T(N) is a normal subgroup of SLa), and called the principal congruence subgroup (of SL,(Z)) of level N. In general, a subgroup of SLJZ) is called a congruence subgroup of SL,(Z) if it contains T(N) for some N. LEMMA 1.38. If f:SLJZ)—.SLaINZ) then the sequence

is defined by f(a)= a mod (N),

21

CONGRUENCE SUBGROUPS OF SL 2 (Z)

1.6

f 1-- 1 (N)—sL 2 (z)—sL 2(ZINz)-1 is exact. The only non-trivial point is the surjectivity of f. We shall prove more generally that the map SL m(Z) .SL m(ZINZ) is surjective for any positive integer m, i. e., if A e Alm(Z) and det (A) m 1 mod (N), then A mB mod (N) for some B e SL m(Z). If m=1, this is obvious. Therefore assume the assertion to be true for m-1, and m >1. Now for such an A, by elementary divisor theory, we can find two elements U and V of SL m(Z) such that UAV is a diagonal matrix. Let a„ •-• , am be the diagonal elements of LIA V, and b=a, ••• a,. Put PROOF.

-

-

-

0 '1 1-a, (ha,

•1 -a 2 10

w=

X=

a,

, A'=

1 •

1

1.

•■

a,

Since a,b= det (A) a 1 mod (N), we see that WUAVX-a A' mod (N). By the induction assumption, there exists an element C of SL m _,(Z) such that a,a, -

^

a3

CE--.

mod (N). am

•=,

-,

Put

B-=U-1 W -1 -

o

Then B has the required property. If N=IIppe is the decomposition of N into the product of powers of distinct primes p, we see that

ZINZ*11 9 (ZIpeZ), GL 2(ZINZ) a. lip GLaipeZ), SL 2(ZINz)•A'llpSL 2(ZIpeZ). Now consider an exact sequence 1 --. X — GL 2(Z/PeZ) . GL 2(Z/PZ) --- 1.

FUCHSIAN GROUPS OF THE FIRST KIND

22

Since X consists of the elements of A gz/pez) which are congruent to 1 2 modulo (p ), the order of X is poe-o. It is well known that the order of cLaipz) is (p 2-1) (p 2—n). Therefore, the order of GL,(Z/P e Z)= P4(e - D (p 2—p)05 2 —1) = P4e( 1— P -1 )(1—P-2) p

the order of SL2(zip ez)=Pee (1—P -2) By Lemma 1.38, we obtain

[I'm: r(N)3= N3 . 11,1 ,(1—p-2). Since —1 2 E r(2) and —1 2 e r(N) for N> 2, we find

(N 3/2) • MI N G P-2)

if

6

if N =2.

—

(1.6.2)

[1(1): r(N)3={

PROPOSITION

1.39. If N >1,

r(N) has

N> 2,

no elliptic element.

of In §1.4, we have seen that every elliptic element conjugate to one of the following elements: —1 -±[0l OI 4 °1 : d i ' PROOF.

r(1) is

4=1 ,].

None of these is congruent to 1 2 modulo (N) if N >1. Since r(N) is a normal subgroup of r(1), we obtain our proposition. Let us now find the ramification indices of the covering r(N)\0*— roAgyk. Let som denote the projection map of 0* to r(N)\0*. By Prop. 1.38, the ramification index at WNW, for Z E 0*, is C1(1) 1 : 1(N),]. If z is an elliptic point of r(1), 1(1), is of order 2 or 3. By the above proposition, F(N)= {1} if N> 1. Therefore the ramification index at sow(z) is 2 or 3 accordingly. Furthermore, putting (1.6.3)

pm

= [1(1):

r(N)],

we see that the number of points on r(v)\0* lying above 9o(z) is tipf/2 or iitv /3 accordingly (if N> 1). If s is a cusp, s is r(1)-equivalent to co. Now we have

1(1)={[

'

Ar-,r(N).= AN) n r(1 ).={[ 01 ii ImEz} ,

23

CONGRUENCE SUBGROUPS OF SL(Z)

1.6 so that Er(1).*.

r(N).] =N. Therefore

r(N)

has exactly pti/N inequivalent

cusps. PROPOSITION 1.40. Let rt be a subgroup of r(1) of index p, and v„ v s the numbers of r'-inequivalent elliptic points of order 2, 3, respectively. Further let vc ., be the number of r'-inequivalent cusps. Then the genus of r1\04, is given by

1)4

g= 1+ 112

3

2•

, e, be the PROOF. Consider the covering F1\0*--, r(1)\0*. Let e„ ramification indices at the points of r 4,4, lying above v1(e2 "3). Then p=e,-F +et , and e t is 1 or 3. The number of i for which et =1 is v s. If t= we have p= v 5 +324, so that E:_s (et -1) = p—t = 214= 2(p—v,)/3. Similarly, if ep is the ramification index at a point P of rw, we have E(ep - 1)=(1-1-1)2)/2

(P lying above io Xi)) ,

E(ep -1)=

(P lying above io,(00)).

Now we have seen that r(1)\0* is of genus 0. assertion from the Hurwitz formula (1.5.1).

Therefore we obtain our

=r ),

In the case r (N we have vs = = 0 if N> 1, and v.= fi N/N. Thus we obtain the formula for the genus gN of roo\o*: gp,=1-Fp N • (N-6)/12N

(1.6.4)

(N> 1).

Let us now determine an explicit set of representatives for the cusps modulo r(N)-equivalence. LEMMA 1.41.

Let a, b, c, d be integers such that (a, b)=1, (c, d) 1, and

{ be la[ dc mod (N). Then there exists an element a of r(N) such that [ ba =

dc

1

(I) Assume [ dc = [ 0 I Then a m- 1 mod (N). Take integers p a Nq Then a has the and q so that ap—bq=(1—a)/N, and Put a= [b 1+'p] required property. (II) In the general case, take integers r and s so that cri-ds= 1, and [ 5a] [ 01 ]. [ dc mod (N), hence put r= Lrcd —si Then r ri' (N) so that mod (N). By the result of (I), we can find an element a of PROOF.

['

o ]=

n. Then rur - ' has the required property.

LEMMA 1.42.

Let s= alb and s' =c/d be cusps of r(N), with integers

FUCHSIAN GROUPS OF THE FIRST KIND

24

a, b, c, d such that (a, b)=1, (c, d)=1. (We understand that +110 = co.) Then

s and s' are equivalent under

r(N) if

and only if +[ ba ]a-- [ dc mod (N).

PROOF. If ±[ ba ]-a- [ dc mod (N), element g = [P (si ] of

r(N)

there exists, by Lemma 1.41,

an

qs][ d c ] = ±[n- If bd 0, we have obviously cr(s'). s. This is true even if bd,- 0, as can be shown by a such that [Pr

simple verification. Conversely, if a(s9= s with g = [ Pr gjE

r(N),

then

alb = (pc-Fqd)I(rc+sd) again under'the assumption bd -,÷- O. Hence there exists a a rational number A such that b [Pr Put A. min with integers qs][2 m and n, which are relatively prime. Then ?g ab ]. z[P q][ c ]. Since r s d (a, 6)=1 and (c, d). 1, we have m = +1 and n = +1, hence A = +1. The verification of the case bd =0 is also easy and may therefore be left to the reader. Thus the r(N)-equivalence classes of cusps are completely determined by Lemma 1.42. For example, if N= 2, there are three inequivalent cusps, represented by 0, 1, co, Let us now study a family of congruence subgroups of SL,(Z), are not normal subgroups of SL 2(Z). Put, for a positive integer N,

(IN = {[ (1.6.5)

r0(N) = A y n,SL,(Z)

db ] e M2(Z)1 c

=A,a db

]

which

0 mod (N)} ,

e SL,(Z) c -,70 mod

(N)} .

Then A N is a subring of Al ,(Z), and r0(v) is a subgroup of r(1) containing 0 r(iv). We see easily that if a = FN LO 1 _I'

ro(N ), cr- '1"(1)a r r(1 ).

(1.6.6) Note that —1

E

ro(4v).

By the map f of Lemma 1.39, ro(N)I r(N) is mapped

to the group of all matrices of the form of order N • c,..(N), where

[r(1):

[g ab _j ] in SL 2(Z/N Z).

This group,

ço is Euler's function. Therefore

r o(N)] =Er(1 ) : ro(N )] = N.

(1+ P-1 ) •

This proves the first assertion of PROPOSITION 1.43.

one has :

Let the notation be as in Prop. 1.40.

If

r ro(N),

iscleary

CONGRUENCE SUBGROUPS OF SL 2(Z)

1.6

(1) p= N • /LIN

(1

+P -1 ) •

if N is divisible by 4,

o (2)

otherwise.

))

v,== IIPIN ( 1± ( 0

if N is divisible by 9,

MIN (1+ (-533—))

otherwise.

(3) vs=

25

where Ç9 is Euler's function.

(4) Lœ = Ed1N,d>0 gAd, NM)), Here we understand that 99(1). 1;

) is the quadratic residue symbol (in the

extended sense), so that

( —3 )— ■ P/

O 1 —1

if

0 1 _1

if if

if if

if

p=2, p l mod (4) , .3 rno d (4) , pF-.. P=3, p:--31 mod (3), 2 mod (3).

PROOF. First we consider all couples (c, d} of positive integers satisfying (*)

(c,d)=1, d1N, O
(or c in any set of representatives for Z modulo (Nid)).

For each couple k, d} we take a and b so that ad—bc=1, and fix them. Then the elements

bci]

for all couples satisfying (*) form a set of rep-

resentatives for F0(N)\I" (1). In fact, it can easily be verified that they are not equivalent under left multiplication by the elements of ro(N), and the number of such couples is exactly p given in (1). Now by Prop. 1.37, the number of double cosets in ro(N)\rmirs for any fixed cusp s. Take s to be O. Then we see that v. is the number of couples ic, d} satisfying (*) modulo the equivalence defined by (c, d}—{c', d' }

if

* *1 1 1 0 *Lc' d'][c [* = dl[m 1] for some m E Z .

If we have the last equality, we have d=d', c1 =c—dm. Therefore, for a fixed d, there are exactly (p((d, Nid)) inequivalent couples, and hence we obtain (4). To determine v3, denote by S, (resp. SD the set of all the elliptic elements 0 —1 of r(i) of order 3 conjugate to z= [ i _ 1 ] (resp. under r(1) (see § 1.4

FUCHSIAN GROUPS OF THE FIRST KIND

26

and Prop. 1.22). Put c = e 2703, A = zu-i,j and

L= Z 2 ={[ax,y e ,

LN={[NXy ] ELI x, y E Z}.

For every a e S1 JS2, consider L as a Z[a]-module. Since Z[a] is isomorphic to A, and A is a principal ideal domain, there exists a Z-linear isomorphism f of A to L such that f(Cx)=af(x) for all xe A. Now let T be the set of all Z-linear isomorphisms of A to L. Then T is a disjoint union of the subsets {_lee

T j f(C4= of(x) with a e Si }

If a e M2(Z) and det (a)= j= f -1(L N). Since

—

1, then JET,

(i= 1, 2) .

af eT2. For any f eT„ put

ro(N) =-- { r

,

we see that the element a satisfying f(Cx)=af(x) belongs to r0(N) if and only if J is an ideal of A. Moreover, since All is isomorphic to Z/NZ, we see that (i) NR,Q(J)=N, where K= Q(), (ii) J is not divisible by any positive integer other than 1. Conversely, if J is such an ideal of A, we can find an element f of T such that f(J) = LN- We may assume that f eT, by changing f for ef with

L = [ 01 —10 ]

if necessary. Then we obtain an element a of

Sint- cm

by

f(x) = of(x). Let us now show that the correspondence between J and the conjugacy class of a in o (N) is one-to-one. Let feT„ f' e T1, f (Cx)= af(x), fl(Cx)= a'f(x), and f(J)=P(J)=LN with the same ideal J. We can find an element r of r (1) so that f =71. Then a= r'er, and TLN=.-- LN, so that a is conjugate to a' in ro(N). Conversely, let f(J)= LN, l'(f)= LN, f(Cx)=of(x), Acx)=T - larf(x) with f eT„ f' e T1, and r e r o(N). Put h=f - 'rff . Then h is a Z-linear automorphism of the module A, and h(Cx)= Ch(x). Put A= h(1). Then h(a+bC)=(a+bC),1 for a, b e Z. It follows that 2 e Ax. Therefore f- '( rLN)= f - x(rfi(f))=2f=f. Thus we have proved that 1,, is the number of all ideals J of Z[C] satisfying the above (i) and (ii). Considering the prime ideal decomposition of J, we see that 1.0, is the number given in (3).

We obtain v2 by applying the same argument to (-1)1 '2 and ri —01 ] instead 0 —1 of and [ As a special case, if N is a prime, 0 and co represent the inequivalent cusps of 1"0(N); the degree of the covering

r °met)*

ro)\ea*

at 0 and oo are N and 1,. the ramification indices (N)]=N+1; 0 is [r(1): r

1.6

CONGRUENCE SUBGROUPS OF SL 2 (Z)

27

respective! y. Define an element r of SL 2(R) by

r=

-1 [.:vO lv — VX10 I

_, 0

—11

= •V N [N Then r2 = —1, and r - T o(N)r z--- r o(N). by

0J •

Therefore we can form a group r:(N)

pok(N).r o(N)u r0(N)v . Then fl(N) is a discrete subgroup of SL,(R), which is commensurable with SL 2(Z), but not necessarily conjugate to a subgroup of SL,(Z). So far, all the examples of r are commensurable with SL2(Z), so that r\ei is not compact. There are of course many r for which TVD is compact, since it is a classical fact that every compact Riemann surface of genus > 1 is holomorphically isomorphic to rvt) with a Fuchsian group r with neither parabolic nor elliptic elements. We shall discuss in § 9.2 some interesting (and actually important) Fuchsian groups r with compact r\t), which are defined in a certain arithmetical way. EXERCISE 1.44. Let r, be a discrete subgroup of SL 2(R) such that r•vt)* is compact, and r a subgroup of r, of index m. Suppose that co is the only 1 cusp of r modulo r-equivalence, and POE, is generated by [,..u 11 ]. Prove that 1 1/m] r: is generated by [0 1 EXERCISE 1.45. Use Ex. 1.44 to prove that no discrete subgroup of SL,(R) contains properly rt,(N), if N is a prime or = 1. (Observe that, if r and rf

are as above,

r, is generated by r and

1 ii m )

[a 1 I

EXERCISE 1.46. Prove that no conjugate of r(N) in SL,(R) is contained in SL 2(Z), if N is a prime.

CHAPTER 2 AUTOMORPHIC FORMS AND FUNCTIONS 2.1. Definition of automorphic forms and functions Hereafter, till the end of §2.6, r will always mean a Fuchsian group of the first kind. As we have seen, T\0* is a compact Riemann surface. It is well known that the set of all meromorphic functions on a compact Riemann surface form a field of algebraic functions of one variable, with the constant field C. Now an automorphic function on 0 with respect to r (or simply, a r-automorphic function) is a function f on 0 of the form f=goio, with a meromorphic function g on r\D*, where p is the natural map of D* to rw. A more general notion, an automorphic form, can be defined as follows.

For every a-,-r a b l e GL 2(R), and Lc

dJ

e C, we put

j(c, z) = cz+d Then, as was shown in §1.2,

j(ar, 4 =3. dz

crkz

r(4) • j (r, 4 ,

uet ( a) . j (u, z)2.

D, we write For every integer k, e e GLR), and a function f on =det (ar f(a(z))- j(cr , 2)- . f Then it is easily verified that f

!Earl =(f CaJ ) ICrik

Let us insert here one word of caution : Two matrices a and —a induce the same transformation on D. However, if k is odd, j(--e,

hence I l [—C] k = —f M k.

—i(a. 4 ,

If k is even, the action of [—a] k is the same as

DEFINITION 2.1. Let k be an integer. A C-valued function f on 0 is called an automorphic form of weight k with respect to r (or simply, a rautomorphic form of weight k), if f satisfies the following three conditions : ( ) f is meromorphic on D ; ( ii) f !Eric=f for all r e f' ;

29

DEFINITION OF AUTOMORPHIC FORMS AND FUNCTIONS

2.1

(iii) f is meromorphic at every cusp of T. The precise meaning of the last condition is as follows. First, the condition must be disregarded if r has no cusp. Suppose r has a cusp s. Take an element p of SL 2(R) so that p(s). co. Putting r,.{ T eri r(s).s}, we have

Pr, P -1 * {.±1} -{ 4- .110 hdmi with a positive real number h. Cal for every e PrsP -1 -

Z}

In view of (ii), flEp - '3, is invariant under

CASE I: k even. Since f Ep - ij k is invariant under z , z+h, there exists a meromorphic function WO in the domain 0
number r, such that f

ce2rizili.)

Then the condition (iii) means that 0 is meromorphic at q=0.

k odd. If r contains —1, the condition (ii) implies f. f, so that there is no automorphic form of weight k other than O. Therefore, we CASE II:

assume that

_r i L0

—

1e

r.

Then pf,p -1 is generated either by [ I 1, or by 01 We say that s is regular or irregular, accordingly. If s is regular, —

1J the condition (iii) should be understood in the same way as in Case I. If s is irregular, g(z)= f Cp -33k satisfies g(z+h). g, hence g(z F2h)=g(z). The condition (iii) means that there is a function meromorphic in the neighborhood of 0 such that f ID" -=V(e"") • —

-

The function V must be an odd function.

REMARK 2.2. The above condition for f at s does not depend on the choice of p. If it is satisfied for some p, then it is so for every p such that p(s). co. The classification of s into regular and irregular ones is also independent of the choice of p. REMARK 2.3. If the above condition is satisfied at a cusp s, then it is satisfied at any cusp equivalent to s under r. The verification of these facts is straightforward, and may therefore be left to the reader.

The expression of f 1[9 -' ] k as a power series in e24."' or in eizih is often called the Fourier expansion of f at s; it has the following form :

f

=

The coefficients c„ are naturally called the Fourier coeicierts.

AUTOMORPHIC FORMS AND FUNCTIONS

30

If k= 0, we see, in view of our definition of complex structure of r\el*, that f satisfies the above conditions if and only if f is essentially a meromorphic function on /W. Thus an automorphic function with respect to r is an automorphic form of weight 0 with respect to r, and vice versa. Let us denote by Ak(r) the set of all automorphic forms of weight k with respect to r, especially by Air) the field of all automorphic functions with respect to r. Further we denote by Gk(r) the set of all f e A k(r) such that f is holomorphic on to and the function 0 or V in the above definition at each cusp is holomorphic at the origin; the latter condition means that the Fourier coefficient c a = 0 for n
f•ge

f e c k(r), fe ck(r) ,

geG(r) =

Ak,,,,(r) , f . g e ck+.(r),

geS„,(r)

f-geS k+„,(r).

PROPOSITION 2.4. Let r, be a subgroup of SL,(R), and a an element of GL(R) such that ara- ' is a subgroup of r, of finite index. Then f.--•fl[a]k gives a C-linear injection of A k(ro (resp. ck(ro,sk(ro) into A k(r) (resp. G,c(r), sk(r)), which is surjective if r, =ara-1. Let (resp. %') denote the set of cusps of r (resp. ro. Then a(%)= W, and our assertion follows immediately from the definition. PROOF.

Put 0*` _-7--Ol.) V. Then we obtain a commutative diagram

a

Ta

with a holomorphic map T a. In particular, if ara-1 = I% Ta is a biregular automorphism of r\o*, which corresponds to the automorphism f--ofo a of the field Air).

DEFINITION OF AUTOMORPHIC FORMS AND FUNCTIONS

2.1

31

Let J be a subgroup of I" of finite index. Identify Air) (resp. il 0 (z.1)) with the field of all meromorphic functions on r\o* (resp. 4\0*). As is observed in § 1.5, JVC)* is a covering of r\to* of degree [1: 3], so that A 0(4) is an algebraic extension of Air) of degree [1: ill Suppose now that j is a normal subgroup of 1, and consider the automorphisms of .4\0*, or of 21.(4), obtained from the elements of r as above (taking tl in place of r). Then we see that A 0(4) is a Galois extension of Air), and Gal(A,,(4)/ A o(r)) is isomorphic to ria 2.5. Let r/ be a subgroup of r of finite index, and subfield of Air'), containing A0(r), with the following Property: (P) If a e r and f 0 a= f for all f E then a e Then PROPOSITION

a

a

a,

a.

Put Z1 = (laerar'a -1 . Then J is a normal subgroup of r of finite index, contained in r/. Identify Gal (210(4)/210(r)) with r/j as above. The property (P) implies that P/2 contains the subgroup of Gal (A0(J)/A 0(F)) corresponding to Since every element of A0(P) is invariant under r/, we obtain A 0(r)c 5 by Galois theory. But we have assumed that hence A„(r/). PROOF.

a.

ac

a.

2.6. Let r/ be a subgroup of I' of finite index. Then Ak(r) fresp. ck(r),sk(r)) is the set of all f in A k (r) fresP. ck(ro,sk(ri)) which are invariant under Erik for all e r. PROPOSITION

The only non-trivial point is about the condition at cusps. be verified in a straightforward way.

But this can

PROPOSITION 2.7. Every r-invariant meromorphic function on algebraic over the field Air), is actually an automorphic function with respect to r. Let g be such a function and let gn-FEM f2 g 2 =0, with f2 E A„(r), be an equation for g over Air). For a cusp s of r, take p and q=e'" 4 as in our definition of automorphic function. Then f2(p - '(z))=0 2(q) and g(p - i(z)) =Yr(q) with meromorphic functions 02 and in the domain 0 < q < r, where r is a positive real number. Since the functions 02 are meromorphic at q = (), we can find a positive integer m so that PROOF.

(1)

lim q 711 • 02(q) = 0 q 0

(2=0, 1, • , n

-

1) .

-

Put V(q)=qmP"(q).

Then

(2) Assume that lim V(q k ). 00 for a sequence of points

WO

which tends to 0_

32

AUTOMORPHIC FORMS AND FUNCTIONS

Then, from (1) and (2), we obtain 1= 0, a contradiction. Therefore V(q) must be bounded in a neighborhood of 0, so that is meromorphic at q = 0, q. c. d. 2.8. Let f eA k (r), and g = (k+1)• (df dz) 2 — k f (d 2fIdz 2). Show that (i) g A2k+4(r); (ii) g S2k4-4(r) if f G k(r). EXERCISE

2.2. Examples of modular forms and functions

and functions. Let Let us now present some examples of modular forms C of rank 2 which is discrete. L be a lattice in C, j. e., a free Z-submodule of e 0. For an even integer k, Take a basis {w„ w2 } of L over Z so that w 1/c02 put E

= Ek(a),, w 2)

=

EGUEZIOlfir k •

This is absolutely convergent for To show this, let P„, denote the parallelogram on the complex plane whose vertices are 4-mco1 + mah. Let r= Min { I zi ; z E P1 } . Then I z> for z E Pm . Since P„,nL has exactly 8m points, we have E.E.F m ra 0.) I -k 8m • (mr) - * Since L {0} is the union of the P„,nL for m = 1, 2, •••, and since is convergent for k> 2, we obtain the absolute convergence of Ek(L). We see easily that 2k Ek(2(0,, 20)2)= Ek((01, w 2), and —

Ek (aw,+ boh,

Ek(wl, (02)

for

m -k+1

Cfcl. db E SL2(Z)

hence

Ek((awl+bw2)/(ccol+do.)2),1)(c - (w 1 /w 2)-1-d)k = Ek(w,1 oh, 1).

This means that, if we put Ek*(z) -= Eir(z, 1), E k* is invariant under [r]k for every r E SL2(Z). Let us now show that E k* is an element of Gk(SL2(Z)), by establishing its Fourier expansion at co :

(2.2.1)

q=

E(z) = 2 • C(k) +2

where C is Riemann's zeta function, and a3(n) denotes the sum of d: for all positive divisors d of n. To see this, we start with a well known formula

(2.2.2)

r • cot @z-z),

-

1

+

which can be found in conventional textbooks on complex analysis. other hand, putting q=e2 , we have

r • cot (rz) = (ir cos (rz))/(sin (rz)) = ri(er"-I-e-7"1)1(CLz—e -'"i) = mi(q +1)1(q —1) =

On the

33

EXAMPLES OF MODULAR FORMS AND FUNCTIONS

2.2

Equating (2.2.2) with the last sum, and differentiating successively with respect to z, we obtain

m) -2 = (2=02 - r;_,nqn ,

(2.2.3) —2-

(2=03 . E7 - in24 4

(-1)k(k— 1) !

my = (2:7i) k • Ez. i nk- le

Therefore, if k is even and

(k

4,

Ek*(z)= E(...)*(0.0)(7rizi-n) -k m,z. Z =22-22:_, = 2 • C(k)+[2 • (2;ri) k/(k-1) J E:-, E7-1 il k- 1 !

-

P

hence the formula (2.2.1). THEOREM 2.9.

Put

g2(z) — 60 • E4*(z) , i(z)= g2(z) 3-27g3(z) 2 ,

g3(z)= 140 • E6*(z) ,

J(z)= 123 g2(z) 3/ i(z).

Then 4(z) is a cusp form of weight 12 with respect to SL,(Z), and J(z) is a modular function of level 1 whose Fourier expansion at 00 is J(4 = q

(1+:7-1 cne)

with integral coefficients c n. Moreover, the field of all modular functions level 1 is the rational function field C(J).

of

PROOF. Let r = suz). Since g2 E G(r) and g3 E we see that E G12 (r) and JE /lo (F). Now if EIT denotes the r-th Bernoulli number, we have

C(2r)= Ean°_, n_ 2, =22,1 /3,70, /(2r)!,

so that 120-

co) = (2701 /12, 280- c(6) = (2r)6/216. X=

217_, sooqn ,

Y=

Put crs(n)qn .

Then

g2(z)=(270 4 C1/12+20X] ,

g3(z) = (270 8 D/216— 7Y/33 ,

(270 -12 4(z) = (5X+7Y)/12+100X 2 +203X 3- 3 72 Y 2 5d 3 +7ds 12 d>0

= E7-1 Edrn

+Zn>lanqn

Now d 5 d' mod (12) for every integer d. Therefore (220 - '2J = Ez.. t b„qn with integral coefficients, and b 1 = 1. It follows that si ,(r), and J has the Fourier expansion described as above. To prove with integers an .

AUTOMORPHIC FORMS AND FUNCTIONS

34

the last assertion, we need the following fact :

4(z)0 for every ze 0 .

(22.4)

We shall prove this in § 4.2. Assuming this, we observe that J(z) is holomorphic on 0. Therefore, the function J, viewed as a function on r\o*, has a pole only at the point corresponding to co, as the Fourier expansion shows. Since this is a simple pole, and r\o* is of genus 0, we see that C(J) must be the whole field of meromorphic functions on r\o*, on account of (3) of Prop. 2.11 below. Let r .SL,(Z) and a e GL2(Q), det (a)> 0. Then C(J, Jo a) is the field of all modular functions with respect to r na - ira. In particular, C(J(z),J(Nz)) (resp. C(J(z),J(z/N))) is the field of all modular functions with respect to r o(N) (resp. Po(N )), where r o(N) is as in (1.6.5), and

PROPOSITION 2.10.

rociv)=-_. { [ a db ]es.i.,2(Z)lb-,7-0 mod (N)} . PROOF. Put r' = r n ara. By Lemma 3.9 below, r, is a subgroup of r of finite index. It is obvious that C(L Joa)c A o(r,). Applying Prop. 2.5 to the present situation, we obtain the first part. The last part is just a 0 special case a.[ N ()] (resp. a= [ 01 ND. 01 In Chapter 6, we shall discuss some generators of A o(r(N)), which can be explicitly written as division values of elliptic functions.

2.3. The Riemann Roch theorem -

The purpose of the next few sections is to compute the dimensions of the vector spaces G k(F) and sk(r) over C, by means of the Riemann-Roch theorem. Therefore let us first recall some elementary facts on the divisors of a compact Riemann surface." For details, see, for instance, Weyl [102], Chevalley [7], Iwasawa [35], Springer [85]. Let Zi3 be a compact Riemann surface, and K the field of all meromorphic functions on U. We identify C with the subfield of K consisting of all constant functions. Then K is an algebraic function field of dimension one 2) All the following definitions, propositions, and theorems are applicable to a complete non-singular algebraic curve V over an algebraically closed field (or rather a universal domain) D of any characteristic. In fact, it is enough to replace V, K, and C by V, I2(V) , and D, where D(V) denotes the field of all (meromorphic) functions on V. The genus of V is defined, for instance, to be l(div (w)) with any differential form <81 on V, or an integer g with which the Riemann-Roch theorem holds. For this, see also Appendix No9.

THE RIEMANN.ROCH THEOREM

2.3

35

over C, i. e., if f e K, C, K is a finite algebraic extension of a rational function field C(f). Let D be the free Z-module generated by the points of Ta, j. e., the module of all formal finite sums ac,P„ with ci, E Z and P„e B. An element of D is ca ll ed a divisor of Z53, or of K. For a divisor A= E 64) we put cp=up(A), and deg (A) pcp. We write A 0 if 11),(14) for all PE 9, and A_?..B if O. For each P 13, the set if E Kif(P)*Go} is a discrete valuation ring with K as its quotient field. Let vp denote the normalized discrete order function K—•ZU fool associated with this valuation ring. If t is a local parameter at P, pp is defined as follows: let f (Q)=E„a„t(Q)

a * 0,

where Q is a variable point in a small neighborhood of P. Then vp(f)=-- veAssociate with each fE Kw a divisor div (f) by div (f)=Epesvp(f)P

Then f.--odiv (f) is a homomorphism of K to D, i.e., div (hf)= div (h)+div( f), div (f -')= —div (f). We put (f ) =E.F.u»o

(f )P,

( f).= - Evp(f)<0 VP(f)P.

Then div (f)=( f) 0 —( f)°,,. PROPOSITION

2.11, For every J E K4, one has:

(1) deg (div ( f))=-- O. (2) div 0 f eC". (3) [K: C( f)j= deg (Do = deg (

provided that feCx.

For a divisor A, we Put L(A)=-_- { f = if E

IC f 0 or div(f)?_.. —A} 1 vp( f)

—i(A) for all Pe M.

Clearly L(A) is a vector space over C. It can be shown that L(A) is finite dimensional over C. Denote its dimension by 1(A). Put {div (nif eKx).

Then DI is a submodule of D. A coset of D modulo D, is called a divisor class. We say that two divisors A and B are linearly equivalent, and write 11,..,B if they belong to the same divisor class. If A.-.B, we have deg(A) = deg (B), and /(A).--,-- 1(B). We can construct a one-dimensional vector space Dif (0) over K together with an additive map d: K Dif (0)

AUTOMORPHIC FORMS AND FUNCTIONS

36

satisfying the following conditions : d(hf)= h • df+f • dh,

(2.3.1)

df=0

4=5

f e Cx .

An element of Dif ( 133 ) is called a (meromorphic) differential form (of degree one) on U. If f e K—C, we have Dif (23) --,- K • df, so that every differential form w on 8 can be written in the form w=h - df with h e K. Then we write h=w1df. In particular, dk/df is a meaningful element of K for every kE K. For each Pe S, we take an element t of K such that p(t)= 1, and put up((o)= vp(wIdt). This is independent of the choice of t. Define div (w) by div((0)= EpEsg vp(w)P . Then div (fw)=div(f)+div(w) for every fE K. Thus the div ((0) for all (0 e Dif (S), #0, form a divisor class, which is called the canonical class of 211 (or of K). We say that w is holomorphic, or of the first kind if div ((0) ._ 0 or (0 =O. THEOREM

2.12 (The Riemann-Roch Theorem). Let g be the genus of Ti3

(see § 1.5), and w a non-zero differential form on Til. Then, for any divisor A of 8, one has 1(A) = deg (A)— g +1+ 1(div (w)— A) .

PROPOSITION 2.13. For every non-zero differential form w on TB, deg (div (w)) = 2g-2.

We see easily that L(0)=C, so that /(0)= 1. Therefore, from Th. 2.12 and Prop. 2.13, we obtain

(2.3.2)

/(div (w))= g.

Fix any non-zero differential form wo. Then

L(div (we)) =-- if E K I div (f)?- — div (w0)}

= If E K1 div (fa)0)_. 0} a.- fo.) E Dif (8) I div (w) > O}. Thus, by (2.3.2), we see that the set of all holomorphic differential forms on 8 is a vector space of dimension g. PROPOSITION

2.14. Let A be a divisor of U. Then:

1 (A)= 0 ; (1) deg (A) <0 1(A)= deg (A)—g+1. (2) deg (A) > 2g-2 = PROOF. If 1(A)> 0, L(A) contains a t least one f # 0 such that d iv (f)-_— A.

37

THE DIVISOR OF AN AUTOMORPHIC FORM

Then deg (A) deg (div(f)) = O. Therefore we obtain (1). If deg (A) > 2g 2, we have deg (div (w) A) < O with any non-zero differential form w on V, so that /(div (0.)—A)= 0 by (1). Then the Riemann-Roch theorem implies that 1(A) = deg (A)— g +1. -

—

2.4. The divisor of an automorphic form We shall now consider the case where 0= /1\0* with a Fuchsian group F of the first kind. Our main interest is in the spaces Gk(r) and Sk (T). suppose —1e r whenever we speak of MT) with odd k, since Ak(r).---if k is odd and —1 e r. If f(t) is a meromorphic function in a complex variable t defined in a neighborhood of 0, we denote by v(f) the order of f at t=0, e., vt(f)= m if f (t)= 5'C„t 4 , Cm # O. Further we put K= A(F), and identify K with the field of all meromorphic functions on S. PROPOSITION 2.15.

Agn# {o}

for every integer k.

On account of (2.1.1), this implies that A k (T) is a one-dimensional vector space over K. PROOF. Take any 0 e K—C. Then 0(1(z))=0(z) for all r r. Take the Then we find Or(z))j(r, z) -2 --, 0'(z) since dr(z)/ dz derivative O' = dØ/dz. j(r, z) -2. At a cusp s, we have 0(p- i(z)) _(q) with a meromorphic function 4) at q = 0, where p and q are as in § 2.1. Then Sb' f CP32 = 45/(P -1 (z))j

(P.- % 4' =0/(q) • (27ri/h) • q ,

hence 0/ E mr). Therefore, by (2.1.1), 0# 0" e A 2 (T) for any integer n, which proves our assertion for even k. The case of odd k will be discussed a little later. For any FE 442 (r), we can view F(z)dz as a differential form on B. In fact, take 0 e K C as above. Since 0'=d0/dze ,42(r), FAY E Ao(r)=--- K. Then we put F (z)dz = (F/00d0. This does not depend on the choice of 0. Conversely, if w e Dif (0), then f= co/d0 EK, fo'e A2(r), and 0.).-----(f0')dz. Therefore the map F ,--*F • dz gives an isomorphism of A 2(F) onto Dif (r\e•*). Now, we can construct an associative (graded) algebra —

z-z-_-_„,Difn (0) over K under the following conditions: (a) Dif° = K, DiP (0) = Dif (0);

(direct sum)

AUTOMORPHIC FORMS AND FUNCTIONS

38

(b) Difn (V), for each n, is a one-dimensional vector space over K; (c) For a EDifm (0) and p E Difn (V), the product ap is defined as an element of Difm" (0), and ap Pa # 0 if a # 0, p#0. The algebra SID is uniquely determined by these conditions. If 0# ap E Dif (0), co" is meaningful and belongs to Dif" (0). Then Difn (Z13)= Kw', so that every element e of Difn (0) is of the form e =fco " with f E K. If e# O. we define div (e) by div (e) = div ( f)+n div (o.)) We see easily that this does not depend on the choice of co, and

div (e;7) = div (e)+div ('i). By Prop. 2.13, denoting by g the genus of T3

(2.4.1)

deg (div (e)) = n(2g 2) -

rw, we have

for 0 # e E Dan (0).

Take 0 as above. For FE Aar), we observe that F/0'nE A 0(.1")= K. and put F (2)(d = (F / 0")(d0) 4. Then F(zXdz)" is a well-defined element of Difn (V), independent of the choice of 0. We see easily that F(2)(d2r is an isomorphism of A 271(r) onto Difn (23). Let F E Ak(r), and PE T3. We define V p(F) as follows. If P corresponds to a point z o of t), take a holomorphic isomorphism A of to onto the unit disc such that 2(20) = 0. If t r e r f r(z a) z o } is of order e, the function t= is the standard local parameter at P (see §1.5). Then put vp(F)= v F)/ „ e. Next, if P corresponds to a cusp s, take p and q =--- exp (22riz/ h) as in § 2.1. Then, as in our definition of automorphic forms, we have -

(q1/2)

F 1[P -1]k =

0(q)

if k is odd and s is irregular, otherwise,

where jl and 0 are meromorphic functions around 0. Then we put V c t3iT)12

vp(F)=

(t = q 112 zz; erizth)

I 10)

accordingly. Note that ii,(0') is odd. Let us put DQ = D Ø. Then we can associate, with each an element div (F) of DQ by

div (F)= Epeyg vp(F)P . It is clear that

div (FIFO = div (F1)+div (F2) (2.4.2)

(F,

Aki(r),

F2

Gk(r)= {FE Ak(r)I div (F). 01,

E

Ak2(r)),

FE Ak(r),

39

THE DIVISOR OF AN AUTOMORPHIC FORM

2.4

j(F e A k (r) div

(2.4.3)

Sk(r).

I {F

ilk(f)

div (F)

(k: even),

(k : odd),

+(1 /2)

where Qi , V., ) are the points of 03 corresponding to the (resP- (21, and deg ( ) regular (resp. irregular) cusps of r. Note that the relation can be extended to DQ in a natural way. PROPOSITION 2.16. Let Pz, •-• , Pr be the points of r\v* corresponding to all the elliptic points of r, of order e1 , ••• ,e r, respectively, and Q,, , (21, ••• ,Q', be as above. Let 0 # Fe A k(r), and if k is even, let v=F(z)(dz)". Then div (F). div (77)-1-(k/2) deg (div (F))=(k/2)- ((2g--2)-1-ET. 1 (1—eT')-Fu+u/}

(k: even), (k even or odd).

PROOF. Let P be a point of 8= r\o*. If P corresponds to a point zo of ef), take t= 2(z)' as above. Then dticlz= e • 2(z)e - l(d,i/dz), and p,(dt/dz)= 1—e -1 . Therefore, assuming k to be even, we have

(2.4.4)

vp(77)= pf,(F • (dz/dt)*12). Vp(F)+(k/2)•

If P corresponds to a cusp s, take p and q = e2 "ih as in p. 29. Then putting 2= p(w), we have F(w)(dur2 = (F I [p- '3,,,)(dz)" = 0(q)(dz/dq) ,,,2 (doka =0 (qx2 :74,11.)-412(dok 2 hence (2.4.5)

vp())=1, 1(0)—k/2 =

Our first formula now follows immediately from (2.4.4) and (2.4.5); the second one for even k from (2.4.1) and the first one. If k is odd, we have div (F) = (1/2)- div (F 2), so that we can derive the desired formula of deg (div(F)) for odd k from that for even k. The above proposition means that we can make calculation of divisors of automorphic forms by putting formally div (c12)= — ET-1 (1- ei-1)P,+EY-tQ,+r,..:,Q;) Q . The number (2g-2)+ET 1 (1—en+u÷u' occurring in the second formula has an important geometric meaning, which will be studied in the next section.

s2(r)

COROLLARY 2.17. is isomorphic to the vector space of all holomorPhic differential forms on Ti3 = r\o*, through the map F. dz. It follows especially that .32 is of dimension g.

(r)

PROOF.

If FE A2(r) and w=F dz, we see, from Prop. 2.16, that div (co)_..0

AUTOMORPHIC FORMS AND FUNCTIONS

40

+7 Q.

if and only if div from (2.4.3).

Therefore our assertion follows

PROOF of Prop. 2.15 for odd k. Take any non-zero differential form w on 8 and any point R o of B. Then deg [div (w)-2(g-1)/2 0]. O. Now it is a classical fact that all the divisor classes of T3 of degree 0 form an abelian group isomorphic to a complex torus of complex dimension g (which is called the jacobian variety of ..).33). Therefore we can find a divisor B on B such that div (w)-2(g-1)R 0,2B, i. e.,

2B—div (w)+2(g-1)R 0 =div(f) K. Put B'= B -1- (g - 1)R 0. by F(z)dz = fw. By Prop. 2.16, we have for some f

E

We can define an element F of

div (F)= 2B' +Ef...1(1 — enPl+

.42(r)

Qi+E7:1Q;.

By Cor. 1.21, the ei are all odd, if — 1 r. Therefore we see that the Hence we can define a function F has an even order at every point of meromorphic function G on .Z) so that G 2 =F. Since FE .42 (r), we have G 1 Cri=X(7)G for every r E r with X(T) = -± 1. Put r, = f r E r ix(r)= 11_ is a subgroup of P of index2. Since FE .42(r), we see that G is Then this settles meromorphic at every cusp of r', so that G E .4 1 (r). If the question, since 0 # Gk E .4,(r) for any integer k. Suppose U": 11'3 =2, and let 1.1 =r'Ure. Since A 0(r) is a quadratic extension of .40 (r), and as is seen in p. 31 (after Prop. 2.4). Gal (4 0(f")/ 4 0 (f')) is isomorphic to r there exists a non-zero element h of 41kr9 such that h(z(z))= — h(z). Then h - G belongs to .4 (1 '1 ) and is invariant under [el, so that h • G e AO') by Prop. 2.6. Therefore 0 # (h • G)k e -4 k(r) for any integer k, q.e. d.

2.5. The measure of r\O For every differential form w on .t) and c e SL,(R), let us denote by 0)0 a the transform of w by a in an obvious sense : if w is of degree 0, and hence a function, w 0 a. is of course meaningful ; in general, d(wûcr).(dw)ocr, (ad\ 1)0 a =(o.).-: a) A ()70 a). PROPOSITION

2.18.

Let 77 be a differential form on

t71

defined by 12

dz,

z x-Fiy. Then : (1) iy 0 c —i= —2i d log [ j(a, z)] for every c e SUR). (2) dl= y-2dx A dy =(i/2312)• dz A d2. (3) Y'dx A dy is invariant under SUR). PROOF.

If a. [P

Qs ]

e SL 2(R), we Inve dz - a j(ï, z) -2dz and by

(1.23),

41

THE MEASURE OF r

2.5 yo o= j(a, z)

y, so that 770a= [(r2+s)/(rz+s):•7: . Therefore

o c—n = [(r2+s)/(rz-l-s)-1]

= —[2ir/(rz+5`:;- dz -= —2i d log (rz+s) .

The formula (2) is obtained in a straightforward way. differentiation of (1), we obtain (3).

Taking the exterior

Now let us define a measure m on D by m(A)= S A

y-2dxdy

for a subset A of D. By (3) of Prop. 2.18, nt is an invariant measure, i. e., m(A) = m(a(A)) for every a E SL 2(R) and for a ;measurable) set .4. We can use this measure to introduce a measure p on r\O*. Let .. put 9: D*--.T\fy, be the projection map, and for L. -T.0*,

ry= fr r I 7(0= ri • For each y, we can find an open neighborhood I: of y such that

ry ={r Erir(u) nu -,=o} r0. Then rv\U may be identified with

an open and r(U) ,---- U for all r neighborhood of it(y) in r\t*. If y is not a cusp and if Pt, is of order e (1, is elliptic if e> 1), we can divide U into e angular sectors U„•-• ,U, such that r(U0= U,4. 1 for 1 i <e and 7(U,), U„ where r is a generator of ft,. Then, for A' Cr,,\U, we can find a set .4 of representatives for A' in U1 , and define tt(A')= m(A). Similarly if y is a cusp, say, at cc, r, is generated 1 h and C may have the form U= ] by an element of the form [ 01 ' {z I im (4> cl . Then, for rl'Cr ,Au,,ve can find a set A of representatives for .4' in the region

{z= x+ ii, y> c, 0

x
(excluding co),

and define ti(A')= m(.4). Now we can cover PO)* by open sets of the form rv \u. If {Mr, is a C'3 -partition of unity subordinate to this covering {W2 }7,,, then for a continuous function f on r\r, we can put

j.

(*)

f dp

=

fh i d .

It can easily be seen that this measure on TVD* does not depend on the choice of {W 2 } and {h 2 ). integral (*).

We shall also write

(f050)y -2 dxdy

for the

118

2.19. If T\D* is compact, then it(r\D*)< co. (In other words, (*) defines a Radon measure.) PROPOSITION

42

AUTOMORPHIC FORMS AND FUNCTIONS

If r\eYk is compact, the covering of the above type can be made by finitely many sets of the form for which the closure of U is compact. co. If v is a cusp, the finiteness If v is not a cusp, then clearly t follows from PROOF.

ruw e(rvw)<

if y -2 dxdy <00 . C
r\e, *,

2.20. Let g be the genus of the compact Riemann surface m the number of inequivalent cusps of and e„ — , e r the orders of the inequivalent elliptic points of Then THEOREM

r,

r.

1

27r

fino y -2 c1xdy=2g-2-1-mi-Ei ,;_i (1-1/e

)

.

rvf-,)*,

It is well-known that being a compact Riemann surface, can be triangulated using piecewise analytic curves, which, without loss of generality, can be chosen so as not to contain any cusps or elliptic points. Using this triangulation, can be represented as a normal form a 1 b 1 aTibT 1 — at b :aUbii consisting of a 4g-sided polygon all of whose vertices are identified, and whose boundary consists of 2g curves a t , b, traced on ce in each direction in the above order, and such that the elliptic points and cusps are in the interior of the polygon. Next, draw non-intersecting piecewise analytic paths connecting one of the vertices of the polygon to the respective elliptic points and cusps. Also draw a small circle around each elliptic point or cusp, whose radius will be made to tend to zero. Cutting the polygon along these paths and small circles, we now get a " polygon " with 4g-1-2m+2r sides, neglecting the small circles. Now take a small open disc in this polygon, and map it into .r, by the inverse of the projection map el*---4-1\el*. This is a holomorphic map, an d can be continued holomorphically to the whole inside of the polygon. Therefore, the polygon can be mapped onto a polygon on which we call H. From our construction, we see that the boundary all of H can be written in the form PROOF.

rvryk

e.,

al/ =

(1)

EL, (S2 - n(S2))+ E7.4ir T,,

(n_—_-2g+m+r) .

Here the T,„ are the curves corresponding to the small circles, the S2 correspond to the "sides ", and 1-2 is a certain element of for each A. The interior of H, when each T„ is shrunk to a point, is certainly a fundamental domain for But the S2 are not necessarily "straight lines" in the sense of non-Euclidean geometry. One can actually construct a fundamental domain for which is a polygon whose sides are straight lines in that sense. For our present need, however, the polygon H, in the loosest sense, is completely adequate.

r

r.

r

THE MEASURE OF r\D

2.5

di

43

Now we consider the differential form y defined in Prop. 2.18. Since y - 2d x dy, we have

y'dxdy

p(['\.*) 11m

(2) by Stokes' theorem. By (1),

air

The limit procedure is taken by shrinking the circles.

fail/

SA

(77-77 0 r2)+Erz

72.

Let F be a non-zero element of t1 2(T). Define a differential form e on D by e=d(log F) = F -1 F/dz. Taking the logarithmic derivative of

F(a(z)) = F(z)j(a, z) 2 we obtain

(a e

e a a = 2 . d(log j(a, z))

(a

,

r).

By (1) of Prop. 2.18, we have

7/0 0--y= —i(eocr—e) '

Therefore 11 77.

(3)

fan,E+Erf

f e.

If T corresponds to an elliptic point y of order e, then clearly

T,

y tends

e, take a holomorphic map r of D on to the unit disc such that r(y)-.-- 0, and put t(z) = r(z)e. Then t is a local parameter, and we may assume that T,, is the image of a small circle C, in the t-plane with origin to O. As for

T„

as its center. This circle should be taken in the negative direction, since the exterior of the circle corresponds to the interior of 17. Therefore putting co= F (z)dz and OW= F(z)(dz/dt), we have =

Cd(log )-1-d(log

(dt/d4)3 = — 27riCvt(0)+ vc(dt/dz)3

= —27riCyp(co)-1-1—e-1 3 = —27ri • 1) p(F) by (2.4.4), where P is the point of rvzyk corresponding to the elliptic point in question. Next, assume that T„ corresponds to a cusp s. Let p be an element of SL 2(R) such that p(s) = co and let q.,---e2riPw/h. Then we may assume that T„ is the image of a small circle C, in the q-plane with origin as its center. Putting w = p(z) and F(p - '(w))j(p', w) 2 =0(q), we have F(z)d z= 0(q)dw. We can take p

0 1s ] if s

co.

Then dw/dz = wz, so that F(z)=0(q)w 2, and

AUTOMORPHIC FORMS AND FUNCTIONS

44

d(log F)=

c [d(log 0(0)+2 d(log w)]

= —22zi • v(ø)_$ "° 2 • d(log w) —27ri • iii(0). —2;ri vp(F)

(w,—.00) .

Here P is the point of r\* corresponding to s_ If s. oo, we can take p to be the identity matrix, and obtain the same result. As for 77, we have = fp(ro 0

=

pero

{ —2i d log 0(p - i, w))1

by (1) of Prop. 2.18. We understand, as above, that p(T) is the segment from w o to w o +h. Then WO- h

=

[dzly 2i • d(log w)] —0 -

(w0—co).

Thus combining all these computations, we obtain, from (2) and (3),

p(r\O*)=

f d(log F)+27r ET., vp,(F)+2r E",". 1 vQ,(F). all

Now (27r0 -1 5 d(log F) is the sum of a 17

(F)

for all P in the interior of the

polygon 11. Therefore we obtain p(F\f"*) = 27r • deg (div (F)), which, together with Prop. 2.16, proves our theorem. From this theorem, we see that (2.5.1)

(1 e») > O.

2g-2+

—

If g> 1, this inequality is trivial. If g= 1, one has m+r 1. If g=0, one has m-1-51T,. 1 >2, hence m+r 3. One can show, without difficulties, that the case g =0, in =0, (e,, e 2, e 3). (2, 3, 7) gives rise to a r with the smallest ke(T\D*). Thus

r

dxdy/y 2 1/42

for every Fuchsian group r of the first kind. Using this fact, it can be shown that the group of all automorphisms of any compact Riemann surface of genus g> 1 has order 84(g-1). For this topic, we refer the reader to Hurwitz, Werke I, pp. 391-430, Fricke-Klein [22, 606-621], and [77, 3.18 ]. It was also shown by Siegel [83] that the converse of Prop. 2.19 is true, j. e., if p(r\)
45

THE DIMENSION OF THE SPACE OF CUSP FORMS

2.6

2.6. The dimension of the space of cusp forms Let Fo be a non-zero element of A k (r), and B =div (F0). Every element F Of 4k(r) can be written in the form F f F. with Je K. Then div (F)_ 0 if and only if div (f) —B. Therefore, by (2.4.2), we have dim

(2.6.1)

ck(r). dim (J E KI

div (f)

—B)

and similarly, by (2.4.3), (2.6.2)

dim

sk(r)= dim

(fEKI div(f)

—B-FE7. 1 Q

);}

where /.1= 1 or 1/2 according as k is even or odd. To compute these dimensions, we are going to apply the Riemann-Roch theorem to the divisors B -1-E7-1Q ) 4- EY:1Q;

—B,

However, these divisors are elements of DQ , but do not necessarily belong to D. To dispose of this difficulty, we consider the "integral part" of an element of DQ . For x E R, let [x] denote the largest integer :5_ x; if p= [x], one has p_x
[A] = EP EC PY3 • LEMMA

2.21. For JE IC' and A E DQ, div (f)

PROOF.

div (f)

—A

div ( f) —[A] .

.:=;.

Put A = Ep c pP. Then —A

vp(f)

— cp [Ce]

vp(f)

f) 5- C P

f

4=)

q. e. d.

,

= div ( f )

[ce]

Let us first suppose that k is even and put n = k 12, Fo = e/(dz)" with E Difn(U). By Prop. 2.16, B =div (Fo )=div (e)+// • (Z,T,, (1-6. 1)/

Q j +NT.-1Q;) .

By (2.6.1) and Lemma 2.21, we have dim ck(r),IcB]). Putting u+ = m, we see that

(2.6.3) LEMMA 2.22.

deg

([

BJ) = n(2g —2+ ln,— v :..,[n(e 1 -1)1 e

Let k E Z, e c Z, e> [k(e — 1)/2e]

.

I f k(e-1) Is even,

(k —2)(e-1)/2e .

PROOF. Put p=[k(e— 1)/2e]. Then R: e — 1)/2e < p+1. so that k(e-1)<2eP 2e. Since both sides of this inequality are even, we have k(e —1) __2eP-4 2e - 2,

46

AUTOMORPHIC FORMS AND FUNCTIONS

so that (k —2)(e —1)5= 2ep, and hence (k-2)(e-1)/2e inequality is false if k(e-1) is odd.)

p, q. e. d. (Note that the

By virtue of Lemma 2.22 and (2.5.1), we obtain, if n >1,

(2.6.4)

deg ŒBJ)—(2g-2) (n-1){(2g-2)+E7,•_ 1 (1.—e)d-m l+m

>m. Therefore, by (2) of Prop. 2.14, I ([B]) = deg ([B]) — g+1. If n=1 and m > 0, we obtain the same result. If n=1 and m=0, we have and Cor. 2.17 answers the question. If n .0, then B=0, so that /([B]). 1. If n < 0, we have

ck(r).sk(r),

deg ([B]) -5 deg (B). n {(2g-2)+E;_ 1 (1 —eT9d-ml < O by (2.5.1), hence, by (1) of Prop. 2.14, 'Tn. O. Thus we have proved

r\vk, m

the number of inequivalent THEOREM 2.23. Let g be the genus of and e1 , , er the order of the inequivalent elliptic elements of r. cusps of Then the dimension of the vector space for an even integer k, is given by

r,

ck(r),

(k-1)(g-1)+(k12). m+E:-.,[k(e,-1)12eJ

g+m-1 dim

(k> 2),

(k.2, m > 0),

ck(r).

(k

2, m.0),

1

(k

0

(k <0).

Applying the same reasoning to the divisor B' = B —E.1 observing that deg([8'])> 2g-2 if n 2 in view of (2.6.4), we obtain THEOREM 2.24. The dimension of the vector space integer k, is given by

S(r),

[k(e,-1)/2e 1 ]

dim S k (1").

and

for an even

(k ). 2),

1

(k=0, m=0),

0

(k=0, m>0),

0

(k <0).

Let us now suppose that k is odd. The notation Fo and B being as above, put 7?=FS/(dz) k. Then )2E Difk (Z3), and by Prop. 2.16,

47

THE DIMENSION OF THE SPACE OF CUSP FORMS

2.6

div (F0 )=(1/2) div (,)+(k/2) - {Er. i (1—enP,-FEI-1Q, -FElf-IQ;}

(2.6.5)

•

From our definition of div (Fe), we see that

1/2 Q'i), 1(P= (in teger)/e i

(P= P2),

mod Z

otherwise.

0

Therefore, from (2.6.5), we obtain

1 1/2 (1/2).

V p(72) 1=-

0

(P=Q),

mod Z

otherwise.

For P.P., put vpi(77)=c,.

This is obvious if P# P.

Then

Ci

Z, and

vpi(F0)=ct/2 +k(ei -1)/2 e5=(eicz+k(e,-1))/2e,. Since ei is odd and e.- vp 1(F0)E Z, c i must be even. Thus we obtain [B] = (1/2) div (0+

[k(ei —1)/2e1]Pid-(k/2) E7.-L Q1+((k -1)/2)

[B—E_ 1 Q—(1/2)

(Y.i] = (11 2) div

Q; ,

[k(e.-1)12e ij P.

+((k-2)/2) E7=1 Q i +((k-1)/2) EtF-1

Q; •

If k< 0, deg (EEC) S deg (B).(k/2). {2g-2+ Er. 1 (1 — ez)+m } < 0 , hence

ck(n=sk(n= {O}.

Suppose that k

deg (CB— E7_, Qi —(1/2)

3. Then, by Lemma 2.22,

Q2) — (2g -2)

.(k-2)(2g-2)/2+u(k-2)12+u1(k-1)12+Er-i [k(e2 -1)/2e2] 12- -2-{2g-2-1-u+u'-FEr.1(1—en} >— = 2

>0. (The e, are all odd, since we are assuming that —1€E Therefore, by (2) of Prop. 2.14, we obtain

r,

see Cor. 1.21.)

2.25. The notation being as in Th. 2.23, suppose that —1eE T. cusps of Let u (resp. u') be the number of inequivalent regular (resp. irregular) THEOREM

r.

Then, for an odd integer k, one has

dim ck(r)=1 dim

sk(r)=1

3),

(k-1)(g-1)+ukl2d-u'(k-1)/2-F=2Ck(e.-1)/2e.]

(k

0

(k <0),

(k-1)(g— 1)-1- u(k —2)/2 -Fuqk —1)/2+

0

Ek(e,-1)/2ej (k 1. 3), (k <0).

AUTOMORPHIC FORMS AND FUNCTIONS

48

We observe that the number u must be even. For an obvious reason, our method is not effective in the case k k =1, we have deg CBI g-1-Fu/2, so that

Gi(r)__ if dim c,(T) =102

(2.6.6)

1. If

dim

(2.6.7)

u > 2g-2 .

Further, we have deg [B—r1- 1 (2 —(1/2) Etf-I

Vi3 = g— 1—u/2 .

Therefore, by (1) of Prop. 2.14, we obtain

sicn= {0}

(2.6.8)

if u> 2g-2.

For example, consider the group TN of (1.6.1) for N> 2. Clearly —1 Ef T N Since every parabolic element of rN is conjugate to a power of [ 01 Ni i under .

we see that every cusp of r N is regular. If pi., = Er, PA, we have u=1.4,1N, and g= 1-i-pp,/12—u/2 as is shown in § 1.6, so that u/2—g+1 = u(1—N/12). Therefore, :

Gx(r Jo= tf,12N

si(r N)= {0 } It is an open problem to determine dim G1(r) and (2.6.9)

dim

and

for 3_.5_. A r._-ç_ 11.

dim

slr)

in a more effec-

tive way. Coming back to even k, if r = sw), we have g = 0, m --= 1, and {e„ e,} -= (2, 31, so that, by an easy calculation, we obtain PROPOSITION

2.26. If r =SL,(Z), for even k_ 2,

dim G,(1')=

(k -.7,- 2 mod (12)),

/ [k/12]

(k * 2 mod (12)),

[k/12] + 1

(k= 2),

0 dim

sk(r)= {[k/12] —1

(k> 2, ks--..-- 2 mod (12)), (k * 2 mod (12)).

[ k/12]

For example, we see that dim Gk(n= 1 and dim sk(r),_ 0 for k= 4, 6, 8, 10. We have seen in § 2.2 that G(I) contains a non trivial element Et Therefore, -

ck(T)=e • E k*

,

sk(n= fol

(k = 4, 6, 8, 10).

For k = 12, dim S„(F)= 1, and dim G„(T)= 2. The form J(z) considered in Th. 2.9 generates s,2(r). As is shown in (2.2.1), E k* is not a cusp form. Therefore G12(r) is spanned by "Az) and EI. By a similar reasoning, we can show that

2.6

THE DIMENSION OF THE SPACE OF CUSP FORMS

s„(n= {0} , sk(n=c • 4- Et-12 .324(r)=c . 4. Eiir+C • 42

49

(k = 16, 18, 20, 22). .

More generally, we have PROPOSITION 2.27. If T = SL,(Z), the space G k(r) is spanni.d over C by the functions g2ag,b with non-negative integers a and b such that 4a+6b= k, and sk(n= 4 • G k_.(r), where 4, g„ and g3 are as in Th. 2.9. PROOF. Put g2(a) 1, o.),)= 60. E4(a) 1 , w 2), g,(o)„ ah)= 140 - E(a) 1, (02) with E4 and E, of § 2.2. Then g2(z)= th(z, 1) and g3(z)=g2(z, 1). We shall later (in § 42) show that g 2(co „ o.) 2) and gs(w i, w2) are algebraically independent over C. It follows from this that the monomials g.(z)ag,(z)b, with 4a+6b=k for a fixed k, are linearly independent over C, since

wi taz)ag.(z)b= fh(o.,„ cf.h)ag 3 (0.)„ a 2)6

(z= a 1/w 2).

Now it can easily be verified that the number of non-negative integral solutions (a, b) of 4a±6b=k is [k/12] or [k/12J+1 according as k a- 2 or *2 mod (12). Therefore we obtain the first assertion in view of Prop. 2.26. Since Az). Gt _„(ncsk(r) and dim s k(r) = dim Gk_„(r) by Prop. 2.26, we obtain the second assertion.

As an example of have

sk(ro

with a congruence subgroup

r, of

SL,(Z), we

EXAMPLE 2.28. Let N be one of the integers 2, 3, 5, and 11, and let k = 24/(N+1). Then sk(ro(N)) is one-dimensional, and spanned by (4(z)4(Nz)) 11'. PROOF. The first assertion follows from Th. 2.24 and Prop. 1.43 by a simple computation. Since ,i(z)* 0 everywhere on eo, we can define 4(zyhit, for any positive integer m, as a holomorphic function on 0. Put g(z)= 4(z)4(Nz). By (1.6.6) and Prop. 2.4, zl(Nz) E sdro(N)), so that ge swr o(N)). Since 4(z)=q0(q) with a holomorphic function 0(q) in q = e2 ' such that 0(0)* 0, we have g(z)=q1"-10(q)0(e), so that g has a zero of order N+1 at the cusp oo. By Prop. 1.43, 0 and oo are the only inequivalent cusps of T o(N), since N is a prime. Put D=N-1/2 r ° —10-11. Then r permutes 0 and oo, and LN

g I Er3k = 4( 1/N44( 1/z)(Nz) - '2z - '2 =4(Nz)4(z)=g(z). -

Since

r [ 01

-

Ii ir_, = [_N 1 O] is a generator of {r E T0 (N) I rn=o1 ,

AUTOMORPHIC FORMS AND FUNCTIONS

50

we see that g has a zero of order N+1 also at the cusp 0. Now let f be a non-zero element of sk(ro(N)). Then both g and fN 4-' belong to su(ro(N)), so that P7+1/g E Since g# 0 on 0, we see that f/g is a holomorphic function on 0. Moreover, since f has zero at 0 and 00, P."-Vg is holomorphic even at cusps. Therefore PI+1/g must be a constant, which completes the proof.

'win.

It is a classical fact that 4(z) has an expression (2 7r)_ "d(z) = q11;.,(1— q7 )"

Actually if we put 7/(z)= e 2 ...z/24

(1 q"), —

v((az+ b)1(cz+ d)) = A • (cz+ d) 1" 2 v(z)

(q = satisfies db ]e SL 2(Z))

with a certain constant A depending on a, b, c, d. On this and other related topics, we refer the reader to Dedekind [8], Hermite [31], Hurwitz [32], Weber [89, pp. 112-130], Siegel [84], and Weil [100 ]. EXERCISE 2.29. Let N be one of the integers 2, 3, 4, 6, 12, and let k= 12/N. Prove that sk(r(N)).= c • 4(z)u 4v. 2.30. We can associate a function so on S1. 2(R) with any element f of Gk(r) by 99(a)= f(a(i))j(a, i)- k for a E 5L 2 (R). Then it can easily be verified that so(r • a) = so(a) for every 7- E r, and ço(a • a(6))= el" • v(a) for cos 0 sin 01 ento , It is often convenient and essential to every a(0) = [—sin 0 cos Oi deal with w instead of f. We shall not, however, pursue this view-point further in this book. REMARK

CHAPTER 3 HECKE OPERATORS AND THE ZETA-FUNCTIONS ASSOCIATED WITH MODULAR FORMS 3.1.

Definition of the Ilecke ring

Let G be a multiplicative group, and r, T" be subgroups of G. Let us write ri, T" if r and r' are commensurable, i. e., if 1 t-1r' is of finite index in r and in r' (see § 1.1, especially Prop. 1.11). Fix a subgroup r of G, and put -,

r

is a subgroup of G containing r, and By (1) of Prop. 1.11, we see that also the center of G. Moreover, if ri is a subgroup of G commensurable the commensurator of r in G. We call with r, then In the following discussion, we shall fix r and a family {r 2 } 2 -,4 of subgroups of G which are commensurable with r, where A is a set of indices. and every A, p e A. Note that ar Acr' ,-,r i, for every a e

r

f". r.

r

PROPOSITION

PROOF.

3.1. If a E

P.

coset decompositions d =Erp :rpna-iria], e=Er,:r„narpa-13.

one has disjoint

rA arp =Uf-irA ce,

with

raar

with

Consider a disjoint coset decomposition

rp =

crpna- T or)a,

2a6,, hence r Actr p = Ui „l a,. If ['2a5 = raaki, Then a-1 r then 451 6,-1 G r p ncr - T aa, and so i.j. This proves the first relation. A similar argument applies to the second one. Now let us consider a Z-module RAF, consisting of all formal finite sums of the form Ek Ck (riakrp) with ck E Z, ak E P. For every rAarp with a E denote by deg (r iarp) the number of cosets r,16 contained in r iar p. Further, for x = Ek ck (rAakrp) e R2p, define deg (x) by deg (x) = Ek Ck • deg (rActkro), and call it the degree of x. (We can actually define another degree by considering cosets 5r p contained in PAarp. This may not be equal to the above one.) We shall now introduce a law of multiplication : R2p x/?0,,--R2v. First consider disjoint coset decompositions

52

HECKE OPERATORS AND THE ZETA-FUNCTIONS

FxF1 =uz (of course with a and p in f). therefore r ,cer,e pr,, is a finite

F91,= uj r„pj

u; r201r „ pi .u,,, Then rA arp union of double cosets of the form Faer„. With u=rocro, y=r,dr„, and w=r,içr, we define the "product " u • v to be an element of R,b,, given by u.v=E u•v; wow,

where the sum is extended over all

w=rAer„cr2art,i3EL„,

and

(3.1.1) m(u ; w) , the number of (i, j) such that I'2 o443; =i-i 2e (for a fixed e). To make this definition meaningful, one has to show that the right hand side of (3.1.1) depends only on u, v, and w, and not on the choice of representatives fai l, {Oi } , and For that purpose, let tt (S) denote the number of elements in a finite set S. We see that r2aipi =r,ie if and only if rAai =r,zePil. Further, for a given j, the last equality holds for exactly one i. Therefore

{(1, j) I raa,d; = rA el = {i I epj

E rAar ti}

=the number of cosets of the form Fs in

r,pr,,nr

The last number is obviously independent of the choice of fai l and {M. Now, if r2ert,---rz r,„ then Z-=.3/773 with a' e r, and 3 e r„ hence

e, ppryn r

F3FIJ n r pa-

203

Therefore the number in question is independent of the choice of After this verification, we can now define the law of multiplication R2fi x4,-, R2, by extending Z-linearly the map (u, y).—)u • y in an obvious way. PROPOSITION

3.2. Let u, v, iv, fai l, Lail, and be as above. Then

deg (w) • m(u • v w) =

r2cri31r= FF„} .

lek

be a disjoint coset decomposition. Then r.laiAirv=r2er, if and only if r2aiPi=r2ere for some k. Observing that the last equality holds for exactly one k, we have therefore PROOF.

Let

4 I (i, j) I FAaPJE

m(uPROPOSITION

r

— EL1#

),

3.3, For every x E RA t, and every y deg (x • y).-- deg (x) - deg (y) .

E

J

2ek} q. e. d.

4,„ one has

DEFINITION OF THE HECKE RING

3.1

53

PROOF. Let the notation be the same as in Prop. 3.2. mation over all w= raerv c rAard3rv, we have deg (u

E

Taking the sum-

deg (w) • m(u y; w).-- the number of all

u,

deg (it) • deg (v) By linearity we obtain the formula in the general Case. PROPOSITION 3.4. The above multiplication law is associative in the sense that (x- y)- z=x-(y- z) for x E R, 2, y E Rap, z PROOF. Let MI, denote the Z-module of all formal finite sums k ck pek with Ck G Z and ek e f . Let u = T aari„-= ui Ta (disjoint). We can assign to u a Z-linear map of Mt, into M2 (which we denote again by u) by means of the action u- Y! k-rte k = Zi,k Ckr Aaiek. It can easily be seen that this does not depend on the choice of tail and {E } . By linearity we obtain a map of R into Horn (Mr, MA). This map is injective. In fact, if Ea ca - 2 a[' r =0 is a non-trivial cancellation, we have raaie = 1 2a2e for some a, and a,. But this implies .F2a1rti -=T2a2rp, hence no such cancellation is possible. Thus we get the injectivity. Now consider disjoint coset decompositions TAart, Ui raai,r,pr =_-ui rppi, and k r21, for each r2er,cr2arpfir.Then 2

a1 p) • qrpier • (r 0)1 =E„.1 rxcip zz,k In(r Ar r pAer „; r2er)l'2ek fi) • (17 ppr)}• rd2 fr

This shows that (y - z)- a = y • (z a) for y e R A,,, z c kw and a c A f. It a = (x • y) (z a). (y (z a)). x - ((y - z)- a) further x c Rid, we have ((x -y) (x- (y z))- a. By the injectivity proved above, we obtain (x- y)- z = x • (y • z),

q. e. d. LEMMA 3.5. Let a G P. Suppose that the number of cosets of the form T2 in Taal; zs equal to the number of cosets of the form 721; in PAT p. Then there exists a common set of representatives fai l such that I 2

(74 = ji airp .

e

PROOF. Let r2e c f l ed; and )2r rlarti. Then E r gyr r2121 t„ e Then rae= r2C, hence -=ave with E pl and 6 e rp. Put C= _-.Cr It , i. e., C is a common representative for T1 and 72r Lt. Our assertion can easily be derived from this fact.

We shall now show that this phenomenon takes place when < oo. As G, we take subgroup of SL,(R) with pC1

r is a discrete

HECKE OPERATORS AND THE ZETA-FUNCTIONS

54

GL(R)= {a

E

GL 2(R)1 det (a)> O).

3.6. Let r1 and ro be commensurable with r, and let a E P. If gravto = (i \), the number of cosets of the form r in Tlar s, is equal to the number of cosets of the form *T o in T laTo. PROPOSITION

Let d=[T ren ce'rla], e =ET a : Tlnar o a -ii. Then e= Ca -1 T l a: a -1T nro j, hence d- (i p\)= itcro n a"T AVE:0= e • = e • 11(1W?). Therefore we have d = e, which proves our assertion on account of Prop. 3.1. PROOF.

Coming back to the general case, we obtain : PROPOSITION 3.7.

Let a

E

P, p E P. Then

(1) r Aapr = ,ar - (r,pr0) if TA= ail ; (2) rApro ,(i,aro)-(ropro) if ro p = pry.

This follows immediately from our definition of the law of multiplication. Let us now fix any semi-group 4 such that rc4ci". Let R(r, 4) denote the Z-module of all formal finite sums E ft cft aT with cft E Z and a t E J. With respect to the law of multiplication introduced above, R(T, 4) becomes an associative ring, which we call the Hecke ring with respect to T and 4. Obviously T = r. 1- r is the identity element. PROPOSITION 3.8. If G has an anti-automorphism a.-0a* such that T*= r and (TaT)*= Tar for every aE 4, then R(r, 4) is commutative. (Here an anti automorphism of G means a one - to - one map of G onto itself satisfying (aP)* = p*a*.) -

Applying * to Tar, we find that the number of right cosets in TaT is the same as the number of left cosets. Therefore, by Lemma 3.5, for any a, p E 4, we can put TaT=U i Ta i =Uta ir and rpr=y1 rp, =y,p,T (all disjoint). Then Tar = Ter = ra: and ij9i =TP*T then rp lai = rp*rcer=(rarpr)* If rarp = t.Je ); 1' p7. = Ue i r. Therefore we have PROOF.

ru,

pr), Ze ce(rer), pr) -(ran=Ee c(ii) same components rer. By Prop. 3.2, we have (Tai)-

with the

Per} = rcr} =

C e - deg (rEr) = # {(i, 3)1 iaj9,i =

((i, j) I rifilatr so that ce = c.

This completes the proof.

(applying *) ci •

deg

wm,

3.2

A FORMAL DIRICHLET SERIES WITH AN EULER PRODUCT

55

So far, no motivation has been given to our discussion. First we take the simplest case as an example. Let F be an algebraic number field of finite degree, J the ring of integers in F, and E= t (see 0.2). For simplicity, let us assume that the class number of F is one. Then to every ideal A= o rl in F, we can associate a coset aE = EaE. Thus in this case we put E=T' and 4 =J—(0) (or J =F— (0)). Our multiplication is just ideal multiplication. If the class number is greater than one, we can make the same type of consideration by means of the ideles. Let us now take a non-commutative (say simple) algebra X over an algebraic number field. Let S be an order in X, i. e., a finitely generated Z-submodule of X of maximal rank, which is a ring with identity. If r =:sx, every left principal ideal Sa is determined by Ta. Since we do not have commutativity, multiplication of ideals does not go so smoothly. Therefore, instead of Ta, we can take the double coset Par which has less variances than Ta. This point of view will be clarified more explicitly in the following sections, by taking X to be a matrix algebra M(Q), especially A if 2(Q). We shall also see in § 7.1 the connection of TaT with algebraic correspondences on algebraic curves. 3.2.

A formal Dirichlet series with an Euler product

Let us confine ourselves to the case G=GL,(Q) and T =SLn(Z). every integer N# 0, put

For

T N = (r E TIT .:-.1 n mod(N)) LEMMA 3.9.

Let

pe Mn(Z), det (13) = b* O. Then r Nb C P - 'r Nig n PrNr.

Put 13' =bp-i. Since 13, e Mn(Z), if rai n mod (Nb), then we have grp F= p, p= b. ln mod (Nb), hence p-i rp-in mod (N). This shows especially that p-irpeM,(Z). If r e T A,b , we have det(p -irp).1, so that p-i rpEr N, hence r e pr N p-i. Similarly r E p-ir Np. PROOF. -

LEMMA

3.10.

r = GLn(Q).

PROOF. If a e GLn(Q), then a = cp with some c E Q and p E Mn(Z). We have ara - ' = pr p-1. By Lemma 3.9, r ni3r fi-t contains Tb with b = det (p). Since [r : r 0 < œ, we have [r: r narcel < 00. Transforming it by the inner automorphism .—)a - '$a, and then substituting a -' for a, we obtain taTa -1 : ara- 'n rj <00, so that a G P.

Put 4= (a e M(Z) i det (a)> 0). Obviously 4 is a semi-group, and TC tic P. We shall now determine the structure of R(r, 4). For n integers 4 to — , a„, let diag [a l , --- , an ] denote the diagonal matrix with diagonal elements a„ •-• , an. By virtue of the theory of elementary divisors (see

HECKE OPERATORS AND THE ZETA.FUNCTIONS

56

Lemma 3.11 below), we know that the representatives for 1"\ZI/1" are given by the diag [a„ , a„J with positive integers a„ , a„ such that a i divides a14.1 . Then we see that the transposition E'--E is an anti-automorphism of G, and Vary= l'ai for every double coset Val with a e G, since we may assume a to be diagonal. By Prop. 3.8, this proves that R(r, 4) is commutative. Our next task is to obtain a sort of multiplication table for the elements of R(r, 4). The main idea is to assign a lattice to each coset Fa, and ta cothnt the number of lattices instead of counting the number of cosets. For that purpose, put

V = (In = the vector space of all n-dimensional row vectors with components in Q, and let G=GL(Q) act on the right of V. We call a submodule L of V a lattice (more specifically a Z-lattice) in V, if L is finitely generated over Z, and V is spanned by L over Q. It can easily be seen that L is a lattice in V if and only if L is a free Z-module of rank n. If a e G and L is a lattice in V, then La is a lattice in V. Note also that if W is a subspace of V and L is a lattice in V, then L nW is a lattice in W. Further, if L and M are lattices in V, then (i) L -FM and L (-)M are lattices in V; (ii) there exists a positive integer c such that cL c M. LEMMA 3.11. Let L and M be lattices in V. Then there exist n elements u„ ..• , u„ of V and n positive rational numbers b„ ••• , b„ such that L= Zu z , Zb,u„ and b", E b,Z. = This is just (a restatement of) the fundamental theorem of elementary divisors. Obviously Mc L if and only if all b i E Z. We call {b,Z, •-• , b„Z} the set of elementary divisors of M relative to L, and write

(L: )i}=(b„ --• ,b„)= {b,Z,

, b„Z} .

If flic L, one has [L : Al] = b, ••• bn. Especially if a = diag [b„ •-• , b„], then (L: La} =(6 1 , ••• ,b„}. Hereafter let us . denote exclusively by L the standard lattice Z. Then

=SL,,(Z)= {a E GI

det (a) > 0} .

For a and i in 4, we have Fa = r iS if and only if La= Li.3. LEMMA 3.12. Let M and N be lattices in V. Then (L: M} = (L: N} if and only if there exists an element a of r such that Ma= N. PROOF.

(L: M}

The "if "-part is obvious. To prove the "only if "-part, let N}= fa„ , a n ). Then there exist 2n elements u, and v, of V

3.2

A FORMAL DIRICHLET SERIES WITH AN EULER PRODUCT

57

such that L= E, Zu, = E i Za ivi. Define an element Zv„ M= E, a of G by u i a=vi for i=1,••• ,n. Then La= L, Ma=N, and det (a) = If det (a)= —1, take —v, in place of vi.

, an be positive integers such that ao., is divisible by a i. Define an element T(a„ , an) of R(r, .1) by Let a l,

T(a„ ••• , an)=rar , As is remarked above, the ring

diag [a„ •-• , a s].

a

R(f", 4) is spanned by the T(a„ ••• , a n) over Z.

3.13. Let Tar =T(a„ ••• , an). Then ['E'—LE gives a one-to-one correspondence between the cosets FE in rai and the lattices M such that {L: M) = (a„ ••• , an ). LEMMA

We may assume that a = diag [a„ , a n]. If re=raa with E we have (L: Le) = {L: La(3)={1,: La)={a1, an). Conversely, if {L: M)= (a, ••• , an ), then, by Lemma 3.12, there exists an element T of f' such that M = Lar. Obviously rarc Far. This correspondence is one-to-one, since rE=r7) if and only if LE=L1. PROOF.

The degree of T(ai ,— , an ) coincides with the number

PROPOSITION 3.14.

of lattices M such that (L: A1)=(a„

,

an).

This is an immediate consequence of Lemma 3.13.

ref'

with ci E Z, then ce is PROPOSITION 3.15. If (rar).(rpr)=Ei ce • the number of lattices M such that (L:M)=(L: Lp) and (M: Le) = ( L: Lac). Let Far = U

PROOF.

(i,

ce

rat and rpr =L), rp, (disjoint). raifi j = re) = # ((i).1)1LaA= Le) .

Then

Here note that i is uniquely determined by $ and j. Assume Lap,= LE and put M=Lpi. Then (L: M)= (L: 4), and (Ai: LE}={Lp i : Laip;}= (L : Lor i ) = (L : La). Conversely, let M be a lattice such that (L: M)= (L: Lp) and (M: LE)=(L: La). By Lemma 3.13, M= L8 1 for one and only one j. Then (L: 1,$;-1 )= (LA: LE)= ( L: La). By Lemma 3.13, Lepil = La i for some and Le= Lap,. Thus each M determines a pair (i,j) and conversely. This proves our assertion.

p be elements of A such that (raT)• (rpn=rag'. In other words,

PROPOSITION 3.16.

to det (6).

Then

T(a t , PROOF.

Let a and

an). T(bi, Let $ E

rarPr.

, bn) = T(aib„ Let

• ab)

det (a) is prime

if (an, b n) = 1 .

M and M' be such that {L : M)= ( L : M')=

HECKE OPERATORS AND THE ZETA•FUNCTIONS

58

= {L : 143} and {M : Le} = {M' : Le} = {L : La}. We have EM-FM' : MJ= [M' : Mn WI The left hand side is a divisor of [L: Ai] = det (p), and the right hand side is a divisor of EL: La] = det (a), since M-FM' C L and LE c M M'. Since det (a) is prime to det (p), we have M-FM' = M and M' = M n M', so that M = M'. On account of Prop. 3.15, this implies that the multiplicity of rEr in (rai) rpr) is one. Now if e rarpr, we can find at least one M as above. Then LecMcL, and Lae is isomorphic to L/MEDM/Le, hence to L/LaeL/Lp, since det (a) is prime to det (p). Therefore the elementary divisors of LE relative to L are completely determined by a and p. This shows that rarpr consists of only one double coset, which is obviously rai, q. e. d. From the above proposition, it follows that every T(a„ •-• , an) is a product of elements of the form T(pei, ••• , pen) with a prime p and exponents 0 e 1 e2 ••• e., and such an expression is unique (so long as we take at most one factor for each prime). For each prime p, let Rr denote the subring of R(r, 4) generated by the T(pei, , pen). Then our question is reduced to the study of the structure of Rr. Before proceeding with this task, we notice a simple fact : PROPOSITION 3.17.

T(c, • , c)T(b„ ..• , b.) = T(cb„ • • , cb.).

This follows immediately from our definition of the multiplication-law in R(r, 4). In particular, we see that T(c, , c) is not a zero divisor in the ring R([', 4). (Actually, we shall see later that R(r , 4) is an integral domain.)

Now we fix a prime p, and will study the structure of Rr. Consider (zipz)n= LIN, as a vector space of dimension n over the prime field ZipZ. PROPOSITION 3.18. (ZIPZ)n. Then Cin)

Let cin) be the number of k-dimensional subspaces of

Cn(n)k

(Pn-1)(Pn—P) (Pn—P k-1 ) (P'- 1 )(Pk —P) •-•

= deg (T(1, • , 1,

(Pk —P k- ')

p, •••

n—k

The equality cin ) = enn2k and the expression of 47" as a rational function in p are well-known. To connect this with deg (T), we use Prop. 3.14. Let M be a lattice in V such that {L : M } = {1, • , 1, p, , pl with n—k l's and k p's. Then pLcMcL, and Avg, is an (n—k)-dimensional subspace of LIpL. Conversely, for every (n—k)-dimensional subspace K of LIpL, we can find such an M uniquely so that mipL= K. This fact together with Prop. 3.14 proves the equality. PROOF.

3.2

A FORMAL DIRICHLET SERIES WITH AN EULER PRODUCT

Define a Z-linear map 0: Rr"

07(1, pai

,

Pan))

by

= T(Pa's

, Pan)) = 0

O(T(Pa°, Pa',

59

, Pa n) , if ao > o.

LEMMA 3.19. 0 is a surjective homomorphism, and Ker(0) coincides with

p,

, P) • R17,1+1'.

PROOF. The surjectivity is obvious; the assertion about Ker(0) follows from Prop. 3.17 and the definition of 0. Therefore, to complete the proof, it suffices to verify the multiplicativity for the elements T(1, pal, , pan). Let us put, for simplicity,

e' = {1,

gat, ••• ,

f' = {1, g

,

= 11 , pei,

pan}

e

,

, en pen ,

pg = m(T(e) • T(f); T(g)), We are going to show that p= p1 .

= m(T(e') • T( f'); T(g')). Let L , zn +I .E7-o Zui , L=Die,,Zu„ Then {L: N} =g, {L' : g', and

N' = Zuo+ Zpaiu,, N=E7-1 Zpciu,. by Prop. 3.15, te,=#{MI{L: M}=f, IM:N}=e), pg,=# {M' I {L' : M'}= f',

: N'} =e'l .

Suppose {L': M'} =f, TM': N') = e'. Then u. E N' C M'. Put M = M' L. N} =e. Conversely, if Then M'= Zu o -FM, and clearly (L: M} =f, M is a lattice in IT = Qui such that (L: M) =1' N) =e, then put M'= Zuo -FM. It can easily be verified that M = M' r' L, (L' : M') = f', (M' : N')= e'. This shows IIg = p Now we have

T(e) • T(f)= T(e')- T(f')= p eT(g')+T(P, •-• , p). x with an element X of Rr". Since sb(T(p, ..• , p))= 0, we see that 0 maps T(e') T( f') to T(e) • T( f). This completes the proof. THEOREM 3.20. The ring RI,n1 is the polynomial ring over Z in n elements

T(1,

P), T(1, -•- 1

1 P/

".

T(p, --• p)

which are algebraically independent. Especially R,;') has no zero-divisors (other than 0). PROOF. We shall use induction on n.

For n=1, our assertion is clear

HECKE OPERATORS AND THE ZETA-FUNCTIONS

60

since T(p6)=T(p)a by Prop. 3.17. Let us therefore assume that n> 1, and the assertion is true for n-1. For every Tar with det (a)=pv, put w(Far) = ii and for X= Ek ck • rak r E R', define w(X) to be the maximum of w(rak r) with non-vanishing ck. Call X homogeneous if the w(ra k r) are the same for all ck *O. In particular T(Pa', ••• , pan) is homogeneous, and w(T(pal, -•• , pan)) +a. The product of two homogeneous elements is clearly homogeneous. Put T")= T(1, , 1, p, ••• , p) with n k l's and k p's. We are going to prove, by induction on w, that every element X of RIT» is a polynomial in Tin), --• ,T?). It is sufficient to consider the elements of the form X= T(pai, , pan). If a,> 0, we have, by Lemma 3.17, ,

—

T(Pa ',

Pan) = T(P,

P)T(Pa1-1 ,

, Pan-1 )

so that the question is reduced to an element with smaller w. (Note that w(X) = 0 if and only if X is a constant, i. e., an element of Z.) Therefore assume a 1 = O. Consider the homomorphism 0: li) .1?') obtained in Lemma 3.19. By the assumption of induction, we have —

(P(X)= T(Pa2)

I Pan) =

Ek Uk

Mk( T in—")

P

where u k e Z, and the Ai k(T1 11-") are monomials in Ti "—" , • •• , T,7.71 ". Note that each M k(T1n --1 )) is homogeneous. Therefore we may assume that w(A ,(T in-")) =w(X) for all k, since there is no cancellation between homogeneous elements with distinct w's. Substituting 7 TP) for Ti" - ", put

Y =Ek uk • 11/1 kg in ) ,

20 •

We see easily that ze(A/ k(Tni))=w(X). Since (P(X—Y)= 0, there exists an element Z of Rn) such that X— Y =T(p,•-• , p). Z. It is clear that w(Z) < w(X). By induction, Z is a polynomial in T,!n), hence X e Z[T11", •-• , T]. To prove the algebraic independence of the Tin) , let P be a polynomial such that P(Tin), ••• , T) =0, P *O. We can express P in the form

,Tn=n k(rin i P,(Tin ) , , TV) , -

where 0 5_1z .5_1, and Pk :* 0. Since r,n) is not a zero-divisor (see Prop. 3.17), we have 0 = Ef..k(T")"Pi(Ti n) , we obtain T;en-21). Applying ••• , T,"_1")= O. By induction, we have P 0, a contradiction. This completes the proof. From Th. 3.20, it follows that the whole ring R(r,, A) is a polynomial ring over Z with infinitely many indeterminates of the form T(1, ••• , 1, p, ••• , p), p being any prime. In particular, R(r, J) is an integral domain. For every positive integer m, let T(m) denote the sum of all rar with a G J and det(a)= in. Now we consider a formal Dirichlet series (with coefficients in R(r, J))

3.2

A FORMAL DIRICHLET SERIES WITH AN EULER PRODUCT

D(s)=

T(m)m 1 =

4ff

61

(ran- det (a) 1 ,

where the last sum is taken over all distinct double cosets Fa ' with a in 4. From Prop. 3.16, we can easily derive (3.2.1)

T(mmi)=T(m)T(mi)

if (m , no =1.

Therefore, D(s) can be (formally) expressed as an infinite product

D(s)= Hp CE7-0 T(Pk)P -kaj where p runs over all primes.

By our definition of T(m), we have

E7-0 T(Pk)Xk = Eoze iz. zen T(P",

, pen)xeli---1-en

with any indeterminate X. We shall now prove that this formal power series is actually a rational expression in X: THEOREM 3.21. Let Tin' =T(1, let X be an indeterminate. Then

p,

, p) with n—i l's and i p's, and

Er_o T(p k )Xk

,

and therefore

ELI T(m)m - ' = H p [E:,_ o (— where the product is extended over all

primes

11, 2 Cn)p-

,

p.

First we prove two lemmas. LEMMA 3.22. Let the integers cl,k) be as in Prop. 3.18. Then

T'X -(E;_ o T(pin)x -)

= E..0 ck ) • {EidI v.. Here we understand that

T(1,

, 1, pdi ,

pdoXdi+---i-dk}

cik) =0 if i> k, and c,"=1.

PROOF. Fix a set of exponents Id„--,d k l, and denote by p(d) the coefficient of T (1 , , 1, pdi , pdk);"d i + -1-cik in the product 7'Xt -(E1°.. 0 7'(pm)Xin). We observe that such a term can occur in T,Yi • T(pin)x- only if i+m = d i + -.. +d k . Fix a lattice N such that {L: N}= {1, •-• , 1 ,Pd ', Pd 9. By Prop. 3.15, p(d)=E

I {L : M}= {1, ••• , 1, p,

,p}, IM:

N } = { L:

where the sum is extended over all Val.' such that det (a)= pm and a e J. (Here and in the following, the number of repetitions of p is always i.) If {L: M} = (1, p, -•- , p} and Nc M, we can find an element a of J such that { M: N} = L : I a}, and obviously det (a)= r. Therefore p(d) is the

HECKE OPERATORS AND THE ZETA-FUNCTIONS

62

number of lattices M such that NC M ,

(*)

p, •-• , p} .

{ L: M} = {1, ---

Take a basis fu i l so that L =

Zu„, and Zu,.+ E!. 1 ZP4un-k+v

•

Then pL+N=E:z- Zu„-i-E!_, Zpun_k+., hence LAPL+N) is isomorphic to (z/pz)k. If M satisfies (*), we have pL+Nc M, and LIAI is isomorphic to (z/pz)i. Therefore p(d)# 0 only if i < k. Assuming i k, we see that M/(pL+N) is a (k—i)-dimensional subspace of L/(pL+N). Conversely, any (k i) dimensional subspace of L/(p1,+N) can be written in the form AmpL+N) with a unique M satisfying (*). We have thus p(d)= c?), which completes the proof. —

-

LEMMA 3.23. Et. 0 ( - 1) ip"i - "12ck) = O if k> O. Put f(X)=TIM(X — pi).

PROOF.

1.

Then we have

f (x)IE f(P')(x — 03

since the right hand side is a polynomial of degree < k which takes the value •-• , p"t. Substitute p4 for X. Then we find 1 at k points p°, 1

clk)(_1)k- t p(k - i)(k- - 1)12

q. e. d. PROOF of Th. 3.21. We simply take the product

Er-0 (-1)'pl" -1

Z'x i] • C

E:-0 T(Pm)Xmj .

By Lemma 3.22, this equals ET-0 ( -1) (p'" -n" EZ-oc k) • {E T(1,

, 1, p",

By Lemma 3.23, only the term with k= 0 is non-vanishing, and that term is just 1, q. e. d. It is worth while restating Th. 3.21 in the special cases n =1, 2.

E: ..1 T(m)m - '

If n = 1,

IL [1— 7p)p - sj ,

and if n=2, (3.2.2)

T(m)m - '=ngi- T(1, p)p - '+T(p,

.

(Note also that T(1, p)=T(p).) Th. 3.21, in the case n .2, is due to Hecke [29], although he did not discuss the abstract ring R([', 4), but its representations in the space of

A FORMAL DIRICHLET SERIES WITH AN EULER PRODUCT

3.2

63

modular forms, see below. The abstract ring R(1-' , .1) was introduced in [71]. The result of Th. 3.21 for arbitrary n is due to Tamagawa [86 ]. THEOREM 3.24. If n = 2, and p denotes a prime, then the following formulas hold.

(1) (2) (3) (4)

T(m)= cd T(a, d). T(1, pk) = T(pk)—T(p, p)T(pk -2) (k.- 2). T(m)T(n)= Edl(m,n) d. T(d, d)T(mn/d 2). T(P)T(p)= Er oPt RY, MR/I +s-2') (r _5_ s), especially T(p)T(P k )=RPk+1)+PRP, PMP") (k> 0). (k =1), (p + 1) T(p, p) (5) T(p)T(1, pk) = T(1, P')+1 (k> 1). PT (p , Pk) (k > 0). (6) deg (T(1, p))= deg (T(pi , p+k))= pk -' (p+1) (7) deg (T(m))= the sum of all positive divisors of m. -

PROOF. The first two relations are obvious. Since 1??,) is a polynomial ring ZET(p), T(p, p)], we can embed R(; ) into a polynomial ring () [ A, B] with two indeterminates A and B so that

1-T(p)x+pRP, pg2 = (1—AX)(1—BX) . Then

EL0T(P ni)X 1".' =[(1—AX) -1 —(1—BX) - 9/(A—B) =E;- 0 (A ffi— Bm)X 111/(A — B) , so that T(Pm) = (A m+1 — Bm'')/(A— B) = Ello An' ' 13' .

Therefore

T(PORY)= EA 8."T(Pr)— P+1 T(P r)]/ (A— B) = (11'+' ET- 0 Ar - T 1 — B"-1 ET- 0 11`13')/(A— B) -=-Er..0 A L Bt(Ar+ s-2L+1 — Br+s -"+')/(A— B) -

=Er-o PL R P` , P`)T(P

2 ),

which proves (4). Observe that (4) is a special case of (3). Therefore (3) follows from (4) and (3.2.1). If k = 1, (5) is a special case of (4). If k> 1, we obtain, from (2) and (4),

T(p)T(1, pk) =R pk+1)+RP , P)[ PT(pk -1)- T(p)T(P k-k )] = To., pk+9+ RP, P)E(P+1)T(pk - ')-T(p)T(pk -2)] . The last term T(p)T(p 2) is given by (3). Then we obtain (5). By Prop. 3.18, we have deg (T(P))= c1 2' = p+ 1, and deg (T(P, p))= 1. Applying Prop. 3.3 to (4), we obtain

64

HECKE OPERATORS AND THE ZETA-FUNCTIONS

(P+1)- deg mpk» = deg (T(pk -"))+p • deg (T(pk - ')) . Then, by induction on k, we can easily verify that (*)

deg (T(pk)) =

1+p+-•• +pk .

From this relation, Prop. 3.3, and (3.2.1), we obtain (7). (*) and (2).

Then (6) follows from

A few remarks are in order concerning the meaning of the Euler product of Th. 3.21. Since we have been working only with the abstract ring R(r , 4), the Euler product is valid only formally. It is not an analytic statement, but rather an arithmetic statement about the properties of the coefficients of the Dirichlet series. The use of the symbol m -8 (so far) involves no analysis ; rather m -8 is just an indeterminate. Now let us introduce some analysis. Suppose that we represent the ring R(r, 4) on some vector space over C. Then the ring elements T(m) act as matrices with complex coefficients. Through such a representation, the above result concerning the Euler product, if it converges, gives an analytic statement about a certain matrix-valued function of a complex variable s, which has certain multiplicative properties. If we diagonalize the matrices T(m) simultaneously, then the diagonal elements of D(s) are ordinary Dirichlet series, each of which has an Euler product. This will actually be done in §§ 3.4, 3.5. As an example, consider the simplest representation

R(r,, 4) Far

Z (see Prop. 3.3).

deg (TaT)

Then we obtain deg (T(m))n = rip EE?-0 (— 1)IP/ 2c1.)P-"] But we have an equality (3.2.3)

( _ i) t p,(t-Iy2crx ,

(1 — )0( 1_px)

(1_pn-ix)

which can easily be proved by induction on n. Therefore deg (T(m))m -3 = Ç(s)C(s

-

1) •• • C(s

—

n +1) ,

with the Riemann zeta-function C. REMARK 3.25. Let F be a local field, i. e., a finite algebraic extension of the p-adic field Qp, or the field of power series in one variable over a finite field. Let r be the maximal compact subring of F, G =GL.(F), F = GL,„(r), and 4 = {a e M(r) I det (a)* 01. In this case, R(T, 4) is essentially a sub-

3.3

THE HECKE RING FOR A CONGRUENCE SUBGROUP

65

algebra of the group algebra of G. To see this, first note that G is locally compact, and r is an open compact subgroup of G. Let R' denote the module of all complex valued continuous functions f with compact support such that f(axb). f(x) for all a E r and b e r. Fix a Haar measure ix of G so that p(r)= 1. For ! and g in R', define the product f * g by

f *g(x)

f (xy - 1)g( y)d p( y) G

(x E G) .

It can easily be verified that f *g e R', and this law of multiplication is associative. Now, to each double coset rar, assign its characteristic function. Extending this correspondence C-linearly, we obtain a C-linear map of R(r,G)0,c onto R', which is actually a ring-isomorphism. Furthermore we can develop a theory of formal Dirichlet series (or formal power series) analogous to the above one. We only have to take, instead of p, the number of elements in the residue field of r modulo the maximal ideal. 3.26. (A) Let { e1, , en } be the standard basis of L. Za, and let L„. Ze,. Prove (by induction on n) that for every a e J, we can find representatives a} so that Far= U. Ta, and L„(x,c L, for v =1, • , n. (B) The notation being as in (A), for every lattice Mc L such that [L: M] is a power of p, put EL,: L n An= pa. and 2(M) = Xyav-av—I . Here X 1, • , X„ are indeterminates, and a=O. For rar= ura, with a e J such that det (a) is a power of p, put 0(rar)= 1(La,), and extend Z-linearly 0 to a map of R' into ZEXi , , Xn3. Prove that 0 is a surjective ring-isomorphism. EXERCISE

{

3.27. Let f be a positive integer, and X a character of (ZIfZ)". Find an expression for EXERCISE

X(m)

deg (T(m))m - g

in terms of the L-function with character X. (Put X(m)=O if m is not prime to !.) EXERCISE 3.27'.

Prove that, if n=2,

T(p)m= EOEVS.,2 where I

1 )]• Pr 7P,

PYT(P m-2r)

1=m!/r!(m—r)L

3.3.

The Hecke ring for a congruence subgroup

Let r, 4, and rN be as in § 3.2. We shall now study R(r, J') with a subgroup r, of r containing rN for some AV, and a certain subset of J.

66

HECKE OPERATORS AND THE ZETA-FUNCTIONS

First we prove a simple LEMMA 3.28. Let a and b be positive integers, and c the greatest common divisor of a and b. Then re = ra- rb. If a E re, there exists an element fi of M(Z) such that p 1 mod (a) and fi a a mod (b) by the Chinese remainder theorem. Then det (13)a-1 mod (ab/c). By Lemma 1.38 (or by its proof), there exists an element r of T such that r is mod (ablc). Then r E ra , r i a E ra, and a r ria, so that recrarb. Since the opposite inclusion is clear, we obtain the equality. PROOF.

Let us fix a positive integer N, and put

4N= {a E M„(Z)1 det (a)> 0, (det (a), N) , 11 , so that zl= d i . Denote by AN the natural map of fix a subgroup r, of r containing rN , and put

M(Z) to M n(ZINZ).

We

0 = {a E .41 AN(r ia)=1N(ar 1)} . We see that P

= 4 N if r' = r N.

LEMMA 3.29.

The notation being as above, let a,

49 e J. Then the following

assertions hold.

(1) T'ar' = {E E rar 2 N(E) e 2N(r'a)} if a E 0. (2) r Nar N = rNierN if and only if rar = r,sr and a mod (N). (3) rar =Tar' =r'ar. (4) T'ar' = T'ar N =r N ar' if a E 0. (5) If a E 0 and r'ar' =U i r' a disjoint union, then rar =U ra t ,

a disjoint union.

is

To show (3), put a = det (a). By Lemma 3.28 and Lemma 3.9, we have = ra T N ca-TarN, so that a - Tar c a - 'rar N. Hence rar C C Tar'. Since the opposite inclusion is obvious, we obtain (3). Next, to see (1), let E rar, and 2N(e) E 2(T'a). Then e ra mod (N) with TETI. By (3), e Tar N, hence e = 3az with 3 E r and z rN. Then r a mod (N). Since rN cri, we see that 3 e hence E ParN C Par'. Conversely if E r'a/-", we have clearly E rar, and by the definition of 0, N(e) E 2N(r'a). This proves (1). At the same time, we have proved that Par' C Since the opposite inclusion is obvious, we obtain (4). The assertion (2) is a special case of (1). Finally, let a E 0, and r'ar' = U i r'a i (disjoint). Then rar = rar' = Ui rai. Assume ra i = raj. Then a, = ra, with r Er. By (1), ai aaj mod (N) with 3 E Then r m 3 mod (N). Since pN cr,, so that r'cr i = r'a i. This proves (5). we have r e PROOF.

rar N

r'ar N.

3.3

THE HECKE RING FOR A CONGRUENCE SUBGROUP

67

PROPOSITION 3.30. Let the notation be as above. Then the correspondence T'ar'.— , raT, with a e 0, defines a homomorphism of R(r, , 0) into R(r, 4).

PROOF. Let a, 49E0, and let T'ar' =U i Ppr, =vi r,fij be disjoint unions. By (5) of Lemma 3.29, rar = U i ra i and rfill = uj rfij are disjoint unions. Put (r'ar')(r' 4911')=I4 (r,$ro with ci E Z. Then rarfir =rarisP=raPisr, =Ue rer, with the same Z's. Moreover, since a, fief), we have 21„,(r'e) = 21,,,(r'a19) for every e E par, isp, so that, by (1) of Lemma 3.29, per, = {C E rer1,1„(0E 2.41 (Pai9)} . It follows that Per, .—,rer is one-to-one. Therefore, put (Tar)(rpr) = Ee ce - (rer) with ce E Z. Then

ce

{(i, j)I rcriP,=ri} ,

=-# {(i, j)

=

.

Therefore it is sufficient to show that r'cr, ie j = pe if and only if rap, Assume ra ifij = re. Then = 1a1i9; with r E r. Since 2,,,(E) E 2,„(Paii3j), we have e da 119i with â E Then 5 mod (N), hence r E r', so that T'e. Since the converse is obvious, this completes the proof. Hereafter we consider only the case n =2. Let t be a positive divisor of N, and b a subgroup of (ZINZ)x. We shall often denote by the same letter b the set of all the integers whose residue classes modulo (N) belong to b. Define semi-groups zit , 4 ,, and a group p by

(3.3.1)

41,={a E 4 I 2N(a)=[ 01 xl with x E (Z/NZ)'} ,

(3.3.1')

J'Af ={[ uiv

(3.3.2)

r, ={[ ca

E 4 u E b,

v-a0(t), wEEO(N), (z,N)=1},

b ]eSL,(Z)i a e 1), b

d

0 (t), c

0 (N)}

For instance, r0(N) and rN are of this type. (But there are some groups between r and r, which can not be transformed to this type of group by any conjugacy in r.) We see easily that 4'N = 4 P=I' 4 tj and 4c 0. PROPOSITION 3.31. The notation being as above, the correspondence Par` with a E 4 defines an isomorphism of R(T', 4,1) onto RU', '1 N ). ,

PROOF. On account of Prop. 3.30, it is sufficient to prove the injectivity and the surjectivity of the map in question. Let 7)E z/ N , and b=det(77). 10 Take an integer c so that bc-a 1 mod (N), and put so = [ u c I Then

HECKE OPERATORS AND THE ZETA-FUNCTIONS

68

By Lemma 1.38, there exists an element r of r such 1 01 rip mod (N). Then r -lv 7,-[ 0 b j mod (N), hence riv e 4, and

det(vio) a-.1 mod (N). -

that r --

rrivi" = rvr.

This proves the surjectivity. To prove the in jectivity, let 1 1 0 a, 19e in and a -. 0c1J' p -. [ 0 d ] mod (N). If Far = r'sr, we have

[0

c -- det (a)= det (A)E-_-- d mod (N), hence a a-, 13 mod (N). Therefore, by (1) of Lemma 3.29, Par, = ri,sri. This proves the injectivity, since R(1-1 ', d'pl) (resp. R(r , 4 N)) is a free Z-module generated by the r'ai' (resp. Tar) with a E 4-1,. Let us now consider a set

4,_{ rac L

(3.3.3)

lb b=0(t), c= _- 0 (N)}. dbiE'il a e -

Then zl' is a semi-group containing I', and d'N. We shall now determine the structure of R(r,, z ). For each prime p, put Ep =GL,(Zp). Then, for every a E 4, the double coset EpaEp is completely determined by the p-part of elementary divisors of a, and vice versa. Further, for a positive integer m, we write m I Noe if all prime factors of m divide N. Then every positive integer can be uniquely written in the form mq with m I N - and (q, N)=1. '

Let a e 4', det (a)= mq, m I Noe, (q, N)=1. following assertions hold. PROPOSITION

3.32.

Then the

(1) r'ai' = {/3 E 4' I det ()= mq, Epi3Ep = EpaEp for all prime factors p of q } . (2) There exists an element e of d' such that det ()= q and EpeEp =EpaEp for all prime factors p of q. 1 0 (3) If is as in (2), and 77= [ 0 m], then

e

Par, _=(r/r/). (rivro=(rivr,)• (4)

The element

e of (2)

can be taken from zit.

Let X(a) denote the set defined by the right hand side of (1). 1 Clearly r'ai' c X(a). To prove the opposite inclusion, let A = La PROOF.

Since a is prime to mN, ae . 1 mod (mN) for some ee Z. By Lemma 1.38, e 01 there exists an element r of SI,(Z) such that r F,_.- [ 0 a j mod (mN). Since -

A e tr,

we see that r e

r ,,

-

ris . [fiiv tbj

mod (mN) with integers b and f. 0 Put 3 = [ r', and 6n3.- [,!) tgb ] mod ( miV) with g e Z. —f 1N 11 Then 5 E and

3.3

THE HECKE RING FOR A CONGRUENCE SUBGROUP

Taking the determinant, we have mq

tb g mod (mN), so that 5rp-- [ 01 mg

tb, c = [1 0 1.1' =5n3e )71

o

Put 7/ = [ 0 n

mod (mN).

-

[ 01 °,2 ]

mod (N), so that

e

69

-

1*

Then det

]

(0= q,

in. Moreover, we see that p î'î'. By

our construction, EpeEp = EpaEp for all p dividing q. This proves (2) and (4). The element e may depend on p. Let us now show that r'evri is determined only by a and independent of the choice of S. To show this, let be an element of 41:, such that det (ei)-= q and Epe,Ep = EpaEp for all p dividing q. Then e and e, have the same set of elementary divisors, hence re['=re,r.

e,

o ] mod (N), we have FNTN=FNClFN by (2) of Lemma 3.29, so that e i =s0e0 with so and 0 in rN . By the Chinese remainder theorem,

Since

e e, [ 0

we can find an element 0 of M 2(Z) so that

1 mod (mN), 01ç7 mod q M2(Z)

for all p dividing

q.

Then det 1 mod (mqN). By Lemma 1.38, we can assume that OE az(Z)Then 0 E IN. Put co , e0770($0 -1. Then, det (co) = 1, and

1 mod N Al,(Z p)

co

co ,- 1 mod M2(Z) -

for all p dividing N, for all p dividing q.

eo77

Therefore w e M2(Z) for all p, so that co E M,(Z), hence co E 1N' Since =weyie-i, we have rie 1 77P=Peov P=r,e)7P. This shows that rierir' is determined only by a. Moreover, we have seen that Par' c X(a)cr'erir'. Then obviously these three sets must coincide, hence (1). Now, for any e as in (2), we see, from our definition of X(a), that both Per/0' and I'I'I' are contained in X(a). Therefore

Par'. per,77P=P7iPcP . To prove that the multiplicity of Par, in show If a1 4', a,

(4)

(r/P).(P7,7p) is 1, we first

e 41 and Pal = ra„ then l'a, = rla2 .

tb, In fact, put a, = ra, with r e I, and 2N(a)= r , 2N(r)=["] LOav x • Then a tb l ] = [zia 2 utb 2 +u'a2 1 1 we have [ l so that v = 0, u = aiaV E b, and t I w,

0

*

va,

hence r e p. This proves (*). Now let E'er' = Ut Pet, r'vr i = U riv i be disjoint unions. By (*), the re, are distinct, and the Iv./ are distinct. Moreover, by Prop. 3.16,

HECKE OPERATORS AND THE ZETA-FUNCTIONS

70

wen • (r 77r) , rEv r =T ar . Therefore the number of (i, j) such that Teivi = Ta is at most one. (Note that TT may contain cosets other than Tv i.) It follows that the number of (i, j) such that T'$ iv i = T'a is at most one, hence the multiplicity of T'ar' in (PCP) (T'72T 1) is one. The product (T'vT 1) • can be treated by the same type of argument.

(per')

PROPOSITION 3.33.

Let a E 4', det

T'ar' =

4'1 det (9) = m}

E

(a)-= m with ml N. Then =

r

m tr ]

(disjoint).

PROOF. The coincidence of the first two sets is a special case of Prop. 3.32. The last union is obviously contained in the second one. Now let p E d', det ( i9)= m. Consider the special case q=1 in the proof of Prop. 3.32. Then 1 tb . we see that orp= e[ 0 m l with an element C of T N, an integer b, and elements ni = r 1 thi r 1 tri r and of p. If b=mh-Fr and 0 6. .r<m, then [16 L O 1 J LO m Therefore p is contained in the last set. To show the disjointness, assume 1 tr a tbi ts with r 0 r < s ni-1 and [0 m] = 01 m Then we r= = l_c [ d JE r i tri = [a ats+tbm] so that r must be 1. This completes the proof. have L 0 mJ c cts-Fdm '

t/

For each positive integer n, let T'(n) denote the sum of all T'ai' with a E 4' and det (a)= n. By Prop. 3.33, we obtain

(3.3.4)

deg (r(m)) = m

if ml N' .

Further, for two positive integers a and d such that

(3.3.5) let

r

(d, N)=1,

r(a, d) denote the element of R(T', 4,) which is sent to T(a, d) = a

[

a ! d,

0]

0 d T by the isomorphism of R(T', 4) onto R(T , 4) of Prop.

3.31. Then

we obtain THEOREM 3.34. (1) R(T', 4') is a polynomial ring over Z of the elements of the form

T'(P) for all primes p dividing N, T'(1, p),

p) for all

primes

p not dividing N.

These elements are algebraically independent. (2) Every element T'ar' with a E is uniquely expressed as a product T'(m)T'(a, d) =T'(a, d)T'(m) with ml N", ald, (d, N) = 1.

3.3

THE HECKE RING FOR A CONGRUENCE SUBGROUP

71

(3) r(m)T'(n)= T'(mn) if m I N -, n I N. (4) T'(n 1 n2)=T'(ni)T'(n 2) if (n„ n2) = 1. (5) R(/', zi')O z Q is generated by the T'(n) for all n over Q. The assertion (2) follows from Prop. 3.32. By Prop. 3.33, if m and nIN', we see that T'(m)T'(n)= cr'(mn) with a positive integer c. By .(3.3.4) and Prop. 3.3, we obtain c = 1, which proves (3). Therefore we see, in view of Prop. 3.31, that R(I-", 49 is generated by the elements listed in (1). The proof of the algebraic independence of these elements is straightforward, and may be left to the reader. Finally, if n= mq with m I N°° and (q, N)= 1, we have T '(n)= T'(m)T'(q) =r(q)T'(m) by (2). Therefore, by Prop. 3.16, Prop. 3.31, and (3), we obtain (4). By (1) and (5) of Th. 3.24, and by Prop. 3.31, we have prcp,p)=T'(P)2—r(P2) PROOF.

for every prime p not dividing N, which together with (1) proves (5). Thus the multiplication of the elements T'(n) can be reduced to that of TI(pk) with a prime p. If p divides N, we have T/(pk)=r(p). If (p, N)= 1, the 'elements r(pk) satisfy the same formulas as in Th. 3.24, on account of Prop. 3.31. We can express these facts as THEOREM 3.35. R(1", 49 is a homomorphic image of RU', J) through the

map T(n).-4T'(n) for all positive integers n, T(P, p)

—,

iv(p, p) for all primes p prime to X,

T(p, p).-, 0 for all primes p dividing N. Therefore, from (3) of Th. 3.24, we obtain

(3.3.6)

T'(m)T'(n)= Ed d • T'(d, d)r(mnId 2) (the summation over all positive divisors d of (m, n) prime to N) .

Moreover, if we define a formal Dirichlet series D'(s) by

(3.3.7)

D'(s)= Er uvr(Par') - det (a)- $ = E,,..., r(n)n-s ,

then, from the above observation, we obtain

(3.3.8)

D'(s)=HpiN [1 -r(P)P 8]' xllp t ,I, [1-T'(P)P - a-FT'(P, P)P -2,] - ' -

By our definition, we have

(3.3.9)

10 Ti(p)=Flo ]i"

for every prime p.

HECKE OPERATORS AND THE ZETA-FUNCTIONS

72

Let us now study T'(q, q) for a positive integer q prime to N. there exists an element ag of S142) such that

0,7 ] .

2p/(aq) = [g -'

(3.3.10)

By Lemma 1.39,

1-1 o

i and ['q. ca.' = T(q, q). Therefore Then 2N(4 • (T)= L 0 4 2f T'(q, q) = ["q. acr . (3.3.11) There is a simple property of r'ari which can be described by means Fa 6 1 of the "main involution" of the matrix algebra. For a = Lc c/J eli12(C) put

a€=rL—C d —b1 =e- t as - i aJ

(e= [—1 ° 4]) .

Then it can easily be verified that

=ac- } -3c , a+ae = tr (a) - 12 ,

(ap = 13' a€ ,

(ca)e = ca

ace= det (a) - 1 2 .

The map e is called the main involution of M 2(C). are stable under it. Let a e ,tc- and det (a)= q. Then 2N(a)-40

(c e C),

Obviously M2(Q) and M2(10

0 ;N(a1)= H d'

q1'

so that a-a cr,a(:.-,. aca, mod (N). Therefore, since a and az have the same set of elementary divisors, by (2) of Lemma 3.29, we have (3.3.12)

- QT' T'aFt=rover'=r'ato

if a e Jt, det (a)=

q.

Moreover, we can easily verify that Pa q = agr', hence, by Prop. 3.7, (3.3.13)

T'ar'=(riagro•(FlaT').(11 'aT')•Gr9 6,P).

From this we obtain (3.3.14)

Par, commutes with ricer' if a E J.

For each positive integer a prime to N, fix an element O a of SLAZ) as in (3.3.10). Then, for every positive integer n, one has PROPOSITION 3.36.

{a E J' I det (a) =

n} = U.0

Pa.-

[g

bdt]

(a > 0, ad= n, (a, N) = 1) ,

and the right hand side is a disjoint union. PROOF.

The right hand side is clearly contained in the left hand side..

ACTION OF DOUBLE COSETS ON AUTOMORPHIC FORMS

3.4

73

To show the disjointness of the right hand side, suppose 70a . [ 0a bdt] u ut'e w . it h = au - [ 0 w ==

7 E T'.

f Put aVra, = [ g h ].

Then

[g e .[2 ][0a bdt]

ru yt] , so that g = O. Since det (a;lra.). 1 and au> 0, we have e= h. 1, LO w

hence a = u, d.w, and yt= bt+fd. Since 7E r,, we have f =Pt with some f' E Z. Then y = 6+ ft d, so that y = b. This proves the disjointness. Now let n= mq with nil Noe and (q, N). 1. Then deg (T'(n)). m • deg (T'(q)). By (7) of Th. 3.24 and by (5) of Lemma 3.29, deg (7''(q))= deg (T(q))= Ecrq>. c. Therefore it is easily seen that deg (T(n)) coincides with the number of the cbsets of our disjoint union. This completes the proof. Action of double cosets on automorphic forms

3.4.

So far our discussion of double cosets has been purely algebraic or arithmetic. Let us now come back to the situation of Chapter 2, and consider the representation of double cosets in the space of automorphic forms, as is indicated at the end of § 3.2. First we recall our notation : j(a, z) = cz-Ed

(z

G

.D, 0-. [

a c

b -] d

GL.,(R))

f i [a]k . det (a)" - f(cr(z))j(cr, z ) k for a function f on f9. Let r1 and r2 be commensurable Fuchsian groups of the first kind, t the commensurator of r, and r2 in GL(R) in the sense of § 3.1, and a E T. For JE Ak(r 1), we put

f 1 Cr lar23k = det (a) k 2-1 - Eli—if I Ea.ik ,

(3.4.1) where

r 1 ar 2 = U(`-1 r,ai, It is clear that f

(disjoint).

I Er l ar2j k is independent

of the choice of the representatives

a. PROPOSITION

3.37. Eriar2jk sends Ak(ro,Gk(r,),sk(ri) into ilk(r), Gk(r 2),

S k (F2), respectively.

Let a E r2. Then {r 1 a(3}„ coincides with fr i ah, as a whole. Therefore, if g= f I Er lar Ak, PROOF.

g I Ca3k= det (a) k/2- ' • E. f I Ca„ajk=det(a)"' • E.f1Ea,,Jk = g . On the other hand, by Prop. 2.4, 1. 1 Cajk

E

Ak(a; 1F ,cr,,). Put

HECKE OPERATORS AND THE ZETA-FUNCTIONS

74

T3 =n.a;T ia.n Pt Then T, is a subgroup of r 2 of finite index, and gE Ak(T3). By Prop. 2.6, we see that g E Ak(T2). The same argument applies to Gk(r) and sk(ro• Consider the module R 12 generated by T 1a1' 2 with a e every X = E c r 1 ar 2 e R„ with ca Z, we define f 1[X]k= E cafIcriarok PROPOSITION PROOF.

(fE

t

(see § 3.1).

For

Akw i» •

3.38. EXYI=EXICYik for every X

E R12

and every YE R23•

It is sufficient to show that

(fIEriaro) I Crogrsik =f I Uriar 2) • (r 213r s)1k Let (T iar a) • (r2Pr3) = E cgriErs) with ce E Z, and let

r1er3 = Uh T l eh

riar,_u,ria,,

be disjoint unions. By our definition of multiplication, we see that r l aiPi = Ee,k Ce • r h •

Therefore

(fiEr,aroolEr2Prok = det (a) 212 '

f

I [ cri 6 1],, = det pycn

=f I U1iar2)- (r2pro],,,

ce • f

I

k

q. e. d.

In particular, fix a Fuchsian group T of the first kind. Then we see that the action of R(T , P) on Ak(T) (resp. ck(r), sk(r)) defines a representation of the ring R(r, We shall now fix our attention to S,(T), and introduce an inner product in the space Sk(r). For two elements f and g of Sk (T), we put (3.4.2)

=f f(z)g(z) • yk'dxdy n

(z= x+iy e .

Here note that f(z)gTilyk and y -2dxdy are invariant under T, on account of Prop. 2.18, and (1.2.3). Therefore the integral is well-defined if it converges. To prove the convergence, it is sufficient to show that f(z)g(z)yk, as a function on /1\0*, is continuous at the points corresponding to cusps. Let s be a cusp of r, p an element of SL 2(R) such that p(s)= co, and r,= f r E T I r(s) = s). Then 1 h

Pr3r'' { ±1} ={±[o 1] 1 7nz} with a positive real number h. Then there are holomorphic functions 0(g)

3.4

ACTION OF DOUBLE COSETS ON AUTOMORPHIC FORMS

75

and P(q) at q =0, such that

f IEP - 9k= 0(e'r 1

")

,

giErlik=?Pïe:‘"") •

Then we have f (w)g (w) 1m (w) k = f (p - I (z))g (p - '(z)) 1m (p - (4)k

= 0(e'"(')F (egzi it) 1m ( z)k

(w p - 1(z))

.

Since 0(0)=T(0)= 0, we see that this function is continuous around the point of T\0* corresponding to s, q. e. d. The inner product < f, g) is of course hermitian and positive definite ; it is called the Petersson inner product (or the Petersson metric) in Sk(T). We shall now determine the adjoint of C11 a121 with respect to the inner product. PROPOSITION

3.39. Let T , and 12 be commensurable Fuchsian groups of

the first kind, and let a e t

.

If det (a)= 1, one has

2=(f,giEr2a- lriJk>1 for every fE sk(r a) and g e sk(ra), where < ,)i denotes the Petersson inner product in sk(r,) for i= 1, 2.

PROOF. First note that, for any a e SL 2(R) and for any measurable set A on 0, we have

(3.4.3)

f.(4) f . g. y k -2 dxdy = (f [a]k) • (g I [a],) • yk -2 dxdy

Now let P be a fundamental domain for TAO. (For example, one can take P to be the polygon 11 on 0 considered in the proof of Th. 2.20.) Let

ra=uerana-T be a disjoint union. we have

Then T 1aT 2 = U

icre„ is a disjoint union.

By (3.4.3),

(f I [T iar 2]k - g • yk -2dxdy ( f I [a e,]k ) • g • yk-zdxdy= E,

(f [a]k)

f [a]) • g y"dxdy sal')

Y k 'dxdy=f amf. (g l[a - l] k) • y"dxdy ,

where Q =U„ e„(P). It can easily be seen that Q is a fundamental domain for r2 r) a - T 1a, hence a(Q) is a fundamental domain for a12a'nr 1 . If < , >' (resp. < , )") denotes the Petersson inner product in Sk(r 2 na - T ia) (resp. Sk(aT 2)), then, we have shown that

HECKE OPERATORS AND THE ZETA-FUNCTIONS

76

Eriarik, g>2 = < f I Calk, g>i = < f, g I Ca - 1>ff

Interchanging f and g, and taking

in place of a, we obtain

i = < f, g 1Ca - lik>" which completes the proof.

c

In view of our definition of j(a, z) and [a]k, we have fi EcI =f for every R', so that

(3.4.4)

fl[r,cr l]k = c"f

(c e Rx).

Therefore, the above proposition needs a modification by a scalar factor if det (a)* 1. However, by means of the main involution e of M 2(R) introduced in § 3.3, we have, for any a E Pi (not necessarily satisfying det (a)= 1), < f I Er1ar2I, g>2 = < f, g

(3.4.5)

Er2air iJk>1.

This can easily be verified, since if a = cfi with c e It* and i E SL,(R), then a' = c Let ro be a normal subgroup of finite index of a Fuchsian group r of the first kind. Then the linear transformation [r oar o]k on sk(ro), with any a e r, is unitary with respect to the Petersson inner product PROPOSITION

on

3.40.

sk(ro). This is an immediate consequence of Prop. 3.7 and Prop. 3.39.

A linear transformation of sk(ri) of the type CriarOk is called a Hecke operator (in a generalized sense). In the next section we shall discuss in detail the Hecke operators in the form by which Hecke originally defined them. Let us now briefly mention that the double coset r 1ar 2 can be interpreted as an " algebraic correspondence ", of which a more detailed discussion will be made in Chapter 7. Let r, =r2 na-irict, and let sop ¶02, and denote the projection maps of 0* to rAvk, r,\ezyk, and rAkyk respectively. We can define two holomorphic maps P1 :

rAtok—

P2

rAtrk

by P, 0 = 0 p2 0 g) , = w2. Note that P, is the natural projection, and P, the composed map of the natural projection of rAvk to (a - ir 1 a)\0* with the isomorphism of (a - ir 1 a)\0* to rl\f,* obtained from z — sa(z). Now let r2=uf.., re, be a disjoint coset decomposition. Then /liar, = (disjoint, see Proof of Prop. 3.1). Therefore if io,(z) is a point of r2\0* with we have

3.5

HECKE OPERATORS AND FOURIER COEFFICIENTS F((02(4) = {40'(e ,(z)) I 1=1,

,

el ,

P 1CP il(s02(z))7 = f491(aet
Since w1(13(z)) depends only on

77

,

el

ro, we can assert

the following: If riar2 = l.)%1 r ,a„ then, through P,oPil, the point w 2(z) corresponds to the points g),(at(z)) for i =1, ••• , e. This is the most primitive form of what we call an algebraic correspondence, especially • a modular correspondence when T's are congruence subgroups of SL 2(Z). As to the historical account of this topic, the reader is referred to Hurwitz, Mathematische Werke I.

3.5. Hecke operators and their connection with Fourier coefficients

r = sL2(z)

and its congruence subgroups. Let us now consider the case of Let N be a fixed positive integer, and let Dtr, and D' be as in (3.3.1-3). From (3.3.14) and (3.4.5), we obtain THEOREM 3.41. The linear transformations [r'or']k on S k(r9, with aeii, are mutually commutative, and normal with respect to the Petersson inner product on

sk(p).

Here we call a linear transformation normal if it commutes with its adjoint with respect to the inner product in question. If N= 1, we have rar= rcer for every a E D, since a and a' have the same elementary divisors. Therefore we obtain, from (3.4.5),

sk(r),

The linear transformations Ercrnk on are mutually commutative, and self-adjoint with respect to the product on

THEOREM 3.42.

with a E J, Petersson inner

sk(r).

It is a well-known fact that mutually commutative normal linear transformations are simultaneously diagonalizable, i. e., there exists a basis of the vector space in question whose members are common eigen- vectors of the transformations. Therefore we can find common eigen-functions of the Er'ar'jk for all a E , which form a basis of Sk(r). In particular, if N.1, the eigen-values are real, since the are self-adjoint. Suppose N=1. Since f IT(p, p)k = pk- 2f, an element of Sk(r) is a common eigen-function of the for all a E z1 if and only if it is a common eigen-function of the T(p) for all primes p. Let f be suth an eigen-function, and let f I T (n) k = p f with p„ E R for each positive integer n. By (3.2.2), we have (formally) ,727-1 pa n - a =Hp 1 pp IF:+pk 1..21

[rank

[ran,:

HECKE OPERATORS AND THE ZETA-FUNCTIONS

78

In the next §, we shall show that this Dirichlet series is convergent on some half plane, and can be holomorphically continued to the whole complex s-plane ; further it will be shown that it satisfies a functional equation analogous to that of the Riemann zeta-function. We shall also prove similar results for congruence subgroups of T. We shall restrict our discussion to Sk(T'). Actually one can consider the Dirichlet series associated with the elements of ck(r). It is known that ok(r) is spanned by sk(ro and the "Eisenstein series" belonging to p, which we studied in §2.2 in the special case N= 1. And it can be shown that the Dirichlet series associated with the Eisenstein series of level N are essentially of the form L(s,X i)L(s—k+1, X 2),

where L(s, X) is an L-function defined by L(s, X) = EZ., X(m)m' with a character X of (Z/NZ)x. For details, see Hecke [27], [29]. Therefore the nature of the coefficients of such Dirichlet series is rather simple. As compared with this, the arithmetical meaning of the Dirichlet series associated with cusp forms is still quite mysterious. Let us now consider r, and d' in a somewhat specialized form. We fix a positive divisor t of N, and consider two extreme cases b=(Z/NZ)* and b= Ill in (3.3.2, 3). Namely we put (3.5.1)

r,;=IT e SL2(Z)12N(T)=[0a atbi]

(3.5.1')

rff = IT e r ,

1 2N(T)= [ 01

44 = {a e zl 14(a) =[

tit)]

with a

E

(Z/NZ)x, b

E

ZiNZ}

,

with b e Z/NZ},

0a tdb ] with a E (Z/NZ)x, bE Z/NZ, d e Z/NZ},

4" =fcr e 412N(a)= [01 tdb] with b e Z/NZ, d e Z/NZ} , where A N is the natural map of M,(Z) to Al,(Z/ N Z). Clearly T" is a normal subgroup of 11, and nip, is isomorphic to (Z/NZ)m. Let 0 be a character of (Z/NZ)x, i. e., a homomorphism of (Z/NZ)x into {z eCl jzf = 1 } . For convenience, we put, for a e Z, 0(a) = I O 1 0(a mod NZ) Further, for

if (a, N) * 1 , if (a, N)=1.

e= r a bi E d, we put a(e)=a z = a. L c c/J

Now we denote by

HECKE OPERATORS AND FOURIER COEFFICIENTS

3.5

sk(n, 0) (3.5.2) If

0q

(35.3)

the set of all elements f of fI

Er] k = 0(ar)- V.

sk(rff)

79

satisfying

for all r e

n.

is an element of SL,(Z) as in (3.3.10), this is equivalent to f lEg qik=0(q)f

for every q Prime to N.

Since Sk(r") may be viewed as a (/-VT")-module, we see that S(r) is the direct sum of the spaces sk(n, 0) for all characters 0 of (Z/NZ)*. We see also that sk(n, 0) = {0} unless 0(-1)= (-1)ft. From Prop. 3.40, we obtain immediately

(3.5.4) The subspaces sk(n, 0) of St([in are mutually orthogonal with respect to the Petersson inner product. V.

Let V', b, and d' be as in (3.3.2) and (3.3.3). We observe that l'ff c r, CI'L zl" c di c z1,ç, and st(p) is the direct sum of the spaces sk(r, 0) for all 0 such that 0(b)= 1. For every a e d„ we can define a linear transformation Ercenik,0 on sk(n, 0) as follows. Take a disjoint decomposition 1-1 a11 =U„ fla,,, and for f e S k(fl,, 0), put

(3.5.5)

fI

marok,0 = det (a)k1-1 E. 0(a(a.)) • f i

It is easy to see that the right hand side does not depend on the choice of

{a„}, and satisfies (3.5.2). Now we have (3.5.6)

Erwrok,0 is

the restriction of

[V' [ ]k to sk(Vl "

0)f or every

g s d', if

In fact, by Prop. 3.36, we can find a disjoint decomposition V' V' =U. r' p,, a tb with elements g,, of the form aa -[ o cd as described there. Then, by the same proposition, we obtain a disjoint decomposition rgr= u ,, ngv. Since 0(a(p))= 1 for g e d' if 0(4)= 1, we obtain (3.5.6). Observe that, for any a e d, there exists an element g of dif such that nan= ngn,. Therefore (3.5.6) implies that f I[T,Çcfr] t,0 belongs to sk(r, 0). Now we see that l'a1',', — )CFa['],,,,A

defines a representation of the ring Rcn, JD on sk(n, 0). Let us denote by T'(a, d)k,0 and T'(n) k ,0 the action of T'(a, d) and T'(n) on sk(n, 0) defined by (3.5.5). By Prop. 3.36, we have

(3.5.7)

fi T'(n) k ,o = re- ' E. EL-4 s.')(a)f ((azi-tb)/d)cr k

(a > 0, ad =n).

80

HECKE OPERATORS AND THE ZETA-FUNCTIONS

(Note that, by virtue of our agreement ib(a)= 0 for (a, N) #1, we can drop the condition (a, N). 1.) By (3.3.11) and (3.4.4), we have

(3.5.8)

f I r(q, 4,0 = q k-2 0(q) • f for f e s k(n, 0) and (q, N)= 1.

Therefore, from (3.3.8) and (3.3.6), we obtain formally

(3.5.9)E77,-1 r(m)k,,,,5 • m -s = HpiN El — r(P)k,or si -1 • Hp N El — r(p)k,op - s +0(p)pk-1-2ii-1 (3.5.10)

r(m)k,v,P(n)k,0= Edi(77100 d k-l sb(d)r(nin 1 d 2)k,o •

Note that, in the last formula, d runs over all the positive divisors of (m, n), since 0(d) , 0 if (d, N) # 1. The convergence of (3.5.9) will be proved in the next section (Lemma 3.62, see also Remark 3.46). Observe that the cusp oo of r" is regular, and the stability subgroup of

1 Let f e sk(r„ 0), and g = f iT1(m) k ,0 with a fixed [0 a positive integer m. Consider the Fourier expansion of f and g at œ:

œis generated by

f (z)=

c(n)e2

",

g(z)=Ez_ i c/(n)ennzlt

By (3.5.7), we obtain

(3.5.11)

g(z)=

Ea

(ad)"cl- kib(a)c(n) e2mn(az-Ftb)lcil

(ad= m) .

Since 71;1, e2r346'6 = d or 0 according as d divides n or not, we obtain, comparing the coefficients of e'z il" of both sides of (3.5.11), e( 1) = Li (1 m) ,ga)ak - cum / (12)

(3.5.12)

3.43. Let f(z)=E7,-Ic(n)e2ir""t be a non-zero element of sk(n) , 0). Suppose that f is a common eigen-function of the T'(n) k ,0 for all n: fi T i(n)k,0= 27, f with 2„ E C. Then c(1) # 0, c(n)= 'Inc(l), and THEOREM

(3.5.13)

An

lip [1_2pp--: +0(p)pk-1-2.11-1

( formally) .

Conversely, if one has formally

(3.5.14)

E7-1 c(n)n' = lip [1 --c(P)rs +0(P)P k-1- "1 -1

then f I Ti(n) k ,o= e(n)f for all n. PROOF.

(3.5.15)

If f

(m) k ,0 = 2„,f, then, taking I to be 1 in (3.5.12), we obtain 2„,c(1) = c(m) .

Therefore c(1)# 0, since f # 0, and (3.5.13) follows from (3.5.9). Conversely, if one has (a.5.14), we see, by the same reasoning as in the proof of Th. 3.24, that

3.5

HECKE OPERATORS AND FOURIER COEFFICIENTS

c(1)c(m) = Ewa

81

ak-1 0(a)c(lm / a 2) .

Therefore, by (3.5.12), we have c'(/) = c(/)c(m), so that fi T 1(m) k,v,= g = c(m)f As an immediate consequence, we obtain

3.44. If two functions of Sk(F, 0) are common eigen-functions of r(n)k ,0 for all n, and belong to the same eigen-values, then they differ only by a constant factor. COROLLARY

Let us fix any basis { f • • ,

L}

of Sk(n,0) over C, where iv = dim

(s

0)).

Put [f i(z)

((z) =

: = E7_, c(n)e2::1-it f.(z)

with complex column vectors c(n). Then the c(n), for n = 1, 2, •-., span the space C' of all x-dimensional complex column vectors. In fact, if they don't, there exists a non-zero C-linear map of C' into C such that (c(n))= 0 for all n. Then we have (f)= 0, which is a contradiction, since f1, — , f, are linearly independent over C. For every positive integer m, define an element A(m) of AMC) by fl Ti(m)= A(m)f • Then, by the same type of computation as in (3.5.11), (3.5.12), and (3.5.15), we obtain

(3.5.16)

A(m)c(1)= c(m).

3.45. If K = dim (Sk(ril, O )) , the linear transformations r(n)k,, for all positive integers n, generate a commutative algebra over C of rank K. Moreover, the identity map of the algebra (or the map r(n)k ,Ø '-34(n)) is equivalent to a regular representation of the algebra. THEOREM

Let A be the subalgebra of M(C) generated by the A(m) for all m. Consider a C-linear map L of A into Ci" defined by L(X)= X • c(1) for X e A. By (3.5.16), L is surjective. If L(X)=0, we have X. c(n)= X A(n)c(1) A(n)Xc(1).0 for every n, so that X = 0, since the c(n) span the whole C'. Therefore L is injective, and hence L gives an A-linear isomorphism of A onto C', q. e. d. PROOF.

2 0( n)e 2rin21t 3.46. Put A(n)=(4(n)) with 20(n)eC, and g ( (formally, for the moment). By the above theorem, the go span, over C, a vector space of rank iv. From (3.5.16), we obtain (g11(z))c(1)= f(z). Since the components of f span sk(n, 0, we see that the gii are actually holomorphic REMARK

82

HECKE OPERATORS AND THE ZETA-FUNCTIONS

sk(rL 0).

Therefore, to prove the convergence of (3.5.9) (or, of Ea„°_, A(n)n), it is sufficient to show the convergence of E7..i cz„n - • for every Eri_i ane 2x.tnrit E sk(r it), 0%) . This will be done in Lemma 3.62. functions and span

To obtain further information on eigen-values, we consider a somewhat general situation. Let A be an arbitrary commutative algebra over a field F, of finite rank, and with an identity element, R the radical of A, and P a unitary A-module. Suppose that the simple components of A/R are all separably algebraic extensions of F, which is always the case if F is of characteristic O. By a theorem of Wedderburn, there exists a semi-simple subalgebra B of A such that A= Be) R. Note that B has the same identity element as A. (This is true even if A is non-commutative. In fact, if 1 = b+r with b E B and rE R, then b=b 2 +6r, so that br= O. Therefore r=br+r 2 =r 2. Since r is nilpotent, r 0, q. e. d.) Let Bz, , B. be the simple components of B, and ei the identity element of Bi. Put Pi = e i P. Then P= PI ED PP,. Since A is commutative, P an A-submodule of P. Since R is nilpotent, we have a finite decreasing sequence of A-submodules -

P, D RP, DR2P,D D 1?iPi

RmiP t = {0}

0).

We understand that R°= A, and m, = 0 if Pi = {0}. Let us now take A to be the algebra generated by the r(n) k ,, for all n over C, and p to be sal 0). In this case every Bi is isomorphic to C. Take a basis {fi, •.• fg} of P=sk (n„ so as to contain a basis of RhP i for every i and every h, and define A(n) as above with respect to this { f, •-• , Then A(n) is clearly a triangular matrix for every n. Let 2 1(n), , 2,(n) be the diagonal elements of A(n). Then we see that, for each I), (3.5.17)

T'(n) k ,sb

2,,(n)

defines a homomorphism of A onto C. Of course these K homomorphisms, as a whole, are independent of the choice of f. Now we have PROPOSITION 3.47. The notation being as above, there exists, for each v, a non-zero element g, of sk(r(i), 0) such that gi,IT/(n) k4 =2„(n)g, for all n; in other words, there exists an element h i, of sk(r, 0) such that h i,(z)= E,-1 `7 2.(n)e27" 1"" In fact, we can find i so that f ,E Pi. Then, for this i, take any non-zero element g, in In view of our definition of Pz , we see that the homomorphism (3.5.17) is the same as the map aj ej +ri—, a,

C, r E R) .

Since (El., a,e,+r)g„=a i g„, the element g,, has the required property.

3.5

HECKE OPERATORS AND FOURIER COEFFICIENTS

83

From Th. 3.45, we see easily that (3.5.18)

Every non-zero C-linear homomorphism of A into C coincides with one of the homomorphisms 2 1, ••• , 2, of (3.5.17).

Let us now consider operators Er' «PI with a group

r, of type

(3.3.2)

with arbitrarily fixed N, t, b. Let 4' be as in (3.3.3), and

E=[ — 1 00 1

We

].

see that Er'. r'e. For every X= E c., - rint„ri e R(P , 4)(oz c with ci,E C, put )(a= E ci, • T'EaE - ir', and 0= {XER(P,z1)1 Xg=X} .

We see that X.—, X is an automorphism of the ring a subring of R(1-', 4). Now let us prove (3.5.19)

R(P, 4),

and hence 0 is

R(T', 40 C 93 .

If aEZIIr, we see that TEac'T =Tar and EaE-1- a mod (N), so that r'EaE - 1-". Par' by (2) of Lemma 3.29. If a is diagonal, we have obviously r/EaE -'1i'=1"''ar'. Therefore we obtain (3.5.19) on account of Prop. 3.32. In the following discussion, we shall denote by End (V, K) the ring of all K-linear endomorphisms of a vector space V over a field K. For each X e R(ri, 4)ez C, let EX3 k denote the element of End (sk(ro,c) corresponding to X. THEOREM 3.48. The notation being as above, let B (resp. B 0) denote the algebra generated by the [X],, for all X E 0 over C (resp. over Q). Suppose that k. 2. Then the following assertions hold. (1) B. is a semi-simple algebra of finite rank. (2) B = B. 0Q C. (3) The characteristic polynomial of [X], for every X E 0 has rational integral coefficients. PROOF. For X = E c%, • T'a,,T ' e R(F', ZI)Oz C with ci, E C, put X* . E ei.. r'cre,ir'• Then we see easily that (X')*=(X1'. BY (3.4.5), EX*I, is the adjoint of [X] k with respect to the Petersson inner product of sk(ro. Therefore tr (EX*X 3k) > 0 unless [X],,= O. Now 0 is stable under the map X.—) X*. Suppose that B. has a nilpotent right ideal N. If [X] k EN, then CXX*3k is contained in N, hence nilpotent, so that tr ([XX*] k)= O. Therefore [X],, = O. Thus B. has no nilpotent right ideal # {0}, which proves (1). Now, for fesk (ro, put f(=f( -2-3. We see easily that f'Esk (ri), and f'! [X],, =(f I [r]x)' for every X E R(r, , 4). Put W. If ESk(r9IP=f1.

HECKE OPERATORS AND THE ZETA-FUNCTIONS

84

skwo=w0,,c.

skwo,

(In fact Then W is an R•linear subspace of and 2f =(f+P) i((in -i- (if)`).) Let B, be the R-linear span of B o. Then we see that W is stable under B 1. Since End (Sk(TO, C)= End (W, R)O R C, we see that elements of B, are linearly independent over R if and only if they are so over C. From this we can conclude that B B i G),, C. Therefore, in order to prove (2), it is sufficient to prove B,=B o OQ R. The proof of this fact and (3) is based on the following statement which we shall prove in § 8.4. —

sk(ro of maximal rank which is

(3.5.20) There is a discrete Z- su bmodule L of stable under the Cr'ar'l for all a e 4.

Assuming this, let V be the Q-linear span of L. Then Sk (P)= V (3)Q R, and hence End R)= End ( V, Q)0(2 R. Therefore elements of B o are linearly independent over Q if and only if they are so over R. This shows that B 1 = 1300Q R, hence (2). Let r be the dimension of S k (TO over C. Then we obtain three faithful representations

(skwo,

Po

: Bo M2(Q) = End ( V, Q) ,

pi

:

A

End (W, R) ,

:

AIT(C)

End (Sk(P), C) .

Restrict p and p, to B o. By Lemma 3.49 below, p 0 is equivalent to the direct sum of p and its complex conjugate. From the above discussion we see that p, is equivalent to p. Since p, is a real representation, we see that po is equivalent to the direct sum of two copies of p. Define po with respect to a basis of L over Z. If e = [X]k with X e then e sends the lattice L into itself, so that Po(e) E M 2,(Z). Therefore the characteristic polynomial of P(E) must have integral coefficients, hence (3). In the above proof we needed the following elementary LEMMA 3.49. Let C? denote the vector space of all r-dimensional complex column vectors, and fx„ , x 2,1 a basis of Cr over R. For every UEM T(C), define an element 2(U)=(Ai(U)) of M2 (R) by Uxi =E1:, )(U)x. Then there [U 0 exists an element Y of GL2,-(C), independent of U, such that Y -1 0 U]Y = 2(U). X PROOF. Let X be the r x 2r matrix whose i-th column is x j. Put Y= [ x Since U X = X2(U), we have CIX=X2(L1), so that [ 01j nY.Y2(U).

Assume

that det (Y)= O. Then there exists a set of 2r complex numbers (a, a21) # (0, • • • , 0) such that aixj = (42, =0. Then ErL t (cat + ca,)x,. 0 for all = 0, c C. Since {x 1 , •-, x 2r ) is a basis of Cr over R, we have a1 = =

a

3.5

HECKE OPERATORS AND FOURIER COEFFICIENTS

85

which is a contradiction. Thus Y is invertible, hence our assertion. 3.50. If we take the ring R ( P, 4) instead of 93, the assertion (1) will remain true, but the assertions (2) and (3) will not. For example, take r(6) to be r', and let k =2. Then .vr(6)) is of dimension 1 over C, and spanned by 4" (see Ex. 2.29). Let a= 1- 1 1 --1. We see easily that L 0 li 4 1'6 1 Crar'I = e2 "6 4". Therefore the Q-linear span of the [X],, for all X e R(T', 4) is two-dimensional over Q, but the C-linear span is obviously one-dimensional. REMARK

3.51. Let r' and 4' be as in (3.3.2 3), r the dimension of Sk(r) over C, and D (resp. Do) the algebra generated by the [T'ar'] k for all a E A' over C (resp. over Q). Suppose that k ___ 2. Then (1) [Do : (1].r. (2) D = D o 0,2 C. (3) The identity injection of D o into End (Sk(P), C) is equivalent to a THEOREM

-

regular representation of D o over Q.

Since R(T', 4') C T, the assertion (2) follows from (2) of Th. 3.48. Let T'(n) k denote the sum of [T'ar'] k with a e zl' and det (a) = n (cf. § 3.3). By (5) of Th. 3.34, Do is generated over Q by the T'(n) k for all n. Now observe that S,.(/'') is the direct sum of the spaces Sk(T'o, 0) for all the characters 0 of (ZINZ)N such that 00) =1, where b is the subgroup of (ZINZY in the definition (3.3.2) of r'. Let 0,, .-• , 01, be all such characters. For each 0„ take a basis Up •-• , fr} of Sg(rI 01), and put PROOF.

fi

c o.,)(n)e2rinvi

[

Or = 1, • - „a) ,

fg

[ fo) = Z7-I a(n)e23""211

f= : f

(p)

with c(n) E C c and a(n) E CT. Define an element w(n) of GL,(C) by fiT'(n) k From (3.5.16), we obtain w(n)a(1) = a(n) for every n. By the same=w(n)f. type of reasoning as in the proof of Th. 3.45, we can show that the map w(n) is equivalent to a regular representation of D over C, and hence [D : C ] . r. This together with (2) proves (1) and (3). 3.52. If I" is as in (3.3.2), and k. 2, then Sk(T') has a basis consisting of cusp forms of which the Fourier coefficients at co are rational THEOREM

integers.

HECKE OPERATORS AND THE ZETA-FUNCTIONS

86

PROOF. Put co(n)=((o po (n)) and f pq= (0„,,(n)e2 1'ns". Since the C-linear span of the w(n) is r-dimensional, we see that the f 14 span a vector space over C of dimension at most r. But we have ( fpo)a(1)= f, so that the fog sk(ro over C, since the components of f form a basis of sk(r). Letspan L be as in (3.5.20), and

E = {e EDo leLC L} . Then Do is spanned by E over Q, and E is a free Z-module of rank r. Define a regular representation 0 of D. over Q with respect to a basis of E over Z. Since E is a subring of Do containing for all a E 4', we see that 0 maps r(n) k into M T(Z). Put 0.= 0(T'(n)k). By (3) of Th. 3.51, there exists an element U of GL T(C) such that Uco(n)U -1 = O n . Put (g 14(z)) = U(f = oneu..tnin. Then sk(P) is spanned by the goo over C, and the gin have integral Fourier coefficients. PROPOSITION 3.53. If f is an element of sk(r) which is a common eigenfunction of the r(n) k for all n, then f belongs to sgr,,,, 0) with a unique character 0 of (Z/NZ)c such that 0 ( )=1, and f is a common eigen-function of all r(n) ft,. PROOF. By (5) of Th. 3.34, f is an eigen-function of T'(q,q) k for every q prime to N, so that, by (3.3.11), f is an eigen-function of [a q]k. Therefore we can define a character 0 of (Z/NZ) by f 157,11=0(0 f Then f esk(n. 0). The last assertion follows from the formula (3.5.6), which implies that T(n)k, is the restriction of r(n) ft to Sk(il cl'). PROPOSITION 3.54.

Let r =

r° '1 Then, for every LN OJ

aE

•

(i'a`i')(E'rTO= PROOF. First note that

ra Lc

and hence ri' = Pr. Then 9

tb r d d] = L —Nb

For a given a E 4

—WI •r, a

put q= det (a), and —tb ] . l mod (N) with b E Z. Put r = [ 01 Then

[g til

,

p=rar-'.

[g 01 ]

a' mod (N). Since q is prime to N, we see that 13 has the same elementary divisors as a. Therefore, by (2) of Lemma 3.29, r'pr' = Prpr, On the other hand, by Prop. 3.7,

=ricer,.

(Ppro(Prr)= pprr,= ["rai' =(T'rr')(T'ar'), q. e. d.

3.5

HECKE OPERATORS AND FOURIER COEFFICIENTS

87

PROPOSITION 3.55.

Let ---= [ °N 0]. Then [r]k (=.(tN)"/ 2 - [rfirrff3k) sends sal 0) onto sk(n, ;,), and Erl = 1. Moreover, for every n prime to IV, one has T'(n)•Er3k=0(n)- Er3k • T'(n)k,PROOF.

fe (*

sk (ro, 0),

Let a E A and det (a)= n. we have

f lEr3kEr'ar"Jk =1 1

)

Er

By Prop. 3.54 and

(3.3.13), if

cer ffiErik

=f I Er'aVPI[r"arff]kErJk =0(n)-1f tEr'arffik [r]k . Take a to be q - al.

Then

f i EriEcrai = gq) -20(q)f I ErI= sl
2= sb(det (a)).T. PROOF.

If q= det (a), and

f lEr'oarOk, o = 2f, we have, by (3.3.13) and

(3.5.6), 2f =0(q)f I [1"'"' aT"I. Denoting by < , > the Petersson inner product on sgr#), we obtain, by (3.4.5),

,T - < f , f> , 2 . = = < f, flEr'crer'lk>=0(q) and hence 2= 0(q)2. PROPOSITION 3.57. Let f E sk(n, 0, f I Ti(n)k6= a. f with a positive integer n prime to N, and g= f I [71. Then g I Ti(n) k ,= d g. This is an immediate consequence of Prop. 3.55 and Prop. 3.56. REMARK 3.58.

ro

Fa tb 1 = rab rt 0 1 i , especially Put a= L 0 1 J ' Then Lc d _I Lk di . Therefore we see that ar'oa- z=r o(tN), and the

—ti = a r L tN o 0 Jv L N OJ map f(z).-4f(tz) sends sk(Il 0) onto

sk(ro(tN), 0).

By means of this map, the discussion of R(p, 4') and the operators Ti(n) k,0 with respect to r,,, can be reduced to the case t=1, by changing the level N for tN. Note that N and tN have the same prime factors, since t divides N. Therefore we could put t = 1 in our definition of 1-', and 4 '0, without losing much generality. (It should of course be noticed that (Z/tNZ)* may have more characters than

(Z/NZ)*.)

es

HECKE OPERATORS AND THE ZETA-FUNCTIONS

3.59. Let p be a prime not dividing N, and f an eigen-function of r(P)k,0 in sk(r o (N ), 0), and let fIT(P)k,o=cpf. Put fin(z)=f (P m z) for m = 0, 1, 2, • • and denote by T"(P)ko, the operator in sk(ro(pw), so), where so(a)= 0(a) for (a, pN)=1. Then we can easily verify REMARK

fi T"(P)k,,, = cp f fm

-1 0(P)f (m = 1, 2,

I P(P)k,w =

It follows that T*(p)k,9, is not semi-simple if I

, 1).

3.

Now let 2 and p be the roots of the quadratic equation

x 2 —c p x-F0(p)p k- i= O. is an eigen-function of TN(p) kw with p as the eigen-value. , [ 0 —1 o-1 Further put r =[N 0 r = pN 0 ]- If f iErJk =g, we obtain easily

Then

f iCrlik=pki2g(pz)

f3 I

[r ' Jk

=p

2g.

Suppose that 0 is the trivial character, and fi[r]k =ef with e=±1. Then ( f —2 f 1) I [r']ft is not an eigen-function of P(p)k ,9 unless cp =pki2(1-Fp -,), which is usually not the case. (At least it contradicts the Ramanujan conjecture, see below.) REMARK 3.60. Let A (resp. AO be the ring generated by all the T'(n)k (resp. T(n) k ,0) over C, and B (resp. BO the subalgebra of A (resp. generated by the T'(n) k (resp. r(n) k ,ç,) for all n prime to N. Then A (resp. can be identified with the direct sum of the algebras 24, (resp. BO for all 0 such that 0 (T )=1. As for A, this follows immediately from Th. 3.45 and Th. 3.51. As for B, take a diagonalization of the T'(n) kx, and define a homomorphism of B onto C by assigning a diagonal element of T'(n) k ,0 to P(n). In view of (3.5.8), one can not obtain the same homomorphism from two distinct 0. This shows that [B:C]=Eo(b) ,,[ap: C], and hence B must be the direct sum of such Bsb. From Th. 3.41 and (3.5.4), we see that B and Bo are commutative semisimple algebras. Moreover, by Prop. 3.54. we have

Er ir (n)Ic[r]k = n2- k r(n, n)k T'(n)k if n is prime to N. Therefore [r]i l B • Cz- Jk = B, and similarly [7];» B0- [7],, = B;. Thus [r] k sends a common eigen-f unction of B (resp. Bo) to a common eigen-function of B (resp. Br). These facts are not necessarily true for A and A 0, as shown in Remark 3.59. However, Hecke proved the following facts: Suppose that t =1, j. e., rg=r,,(N). Then (at least) in the following

THE FUNCTIONAL EQUATIONS

3.6

89

two cases. (I) 0= 1, N is a prime, and S k(r(1))= O. (By Prop. 2.26, the last condition is satisfied if and only if k < 12 or k =14.) (II) 0 is a primitive character modulo (N). For details, see [30, Satz 22, Satz 24a]. Historically, the connection of a cusp form with an Euler-product was first mentioned by Ramanujan [60]. He considered the Fourier coefficients c,, of the function

(2r)- '24(z) = q TM, (1

qn)24

= n.3.1 cit

e

7 ==. e22)

(1

and made two conjectures :

E7..1 can' --= lip (1—c pr s+p11-2s)-1; cn = 0(n1112+t ) for any 6> 0.

(X) (Y)

The latter is equivalent to the inequality

I cpI 2P1112 for all primes p.

(Z)

The first conjecture (X) was proved by Mordell [51]. Since si,(r) is onedimensional and spanned by 4, 4 must be a common eigen-function of all Hecke operators, and hence (X) follows from Th. 3.43. It was Hecke who made the first systematic investigation of the relation of modular forms with Dirichlet series having Euler-product, in its full scope. In the above, we have presented the easier part of Hecke's theory [29], [301 along with some new results. The idea of diagonalization of the Hecke operators by means of the inner product in the space of cusp forms is die to Petersson [55]. He also generalized the conjecture (Z) in the following form :

(Z') Every eigen-value ,R p of T(p),,,k, for any prime p not dividing the level N, satisfies I Api 2p(k-1)/2. If k = 2, we shall be able to prove, in § 7.4, that (Z') is true for almost all p. In the general case, it was shown by Rankin [61] that c7, =0(nk/2-1/5) for every E;7_, Cn e27: Various methods for the estimate of cn E sk(r(N)). are discussed in Selberg [64]. ‘ n"N

3.6.

The functional equations of the zeta-functions associated with modular forms

Let us first prove two fundamental lemmas group r of the fii .st kind.

for an arbitrary Fuchsian

90

HECKE OPERATORS AND THE ZETA-FUNCTIONS LEMMA

3.61. If f E sk(r), one has I f(x+iy)1 My ''2 with a constant M

independent of x. Conversely, if an element f of k (V ) is holomorphic on 0, and I f(x-Fiy)I My -k " with a constant M independent of x, then f E sk(r). For any holomorphic element f of A k(r), define a real valued function h on f-,) by h(z) , h(x Fiy) , f(z)I y". Since Im (1-(4) , Im (z)I j(r, z) I -2 for r e SL 2(R), we see that his r-invariant. If s is a cusp of I", take p and q.e 24.1zIA (or = e"111) as in p. 29. Then f I[p - '12 = 0(q) with a holomorphic function 0 in the domain 0< II
-

LEMMA 3.62. suppose that co is a cusp of

r,

and let

t r E r • {-±-1}1 7-(00)= col = {±-1} • {[ (1) with a positive real number h. cn einzlh

1 (See § 2.1). for all n. PROOF.

7:12-1

cne

2r.winsilt,

Let f

Esk(r),

ri m E

and

if k is odd and co is irregular,

otherwise.

Then there is a constant B independent of n such that Icn i< B. n"

If k is even, put q= e 2r 1"4, and F(q)=

cnqn. Then

c7,=(27ri) 1 f F(q)q- n - idq,

where the integral is taken on the circle I ql = r in the positive direction, for a small r > O. If Im (z)=y=h/27171, then I e"triAl=e -lin. By Lemma 3.61, My - k12 with a constant M. Therefore, taking r to be cun, we have ic.1 Me- (h/27rn)k' 2. The case of odd k can be treated in a similar way. Our task is to prove a functional equation for the Dirichlet series a n n - ' attached to any 1.(z)=E;_ 1 a nenlig of sk (V ', 0). For the reason explained in Remark 3.58, it is sufficient to consider the case t = 1, 1. e., = I-1 0(N). We shall generalize our question by considering Ez., X(n)a n n -3 with any character X of (Z/rZ)w, where r is a positive integer prime to N. Therefore let us first recall a few elementary facts on the Gauss sum associated with X.

91

THE FUNCTIONAL EQUATIONS

3.6

Let us fix a positive integer r, and X a character of (Z/rZ)x, j. e., a homomorphism of (Z/rZ)x into Cm. We assume that X is a primitive character modulo (r), by which we mean that there is no character of (Z/sZ)x with a proper divisor s of r satisfying e(x). X(x) for (x, r). 1. Then we put, for C E Z, if (c, r) = 1 , X(c mod rZ) X(c) = if (c, r) # 1 , 0 and define the Gauss sum W(X) by W(X) =D:4 X(c)Cc, LEMMA 3.63.

C=

.

The notation being as above, one has :

(1) E;=4 X(c) Cbc = X(b)W (X) for every b E Z; (2) W(X)W(X). X(-1)r ; (3) 1 W(X )1 2 = r ; (4) 'WM= X(-1)W(X). If (b, r)= 1, denoting by

PROOF.

the inverse of b mod rZ, we have

X(c)Ce = 111 X(V i a)(a = X(b 1) a x(a)ca= X(b - ')W(X) . Suppose that s = r/(r, b)< r, and put H = {a E (Z/rZ)' I aal mod sZ } , and let (Z/rZ)x = U vey Hy be a disjoint decomposition. Since bs --= 0 mod (r), we have xb-_.b mod (r) for x e H. Further, since X is a primitive character modulo (r), X can not be trivial on H. Therefore

X(c)C = EvEY ExEll X( YX)Cv" =

(Pb( y)

X(X) = O.

Now, by (1), W(X)W(X)= L W(X)i(c)Ce = L ,C X(b)c= since E, C" = r or 0 according as a Therefore

X(b)E,Cca'"*" = X( —1)r ,

0 or a *0 mod (r). Note that X(-1) = +1.

Er i(c)C - c= Ec fo—oce= i(-1)K1)= -7(T)= i4 and W(X)WCT) = W( X) W( 2)X( —1) = r. Let us also recall a definition of the 1-function : " r(s)=

e—txt - idx

(s E C).

3) We have two gammas: one for a discrete subgroup of SL2(R), and the other for the gamma function. Since the distinction will be clear from the context, we use the same letter for both objects.

92

HECKE OPERATORS AND THE ZETA-FUNCTIONS

Substituting ax for x, we obtain

T(s),

(3.6.1)

0

e - "x 3-1 dx

(s C, a e R, a > 0) .

PROPOSITION 3.64. Let N and r be positive integers, s a positive divisor of N, and Al the least common multiple of N, r 2, and rs. Let X (resp. 0) be a primitive character of (27rZ) (resP. (Z7 sZ)x). Further let f(z)=E7-iae 217"7 be an element of S k(ru (N), 0). Then h(z) = X(n)a neni belongs to

& (ro(M ), ox2). PROOF.

1 and a=[ 0

=

Put

'47-1

J

for u e Z. Then cnuane2n%nz

f I Caulk = EZ-2 so that, by (1) of Lemma 3.63,

Tc(u)f I Caul,

147 a0h(z) =

By Prop. 2.4 and Lemma 3.9, we see that h e Sk(r(r 2N)). Therefore, to prove our assertion, it is sufficient to check the behavior of h under an element

r = [ :talc db ] of

Put

r").

a' = a+cuAl Ir , b' = b+du(1—ad)/r—cd2 u 2M/r 2 , d'.d—cd 2 u.11/r. Then a, b, c, d are integers, dEE d' mod (s), and ilo L

ul.rir a bi_r a' 1 JLAic d L

blirl d'u/ri dJL O 1 J-

Therefore, putting v = d'u, we have f Ecrur]k = ib(d)f I h I [Tic =

CavJk,

so that

rx)- i ib(d)x(v) E„ X(v)f I [crv]k = 0(d)X(c1 2)1z ,

PROPOSITION 3.65.

The notation being as in Prop. 3.64, suppose that r is

prime to N, and put r =

Then

q. e. d.

r 0 —1 1 , r 0 LN

0J ,

r

= L r 2N

—11

0 _I, and fi Erik =

h Erik = 0(r)X(N)1V(X) 2 7-1

I

b ne2z

X(n)b ne' .

PROOF.

Let us use the same notation as in the above proof. Suppose that (u, r)-, 1. Then we can find two integers d and w so that dr — Nuw= 1.

Then a„.:-. 1 =

r — Aru

—w d _law'

Put g =fi[r] k.

Then

3.6

THE FUNCTIONAL EQUATIONS

W(X)h 1E1'1 = E. x(u)f I [crue ]k =

93

Tc(u)0(r)g I[awl

= (P(r) E. X(— N w)g I [a ,J = (P(r)X(— N)W(X)EZ., X(n)b„e 2 " . This together with (2) of Lemma 3.63 proves our assertion. THEOREM 3.66. Let r be a positive integer prime to N, X a primitive character of (Z/rZ)x , and 0 an arbitrary character of (Z/NZ)x . For every f(z) =E;;3-i ane"" of skw 0(N), 0, Put L(s, f, X) = r„°_,X(n)an - s , R(s, f, X) = (r2N) 2(270 -' r(s)L(s, f, X). Then L(s, f, X) is absolutely convergent for Re (s)> 1+(k/2), and can be holomorphically continued to the whole s- plane. Moreover, it satisfies a functional equation

R(s, f, X) , i k 0(r)X(N)W(X) 2r- 'R(k where r

s, f I [r]k ,

,

r ° —11 0 J.

L

In view of Prop. 3.64 and Prop. 3.65, it is sufficient to treat the case r = 1, and X = 1. The absolute convergence of L(s, f, 1) for Re (s)> k12 -1-- 1 follows from Lemma 3.62. By (3.6.1), we obtain formally PROOF.

(*)

0

f (iy)ys'dy= E;;'., a,,

e-2"nvys - idy

0

(27)-: r(s)L(s, f , 1) .

To see that this formal computation is actually valid, we note that

I J.: f(iY).Ys - IdYi

5,4f 1 Y-kavkady

o

if Re (s) > k/2+1, by virtue of Lemma 3.61, and

f:f ( w ys idy I -

B

e - 2=11y Reo) - dy 0

for any s e C. (A and B are constants.) E

c

f(iy)ys - idy =

a„

Now we have e - "nvy'dy,,

since En a n e -2"V is uniformly convergent for y can take Ai so large that

I

n ‘ irs>A1 tAn —

_2

v is - 1

„ '4%.1

I

(E —co)

E.

For any small ./ > O. we

-1 Erz>m la a l j‘ e-2a'nv Y dy

= r(a)(27)— E„, m

< )2

(Re (s)=--

94

HECKE OPERATORS AND THE ZETA-FUNCTIONS

Therefore we see that

IS: f(i)0Y -1 0-ET-, an f'e 0 - 27:nvys-idyl Ç e27y1dy j < 7.

f (iy)ys - icly —Zit., a

This proves the validity of (*) for Re (s)> k/2+1. For the same reason, if g=f [r]k, we obtain

I

(**)

0

g(iy)y' - 'dy = r(s)(270-: L(s, g, 1).

Put A = N-112. Then So

A

fy

dY —f

00

-•. f(iy)y S1 dy+f fuy)y - idy. A

As is seen above, the first term is convergent for Re(s)> k/2+1, and the second term is convergent for any s. Changing y for 1/Ny, we obtain, since f(i/Ny) y yt g

f (iy)yt

WATy)AT- Sy - t- 1 dy ikNk/2-

‘j°

A

0

The last integral is convergent for any s. co

f (iy ) y-'1dy

=

g (iy)yk

1

idy

A

ik Nk12-•S

A

d.

A

fy ( j a

,)

Similarly dy

0 5"

(Re (s) > k/2+1) .

Therefore if we put R'(s, f)= r(s)(27r)-gL(s, f, 1), we see that R'(s, f) can be holomorphically continued to the whole s-plane, and R1( s, f) = ik • T ki2t

Note that

R'(k—s, g) .

r(s) 1 is an entire function. Therefore we obtain our theorem.

In the above discussion we have obtained a Dirichlet series L(s)=E;`,3_, = a ne 2,:n nt by means of the " Mellin inverse transfrom a function f(z)E formation "

7( 1:3)Y

= r(s)(27 ) -8 L(s)= R(s) -

One can actually obtain f(z) from L(s) by the " Mellin transformation"

f (i y) = (2x1) - ' R(s)x - 'ds , where the integral is taken on the vertical line Re (s)= o for some a > O. Hecke employed this correspondence between f(z) and L(s) to prove that R(s) satisfies a functional equation of the above type if and only if f(z) is an

3.6

THE FUNCTIONAL EQUATIONS

95

automorphic form with respect to a certain discrete subgroup r of SL2(R). This result is not completely satisfactory, since r \0* is often non-compact. A more complete result was recently obtained by Weil, who showed that if one assumes the functional equations for 117_,X(n)an' for sufficiently many characters X, then f belongs to Sk(r o(N), 0) for some N and çb. For details of these results, we refer the reader to [28], [98], [101]. In our treatment, we have defined an automorphic form to be a complex analytic function. More generally, Maass considered real analytic automorphic forms on 0 which are eigen-functions of some invariant differential operators. For such forms, he developed the theory of Hecke operators and generalized the above correspondence between f(r) and R(s). Here we content ourselves with mentioning only [44], [45], [46] among his numerous papers on this subject. There are also (at least) three important topics which we do not touch in this book. The first one is the connection of modular forms with quadratic forms. If P(x). —1ZiZ.M2k PoX,Xi is a positive definite quadratic form with p zi in Z, then E.Ezne ziz , called a theta series, is a modular form of weight k with respect to some congruence subgroup of SL,(Z). Here the Eisenstein series play an essential role. The reader may be referred to Hecke [26], [30], and Schoeneberg [62]. One should also note many of Siegel's works on quadratic forms, and its generalizations, which are now accessible in three volumes of his collected works. A treatise of this topic, in the adele language, is given in Weil [97 ]. For this see also Shalika and Tanaka [67]. The second is the explicit computation of the trace of Hecke operators, for which we only mention Selberg [63], Eichler [17], [18], [19], [20], and Shimizu [68]. Finally there is an aspect in which the theory of group representations plays an essential role. For this the reader is referred to a recent work of Jacquet and Langlands [37], and also to the earlier works quoted in the volume. Although we mention these topics separately, they are closely connected with each other, and with what we consider in this book. Our discussion has been restricted to the case of congruence subgroups of SL4Z). Actually one can construct zeta-functions from automorphic forms with respect to a unit group of a simple algebra over a number field. They have Euler products of the form of Th. 3.21. For details, the reader is referred to Maass [45], Godement [23], Tamagawa [86], Shimura [74], Shimizu [68], Weil [101], and Jacquet and Langlands [37]. Simple division algebras of an arbitrary degree are treated in [23] and [86], while the remaining articles are concerned with quaternion algebras (in the general sense, including matrix algebras of degree 2). -

CHAPTER 4 ELLIPTIC CURVES 4.1.

Elliptic curves over an arbitrary field

In this section we give a brief review of a few elementary facts about elliptic curves (without detailed proofs) 4 . An elliptic curve is an abelian variety (a projective non-singular variety with a structure of algebraic group, necessarily commutative) of dimension one, or what amounts to the same, a projective non-singular curve of genus one with a specific point, called the origin or the neutral element. If the curve is defined over a field k, and the origin is rational over k, then the group law is automatically defined over k. Therefore, when we speak of an elliptic curve defined over k, we understand that the curve and the origin are rational over k. Let E and E" be elliptic curves defined over k. By a homomorphism of E into E' (defined over k), we mean a rational map (defined over k) of E into E' that is a group homomorphism. The module of all homomorphisms of E into E' is denoted by Hom (E, E'). Any rational map of E into E' transforming the origin of E to the origin of E' is automatically a homomorphism. An element A of Horn (E, E') is called an isogeny, if it satisfies the following mutually equivalent conditions : (i) 2* O; (ii) Ker (,Z) is finite ; (iii) A is surjective. (Note that we always identify E with the set of all points on the curve rational over the universal domain, see Appendix.) If there exists an isogeny of E to E", then there exists an isogeny of E' to E, and we say that E and E' are isogenous. This is an equivalence relation. Now we define End (E) =the ring of all endomorphisms of E (over the universal domain) = Hom (E, E), End Q (E) = End (E)Oz Q . 4) As for the terminology and notation concerning algebraic geometry, see Appendix. Although our discussion in this and next chapters is restricted to elliptic curves, the theory cannot be fully understood unless one considers them as special cases of abelian varieties. Therefore the reader is advised (though not required) to have some acquaintance with the definition and elementary properties of abelian varieties, as given in (the easier part of) Weil [92], L951, and Lang [43]. See also Appendix Nos 10-13. We borrowed, for example, the construction of the roots of unity e N in § 4.3 from [92, pp. 150-153 ]. For a detailed discussion of abelian varieties with complex multiplications, see F 81].

4.1

ELLIPTIC CURVES OVER AN ARBITRARY FIELD

97

For k =C it will be shown that End (E) is a free Z-module of finite rank, and EndQ (E) is a division ring of finite rank over Q. The same is known to be true for all k. All possible types of EndQ (E), and even of End (E), have been determined by Deuring [10] : End Q (E) is isomorphic to either Q, or an imaginary quadratic field, or a quaternion algebra over Q ramified at a prime p and oo ; the last case can occur only when the characteristic of the universal domain is p. But we shall not discuss this topic in full generality ; we shall treat only the case of characteristic 0 in § 4.4. From now on, we shall assume that the characteristic is not 2 or 3. Then an elliptic curve defined over a field k is always isomorphic, over k, to a projective curve

(4.1.1)

E: Y 2Z =4X 3 —g3XZ 2 —goZ s

with gi in k, and d = gq- 27g i # 0. (The non-vanishing of d is equivalent to the non-singularity of the curve defined by the equation.) We can take the point (X, Y, Z)= (0, 1, 0) as the origin. Conversely, every curve of this form with d * 0 is an elliptic curve. Hereafter, for convenience, we shall write the equation in the affine form

(4.1.2)

E: y 2 .4x 3 —g,x—go ,

but always regard it as a complete curve, by adjoining the point (x, y) = (co , co), which is the origin of E. Then the map (x, y).(x, —y) gives the automorphism —1 of E. Now such a curve is characterized by its invariant

j(E)=j E =g3/t1 (or JE=2633:1E which has nicer integrality properties) in the sense that two curves E and E' defined respectively by the equations y 3 = 4x3—g2x—g3 and y2 = 4.70—gx—g; are isomorphic over the universal domain if and only if jE = glAgi --27g I) = g?1(g?-2 7 g'32)= j E, . One can state this fact in a somewhat stronger form : PROPOSITION 4.1. Let E and E' be defined by y 2 = 4x3 —gox— go and y 2 = 4,t 3—g;x—g, respectively, and let 2 be an isomorphism of E onto E". Then there exists an element p such that g2 =

t g2 ,

,g= tie g, ,

2(x, y) = (p2x, p' y) .

Observe that j E belongs to any field of definition for E. Let k o be the prime field. Then, for any j in the universal domain, there exists an elliptic curve E defined over k 0(j) with invariant j:

98

ELLIPTIC CURVES

for 1= 0, take g2 = 0 and g,. 1 ; for j= 1, take g2 = 1 and g 3 = 0 ; for 1* 0, 1, solve gl(g-27)= j, and take g2 = g,= g (g=27i/(i—l)e ko(i)). (This is just one of many possible choices, and therefore should not be regarded as standard.) For E as above, and for an automorphism a of the universal domain, we define an elliptic curve E° by E° : y 2 = 4x 3 —gf x-gf . Then clearly j(EQ). j(E)°. Therefore E is isomorphic to E° if and only if a is the identity map on Iz o(j E). The field k o(j E) is characterized by this property if the characteristic is 0, and called the field of moduli of E. We have just shown that E has a model defined over its field of moduli. One can define the field of moduli for any " polarized " abelian variety, see § 5.4. However, it is an open question to know whether any polarized abelian Variety has a model defined over its field of moduli.

4.2. Elliptic curves over C Let us now consider the case where the universal domain is the complex number field C. Every elliptic curve defined over (a subfield of) C, as a complex analytic manifold, is isomorphic to a one-dimensional complex torus CIL, where L is a lattice in C, by which we mean a discrete submodule of C of rank 2 over Z. Conversely, let L be an arbitrary lattice in C. Then, an elliptic function with periods in L is, by definition, a meromorphic function on C invariant under the translation by the elements of L; we can regard such a function as a meromorphic function on CIL and vice versa. Let FL denote the field of all elliptic functions with periods in L. It is known that FL is generated by the Weierstrass functions P and P', defined by P(u). P(u ; L)= u-2+E.EL, [(u—ai) -2 — co -2] ,

d

" u) = W P(u) =

2u -3 -2 E EL, (u—w) -3

(V= L—{0}).

—

(It is easy to see that P and P' are contained in FL , and have a pole only at u = 0 (modulo L), of degree 2 and 3, respectively. Therefore, by (3) of Prop. 2.11, we have [FL :C(13 ) ] =2, and [FL: C(P')]=3, hence FL =C(P, P') as asserted.) The Laurent expansions of P and P' at u = 0 have the form :

P(u)=u -2 + EZ 2 (2n-1)G 2„(L)u 2n -2 , P '(u) = — 2 u -3 + EZ-2 (2n-1)(2n —2)G2„(L)u 2 n -3 , C' (T\ — Ewev w -2n . ■

99

ELLIPTIC CURVES OVER C

4.2

Then we have an equality P'2 = 4:33 — g,(L)P —g 3(L)

(4.2.1) with

g2(L) = 60 . G,(L) ,

g,(L)= 140. G,(L).

(The difference P'2 —(4P 3 —g2(L)P—g 3(L)) is holomorphic on CIL except at 0; but from the expansions given above, we see that it is holomorphic and vanishes at 0; hence it must be identically equal to 0.) Since FL = C(P, P') is a function field of genus 1, we have g2(L) 2 -27 g,(L) 2 # 0 .

(4.2.2)

In fact, if this is 0, the equation (4.2.1) defines a curve of genus 0. For a given L, define an elliptic curve E by E: y2 = 4x3 —g 2(L)x—g 3(L).

(4.2.3)

Then the map u .--, (P(u), P'(u)) gives an isomorphism of CIL onto E. Let 0 denote the complex upper half plane as before. For two complex numbers co l and ah such that (1)1(1) 2 e 0, we obtain a lattice L= Zah+Zah. Conversely, any lattice in C can be given in this form. We then write P(u ; w„ 0)0= P(u ; L) , 4 (w1,

(DO= g2(w1, (DO' —27.g3(wi, (02) 2 ,

g2(a)1, ah)= g,(L) ,

aw l , (DO=g3(L)

(L = Zv i + Zah) •

In § 2.2, we defined a modular form J(z) and a modular function j(z) by J(z). J(z, 1) = g2(z, 1) 3 -27 g,(z, 1)2, j(z) = g2(0)1, w2) 31(g2(w1, w2) 3-27.g2(w1, w2 )2 )

(z = 0)11(02)

(or rather J(z) = 2633 • j(z)), and proved some fundamental properties of these functions. We observe that j(z) is the invariant jE of the elliptic curve (4.2.3), which is isomorphic to CI(Za) 1 -EZw 2). We shall later discuss the connection of modular functions of higher level with the points of finite order on E. Now (4.2.2) shows the non-vanishing of J(z) on 0, which was stated but not proved in § 2.2. Let us now show that, for any r, s e C such that r 3 -27s2 # 0, there exists a lattice L in C satisfying g2(L). r and g,(L)= s. To see this, consider an elliptic curve E : y2 = 4x 3 —rx—s. Then E is isomorphic to a torus CIL' with a suitable lattice L', and hence isomorphic to the curve y 2 = 4x3 —g,(L')x—g,(1!). By Prop. 4.1, we have g2(L'). p 4 r and g,(L')= p's for some p e C. Then the lattice L= pL' has the desired property. This implies especially that g2(w 1, w 2) and gs(oh, (00 are algebraically independent over C, which we needed in the proof of Prop. 2.27.

100

ELLIPTIC CURVES

4.3. Points of finite order on an elliptic curve and the roots of nnity Let E be an elliptic curve defined over a field of characteristic p (which may be 0), and N a positive integer. Put g(N)=g(N, E)= {t

E

E I Nt= 0} .

It can be shown that g(N) is isomorphic to a subgroup of (ZINZ) 2, the product of two copies of Z/NZ. Especially if p does not divide N, g(N) is isomorphic to (ZINZ) 2. (This is obvious if the universal domain is C, since E is then isomorphic to a complex torus.) It should also be remembered that

(4.3.1) If E is defined over k, then the coordinates of every point of E of finite order are algebraic over k. This is obvious, since the number of the images of such a point t under N 2, if t e g(N). isomorphisms over k is Now fix a rational prime 1, and put ga) = U,7-1 g(in) . If p does not divide 1, it can be shown that g") is isomorphic to (Q1/ZI)2, where Q1 denotes the 1-adic number field, and Z1 the ring of /-adic integers. Let a e End (E). Then a induces an endomorphism of g"). Since every endomorphism of (QI/Z1)2 is represented by an element of M,(Z i) in an obvious way, we thus obtain an injective homomorphism of End (E) into M 2(Z1), which can be extended to an injective homomorphism R 1 of End Q (E) into MAO. We call R 1 an /-adic representation of End,2 (E). It can be shown that the characteristic polynomial of R 1(a), for any a E End,2 (E), has rational coefficients (integral coefficients if a E End (E)), and is independent of 1. We shall now associate an N-th root of unity eN (s, t) with two elements s and t of g(N). Let Do be the module of all divisors of degree 0 on E, and DR the submodule of D o consisting of the divisors of all functions on E, so that Do/DE is the module of all divisor classes of E of degree O. For each t E E, let (t) denote the divisor associated to the point t. It is a well-known fact that the map t e Do defines an isomorphism of E onto D0/D. (Actually the group law on E is defined by means of this one-to-one correspondence between E and Do/DR .) Therefore we have (4.3.2)

If t 1 ••• , t„, e E, c,, ••• , c„, ,

Elt i CPO

E

Z, nt i ci = 0, and

Efli ci tt = 0, then

E

If t E g(N), we see that N ((t)—(0)) E DR, hence N ((t)—(0))= div ( f) with a function f on E. Take a point t' on E so that Nt'= t. By (4.3.2), there exists a function g on E such that

4.3

POINTS OF FINITE ORDER AND THE ROOTS OF UNITY

--div (g) = ET,E„N) ''-U

101

(u) .

We see easily that the functions f(N x) and g(x)N (xe E) have the same divisor. Replacing f by a suitable constant multiple, we thus obtain two functions f and g which are characterized, up to constant factors, by the properties

div (f). N • ((t)— (0)) ,

g(x)N = f(Nx)

(x

E

E).

If s e g(N), we see that g(x+s)N = g(x)N, hence g(x+s)= e,v(s, Og(x)

with an N-th root of unity eN(s, t). PROPOSITION 4.2. Suppose that N is prime to the characteristic of the universal domain. Then the function eN(s, t) on g(N)xg(N) has the following properties: (1) eN(s i -Es 2, t)= e N(si , OeN(s2, t); (2) eN(s, t 1+4) = eN(s, ti)eN(s , t2) ; (3) e N(t, s)= eN(s, t) - ' ; (4) eN(s, t) is non-degenerate, i. e., if eN (s,t) =1 for all seg(N), then t= 0; (5) if t is of order N, e N(s, t) is a primitive N-th root of unity for some s E g(N); (6) for every automorphism a of the universal domain over a field of definition for E, eN (s, = eN (sa, t"). PROOF. The first and last properties are obvious from our definition. To show (2), put t3 = t1 +4, and let ft and g, be functions with the above properties for tt, for i= 1, 2, 3. Since t1 -Ft2 —4-0=0, by (4.3.2), there exists a function h on E such that div(h)=(t)+(t 2)—(4)—(0). Then div(f1f2f = div (V ), so that f1 f2f =chN with a constant c. Therefore (g1 g2 gV)(x) =c'h(Nx) with a constant c', from which we obtain (2). To prove (3), observe that div (M1 f (x— it)) = N • EiY.:5 1 ((it +t)--(it)) = O, hence

1Tiv..7, 1 f(x—it) is a constant.

114-0 ' g(x—it') must be a constant.

Therefore, if Nt' = t, we see that Substituting x—t' for x, we obtain

g(x)g(x—t')•-• g(x—(N-1)n= g(x—t') ••• g(x—(N-1)t9g(x—t),

so that g(x)= g(x—t), which implies eN (t, t)= 1, hence (3). If e N(s, t)= 1 for all s e g(N), then g(x+s)= g(x) for all seg(N). Therefore g(x)=p(Nx) for some function p on E. It follows that f(x)=p(x)N, hence div(P)=(t)—(0), which is possible only when t =0, since E is of genus one. This proves (4). Finally, to prove (5), let t be of order N, and let T y be the group of all N-th

102

ELLIPTIC CURVES

roots of unity. Then s .—■ e N (t , s) is a homomorphism of g(N) into TN. If this is not surjective, there exists a positive divisor M of N smaller than N such that e N(t,$)Af= 1 for all s e g(N). This implies, by virtue of (4), that Mt= 0, which is a contradiction. This completes the proof.

4.4. Isogenies and endomorphisms of elliptic curves over C

Let E and E' be elliptic curves isomorphic to CIL and CIL', respectively, with lattices L and L' in C. Then every homomorphism of E into E' corresponds to a complex analytic homomorphism of CIL into CIL', and vice versa. Now every complex analytic homomorphism of CIL into CIE.' is given by a linear map u . pu with a complex number p such that pLc L'. Therefore

Hom (E, E')..._. Hom(C/ L, C / L')= {p e C I pL C L'} . Especially End (E)7,- End (C/ L)= {pE CI pL C L} ,

End,2 (E) z End,2 (C/L)={peCI 11 ' ( QL)CQL} . Here QL denotes the Q-linear span of L. complex multiplications if End (E)* Z.

We say that an elliptic curve E has

PROPOSITION 4.3. Let L= Zco,+.7,(.1, 2 and L' = 7,0);-FZco with z= 1/(02 E .D, z' = a4/a4 E .D. Then CIL and CI L' are isogenous (resp. isomorphic) if and only if there exists an element a of GL(Q) (resP. SL 2(Z)) such that a(z0= z, where GLt(Q)= fEEGLA) I det (E) > 0}. a b] PROOF. If 0 *p e C is such that pL c L', we obtain an element a =[ c d of M 2(Z) n G L2(Q) such that (0

P[::] = [

n[acui],

hence det (a) > 0 and z = a(z'). Conversely, if a(z0= z for a = [ ca db ] e M 2 (Z)n GU(Q), Put 2= cz'-Fd. Then we see that 2* 0, and (4.4.1)

Ar z1 _r a L1J — Lc

bir z , -I cliL 1 _1'

or

(2°41 w a) [: 21 ] = [ ac

bd][ (4] '

hence pL c L' with p= 2(141(0 2 . Especially pL = L' if and only if a E SL2(Z). PROPOSITION 4.4. Let L= Zco i +Zw 2 with z = 0)1(0 2 E .V. Then CIL has complex multiplications if and only if there exists a non-scalar element a of GU(Q) such that a(z)= z.

ISOGENIES AND ENDOMORPHISMS OF ELLIPTIC CURVES

4.4

103

PROOF. Repeat the proof of Prop. 4.3 with z = z' and w i =a4 Then we a= ra bi see that every p# 0 satisfying pLc L corresponds to an element L c dJ of A/1 2(Z)nGL.,*(Q) through the relation (4.4.1) with ,i=2 and z=zi. From (4.4.1), we obtain

(4.4.2)

rz

L1

2irp 1J1.-0

Oi_ra /.1_1 — Lc

birz dJL1

We see easily that pe Z if and only if a is a scalar matrix, hence our assertion. PROPOSITION 4.5. Let L and z be as in Prop. 4.4. Then CIL has complex multiplications if and only if Q(z) is an imaginary quadratic field. If that is so, EndQ (C/L) is isomorphic to Q(z). PROOF.

In the relation (4.4.2), if a= La

is not a scalar matrix, p d cannot be real; moreover p and p are the characteristic roots of a, and therefore satisfy a quadratic equation over Q. Since p = cz-Ed and c 0, we have Q(z)=Q(p), so that Q(z) must be imaginary quadratic. Conversely, if K.Q(z) is imaginary quadratic, we have QL= ah • (Qz+Q). ahK, so that

(4.4.3)

End,2 (CIL).

E

Cl pQLCQL}. fp E C I pKc 10.K.

PROPOSITION 4.6. Let L and z be as in Prop. 4.4. Suppose that CIL has complex multiplications, and let K=Q(z). Then there is an infective homomorphism (or simply, an embedding) q of K into M 2(Q) such that

(4.4.4) PROOF.

(4.4.5)

q(K x) = {a e G

I a(z) = z} .

In view of (4.4.2) and (4.4.3), we can define q(p) for p e K by P

=q P E d • (

)

Then our assertion is obvious from (4.4.3) and what we said in the proof of Prop. 4.4. PROPOSITION 4.7. Let K be an imaginary qudratic field, and q an embedding of K into 111 2(Q). Then there exists a point z on for which the relation (4.4.4) holds. PROOF. Let 2 e K —Q, and a= q(2). Then det(a)=NK,Q(2)=2,T > 0, and a has 2 and 2-- as its characteristic roots. Therefore, a, as a transformation on is elliptic, and has a fixed point z in 0. If we write the relation (4.4.2) for the present a and z, then 2 =p or A =ft. In any case Q(z)=Q(2). K. If q' denotes the embedding of K into M2(Q) defined by p[zd=q'(p)[fl,

104

ELLIPTIC CURVES

we see that q(2)= q'(2) or q(2)= q'(). Since K =Q(2), this implies either q(p)=q'(p) for all p E K, or q(p)=q'(fl) for all p E K. Therefore we obtain our assertion from Prop. 4.6. We have also seen that there are exactly two embeddings of K into 1112(Q) with the property (4.4.4) for a fixed point z. We call an embedding q normalized if it is defined by (4.4.5). The other one is defined by (4.4.5) with 2 in place of z. Let q and q' be arbitrary embeddings of the same K into A12(Q). Then there exists an element p of GL 2(Q) such that (7/(1)= pq(p)13 -1 for all p G K. (This is well-known, and can be proved as follows. Through the embedding q (resp. q'), regard Q 2 as a one-dimensional vector space V (resp. V') over K. Then V and V' must be isomorphic over K; this means the existence of a Q-linear automorphism 13 of Q 2 such that q/(p)13= pq(p).) Let z (resp. z') be the fixed point of q(Kx) (resp. q'(Km)) on 0. Then 13(z)=z' or 2', since z' and 2' are the only fixed points of (AK") on C. Therefore

p[f]=1'] or =c[ 1

with a non-zero complex number c. It follows that if both q and q' are normalized, det (13) must be positive. Let us now fix an imaginary quadratic field K (always considered as a subfield of C), and determine all isomorphism classes of elliptic curves E such that EndQ (E) is isomorphic to K. We first observe that End (E) is an order in End Q (E). In general, by an order in an algebraic number field F of finite degree, we mean a subring of F, containing Z, which is a free Z-module of rank [ F: Q]. Every order in F is contained in the ring of all algebraic integers in F, which is called the maximal order in F. By a lattice (or Z-lattice) in F, we mean a free Z-submodule of F of rank [F: Q]. For a Z-lattice a in F, if we put o= {p E FI pa ca), then o is an order in F. We call o the order of a, and a a proper o-ideal. We can classify all the proper o-ideals, for a fixed o, with respect to multiplication by the elements of F', as we usually do for the fractional ideals in F. Coming back to the imaginary quadratic field K, let a be a Z-lattice in K. If we consider a as a submodule of C, it is a lattice in C, so that C/a is a complex torus. Then we have

(4.4.6)

End (C/a)= {p

Pa c a} = {PeKItlac ct}

Let E be an elliptic curve defined over C such that End Q (E) is isomorphic to K, and o an order in K corresponding to End (E). Then E is isomorphic to C/a with a proper o-ideal a. Conversely, for any proper o-ideal a, End (C/a) is isomorphic to o. Moreover the class of proper o-ideals a is uniquely determined by the isomorphism class of C/a. In other words, C/a is isomorphic to CA if and only if pa= b for some p e K v. PROPOSITION

4.8.

E

ISOGENIES AND ENDOMORPHISMS OF ELLIPTIC CURVES

4.4

105

PROOF. Since there are two isomorphisms of K onto End Q (E), o may depend on the choice of isomorphism. But, if a e o, we have a+aE Zco, so that d e o. This shows 6=o, hence o is independent of the choice of the isomorphism of K to End G (E). Now E is isomorphic to a torus of the form C/(Zz+Z) with z e K. Put a=Zz+Z. Then a must be a proper o-ideal by (4.4.6). The converse part is just a restatement of (4.4.6). The last assertion can be verified in a straightforward way.

From this result, we obtain the following two propositions :

Let E and E' be elliptic curves defined over C. Suppose that E has complex multiplications. Then E' is isogenous to E if and only if EndQ (E') is isomorphic to EndQ (E). PROPOSITION 4.9.

For an order o in K, the number of classes of proper o-ideals is exactly the number of isomorphism classes of elliptic curves E such that End (E) is isomorphic to o. Especially if o is the maximal order in K, the number is nothing but the class number of K. PROPOSITION 4.10.

Let o ic be the maximal order in K, and o an order in K. Then there is a unique positive integer c such that o = Z-i-co K. Further, for every proper o-ideal a, there exists an element p of If' such that pa+co.o. Moreover, for two proper o-ideals a and b, let ab denote the Z-module generated by the elements xy with x E a and y E b. Then all the proper o-ideals form a group with respect to this law of multiplication, with o as the identity element. PROPOSITION 4.11.

It is well-known that 0 ic =Z+Z2 with an element A. We can put 0 n ZA=Zc2 with a positive integer c. Then Z EcoK =Z+ZaCc. If r±s2 E o with r and s in Z, then s2eo, so that secZ. Therefore o= Z-Fco ic. The uniqueness of c is obvious. For a Z-lattice a in K, put PROOF.

-

a* = fpE K 1Tr K,Q (pa) CZ). Then we see easily that a* is a Z-lattice in K, (a*)* . a, and a* c b* if bc a. Moreover, if oa c a, we have oa* c a*. Therefore if o (resp. o') is the order of a (resp. a*), we have 0 cc', and o' co since a**. a, so that o=o'. We can verify in a straightforward way that o*.gri(c2) - lo if g(x)=0 is the monic irreducible equation for a over Q. Let a be a proper o-ideal. If e e ((te )* , then Tricm (eaa*)c Z, so that ea* c a*, hence e e o. It follows that (aa*)*C o, hence o*c aa*. On the other hand, Tr (aa*o)=Tr(aa*)c Z, hence oa* Co*. Therefore we have aa*=o*, so that a - (g'(c2)a*)=13.. Thus we have shown the existence of inverse in the semi-group of proper o-ideals, hence the last assertion. Put b = gi(c2)a*. Define a Q-linear map f of K into Q by f(r±s2) =r for r and s in Q. Then f(ba) , f(o).Z. Therefore, for every rational

106

ELLIPTIC CURVES

prime p, there exists an element pp of b such that f( p a) is not contained in PZ. Then we can find an element p of b so that p --p p modpb for all prime factors p of c. Then f(pa) is not contained in pz for all such p. Hence Apa)= mZ with a positive integer m prime to c. Then f(pa+co K )=mZ±cZ=Z. If a E o, we have f (a) , f (is) for some p E pa+co K . Then a— J9 E Zc2 c co K , so that a =(a—p)-i-p E pa+co K . This shows that D =pa±co,c. Since both pa and coif are ideals of D, we have o=oo=(pa±co K)(pa±co K )c pa±c 2oir c pad co, so that D = pad co. -

-

The integer c (or the ideal COO is called the conductor of D. It can easily be seen that cox = {a E K1ap ic c a } . In (5.4.2), we shall show that every proper o-ideal is "locally principal ". As our argument shows, aa*= D* holds for any proper c-ideal a with any order D in K, even if [K:Q]> 2. If o = Z[r] with an element r satisfying a monic irreducible equation g(x)=0 over Q, then D*=,t(n)'o, so that every proper D-ideal a is invertible.

EXERCISE 4.12. given by

Prove that the number of classes of proper D-ideals is h c • [pi :

-11,,Ic[1 — (1::

where h is the class number of K; (

-

--) is 1,

4

—

1, or 0, according as the

prime p decomposes in K, remains prime in K, or is ramified in K. EXERCISE 4.13. Let F be an algebraic number field of finite degree, K a quadratic extension of F, and of (resp. D ic) the maximal order in F (resp. K). Generalize Prop. 4.11 to the case of an order in K containing o f. (Although this can be done globally, it may be easier to treat, at first, the corresponding problem for local fields. The assertion (5.4.2) can also be generalized.)

4.5. Automorphisms of an elliptic curve Let Aut (E) denote the group of all automorphisms of an elliptic curve E defined over C. If E has no complex multiplication, Aut (E) consists only of -±-1. Therefore suppose that E has complex multiplications, and let D and K be isomorphic to End (E) and End 4, (E) as in Prop. 4 8. Then Aut (E) is isomorphic to DA. Since K is imaginary quadratic, as is well known, DA contains more than -±-1 only in the following two cases :

K =Q(1./ —1), o = Z[1./ ox = {±1, ±1/-1 }. (B) K= Q(), C= e"'", o=ZEQ, o = {±1, ±C, -±--C 2 1. In these two cases, o is the maximal order in K, and the class number of K (A)

INTEGRALITY PROPERTIES OF THE INVARIANT

4.6

J

107

is one, so that, by Prop. 4.10, in each case, there is one and only one elliptic curve E, up to isomorphisms over C, such that End (E) is isomorphic to o. Let E be defined by y 2 = 4x3—c 2x—c3 with c, and c, in C. We observe that

(4.5.1)

JE

=1

c, = ;

JE

=O

C2 - = 0 .

Now, if c, = 0, Aut (E) contains at least 4 elements : (x, y).-0 (x, -±y), (—x, y) ; if 0, Aut (E) contains at least 6 elements : (x, y).—,(Cv x, -±-y), v =0, 1, 2, with = 0'3. Therefore, from (4.5.1), we obtain

(4.5.2)

E belongs to the case (A) resp. (B) if and only if JE =1 resp. JE= 0 .

Moreover, we see that Aut (E) consists of those 4 or 6 elements. Hereafter we denote by e the set of all elliptic curves E of the form 312 = 4x 3 —c2x—c 2 with c2 and c3 in C. We classify e into three classes e, with i =1, 2, 3 according to the number 2i of automorphisms. Thus e, and e3 consist of the members of e of the type (A) and (B) respectively, and c, contains all the remaining members of C. For any elliptic curve E: y 2 = 4x 3 —c,x—c„ we define three functions hiE on E by

h'E((x , y)) = (c2c2 / 4) • X, h2E((x, y)) = (ci/ 4 ) x 2

(4 =c3-27c ) ,

h2E((x, y )) = (cs/d). x 3 . They are obviously defined over any field of definition for E. If EEC„ we have 4=4=0 and h2E((x, y)) = cx 2 ; if E E es , we have h'E = 4=0, and h2E((x, y)) = (-27 c a) - x3 . By means of the explicit form of the elements of Aut (E) mentioned above, we can easily verify

(4.5.3) When E e„ one has 4(0 = hiE(t') if and only if t = at' for some a E Aut (E) . (4.5.4)

Let E and E' be members of e, and y an isomorphism of E to E'. Then hiE = hiE. y for i =1, 2, 3.

In fact, if E is as above, and E' is defined by y2 = 4x 5 —c'x—cL then, by Prop. 4.1, Cx, y»=(1- 12x, P3Y), c =ti4c2, c;=116 cs with an element ti of C Therefore we obtain (4.5.4) from our definition of hiE .

4.6. Integrality properties of the invariant In Th. 2.9, we proved that the modular function

J(z)=123j(z)=12‘g,(z)3/4(z)

J

108

ELLIPTIC CURVES

has a Fourier expansion of the form (4.6.1) with c„ e Z.

e 2z

./(z)= q -1(1+ .11,7-icAn)

Let us now prove

4.14. If z belongs to an imaginary quadratic field and 1m (2)> 0, J(z) is an algebraic integer. THEOREM

We shall give here an analytic proof of this fact, although a more intrinsic algebraic proof is now possible by virtue of the Néron minimal model [53], see Deuring [10], Serre and Tate [66 ]. The fact that f(z) is an algebraic number can easily be seen as follows. Let N=Q(z), L=Zz+Z, and let E be an elliptic curve isomorphic to CIL. Observe that, for any a e Aut (C), EndQ (E0) is isomorphic to K. Now there are only countably many isomorphism classes of elliptic curves whose endomorphism algebras are isomorphic to K. Since j(EQ)=j(E)°, it follows that {j(E)0 a E Aut (C)) is a countable set, hence j(E) must be algebraic. The fact that f(z) is integral is much deeper, and requires a more elaborate argument (whatever method one uses). PROPOSITION

4.15. Suppose that an equality ET-0 a k .1( 2)k =

= e"")

b nq n

holds for all z ea), with constants ak and b a in C. ring generated by the b„ over Z.

Then the a, belong to the

Substitute the expression (1' 1(1+1:77 c nqn) for J(z) in E'kn_o a ,f(z)k Then we obtain PROOF.

b2 _,,, ,=(m(m —1)/2) Since c„ e Z, our assertion is obvious. Let us call an element a

. r a bi of 1t1 2(Z) primitive if a, b, c, d have L c dJ

no common divisors other than +1. only if a e r -[on

If det (a)= > 0, a is primitive if and

T, where r =SL,(Z).

r .[on

By Prop. 3.36, we have

r =UeleAra

a with the set .4 of all the matrices a =[ 0

b] u (l nder the conditions d>0,

INTEGRALITY PROPERTIES OF THE INVARIANT .1

4.6

109

ad= n, 0 1, and consider the polynomial

ILEA (X-

a)

= n=o sinX m

with an indeterminate X, where the s„ are the elementary symmetric functions in the T. a, hence are holomorphic functions on 0, which have Fourier expansions in q". For every r e r, we have UL,E4 rar =u,„.4 Ta, so that

{»,aor I ae

=fica I ae AI.

It follows that si,, -r = s„„ hence s n, is a modular function of level 1. Since s,„ is holomorphic on 0, s„ is a polynomial in J, say sm = Sm (f). = f- a bi e A, the q-expansion of Jo a is of the form For a L 0 dJ

(4.6.2)

i(a(z))=

E:=1 cmCrqmai

dj

Thus the coefficients are algebraic integers of Q( ( „). Let a be an automorphism of Q(C„), such that C„'. for some t with (t, n). 1. Transforming 43. ra b' eA, the coefficients of jocr by a, we obtain joi3 with bt mod (d). LO

dJ

Since a /3 gives a permutation of the set .4, we can conclude that the q-expansion of s„ has coefficients in Z. Applying Prop. 4.15 to S„,(J), we see that the polynomial S 71. has integral coefficients. Thus we obtain a polynomial

F.(X, B=

(4.6.3)

ILEA

(X—Jo

= Sm(i)X m

belonging to Z[X, PROPOSITION

4.16. For any e e GLA) with det (e) > 0, P.; is integral

over ZEJJ.

Multiplying e by a suitable rational number, we may assume that e is a primitive element of Al,(Z), since this does not change Joe. If 0 det (e) = n> 1, e E T •[ on i ]• T, so that TÇ=Ta for some a e A. Then PROOF.

we have pi- e= Jo a, so that F.(PDE, .1)= 0, hence our assertion. Let us now put

lin(J)= F.(1 J)=ILEA (J-J"' a) Then H„ is a polynomial in J with coefficients in Z. If n is not a square, the highest coefficient of the polynomial H,i(J) is +1. PROPOSITION

PROOF.

4.17.

If n is not a square, we have o/d * 1 in (4.6.2), hence the leading

110

ELLIPTIC CURVES

coefficient of the q-expansion of j—joa is a root of unity, and so is the leading coefficient of the q-expansion of I-1„( .1). This coefficient is equal to the highest coefficient of the polynomial H., which is rational, so that it must be +1. Now suppose that K=Q(z) is imaginary quadratic, L=Z+Zz, and let o be the order in K isomorphic to End (CIL). First assume that o is the maximal order in K. Then we can find an element p of o such that NK,Q(P) is a square-free integer n> 1. (In fact, if K=Q(/ 7-7.), take p= 1 , and if K= Q(/), m>1 and square-free, take u= v') Define an element e of M 2(Z) by

ti[n= e[1]

= q(p)

with the notation of (4.4.5)) .

Then det (e). n, and e is primitive, since n is square-free. Therefore joe= joa for some a E A as in the proof of Prop. 4.16. Since e(z)=- z, we have J(z)=-J(e(z))=-J(a(z)), so that 1-1.(j(z))= O. By Prop. 4.17, this shows that J(z) is an algebraic integer. Next consider the case where o is not the maximal order. By Prop. 4.3, there exists an element p of GLt(Q) such that End (C/(Zz'-i-Z)), with z' = Az), is the maximal order. By Prop. 4.16, J(z) is integral over Z[J(e)]. Since J(e) is integral, this completes the proof of Th. 4.14. Actually an arbitrary order o in K contains an element p such that NK,v(p) is a prime. In fact, take a positive integer h so that ho x c o. By the generalized Dirichlet theorem, there is an element p of K such that p 1 mod hox, and NK,Q(p) is a prime. Then p E O. Applying the above argument to this p, we can show that J(z) is integral, without reducing the question to the case of maximal order. The equation H.( J)=0 is called the modular equation for the degree n. For the classical treatment of this and other related topics, the reader is referred to Fricke [21 ], Hurwitz [32], and Weber [89].

CHAPTER 5 ABELIAN EXTENSIONS OF IMAGINARY QUADRATIC FIELDS AND COMPLEX MULTIPLICATION OF ELLIPTIC CURVES The purpose of this chapter is to study the behavior of an elliptic curve E with complex multiplications under Gal (Ka b /K), where K is an imaginary quadratic field isomorphic to End Q (E), and Ka b the maximal abelian extension of K. The reader will be required to have some knowledge of class field theory. We shall state the main theorem 5.4 in the adelic language, and derive from it the classical result on the construction of Kab by means of special values of elliptic or elliptic modular functions. This topic will be taken up again in § 6.8, in a different formulation without elliptic curves. 5.1.

Preliminary considerations

There is a simple principle concerning the field of rationality, which we shall often make use of in this and next chapters. Let X be an algebrogeometric object, defined in the universal domain C, such that X° is meaningfully defined for every automorphism a of C. Thus X may be a variety, a rational map, or a differential form on a variety (see Appendix). Then our principle is as follows : Let k be a subfield of C. If .3(0 = X for all a E Aut (Clk), then X is rational over k. Or equivalently, if X 0, far a e Aut (C/k), depends only on the restriction of a to k, then X is rational over k.

This is not a completely rigorous statement if X is defined with respect to some other algebro-geometric objects. For example, if X is a rational map of a variety U into a variety V, it is better to assume that U and V are defined over k. The same remark applies to a differential form. We can state a similar fact for two subfields of C: Let k and k' be subfields of C with countably many elements. Suppose that k' is stable under Aut (Clk). Then the composite kk' is a (finite or an infinite) Galois ex tension of k. Moreover, if every element of Aut (C/k) induces the identity map on k', then k' c k.

Now let us consider a projective non-singular curve V defined over a field k of any characteristic. We shall denote by k(V) the field of all

112

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

functions on V rational over k (see Appendix NO 4). Let W be a projective non-singular curve, and A a rational map of V into W, both defined over k. Then it is well-known that A is a morphism, j. e., defined everywhere on V. Suppose that A is not a constant map. Then f f 02 defines an isomorphism of k(W) into k(V). Let1 k(W)02 denote the image of k(W) by this isomorphism. We say that A is separable, inseparable, or purely inseparable, according as k(V) is separable, inseparable, or purely inseparable over k(W)a2. Further we put deg (2). [k(V ): k(W)o2], and call it the degree of ; this does not depend on the choice of k. Let Dif ( V) denote the set of all differential forms on V, and .W(V) the set of all holomorphic elements of Dif (V), i. e., all differential forms of the first kind on V. Further let .9)(V ; k) denote the set of all elements of 2(V) rational over k (see Appendix N°s 8, 9). If A and W are as above, for every co= h df E Dif (W; k) with f and h in k(W), we can define an element coo2 of Dif (V; k) by co 2 =(h a A) d( fa 2) . If w e 2(W), then wo PROPOSITION 5.1. Let V, W, 2, and k be as above, and let Then 0,02# 0 if and only if A is separable.

k).

PROOF. The differential form df has the property (5.1.0)

df # 0 if and only if k(W) is separably algebraic orer k(f).

(See Appendix N os 8, 9.) Put w=h-df with h and f in k(W). Since w 0, k(W) is separable over k(f), so that k(W)oA is separable over k(f 2). Applying (5.1.0) to d(f02), we see that k(V) is separable over k(f02) if and only if (002 0, hence our assertion. PROPOSITION 5.2. Let V, W, 2, and k be as above. If A is purely inseparable and g= deg (A), then there exists a biregular isomorphism p of W to Vq, rational over k, such that poA is the g-th power morphisin of V to Vq, where Vq denotes the transform of V by the g-th power automorphism of the universal domain.

PROOF. Let y be a generic point of V over k, and let w = 2(v), K = k(v), L = k(w). Our assertion is equivalent to (or, at least follows from) the equality L = kwhere Kg = fag la E KI. In fact, if k- = L, we have k(v)= k(w). Since yq is a generic point of Vg over k, we can define a birational map p of W to 179 by p(w) = vq. Since W and Vg are projective non-singular, itt is biregular. Then p(2(v)) = vq , so that po2 is the q-th power morphism of V

PRELIMINARY CONSIDERATIONS

5.1

113

to Vq. Thus our question is reduced to show that k•Kg=L. By our assumption, K is purely inseparable over L, and [K: L] = q, so that k Kg CL. Therefore it is sufficient to show that [IC: k • 10 ] = q. Since K is a regular extension of k, there exists an element x of K such that K is separably algebraic over k(x). Then k Kg is separable over k(xq). Now K is separable over k(x), and purely inseparable over k Kg, so that K is the composite of k(x) and k • K. Since k(x) is purely inseparable over k(xg), and k Kg is separable over k(xq), we have [K: k 10]= [k(x): k( x )3 =q ,

which completes the proof. Let E, and E2 be elliptic curves defined over a subfield k of C, and Te the algebraic closure of k in C. Then every element of Horn (E„ E2) is defined over Tz. Moreover, if End (El ) is isomorphic to Z, and A e Horn (E„ ED, then Al = +A for every automorphism a of -12- over k. PROPOSITION 5.3.

If 2 E Horn (E„ E2) and a is an automorphism of C over k, then 2°E Horn (E„ E2). Since Horn (E1 , E2 ) is at most a countable set, there are at most countably many 2 .1, so that A must be defined over Z. If End(E 1 ) is isomorphic to Z and A # 0, we see that Horn (E„ E2) is isomorphic to Z, so that mit'1 = n2 with non-zero integers m and n. Then m2 • deg (2")= deg (m2') = deg (71,0= n 2 deg (A). Since deg (2")= deg (2), we obtain m= ±n, so that 2.1 = +2. PROOF.

Let us now consider an elliptic curve E over C such that Endcl (E) is isomorphic to an imaginary quadratic field K. We shall now show a way of choosing a canonical one among the two isomorphisms of K onto End Q (E). First observe that the vector space 0(E) of holomorphic differential forms on E is one-dimensional over C. Let 0 w E .0(E). For every a E End (E), we have wo a E 2(E), so that woct= paw with an element pet of C. If E is identified with a complex torus CIL, with a lattice L in C, and if u denotes the variable on C, then w = c du with CEC. Therefore, if a corresponds to the linear map u pu as in § 4.4, we have woa=c-d(pu)=cp• du = p • co, so that p= pa.. Thus we can choose an isomorphism 0 of K onto End() (E) which is completely characterized by the condition (5.1.1)

WO

0(p)=pw

(p

E

K, 0(p)E End (E)).

Observe that this condition does not depend on the choice of w. We say that (E, 0) (or simply 0) is normalized if this condition is satisfied. If (E', 0') is another normalized couple with the same K, then every isogeny A of E to E' satisfies

114

(5.1.2)

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

2 o 0(p) = 0' (p)0 2

(p E K) .

In fact, if w (resp. w') is a differential form on E (resp. E') as considered above, we have w'o2 =bw with a constant b, so that w'o,20 0(te). bpw = w'o 6P(p)o 2, hence (5.1.2). As another application of this idea, we can prove (5.1.3) If E is defined over a field k, every element of End (E) is rational over kK. To show this, observe that we can take w rational over k. Let a E Aut (CAK), p E K, 0(p) e End (E). Since E, w, and p are invariant under a, we have (Do 0(te)C = (w o 0(p))° = (pw)° = pw = wo 0(p) (see Appendix N° 8), so that 0(p)° = 0(p). This implies that 0(p) is rational over kK. The couple (E, 0) and K being as above, suppose now that E is defined by y' .4x 8 -c2x -c, with c, and c, in an algebraic number field k of finite degree, containing K. (In view of Th. 4.14, we can always find such a model E among a given isomorphism class of curves.) Take a prime ideal p in k, prime to 2 and 3, for which E has good reduction modulo p. 6' By this we mean that c, and c, are p-integers, and cl-27c1 is a p-unit. Then E modulo p is, by definition, an elliptic curve y 2 . 4x 3 e2x E,, -

-

where the tilde means the residue class modulo p. We denote this curve by p(E), or E when p is fixed. Obviously j(f) is the residue class of j(E) modulo p. For a point t on E rational over k, we can define p(t)=I .(t modulo p) as a point on E in a natural way. It can be shown that t p(t) is a homomorphism. Furthermore, we have -,

(5.1.4) If p(t)=0 and Nt=0 with an integer N prime to p, then t= 0 . An elementary proof is given in Lutz, "Sur l'équation y 2 =x 2 Ax B dans les corps p-adiques," J. Reine Angew. Math. 177 (1937), 238-247. See also [81, § 11, Prop. 13], where a corresponding fact for higher dimensional abelian varieties is proved. Now consider another elliptic curve E' defined over k which has also good reduction modulo p. Let 2 be an element of Hom (E, E') rational over k. -

-

5) As to the general theory of reduction modulo p of algebraic varieties, especially abelian varieties, see Shimura [69], Shimura and Taniyama [81, Ch. III]. Néron [53] established a model of an abelian variety with the best behavior for reduction modulo For further study of this topic, especially a criterion for good reduction, see Serre and Tate [66 ].

p.

5.2

CLASS FIELD THEORY IN THE ADELIC LANGUAGE

115

Then we can define ; = p(2) in a natural way as an element of Hom (E', Ê ). It can be shown that 2. ,p(2) defines an injective homomorphism of Horn (E', E) into Horn (k, f), and deg (;). deg (2) (see [81, § 11.1, p. 94, Prop. 12]). Especially, when E=E', we obtain an injective ring-homomorphism of End (E) into End (f). Therefore we can define an injective map –

0 : K , End Q (2) by J(p) =1)(0(p)) for ti e K, 0(p) e End (E). The image d(K) does not necessarily coincide with End Q (f). We have, however, the following assertion : (5.1.5) Every element of EndQ (f) commuting with all the elements of J(K) belongs to J(K), i. e., the commutor of J(K) in End Q (f) is J(K). This follows immediately from the fact that EndQ (f) is either a quadratic field or a quaternion algebra over Q. Another way is to consider an /-adic representation of EndQ (E); this method is applicable to the higher dimensional case, see [81, § 5.1, Prop. 1]. If w =dxly, we can define p(w) = ii; in a natural way as a differential form on f, different from 0. If c is a p-integer, we put p(cco)=Eill. We can then verify the formula p(a) o 2) = cZo ; for every 2 e Hom (E', E) rational over k. (See [81, § 10.4].) 5.2.

Class field theory in the adelic language

Before going further with (E, 0), we recall some elementary properties of the idele group of an algebraic number field and fundamental facts of class field theory» For an algebraic number field K of finite degree, we denote by K; the idele group of K, by K:, the archimedean part of K;, and by K om„+ the connected component of the identity element of K. Further we denote by Kab the maximal abelian extension of K. Then there is a canonical exact sequence (5.2.1)

1 -- Kx1C:„.-- K;--- Gal (Kap/K) — 1,

4 denotes the closure of KxKL,.." We shall denote by [s, K] the where fr-7177 element of Gal (Kab /K) corresponding to an element s of K. For an element x of K; and for a finite prime p of K, we denote by xp the p-component of X. Then we define a fractional ideal i/(x) in K by il(x) ;,= xpop for all p, where Do -

6) As for these, we refer the reader to Cassels and Freohlich [6] and Weil [99]. We follow, for the most part, the notation of the latter. 7) One can easily verify that, if K is either Q or an imaginary quadratic field, then K " K : 4. itself is closed. This is because in both cases the group of units of K is finite.

116

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

denotes the maximal compact subring of the completion Kp of K at p. Put U(1)= x EK xp o; for all finite primes 13 of K1 ,

and for every integral ideal c in K, W(c)= {xe K I xp

-

1 e cop for all

1.1

dividing c}

1/(c) = U(1)nw(c). Since KNU(c) is an open subgroup of K; containing Kx1f1 +, there is a finite abelian extension F, of K characterized by

Fc = {a

E

Ka i acsx = a for all s e U(c)} .

We call F, the maximal ray class field modulo c over K It is the maximal one among the class fields whose conductors divide c. Let u E W(C). Then

(u) is prime to c, and [ u, K] coincides with the Artin symbol il

F /K (u)

on F.

In particular, if q is a prime ideal in K prime to c, and if 14 4 is a prime element of og and u, = 1 for all finite then Eu, IQ induces the Frobenius element of Gal (FIK) for q. Let a be an arbitrary Z-lattice in K, which is not necessarily a fractional ideal. For each rational prime p, put Kr = KOQ Qp, and ap = atO z Zr. Then ap is a Zr-lattice in Kp . For every x e K:t , we can speak of the p-component xp of x, belonging to K;, since KA = If& A. Observe that xpap is a Zr -lattice in Kp . By a well-known principle, there exists a Z-lattice b in K such that bp = xrap for all p. We denote b simply by xa. In other words, xa is a unique Z-lattice in K characterized by the property (xa) p = xpap for all p. We can now associate with x an isomorphism of K/a onto K/xa. To do this, first observe that K/a is canonically isomorphic to the direct sum of Kp/ap for all p. (In fact, Q/Z is the direct sum of Qp/Zp for all p, and K/a is isomorphic to QVZ 2.) Then multiplication by xp defines an isomorphism of Kr/a p onto Ifp/xpap. Combining these isomorphisms together for all p, we obtain an isomorphism of K/a onto Klxa. We shall denote by xw the image of an element w of K/a by this isomorphism. The situation is explained by the commutative diagram Kp /a p

Kp /xpap

(5.2.2) K/a

K/xa

where the vertical arrows are canonical injections. In other words, if u e K,

5.3

MAIN THEOREM OF COMPLEX MULTIPLICATION

we take an element v of

117

K such that v xpu mod xpap for all p, and put x - (u mod a) , v mod xa .

We shall write this element also xu mod xa. Although xu itself is meaningless, the notation may be justified, since the p-component of x. (u mod a) in Kp lxpap is exactly xpu mod xpap. It should be remembered that we have been discussing the localization with respect to rational primes. However, if a is a fractional ideal, Kla is canonically isomorphic to the direct sum of the modules Kplap for all prime ideals p in K. Therefore we can define, in such a case, the above homomorphism of K/a to Klxa by means of the commutative diagram similar to (5.22) with prime ideals p in place of rational primes p. 5.3.

Main theorem of complex multiplication of elliptic curves

Let us come back to a normalized couple (E, 0) and an imaginary quadratic field K. By Prop. 4.8, we can find a Z-lattice a in K so that C/a is isomorphic to E. Fix an isomorphism e of C/a to E. Since 0 is normalized, we have e(av). 0(a)(e(v)) for any a in K satisfying aa ca. Observe that e(K/a) is the set of all points of E of finite order. We are now ready to state the main theorem of complex multiplication. THEOREM 5.4. 6) Let K, (E, 0), a, and e be as above. Let a be an automorphism of C over K, and s an element of K; such that a.[s, K ] on Kaz,. Then there is an isomorphism

E': C/s'a— Ea such that e(u)a .e'(s - lu) for every u mutative.

Kla

E

K/a, j. e., the following diagram is com-

e

s -1 K/s'a

e'

Obviously V is uniquely determined by the above property, once e is fixed. PROOF. In the above statement, we have not assumed that E is defined over an algebraic number field. Actually if we can prove the theorem for a 8)

This theorem was originally given (in the lectures at Princeton University) in terms of a finite number of points on E, as in [80, Th. 4.3]. The present formulation for all points of E has been suggested by A. Robert.

118

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

curve which is isomorphic to E (whether or not defined over an algebraic number field), then we can easily derive from it our assertion for E. Therefore it is sufficient to prove our assertion for a specially chosen curve in a given isomorphism class of elliptic curves. In the next place, let us reduce the proof to the case End (E)=8(oK ) with the maximal order oic in K. Take any fractional ideal b in K contained in a, and let E 1 be an elliptic curve with an isomorphism C/b—.E,. Let 2: be the isogeny for which the diagram

e,:

C

C/6

C

C/a

e1

E,

IA

id.

e

E

is commutative. Assuming our assertion to be true for E„ we obtain an isomorphism e; : Cis - lb --, .Ef and a commutative diagram : Klb

el

E, a

s -1 K/s - lb

e;

Er

Now we have Ker (2)=e ,(alb), so that Ker (2°). Ker (2)6= e,(a/by'=e;(s'a/s - ib). Since s - lb c r'a, we can find an elliptic curve E' and an isogeny 2' of Ef to E' such that the diagram

E'

C /s -"'b id. I

C I via

27

E'

is commutative. Then Ker (2') = e(s'al s - lb) = Ker (2°). Therefore we can find an isomorphism Lof E' to E" so that ao 2' = 2". Putting e' = Ear), we obtain a commutative diagram :

C/s - lb id.

ef A°

C /s'a

E"

5.3

MAIN THEOREM OF COMPLEX MULTIPLICATION

119

Then we have, for u E K, e(u mod a )' = ,e(e,(u mod b )' ) = 2g(el(s -l u mod s' 'f))) = e' (s- 1 u mod a) ,

which proves our assertion for E. Thus we may assume that a is a fractional ideal in K, so that O(o) = End (E). Furthermore, as is remarked at the beginning of our proof, we may take E to be defined over Q(i E). Now let h be the class number of K, and {j„•-• , j k } be the set of all invariants of elliptic curves whose endomorphism rings are isomorphic to oK (see Prop. 4.10). For each j i, we take an elliptic curve E, such that j(E,)=j, defined over Q(j) (see § 4.1). We put E=E„ Take any positive integer m >2, which we shall make large afterwards. Since ok is a finite group, if C e ok and C--71 mod moK, then C•=- 1. Define an abelian extension F„, of K as in §5.2 with moK as c. We can find a finite Galois extension L of K so that F„,c L, j1, • - • , e L, and every point of order m on E is rational over L. Further, for the given automorphism a of C over K, we take a prime ideal F13 in L so that the following conditions (i—v) are satisfied: ( i ) The restriction of a to L is a Frobenius element of Gal (LIK) for 13. (ii) If p=Fl3nK, then N(p) is a rational prime, and 17 is unramified in L. (iii) q3 does not divide 6m. (iv) The curves ET have good reduction modulo 13 for every r e Gal (L1K). (y) The residue classes of j„ -.• ,j modulo fl are different from each other. The existence of such a 13 is ensured by the Tchebotarev density theorem. Note also that the conditions (iii—v) exclude only finitely many primes. Put p=N(p). Now C/ - 'a is isomorphic to Ei for a unique i. Fix an isomorphism 77 of C/p - ia to E,. Take an integral ideal g in K prime to p so that g17= aoK with a e oK. Since a C 13 - 'ci and ap - 'aca, we obtain a com-

mutative diagram C/a id. C/V i a

(*)

1

7

a C/a

with isogenies A and p. Then we have po)=0(a). By Prop. 5.3, A and ge.e are defined over a finite algebraic extension L' of L. Take a prime ideal ,ED

120

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

in L' which divides 13. We shall now consider reduction modulo e and indicate the reduced objects by putting tildes (see § 5.1). Take a holomorphic differential form w on E rational over L for which 0, as mentioned in § 5.1. Then (Do o = ID(wo 0(a)) = 0(ace) = crib- =0, since a Ep. It follows that the isogeny /Jo is inseparable, by Prop. 5.1. The above diagram shows that Ker(p)= 72(a'a/ir 1a)= 7)(g - 13 -Ja/p - 'a), which is of order N(s). Since g is prime to p, this shows that ti is separable, hence must be inseparable. Since Ker (2) -= e(rla/a) is of order N(p)=p, we see that deg (I). deg (2). p, and is purely inseparable. Let p denote the p-th power automorphism of the universal domain of characteristic p, and r the p th power morphism of to Es. By Prop. 5.2, there is an isomorphism of E. to f9 such that so I = - have the same invariant. Therefore we It follows especially that E, and Es' have 7, =79 =13(?), in view of the condition (i). Now both j i and ja belong ,jh } . By (v), we have j, =j°, so that Et is isomorphic to E°. to {j1, Therefore we can replace E, by E° in the above diagram, and repeat the above reasoning, (possibly changing L' and C). Since 13(E 0 )= both I and lr are isogenies of E to f9, so that e is an automorphism of f9. Since a = id. on K, we have coco 0(a)d = (w 0(a))'1 (aco)g ace' for every a E so that (E", 0°) is normalized in the sense of § 5.1. Therefore, by (5.1.2), we have 3 0(a). 06(a)c A so that 1:g a) , 0(09 o for all a E oK • Now the isogeny ir has the same property r g (a) = (a) r (see Appendix (7.1)), hence 6o8(a)9 =a(a)''3c for all a e o K . By (5.1.5), s =0(r)" with an element r of o K, which must be a unit of oK, since e is an automorphism. Put ic -= 0(7) 10 2, e* = 0(r)" ,D7). Then K is an isogeny of E to EL', and = r. Now by replacing A, 7) by N., e*, the upper part of our diagram (*) becomes as follows : -

(

-

e* This is still commutative. Let t be an element of E such that ml =0. Then q3(t-)_= ri = C.(Kt). Since m is prime to P, we have t" = Kt by (5.1.4). For u E ,n'a, put 1/ 1 = u mod a, and u, = u mod ir 'a. Then e(u,)°= r(e(u,))=e*(u 2). Now let c be an element of K ; such that cp is a prime element of K0 and c, = 1 for all #1.). Then the restriction of a to F„, is Es, KJ = [c, K], so that c= sde with some de Kx and e E U(MOK), where U(moK ) is as in §5.2 (with inoic as c). Since 1.) - Ia = c - la = d 1 sa, we can extend (**) to a commutative diagram

CONSTRUCTION OF CLASS FIELDS

5.4

121

e

C/a id. I

C/ - la

C

E*

dl C

Cis- la

e

'

with a suitable choice of an isomorphism e'. Then, for u, u 1 , and u, as above, we have E(iii) ° = e*(u2)= V (du mod s'a). We have mu e a, e e U(mo K ), and d = s- ice- '. Let q be a prime ideal in K. If q #13, we have c, .1, so that

du= sVe'u= su mod sq- laq ; if ci . p, we have u e al, so that

du = s,Vcv e,- zu

E

s;' cap = s,- '0x0t, .

These relations imply that

du mod s - 'a . s - Ju mod sa . Therefore we obtain

e(u mod a)u .. e, '(s - 'u mod sa) for every u e m'a. Now taking any multiple n of m in place of m, we obtain an isomorphism e" of C I s- 'a to EJ such that e(y)"= e"(s - 'y) for every y E n'a/a. Since e"oe' - ' is an automorphism of E°, we have .06(C)oe' with a unit C of cl ic satisfying ca c a. Then, for every v e ma/a, we have

e(Cvr = e(C)(e(v)°) = 0( C) c (V(s- 1 v)) = e''(s" ' 14 = so that Cy .y for every VE ma/a. It follows that C --=1 mod mo K . Since m > 2, as is remarked above, we have C .1, hence e' . C. This implies that e(y)" . e'(s -1 y) for every V E na/a, for every multiple n of m. Thus e' has the required property, and the proof is completed.

5.4. Construction of class fields over an imaginary quadratic field Let us now derive from the above theorem a few classical results of complex multiplication due to Kronecker, Weber, Takagi, and Hasse. First let us denote by i(a) the invariant of an elliptic curve isomorphic to C/a for a Z-lattice a in K. Then a and s being as in Th. 5.4, we have j (a) = j (s- 'a). This means that i Or depends only on the restriction of a to Kab. Thus we obtain

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

122

(5.4.1) For every Z-lattice a in K, one has j(a)e Kab, and j(ar.c=j(s'a)for all s K; . We shall now prove (5.4.2) For an order o in K, a Z-lattice a is a proper o-ideal if and only if a= xo for some x of K. The "if "-part is obvious. To prove the converse, let c be the conductor of o. If a is a proper o-ideal, there exists, by Prop. 4.11, an element ti of K such that pa+co=o. Let p be a rational prime. If ptc, we have op = (ox)p, so that a p is a principal o r-ideal. If plc, we have cop c pop, so that pap ±pop =or. Then op = pap +p(pap -f-po p)= pap ±p2or. Similarly, by induction, we can show that o p = pap +rop for every positive integer m. But we can find m so that rOp c pap. Therefore pap = op. Thus ap is a principal op-ideal for all p, hence

(5.4.2). 5.5. Let K, E, a, and e be as in Th. 5.4, and hiE the function on E defined in § 4.5. Let u be an element of K/a, and THEOREM

W= (s

E

K;I sa= a, su= .

Suppose that E belongs to 6.1. Then the field KU E, h(e(u))) is the subfield of K a b corresponding to the subgroup Kx W of K. Observe that W is an open subgroup of K; containing K. Let F denote the subfield of Kat, corresponding to K x W. Let a E Aut (C/K). Take s E K; so that a =Es, K] on Ka, and take e' as in Th. 5.4. Put t= (I) Suppose that a is the identity map on F. Then we can take s from W, so that sa=a. It follows that E° is isomorphic to E, hence j`i=is.. Further we can find an isomorphism s of E° to E so that soe' =e. By (4.5.4), we have hi<st`i)= h', (t°) = hi E(0° . Since sr = s(e(u)")=s(e/(s - '11))=e(u)= t, we have hiE (t) = MEW°. This means that a is the identity map on KU R , 1iLE(0), hence K(j E , 4(0) c F. (II) Conversely, suppose that a= id. on K( j E, iztE(0). Then j(E)=j(E)a =j(Eu), so that there exists an isomorphism (3 of E° to E. By Prop. 4.8, there exists an element p of KJ` such that ps'a=a. Choosing 3 suitably, we obtain a commutative diagram : PROOF.

C/s - la

C/a

5.4

CONSTRUCTION OF CLASS FIELDS

123

By (4.5.4), we have hiE (43r) = ha (r) = hig(t)° = h(t). Hence, by (4.5.3), there exists an element C of K such that Ca = a and 0(C)3t° = t. On the other hand, at° =3(e(u)r)=3(e'(s - '21))=e(tes -1 u), hence e(u)=e(Cps - 'u). Putting Cps' = s', we see that s'a = a and s'u = u, hence s' E W, and s E Km W. Therefore a =. id. on F. This shows that FcK(j E , 4(0), and our proof is completed. 5.6. Let E be an elliptic curve belonging to eL. Then K ab is generated over K by JE and the values h(t) for all points t of finite order on E. COROLLARY

This follows immediately from the easy fact that Kw ICI (closed by itself) is the intersection of the KmW, with the subgroups W of the type described in Th. 5.5, for all choices of u. THEOREM

5.7. Let o be an order in K, and a a proper o-ideal.

Then the

following assertions hold.

(i) Gal(K(j(a))IK) is isomorphic to the group of all classes of proper o-ideals, through the correspondence a —4 b such that j(a) * =j(b'a). (ii) [K( : KJ = (a)) QJ(iii) If a 1 , ••• , an are representatives for the classes of proper o-ideals, then i(an) form a complete set of conjugates of j(a) over Q, and over K. (iv) If o=oE, and hence a is a fractional ideal in K, then K(j(a)) is the maximal unramified abelian extension of K, and j(a)'' =j(b - la) for a=

(K(I (a))/ K b

), b any fractional ideal

in K.

The notation being as in Th. 5.5, put u = 0 (or disregard u). Then W= JCL- H,, o;,. On account of (5.4.2), we see easily that 1(;`, D s so gives an isomorphism of KIK 4 W onto the group of classes of proper o-ideals. Therefore we obtain (i) from Th. 5.5 and (5.4.1). If o = o K, the class field F over K corresponding to IC W is the maximal unramified abelian extension FIK b--) on F. Therefore we of K. Further, if b = soK , we have [s, K]=(-PROOF.

obtain (iv). The assertion of (iii) with the basic field K follows from (i). Let E be an elliptic curve isomorphic to C/a, and let Cf E Aut (C/Q). Then End (E°) is isomorphic to End (E), and hence to o. By Prop. 4.8, E 6 is isomorphic to C/a„ for some v. Then j(a)° =j(EQ)=j(a„). This shows that EQ(.i Q] n = [K (j (a)) : KJ. Since the inequality in the opposite direction is obvious, we obtain (ii) and the assertion of (iii) over Q. Since the Fourier expansion of j(z) (see (4.6.1) and Th. 2.9) has rational Fourier coefficients, we see that j(-2)=T(Z3 for all z E 0. Therefore, if a = Zw z +Zah and WICO2 E, we have d = Z- (-05!)-1-Zo5 2, so that AEI) =R - 051105 2) = j(co i/co0= j(a). This implies that j(a) is real if and only if a and a belong to the same class of proper o-ideals. On account of (5.4.2), we can easily show

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

124

that ad is a principal o-ideal.

(5.4.3)

Therefore we obtain

Let o be an order in K, and a a proper o-ideal. Then j(a) is real if and only if a' is a principal o-ideal.

5.8. For o and a as above, prove that K(j(a)) is normal over Q, and study the structure of Gal (K(j(a))/Q). Further prove that the following three statements are equivalent to each other : (i) Q(j(a)) is normal over Q; is totally real ; (iii) the group of all classes of proper o-ideals is a (i)Qja product of cyclic groups of order 2. EXERCISE

5.9. Let F' be the subfield generated over K by the values j(z) for all zE K such that 1m (z)> 0. Prove that F' is the subfield of Kab corresponding to QK 4 K. (Observe that (2:4 1('`K:.` =IIpZ;Kx K1-) EXERCISE

5.10. Let E be an elliptic curve belonging to e, such that End (E) is isomorphic to the maximal order oK . Prove the following assertions : (1) For any integral ideal c in K, there exists a point t on E such that EXERCISE

a E oic , 0(0 = 0

r=:.

ac,

where 0 is the normalized isomorphism of K onto EndQ (E). (2) For any such point t, the field K(j E , hi(t)) is the maximal ray class field modulo c over K, defined in § 5.2. Complex multiplication of elliptic functions may be a fascinating subject of the history of mathematics. But we refrain from making any historical comments, and mention only a few classical and modern works : Weber [89], Hasse [25], Deuring [11], [13], Ramachandra [59]. Further references can be found in these articles. In § 6.8, we shall discuss another formulation of complex multiplication in terms of modular functions of arbitrary level.

5.5. Complex multiplication of abelian varieties of higher dimension

We shall now briefly explain how the results of the previous section can be generalized to the higher dimensional case. Here we must assume that the reader is familiar with abelian varieties (over the complex number field). For the terminology and notation, see Appendix. Except for the notion of a CM-field (see below), the results of this section will be used only in § 7.8. A. Algebraic preliminaries In this section we denote by xt' the complex conjugate of a complex number x. By an algebraic number field, we always mean a subfield of C algebraic over Q of finite degree. By a CM-field, we understand a totally

COMPLEX MULTIPLICATION OF ABELIAN VARIETIES

5.5

125

imaginary quadratic extension of a totally real algebraic number field. PROPOSITION 5.11. An algebraic number field K is a CM-field if and only if the following two conditions are satisfied. (1) p induces a non-trivial automorphism of K. (2) pr = rp for every isomorphism r of K into C. The proof is straightforward, and left to the reader as an exercise. As an application, we obtain PROPOSITION 5.12. The composite of a finite number of CM-fields is a CM-field. If K is a CM-field, then every conjugate of K over Q and the smallest Galois extension of Q containing K are CM-fields. Let K be a CM-field, and 0 an absolute equivalence class of Q-linear representations of K by complex matrices. We shall often denote by the same letter 0 any representation of K in the class 0. We call (K, 0) a CM-type if the following condition is satisfied : (5.51)

The direct sum of 0 and its complex conjugate is the equivalence class of regular representations of K over Q.

Under this assumption, if [K: Q] = 2n, 0 is the direct sum of n isomorphisms • , çon of K into C such that , 0,p} is the set of all isomorphisms of K into C; in (5.5.2) { ço„ •-• , ço n , so i p, other words, p i, •-• , p„ correspond to all distinct archimedean valuations of K.

We write then 0 =

ço i , and tr 0(x) =

det 0(x) = IT= -T91

fPz

(X

e K).

Let us now construct another CM-type (K*, 0*) from a given CM-type (K, 0). First let K* be the field generated by tr 0(x) over Q for all x e K. Then, for any c E Aut (0), we have, by Prop. 5.11, xd = tr 0(xri , x" = X 9V1P = tr 0(x)°P= so that crp= pa on K*, j. e., K* satisfies (2) of Prop. 5.11. Since tr 0(x)" = tr 0(x 0 ), p induces an automorphism of K*. If p= id. on K*, we have tr 0(x)= tr 0(x)" for all x e K, so that 0 is equivalent to 0P, a contradiction. Therefore, by Prop. 5.11, K* is a CM-field. Let F be the smallest Galois extension of Q containing K, and let G=Gal(F/Q). Denote by H (resp. H*) the subgroup of G . corresponding to K (resp. K*). Extend sot to an element of G, and denote it again by so,. Put S= JI Hço. Then we see easily that H* =

{ r

E

G I Sr = S} .

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

126

Therefore S' = {a -1 1 E S } is a union of cosets with respect to H*. We have thus S - ' H *0, with elements O j of G. By Prop. 5.12, F is a CM-field, so that by Prop. 5.11, the restriction of p to F belongs to the center of G. In view of (5.5.2), we have G=SUSp, so that G=S - JUS - Ip, which shows , ç1} satisfies (5.5.2). Therefore we that [K. * : Q] = [G: H*] . 2m, and {0„ obtain a CM-type (K*, 0*) with 0*. D't, 0,. We call (K*, 0*) the reflex of (K, 0). 9' Since S - '1= for r E H, we see that det 0*(x) eK for every xe K*. Consider the idele groups K ,x4 and Kr of K and K*. Then the map KX

det 0* : K*x

can be extended to a continuous homomorphism of K A to K. For simplicity, we put 7)(x). det 0*(x)

(5.5.3) B.

(x E Kr).

Abelian varieties with many complex multiplications

Let .4 be an abelian variety of dimension n, defined over (a subfield of) C. Take a complex torus CVL with a lattice L in C'z, isomorphic to A, or rather,

consider an exact sequence (5.5.4) with a holomorphic map e. Then every element of End Q (A) corresponds to a C-linear transformation of C. Thus we obtain a Q-linear isomorphism 0, of End Q (A) into .1/„(C) by e 01(2) =*4: for A E End (A). Observe that (5.5.5)

01(2) maps QL to QL for every A

Since RL (55.6)

E

EndQ (A) .

Cn, one can easily show that

The direct sum of 0, and its complex conjugate is equivalent to a rational representation of End Q (A) (see Appendix No 11) .

Now we impose the condition that End Q (A) has a subalgebra isomorphic to an algebraic number field K of degree 2n. This is a generalization of an elliptic curve with complex multiplications. It is convenient to discuss a couple (A, 0) with a fixed isomorphism 0 of K into EndG (A) for the following two reasons : (i) there may be many isomorphisms of K into End Q (A); (ii) one has to deal with various A's with the same K. It can be shown that 0 9) In [81, §8.3 1 , we called (K*, 0*) the dual of (K, 0). The notion of reflex can be defined for a couple (K, 0) with any algebraic number field K and any representation class 0. For details, see F751,, [77]. A more intrinsic definition of reflex without the extension F is given in F801.

COMPLEX MULTIPLICATION OF ABELIAN VARIETIES

5.5

127

maps the identity element of K to the identity element of End,2 (A) [81, p. 39, Prop. 1]. Put 0 = 0, o O. Then 0 is a Q-linear isomorphism of K into M„(C), and 0(1) = 1,, Therefore, we can find n isomorphisms so„ , go. of K into C such that 0 is equivalent to the direct sum of soi , , son. We say that (A, 6') is of type (K, 0) or (K, {v,}). From (5.5.6), we see that 0 satisfies (5.5.1). In view of (5.5.5), we can consider QL as a K-module, through 0. Since [QL Q]= 2n = [K: Q], we can find an element w of Cn such that QL=0(K)zu. Changing the coordinate system of Cn, we may assume

(aEK).

(5.5.7)

If

,

w=

we have

w„ QL

f[

ainw, 1 a

E

arnw„

Since RL=Cn, none of the w, can be O. Therefore, changing again the coordinate system by the matrix

, and putting

(el (a E K) ,

u(a)=[

(5.5.8)

ai"'

we see that u is an isomorphism of K onto QL, and can be extended to an R-linear isomorphism of KR=KOQ R onto RL=C", which we write again u. Put a = u - i(L). Then we obtain a commutative diagram 0

K R/a

- a

(exact)

tu

(5.5.9) O

L

C12

A

0

(exact).

In other words, A is obtained as K R/a with a Z-lattice a in IC; the complex structure of A is determined by u; and : K End() (A) is obtained by (5.5.7). This implies especially PROPOSITION 5.13.

Any two (A, 8) of the same type (K, 0) are isogenous.

Let us now take a polarization C of A and consider a triple (A, C, 8).

128

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

Let r denote the involution of EndQ (A) determined by We now impose the following condition on (A, C, 0).

(5.5.10)

c (see Appendix No 13).

0(K)r = 0(K) .

This holds whenever A is simple, since 0(K)= EndQ (A) if A is simple (see [81, p. 42, Prop. 6 ] ). Under the assumption (5.5.10), it can be shown that K is a CM-field, so that (K, 0) is a CM-type. The condition (5.5.10) implies

0(aP)= 0(a)r for every a E K.

(5.5.11)

Now take a basic polar divisor in C, and consider its Riemann form E(x, y) on with respect to (5.5.4) (see Appendix Nos 11-13). Then (5.5.11) is equivalent to

en

(5.5.12)

E(0(a)x, y) = E(x, 0(a9)y) .

Put f(a)= E(u(a), u(1)) for a E K. Then f is a Q-linear map of K into Q, so that f(a)= Tric.,Q (Ca) with an element C of K. Then we have

E(u(a), u(b))= E(u(a), 0(b)u(1)) = E(0(bP)u(a), u(1))= E(u(bPa), u(1)) , so that we obtain

(5_5.13)

E(u(a), u(b)) = Tr" (CabP)

(aEK,bEK).

Since E is alternating, we have

(5.5.14)

Ca

= —CC.

Now we can show

(5.5.15)

E(z, w)= E'„'=_, C9v Zail„

for

zE

C

,

IV

e

CT'

where z „ and w,, denote the components of z and w, respectively. In fact, (5.5.13) shows that (5.5.15) is true for z, w E u(K). Since u(K) is dense in Cn, we obtain (5.5.15). Now E, being a Riemann form of a positive non-degenerate divisor, has the property that E(z, -1.1L-1- w) is symmetric and positive definite. This holds if and only if

(5.5.16)

1m ((Pp) > a

for

p =1, •- , -

n.

Thus, from a given (A, C, 0), we have obtained a CM-type (K, 0), a Z-lattice a in K, and an element C of K satisfying (5.5.14) and (5.5.16). Conversely, we can construct (A, C, 0) from these data. In fact, let (K, 0) be a CH-type, and a a Z-lattice in K. Then we define u by (5.5.8), and form a complex torus A= Cn IL so that (5.5.9) holds. Define 0(a) for a E K by 0(a)o e = e o 0(a). Take an element C satisfying (5.5.14) and (5.5.16). (The

5.5

COMPLEX MULTIPLICATION OF ABELIAN VARIETIES

129

existence of such a is clear.) Define E by (5.5.15). Then it can easily be verified that E is a Riemann form so that A has a structure of an abelian variety with a specified polarization. This shows also that the isomorphism class of (A, C, 0) is completely determined by the data (K, O ; a, O. We say that (A, C, 0) is of type (K, 0; a, C) (with respect to e) in this situation. Observe that (a, C) depends on the choice of the map e of (5.5.9). C. Main theorem Let (A, C, 0) be as above, and let a E Aut (C). Then C° is naturally defined as a polarization of A 0. We define 00: K-->End ci (Al by 0°(a) = 0(0' for a E K, 0(a) E End (A). By our definition, if (A, 0) is of type (K, {so}), we can find n linearly independent holomorphic differential forms ah, , ah, of degree 1 on A such that

, 0 0(a). as-%, w

t

(a E K, 0(a) E End (A), v=1,

, n).

Then we have 0,`, 0 V(a) aircof, so that

(5.5.17) (A°, 0°) is of type (K, 0°) . PROPOSITION 5.14. Let (K*, 0*) be the reflex of (K, 0). (A', 0°) is of type (K, 0), and isoge nous to (A, 0).

If a= id. on K*,

This follows immediately from the definition of K* and Prop. 5.13. Now the relation of (A, C, 0) with (A°, Ca, 0 0 ) is given by the following main theorem, which is a generalization of Th. 5.4. THEOREM 5.15. Let (K, 0) be a CM-type, (K*, 0*) the reflex of (K, 0), a a Z-lattice in K, and C an element of K satisfying (5.5.14, 16). Let (A, C, 0) be of type (K, ; a, C), u the map defined by (5.5.8), and e a map such that (5.5.9) holds and C corresponds to C through C. Further let a be an element of Aut (C/K*), and s an element of KV such that a =[s, K*] on KL. Define 72 by (5.5.3). Then there is an exact sequence U(*(S) - l a)

ef -4*

AG —0

with the following properties: (i) (A°, C°, 0 0 ) is of type (K, 0; 72(s)- ia,Ci) with respect to e', where C' = N(il(s))C. (For the symbol il(s), see § 5.2.) (ii) e(u(a))(1 =e'(u(72(s)-- ia)) for all a E Kla. A proof in a more general setting is given in [80, 4.3]. If the reader is familiar with the results of [81], especially with the prime ideal decomposition of the Frobenius endomorphism [81, § 13, Th. 1], then he will be able to give

130

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

a proof exactly in the same manner as has been done for Th. 5.4. In the above theorem, we have imposed no condition upon the field of definition for (A, C, 8). Actually there exists a model of (A, C, 8) defined over an algebraic number field, see [81, p. 109, Prop. 26 ] . Let t„ , t r be points of A (of finite or infinite order). One can prove that there exists a subfield k of C which is uniquely characterized by the following condition :

(5.5.18)

An automorphism a of C is the identity map on k if and only if there is an isomorphism 2 of A to Aa such that 2(C). Cc', 'it,. tf for i=1,•••,r, and 206(a) , 0°(a)0 A for all a e K. (Such a A is called an isomorphism of (A, C, O ; t1 , , t r) to (A°, 07 , 8"; ti', • ,

We call k the field of moduli of (A, C, ; ti , ••• , t,.). (For the proof of the existence of k, see [72 ], [75, II].) With this concept, the following result can easily be derived from the above theorem : COROLLARY 5.16. The notation and the assumption being as in Th. 5.14, let , y,. be elements of Kla, and let T be the set of all the elements s of Kr such that q qP N(il (s)) = 1, qv(s)a = a, v2(s)v, = v,

(i = 1, • • • ,

for some q e Kx. Then the field of moduli of (A, C, ; e(u(v,)), ••• , e(u(vT))) is the subfield of Kt corresponding to the subgroup T of Kr.

Let k, be the field of moduli of (A, C, 8), and G the group of all automorphisms of (A, C, 8). Then G is isomorphic to the group of all units of K, and one can construct a quotient variety W of A by G and a projection map p: A. W satisfying the following conditions : ( i) W is defined over le,. (ii) If a e Aut (Clk,), and f is an isomorphism of (A, C, 8) to (A", C7, 8"), then p= pc o f. (Observe that such an f exists for any a e Aut (C/k,), on account of the definition of the field of moduli.) Then one can easily show that

(5.5.19)

For every point t of A, the field of moduli of (A, C, 8; t) is k,(p(t)).

It may be worth while noting that, if A is simple, G coincides with the group of all automorphisms of (A, C). If A is an elliptic curve E, we see easily that k, = Q(iE), and G = Aut (E). Thus the map p is a generalization of hiE , and hence the combination of (5.5.19) with Cor. 5.16 yields a generalization of Th. 5.5. There remains the question of finding a generalization of the function j(z) in the higher dimensional case. But this is settled in the following way.

5.5

131

COMPLEX MULTIPLICATION OF ABELIAN VARIETIES

The polarized abelian variety (A, C) determines a point z in the Siegel upper half space t)o of degree n, modulo a certain discrete subgroup of SP(n, R) commensurable with Sp(n, depends on the type of C.) There exists a r-invariant holomorphic map so of 0„ into a complex projective space, such that Q(ço(z)) is the field of moduli of (A, C) for any (A, C) with a polarization C whose type determines r. One can also formulate a similar result by using the Hilbert modular group instead of the Siegel modular group. For details, see [77 ], [78], [80]. Finally let us make a few remarks about the relation of the field of moduli of (A, C, 0) and that of (A, C, 00, where 0' is the restriction of 0 to any subfield F of K. The field of moduli of (A, C, 0') is a unique subfield k of C satisfying (5.5.18) with the following modification: the points ti are disregarded; o 0(a) = 60(a) o A is required only for a e F. If F =Q, k is the field of moduli of (A, C).

z). (r

r

PROPOSITION 5.17. Let K, K*, and (A, C, 0) be as in Th. 5.15, F a sid)field of K, 0' the restriction of 8 to F, and k o the field of moduli of (A, C, 0'). Suppose that A is simple. Then the following assertions hold: (1) k oK* is the field of moduli of (A, C, 0). (2) K* is normal over ko n K*. (3) ko K* is normal over ko. (4) Gal (koK*/k o) is isomorphic to a subgroup of Aut (K/F). (5) k o contains the smallest subfield of K* over which K* is normal. PROOF. Let a E Aut (C). If there exists an isomorphism of (A, C, 8) to (A 0, CO 3 80), we see, by (5.5.17), that 06 is equivalent to 0, and hence a is the identity on K*. This shows that k oK* is contained in the field of moduli of (A, C, 0). Let r E Aut(C/k0). Then there exists an isomorphism f of (A, C, 0') to (Ar, C7, 01. Since A is simple, we have 0(K)= EndQ (A) by [81, § 5.1, p. 42, Prop. 6]. Therefore we can define an element p of Aut (K/F) such that Or(a)0 f = f o 0(0) for all a e K. Then ø(a) r and ø(a) have the same set of characteristic roots, so that (soir, , (Par} coincides with {p49 1, tiçon} as a whole. Therefore we have (E i a ,)r=E, ar191 for all a e K, which shows that K *r = K*. This proves (3). If M= {x K* I x 6 =x for all a E Aut (K*)} ,

we have r = id. on M for every r e Aut (C/k o), so that Mc ko. This proves (5) and (2). Now, if r = id. on K*, {pSol, ••• pson} coincides with fçoi, P.1 as a whole. Let F, G, H, H*, and S be as in the definition of (K*, 029 in the paragraph A. Take elements of G which coincide with p, Sol, ••• 9rit and denote them again by the same letters. Then pH= Hp, and pS= U t plisoi = Hpç = Hso t = S. By [81, § 8.2, p. 69, Prop. 26], we have p e H, so

132

COMPLEX MULTIPLICATION OF ELLIPTIC CURVES

that p= id. on K, hence f is an isomorphism of (A, C, 0) to (Ar, Cr, Or). This shows that koK* contains the field of moduli of (A, C, 0), and hence (1). We have also seen that r= id. on koK* if and only if p= id. on K. Therefore, assigning p to r, we obtain an isomorphism of Gal (k oK*Ik o) into A ut(K/F). For example, if K is not normal over Q and [K: Q] =4, then K* is also a field of the same type, and A is simple (see [81, § 8.4, (2), c), p. 74]). Therefore, in this case, taking F to be Q, we know, by (5), that the field of moduli of (A, C) contains the real quadratic subfield of K*.

CHAPTER 6 MODULAR FUNCTIONS OF HIGHER LEVEL 6.1.

Modular functions of level N obtained by division of elliptic curves A. The functions ffi(z)

rN

Let N be a positive integer, and =T(N) the principal congruence subgroup of r,=SL 2(Z) of level N, which is defined by

r „, = {r e SL 2(Z) I r 12 mod (A)}

(see §1.6).

We shall now construct some functions which generate the field of all modular functions of level N, and which behave nicely under the transformations of F 1 . The main idea is to consider the points of finite order on the elliptic curve

EL : y 2 = 4.e—g2(L)x— gs(L)

(6.1.1)

with variable L. If L = Zoh+Zoh, we see that every point of finite order on EL can be written as ( P( a[w i

W2

];

L), P' (a[ W1 ] ; L)) W2

with a E Q 2, and conversely such a point is of finite order for any a e Q2 . Here we consider a as a row vector. In view of the definition of P, we see that 11 (a[ il , w i , oh) is a homogeneous function of degree —2 in cop ah. ah Therefore we can define three types of functions fa=f41, 11 Al on 0 by fa(Z)=.0X2)=

g2(w1 , (02),g3(w1,

oh)

Awl , (0 2)

f ,, (2, ) _. _ g_2(a) 1,

w 2) P (a[ °1-' w 1' (02

(00_2 p r wi 1 . (a

\ L (0 2 -1 '

ZINI, w2)

nr, z = _gs(ed p w 2) " 4(w, (02)

9

w w )2 11 - 2 /

' 3

a[ (1) I ] ; 0) 1, (02)

P (

W2

(z= ah/w2 E ; a e Q2, e Z2 ) . Especially we can substitute (z, 1) for (ah, ah). We see then that these functions are holomorphic on D. Since j(z)=g/4, we have j(z)-1= 27 -g3/4, so that

134

MODULAR FUNCTIONS OF HIGHER LEVEL

.a(z)= 27 (i (2) -1 ) - Va(2)2 ,

(6.1.2)

fa3(z). 27 j (2) - '( j(z)-1) -1fa(z)3 . The functions fl and f 2 are rather auxiliary, and will be put to use only in § 6.8. Let W a) I TE ri, z=c0i/(02, I_ co; J = TL a), ] , z' . oa co , and aT -=- a' .

r l1

r

Then - co l 1 r ail 1 a/ I_,,,-2j = aL 0.12 ,j,

z' = r(z) ,

and

Zah-f-Zw. = Zola -{-Zo4 .

Therefore substituting z' for z in f (z), we obtain (6.1.3) fo 7- = flr for every T e

r,

and every a

e (22, e Z 2 .

Since P(u; L)= P(v ; L) if and only if u m + y mod L, we see that (6.1.4)

a

(6.1.5)

fa =fb

-±-b mod Z2 ,:=

f=f. a a ±b mod Z 2 .

From (6.1.3, 4) we obtain (6.1.6) If Na e Z2, then Li, 0 r ,f,f for all T e r !„ • { -±-1} . Therefore, in order to ensure that fl, with a e N -1Z 2, is a modular function of level N, it is sufficient, by virtue of Prop. 2.7 and (6.1.2), to show that fa is algebraic over C (j). We shall actually prove in Th. 6.6 that fa is algebraic over Q(j). (Also, the Fourier expansion of fa will be explicitly given in the proof of Prop. 6.9.) Assuming this result, we obtain PROPOSITION 6.1. For every positive integer N, C (j , fa I a e N- Z 2, e Z2) is the field of all modular functions of level N. PROOF.

Let

RN

denote the fi eld of all modular functions of level N.

Then

CO)CC(i, fal a e N'Z', e Z2) c RN. Now RN is a Galois extension of C(j), whose Galois group is r,/r N - {±1}. Therefore, to prove our proposition, it is sufficient to show that if T e r, and ho r = fa for all a E N - JZ 2, e Z 2, then 7- e rN • (+1). But this follows immediately from (6.1.3, 5) and the following 6.2. Let a be an automorphism of the module (Z/ NZ) 2 such that au= so for every u e(Z/NZ) 2 with su = +1. Then a= +1. LEMMA

The proof is very easy and may therefore be left to the reader.

MODULAR FUNCTIONS OF LEVEL N

6.1

135

B. The field generated by the points of finite order on an elliptic curve Let us now discuss the points of finite order on an elliptic curve in a more intrinsic way, without any reference to complex tori or tt). Consider an elliptic curve E: y 2 = 4xs—c,x—c. with c2 and c, in C, such that Aut (E)= {±1}, and functions hiE , for i =1, 2, 3, defined in § 4.5. For simplicity we write h for hig . For a positive integer N, we put

gN ={teEiNt=0}, and consider the field FN = Q(JE, h(t)Iteg N) •

In view of (4.5.4), we see that the field FN depends only on N and the isomorphism class of E. Therefore, to study the structure of FN, we can assume that E is defined over Q(j E), by changing E for a suitable curve isomorphic to E. Assuming this, let a be an automorphism of C over Q(I E ). Then Ei =E, and t —, r gives an automorphism of the module gN. Since g N is isomorphic to (Z/NZ) 2 , the group of all automorphisms of g N is isomorphic to GL,(ZINZ). Since h is rational over Q(JE), we have h(0° = h(r), so that FN is stable under a. Therefore FN is a Galois extension of Q(JE). If a = id. on FN, we have h(t)= h(t), so that by (4.5.3), r =e,t with et = ± 1. By Lemma 62, e, is independent of t. Thus a induces an automorphism ±1 on g N if a = id. on FN. Therefore we obtain an injective homomorphism Gal (FN/Q(./E)) — GL2(Z/NZ)/{ ±12} •

(6.1.7)

More explicitly, take two elements t i and t2 of gN so that g N =Zt i +Zt 2. an automorphism a of C over Q(J E), put

For

tf =pti --fqt 2 ,

(6.1.8)

4 = rt 1 i-st 2 ,

with an element p = { Pr qs ] of M,(Z).

Then det (p) is prime to N, and the

restriction of a to FN corresponds to ±p mod (N).

(6.1.9)

kat 1i-bt2)°=h(ait1 -Fbit2)

PROPOSITION

6.3.

if

(a

We have clearly

op = (a, b').

The notation being as above, the following assertions

hold:

(1) FN contains a primitive N-th root of unity, say C.

136

MODULAR FUNCTIONS OF HIGHER LEVEL

If an element r of Gal(FN/QUED corresponds to an element a of GLaINZ), then Cr = Cdet (a). (Note that Cdet (a) i s meaningful.) (3) If 2 is an isogeny of E onto an elliptic curve E' such that Ker (2)C gm, then j(E') e FN. Moreover, if End (E)=Z, then, for ge Aut (CIQ(jE)), one has j(E')°=j(E') if and only if Ker (2)° = Ker (2). (2)

PROOF.

Consider the symbol eN (s, t) of § 4.3. For an automorphism a of

C over Q(f E), define /3 =[Pr qs ] as above. Put C = eN (ti, t2). By Prop. 4.2, we have

c- = eNctr, to = eN (pt i+qt2, rt1 ±st2)= eN (t„ t2)P8-gr = Cdet (49) . For every u and y in Z, we have

eN(t„ uti + vt2) = eN(t„ toy , so that, by (5) of Prop. 4.2, C = eN (t, t 2) must be a primitive N-th root of unity. If a =id. on FN, we have p -.7.. -4-1 mod (N), so that C° = C. This shows that C E FN, hence (1) and (2). Let 2 and E' be as in (3), and again a an automorphism of C over Q(jE). Then 2° is an isogeny of E onto E'°. If Ker (2)° = Ker (2), E'° is isomorphic to E', so that j(El)°=j(E0. This is so especially if a= id. on FN, since one has then t°=+t for all teg N . This proves that i(E')e FN. Suppose conversely that j(E')° =j(E'), and further End (E)=Z. Then there exists an isomorphism p of E' onto E'°. Observe that p32 and 2' are elements of Hom (E, E'°) of the same degree (cf. § 5.1). Since Hom (E, E'°) is isomorphic to Z, we have po2=+2°, so that Ker (2) = Ker (2°) = Ker (2)4. This completes the proof of (3). 6.2.

The field of modular functions of level N rational over Q(e"ti')

We are going to connect together the results of Parts A and B of the preceding section, by means of the following two lemmas.

LEMMA 6.4. Let L=Zw i -i-Zw 2, and let E be a member of e isomorphic to CIL (see § 4.5). Then, for any isomorphism e of CIL onto E, we have") h( , (a[ w l ])) ----fl(wi/wo

(a e Q2 , Œ Z 2 ; i= 1, 2, 3) .

,

PROOF. Let E' be defined by 3/2 = 4x 3—g 2(L)x—g 3(L), and let e' be an isomorphism of CIL to E' defined by 10) One should actually write hereafter use the abbreviated form

(u mod L) instead of f (u) for u E C. e(u) if there is no fear of confusion.

But we shall

THE FIELD OF MODULAR FUNCTIONS OF LEVEL N

6.2

137

et(u) = (P(u ; L), P t(u ; L)) .

Put 72 = Ç'oe'. Since 72 is an isomorphism of E onto E', we have h(e(u)) h'(e'(u)) by (4.5.4). From our definition of hiE and fL we obtain 4,(e/(u))

= il(a),/w 2) if u = CD' I hence our assertion. LEMMA 6.5. Let fa I a E Al be a set of meromorphic functions in a connected open subset D of Cd, indexed by an at most countable set A. Let k be a subfield of C with only countably many elements. Then there exists a point z. of D such that the specialization {fa}aEA—°{.f.(z0)}aEA defines an isomorphism of the field k(f cr i a E A) onto k(f„(z 0)Iae A) over k. We call such a point z o generic over k for the functions f oe Actually we need this lemma only in the special case d= 1, where the proof is much

simpler. PROOF. We may assume that A -= {1,2,3, ••-} (finite or not). By induction, we see that there exists a subset B = 1) v —} of A such that : (i) vi < bi , f 2 , are algebraically independent over k; and (iii) f i,—, f„ are Let S m be the set of all polynomials algebraic over k(f,,IvE B, Lin). P(X„ ••• , X.)# 0 in m indeterminates with coe fficients in k, and W, the set of the points of D where h is not holomorphic. Put, for each PE Sm ,

Fp = {z D — Ur=1 1 I P( f.x(z),

.(z)). 0} .

The closure of Fp in D has no interior point of D. Now observe that S„, has only countably many elements. By Lemma 1.2, there exists a point z o of D not belonging to the countable union (U.E4 (U7,-1 Upes „, Fe). Then, by virtue of our construction, k(f,,-- ,f„) has the same transcendence degree as k(f 1(z 0), , j,,(z 0)) over k for every n. Therefore the specialization over k defines an isomorphism of these fields, hence our assertion. Now let us put, for a positive integer N, =Q(:), fala

N -1 V, e Z2) .

We have seen in Prop. 6.1 that C N is the field of all modular functions of level N. We call (by abuse of language) an element of a modular function of level N rational over (2(e2ru The following theorem will justify this

definition. THEOREM 6.6. The field

has the following properties. (1) N is a Galois extension of Q(I). (2) For every fi E GL,(Z/NZ), fa '—f gives an element of Gal ( N/Q(j)), which we write r(13). Then 13-0 r(fi) gives an isomorphism of GL2(Z/NZ)/{±1}

MODULAR FUNCTIONS OF HIGHER LEVEL

138

to Gal ( N/Q( j)). (3) If C is a primitive N-th root of unity, then CE N , and Cro ) =Cd 4 0. (4) Q(C) is algebraically closed in (5) oN contains the functions jo a for all a E M 2 (Z) such that det (a) = N.

By Lemma 6.5, we can find a point zo of 0 generic for the functions j, f , foci for all a e N 1 Z 2, e Z 2, and for all a E M 2 (Z) such that det (a). N. Since the substitution of z,, for z gives an isomorphism, it is sufficient to prove our assertions for i(z0), a(zo), (a(zo)) instead of j , fa, foci. Obviously j(z 0) is transcendental. Take c EC so that c/(c 27). j(z o), and consider an elliptic curve E: y 2 .4x 3 cx c. Then j E =j(z 0), so that there exists an isomorphism E of CAZz o +Z) onto E. Consider gN, h= ME, and FN of § 6.1, Part B, with respect to the present elliptic curve E. Put PROOF.

-

—

—

v(a) =(a[z7°]) for a E Q 2 . By Lemma 6.4, we have h(t) = f(z) if t = so that FN = QU(z0), fa(2o)

ae

12,2, e 2,2)

Then the assertion (1) follows from the fact that FN is a Galois extension of Q(JE).

Put t 1 = 71((N - ', 0)), t 2 =72((0, N -

)).

If a and fi = [Pr 'is ] are defined

with respect to these t, and t, as in (6.1.8, 9), then )7(a)a = )2(0) for all a E N -1Z2, so that fa(zor = h()2(a)) = h()2(0))= fo(20). Therefore we obtain (2) and (3) from Prop. 6.3, if we could prove the surjectivity of the map (6.1.7) in the present case. Let A be the image of the map (6.1.7). Let r E SL,a). Since ja r =.fao r by (6.1.3), we see that f a —*far defines an automorphism of SN over Q(j). Transferring this result to FN, we can conclude that SL,,(Z /NZ)I{ ±1} c A. Identifying A with Gal (FISQUED, let B denote the subgroup of A corresponding to Q(, JE). By Galois theory, we obtain -

[A: B7= [WC, jE):Q(JE)1=[(Z/NZ)x :1].

By (2) of Prop. 6.3, we have SL2 (Z/NZ)/{±1}C B, so that A=GL 2 (ZINZ)1{ ±1}, and B =SL 2(Z/NZ)/{±1}. To prove (4), put k =Cr\p/. Then every element of k is invariant under SL,(ZINZ), since, as is shown above, the action of SL,,(ZI NZ) is obtained from the substitution z.-.4r(z) with r E SL,(Z). Moreover, we have seen that Q(C, j) is the subfield of N corresponding to SL 2(Z/NZ). Therefore k c Q(C, j), so that kc Q(C). This proves (4). To -

prove (5), let a e M2 (Z), det (a)= N, a[7.°]=[ (70 ], and let E' be an elliptic curve isomorphic to C/(Za4+Za4). Since Na i EM,(Z), CZa);± Za2. Therefore we obtain an isogeny A of 2(e(u))= E/(Nu) for u E C, where E' is an isomorphism E'. Then Ker (2) = (1‘,1- '(Zu);+ Za4))c $(1\1 -1(Zz 0 + Z))

we see that N(Zz o+Z) E onto E' such that of C/(Za4--FZco4 onto g N . Now we have

THE FIELD OF MODULAR FUNCTIONS OF LEVEL N

6.2

139

j(a(zo)) = j(ald a4) = j (E'), and j(E9EFN by (3) of Prop. 6.3. This proves (5). The Galois theoretical correspondence between fields and groups in the above theorem can best be described by the following diagram, in which we put kN =Q(e2iri/N ) . RN denotes the field of all modular functions of level W. P.,

1±1

GL,(Z/NZ)

IIJ

(Z/NZ)

6.7. The notation E and gN being as above, we see easily that QUE, t It E gN) is a Galois extension of Q(JE) whose Galois group is isomorphic to a subgroup H of GL,(ZINZ). The above result implies that H- {-1-1} =G1,(ZINZ). Take an element r of SL,(ZINZ) so that r2=-1, say 0—1 r=[i 0 1 Then either r or - r is contained in H, hence —1= 7-2 e H. REMARK

Therefore we have H=GL2 (Z/NZ). the rest of the book.

We shall make no use of this result in

PROPOSITION 6.8. Let ZN denote the field generated over Q(j) by the functions of the form joa with a E Ma), det (a) = N for a fixed N. Then TN is the subfield of 5N corresponding to the subgroup

{[ao of

a E (Z/NZ)x

{ 1}

GL2(Z/NZ)/ ± 1 }

Let 20, E, gN, t, t2, and FN be as in the proof of Th. 6.6. As is shown in the proof, every a E M2(Z) such that det(a)=N corresponds to an isogeny 2 of E to an elliptic curve E' such that Ker(2)c gN . Especially, ri r 1 01 N 01 Ker (2)= Zt„ Z4, or Z(ti-Ft2) according as a= LO NJ' LO 1J' or LO Ni• We then have j(E)=AzolN), .i(Nzo), or Mz 0 -F1)/N), accordingly. Let a be PROOF.

140

MODULAR FUNCTIONS OF HIGHER LEVEL

an automorphism of C over Q(JE) whose restriction to Fly corresponds to an element A of GL,(Z/ NZ). By (3) of Prop. 6.3, if a leaves f(a(z o))= j(E') invariant for all such a, then Ker (2) for all corresponding 2 must be stable under a. Especially Zt v Zt„ and Z(t 1 +t2) must be stable under a. Then we IS = r a 01 see easily that A is of the form A= r a '31 . Conversely, if then L 0 aJ LO a J ' every subgroup of gN is stable under a, so that, by (3) of Prop. 6.3, j(E') =j(E) for any E' as above, hence a = id. on ON. REMARK. As the above proof shows, the conclusion of Prop. 6.8 is true even if we restrict a to the elements whose elementary divisors are 1 and N.

aN

PROPOSITION 6.9. (1) coincides with the field of all the modular functions of level N whose Fourier expansions with respect to e27"1"' have coefficients in km = we 21.v) . (2) The field Q(j(z), j(Nz), f a, (z)), with a i = (N- ', 0), coincides with the field of all the modular functions of level N whose Fourier expansions with respect to e2 '"N have rational coefficients. (3) The field of (2) corresponds to the subgroup rr±1 0 7 1 1L 0 x j I xe (Z/NZ)x}/{±1} of GI,(Z/ NZ)/1 ±1}.

These results will be needed only in the proof of Prop. 6.35 and Ex, 6.26. PROOF. To prove (3), let z o, E, and Fp, be as in the proofs of Th. 6.6 and Prop. 6.8. We have seen that there exists an isogeny 2 of E onto E' such that j(Nz o) = j(E'), and Ker GO= Zt 2. Let a be an automorphism of C over Q(j(;)), and let i3= [ P r qs ] be an element of GL 2(Z/ NZ) corresponding to the restriction of a to F,,. Then a= id. on QU(z0),../(Nzo),.f.,(z0)) if and only if Ker (2)G= Ker (2) and a i p a +a, mod Z2. This is so if and only if

A =[±10 °J,

hence (3).

To prove (1) and (2), we consider the Fourier expansion of fa. y = u/co, and z = w,/w 2 , we have co3 . P(u ; w j , co2) ----- v -2 +E'[(v—mz—n) -2 — (mz -1 0 -23 -

Putting

am, n)* (0, 0))

= —2 E7=1 n -2 -2 E7, = , E7., (mz+n) -2 -i-EZ=,(v-I-n) -2 EL-. [(v+mz+n) -2 4-(—v-Fmz+n) - 2] • By virtue of (2.2.3), this is equal to

6.3

A GENERALIZATION OF GALOIS THEORY

2/3+87r2 -

'47r2

not

n • e 2 7"n" —470 Zen • ezrinv •

Ern.'

n.

141

Therefore, putting u.(rco 1 1--sw 2)/N with integers r and s, C=e2'"N, q = and gpf = e21 , we obtain -

(6.2.1)

(a)2/270 2P((rco1 -f-sw2)/N; w, w 2) = —(1/12)+2 Ez. / nqn/(1 q) —

— VG/(1 --Cs qrN) 2 — E7 =1 W s Tirï

" qii nr) nqn / (1— qn) , r < N, (r, s) e NZ2).

This, together with the results of § 2.2, shows that the Fourier coefficients of fa belong to k N for every a e N -1 Z 2 e Z 2 . Let X (resp. X') denote the field of all the modular functions of level N whose Fourier coefficients with respect to am belong to Q (resp. k m). Then X (resp. X') and C are linearly disjoint over Q (resp. k N ). In fact, let p i, , p„, be elements of C linearly independent over Q. Suppose E71 1 p i g,. 0 with g , in X. Let gi = E n c,„q1 with c i „ e Q. Then E i p,c in = 0 for every n, so that c,„ =0 for all i and n, hence g,=•-•=g,„.O. The same argument applies to X' and k m . Now, since a A,C X' c Cap', we obtain, from the linear disjointness, To prove (2), Put Y = QU(z), i(Nz), f a ,(4). From the above formula (6.2.1), we see that fa, e X, so that Yc X. By our assertion (3) which is already proved, and by (3) of Th. 6.6, we see that only the identity element of Gal (aN/Q(J)) can leave the elements of Y(C) invariant, hence = Y ). Thus Yc Xc Y(C). From the linear disjointness of X and Q(C) over Q, we obtain Y = X. This completes the proof. ,

—

a,

(

6.3. A generalization of Galois theory Let k be a field, and K an arbitrary extension of k. We shall now make a few elementary observations about the Galois-like correspondence between the subgroups of Aut (K/k) and the subfields of K. In later sections, our results will be applied to the field of all modular functions rational over cyclotomic fields, i. e., the composite of the for all N. In this section, for simplicity, we fix the fields k and K, and put 91= Aut (K/k). For a subfield F of K containing k, we put

aN

g(F) = Aut (K/F)= {a

9t I x.= x for all x e F) ,

and for every subgroup S of 91, f(S) = {xeKlx"=x for all aeS).

142

MODULAR FUNCTIONS OF HIGHER LEVEL

We can make 91 a Hausdorff topological group by taking, as a basis of neighborhoods of the identity element, all subgroups of the form {a e

xid = x„

, x„d = x„}

, x,} of elements of K. We observe that the topology for any finite set {x1 , of Aut (K/F)= g(F) is the same as that induced from the topology of 91. The following proposition is fundamental and well-known. PROPOSITION 6.10. If K is a (finite or an infinite) Galois extension of k, then 91 is compact, g(f(S))= S for every closed subgroup S of 91, and f(g(F))= F for every subfield F of K containing k. (In this case, of course 91= Gal (K/k).) In a more general case, we have PROPOSITION 6.11. Let I denote the set of all compact subgroups of 91, and 0 the set of all subfields of K containing k, over which K is a (finite or an infinite) Galois extension. Then g((S))=S and f(S) E 0 for every Se f(g(F))= F and g(F)E I for every Fe O. Thus there is a one-to-one correspondence between I

and O.

The fact that f(g(F))= F and g(F) E E for every Fe 0 follows immediately from Prop. 6.10. To prove the remaining part, let Se I, and a E K. Obviously S = UbeK {cc S ad = b}. Since S is compact, S is covered by a finite number of the sets of the form {CE SI a = b } . This shows that {au I u e SI is a finite set, say {a 1 , •-• , a n } . Then the polynomial 117,i.,(X—a i) has coefficients in f(S). This shows that every element a of K is algebraic over f(S), and an irreducible equation for a over f(S) splits completely in K. Therefore K is a Galois extension of f(S). Now S is a closed subgroup of g(f(S))= Gal (K/f(S)). Applying Prop. 6.10 to S, we obtain S= g(f(S)). PROOF.

6.12.

The notation being as in Prop. 6.11, let E' be the set of all open compact subgroups of %, and 0' the subset of 0 consisting of all F e 0 which are finitely generated over f(1). Suppose that O' is not empty. Then 91 is locally compact, and the one-to-one correspondence between E and 0 induces a one-to-one correspondence between E' and 0'. PROPOSITION

Put k.= f(%). Suppose that a member M of 0 is generated by a finite number of elements x„ •-• , x, over k.. Then PROOF.

g(M)=

e 911 x= x i, , x: = x}.

11) The fact that every compact subgroup S corresponds to a member of 0 is mentioned in N. Jacobson, Lectures in abstract algebra, vol. III (1964), p. 151, Ex. 5. See also Pjateckii-Shapiro and Shafarevic [58] and Ihara [34 ].

THE ADELIZATION OF GL 2

6.4

143

Therefore g(M) is open, hence g(M) e E'. It follows that /I is locally compact. Conversely, let S E E', and F = f(S). Then we have g(MF). g(M) ng(F),. which is open and compact, hence [MF: M]=Eg(M): g(MF)]< co. It follows that MFE 0 1, hence Fe V. PROPOSITION 6.13. Let S be a subgroup of 1t, F=f(S), and F, the algebraic closure of F in K. Then Fi is a Galois extension of F. If moreover g(F)=S, then g(F1) is a normal subgroup of S, and S/g(F,), as an abstract group, is canonically isomorphic to a dense subgroup of Gal (FIF). If u e Fi, O E S} is obviously a finite set, say {u1, •-• 11 . } Then mi (X u 1) has coefficients in F, so that Fi is Galois over F. If g(F)= S, then g(F1) c S, and Ff =F, for every a e S, hence g(F,)= g(F 1`)= Ig(F,)cr for every a e S. Now SIg(Fi) can be identified with a subgroup of Gal (F,/F) in a natural way. Since F is the fixed subfield of F, for this subgroup, we obtain the last assertion. PROOF.

—

If f(g(F))=F for a subfield F of K containing k, then f(g(M))= M for every finite algebraic extension M of F contained in K.

PROPOSITION 6.14.

Put S = g(F) and T = g(M). Let F, be the algebraic closure of F in K. Considering the restriction of the elements of S to M, we find [S: T][M: F]. If f(S) = F, we see from Prop. 6.13 that every isomorphism of M into F, over F can be obtained from an element of S. Therefore ES : T]=[M : F]. Let S =UgER Ta be a disjoint union. We see that, for every y e f(T), 11„,t (X-1/°) has coefficients in F, hence f(T)c Fi. Now for every finite extension M' of M contained in f(T), we have g(M'). T. Taking M' in place of M, we obtain [S: T]= [ M' : F], so that M = M'. This proves M= f(T), q. e. d. PROOF.

6.4. The adelization of GL2

Throughout the rest of this chapter, we denote by G the group GL 2, viewed as an algebraic group defined over Q. We are going to define the adelization G A of G, the suffix A denoting the adeles of Q.") First put G,=GL2(Qp)

(p : rational prime),

G.= GL2(R), Go„+= {xe G. I det (x)> 0} .

Then GA is, by definition, the group consisting of all elements x=(--, xp , 12)

For the general theory of adelization of algebraic groups, see Weil [96].

xé„,)

144

MODULAR FUNCTIONS OF HIGHER LEVEL

of Hp G p xG c., such that xp E GL,(Z p) for all except a finite number of p. can be identified with G L 2(A). Put

GA

U= Hp G L,(Zp )x G.. . Then U is a subgroup of G A , locally compact with respect to the usual product topology. We define the topology of GA by taking U to be an open subgroup of GA. Put GQ = G L 2(Q), and consider it a subgroup of GA by the diagonal embedding a -4 (a, a, a, --)e GA. We denote by G o the non-archimedean part of GA ) i. e., the set of all elements of GA whose co-component is 1. Then we put G A+

= G0G40+

I

GQ+ = GQ nG,4÷

= {a e GL2(Q) I det (a) > 0) .

Observe that the map x.---)det(x) defines a continuous homomorphism of GA into Q. We define a homomorphism

(6.4.1)

a: GA- , Gal (Qab/Q)

by a(x)=[ciet(x)", (2]

(x E G A ) .

(For the notation [s, Q] with s E Q, see § 5.2.) Note that a(x)=1 if XE GQGoei.. For any positive integer N, we put

(6.4.2)

UN= {x= (x)

E

U 1 xp m

1 mod N- M 2(Zp)) •

Obviously U = U„ and UN is an open subgroup of GA.

We also observe :

(6.4.3) Every open subgroup of GA contains UN for some N. For every open subgroup S of GA, we see that det (S) is open in Q. Therefore the subgroup Q't - det (S) of Q ,; corresponds to a finite abelian extension of Q, which we write k s = k(S). It is easy to see that k(U N)= k N = Q(e2 tr"), and

(6.4.4) (6.4 5)

k(S)=k(xSx") for every X E GA, ScT =

k r ck s .

Let g be a Z-lattice in Q2. We can then define the action of an element of GA on Q2/g in the same manner as in § 5.2. To do this, first let gp denote the closure of g in (Pp , and identify Q2/g with the direct sum of the modules C,/g p for all p. For every c=(c) E GA, we define gc to be the Z-lattice in (2 2 characterized by the property (gc)p =gpcp. Then right multiplication by c p defines an isomorphism of (4,/gp onto Q7,/x,pcp, hence an isomorphism of Q2/g to Q2/c. We shall denote by wc the image of an element w of Q2/g by this isomorphism. The situation is explained by the commutative diagram

THE ADELIZATION OF GL2

6.4

Cp

Q2/gpCp

Q2p/ gp 4■40.•■••■•■••■..-,

1

145

C

Q21g

where the vertical arrows are canonical injections. In particular, every element of U gives an automorphism of Q 2/Z 2. We also note that

U= c E GA + I Vc = Z 2 } .

(6.4.6)

{

Let us now prove a few useful lemmas.

SL 2(A)={x E LEMMA

GA

Put

I det (x)=1} .

6.15. For every open subgroup S of GA, one has SL2(A)=SL2(Q) - (S n SL 2(A)) = (S n SL2(A)) - SL2(Q) •

This is the simplest case of the "strong approximation theorem" for semi-simple algebraic groups. In the present case, it is merely a reformulation of Lemma 1.38. Let g = Z2, and let ce GA. Then we can find an element a of Ge such that gc=ga. By (6.4.6), we see that ac'EUG., (which proves the equality GA= U • GO. If C E 5L2(A), we have de“a) E det (UG.)n Qx = {±1}. Take an element c of G4, such that ge = g and det (e) = det (a). Then gc =gza, so that c • (ea) - ' belongs to U r\SL 2(A). This proves PROOF.

(6.4.7)

SL(A) = (U

na,(A)) • SL,(Q).

In view of (6.4.3), it is sufficient to prove our lemma in the special case S= UN. By virtue of (6.4.7), the question is reduced to showing that (6.4.8)

u na2 (A)cui NnsL2(0- sL2(z).

Let y e U r)SL 2(A). We can find an element p of M 2(Z) such that p-m vp mod N- M 2(Z) for all p. Then det (i9) -_a- 1 mod (N). By Lemma 1.38, there exists an element r of SL 2(Z) such that r -mp mod (N). Then vr i e UN nSL 2(A), hence (6.4.8), q. e. d. -

LEMMA 6.16.

The restriction of a to G A+ is surjective.

Since GA = GA+GQ, we have a(G, 4+)=a(G 4)= [clet (G A ), Q]. easy to see that det (G A)=Q;(4, hence our assertion. PROOF.

Let S be an open subgroup of GA+. (i) SG Q += GQ+S = { x E GA+ I a(x)=id. on ks),

LEMMA 6.17.

Then

It is

146

MODULAR FUNCTIONS OF HIGHER LEVEL

(ii) for y E G A+, one has SG Q +y= E G A+ I a(x)= o(y) on k 3}; the product SG Q,y can be taken in any order of S, G Q+, and y. By our definition of k s, a(s). id. on k s for sE S. Therefore it is sufficient to show that if a(x) = id. on k s for X E G, then x E SG Q+ and X E GQ + S. But the hypothesis implies that det (x) E Q X det(S), hence det (x). det (a) det(s) for some a E GQ and s E S. Then det (a) > 0, and det (a'xs- ')=1. By Lemma 6.15, cr- 'xs - i= IV with p E SL,(Q) and t E S, hence X = ap • ts E GQ+S, and similarly XE SGQ +. (ii) This is only an obvious generalization of (i). Indeed, from (i), we obtain PROOF.

(i)

SG Q+ Y=ySG Q+ =GQ+Sy. yG Q+S. {xE G A+

6r (X) = Cr ( 31 )

on ks)

Further, since k T =ks if T=y-1Sy, we have y - LSyG Q+ =SGQ+ by (i), so that SyG Q+ = ySGQ+. Similarly G Q4.yS= GQ+Sy.

6.18. Let S be an open subgroup of G A+. morphism of G A+/SG Q+ onto Gal (k3 #2), and LEMMA

Then a induces an iso-

[G A .,.: SGQ+J= [ks: Q]= [QI: Qx • det (S)].

This is an immediate consequence of Lemmas 6.16 and 6.17. LEMMA

6.19. G A+ = GQ+U = UGQ+.

This follows immediately from Lemma 6.17, since k u =Q. More directly, in the proof of Lemma 6.16, we have seen that GA= UGQ, hence GA.,. = GQ+U.

6.20. Prove : ) The normalizer of UN in G A+ is UQ:4, and UQ:4 = UQx ; (ii) If G* denotes the closure of G Q+ G.,,, then EXERCISE

G* =GQ+Gœ+SL2(A)=

E

A+

1 det (x) E Q x Q1+1

6.5. The action of U on

a

Let us now come back to the field ON defined in § 6.2. Therefore, if we put aNCa<W if M is a multiple of N.

We see easily that

= UN.1ON

then O is a Galois extension of O„ and C • O is the field of all modular functions of all levels. Further we see that Qab is the algebraic closure of Q in a, so that O and C are linearly disjoint over Q. Our next goal is the determination of Aut (0) (Th. 6.23). In this section, we study the part of Aut (0) obtained from the elements of U, and its relation with the substitution

THE ACTION OF U ON

6.5

a

147

2 a(z) for any a E GQ + For convenience, we understand that the suffix a in the notation fa indicates also an element of Q2 /Z 2, since fa depends only on the class of a mod Z2. (It is also convenient to put fa = j. But we shall not do this for fear of confusion.) .

PROPOSITION 6.21. For every u E U, one can define an element r(u) of Gal (ial) by fl`u ) = fa u for all a E Qt/Z 2, * O. Moreover, r(u) has the following properties : (1) The sequence 1–.{-±1} • G e.,–.0 (a/ 35 ,)--.1 is exact. (2) r(u)= a(u) on Qab• (3) hr(o= hor for all h E and rE SL2(Z).

a

For every U E U and every N, there exists an element a of M2(Z) n GQ+ such that u p a mod N. M2(Z9) for all p. Then au= aa for every a e Z 2 / Z 2. Therefore, by Th. 6.6, fa fa i defines an element of Gal (aN/ai), hence an element of Gal (am). Call it r(u). By (2) of Th. 6.6, we see that the restriction of r(u) to a N defines an exact sequence PROOF.

1—' { -±-1} • UN --0 U —. Gal (

(6.5.1)

N /)—'

1.

Therefore Ker (r)=()%,{ 1 1 } • UN= {± 1} • G.+ ; r is a continuous homomorphism of U to Gal (am ) ; and r(U) is dense in Gal (a/al). Since UIG.4 is compact, we obtain the assertion (1). To see (2), let u and a be as above. Define two elements c and c' of Q:4 by - -

det (a)

for PI N

1

for

Cp=

p N or p= co ,

and cc' =det (a). Then we have, by (3) of Th. 6.6,

«u)) = dkeNt /(Q co = [c', Q] = [det (a) -1 e, Q]=[c - ', Q] = [det

Q] = a(u)

on k N ,

so that r(u)= a(u) on kN for every N, hence (2). If u= rE SL 2(Z), we can take r as the above a, so that fcrui =far =fa o r by (6.1.3), hence (3). PROPOSITION 6.22. (1) For every a e G Q, and for every h e a, the function ho a belongs to a. (2) If a E GQ+, p E GQ + , U E U, y E U, and au= v, then (joa)r(u) =j4, and ( fa o a)") = fay ° p for every a E Q2 /Z2 , * O. Let a, be the field generated over Q by the functions h oa for all hE a and all a E G Q ,.. By Lemma 6.5, there exists a point 22 of it) such that g' g(z 0) defines an isomorphism of a, onto the subfield PROOF.

–

148

MODULAR FUNCTIONS OF HIGHER LEVEL

a)

Q(h(a(z o)) I a e G Q4., h E of C. Therefore, it is sufficient to prove the assertions corresponding to (1) and (2) on the field N. To prove (1) and (2), taking suitable scalar multiples of a and p instead of a and p, we may assume that a' 19' belong to M,(Z). Now, for every z E 0, put L(z) = Zz+ Z. Define c and E by

E: y2 = 4x 3—cx—c ,

c/(c-27) , j(z 0).

Take any isomorphism e of C/L(z o) to E, and put

t (a) _ e (a[zio])

e Q2/z2)

By Lemma 6.4, fa(z0)= hi(t(a)). To simplify our notation, put a= al , and u4= a1(z0) for i=1, 2. Then there exist p, E Cx such that

[ 2114 =

W( 1 ]

(1= 1, 2) ,

and multiplication by ta t defines an isogeny of C/L(z o) to C L(w f). and Ei, for i=1, 2, by

Ei : 312 = 4x 3 c,x c,, —

Let

—

p= a2,

Define ct

c1 /(c1 27)=j(wd -

be an isomorphism of C I L(w) to Ei, and put s ,(a) = ,(a[u;:.

])

E Q 2 /Z 2 ;

= 1, 2) .

Then there exists an isogeny 2, of E onto E, such that the following diagram is commutative.

11i

1

• E,

Then 2,(t(a)) = 2,(e(a[ z1°D)= )2,(a[ zi° ]p,)=72,(aaTi lvi'D = s,(aaTi) for every a E Q 2/.Z 2. Therefore Ker (A i) = t(Z2ai /Z2). Now consider the automorphism a of Q(h(z.) I h over Q(c) such that fa(zor=fa.(zo) for all a E Q 2 /Z 2 , 0 (this corresponds to r(u)). Extend it to an automorphism of C, and denote it again by a. Then we see that E'=E, and by (4.5.3), t(a)° = ±t(au), since fa(z0). hIE (t(a)). It follows that

Ker (2r) = Ker (2,)° = t(Z 2au/ Z 2). t(Z 2 v1 e / Z 2)= t(Z 2 13/ Z 2)= Ker (22) .

THE STRUCTURE OF Aut

6.6

(a)

149

Therefore Er is isomorphic to E2, so that j(wi)' =i(w2), hence cf =c2 , and E,' = E2. Both 2f and 22 are isogenies of E onto E2 with the same kernel, so that 2`;` =e22 with an automorphism s of E2. Since j(;) is transcendental, E, Ez , E2 have no complex multiplication, hence s = +1, so that 2r = ± A,. Therefore, for every a E Q2/Z 2, we have

s,(a

= i(t(a)))° = 2r(t(a)a) = ±22 (±4au))= ±22(t(au)) = ± s2(au9- ') .

Let b = act'. Then aup - ' = bauf1 -1 = bu. (In fact, let a be an element of Q2 which represents a, and 6 = da- '. Then aupP -i = 6aupP - = 6vp, which shows aup - ' = bv.) Since a aa- is a surjective endomorphism of Q2/ .Z 2, we .obtain s 1 (b).1 = s2(bv) for every b e Q2/ Z 2. By Lemma 6.4, we have, for every E Q2/Z2,

f b(wi)° = hiE,(si(b))a = Iz IE2 (s2(b0)= ft, u(w2) .

Thus we have proved <6.5.2)

j (a(z

=j(49 (z ) )

f b(a(z

= f tiv((zo))

(1) E Q2/Z2) •

This applies to any automorphism a of C over Q(j(z o)) such that f a(z 0)6 = fazo). Then by (1) of Suppose especially that a is the identity on Q(h(z0)1 h e Prop. 6.21, u e { +1} - G.+, and we can apply the formula (6.5.2) to the case a = 13, v= autr- ' E s 4 11 • G.+. Then we see that a leaves the elements j(a(z0)) and f b (a(z 0)) invariant. It follows that these elements belong to the field By virtue of our choice of z 0, this proves (1) of our proposition. Q(h(zo) I h E Then the assertion (2) follows from (6.5.2).

a).

-

a).

6.6. The structure of Aut () We shall now define a homomorphism r:

A+

Aut a •

By Lemma 6.19, we have GA+ = UGQ+=GQ+U. For u e U, we define r(u) to be the same as the element r(u) of Gal (W,) defined in Prop. 6.21. As for a E GQ ± , we define r(a) by (6.6.1)

h r(a) =hoa

for all h e J .

Obviously this defines a homomorphism of GQ , into Aut (a). Thus the symbol r is defined on SL2(Z)= U nGQ + in two different ways, but, both definitions coincide by virtue of (3) of Prop. 6.21. Now, for x= ua E GA+ with 21 E U and a e G Q+ , we put r(x) = r(u)r(a) so that Jr(x) = 0 a

g(r) =f„0 a .

150

MODULAR FUNCTIONS OF HIGHER LEVEL

If x = u'a' is another expression with u' E 1.1 and a' e G 9+, then 24 -1 2e=aa' -1 Therefore, putting 3 = u -1 u', we have eSL,(Z). r(u0r(d) = r(ucl)r(5 - ia). r(u)r(d)r(a)'r(a)=r(u)r(a) ,

since r is multiplicative on U and on GQ+. Thus the symbol r(x) is well defined independently of the choice of u and a. We have to show that r is actually a homomorphism. To see this, let x = ua and y = vp with u e U, v e U, a e G Q .„ p e GQ+. Since GA+ = UG Q+ , there exist elements w E U and 7- e G e+ such that av =wr. By our definition, r(xy)=r(uw)r(n9)= r(u)r(w)r(r)r(P)

and

r(x)y(y)= r(u)r(a)r(v)r(P) .

Therefore it is sufficient to show that r(w)r(r)= r(a)r(u). But this is nothing but the assertion (2) of Prop. 6.22. Since both r(a) and c(a) are trivial on Qab if a e Go+, we obtain, from (2) of Prop. 6.21,

(6.6.2)

r(x)= c(x) on

Qab

for every x E

GA+

•

Let us now prove

(6.6.3)

4:1' UN

= {x e G4.1.

r(X)=

id. on 5N}.

The inclusion c is obvious in view of (6.5.1). Let x e GA+, and suppose that r(x) = id. on 5N. By (6.6.2), c(x) = id. on k N. Therefore, by Lemma 6.17, x= ua with u E UN and a E G e+ . Then r(a) = id. on 5 N, hence a e Q'T ph so that X E (rU N, which completes the proof of (6.6.3). From (6.6.3), we obtain

Ker (r)= mi 42' UN = the closure of Cra., + =QxG., (since Qxa:,+ is closed in Q. The relation (6.6.3) shows also that r is continuous, and moreover, r induces an open map of GA+/QxG.., to r(G A+ ). Therefore r induces a topological isomorphism of G A SWG...., onto r(GA+). By (1) of Prop. 6.21, r(U) = Gal (/1). Since Gal (/N is open in Aut (a), it follows that r(G A+ ) is open, and hence closed, in Aut (W.") Now we have one of the main theorems of our theory : THEOREM 6.23.

The sequence

r

1—e (rGc., + --• G. —• Aut (a) —1 is exact, so that Aut (a) is isomorphic to G A+ KrG, as a topological group. 13) That T(G 4 is closed can be shown also as follows. Since r(G A+ is homeomorphic to G A+ /QxG. +, it is locally compact, and hence closed in Aut (g), by virtue of Prop. 1.4. )

)

151

THE STRUCTURE OF Aut (g)

6.6

Since r(G A .,) is closed in Aut (g), it is sufficient to show that r(G A .,) is dense in Aut (g). Let C e Aut (g). By Lemma 6.16, there exists an element y of G A+ such that a(y)=C on (r:jab. Put r = C • r( y) -1. Then r is the identity map on Q co. Since a and C are linearly disjoint over (li b, we can extend r to an automorphism of Cg over C, which we denote again by r. Take and fix any positive integer N> 2. We can find two positive integers M and M' such that N<M <M', 5.V 1 C.r, and glirc ,,,.. Then we have cg,v cenc ca,v„ so that there exists a subgroup Li of F N containing Fm . such that en is the field of all modular functions with respect to J. Let 0* be the union of 0 and the cusps of /',. Put V =0*/r m and V'=0*/J, and denote by so (resp. so') the projection map of 0* to V (resp. V/). Then V and V' are compact Riemann surfaces, and CM (resp. can be identified with the field C(V) (resp. C(VO) of all meromorphic functions on V (resp. VO, through the map C(V) j".—..f09) (resp. C(VO f .fov). Since r is an isomorphism of Cgm onto Cgs,1 over C, there exists a biregular isomorphism ri of V' onto V such that (foso): =to)? ow' for every f e C(V). Put Vo = p(0), and V,;= w'(0). We are going to show that 71(VD= Vo. Let p e V and assume that 71.(p) OE v o, i. e., 0) =c(s) with a cusp s of T if. If y is the discrete valuation of cg; corresponding to the point p, then y is unramified in Cg, since p=so'(z) with a point z on 0 which is not elliptic. (Here observe that neither "M nor J has elliptic elements, since N> 2.) Now define a valuation y* of CM by v*(h)=L(h 7) for he Cgm. Since r is an automorphism of eg, v* must be unramified in C. On the other hand, y* is the discrete valuation of Cgm corresponding to the point 77(p) = so(s). Since s is a cusp, v* is ramified in C. (In fact, if L is a multiple of M, the ramification index of v* in C& is L/M, see Prop. 1.37 and § 1.6.) Thus we get a contradiction, hence 77(p) must be contained in V o. Similarly we can prove that 77- ' maps Vo into VL hence 7i gives a biregular isomorphism of V,Ç onto Vo. Since 1/=0/ 4 , V 0 =0/r m , and neither J nor has elliptic elements, we can find an element p of SL,(R) such that v 0 19= rioso', and 49 - 1({ ± 11 .rm),9 = { ± 1 . J. Observe that spans MA) over Q, for [1 0 dj, [1 d 01 ], [d2 d-f- 1 di ], every positive integer d. (In fact, four elements r1 d 1 Ld d2±1] of I'd are linearly independent over Q.) Therefore we have PROOF.

cap

rm

}

rd

A-1 M2(Q)19 =M2(Q), so that x —. I3 -1 xp is an automorphism of M2(Q). By a well known theorem, there exists an element a of GL 2(Q) such that p - xxiS = ceixa for all x E M2(Q). Then ap - lx= xap - 1 for all x E M2(Q), SO that a 19 -2 = c- 12 with C E Rx. It follows that a= cp, det (a). c2 >0, and so oa = so o i9 =7/ 050/, hence (f0TY:=foriose=foço oa for every f e C(V), i. e., iir = h o a for every h E C if. Therefore we have 7 = r(a) on gm, so that C = r - r( y) = r(ay) on am-

MODULAR FUNCTIONS OF HIGHER LEVEL

152

Since M can be taken arbitrarily large, this shows that r(G A+) is dense in Aut(), and completes the proof. There is an obvious analogy between the above theorem and the fundamental exact sequence (5.2.1) of class field theory. Actually they are not only analogous, but also closely connected with each other by a certain explicit formula, which describes the behavior of the values of the functions of at special points belonging to imaginary quadratic fields. We shall discuss this in §6.8.

6.24. Let be the subfield of J generated over Q by the functions j o a for all a e G Q+ . Prove : (i) The subgroup of G A+ corresponding to (in the sense of Prop. 6.11) is Q- ; (i i) Qabr1' is the composite of all quadratic extensions of Q; (iii) The subgroup of G A+ corresponding to I det (x) G QxQ:.+} ; (iv) Every element of Aut (W) is exQabW is {X e tensible to an element of Aut (); (v) Aut (al is (canonically) isomorphic to G A+/Q:4G.+ (cf. Prop. 6.8). EXERCISE

6.25. Show that every automorphism of 5 N extensible to an element of Aut (a) must belong to Gal ( N/ , i. e., it is the restriction of an other than the element of r(U) to N. Especially, no automorphism of identity map is extensible to an automorphism of EXERCISE

)

EXERCISE

6.26.

Let

be the subfield of

consisting of the elements

(with d e Q). Prove : (i) = Qabo; invariant under r(x) for all x= rLOi °1 dJ U is generated over Q by the functions j(N2) and fa ,,(-)Qab= Q; is the field of all modular with a = (1/N, 0) for all positive integers N; (iv) functions (of any level) with rational Fourier coefficients at co (with respect to e27:1" for some N) (cf. Prop. 6.9).

6.7.

The canonical system of models of f \* for all congruence subgroups r of GL2(Q)

Before discussing the main topic of this section, let us first introduce the notion of a model of rw, where r is a Fuchsian group of the first kind, and D* is the union of D and the cusps of T. (T may be a subgroup of SL,(R), SL2(R)/{±1} , G.+, or Gœ +/Rx.) Since rvt)* is a compact Riemann surface, as shown in § 1.5, there exists a projective non-singular algebraic curve V, defined over (a subfield of)C, biregularly isomorphic to F\ *. It is often convenient to specify a r-invariant holomorphic map p of D* to V which gives a biregular isomorphism of r\o* to V. If V and io are in that situation, we call (V, 9) a model of r\tro. For example, if r=suz) and

THE CANONICAL SYSTEM OF MODELS OF rw

6.7

153

PI denotes the projective line, (P', j) is a model of

Coming back to the general case, let ri be another Fuchsian group of the first kind, 0*' the union of 0 and the cusps of ri, and (V', V) a model of rAtv". Suppose that ara - 'c ri with an element a of G.,. Then, as is shown in § 2.1, we can define a rational map T of V to V' by T(p(z)). se(a(z)), i. e., by the following commutative diagram : a

0*

This includes, as special cases, the following two types of maps : CASE a: a = 1, hence CASE b:

ara-l=ri.

rcri.

Then T is the usual projection map.

Then T is a biregular isomorphism of V to V'.

Now the purpose of this section is to discuss the following question, which is actually somewhat too naive a problem setting, though, so that a modification will be made afterwards. To any Fuchsian group r which is contained in G Q , and contains r N for some N, associate, once for all, a model (Vr, Tr ) of r\o*, and an algebraic number field k r in such a way that the following conditions are satisfied: (1) Vr is defined over k r . (2) If a e GQ+ is such that ara - ' C z/, then lej C k r , and the rational map T of V r to V4 defined by Tov r =soj oa is rational over k r. Here and henceforth 0* means of course Ol../QU{co}. Suppose we could find such a system of (V 1, çor) and kr . Then consider a field fo çor I f E k 1( V 1)} where k 1(V 1) denotes the field of functions on V r rational over k r , see Appendix No 4. It is natural to assume that azi=aN if zi=rN. By our assumption, r contains rN for some N. In view of the condition (2), we see that k r c k N , and 51C 5N. Therefore ar is a subfield of a. Then, (assuming that a is a Galois extension of ar,) ar corresponds to an open compact subgroup of Aut (a) by Prop. 6.12. Now Aut is isomorphic to G A ,./WG., Therefore, it seems reasonable to consider, instead of the family of r, the family of all open compact subgroups of G A+ /QxG. +, or the subgroups of GA+ corresponding to them. Thus we are led to consider the set 2 of all open subgroups S of GA+ containing (2"Gœ+ such that S/QxG., is compact. We see easily that L has

(a)

-

154

MODULAR FUNCTIONS OF HIGHER LEVEL

the following properties:

(6.7.1) If S e 2 and Te 2, then S and T are commensurable, and SnTE.Z. (6.7.2) If SE 2 and

XE G A ,

then xSx -I

E

Z.

Put, for each SEZ,

rs =SnGQ÷ , &= {h E By Prop. 6.12, and

(6.7.3)

I hr(z) = h for all x E S).

s is finitely generated over Q,

is a Galois extension of

s,

S= {x e G A+ 1 r(x)= id. on 15}, i. e., r(S) = Gal (15/.$) •

For example, if S = Q 4 UN, we have rs =(Q4uN)r) GQ+ = Q.(uN n GQ+) = QrrN, so that the group Is (or rather I' s/Q1, as a transformation group on .V, is the same as T N. Moreover, 35 s=-N, by (6.6.3) and Prop. 6.11. In general, we have the following PROPOSITION 6.27. For any S E 2, rs is commensurable with (r1" 1 , (so that I s/Q- is a Fuchsian group of the first kind commensurable with 1,/{-1-1 } ), and Ci s is the field of all automorphic frInctions with respect to Is. Furthermore, k s is algebraically closed in s , where k s is as in § 6.4, p. 144. PROOF. By (6.4.3), S contains CI* UN for some N. Put T= Cr UN. By (6.7.1), we have [S: T]
Conversely, by Lemma 6.17 and (6.7.3), we obtain R CS (1(G Q -,T)=(S nG,H.)T = r sT ,

so that

rsT= R.

Therefore krs corresponds to

rsT,

hence

Mr : kr.si=[TsT: T]= [Ts : rr ] . Since C and & are linearly disjoint over kr, we have

[C&: C& ] = E& : krV 3 3 = Efts : ri].

THE CANONICAL SYSTEM OF MODELS OF

6.7

r\o.

155

Let R s be the field of all automorphic functions with respect to Is. Then c s C Os, and C& = gnr, hence EC& : Ins] = [Ur : Vs] = Ers :

rT3=cc&:Cs3.

This proves that alls =C s. REMARK 6.28. It can happen that S#T even if Take for example

S=Qx -{xeUlxp m

[ oa01] mod N- A/I,(Zp)

r 01 T=Q.•{xe Ulxp=1_0 dJ mod N. A/12(4)

rs=rr and k s =k r . (ae 4)} (d E

4)1.

Then rs=r r =QxrN, ks=kr=Q, but S#T if N> 2. Nevertheless, we have: LEMMA 6.29. Let SEZ, Te Z.

If r s =r r , ks=k r , and ScT, then S=T.

PROOF. By (i) of Lemma 6.17, the assumption ks =k r implies GQ+S=G Q+71, so that if SC T, we have TC(G Q÷ nT)S=T rS. Therefore rr = r s C S furnishes the opposite inclusion TCS, so that T = S. PROPOSITION 630. Let r' be a discrete subgroup of G oe,+/Rx commensurable with Q.' r igr, and containing rN for some N. Then I"' r s/Q* for some S E Z. PROOF. Let p be an element of G, which represents an element of r/, and let r/f=r1n pr,p-1. Since Er1 : 1 if3
(6.7.4) (6.7.5)

V s is defined over ks

={ f (Dios f

ks( Vs)}

•

156

MODULAR FUNCTIONS OF HIGHER LEVEL

We fix (V s, v s) for each S e 2 once for all. Let S e 2, T e 2, and xe G„• Suppose that xS.t -' c T. Then r(x) gives an isomorphism of f T to a subfield of Replacing J 3 and /.. by k s(V s) and k(V), we obtain an isomorphism r'(x) of k(V) into k 3(V 3) such that ft. ') osos =-- (fa sor)' ) for fe kr (V T). Therefore by Appendix N. 6, we find a unique biregular morphism J3(x) of V s to V 4..(.t) such that f'cirs(4=7 (z) for f E k r(V T), i. e.,

as.

(61.6)

f

0 iTs(x)0

vs= ( f 0 çor)?(x)

for

J E kT(VT).

It can easily be verified that frs(x) has the following properties : (6.7.7)

J3(x) is rational over k3 ;

(63-8)

irs(x) °(v) oisR( Y) = ITR (xY) ;

(6.7.9) iss(x) = id. if

X

e S;

(6.7.10) irs(a)[Vs(z)] = VT(a(z)) if a s G Q .4. and T=aSa - '.

Especially, if S C T, j r3 (1) is defined, and irs(1)[ços(z)] = Sor(z) •

Therefore i r3 (1) corresponds to the natural projection map of rs\•ryic to If xSx- ' = T, both frs(x) and jsr(x-1) are meaningful, and Jsr (x- i)°cr'of Ts(x)= id., so that frs(x) is a biregular isomorphism of Vs to lq(z). In the most general situation xSx' c T, we have jr s(X) =.-- Jr RaY") C jRs(X)

= jr p(x)ofp3 (1)

(R=xSx'), (P=x - iTx),

so that J3(x) is a composed map of a biregular isomorphism and a projection map, in either order. As an illustration, fix a positive integer N, and let us consider a member S of .S defined as follows : S=QKU' , U'

= { x e Ulxp e U;

u;={[

for all finite p}, mod NZ} .

Put a = [(1)V °1 ]. Then we see easily that r s=Qx(U' n GO = Qx ro(N ), where r0(N) is as in (1.6.5), and Qx • det (S)= Qx • det (U)= (4, so that ks= QFurther we have CtU N cS=QxUnQxa'Ua, so that the functions j and jo a are contained in the field s , and sCaN. Observe that j(a(2))=j(Nz). By

6.8

AN EXPLICIT RECIPROCITY-LAW AT THE FIXED POINTS

157

Prop. 6.27 and Prop. 2.10, we have Cs=CC/(z), i(Nz)). Since QU(z), j(Nz)) and C is linearly disjoint with aS over ks=Q, we obtain s= (?(..1(z), l(Nz)) Consider, as another example, the groups S and T of Remark 6.28. Since [17. Ts = = WrN, we have C&= C . = ■■/ by Prop. 6.27. Thus V s and V T are models of [1N\.*, defined over Q, but an obvious biregular map Y: VT —0 1 defined by Yoso r =sos is not rational over Q if N> 2. It can be shown that Y is defined over Q(CN ±C7,19, where CN = e 2ni/N.

6.8. An explicit reciprocity-law at the fixed points of GQ, on Let K be an imaginary quadratic field, and q a normalized embedding of K into MA) in the sense of § 4.4, and z the fixed point of q(Kx) on .D (see Prop. 4.6 and Prop. 4.7). In § 4.4, we have shown that every non-trivial fixed point of an element of G Q, on ti is obtained as such a point z. The purpose of this section is to study the nature of the values of functions in the field at z. First we observe that the embedding q defines a continuous homomorphism of K; into G„; we denote it again by q. THEOREM 6.31. The symbols K, q, and z being as above, the following assertions hold. defined and finite at z, the value h(z) belongs to Katt, (i) For every h e and h ( z) [3 . 1rj = for every s K;. (ii) For any S one has

E S,

the point ç 3(z) is rational over Kato and for every seK, s(z) c"n = 5r(q(s)-1 )(ço7(z))

where T =q(s)Sq(s) -1.

One may notice that the relation (i) explains the deep arithmetic meaning of the map r, exactly similar to the fact that the canonical map of K A' to Gal (Ka a K) is defined locally by the Frobenius automorphisms. Thus our two theorems 6.23 and 6.31 provide an analogue of class field theory for the field B; which is of Kroneckerian dimension 2. It should also be observed that the relations (i) and (ii) are generalizations of (5.4.1). PROOF. As is seen in § 4.4, z belongs to K. Define a Q-linear isomorphism q(te k = tz : Q 2 ---.K by cz (a)= a[n for a E Q2 (row vector !). Since [ I] for ye K" (see (4.4.5)), the diagram

158

MODULAR FUNCTIONS OF HIGHER LEVEL

I

Q2

• q(P)

t,

Q2

is commutative. If we put at = Zz - F Z, then r, induces an isomorphism (12/ Z 2 onto K/a., which we also denote by cz. Let e be an isomorphism Cla, to an elliptic curve E ee. Let a be an element of Aut (CIK), and s element of K ; such that a=[s, KJ on Kati . Take an isomorphism e' C / ' a, onto E° as in Th. 5.4 for these a and s, so that

(1)

e(x)° = e/(s- lx)

(xe

of of an of

K/a z) .

By Lemma 6.19, we can find an element y of U and an element a of G q ^ so that q(s) - = y a - Then Z 2q(s)- = Z 2 cr -1 . Put w=a - '(z). Then we find an element A of K' such that a-i [ 1]

[2w 2

Observe that

a: = cz(r),

sa z=

c,(Z 2 q(s) - ') = r,(Z 2a - ') = A c(Z 2) = itav,

Therefore we have a commutative diagram.

E

"

with a suitable choice of an isomorphism E. Let a e (12 / V, and u = By Lemma 6.4, we have (2)

fl,(z)= hiE(E(u))

(i=1, 2, 3),

and s"u =5 • c 2 (a) = r,(a - q(s) - ')= ,(aya -') = 2.c(ay) (mod s - 'a, = 2a„,). fore we have ei(s - iu)=e''(( au(ay)), so that hi4(eqs - '24))=f(w)

(1= 1, 2,3)

by Lemma 6.4, hence from (1) and (2), we obtain

(3)

= hiz.,,(E(u)°) Eiv (a - (z))

(1 = 1, 2, 3) .

Also we have

(4)

(z)' =

=j(w) =i (a -1(2)) •

There-

6.8

AN EXPLICIT RECIPROCITY-LAW AT THE FIXED POINTS

Now fix a positive integer N> 2, and let N- lZ 2/Z 2 Further let V'N be the locus of

40/61)= Ci f ga), f KO,

,

,

—

159

{0}={a,b,.--}.

, f , f g(3),

in the affine space of dimension 3(111 2 1)+1, where g denotes the variable on If P= QXUN, VP is birationally equivalent to ViN , and there exists a birational map X of V, to 1PN, rational over k,= k N, such that Xosop=ie. Since Vp is non-singular. X is defined at every point of sop(); X is not biregular, but we see that X is one-to-one in the following sense : -

(5) If z, e

z2 e .0,

and go'(zi)=40'(z2) , then pp(z i)= Pp(z2)-

In fact, if so'(z,)= p'(;), we have Az 2)=1(z 2), so that there exists an element T of 1', such that

r(z 2)= z2. Put L 1 = Zz,+Z, and e(a)=a[ zI] for a e W.

Denote by the same letter e the map of 10/Z2 onto CIL, obtained from e. Let e i be an isomorphism of CIL, to an elliptic curve E i ee. If x=c(a), a e N -'Z2/Z 2 —{0}, and y= c(a), then we have, by (6.1.3) and Lemma 6.4,

hiE,(ei(x)) =f(z 1 ) = f(z2) =flx(r(zi)) = f(z) = 41(e1(Y))

(i= 1, 2, 3) .

Therefore, by (4.5.3), 6.(ei(x))= fi(y) with an automorphism ed of E1. If E, e e l , we have Ca = +1, and caa=ar for all a e N-'Z 2/Z2 —{0}. By Lemma 6.2, we have r e T N • {±- 11, so that iop(z2)= sop(r(zi)) = Wp(zi), which proves (5) in the case E,Eei . Suppose that E, e e2 then L1 is a fractional ideal in K Q(N/ —1), and sa is identified with (multiplication by) one of the four units +1, +-V=1. For e e { +1, +-V=1 } , we can define an element e* of r, by ;

zi = ez, so that c(bc*) = E • e(b) for b e (r/Z 2 . Then we have ar = act L 1 J L e for every a e N-JZ2/Z2 —{ O}. Now we need the following LEMMA 6.32. Let N be a positive integer >2, r an element of I's , z, an elliptic point of r„ and d = {6 e r,ia(zi)=z,}• Suppose that, for every u e Z 2, there exists an element 3. of d such that urmu6„ mod (N). Then r e Ar y . Since every elliptic point of r, is r,•equivalent to N/=-1 or e21", it is sufficient to prove our assertion in the cases z 1 = -../-71 and z1 = e1. If z1= e2'2 ", we have, by the result of §1.4, PROOF.

={±-12 , -421 11 -±[1 11 01 ]}. r1 0 1 LP 4 J mod (N) with some integers p and q. Since det (ro = 1, we have q a- 1 mod (N), 10 so that Is [ mod (N). On the other hand we have (p, a- cr' c6 P1 Put b = (1, 0), c = (0, 1), and la = r6v.

Then bid

b mod (N), hence

160

MODULAR FUNCTIONS OF HIGHER LEVEL

mod (N) for some 3

. r o 1 1. L —1 1 _I

E

4.

Looking at the elements of 4, we see that 3 =1 2 or

i 1 But if 3=[ _ 0 1 I we have

r,...[_ 11 io ]

mod (N), so that

(b+c)r' m- (0, 1)* (b+c)c mod (N) for any element 6 of 4, a contradiction. -.77--1 can be Therefore 3= 1 2 , so that r, e rN, hence r e r N zl. The case z,=treated in a similar and simpler way. Applying this lemma to the present situation, we see that ra e Pis, for some 3 of re, such that (3(z 1). z,. Then i.op(z2)= wp(r3(zi)) =9p(zi), which proves (5) in the case E E 6.2. The remaining case E e 6.3 can be treated by the same argument, on account of Lemma 6.32. Coming back to the original z, s, a, a, y and P=QxU N, we see that a(y). a(q(s) - ')=[s, K]=a on Q. Now A! .--, fl y defines an automorphism of N over I which induces a birational map f of V'Ar to VV), which is obviously defined everywhere on IPA,, and satisfies fo X= XG0Jpp(y). From (3) and (4) we obtain ç' (z) . f[so'(a- (z))], so that X ° [wp(z)°] =PCX[9p(a - '(z))33 = )( G C/pp( Y)E9p(cc l (z))3] By (5), we have 9 p(Z)° = IPP( 3)tç ° 14a -1 W)]. obtain

Putting R = aPa-l = q(s)Pq(s)- ', we

WP(zY =ipp(Y)[./pR(a - 9[WR(z)]i

=JpR(9(s)- 9[WR(43 •

This formula holds for P=QmUN with any N> 2. Now for every S e 2, we can find a positive integer N> 2 such that QxU Nc S. Therefore, putting P=QxU N and T.q(s)Sq(s) - ', we have (6)

9.3(z) ° =f.sp(1) 6 E9e(z)° 3 =./.9p(1)°[./pR(q(s) - 9[9R(4 3 ]

=./.3R(9(s) - E9R(z)3 =jsr(9(s) - X/TR(1)E9R(413 =./.97-(q(s) -1 )[9,(z)] . ')

Let h be an element of defined and finite at z. Then h=foços for some S E 2 and some function f on Vs, which is rational over k s, and defined at 9s(z). Therefore, by (6.7.6), (7)

h(z)" = fa(9 s(z)°)=f°(J sT (q(s) - 1 )[9r(z)]) = f '"`"-1 )(z) .

In both formulas (6) and (7), we observe that Ws(z)° and h(z)° depend only on s, 1. e., they depend only on the restriction of a to Kai,. Therefore 9s(z) and h(z) are rational over Ka,, so that we can replace a by [ s, K] in (6) and (7), and thus obtain (ii) and (i) of our theorem. PROPOSITION 6.33.

The notation being as in Th. 6.31, let S

E

Z, and

W={sEK:4 1q(s)ES}. Then IC- k 5(95(z)) is the subfield of K ai, corresponding to the subgroup K''W of K.

AN EXPLICIT RECIPROCITY-LAW AT THE FIXED POINTS

6.8

161

PROOF. Let s E K, and r =[s, K ] . Then r = a(q(s) -') on Qab. If s E W, we see that 7r= id. on k s. By (6.7.9) and (ii) of Th. 6.31, we have w s(zr =9 s(z), Conversely suppose that r= id. on ks(ç0 s(z)). so that 7r= id. on ksWs(z)). By (i) of Lemma 6.17, we have q(s) - ' = ta with tE S and a E G Q+ . Putting T= q(s)Sq(s) -i , by (ii) of Th. 6.31, and (6.7.10), we have

Sos(z)= W.s(z)' =Jsr(ta)[97-(4]=9.s(a(z)), so that z =ra(z) with r E rs. By (4.4.4), ra =q(b) with b q(bs) - i=tri E S, so that sElfx W, which completes the proof.

E

Kx.

Then

We now specialize the formula (i) of Th. 6.31 by taking h to be a more explicitly given function. First we take the function LI as h. Although the result in this case is essentially the same as (3) of the proof of Th. 6.31, we formulate it in a somewhat different way. PROPOSITION 6.34. Let a be a fractional ideal in K, and fat„ (1, 2) be a basis of a over Z, with zo = (.01 /(02 E el. Further let N be a positive integer, CN the maximal ray class field over K modulo N, and b a fractional ideal in K, prime to N. Then, for every a E N-1 Z 2, EE Z 2, the value f(z 0) belongs to C A,. Moreover, if (.i4

0.= / CN/K\ l= Zco+Zco'l , k, b ) ' ab with $

E

[ w' (0 2

] = E C cE4 1

G e+ , then one has igz or = i 11, (4

(i= 1, 2, 3) ,

where b is an element of N'Z' such that bm--. ae mod Z; for all prime factors p of N.

PROOF. From Prop. 6.33, we see that h(z0) E C. for every h E IJ N, which proves the first assertion. To prove the second one, let s be an element of K ; such that soK = b. We can take s so that sp = 1 for all prime factors p of N. Then Es, KJ =a on C. Define an embedding q: K—M,(Q) by [

P'' ]= q(p)[ (1'1 ] for p E K. For every rational prime p, we have co, tew2 , , r co s - t ] = Z;q(s;, 1)[ W1 ] = ZO(s - )e[ w i 91- oh 1 I ' .)! f.d.]• P ,L ,Z2 i = (ab) = ap si,'' = ZJI co,sp 02 ce2 -

(Each term is a lattice in K p = KC& Qp = Qp ce 1 -FQ,c02 ; the p-component s p of s is an element of Kp*.) Therefore Z;= 4,q(siDe for all p, so that q(s 1) e =t with an element t of U. By (i) of Th. 6.31, we have

igzor = (ir" ) (eAzoD= (f Dr")(4 •

162

MODULAR FUNCTIONS OF HIGHER LEVEL

Since sp = 1 for all p dividing N, we see that at m ae mod Z1, for all p dividing N. Since b is prime to N, we see that e E GL,(Z p) for all such p. Therefore we have ( A)" = f with b as described above. This completes the proof. REMARK. If b is an integral ideal, we have a c ab - ', so that In this case we can put b = ae, so that the formula becomes

(6.8.1)

Ma).

fl(zor =fWe -i(zo)) •

Next we consider a modular function which is obtained from automorphic forms with rational Fourier coefficients : PROPOSITION 6.35. Let g, and g, be automorphic forms of weight k with respect to other than 0, and a an element of GQ+. Put S=Q 4 (a - 'LlanU), and h=(g1 I[a]k)/g2, where

r=suz),

gi f Eajk = det (a) k/2g-1 (a(z))j(a, z) -k (see g 2.1), and U is as in § 6.4. Suppose that the Fourier expansions of g, and g2 with respect to e'''' have rational coefficients. Then he s. Note that the weight k must be even, since there is no non-zero automorphic form of an odd weight with respect to I' (see g 2.1).

r

PROOF. We can find two elements r and (3 of so that a=rficl, r rm 01 d9=L 0 r jwith reQ, meZ. Then we have (g1 1[19] k)/g2 = ho 3', and

Qx( 18- 'U19 n ti)= 5.35'. Therefore it is sufficient to prove our assertion for p. r rm 01 In other words, we may assume that a= Then a - Tanr=r o (m), L 0 r and h(z)= mki 2g1(mz)Ig 2 (z). Therefore h is invariant under I' o(m), and has rational Fourier coefficients, so that h belongs to the field Wm = Q(j, j(mz), fa) considered in (2) of Prop. 6.9. By virtue of that proposition, and through the isomorphism of U/U„, onto GL 2(Z/mZ), we see that =T with 10 T =CP - fx E U I xp =[ 13 d ] mod m • M2 (Zp) (d E Z;)} . Now it can easily be verified that a - ilia nu c To(m) • T. Since h is invariant under both T o(m) and T, we have he &, q. e. d. PROPOSITION 6.36. Let g1 ,g2 ,a, and h be as in Prop. 6.35, and N, a, b, coi, (02, z o, a, and CN be as in Prop. 6.34. Suppose that det(a)= N and ace M,(Z). Then h(z o)e CN. Moreover there exists an element 72 of G Q+ satisfying the following condition:

(*) 72[ (D1 ] is a basis of ab i-over Z, and ava - lE GL,(Z p) for all p dividing N. 2

6.9

THE ACTION OF AN ELEMENT WITH NEGATIVE DETERMINANT

163

If 72 satisfies (*), then h(z 0 )g=h(7)(z o)).

PROOF. Put S=Qx(a - lUan Li). By Prop. 6.35, h E s . Observe that UN CCC'UanU, hence he N . Therefore h(z 0) E as is remarked at the beginning of the proof of Prop. 6.34. Take co'„ co;,, s, and t as in Prop. 6.34 and its proof. Put L= P. Since t E U, we see that L/ Lat is isomorphic to Laa, so that Lat = Lar for some TE I" by Lemma 3.12. Then art"a"E U.

c„,

Put ;2= g- i. Then 72[ w' ]= r[ (14,1 Since tp = $ for all p dividing N, we (02 co: have ava"E GL,(Z p) for all such p. Thus we have shown the existence of 72 satisfying (*). Next let ri be any element satisfying (*). Take 71 -1 as E in the proof of Prop. 6.34. Then we have q(s - 977 -1 = t with t E U, as is proved there. Since sp = 1 for all p dividing N, we have 72 - '=t,, for all such p, so that at pa' E GL,(Z p). The last inclusion holds also for all p not dividing N, since det(a)=N and a E M 2(Z). Therefore ata - ' E U, hence t E a'Ua nu cs. By (i) of Th. 6.31, we have h(z or = hra 0(4)= hr")(72(z 0))= since h E

s.

This completes the proof.

EXERCISE 6.37. Generalize Propositions 6.34 and 6.36 to the case where the order of a= Zah+Zw, is not necessarily maximal (cf. Prop. 4.11, (5.4.2)).

6.9. The action of an element of

GQ with negative determinant

For every x E GA, let xo denote its projection to Go. If a E G0+ , the element r(a)=r(a,,) is defined by hr(a) = ho a for h E 5. If a E G G, and det(a)
$'(u)=M-3. Put Therefore

o=[ 011 Then

4n =[ 21 ], 6(2)=112 E 0, and T.= Z+ Z2.

164

MODULAR FUNCTIONS OF HIGHER LEVEL

(1)

(f)= (1/ 2) = (ô(2))

(z)=

For every a E Q 2 we have

e/(42[12- ]) = 4a( ]) so that, by Lemma 6.4,

(2)

]))= f7TY.

fl(3(2))= iz€2(e/(a[; ])) =

Since (3 0 E U, we obtain, from (1),

P(4°) (4 =/(z) = TOW , which, together with (2), implies

h0(z) = h(c3(2))

for all

hE .

If a and a0 are as in our theorem, we have cur' E G Q+ h' = ea" ) = h o a -4 we have

,

so that, putting

,

h'°°(z) = hr (aa -mr`a0) (z)= h'rcao(z) = h'(6(2))= which proves the assertion (i). from (i) and (6.7.6).

The second assertion follows immediately

COROLLARY 6.39. Let K be an imaginary quadratic field, q a normalized embedding of K into It 2(Q), and z the fixed point of q(Kx) on to. Further let 9Z be the normalizer of q(K 1 ) in G Q. Then (1) [%: q(K x)] =2 ; (2) det (a) < 0 and a(z)= 2 for every a E 91-q(K x (3) h(z)= hrcao(z) for every hE and every a E 91- q(K );

PROOF. By the discussion of g 4.4, there exists an element i3 of GQ such that det(i3)< 0, A(z)= 2, and q(ei)= 13 - lq(a)13 for all a E K. Then A E q(Kx). Let a E %. Since cc'q(K)a= q(K), we can define an automorphism a of K by (1(ce)= a lq(a)a for a E K. If a = id., a must be contained in q(K), since q(K) is the commutor of q(K) itself in M2(Q). Therefore, if a E q(K), we have ce= ã for all a e K, so that a1S-"q(a).q(a)a 1S -1 for all a E K. Then c 43 -1 Eq(K). This shows that 91= q(K x)U q(K ")A, hence the assertions (1) and (2). The last assertion follows from (i) of Th. 6.38 and (2). REMARK 6.40. Since G A /Qxa., is naturally isomorphic to GA+IWG.+, we can define a homomorphism r' of GA to Aut () with kernel Qxa. so that r = r' on GA+. However, such an extension of r does not keep one of the fundamental properties (6.6.2). To see this, take a and a0 as in Th. 6.38, and

6.9

THE ACTION OF AN ELEMENT WITH NEGATIVE DETERMINANT

165

put a = aoa.. By (i) of Th. 6.38, r(a o) coincides with the complex conjugation on Qa b. Since cr(a) = id., we have

= a(a0)-1 = r(a o ) - ' = complex conjugation

(on Qao).

On the other hand, e(a.) = id. by our definition, so that ri(cr.)* . cr(a.) on (lab' Therefore, in order to discuss the whole GA, it is necessary and natural to consider more functions than those of . This can be done in the following way. Let 0- denote the lower half complex plane, i. e.,

0- = {z e C 1 Im (z) < 0} . For every complex valued function f defined either in 0 or in 0-, define f* by f *(z) . f(2). Put

a*=1/*IfEW,

Then * is a field of meromorphic functions on 0-, and R can be regarded as a ring of meromorphic functions on 0U0-. Let Aut (R) denote the group of all automorphisms of the ring R. Define A: G A --* Atit

OR)

as follows : (

f, h*)2Cr) _.-= ( f r(z) , (hz-(x))*)

( f, 10

)1

(x) = 00=0),( fr(z0))*)

hem, f E a, h E 'a)

GrEGA+JEa, (X

e GA — GA+

;

•

Then it can easily be verified that A is a homomorphism, and

(6.9.1)

Ker (A).

(6.9.2)

(a, a)2(x) = (a`") , aa ( z))

(x E GA, a

(6.9.3)

rIca)=roa

(r E51, a e GO .

E Qab) 1

The last formula follows from (6.6.1) and (i) of Th. 6.38. If we define an injection e : a --.51 by e( f)= ( f, f'), then

(6.9.4)

e( frn

= e( f)"

( f e, x e GA+)

.

Further, by a straightforward argument, we can show

(6.9.5)

2(G A) is the commutor of A(G.) in Aut (m).

REMARK 6.41. Let K, q, z, and It be as in Cor. 6.39. We see that K ab is a Galois extension of Q, and Gal (Kao/Q) is a non-abelian group with Gal (Kabilf ) as a subgroup of index 2. Put

166

MODULAR FUNCTIONS OF HIGHER LEVEL

9:1t=q(K,24)91= q(K;)U q(K)p with an element /3 of 91—q(K x).

Then we can define a map

p: Tit

—

Gal (ifab/Q)

as follows :

p(q(s)) = Es -1, KJ

for s e K,,

p(p) = complex conjugation, p(xp). p(x)p(p)

for

x e q(K ,".).

By (i) of Th. 6.31, (3) of Cor. 6.39, and (6.9.3, 4), we have, with the fixed point

= ra(v )(z)

2.

(r e t(), y e 1R).

It follows that p is a homomorphism. We then obtain a commutative diagram..

1 ------ q(K ai)

un

9n/q(K)

IP 1

1

Gal (K„/K)

- Gal (K„/Q)

Gal

1 (K IQ)

1

(exact)

1

(exact)

CHAPTER 7 ZETA-FUNCTIONS OF ALGEBRAIC CURVES AND ABELIAN VARIETIES 7.1. Definition of the zeta-functions of algebraic curves and abelian varieties; the aim of this chapter Let V be a projective non-singular curve of genus g, defined over an algebraic number field k of finite degree. For every prime ideal p in k, let p(V) denote the curve obtained from V by reduction modulo p. There exists a finite set Z1 of prime ideals in k such that p(V) is a non-singular curve (of multiplicity one) if p EE b. It can be shown that p(V) is of genus g for such a p [81, § 10.4, Prop. 11 ]. Now the zeta-function Z(u ; p(V)) of p(V) over the residue field Kp of p has the following form : Z(u ; p( V )) = Fp(u)/[(1— u)(1—N(p)u)] .

Here u is an indeterminate, N(p) is the number of elements of xy, and Fp is a polynomial of degree 2g of which the constant term is 1. Then the zetaf unction of V over k is defined (formally) as an infinite product" )

as ; V / k) = lbee Fo(N(p) - s) - i , with a complex variable s. One can actually define in a similar way the zeta-function(s) of an arbitrary (non-singular projective) algebraic variety over k. But we shall consider here only the zeta-functions of curves and abelian varieties. To define the zeta-function of an abelian variety A defined over k, we first observe that there exists a finite set Z1' of prime ideals in k such that, for every pe A has good reduction modulo p in the sense of [66], or equivalently, A has no defect for p in the sense of [81, §11]. Let p(A) denote the abelian variety obtained from A by reduction modulo p, Try the Frobenius endomorphism of p(A) of degree N(p), and R I an /-adic representation of End (p(A)) for a rational prime I which is prime to p. Then (the one-dimensional part of) the zeta-function of p(A) over ICD is given by F(u) = det [1—R i(rp)u] .

Therefore we define the zeta-function of A over k to be an infinite product 14)

Although we are neglecting the " bad" primes p in our discussion, it is actually important to consider the Euler factors also for them. See § 7.9, C.

168

ZETA-FUNCTIONS OF ALGEBRAIC CURVES as ; ilik)= Mew F4', 070) -1)-1 .

If A is the jacobian variety of the curve V, then A can be defined over the same field k of definition for V. Moreover, as Igusa has shown, we can choose a model of A so that Z' c23. Then, for every p €E 23, we have F=F by Weil [92], so that (s; A/k) is essentially the same as as; V1k). Coming back to the general case, we can now state, in a somewhat specialized form, THE CONJECTURE OF HASSE AND WEIL. Each of the functions C(s; VIk) and as; Alk) can be holomorphically continued to the whole complex s-plane, and satisfies a functional equation. The zeta-function of a variety has been determined, and hence the conjecture has been verified, in the following cases : (I.) Algebraic curves of type ax 74 +pyn+r = 0 (Weil [93]). (l b ) Elliptic curves with complex multiplications (Deuring [12]). ( lc ) Abelian varieties with sufficiently many complex multiplications (Taniyama [87]). (II.) Algebraic curves isomorphic to r\D* with certain congruence subgroups r of SL,(Z) (Eichler [16], Shimura [70]). (11b ) Algebraic curves isomorphic to r\D* with arithmetic Fuchsian groups r obtained from quaternion algebras (Shimura [73], [77]). (H e ) Certain fibre varieties of which the base is a curve of type (II..b), and the fibres are abelian varieties (especially elliptic curves) (Kuga and Shimura [42], lhara [33], Deligne [9]). The result of (Ie ) generalizes that of (Ib), and in essence, of (I.). Similarly (lib) includes (II.) as a special case. The zeta-function in the cases (Ia.b.c) is a product of several Hecke L-functions with Graen-characters of totally imaginary fields. On the other hand, the zeta-function in the cases (Ila.b.c) is a product of Dirichlet series of the type of Ch. 3, or their generalizations. In this chapter, we shall discuss the cases (le ) and (II.), with more stress on the latter case than the former. More specifically, we shall verify the above conjecture for the curves V s defined in § 6.7, and also for the abelian varieties of CM-type considered in § 5.5. Further in § 7.7, we shall investigate some class fields over real quadratic fields which are closely connected with the zeta-functions of V s.

7.2. Algebraic correspondences on algebraic curves Let us first recall some elementary properties of algebraic correspondences on curves. For a systematic treatment of this topic, the reader is referred

7.2

ALGEBRAIC CORRESPONDENCES ON ALGEBRAIC CURVES

169

to Weil [90, Ch. VIII] and [91]. Let U and V be projective non-singular curves defined over a field k. By an algebraic 1 cycle, simply a 1 cycle, or an algebraic correspondence, on Ux V, we understand a formal finite sum X= E i niDi with ni E Z and one-dimensional subvarieties Di of Ux V. We denote by 'X the 1-cycle on V x U, which is the transform of X by the map (u, v) u) of Ux V to Vx U. By a 0 cycle, or a divisor on U, we understand a formal finite sum c=Ei m tbi with m, E Z and bi E U. We put deg (c)=E i mi. For such X and c, we can define a 0-cycle X[c] on V by -

-

-

X [c] = prv [X • (cx V)], where pr y denotes the projection of U x V to V, and X (cx V) the intersection product of X and cx V. Define two integers d(X) and d'(X) by

d(X)U =pru (X) ,

d'(X)U = prv (X) .

(See [91]. Roughly speaking, d(X) (resp. d'(X)) is the number of sheets of X, viewed as a covering of U (resp. V)) Then we have deg (X[c])=d(X)- deg (c). Let W be another projective non-singular curve, and Y a 1-cycle on V x W. Then we can define a 1-cycle Z= Yo X on UxW by

Z =pr u . w [(X x W) (Ux n] Z is uniquely characterized by the property

(7.2.1) Z(b)=Y[X[b]] for every b E U, and `Z[c]=t)CCI Y[c]] for every c

W.

Therefore '2= tXotY. We call X proper, if X has no component of the form ax V with a e U or Uxb with b E V. We see easily that Y X is proper if X and Y are proper. Moreover, if X and X' are proper 1-cycles on Ux V and X(a)=X/(a) for a generic point a of U over a field of rationality for U, V, X, and X', then X= X'. Let A u (resp. A v ) be the jacobian variety of U (resp. V), and fu (res11 fi, ) a canonical map of U into A u (resp. V into AO. With every 1-cycle X on Ux V, we can associate an element e of Hom (A u, A v ) such that, if X[u] = E, v, with u E U and vi e V, then

euu(o) = n f v(v)- - c with a point c of Av independent of u. If k is a field of rationality for U, V, and X, then A u and A v can be chosen so as to be rational over k. The maps fu and fv may not be rational over k, but it crn easily be shown that is rational over k.

170

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

For a projective variety Z, let .0(Z) denote the vector space of holomorphic differential forms of degree one on Z. If U, V, and X are as above, we can associate with X a linear map O X of Z(V) into Z(U) as follows. By linearity, it is sufficient to consider the case where X is irreducible. Let k be a field of rationality for U, V, and X. Take a generic point u of U over k, and put X[u] = Di.t yi with vt E V. Let W be a projective non-singular curve with a generic point w over the algebraic closure k, of k such that MO = ki(u, 2)1 , .•- , ye). Let p (resp. qt ) be the morphism of W into U (resp. V) defined by p(w)= u (resp. q i(w)= yi) over k,. For s Eg(V), one can show the unique existence of an element so X (also denoted by (3X(e)) of Z(U) such that

(s X) op =

cog,.

(For the notation so q i, see § 5.1 and Appendix No 8.) above, the map

If A u , fu, A v , fv are as

is an isomorphism, and

(wof v) oX =(woe)of u with the element e of Hom (A u, A v) associated with X. (For details of the proof of these facts, see [81, § 2.9, Prop. 9].) In other words, the diagram

g(Av) (7.2.2)

3e

2(A u )

01

aft!

2(V)

(SX

2(U)

is commutative, where 3 indicates the action of the map (or correspondence) on differential forms (see Appendix No 8). We shall now discuss a special type of correspondence for the curves which are models of the upper half plane modulo Fuchsian groups of the first kind. We fix a family g = frl i 2 e AI of mutually commensurable subgroups of SL,(R) which are Fuchsian groups of the first kind, and denote by P the set of all elements a of GL(R) such that arct -' is commensurable with for a member I of g (see § 3.1). Note that P does not depend on the choice of r, and all members of g have the same set of cusps (see Prop. 1.30). Let 0* denote the union of and the cusps. For each 12 E g, fix a model (1,1 , WI) of r2\0* in the sense of § 6.7. Now, for T 2,1; E g, and a e P , put

(7.2.3)

X=

xr2a1p) =

ç t,(z) x q)2(a(z)) z E k)* }

(Z Vp X V2) .

It can easily be verified that X(T 2a1;) is a proper 1-cycle, and actually an

ALGEBRAIC CORRESPONDENCES ON ALGEBRAIC CURVES

7.2

171

absolutely irreducible curve, on V,. x V2; it depends only on the coset rgrr„, is a disjoint union, and and not on the choice of a. Ifr 2 cr..r p = pel r

11n{±1}=rpn{±1}, then XEçop(z)]=

(7.2.4)

2 (a t (z)) .

Therefore, if we define deg (12a1'p) as in § 3.1, then d(X(T 2a1" p p= e. deg (12a1,.) .

(7.2.5)

Further we see easily that

`X(T 2ar p). X(r pa-1 12)= x(r pot,

r2

)

,

where c denotes the main involution of M2(R) (see §3.3). PROPOSITION 7.1.

Suppose that

and

(r,otro).(rJr,)=E with

ce e Z

•

r 2er„

in the sense of the multiplication-law defined in § 3.1.

X(r lar 0)0 X(I-'p fir,) = Ze

Then

ce x(rier,). -

This can easily be verified by applying the correspondences on 9„(z), on account of (7.2.1) and (7.2.4). Let S2(111) be the vector space of all cusp forms of weight 2 with respect to TA (see § 2.1). In Cor. 2.17, we have seen that the map f(z).--, f(z)dz is an isomorphism of S2 (T2 ) onto 2(12\0*). More precisely, if we distinguish from TAO*, the isomorphism S 2 (1 2 ) f'— VV.) is obtained by the relation f(z)dz = o so 2. Now let us show that

Crictr,i2 (7.2.6)

5X (I' Acrr is a commutative diagram, where [11 l a p ]2 is the map defined in §3.4. Put r=fl1 aj1 1cr with the elements a t as above. Let (W, 0) be a model of r\*. We can define morphisms p: W—V p and qi : W 411 by poo = so, and q t o çb = v2 o at. If we put u = y,p(z) and vi = v2(cri(z)), then we see that the present symbols are exactly in the same situation as in the definition of (SX. Therefore if f e s2(r2) and e e V2) are such that f(z)dz = cow l, then

nr

-

172

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

soXoçoii , ( 60X0P)00 -=E:=160q,00=E:=160W2ccei

=

f (z)dz) 0 a, = El=1 f(a i(z)) det(aJj(a s, z) -2

= f1ErAar p12,

which proves the commutativity of (7.2.6). Especially if A =p, by virtue of (7.2.6) and (7.2.2), we see that the eigen-values of [r2cer2] 2 coincide with those of the endomorphism e of AA associated with X(r 2ar 1). It follows that (7.2.7) The eigen values of Er l ar 2], are algebraic integers. -

7.3.

Modular correspondences on the curves V 3

We shall now specialize our discussion to the groups Ps defined in § 6.7. Let 2 be as in § 6.7, and let r.'5 =RxrsnS4(R). Then the transformation group r s/Q' on 0 can be identified with T'/{±1}. Let (Vs, gosirs(x), (S,Te.Z, xeG„)} be as in § 6.7. We can consider J3(x) as a proper 1-cycle on Vs x VP-r ) rational over ks. Let SE 2, TE.Z, and x E G A+ . Put W =SnxiTz. Then we can define a proper 1-cycle X 3(x) on V s x V!'" by (7.3.1)

XTs(x)=./(x) O V3w (1) .

Then we see that (7.3.2)

d(XTs(x))= Crs

few],

d'(XTs(x)) =

: rw]

Note also that Xs(x) is the image of Vw by the map (J3(1), J(')) e., the locus of jsw(1)(v)x jrw (x)(v) with a generic point v of Vi,.. The cycles XTs(x) have the following properties. (1) Xs(x) is absolutely irreducible, and rational over k w , where W=Snx - 1Tx. (2) Xs(x) =.i.r.s(x) if S c x'Tx. (3) Xrs(x) = `.1.sr(x - 9° if x'Tx c S. (4) Xrs(x) depends only on Txr s. (5) If ks= k w for W = Snx - iTx, X Ts(x) depends only on TzS. (6) Xrs(a)= X(rkzr' s) if a E GQ+. (7) XrR(x3)= Xr 3(4 0( y ) ° isR( 3') if y e GA+ and R = y'Sy. (8) XRs(Yx)= iRr(Y) °w ° Xrs(x) if y e GA+ and R = (9) t il(rs(x)= PROPOSITION 7.2.

PROOF. The assertions (1), (2), (6), (8) follow from our definition in a straightforward way. To show (9), put W=Snx -1 Tx and P=xWx 1=TnxSx 1. -

-

173

MODULAR CORRESPONDENCES ON THE CURVES V

7.3

Then X 57.(xl' x' =J8p (x -1)°`'0`.1Tp( 1 )"

=.1sw( 1 )0.1w p (x - 7" )0 ljr poycr,

= isw(1)07pw(x)0

1.1rp(Vicr )

_Ï f1\

= ,,Y rs(x). We obtain (7) from (8) and (9). (3) from (2) and (9).

Then (4) and (5) follow from (7) and (8);

PROPOSITION 7.3. Let Se 2, T E 2, xE GA+, and W = Sn x- Tx. Then the following three conditions are equivalent to each other. (1) k w = k s. (2) S= w!' 8. (3) TxS=TxT s. Moreover, if these conditions are satisfied, d(XTs(x)) =[S: W]=C[' s: !'w].

By Lemma 6.17, we have k w =ks if and only if WG Q .=SG Q+ . Therefore the first two conditions are equivalent. Next, if S=WT s, we have Sc x'Txrs, so that x - iTxS= x -1 Txr8, hence TxS=TxT s . Conversely, if TxS=TxT s, we have x''TxS= x - 'TxT s, so that S5n(x-'7'xr s)= WI's, hence S=WT s. Since T w = Ts n W, we have [T s: TIO =[S : W] if S=W r s• Therefore we obtain the last assertion from (7.3.2). PROOF.

PROPOSITION 7.4. Let R, S, TEL, and x, y e GA+. Suppose that TxS =Txrs, SyR= Syr R , and TwR =Twr R for every WE TxSyR. Let (TxS)-(SyR) = E c, • (TwR) with c,„ EZ in the sense of the multiplication-law of § 3.1. Then Xrs(x)°v ) 0 XsR( 37) = E 4, • XTR(w) .

PROOF. Put Q=y- 'Sy, P=y- ix - Txy, M=x -1 Tx, Rr\Q=W, QnP=Z.

By Prop. 7.3, kR = k w, and kQ = kz, hence Q R = (2/-R , and PQ = prQ Therefore we have (PQ) • (QR)= E cr . (PTR) with Cr E Z and elements r of /"Q. Then we can show, in a straightforward way, that .

(TxS)- (SyR)= E c r • TxyrR .

By (8) of Prop. 7.2, we have

X 8(x) = J(x) oX ms (1) hence

and X sR (y)= J3,2 ( y) o X QR (1) ,

174

ZETA-FUNCTIONS OF ALGEBRAIC CURVES X r Axr(v) 0 X sR(Y)=

I rm(x)a(v) 0

X mQ( 31) 0 X QR (1)

= rm(x)°") 0 m p( 0 X pQ (1) 0 XQ R(1) = Ir p(xY) 0 X p Q(1) 0 X QR (1) .

Now let rR = u, rwis, and r,Q=u,rza, be disjoint unions. Then cm=uiQpi =r w n Rx = Qx , and PQ= U, Pat are disjoint unions. Let zee'. Since rR n we have t hw (1)[9oR (z)] = Ei yow(Aj(z)), so that XQR(1)[4 0 R(z)] = Ei 40Q(fii(z)). For the same reason, we have XpQ(1)[XQR(l)[yoR(z)ii = Eid Wp(a ilei (z)). From our definition of (PQ) (QR) (see § 3.1), it follows that Xp Q(1) 0 X0(1) = E Cr • XPR(T), hence Xr s (x )a( m) 0 XsR( = E Cr • IT(XY) 0 XPR(r) = E cr - XTR(xYr) • -

By our assumption and (5) of Prop. 7.2, XTR(w) depends only on TwR, for every w e TxSyR. Therefore we obtain our proposition. 7.5. Let W =W.G.., and W' =W,ÇG..., with compact subgroups W. and 141 ,Ç of G.. Then PROPOSITION

QKW PROOF.

=Qm({±1}W

{±1}W').

Let ax = by with a e Q, b e Q, xe W, and y e W'. Then

a21b2 = det ( yx - ') e Qx ç det (W,,MG., +)= ±1} , so that a = ±b, hence x= +Y. This proves our proposition. Put Up = GL,(Z p) for each rational prime

U=

p,

and

Up,

g=RxripZpr

(7.3.3)

gx = Rx xi-1p

R-;`-x11, Z;

(R.;-= {xeRlx>0})•

For elements a = (a p) and b = (b p) of g, and for a positive integer s, we write a L-E b mod (s) if ap E SZp for all p. Let us fix a positive integer N, a positive divisor t of N, and a subgroup fi* of gx such that

(7.3.4)

gx laa- 1 mod (N)}Cb*.

Since the left side is open in gx, so is b*, and b* = Rw • h with an open subgroup Ij of Hp Zi),`. Given N, t, and b*, define U' and S by (7.3.5) u,

[ ca di)]

S = QXUI.

e UId e

c

0 mod (N), b

0 mod 04 ,

175.

MODULAR CORRESPONDENCES ON THE CURVES Vs

7.3

det (U') = Then Se GQ nu'. Then

and Qx • det (S)= Qx

I

(L, so that k s =Q.

(7.3.6) r'={[ ca di) ] E SL2(Z) a e b*, d e b*, c

0 mod

(N), bEE

Put

0 mod (t)} ,

rs=Qxr`. Therefore I , is exactly the group defined by (3.3.2). Note that the present b* corresponds uniquely to a subgroup of (Z/NZ)m, which we wrote b in (3.3.2). We consider also a semi-group

(7.3.7)

db l E ma)

=

GQ,

Ia

C

E

0 mod (N), bEE 0 mod (01

which is the same as (3.3.3). LEMMA 7.6. det (Up n x - iu px)= PROOF.

a

z

for every x

e GL2(Qp).

We can find elements y and z of Up so that yxz=a-

e Qpx and b

[ 01 bO] with

Zp. Put v = [ 01 (3;) ]. Then [01 0]

e

up nv - iu

z - i(u p nx- iupx)z

for every C E Zp", q. e. d. PROPOSITION 7.7. The notation being as above, X ss(a) is rational over Q for every a e J'. PROOF. By (3) of Prop. 3.32, if a e J', we have

Par,

ri E'Lo

=crerocrvo,

O] mod (N),

=[01

],

(q, N)=1,

with positive integers q and m. By Prop. 7.1 and (6) of Prop. 7.2, we have X ss (a)= X58(E) 0 Xss(77). Therefore, it is sufficient to prove our assertion for

e and 72. As for 7), observe that [ oa E

nrriU'r) for every a E g).1., hence

det (Sn )2 - ',37)) = Q. By (1) of Prop. 7.2, X 83(i1) is defined over Q. As for e, if p does not divide N, then det (1.1 pne - iu„$)=4; by Lemma 7.6. If p

(Ix •

divides N, and U'p denotes the projection of U' to G p, then contains [ a CI ] for every a e 4.

01

Qx •

uiug

Therefore det (U'n e'U'e) = g.1,, so that

det (Sn e'Se)=Q:c By (1) of Prop. 7.2, Xss(e) is rational over Q, which

completes the proof.

176

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

Let q be an integer prime to N, and aq be an element of SL 2 (Z) such that I. 0 qaq a[o q2 ] mod (N)

(7.3.8)

(see (3.3.10)) .

We see that aq Saq- ' = S, so that j ss(cr q ) is meaningful, and rational over k s = Q. Note also that iss(a,7) depends only on the residue class of q modulo (N), and not on the choice of aq . and b,,= ({±1} • b*) n(H, 4). Let k ] denote the subfield of Q a b corresponding to the subgroup RA; of Q. Then X ss(r) is a birational automorphism of V s rational over k. Moreover, if km Q(e 27.,/N) , e27, ./N, and p(q) is the element of Gal (ksIQ) such that C.'") = C17, then PROPOSITION

7.8. Let 7 =[N ° 1

X SS(r) = X SAT) " ° SS(a q) •

PROOF.

Put b' = R N, and

W={[ ca db ]eUlaebi, deb', b

0 mod (0,

0 mod (N)}.

By Prop. 7.5, we have Sn r'Sr =Grar{_±1} nr - itPz-{±1})=Qx W.

Since det (W)= R, X 33(r) is rational over k, on account of (1) of Prop. 7.2. Further we see that r'Sr Qxr', and 7 - ir'7=-P. Therefore Xss(r) = X(r'rr') is a birational automorphism of V s. Let y=(y p) be an element of Go such that y p =

q° ] or 1 according as p divides N or not. Then we

see that aVy E U', so that Iss(aq )=./ss(Y). Now a(y)=Ddet(Y)% QJ = P(q) on k N , 7.y= yer and y` e U', so that, by (4) and (7) of Prop. 7.2, Xss(r) = Xss(rY) = Xss(r)P(q) ohs()) = Xss(r)" 0 /3s(aq)

The algebraic correspondence X ss(a) with a e GQ , is often called a modular correspondence of level N. If S= U (so that the level is 1), and a is a primitive element of M2(Z) of determinant n in the sense of § 4.6, the modular correspondence X uu(a) can be represented by the equation F„(X, J) =0, with the polynomial F, of (4.6.3). 7.4. Congruence relations for modular correspondences Let p be a rational prime, and 13 a prime divisor of e) which divides p. If X is a variety, or a cycle etc., rational over 0, we shall denote by or 13(X) the object obtained from X by reduction modulo 43. Let U, V, W be

ri

CONGRUENCE RELATIONS FOR MODULAR CORRESPONDENCES

7.4

177

projective non-singular curves, X a proper positive 1-cycle on Ux V, and Y a proper positive 1-cycle on V x W, all rational over 0. Suppose that 0, g W are non-singular curves, and k.', k are proper. Then we have

13(X 0 Y) = g 0 i'' . (For a general theory of reduction modulo F13, we refer the reader to [69], [81, Ch. 111 ]. ) Let us now fix a member S of 2 of the form S=QxU' with an open subgroup U' of U, where U is as in (7.2.3). We note that there is a finite set Z s of rational primes such that the following statements hold for every p not contained in Zs.

(7.4.1)

Up C U'.

(7.4.2) q3( vp is non-singular for every a e Gal (k sIQ), and every prime divisor 13 of 0 which divides p. (7.4.3) 13(Jss(c)) is a biregular isomorphism of 13(V 3 ) to $(11,4") for every ce QI and every prime divisor of 0 which divides p. (Note that there are only finitely many iss(c), since Q1/(Q.1 n S) is finite.)

(7.4.4) If S i =QxU, 13(J(1)) is a surjective morphism of 13(V 31) to 13(V 3) for every 13 which divides p. Here we take V s, to be the projective straight line, and Ts , to be the modular function J of Th. 2.9. Now fix a rational prime p not contained in Zs, and a prime divisor q3 of 0 which divides p. Let 7 denote the p-th power automorphism of the universal domain containing the residue field of 0 modulo 13. We denote by O s the Frobenius correspondence on 17s x N', i. e., the locus of axa on 73 X Vs'r with a e .

7.9. The notation and assumptions being as above, let w l, be an element of Al 2(4,) such that det (w,,) =p, and w the element of GA of which the p-component is wp , and other components are all equal to 1. If p EETs, then THEOREM

X 35 (w 1 ) is rational over ks, and one has X 53(w 1 ) = 0 s+z (Noiss(det (w) - `) Let U" be the projection of U' to 111#73 Ut. Then U'=U p Uir, so that wu ,w - i nu, = ( v p u pivi,- 1 nup)uff. On account of Lemma 7.6, det (wU'w - ln U0 = det (P). It follows that ks=k y if Y =wSw - inS. By (1) of Prop. 7.2, X ss(w - ') is rational over k s. Now we can find infinitely many imaginary PROOF.

178

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

quadratic fields K such that p decomposes in K. Take such a K, and a normalized embedding q of K into A1,(Q) such that q(o K)C M,(Z), where off denotes the ring of algebraic integers in K. Let z be the fixed point of q(K*) on 0. Let p=13nK. By Prop. 6.33, K • k s(w s(z)) is the subfield of K 32, corresponding to K x (s E K;lq(s) E S). In view of (7.4.1), we see that p is unramified in I• ks(Ws(z)). Let Kp be the completion of K at p, up a prime element of Kp, and u an element of K; of which the p-component is up, and other components are all equal to 1. Let p be a Frobenius automorphism of over K with respect to q3 and p. Since p=[u, K] on IC • ks(Ws(z)), we have, by (ii) of Th. 6.31, sos(z)P=Jsr(q(u)'- ')Eçor(z)], where T =q(u)Sq(u) - '. Since q(oK)c A/2(Z), we see that q(u) p is contained in M2(Zp), and has the same elementary divisors as w p . Therefore, by (7.4.1), we have Sq(u) - 'S =Sw - lS. We have seen in the above that k 3 = k 7 if Y = SnwSw -i. By (5) of Prop. 7.2, this implies that X ss(w -i) depends only on Sw'S. Therefore, putting R=Srg, we have xss(ur- ') = xss(q (u)- l) = .IsR(q(u )- 3 ) 0 'AR (1)

= i3r(q(u) -1) ° ./TR( 1) ° 'ARM Since so7-(z)X Ws(z) Ehr(q(U) -1) as shown above, we see that ws(z)x9o s(z)P Xs s(w - '). Put a = g3(v)s(z)). Then a xcere it'33(w -1). Now we have J31s(1)(a) = sosi(z)=J(z). If E is an elliptic curve isomorphic to C/(Zz+Z), f(i) is the invariant of f. Since p decomposes in K, End,a (f) must be isomorphic to K. (This result is due to Deuring. For the proof, see [10] or [81, p. 114, Th. 2].) Taking infinitely many distinct fields K," we obtain infinitely many distinct

Rig)

and hence in view of (7.4.4), infinitely many distinct points a on Vs such that a x a' e :ets(w -i). It follows that O s Cjess(w -1). By (9) of Prop. 7.2, we have Xss(w)= t Xs3(w - 9'w ) , hence jess(wr =qss(w -'). Put c=det (w), wr=cw - '. Since wi, and w p have the same elementary divisors, we have SwcS=SwS, so that Sw - iS=Swcc - lS=Swc -'S, hence X88(w - 9= XsAwr e-1) °./ss(c -1) by (7) of Prop. 7.2. Since a(c-1)= [ e, Q]=p2 on ks, we have ,

Zss(w -1) = k'ss(w) 2 oiss(c-i)

)

= t k'ss(w -I)'ojss(c -1)D s Coj.ss(c - ')

.

It can easily be seen that d(03)= d'( t 0; 0/s3(c -1 ))=1, and d'(08 )= d(1);(3.iss(c -2)) = p. Since O s and t Colgs(c-') are irreducible and distinct, we obtain (*)

0 s -F eCoiss(c -1)Css(w -l) •

On the other hand, we see that

7.5

ZETA-FUNCTIONS OF V s AND THE FACTORS OF THE JACOBIAN 179 CS :wSw - i nSJ

p:

wpU pw,-,' nUpj=p+i ,

hence, by Prop. 7.3, d(k'ss(w -1)). d(X ss(w - 9) p+1. Comparing d and d' of both sides of (*), we conclude that the inclusion must actually be an equality. This completes the proof. COROLLARY 7.10.

Let S and U' be defined by (7.3.5). Let ap be an element [ 01 p0 ], r= [ ON of SL,(Z) satisfying (7.3.8), and let a. pEE0 s, one has

il?ss(a)= 0s-Ft0sojss( 0 p),

(1) t Os

(2)

Vss(ap) = qss(r)° t OS° li.i;Ss(r)

-

We first observe that Up C U' if and only if p does not divide N. Therefore, if p 93s, p does not divide N. Let y be the element of G, of which the p-component is 1, and other components are all equal to a. Put a = wy, c= det (w). Then ye E U', y'Sy = S, and a(y). a(w - i). By (7) and (9) of Prop. 7.2, we have PROOF.

(*

)

X 33 (a) = Xss(1074) ohs(

= t Xss(w -1) 0 Jss( 3')

Now we see that 3,ap- ' E U', and p= det (a). cyy€.

Therefore

J33(c -1)=JsA Y.Y0 =Iss(Y)=iss(orp) -

From the above theorem and (*), we obtain il?ss(a) = qs3(w -1) ohs( y) = t ø s ofss(ap) -F ILS(ap)

°°

Here note that ks=Q, hence 0';= Os. Since Jss(ap ) is rational over ks= Q, O s commutes with iss(Grp), by virtue of (7.1) of Appendix. Thus we obtain (1). The formula (2) follows immediately from Prop. 7.8 and (7.1) of Appendix.

7.5.

Zeta-functions of V 3 and the factors of the jacobian variety of Vs

We shall now determine the zeta-functions of the curves V s with members S of 2 of the type (7.3.5), and the zeta-functions of some abelian varieties occurring as factors of the jacobian variety of Vs. The main idea is to connect the Frobenius morphism with Hecke operators by means of the congruence relations of Th. 7.9, or Cor. 7.10. THEOREM 7.11. Let U', S, be as in (7.3.5) and (7.3.6), and 93s be as in § 7.4. Let ap, for a prime p EE0 s , be an element of SL 2(Z) satisfying (7.3.8) (see also (3.3.10)), and ap=[01

]. Further let 52(P) be the vector space of

180

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

r,,

all cusp forms of weight 2 with respect to and Er/aril the action of T'ai' on .32 (TI) defined by (3.4.1). Then, for every p not contained in T s, the zeta-function of P(V s) over the prime field is given by

Z(u -,p(V s))= [(1-0(1—pu

)]

i- det (1—[T'a pri],u+p • [riaprl2 u2).

Fix a prime p Er Zs, and a prime divisor 43 of 0 which divides p. Denote, as before, by ik. or q3(X) the object obtained from X by reduction modulo 43. Let A s be the jacobian variety of V s, and R i (resp. RD an /-adic representation of End (A s ) (resp. End (ils)) for a rational prime 1 different from p. By [81, § 11, Prop. 14], we can assume RIM= RV) for every 2 e End (As). Let zp denote the p-th power endomorphism of lis. Then there exists an element 4 of End (As) which is associated with gO s and satisfies x•plr: = p. Let ep , vp, and A be the elements of End (A s) associated with Xss(ap), h(a), and X ss(r), respectively. From Cor. 7.10, we obtain PROOF.

(7.5.1)

È- P = 2r P+ Z41;

(7.5.2)

4 13 P = I -17r; g •

Therefore, if u is an indeterminate, we have [1 — u - nzp)j [1 — u • RK,g -2 4 1-3')J = 1—u • RKÈ-p)+Pu 2 - R'Op)

= 1 u - R i(e p)+ pu2 • —

Since z p and 2r; have the same characteristic polynomial, we have det [1—u • RKzp)j 2 = det [1—u • Ri(ep)+Pu 2 - Ri( 7203 •

Now the representation R i is equivalent to the representation R° of End Q (As) on the first cohomology group of A s. If R denotes the representation of End Q (A s) on ..0(A 3), then R° is equivalent to the direct sum of R and its complex conjugate (see Appendix N. 11). In § 7.3, we have shown that X 33(a) and jss(ap) are rational over Q, so that ep and i rational over Q. Therefore, taking a basis of .0(A 3) over Q, we can assume that R(E) and R(72 p) are rational matrices. Therefore we obtain, through those two equivalences of representations, det [1—u • R(r)J 2 = det [1—u - R(ep)-F-Pu 2 • RO2pn 2 ,

so that det [1—u - nzp)]=det [1—u - R(E p)-1-pu 2 • R(i,)] .

By virtue of the commutativity of the diagrams (7.2.2) and (7.2.6), we may put R(ep)=Eriapr'32, R(i2p)=Erierpr'32. This completes the proof. THEOREM 7.12.

Let T,(n)kop be as in § 3.5.

Then, for almost all primes th

7.5

ZETA-FUNCTIONS OF Vs AND THE FACTORS OF THE JACOBIAN 181

&very eigen-value

Ap

of P(p)2,0 satisfies I

2p".

PROOF. Take b* to be {a egX a 1 mod (N)}. Then r/ coincides with the group of (3.5.1'). By (3.5.6), T'(P) 2,0 is the restriction of [P'aprfi]2 to s2(ri;, 0). Since zp commutes with 711 )2p, we see, from (7.5.1) and (7.5.2), that any characteristic root of Ri(ep)= /?;( p) is of the form p-Fp' with a characteristic root p of Rz p) and a characteristic root p' of IiI(71). By the Weil theorem, f pI=I p' I =p"2. Since R(e p)=EP'a prfij„ we obtain our assertion for all p not contained in Zs.

A prototype of Th. 7.9 is already in the works of Kronecker. The relation (7.5.1), in the present formulation, was first proved by Eichler [16] for 10(N) and its subgroups r/ of index 2; he then obtained Th. 7.12 for such groups and Th. 7.11 for I' o(N). The generalization to the present form and the formula (7.5.2) were given in [70]. Actually in [70], Th. 7.9, or rather Cor. 7.10, was proved by means of the congruence relations for an elliptic curve with a variable modulus. This method is simpler than the above proof of Th. 7.9 in the sense that it does not require any result of complex multiplication. It was shown by Igusa [36] that the set of primes Zs is contained in the set of all prime factors of N. But we shall not discuss this point in the present book. Let 4' be defined by (7.3.7). Let P(n ) and T'(a, d) be the elements of 40 as in § 3.3 (see especially Th. 3.34). Denote by T'(n) 2 and Tqa, d)2 the action of Tf(n) and T'(a, d) on S,(P), respectively (see §§ 3.4-5). Then we have Er'a p rlj,= T'(P) 2 , and Er'apr']2= P(P, P)2 by (3.4.4) and (3.3.11). Therefore, by virtue of Th. 7.11, the zeta-function of V s over Q has the form

as ; V 10= nPF-Ms det [1— r(P)2P - s+P(P, P)2P' 2'] - ' This is, up to finitely many Euler factors, exactly a Dirichlet series of the type discussed in §§ 3.3, 3.5, 3.6. More precisely, let sk(P, 0) and T'(n) k,0 be as in §3.5. Then sk (r/) is the direct sum of all the sk(rr„ 0) such that 4'()= 1. Put =E7.1 T'(n)k, sb-

.

Then «s; V /Q) coincides, up to a finite number of Euler factors, with the product (7.5.3)

D(s)= no(1 ) = 1 det [D2,46(s)] •

For each element f(z) =E7. 1 a,,e2'in"` of <7.5.4)

sk(rif), put

L(s, f) =

By virtue of the discussion of § 3.5, especially of Prop. 3.47, we can find a set

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

182

of elements

(7.5.4')

fh„ ••• ,

h,) of

sk(n, 0)

such that = 1, • • • , 1c)

h„(z)=Er,. 1 2(n)e2'":" , h 1 IT'(n) k,s6 = 21(n)h„

det (Dk ,se,(s))= H. 1 L(s, h„). ({h„ •-• , h,) is not necessarily a basis of from Th. 3.66 and Remark 3.58,

sal 0).)

Therefore we obtain,

THEOREM 7.13. The zeta-function as ; V 8/Q) of Vs over Q is an entire function, and satisfies a functional equation.

By Remark 3.58 and Th. 3.66, the functional equation of L(s, f) is given by

R(s, f)= (tN)" 2(2z) -11(s)L(s, f)= ik • R(k—s, f where r=[L N 1 . The above hi, may not be an eigen-function of [r] k . However, on account of Prop. 3.57 and the result of Hecke mentioned in Remark 3.60, if N is a prime and t = 1, we see that hv [ r]z = euh„ for the basis {11 1 , ••• , h,) of S2(fl, 0) with Si, = ± 1, for a real character 0 ; if 0 is not real, [r] 2 sends a common eigen-function of the P(n) 2,0 to a common eigenfunction of the r(n) 2,176. Therefore the Dirichlet series D(s) of (7.5.3) satisfies the functional equation

(7.5.5)

R(s)=[N" 2(270- T(s)1 1D(s)= p • R(2 — s)

with p= +1, where g is the genus of V 3. For example, assume t =1, and b= gx, so that = 1= r 0 (N). By Prop. 1.40 and 1.43, V 3 is of genus 1 for the following 12 values of N:

(7.5.6)

11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49.

In these cases, the zeta-function of V s is (up to possible bad factors) (s; V IQ) = L(s, h)= EZ.Icnn -s =ll piN (1—c,,P - ) 1 • 11„ N (1—cpP- 8+ p1-2s)— 1 ,

a)).

with an element h(z)=E7_, ce" of s2(r It can be shown that h I [r].= — h, and hence the functional equation has the form R(s, h)= No(270 - 1(s)L(s, h)= R(2 — s, h) . As Fricke [21] observed, the elliptic curve T 0 (N)\0* has no complex multiplications for the first 8 values of (7.5.6). For further examples, see Ex. 7.26 below. In the next place, we shall consider the zeta-functions of abelian varieties which occur as " factors " of the jacobian A s of V. We denote by en, for

ZETA-FUNCTIONS OF Vs AND THE FACTORS OF THE JACOBIAN

7.5

183

each positive integer n, the element of End (A s) corresponding to the sum of X33(a) for all Par' such that a e J' and det (a) = n. Then we see, from Prop. 7.7, that is rational over Q. Moreover, through the diagrams (7.2.2) corresponds to T/(n) 2. For simplicity, we assume hereafter and (7.2.6),

en

en

t =1,

(7.5.7)

without losing much generality, in view of Remark 3.58. THEOREM 7.14. Let f(z) be an element of S2(P) which is a common eigenfunction of T'(n) 2 for all n, and let f Ir(n) 2 = ct n .': Let K be the subfield of C generated over Q by the complex numbers a n for all n. Then there exists an abelian subvariety A of A s and an isomorphism 6 of K into End4, (A) with the following properties:

(1) dim (A)=[K : (2]; (2) 6(a„) is the restriction of e„ to A for all n; (3) A is defined over Q. The couple (A, 6) is uniquely determined by (1) and (2). Moreover, for every isomorphism a of K into C, there exists an element f0 of such that f.IP(n)2=a:f. for all n, and f.(z)= De=i ge2rInt

s2(r,)

.

We may and do assume, for simplicity, f(z)= E7=1 ane' (see Th. 3.43). THEOREM 7.15. The notation being as in Th. 7.14, suppose that froc = g* in the definition (7.3.5) of U' and S. (This implies that

r, =ro(N)={[ ac bd ]

SL,(Z)1cm- 0 mod (N)} . )

Then the zeta-function of A over Q coincides, up to a finite number of Euler factors, with the product

L(s, fa) = H.

(n3

=1

an) ,

where the product is taken over all the isomorphisms

a of K into C.

PROOF of Th. 7.14 and Th. 7.15. Let Z be the subalgebra of End Q (As) generated by the for all n. If A s is of dimension g, is a commutative algebra of rank g over Q, by Th. 3.51. Let R be the radical of Z. By a theorem of Wedderburn, there exists a semi-simple subalgebra of Z such that Z = Se T. Let R„ , R,. be the simple components of S. Now the map a„ defines a homomorphism p of Z onto K. Therefore p( )?)= 10), and p ( ?) # 10} for one and only one R i, say Q. Then p gives an isomorphism of R, to K. Denote by p' the inverse map of this isomorphism. Then p'op is the projection map of Z to S?„ so that p/(a) is the projection of E„ to R1.

en

en

.,

184

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

Take an integer 0 so that 2,9P * {0} and R,Rg+i = {0}, where we understand that 9V= Z. Let V be an irreducible R1-submodule of RI . Then WI is a minimal ideal of Z. Put 1I1„ = Tin End (A s), and A=ITI.A s. Since every element of V. is defined over Q, A is an abelian subvariety of A s defined over Q. Since the action of Z on s,(p) is equivalent to a regular representation : Q] = [R, : Q] = [K: Q]. For of Z (see Th. 3.51), we see that dim (A). each a e K such that p'(a) E End (A s), denote by 0(a) the restriction of p'(a) to A. Then ê can be extended to an isomorphism of K into EndQ (A). It is clear that 8(a,i) is the restriction of e n to A. To prove the uniqueness of (A, 0), let (A', 0') be a couple satisfying (1) and (2). Consider A s as a complex torus Cg/L with a lattice L in Cg. We may assume that cg = s2(r) and en is represented by the operator T'(n) 2 on s2(ro. Let W be the subspace of sa.' ,) corresponding to A'. Since 8'(an) is the restriction of en to A', 0'(p(E))is the restriction of e to A' for every e E Z. Therefore, A' (and hence W) is annihilated by and Ri for i> 1. Consider W as a module over 21 041 C. or over KOQ C. Then we find a basis ( f„ , fnj of W over C such that f„ is sent to a°,..f„ by 0'(a) for every a e K, where a„ is an isomorphism of K into C for each v. Here m = dim (A') = [K:Q]. Then

(*

)

LIT/(02=41.

(1».7n;1n<00).

By Prop. 3.53 and Cor. 3.44, f „ is uniquely determined by (*) up to constant factors. Therefore we see that a1, , a,, are distinct, and W is uniquely determined by f. This implies that A' is unique, and hence A = .4'. This completes the proof of Th. 7.14. Suppose that I)* = gx• Let p be a rational prime not contained in Zs. By [40] or [66 ], A has no defect for p. Since aj,E p, we have Th, = 1, so that the relation (7.5.1) becomes = Irp+11, . Let R;' denote the /-adic represendegree p. By tation of End (A), and Tql the Frobenius endomorphism of A the same reasoning as in the proof of Th. 7.11, we have

det [1—u Rf(7q)] = det [1— T iv(P)2u+Pu 2] , where Tv(p)2 means the restriction of T'(P)2 to W. Therefore, in view of (*), we obtain Th. 7.15. To obtain further information, particularly in the case I)** gx, we first observe, in view of Prop. 3.53, that f belongs to sal, 0) with a unique character 0 of (Z/NZ)K such that 0(b)=1, where b is the subgroup of (Z/NZ)x corresponding to b*. The values 0(n) belong to the number field K, on account of (3.5.8) and (5) of Th. 3.34. Let 2 denote the set of all isomorphisms of K into C. Then, for every a E 2, f, belongs to sal 0"), again on account of (3.5.8) and the equality pr(p,p).r(p)2—T ,(p2). Now suppose that the

7.5

ZETA.FUNCTIONS OF Vs AND THE FACTORS OF THE JACOBIAN 185

following condition is satisfied : (7.5.8) For every a E 3, all the T'(n) 2,s5, belong to the algebra generated over Q by the T'(n) 2,0, for all n prime to N. We obtain then (7.5.9) K is generated by the an over Q for all n prime to N. By virtue of the result of Hecke [30] quoted in Remark 3.60, (7.5.8) is satisfied 0 —1 if 0 is a primitive character modulo (N). Let r = [N 0 ], and let is be the element of End (As) associated with Xss(r). conjugation. By Prop. 3.55, we have

Further let p denote the complex

(7.5.10) 4, = 0(n)acz,IP for every a e 2 and for every n prime to N. Therefore we see that KP =K and a: P= af,° for every a e 3, so that K is totally real if p is the identity map on K. If it is not, K must be a CM field. in the sense of § 5.5, on account of Prop. 5.11. By Prop. 3.57, we have f • i [r]2T'(02 = erf,Pf,, I CrI for all n prime to N. Therefore, by (7.5.8) and Cor.. 3.44, we see that -

(7.5.11) For every a e

fo I Cri is a constant multiple of

fop.

It follows that [r] 2 sends W onto itself, and hence A is stable under is. Therefore, by means of the same argument as in the proof of Th. 7.11, we obtain the first part of the following THEOREM 7.16. The notation being as in Th. 7.14, suppose that the conditioa (7.5.8) is satisfied. Then the zeta-function of A over Q coincides, up to a finite number of Euler factors, with H._ L(s, f.). Moreover, if 0 is the trivial character, K is totally real. If 0 is not trivial, then K is a totally imaginary quadratic extension of a totally real algebraic number field K', and there exists an abelian variety A' such that A is isogenous to A' X A', and EndQ (A') contains an isomorphic image of K'.

If 0 is trivial, K must be totally real on account of (7.5.10). Suppose that 0 is not trivial. For every q prime to N, let aq be as in (7.3.8), and vg the element of End (A s) associated with iss(a q). By Prop. 7.8, we have PROOF.

(7.5.12)

P= l31777,1 if C° =Cq for C=e2 N

Since 0 is non-trivial, we see that 71q# id. on A. Let p be the restriction of is to A. Then p° # p for some a E Gal (Q()/Q), so that p is different from ±1 on A. Put A' = (1+ p)A. Then .4'# 0, and since p2 = 1, dim (A') < dim (A) [K: Q]. From (7.5.11), we obtain

186

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

p6(a)=6(ao)p for every a E K.

(7.5.13)

Assume that K is totally real. Then 6(a) gives an endomorphism of A' for every ac K. Therefore K is embeddable into End Q (A'). But this is impossible on account of the following LEMMA 7.17. Let K be a totally real algebraic number field, and A' an abelian variety defined over (a subfield of) C. If there exists an isomorphism of K into End 4, (A') which maps the identity element of K to the identity element of End (A), then [K: CI] divides dim (A'). PROOF. Let R and R° denote respectively the representations of End Q (A) on 2(A) and on the first cohomology group of A. Then R° is equivalent to the direct sum of R and its complex conjugate R. Restrict R and R° to the image of K in End Q (A), which we identify with K. Then R is equivalent to a direct sum of several isomorphisms of K into C. Since K is totally real, we see that R is equivalent to R. On the other hand, since R° is a rational representation, we see that tr (R(a)) = tr (R°(a))/2 E Q for every a e K. Therefore the degree of R must be a multiple of [ K: Q]. Since dim (A) is exactly the degree of R, we obtain our assertion. Coming back to the proof of Th. 7.16, we can therefore conclude that K is a CM-field in the sense of § 5.5. Take an element b of K so that 0* b = —6 and 8(b) E End (A). By (7.5.13), we have 6(b)A' = (1— p)6(b)A = (1— p)A. Therefore A = (1+ p)A-F(1— p)A= A'+6(b)A', and hence A is isogenous to A' x A'. Further, if a EK and ao = a, we have 6(a)A' C A' by (7.5.13), so that End Q (A') contains the isomorphic image of {a e K I ao = a}. This completes the proof of Th. 7.16. Let k be the subfield of Q(e27:00 defined in Prop. 7.8. Then A is defined over k by that proposition, and hence A' is defined over k. Moreover, let K'. {a e Ki aP =a}, and let 61(a) be the restriction of 6(a) to A' for every a e K'. Then 6' is an isomorphism of K' into End Q (A'), and 6'(a) is defined over k for every a E K'. Now it is natural to consider the zeta-function of A' over k. Since a general discussion of this question has been made in T. Miyake [50]'5 ), we shall only treat here the simplest case, in a somewhat different formulation. Besides (7.5.8), let us make the following assumptions : (7.5.14) 0 is a character of (Z/NZ)m of order 2 such that 0(-1) = 1. (7.5.15) b* corresponds to the kernel of sb, so that [(is: b*] = 2, and hence a b 1] E SL2(Z) 0(a)=1, cE--_- 0 mod (N)} •

r, =1[c (

15)

I

This paper treats also V s and A s for a more general type of S than (7.3.5).

7.5

ZETA-FUNCTIONS OF

v s AND THE FACTORS OF THE

JACOBIAN 187

7.18. The notation being as in Th. 7.14 and Th. 7.16, suppose that the conditions (7.5.8), (7.5.14), and (7.5.15) are satisfied. Let k be the quadratic extension of Q corresponding to 0. Then A' is defined over k, and A" is isogenous to A' over k for the generator s of Gal (k/Q). Moreover, the zeta-function of A' over k coincides, up to a fiftite number of Euler factors, with the product ilde4 Us, fd). THEOREM

Note that k is real, since 0(-1)= 1. PROOF. As is mentioned above, i5 (and hence its restriction 1-1 to A) is defined over k, so that A' =(1 Fp)A is defined over k. Let q be a positive integer such that 0(q)= —1. Let 7h be as in the proof of Th. 7.16. Then 7hz = —1 on A, since the cusp forms f., for all a e a, are contained in .S2(ni, 0)Therefore, from (7.5.12) we obtain, if s is the generator of Gal (k/Q), -

(7.5.16)

te = — ti -

It follows that A" =Ca F OAT -=(1 p)A= 8(b)A 1, where b is an element of K considered in the proof of Th. 7.16. Therefore A" is isogenous to A' over k. For every prime ideal 13 in k, let vp denote the Frobenius endomorphism of A =1)(A) of degree NQ), and Ri the /-adic representation of End (A). From (7.5.1) we obtain J(ap)= Ép = irp+O(P)71 on A for every rational prime p not contained in Zs. Therefore, if N(1))=p 2, we have sal, =4, so that -

(

*)

—

det [1 — u2R 1(S01, )] = det [1—u - Ré(xp)] - det [1+u • RiOrp)] = det [1—u - RK 7rp)] • det [1-0(P)u - Ri(en] = det [1—u - Ri(jp)+O(P)Pu9 -

Let Tw(P): be the restriction of r(P)2 to W. By the same reasoning as in the proof of Th. 7.11, we see that the right hand side of (*) coincides with

(,0,0)

det [1—Tw(p) 2u+O(p)pu 2j2 .

On the other hand, if (p) = pp, in k, both so, and sop, can be identified with 7rp, so that

det [1—u - RI(S0 1,)] - det [1—u - Ri(VP-)] = det [1—u - R;(2 rp)]2 = det [1—u - nr p)] • det [1-0(p)u. Ri(r'; )] = det [1—u - R'Lap)+41)(P)Pu9 , which coincides with (**), for the same reason as above. It follows that (s; A/k) is, up to a finite number of Euler factors, given by the product

ZETA FUNCTIONS OF ALGEBRAIC CURVES

188

-

ri 3 _ 3 L(s, f4)2.

Now A is isogenous to A' x A' over k. Therefore, if 95.1), denotes the restriction of 9p to p(A'), and R2 the /-adic representation of End (p(A0), we have det [1—u R7(44)]2= det [1—u Ri(9p)] so that

det [1— Tw(P)2u+Sb(P)Pu 23

N(13) = p2,

det [1-0 • Rini

if

clet [1— u R2(951,I)] - det [1—u. R2(90]

if (P) =W .

Therefore we obtain our assertion for C(s ; A'/k). We also notice that

(7.5.17)

det [1— u R2(9)] -= det [ 1 — u • R2(4)]

if

(P) =

,

since det RK2rp)] equals both sides squared, or since A' is isogenous to A" over k. Now we have [r] i= 1, so that, by (7.5.11), we obtain

(7.5.18)

h I Cri2 = rhp,

hp

1 CrI

with a constant r. (Prop. 3.40 implies that Iii = 1, though we do not need this fact.) Therefore, if we put gs, L(s, m= [K: Q], and R(s, A')= r(s)m(2r) - n"Nm" 2 L(s, A'), then

(7.5.19)

R(s, A')= R(2—s,

As an example, let us consider the case where

(7.5.20) If

r

dim

(s2(n, 0)) = 2 .

is as in (7.5.15), we have

S2(1-'')=S2((',)-Fs2(r,

0) •

Therefore, by Th. 3.51, the T'(n)2,95 form an algebra 91 of rank 2 over Q. Under the assumption (7.5.8), 91 must be semi-simple, so that 91 is isomorphic to a quadratic field, or to (KM. Th. 7.16 implies that the latter case is impossible, since Q is not totally imaginary. Thus 91 must be isomorphic to a quadratic extension K of Q. Then, by Th. 7.14, we find an abelian subvariety A of A s of dimension 2, and an isomorphism 0 of K into End Q (A). By Th. 7.16 and Th. 7.18, K must be imaginary, and A is isogenous to Ex E with an elliptic curve E defined over a real quadratic field k. If A o denotes the jacobian variety of ni\o*, then the jacobian A s of 1-'1\* is isogenous to Ax A o. An elliptic curve E of this type has very interesting properties, which we shall discuss in § 7.7 along with some more examples of A and A'.

7.6

1-ADIC REPRESENTATIONS

189

In the above, we have started our discussion from a common eigen-function f of the Hecke operators on s2(ri) and obtained an abelian variety A. Instead, we can start from an abelian subvariety of A s as follows. PROPOSITION 7.19. Let S and I" be as in (7.3.5-6) with 1)* = g , t 1, and A s the jacobian variety of V s as before. Further let en be the endomorphistrz of A s corresponding to the Hecke operator T'(n), on S2(1'). If A is an abelian subvariety of A s rational over Q, then A is stable under for all n prime to N. Moreover, if X denotes the subspace of s2(ri) corresponding to A. and Tx(n) the restriction of 7i(n) 2 to X, then (s; AlQ) coincides, up to a finite number of Euler factors, with det (E (fl„ N)=, Tx(n)n - ').

en

PROOF. Let p be a rational prime not dividing N. To prove the first assertion, it is sufficient to show that ep (A)c A. Suppose that ep (A)cE A, and put A* = e p (A)± A. Then A* is an abelian subvariety of A s, and dim (A) < dim (A*). Let tilde denote reduction modulo p. If rp and r are as in the proof of Th. 7.11, we have rp(A). al (A). A, since A is rational over Q, so that c A by (7.5.1). (Note that 77/, = id. on account of the assumption 1)*= gx.) But ;14, -.4(A)+,4 has the same dimension as A* by the general theory of reduction modulo p (see [69 ]) , which is a contradiction. Therefore e p(A) C A. Now consider A s as a complex torus sxrwL as in the proof of Th. 7.14. Then A corresponds to a vector subspace X of s2(P) stable under the r(n) 2 for all n prime to N. Then we obtain the last assertion by means of the same argument as in the proof of Th. 7.11. Since the T x(n) for all n prime to N form a commutative semi-simple algebra, we can find a basis { f„--- , fr } of X over C formed by common eigenfunctions of all such Tx(n). Put h I Tx(n). a,, nf t. with a un e C. Then, for each fixed 1J, {gE

s2(r)l g iT'(n)2 = ag

for all n prime to N}

is stable under the T'(n), for all n (not necessarily prime to N). Therefore we can find a common eigen-function g„ of all r(n) 2 such that g„IT1(n)2 = k n gi, with b„ n = ap„ if (n, N)= 1. By Th. 3.43, we may assume that g(z) jne 2..invt . This shows that C(s; A/Q) for the above A coincides, up to a finite number of Euler factors, with the product Hr.. , L(s, g ia). The functions gi, may not be contained in X. It should also be noted that 7 , N)= b belongs to .vrousit» (cf. Hecke [30, Satz 19 ] ).

7.6. 1-adic representations We shall first generalize the notion of /-adic coordinate system of an abelian variety A, by considering everything relative to an algebraic number

190

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

field embedded in End ci (A). Let A be an abelian variety defined over any field, F an algebraic number field of finite degree, and 0 an isomorphism of F into End Q (A) which maps the identity element of F to the identity element of End (A). Put g = dim (A), and h =[F : Q]. By [81, § 5.1, Prop. 2], 2g is a multiple of d; put 2g = dh. The same proposition asserts : (7.6.1)

The characteristic polynomial of 0(x), for every x e F, is the d-th power of the principal polynomial of x over Q; especially deg (0(x))= N,1 (X) 4.

(See Appendix No 10 for the notation deg ( ).) Let o denote the maximal order in F. Suppose that (7.6.2)

0(o)c End (A) .

For an integral ideal (or an integer) a in F, put (7.6.3)

A[a] =ft E Al O(a)t=0} ,

AEal ,uz "Kan]

It can easily be seen that A[nti]= A[a]+ A[b] if a is prime to b (cf. [81, p. 61, Prop. 18,3).

PROPOSITION 7.20. If A is defined over a field whose characteristic is either 0 or prime to a, then A[a] is isomorphic to the direct sum of d copies of o/a. It is sufficient to prove the assertion in the case where a is a power of a prime ideal I. By elementary divisor theory, il[1 4], for a positive integer n, is isomorphic to PROOF.

(*)

ori e... eon

(0 < m1

n).

In [81, p. 56, Prop. 10], we have proved : (7.6.4)

A[a] is of order N(a)4 under the assumption on the characteristic of the field of definition for A.

Therefore we have m 1 -j- ••• -Fm,= nd. On the other hand, (*) implies that A[1] is isomorphic to (o/1)', hence by (7.6.4), we have s=d. Since 774 n, we obtain m, = = m: = n, q. e. d. Now, for every prime ideal 1 in F, let F1 (resp. (0 denote the 1-adic completion of F (resp. o). We fix a vector space W over F of dimension d, and an o - lattice D in W. (An o- lattice in W is a finitely generated o-submodule of W which spans W over F.) We put W1 = W®FFI , and D, =DO. or . From the above proposition, we obtain easily (7.6.5)

If A is defined over a field whose characteristic is either I, then there exists an exact sequence

0, or prime to

191

I-ADIC REPRESENTATIONS

7.6 0

—

t D, ---- W I — ACIoej — O.

(In brief, A[l] is isomorphic to (Flol)d.)

We call such an exact sequence, or the map t, an I-adic coordinate system of A. Let Y be the subring of EndQ (A) consisting of all the elements which commute with the elements of 0(F). Every element e of Yn End (A) induces an endomorphism of A[l], which is obtained from a unique element Ri(e) of End (WI, F1 ) stable on Di . In this way we ohtain an F-linear homomorphism R 1 : Y ----. End (W,, F,)

(2,_ Md(F1)).

If K=Q and I./Z with a rational prime I, this is the /-adic representation of Weil [92, No 31]. PROPOSITION 7.21. For every E e Y, the characteristic polynomial h of R I(E) has coefficients in F, and is independent of I. Moreover, NFIQ (fe) (understood in an obvious sense) is exactly the characteristic polynomial of e in the sense of [92, No 67]. PROOF is given in [78, 11.9]. PROPOSITION 7.22. The restriction of R, to any simple subalgebra Z of Y, containing 0(F), is faithful, and equivalent to the direct sum of a multiple of the reduced representation of Z over F and (possibly) a zero-representation. Moreover, the restriction of R, to Z can be extended to an F,-linear representation

Z OF Ft

-,

Md (F,)

which is equivalent to a multiple of the reduced representation of F, modulo a zero-representation.

ZO F F, over

This follows from Prop. 7.21, by means of the same reasoning as in [81, § 5.1, Lemma 1]. Suppose now that A and the elements of O(F)n End (A) are defined over an algebraic number field k of finite degree. Then Gal (0/k) acts on A[1–].. Therefore we obtain a representation R,: Gal (Q/ k) — End (D I , o l)x

(.;-.-.. G Ld(o1)) .

Let B be the set of all the prime ideals in k for which A has defect. Take a prime ideal 1) in k which does not belong to B and which is prime to N(I). Let q3 be a prime divisor of 0 which divides 0, and a a Frobenius element of Gal (0/k) with respect to 93. Further let Â denote the abelian variety

192

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

obtained from A by reduction modulo p. Then we can define an isomorphism : F—' End Q (A) by g(a)=1)(0(a)) for every a e K such that 0(a) e End (A). Therefore an 1 - adic representation Rf of the commutor of g(F) in EndQ (A) can be defined as above. Let sop denote the Frobenius endomorphism of A of degree NQ). By [81, § 11.1, Prop. 14], we have a commutative diagram :

o

AD1

Wx/DI

(7.6.6)

id.

1

reduction modulo 13 Â[r]

WI /D i

If t e il[1° ], we have S43(ta) . so(f). Therefore, if we define IR / and .1i. with respect to the horizontal arrows of (7.6.6), we obtain

(7.6.7)

R1 (c) =--- Rf (sop) -

(Note that so, belongs to the commutor of g(F).) This shows that IR,(a) is uniquely determined by $, hence we obtain the first part of the following PROPOSITION 7.23. Let 2(1) denote the subfield of 0 corresponding to the kernel of s.R I . Then a prime ideal p in k is unramified in 2(1) if p does not belong to B and is prime to N ). Moreover, for such a prime ideal p, let a be a Frobenius element of Gal (01k) with respect to any prime divisor 13 of p in Then the characteristic polynomial of sna) has coefficients in o, and depends only on p (j. e., it is independent of the choice of I and 13). (

a.

This is a generalization of [81, § 18.5, Prop. 18]. The assertion concerning tR,(a) follows from (7.6.7) and Prop. 7.21. The notation being as above, let I/0(u) denote the characteristic polynomial of 1 (a). Then we can define the zeta-function of A over k relative to 0 : F—.End Q (A) by as; Alk, F)=IIDEB

N(p)ds - Hp(N(p)s) — ' .

If F=Q, this is exactly as ; Alk) defined in § 7.5. It is of course natural to extend the Hasse-Weil conjecture to as ; Alk, F). Observe that as ; Alk, F) depends only on the isogeny class of A over k. Therefore the assumption (7.6.2) is inessential, since for a given (A, 0), we can always find another (A', 01 satisfying (7.6.2) by an isogeny rational over k (see [81, § 7.1, Prop. 7]). Now, since H, is the characteristic polynomial of sai(a), we see that as; Alk, F) is analogous to the Artin L-functions of finite normal extensions of algebraic number fields. Therefore the determination of as; Alk, F)

193

1-ADIC REPRESENTATIONS

7.6

not provides a certain reciprocity law for the extensions R(1) of k, which are necessarily abelian, as we already emphasized in [81, § 18.5] and in [73, § 6.3]. For further discussion of this topic, we refer the reader to Taniyama [87], the author [76], [78], [79], [80], and Serre [65]. Coming back to (A, 0) which is not necessarily defined over k, but over any field, suppose that A has a polarization C with the following property : (7.6.8) If * denotes the involution of End Q (A) defined by C (see Appendix N°13), then 0(a)* =0(a) for every a e F.

Since * is a positive involution of End Q (A), F must be totally real. For the rational prime 1 divisible by 1, Put WL = WOQ Q1, and Di = DOz Z. Then we obtain an /-adic coordinate system 0 --- D1 --- 1471 -------4 A[I] ----'' 0

(exact) .

Take a divisor X in C. By Weil [92, No 76], we can associate with X a nondegenerate alternating form Et :

W 1 x1471 --°Cli

such that Ec (x, y) e Z, for every (x, y) e Dt x Di, and E 1(R 1 (2)x, y) = E ,(x, R 1 (2*)y) (7.6.9) for every 2 e End Q (A), where R1 is the /-adic representation of EndQ (A). Now W I can be identified with a subspace of W 1 in a natural way. Restrict E, to WI x Wt . By [75, I, Lemma 1.2], we can find a non- degenerate alternating form W i x W I --. F1 SI :

such that

El(x, y)=Trrlirch (S i(x, y))

((x, y) e WI x W I).

Suppose now that A, X, and the elements of 0(F) End (A) are all rational over a finite field with q elements. Let so be the Frobenius endomorphism of A of degree q. Then

S I(Rt (w)x, y) = Si(x, R t (so*)y) . It follows that R I M and R i(v*) have the same characteristic polynomial. We shall now consider A s, f, A, K, and 0 as in Th. 7.14. Let Cs be the canonical polarization of As, and * the involution of End() (A s) defined by C 3. Consider A s as a complex torus Cg/L, and take a Riemann form Es on Cir corresponding to a divisor in C s. Cg can be identified with sicro. By (6) and (9) of Prop. 7.2, we have 'Xss(a)= Xss(ot -')= X ss(a`) for every a e J', so that

194

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

Es([T'arl.x, y). Es(x, [PaT']2Y)

((x, y)

E

C g X Ce)

.

In view of (3.3.13), this implies especially

Œct = NE: if q is prime to N.

(7.6.10)

If we denote by E the restriction of Es to the subspace of Cg corresponding to A, then we have, by (7.5.10),

(7.6.11)

E(0(ag)x, y)= E(x, 0(4)y) .

Let Y be a divisor on A corresponding to E, and ry the isogeny of A to its Picard variety associated to Y. Then (7.6.11) implies that `0(aq)ioy =r y 0(4). Changing Y by algebraic equivalence, we may assume that Y is rational over 0. Then to(ae)wy,=pr.- 13(a ) for every a e Gal (0/Q). Let X be the sum of all distinct Y° with o e Gal(Q/Q). Then X determines a polarization C of A rational over Q, and t O(ag )rx =s0x0(4). If we denote by * the involution of EndQ (A) defined by C, then we have

0(4= 0(;)* if q is prime to N.

(7.6.12)

The corresponding relation holds on (A, 67), by reduction modulo p. Now we can re-formulate Theorems 7.15 and 7.18 as follows. THEOREM 7.24. Let f, A, K, and 0 be as in Th. 7.14. Suppose that r' = ro (N and (7.5.9) is satisfied. Then, up to a finite number of Euler factors, C(s; A/Q, K) coincides with L(s, f). ),

PROOF. We have [K: (2 ] = dim (A), so that d =2. By our assumption, K is totally real, and 0(a)* = 0(a) for every a e K. Let R, and RI denote the 1-adic representations of End(A) and End (A), respectively. From (7.5.1) we obtain

lOrp+4)= R,(0(ap))=ap12 , so that

det [1—u • Rf(ir p)] det [1—u • R',(71,)]. (1--au -1-pu 2)2 Since nrp) and R;(71) have the same characteristic polynomial as remarked above, we obtain det [1—u • nirp)] = 1 —au +PO , which proves our theorem. THEOREM 7.25. Under A', K', and k be as in Th. Euler factors, C(s; A'/k, K') conjugation, and fp is as in

the assumptions (7.5.8), (7.5.14), and (7.5.15), let 7.16 and Th. 7.18. Then, up to a finite number of coincides with L(s, f)L(s, fp), where p is the complex Th. 7.14.

195

t-ADIC REPRESENTATIONS

7.6

PROOF. Let Pp and 0,', be as in the proof of Th. 7.18, I a prime ideal in K', and R I (resp. .1?, R?) the t-adic representation of End (A) (resp. End (A), End (A')). Then, as in the proof of Th. 7.18, we have

(7.6.13)

det [1 2 —u - R?(4)] 2 = clet [14—u • R(400)] ,

(7.6.14)

det [14—u • Ri(e(ap))+P • 4b(P)u 21 4] if N(p) =p 2 ,

1. det [1 4 u 2npl,)] —

.

det [14 —u • g(pp)] - det [1 4 —u - Ri(wip)]

if (p) = PP' •

By Prop. 7.21 and Prop. 7.22, R 1 (0(ap)) has a p and a: as characteristic roots, both with multiplicity 2. Therefore the left hand side of (7.6.14) equals

(1— ap u +p • s1i(p)u2)2(1—ar,u—Fp •

no)

and R?(4) have the same Further, for the same reason as in (7.5.17), characteristic polynomial if (P)=W. Therefore we obtain (7.6.15) det [1 2 —u - R(P)] = det [1 2 u. RN2g)] = 1 apu +PO if (p) = NJ' , —

—

det [1 2-0M(0)] = (1— apu+ Pu2)(1 af,u pie) —

—

if NW= p2 ,

hence our assertion. Consider r0(N) for N= 23, 29, 31. Then the genus of V 5 (= ro(N)\0*) is 2. It has been shown by Doi [14] that A s is simple, and End Q (A s) is isomorphic to Q(N/5), Q(s/2), Q(VS), respectively ; moreover, End (A s) is isomorphic to the maximal order in these fields. Therefore we can put A= A s with these fields as K. There exists an element f(z) = E7.1a„e 2sing of S2(r 0(N)) such that fi T'(n)2= a. f for all n, with a. in K. f Then Th. 7.23 implies that (s ; A s/Q, K) is essentially L(s, f)= Ez= 1 an we use this f to define 0 : K ' EndQ (A)); (s; A 5/Q) is essentially L(s, f)L(s, L where fo(z)= Ez., gel'i, and a is the generator of Gal (K/Q). EXAMPLE 7.26.

—

REMARK 7.27.

),

(A) In the setting of Th. 7.25, we have

L(s, f)L(s. f) = H E I N (1 — a pP ")-1 (1 — ar,P -8 ) -1 x np # N(1 —app -2 +0(p)r 2s) -1(1—af,p-t +sb(p)p1-20-1. Put R(s). N 2(24 -21r(s)2 L(s, f)L(s, fp). By Th. 3.66, we obtain R(s). R(2—s). If N is a prime, we have k= Q('.. /1V), and N 1 mod (4). Moreover, aNaliN =N by virtue of a result of Hecke [30, Satz 61 ]. -

It may be conjectured that, if N is a prime, the abelian variety A' has good reduction for the prime ideal (VAT), hence for all prime ideals in Q(N/N),

196

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

and that the factor (1—aNN- )-1 (1-4,,N-1) - ' is the Euler factor (s; A' /k, K') for (VN); this means that one has exactly C(s; A' /k, KO= L(s, PL(s, fp) without worrying about bad primes. We shall come back again to this question at the end of the next section. 7.27. (B) The jacobian variety A s of V s has often some points of finite order, rational over Q, which are obtained in the following way. For simplicity, let us assume that REMARK

S =Q4 • U' ,

u ,_fra b i

c

—11_c

0 mod (N)} ,

so that r s = Qx • 1 0(N). Let 0 be a character of (ZINZ)x of order r such that 0(-1). 1, and f the subgroup of gx corresponding to the kernel of 0 in an obvious sense. Put T=W .

{[ca bd] E P l aEf}

Then we have

rr . Q. •

{

[

d b

]

I

ro(N) 0(a) = 11 ,

and VT is a cyclic covering of V s of order r; every element of Gal ( V T / V s), as a birational automorphism of VT, is defined over Q. Suppose that (7.6.16)

VT is unramified over V s .

Let P be the projection map of VT to V 3. Put F =Q(e27 ). Then the function field F( V T ) is generated over F(V s)oP by an element h such that hr E F(Vs)oP, and hoÀ=e 2 "th, where A is a generator of Gal (V T / V s). (For the notation F(V s), see Appendix No 4.) Let div T (resp. divs) denote the divisor of a function on VT (resp. V s). Then P(divr (h))=ra with a divisor a of V s rational over F, and ra is linearly equivalent to O. For every a e Gal (FIQ), (h°)T E F(V 3)0 P, so that le = fh with an element f of F (V s)o P. Then we see that a° is linearly equivalent to a. Therefore, if t denotes the point of A s corresponding to the divisor class of a, then t is rational over Q, and rt= O. If ct= 0 for a positive integer c < r, then ca = div s (g) with g EF(Vs), hence P(div T (he))=r. div s (g). It follows that div T (he)=divr (goP), so that he E F(V s)o P, a contradiction. Therefore t must be of order r. Thus we have shown (7.6.17) The jacobian A s of V s has a point of order r rational over Q under the assumption (7.6.16) . The verification of (7.6.16) in each case can easily be done by checking which parabolic and elliptic elements of r.(N) are contained in rT . For

197

CLASS FIELDS OVER REAL QUADRATIC FIELDS

7.7

instance, VT is unramified over V s at the cusps, if N is square free. -

Let, for example, N = 23, 29, 31. By the above principle, we can find a point t of A s of order r = 11, 7, 5, respectively, rational over Q. Let aneninz, 5), Q(/T), Q(i/-5), as considered in Ex. 7.26, and let f(Z)=E7.1 K= Define an with a, = 1, be a common eigen-function of all r(n)2 on isomorphism 0 of K onto EndQ (A s) with respect to f, as in Th. 7.14. Consider reduction modulo p of A and t for a prime p not dividing Nr. If rp and are as above, we have gpi = E and r E = pf, so that [1+p—é(ap)]i =0 by (7.5.1). Let a be the integral ideal in K generated by r and 1+p—a, for all such P. Then 0-. (a)i = O. Since j is of order r, a can not be the identity ideal. Therefore we obtain

s2(r ow».

( 7.6.18) 1—ap +p 0 mod (I) for every prime p not dividing Nr, where I is a prime ideal in K dividing r. (Actually the congruence is true also for p= r.) Similarly, let N= 11, 14, 15, 17, 19, 21. In these cases A s is an elliptic curve. The above principle ensures the existence of a point of A s, rational over Q, of order r =5, 3, 4, 4, 3, 2, respectively. Then we obtain, by a similar and simpler argument,

(7.6.19) If f(z)=E; =,a„e"'", with a,=1, is a generator of sjr,(N)) for these values of N, then 1—a p ±pL=_ 0 mod (r) for every prime p not dividing Nr. 7.7. Construction of class fields over real quadratic fields We shall now show that certain points of finite order on the abelian variety A' of Th. 7.16 and Th. 7.18 generate non-cyclotomic abelian extensions of real quadratic fields. Let us first recall the properties of A' and other

symbols. N: a positive integer. cfr : a character of (ZINZ)m of order 2 such that 0(-1) = 1. f: an element of s2cr o(N 0), j. e., a cusp form of level N satisfying ),

f ((az+b)(cz +d) - ')(cz+ d)-2 = 0(d) f(z)

for all La db ]e ro(N).

We assume that f is a common eigen-f unction of all the Hecke operators T'(n), x, on .32(r 0(N), 0 and ,

f(z)=E:)..i a Tie2'

f I r(7)2 1 0 = af

(cf. Th. 3.43) .

Further we assume that the algebra of all r(n)2,0 can be generated only by the subset {T'(n) 210 I n prime to N}. (See Remark 3.60, especially the results

198

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

of Hecke mentioned there.) k: the real quadratic field corresponding to the kernel of il). e: the generator of Gal (k/Q). K: the field generated by the numbers a n over Q for all n. p: the complex conjugation. K'= Ix E K I xP . xl. We have shown that K' is totally real, and K is totally imaginary. Further we have (7.7.1)

aft .0(n)a„ if n is prime to N.

In Th. 7.14, wi have obtained an abelian variety A and an isomorphism 6 of K into End ci (r ); A and 6(a) for all a e K are rational over Q; the couple (A, 5) is characterized by (1) and (2) of Th. 7.14. Further A has an automorphism p, rational over k, such that (7.7.2) (7.7.3)

p2 = 1 ,

p . 0(a) = e(a0)/1

(a

E

K),

IL` = —ft.

We put (7.7.4)

A' .(1+1t)A.

Then we have seen that A' is an abelian subvariety of A rational over k, and (7.7.5)

A= A'±A",

A ,' . (1 p)A . —

Denote by 6'(a) the restriction of 6(a) to A' for every a E K'. Then 6' is an isomorphism of K' into End Q (AO. We can also define an isomorphism 6" of K' into EndQ (A") by 6"(a). 6'(a)'. Obviously 5"(a) is the restriction of 6(a) to A". Let p be a rational prime not dividing N, and p a prime ideal in k, dividing p. As is noted in § 7.5, (7.7.6)

A and A' have good reduction modulo p.

Let the tilde denote reduction modulo p, and np the Frobenius endomorphism of A of degree p. Further let 4 be the element of End (A) such that 74 r p .p. From (7.5.1) we obtain (7.7.7)

irp+O(Orp 157(ap).

Let o and o' be the maximal orders in K and K', respectively. In genera/ 0(o) may not be contained in End (A). However, by changing A by a suitable isogeny over Q, we may assume that condition. In fact, let m be a positive

7.7

CLASS FIELDS OVER REAL QUADRATIC FIELDS

199

integer such that 0(mo)c End (A). By [81, § 7.1, Prop. 7, Prop. 8], we can find an abelian variety A„ an isomorphism 01 of K into End,a (AO, and an isogeny A of A to A, such that : (i) A • 0(a) = 0i(a)2 for all a E K; (ii) 0,(o) c End (A1) ; (iii) Ker (2). {t e A10(mo)t = O}; (iv) A„ 2, and 01(a) for every a eo are rational over Q. Observe that p maps Ker GO onto itself. Therefore we can define an automorphism p, of A 1, rational over k, by p12 =21/. Change A, 0, p for A„ 0 1, P1 write them again A, 0, p. This change does not disturb (7.7.2, 3). Defining again A' by (7.7.4), we have still (7.7.5,7). Of course the new A may no longer be a subvariety of the jacobian A s of the curve V s, which we do not need in the following discussion. All what we need is (A, 0), (A', 00, an automorphism p of A, and a cusp form f(z)=E7. i aft ev% which satisfy the above (7.7.2,7) and 0'00 c End (A') .

0(o) c End (A),

(7.7.8)

In other words, these conditions and (7.7.9) below are the " axioms " of our theory, for which we have (not yet, so far as (7.7.9) is concerned) shown the existence of the objects satisfying them.18' Let b denote the different of K relative to K'. Put 6.= {x E 0

XP

=

—

x}.

Let 6 be the ideal of o generated by 6, . Then b C 6. Observe that an E 6 , if PROPOSITION

7.28.

For every x

E 60,

one has —Nx/Q(x)ENk,,Q(kg). Moreover,

—N(6) e Nk/Q(kl•

PROOF. Let X E 60. Then x- '6 0 c IC'. Let e be the fractional ideal in K' generated by x- '60 over o'. Then we see that 6 = xoe, hence N(b)= NvQ(x)N(e) 2. Tberefore, to prove our proposition, it is sufficient to prove —Nic,Q(x) eNk,Q(kx). Take a basis {ah, ••• , con} of g(A0 rational over k. Then { 0, ••• , co: } is a basis of g(A"). By (7.7.2) and (7.7.5), we see that 0(x) maps A' to A", and A" to A'. We can put (.4 o 0(x)= Yhialt (h =1, , n) with 3° hi in k. Applying € to this relation, we obtain coho 0(x) =E t yl,a4 , hence (o no 0(x2) = E id Ai yij coi. Now the representation of K' on g(A0 is equivalent to the regular representation of K' over Q. Therefore —NK ,Q(x)= N K .,Q (x 2)= det (

Yh.t) •

det (y,)' E

Nic ,Q(k x ),

q. e. d. 16) Probably it is not always best to assume (7.7.8). The reader should consider the condition and the change of A, 8, p rather tentative, or made just for the sake of simplicity. The same remark applies to (7.7.9, 15) below.

200

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

7.28'. Hecke [30, Satz 61] proved that (4,4= N if N is a prime_ Therefore we have N e NK /K.(Kx). It is noteworthy that this is reciprocal to Prop. 7.28. REMARK

Let us now make the following assumption :

(7.7.9) b is prime to 2. The fields K and K' are determined by f, so that (7.7.9) can be viewed as a condition on f. If q is a prime ideal in K prime to 2, then b is not divisible by q2, by a well-known property of the different. Therefore, under the assumption (7.7.9), b is a square-free ideal in K, and b P =b. Put c = Nicl(b).. Then c is a square-free integral ideal in K', and co = b2. Put

X= it e A 1 6(b)t = 0} = A[b] ,

1)--xn A',

3=gnA".

We see easily that u and 3 are o'-modules, and g is an o-module. PROPOSITION 7.29. The o'-modules u and

a

are isomorphic to

o' lc, and

PROOF. By Prop. 7.20, we see that g is isomorphic to OM'. Observe that o/6 is isomorphic to o'/c. From our definition of b, we see that g is stable under p. Therefore (l+p)g c » and (1 —p)C3. Since b is prime to 2, every element t of g can be written as t =2s with S E g, so that t =(1+p)s+(1— p)s E 1)+8. If u E tm 3, we have 2u=0, so that u= O. This proves that g = %pea. Extend 6 to an automorphism 3 of 0. Then 3 gives an o'-isomorphism of u. to 3. Therefore both u and 3 must be isomorphic to o'/c. Let F denote the field generated over k by the coordinates of the points of g. By the above proposition, we can find an element s of u and an element t of 3 so that

(7.7.10)

u = 8(o')s ,

3= OM/ .

Then we have F= k(s, t). It is this extension F of k whose class-fieldtheoretical structure we are going to investigate. THEOREM 7.30.

The field F is an abelian extension of k, which is unramified at every finite prime of k not dividing N(c)N. Moreover, if m is a positive FA rational integer prime to N(c)N, and a = (--), then x° = mx for every xeg.

(m)

PROOF. By Prop. 7.23, every prime ideal in k, prime to N(c)N, is unramified in F. Observe that the map a —, 13(a)s defines an isomorphism of

CLASS FIELDS OVER REAL QUADRATIC FIELDS

7.7

201

o'ic onto ». Let r e Gal (0/k). By our definition of u and 3, r maps t) and b onto themselves. Therefore F is a Galois extension of k, and one has (7.7.11)

sr = 8(gr)s ,

tr =

with elements gr and It, of o' prime to c. We see easily that r ,---*(gr, hi) defines an isomorphism of Gal (F/ k) to a subgroup of (o/c)-x 2, and hence F is abelian over k. Let p be a rational prime not dividing N(c)N, p a prime ideal in k dividing p, and q3 a prime divisor of 0 which extends p. Let k. or 13(X) denote the object obtained from X by reduction modulo 13. Further let rp and ei; be as in (7.7.7), and a a Frobenius element of Gal (0/k) for 13. Then. Za) on F. P (I) First consider the case 0(p) = -1, so that N(p)=p2. From (7.7.7) we obtain R-;,—p= r p(zp -4)= rp • e7(ao. By (7.7.1), we have ap e bo, so that (ir;,—p)1=0 for all x E g. Since ir2p x= 13(x ) , we have 13(x°—px)= 0 for all x E g. By [81, § 11.1, Prop. 13], reduction modulo 13 defines an isomorphism of g onto 13(g). Therefore we have x° =px for all x e g. (II) Next suppose that o(p) = 1. Then (p)= ppe in k. Let 3 be an element a

—

(

of Gal (0/Q) which coincides with e on k.

Then Fa = F, and 3 - ia3 = (

on F. Now, for every prime ideal 1 in K' dividing c, consider I-adic coordinate systems on A' and on A's. We have fo= 2 2 with a prime ideal 2 in K. By our definition of b, we can find an element 2 of bo such that 2 is divisible by 2 but not by 22. By (7.7.2), we have 0(2)p = p0(2). Then we see that 0(2> maps il/[1] into A"[I], and A"[1] into AV]. (For the notation AV], see (7.6.3)) Moreover, we have —

Ker (61(2))n AM = A[2]---=g n A[i], so that Ker (0(2))r\ ATI] =1) n AV] Ker (OM) n A"[1]= 3 r\

Am] --• °VI.

Put q, =1)(-A 24/[1] and al = an A"[I]. Since Abc co, we see that 0(2)(A VD c di ,

8 (2)(A"10 C Ill •

Comparing the order of the modules, we obtain exact sequences 8(2)

0 — », — 0

—

&

—

8(2) A"[1] --- 0, — 0 -

202

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

Take elements u and y so that th = 0(00u ,

AV] =fli+O(V)y .

al =----0(0')145 ,

A/TO = al -F0(0y 8 .

Applying (3, we obtain

F/k The symbols cr=(— ) and gr„, h, being as above, we have V= 0( gg)u, P ziaci = 0(h 4)ua . Put v° = 0(c)u+ - 0(d)v with c and d in o'. Then (0(2)v)° = 0(2)u° = 0(d)0(2)v. On the other hand, since 0(2)v e a, we have (0(2)v)° = 0(h 4)0(2)v. Since 0(2)v generates at over 0(o0, we see that d -- h, mod L Thus we obtain ya= 0(c)u- F 0(14)v with c e o'. Similarly v ""= 0(e)ze +0 (g,)v 8 with e e o'. In other words, if we define I-adic representations RI and 311 of Gal(4)/k) on A' and on A's as in § 7.6, then

na) ..-- [ti

e1 gr., -1

R(o) .-... [0124

mod I.

By (7.6.7) and (7.6.15), we have ap Egu -i-ha ,

p -E. gah,

mod I.

This holds for all prime factors I of c. Therefore (7.7.12)

ap a-- ga l h„,, - -

p .--_ gah,

mod c.

Now we have v4g5-1 = 0(e)u--1-60(g4)v ,

u"8-1 =0(h,)u , (u3)8-1 " = 0(g a)ua ,

(1/;)" 1" = 0(c)w 3 + 60(h,)va ,

so that

(7.7.13)

Rf (30.3 - 9 .:.-s

c) ,

M(3 -1 o.c1)-_,..

(o-)

mod I.

Therefore

RI(o- • (3cr3 -1) -- [to)

pl

mod I,

1)if(a• (3 - Icr(3).E[ 10)

p* ]

mod I.

Flk File Since (3cra'=(3 - icra=(—) on F, we have, if r=(—)n the r =.7-5475 - ' (P) ' Pi = a • 3- '473, so that xr=px for every xetH--a=g. (III) Let m be a positive integer prime to N(c)N, and a=( F/k ) We (m) •

CLASS FIELDS OVER REAL QUADRATIC FIELDS

7.7

203

can find a rational prime p, prime to N(c)N , such that p-- m mod N(c)N kle (f), where f is the finite part of the conductor of F over k. (Note that Nkm(t) is divisible only by the prime factors of N(c)N, since every prime ideal in k prime to N(c)N is unramified in F.) Then a = f Fne ), and hence, by the

k (P)

results of (I) and (II), x° =px= mx for every xe g. This completes the proof. COROLLARY 7.31. Let cnZ= qZ with a positive integer q, and let C be a primitive q-th root of unity. Then C e F, and F # k(C).

Flk )= id., P we have g‘, -- h„ -- 1 mod c, so that N(p) -- 1 mod qZ. By class field theory, this shows that k(C) c F. Take a rational prime p not dividing N such that PROOF.

Let 13 be a prime ideal in k not dividing qN. If a= (

p .. —1 mod qZ, and put r = ( F05/ )k ). Then r = id. on k(C), but sr = —s by -

Th. 7.30, so that r # id. on F. Therefore F # k(C)Let r denote the ring of all algebraic integers in k, and rp, for a prime ideal p in k, the p-completion of r. For every integral ideal a in k, define a subgroup u(a) of the idele group /el of k by

(7.7.14)

u(a) = {(xp) e Hp r; i xp — 1 e rpa for all pl .

7.32. Let F be an abelian extension of k, and iv the subgroup of k74 corresponding to F. Suppose that u(aln) C in with an integral ideal a in k, a prime ideal I in k, and an integer n> 1. Further suppose that [F: k] is prime to N(I). Then u(al)Cm. LEMMA

PROOF. Our assertion follows immediately from the equalities [F: kJ = [ le:4 : w]

and

[u(al) : u(aI")] = N(t)n - 1 .

Put q = N(c). We shall obtain further information on the conductor of F under the following set of assumptions :

(7.7.15) (i) q is a prime; (ii) N is square-free; (iii) N is prime to q(q-1); (iv) sb(a) = (*),

Then k=QWN). Since 0(-1)= 1, we have 1‘1..-1 mod (4). By Prop. 7.28, we have qr= qqg with distinct (principal) prime ideals q and ci` in k. Observe that o'ic is canonically isomorphic to Z/9Z. Therefore the numbers gr„, h, of (7.7.11) can be taken from Z modulo qZ. Thus we have an injective homomorphism

(7.7.16)

Gal (Flk) s r — (gr, k)e (ZIqZ)".

204

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

It follows that [P.': k] divides (q 1) 2. -

k.

THEOREM 7.33. Let i denote the product of the two archimedean primes of Then the conductor of F over k is exactly qq`i.

PROOF. Let f denote the conductor of F over k. f must be of the form b f = j • qa(q`) • MIN ma

On account of Cor. 7.31,

with integers a > 0, b> 0, c O (e depending on m), where m runs over all prime ideals in k dividing N. Since [F: k] divides (q-1) 2, we see, in view of (7.7.15) and Lemma 7.32, that a, b, c are j. e., f is of the form iggan with a square-free ideal n dividing N. To show that n= r, take any prime ideal m dividing N, and let m' be the ideal such that mm'. Ari— V• r. Let xeq. Since rimmt is isomorphic to Z/NZ, we can find a positive rational integer y F/k such that mod m, and mod gm'. By Th. 7.30, we have (--) =1. (Y) Let x' denote the image of x by the natural injection r— k. Then fe]

ii ( y /kix9

("If)

1

(see § 5.2) .

This shows that r; is contained in the subgroup of k; corresponding to F, so that m is unramified in F. This completes the proof. Our next task is to investigate whether F is actually the maximal rayclass-field of conductor iqqa. Let 24 0 denote the fundamental unit of k, and I)„ the smallest positive integer such that u;,"n is totally positive and utn 1 mod (qq9n. Further let F„ denote the maximal ray class field modulo i • (qq4)n over k, i. e., the subfield of ke, corresponding to the subgroup le C., • u((qq9n) of le:4 , where ) is as in (7.7.14). If ck denotes the class number of k, one has (7.7.17)

[Fa : k] ,-2ck(q-1) 2q" -2/v„.

In the numerical examples for small N (see below), we notice that u 0 -1 or u4-1 is divisible by q, according as u, is totally positive or not. Observe that :4-1= u o • Tr k/Q (u o), if Nk,Q(uo)= —1. PROPOSITION 7.34. Suppose that N k,, Q(u 0)= —1, and 14-1 is divisible by q. Let et be the highest power of q which divides Tr k/Q (uo), and g the union of the fields F„ for all n. (In other words, g is the largest abelian extension of k in which only q, q`, and the archimedean primes are ramified.) Then g is generated over F„, by the qn-th roots of unity for all n. PROOF. We have 4.1-1-ey with an algebraic integer y prime to q. Therefore it can easily be shown, by induction on n, that v = 2qn, and

CLASS FIELDS OVER REAL QUADRATIC FIELDS

7.7

205

by (7.7.17). Put C„ -= exp [27ri/qn]. Then hence EF„.1.m : k3 = ck (q k(. 4.,,,) k( m). In fact, if k(C A4.„,)n Fm is larger than k( m), then Fm must contain k(C„, 4.1). Take a rational prime p so that p 1+qm mod (r+ 1) and «P)= 1. Then the prime ideal pr in k decomposes completely in F„, but not in k( m.1.1), since p2 *1 mod (qm•"). Thus k( a.4.„,) and Fm are linearly disjoint over k(„,), so that 1)2qn+2m-2

-

—

[F„,(C„4„,) k] = qn [Fm : =

Since F„,(Cn+m)C

F n+m,

we obtain

F m(n+m) = F n+m,

k] . which proves our proposition.

Observe that every element of (r/qr)x is represented by a totally positive element of r. For every totally positive element a of r prime to q, consider =( Fik and then the element (g„, kJ of (c'/c)' 2 as in (7.7.11), or of (Z/qZ)" ar as in (7.7.16). Observe also that r/qr is isomorphic to (ZIqZ) 2. Thus we obtain a sequence of homomorphisms (7.7.18)

(r/qr) — Gal (F/k)—(c//c))12 —(Z/qZ)",

(Z/qZ)x 2

al—*

a= F/k ar

The first and last arrows are isomorphisms, determined up to the change of (g., 14) does not depend on the choice factors. Recall also that the map a of the points s and t. For every (x, y) e (Z/qZ)T 2 with x and y in (ZIqZ)x denote by (g(x, y), h(x, y)) the element of (Z1qZ)' corresponding to (x, y) through the composed map of the homomorphisms of (7.7.18). In this way, we obtain a homomorphism

(x,

(7.7.19)

(g(x, y), h(x, y))

of (Z/qZ)x2 into itself. We can naturally ask the following question (if k is of class number one) : (7.7.20)

Is the map (7.7.19) the identity map, up to the change of x and y?

We shall later show that this is the case at least for N=29, 53, 61, 73, 89, 97. First we prove a few simple propositions. PROPOSITION 7.35. If [F: k ] =(q-1) 2, N"(u0). —1, and the class number of k is one, then F=F„ and 4 1 is divisible by q. -

PROOF.

Since F c F, and 1), 2, we have [F:

Therefore, if [F. : k].(q

(q 1)2 . -

-

1) 2, we have I), = 2 and F=Fi.

206

ZETA-FUNCTIONS OF ALGEBRAIC CURVES PROPOSITION

7.36. The map (7.7.19) has the following properties :

(a) g (x, x). h(x, x)= X; (b) g(x, y)h(x, y)= xy; (c) g(x, y)= h(y, x).

PROOF. The first relation follows from Th. 7.30. In (7.7.18), we can take a so that Nk,e(a) is a rational prime p not dividing qN. If x a mod q and y a mod cr, then y a' mod q, so that xy.p mod qZ. From (7.7.12) we obtain p-m g(x, y)h(x, y) mod qZ, hence (b). The proof of Th. 7.30 shows that F/k

i f a=-() and r =--- ( --) then 134 i s PROPOSITION 7.37.

ha m g r mod c. This proves (c).

There exists a rational integer b such that g(x, y)=

50

h(x, y)=

X 1-1

PROOF. Since (ZIqZ)* is cyclic, we have g(x, y)= xay°, h(x, y)=,eya with integers a, b, on account of (c) of Prop. 7.36. From (a) of Prop. 7.36, we obtain a-Fbm 1 mod (q-1), hence our assertion. PROPOSITION 7.38.

The answer to the question (7.7.20) is affirmative if and

only if

(7.7.21)

ap Trk/Q (a) mod c

for every rational prime p not dividing qN, and for every totally positive element a of r such that o(p)= 1 and NklQ (a)=p. Moreover, (7.7.21) is satisfied by all such p and a, if it is satisfied by at least one a such that alaz generates

(By virtue of the generalized Dirichlet theorem, one can always find an element a of r such that N klaa) is a rational prime not dividing qN, and a/a' generates (r/q)x.) PROOF.

Let NvQ (a)=p and

o(P)= 1

with a totally positive element a of

and a r and a rational prime p. Further let a = (--), ar

xo, cr'==-y 0 mod q

with rational integers xo and y 0. Then Trk/Q(a)=- x0 -1- Y0 mod qZ, and g(x o, yo)+11(x o, Y

ga +h. ap mod c

by (7.7.12). Therefore, if (7.7.19) is the identity map, we obtain (7.7.21). Conversely, suppose that (7.7.21) is true for a, and a/a' is of order q-1 in (r/q)x. Then g(xo, y 0)-1-h(x 0 ,y 0) mod qZ . By (b) of Prop. 7.36, we may assume, changing x and y if necessary, that

207

CLASS FIELDS OVER REAL QUADRATIC FIELDS

7.7

[K: Q]

N

29

37

41

53

61

73 89

97

2

K'

Q

2

2

4

4

4 6

6

Q

Q

Q(/) 2)

Q(V3)

up

N(c)

5+Vff 2

5

1 6+ V37

2 32+5V-41

7

13

7 +N/6-8-

2

39+5V-fil 2

p

(1:15 )

2 3 5 7 11 13 17

— — + +

V_ 5

+ —

—1 —2V=5

2 3 5 7 11 13 17

— + — + + — —

2i —1 —2i 3 —3 —6i 2i

2 3 5 7 11 13 2 7 11 13 17 29

+ — + — — —

—1 2V-22 —2V---2. 2V-2

2 3 5 2 3 5

— + + + + —

(-1+v')/2 VT)/ 2 (1+

— + + + + +

ap

—V-5 —3 2

—2—VT 3V21-2V 2 —3

—3+3V2 —1— VIV3

Q(V5)

89 1068+1251/n

Q(a2)

5 500+53A/89

2 3

+ _

as + a2 -3a-1 = 0 a6+17 a4 +83a2+ 125 = 0

467 5604+ 569.7g7

2 3 5

± + —

a 3-3a— 1 = 0 a3 -3a— 1 = 0 a6+27a4+204a2 +467 = 0

Q(a2)

208

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

xo g(x o, y o),

yo h(x o, y o)

mod qZ .

If b is as in Prop. 7.37, we have x„:---E xr'yt mod qZ, so that (x oly o)b 1 mod qZ, hence b 0 mod (q - 1). Therefore we have g(x, y)= x, h(x, y)= y for all x and y in (ZIqZ)x. This completes the proof.

The table in p. 207 gives the Fourier coefficients a p for every prime level N 97 with 0(x)= ( -7x\7-). Note that sjro(N ), 0)= {0} for all primes N < 29. These values az, have been computed by Doi and Naganuma (by hand) and by Trotter (by computer) by means of the trace-formula of Eichler and Selberg. For each N in the table, the Hecke operators Ti(n) 24,5 generate, over Q, a field whose degree equals the dimension of sjro(N ), 0) over C. (This is not necessarily true for larger N.) Therefore we find unique (A, 0), K, f belonging to N and 0. The uniqueness should be understood as follows : K is unique up to the conjugacy over Q; f =E7,3 1 ae2 t" is unique up to the conjugacy of the an over Q. We have [K :Q]=2 [I(' : (1] = dim

(s,cro(N ), OD •

In the table, the values of ap, or the irreducible equations for them, are given with respect to a fixed conjugate of K, and a fixed isomorphism O. We observe that :4 - 1 = u o . TrklQ(uo) is divisible by N(c) for all these N. For example, if N =61, we have a z = so that c = (-4-1/ -3-) and N(c) = 13. On the other hand, u.=(39 -1- 5-V 61)/2, hence Nt ,Q(240= 39. Now we see that (7.7.9) and (7.7.15) are satisfied for N= 29, 53, 61, 73, 89, 97. For these N, the class number of k (kV N) is one. ,

THEOREM 7.39. The following assertions hold (at least) for N=29, 53, 61, 73, 89, 97. (1) The map (7.7.19) is the identity map, up to the change of x and y. (2) F = F1 , j. e., F is the maximal class field over k of conductor (3) There is no abelian variety defined over Q which is isogenous to A' over (4) A' is simple, and End Q (Al= O'(K/). (This assertion holds also for N=37, 41.) We shall discuss here only the case N =97 ; the other cases can be treated in a similar and simpler way. If N=97, the table shows that q= 467. The number r satisfies the equation K'=Q(r), r= (co+co- ), co = PROOF.

(*

)

X 3 -3X-1= 0

which has r, 2— r2 = -(w2 d-w -2),

—(a) 4 +0)-4 ) as its roots. Since

7.7

CLASS FIELDS OVER REAL QUADRATIC FIELDS

209

NK.,Q(20—r) = 17.467, there is a unique prime ideal which is a common divisor of 20—r and 467. Choosing 0 suitably, we may take this prime ideal as c. The table shows that a, is a root of (*). Now one has r -7=- 20,

2—r 2 z-_-_- 69,

r2 —r-2 z-_-_- 378 mod c.

Our theory tells that X 2 —a p X+p :---.- 0 mod c has roots in oqc if gp)=1. Therefore a, must be r, since the congruence has no solutions if a, is 2—r 2 or r2 r -2. Solving the congruence X 2 -20X+3=- 0 mod (467), we obtain (x,y). (97, 390) as solutions. One has 3 = aag with a -,- 10-HAT, which is totally positive, and Tr k,Q(a)--= 20 ,. -_- a, mod c. But 97/390 is of order 233 modulo 467. Therefore the argument of the proof of Prop. 7.38 shows that the exponent b of Prop. 7.37 must be divisible by 233, i. e., —

(**)

g(x, y) = ± x ,

h(x, y). ±y .

Similarly, checking the solvability of X 2 — a2 X +2:--,-- 0 mod c, we find that a2 is either 2—r 2 or r2 —r-2. If a 2 = r2 —r-2, the congruence has solutions (x, y).(197, 339). On the other hand, 2 =PP' with p=-(69+7Vgn/2 which is totally positive. Then we observe that (197, 339) does not fit the relation (**). Therefore we must have a2 = 2—r2, so that Tr k,Q(P) --= 69 -7,- a, mod c. The congruence X 2 -69X+2 - ,7 0 mod (467) has 412 and 433 as solutions. Since 412/433 is a primitive root modulo 467, we obtain the first assertion by virtue of Prop. 7.38. Then, by Prop. 7.35, we have F----F,. To study the structure of End Q (A'), we consider the p-th power Frobenius endomorphism 9p of A' modulo p, where p is a prime ideal in k=q-V97 ) such that N(p)= p, gm= 1. On account of (7.7.7), sop satisfies the equation X 2 apX Fp= 0, where we identify a p with a,(09). If p= 2, we have a 2 =2—r2 . Then we see easily that p =2 remains prime in K' and decomposes into two prime ideals F43 and 11 in the field K'(92). It can easily be seen that K'(V2) is not Galois over Q, and K' is the only non-trivial subfield of K'(92). Since ç'2 is divisible by 13 or 1-3, but not by 431-3= (2), we see that If'(v 2)= Q(93) for every positive integer n. Now every element of End ci (A' mod p) commutes with V; for sufficiently large n. Therefore, by [81, § 5.1, Prop. 1], we obtain —

-

(***)

End Q (A' mod p)= Q(91)= K/(9p)

for p = 2. If p =3, we have a prime ideal 1=((1—w)(1—co - 9) in K' such that t' = (3), and t decomposes into two prime ideals in K'(9.3). Then, by the same reasoning as above we obtain (***) for p =3. But a2 = (0 1--w -2)a-_ I mod t, so that X 2 a,X +27.7 0 mod t is irreducible. It follows that t remains prime in K'(92). Therefore K/(9 2) is not isomorphic to K'(9,). This shows that Endci (A') =K', and hence A' is simple. —

—

-

-

-

210

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

To prove (3), let B be an abelian variety defined over Q, and A an isogeny of A' to B, rational over Q. Then we can find an isogeny 2' of B to A' such that 2'2= deg (A). id A ., where id A , denotes the identity map of A'. Let e denote the restriction of 0(a5) to A'. Since a, E 50, and NK,o(as ) = 467, e is an isogeny of A' to A" of degree 467. Extend e to an automorphism (3 of Then =0'(e) with an element e of o'. .I'Ve is an endomorphism of A', so that 2' By (7.6.1), we have

NKvQ (e)2 = deg (0'(e)) = deg (2'24e) = 467 deg (A)s , which is a contradiction, since 467 is a prime. This completes the proof.

In the case N =89, there is a possibility that one should take 53 instead of q = 5, and consider the congruence a p a Tr (a)

mod c3

instead of (7.7.21). Then the coordinates of some points of order 53 on A would generate F, over k. Recently W. Casselman [4] has proved that the abelian varieties A' for N =29, 53, 61, 73, 89, 97 have good reduction for all prime ideals in k =Q(1/ N). As a consequence of this result, we shall now show that (s ; A'/k) is exactly L(s, f )L(s, = (E7 (Inn' s)(E7,3 an') if N =29. In Th. 7.25, we have seen the coincidence of the Euler factors of C(s ; A' /k, K') and L(s, f)L(s, fp) for all the prime ideals in k other than n = N/N • r. Now we see that A' and A" have the same reduction modulo n. If a E o and aP = a, then 0(a) defines an isogeny of A' to A". Taking this modulo n, we obtain an element of End (n(A0), which, together with #'(If'), generates a subfield S? of Endo (n(A0) isomorphic to K. Let r be the Frobenius endomorphism of n(A') of degree N. Since the elements of Si' n End (n(A0) are defined over the prime field, so commutes with those elements, so that so is contained in S? by virtue of [81, p. 39, Prop. 1]. (If N =29, one has dim (A') = 1, so that the assertion follows from (5.1.5).) Therefore so has an element ço„ of K as an eigen-value. By the Weil theorem, we have 141 2 = N for every isomorphism r of K into C. If N= 29, we have K= Q(/), so that the condition I so,1 2 = 29 implies —

). Then a Wo= ±3 4- 2. Put a = (29+5N/29 )/2 = V2g• u o, and a = is totally positive, and a 2 mod Sr. so that (gu, Um (2, 2) mod (5). By virtue of (7.6.7) with (5) as I, we obtain Tricto(so„) -- gu +11, -. 4 mod (5). It follows that . This agrees with the Fourier coefficient aN of our cusp Wo= form f , given by Hecke [30, pp. 904-905]. Thus we have the exact equality (s ; A' /k)= L(s, f)L(s, f D) for the elliptic curve A' in the case N =29. We can also verify that End o (n(A0) is actually isomorphic to K for the

THE ZETA-FUNCTION OF AN ABELIAN VARIETY OF CM-TYPE

7,8

211

above six values of N. It need hardly be said that, in this section, we have merely hegun the investigation of the mysterious connection between real quadratic fields and the cusp forms of Hecke's " Nebentypus". Also, while the above discussion has been restricted to the case of weight 2, there is some numerical evidence connecting cusp forms of weight > 2 with real quadratic fields in a similar way. The author hopes to treat this question on some other occasion. 7.8. The zeta function of an abelian variety of CM type -

-

Let A be an abelian variety of dimension n, defined over an algebraic number field k, such that End o (A) is isomorphic to a CM-field K of degree 2n. We shall now determine the zeta-function of A over k.rn We fix a polarization C of A and an isomorphism 0 of K into Endo (A), and define couples (K, 0), (K*, 0*) as in § 5.5. Further we assume (5.510) and the following two conditions :

(7.8.1)

All the elements of O(K)r\ End (A) and

(7.8.2)

K* C k.

C are rational over

k;

(Actually (7.8.2) follows from (7.8.1), and if A is simple, the converse is true, see [81, § 8.5, Prop. 30] ; cf. (5.1.3) when A is an elliptic curve.) Take a Zlattice a in K and an isomorphism e of C'/u(a) onto A as in (5.5.9), where u is defined by (5.5.8). Put 77( y) = det (0*( y)) = 77(Nk/x-(x))

(yEKr), (x 14) .

Recall that 72(y) E K1 for every y E K r. Since (K*, 01 is a CM-type, if we denote by p the complex conjugation in Kx and its obvious extension to K1, we have (7.8.3)

p(x)p(x)P = Nkm(x)

(x E k;).

PROPOSITION 7.40. (1) Every point of finite order on A is rational over kab. (2) For x E kl there exists a unique element a of If' such that a• p(xra=a, aaP = N(il(x)), and e(u(v))rx ,k1 = e(u(a - p(x) - 'v)) for all v E Kla.

The map x.—) a defines obviously a homomorphism of kl into le. 17) If the reader is interested only in the one-dimensional case, he can simplify the whole discussion by assuming that A is an elliptic curve, K is an imaginary quadratic field, 6 is normalized in the sense of § 5.1, u(a) = a, K* = K, 0 = 0* = id., and p(x) = NkiK(x). The polarization can be disregarded ; Th. 5.4 can be used instead of Th. 5.15.

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

212

Let k' be the field generated over k by the coordinates of all the points of finite order on A, i. e., the points e(u(v)) for all v e K /a. For x e k; , let r be an element of Gal (k1k) such that r= [x, k] on k' fl k ab. Put y= NkIK .(x). Then r= [ y, K*] on k'nKL. By Th. 5.15, there exists an isomorphism el of Cn/u(p(x) -1a) onto Ar such that (Ar, Cr, Or) is of type (K, 0; p(x) -1a, N(il(y))C) with respect to e', and e(u(v))r =E'(u(p(x) - iv)) for all v E K / a, where C is as in Th. 5.15. Since r = id. on k, we have (Ar, Cr, Or) = (A, C, 0). Therefore we can find a linear transformation T in Cn such that : (i) T(u(p(x) - la)) = u(a) ; (ii) E' = eo T; (iii) T commutes with the elements of 0(K); (iv) T sends the Riemann form E of (5.5.15) to N(il(y))- E. Then T =0(a) with an element a of Kr, so that a • p(x) - la= a. On account of (iv), we have aaP = N(il(y)) = N(il(x)). From (ii), we obtain PROOF.

e(u(vpr = equ(p(x)' - 'v)) = e(u(a • p(x) 1 v))

(1) e K/a).

Put, for every positive integer n,

(7.8.4)

W. = Ilv e K; I wa =a, wv= v for all v E n -la/a} .

Let W', be the projection of W, to the non-archimedean part of K;, and let z be the non-archimedean part of a • p(x) - '. Then 2 E W; and E(u(v))' = E(u(zv)) for all v E K/a. Obviously the element z of IC is uniquely determined by the last equality. Moreover, we see easily that the map T ,— Z defines a homomorphism of Gal (k'/k) into W. This is injective, since k' is generated by the coordinates of e(u(v)), and hence r is completely determined by e(u(vDr. It follows that Gal (k' /k) is abelian, which proves (1). Then the above a has the property of (2). The uniqueness of such an a is obvious. PROPOSITION 7.41.

For 2=1, ••, n, define a Cr-valued function çb2 on k ; by

0 2(x) = (a/ p(x)) 2

(x E k)

with the element a of (2) of Prop. 7.40, which is unique for x, where ( )2 denotes the component of an idele at the 2-th archimedean prime of K (with any ordering). Then Oi is a continuous homomorphism of k; into C" trivial on k" (1. e., 01 is a GrOssen-character of k).

PROOF. It is obvious that 02 is a homomorphism. If x e kt, we have [x, k] = id. in (2) of Prop. 7.40, so that we can put a = p(x), hence 02(x)= 1. If x e kœ', we have again [ x, k]= id., and a =1, so that 02(x)= (p(x) - 92. Now take a positive integer n > 2, and let en) be the field generated over k by the coordinates of e(u(v)) for all v e na/a. Since kcn ) c k at„ kc" corresponds, by class field theory, to an open subgroup Y of k; containing kx14. Let x be an element of Y such that p(x) E W„ and xœ = 1. Let a =[x, k], and let a be as in (2) of Prop. 7.40. Then a= aa, aaP =1, and if v E n - la/a, we have

7.8

THE ZETA-FUNCTION OF AN ABELIAN VARIETY OF CM-TYPE

213

e(u(r4)= e(u(v))° = E(u(a • p(x)- '1))) = E(u(av)) , so that (a-1)a c na. Observe that a is a unit of K, and 1 cr'l = 1 for every isomorphism r of K into C, on account of (2) of Prop. 5.11. Therefore a must be a root of unity. Since n > 2, we have a =1, so that 02(x). 1. This proves the continuity of Oi (and also that the kernel of the map x --. a is open), and completes the proof. We can now attach to OA an L-function of k as follows. (For detailed discussions about such L-functions, the reader is referred to [6] and [99 .) For every prime ideal p in k, let k o denote the ,-completion of k, and op the maximal compact subring of k t,. Consider kp a subgroup of le:, in a natural way. Then we say that Oa is unramified at p if 02(e) .1. This is so for all except a finite number of p. Then we define the L-function L(s, 0 2) by ]

L(s, 0 2)= np [1-02(cp)/s/M - ij - i , where the product is taken over all p where 02 is unramified, and cp is a prime element of kt,. Observe that sbAcp) does not depend on the choice of cp. It is a classical fact, first proved by Hecke, that L(s, 0 2) can be holomorphically continued to the whole s-plane and satisfies a functional equation. THEOREM 7.42. The notation being as above, 02 is unramified at p if and only if A has good reduction modulo p. Further the zeta-function of A over k coincides exactly with the product

Ill., L(s, 0 2)L(s, çTi i). PROOF. Let p and cp be as above, and a =Ecp, kJ. Suppose that A has good reduction modulo p. Define pp, R't , A, and 2(1) for every rational prime 2 as in § 7.6 (with Q and 1 as F and 1). Suppose that p is prime to 1. By Prop. 7.23, p is unramified in RM. Since 2(1)c k b, we see that a induces a Frobenius element of Gal (2(1)/ k) for p. Therefore we have 511(a). niot,) by (7.6.7). If a is defined for cp by (2) of Prop. 7.40, we have e(u(v)? = e(u(a - p(c t,) - iv)) for all v e K /a. Since the /-component of ct, is 1, we have = 6(a) • e(u(v)) for all V E Ina/a, n = 1, 2, --.. It follows that pp = Therefore, if X is an indeterminate,

det [1—Ri(pp)X] = M= 1 (1 — (a) AX)(1 — (d) 2 X) = M.1(1— 0 (c 0) X )( I— y7)i(cp) X ) ,

hence our proof is completed if we prove the first assertion. For that purpose, we use the result due to Serre and Tate [66]:

(7.8.5) p is unramified in 2(1) if and only if A has good reduction

modulo p.

214

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

Let y iv, and let a be an element of Kx determined for this y as in Prop. 7.40. Further let Hi = U:=0 1-11`a. Suppose that A has good reduction modulo p. By (7.8.5), we have [y, le] , id. on Q(I), so that e(u(y)).e(u(y))cir.k 3 = e(u(a • p(y) - 'v)) for all y E Hi/a. Then the 1-component of a • p(y)i is equal to 1, so that a-= 1, hence 02(y)=1. Conversely, if 02 is unramified at I), then 02(30 =1, so that a= p(y)A --=1. Therefore e(u(y))cibe=e(u(y)) for all y [ y, le]. id. on .9(1) for all y iv. By (7.8.5), A has good reduction j.e, modulo p. This completes the proof. For further discussion about the conductor of sbl, the reader is referred to Deuring [12 ] (in the one-dimensional case), and Serre and Tate [66] (in the general case). THEOREM 7.43. The notation being as above, let F be the maximal real subfield of K, and op the maximal order of F. Suppose that O(DF) C End (A), and the natural injection K coincides with the map a (a), of Prop. 7.41. Then C(s ; A/ k, F). L(s, 0 1)L(s, sb,). For the definition of (s; Alk, F), see § 7.6. The assumption about 6(op) is not essential, since we can always find a model satisfying this condition by changing A by an isogeny over k. With the same notation as in the above proof, take a prime ideal in F dividing I, and define R; as in § 7.6. Since 0=0(a), we have, on account of Prop. 7.21, PROOF.

(7.8.6)

det [1 —R(ç:) 1,),C]

(1—aX)(1—eiX) ,

[1

-

0,(cp)X][1

—

sb,(cp)X],

hence our theorem. Let k' be a subfield of k containing K*. If (A, C, 0) is rational over k', we can define characters çb of 11; in the same manner as above. Then it is easy to verify

(7.8.7)

Oa= ibIjo N. •

We can actually prove a stronger result as follows: THEOREM 7.44. The notation being as above, let M be a subfield of k containing K*. Then the following two conditions are equivalent: (1) There exists a continuous homomorphism w of M; into CIN trivial on Mx (i.e., a Grnen-character of M) such that 02= VON klM• (2) All the points of A of finite order are rational over Mab • k. Moreover, if these conditions are satisfied, the number of characters w as in (1),

7.8

THE ZETA-FUNCTION OF AN ABELIAN VARIETY OF CM-TYPE

215

for a fixed 2, is exactly [Mab nk: M]. We note that the case M=K* is most interesting, see the discussion after the proof. PROOF. Assume the existence of io as in (1). Let a e Aut (C/A", • k). Take z E /Z AI so that a = [z, le] on kab, and put s= Nklar(z). Since a = id. on Map s is contained in the closure of MxM:... We can find an open subgroup T of the finite part of M; so that v(T)= 1. Then s e MxMLT, so that s = Art with p E Mx, r e Mc:, and t e T. Since Ark,m(kl)= MI, we have r = Nkrm(Y) for some y e kL. Put x = zy - ', and define a as in Prop. 7.40 for this x. Then (a/p(xDA = 02(x) = v(At) = 1, since w(APT)= 1. Put for a

pi(a)= 7)(A TKIK-(a))

E

M.

Then p(x)= pi(N kim(x))= giet). Since 12/(02 =1 and p/(P) e K", we obtain p'(A)= a. Therefore a/p(x). p'(0 -1 . Since a =[x, k] on k ab , we have (u(y))'' = E(u(g(t) -1 1))) for all V E K/ a. Now we can replace T by any of its open subgroups, especially by

Tv = (w

E

T p/(w)y =y)

for any fixed V G K/a. Then we see that e(u(y)) is invariant under a for every y e K/a, which implies (2). Conversely, suppose that (2) is satisfied, and put S= M"- N k,m(k;). Then S is the subgroup of M; corresponding to Ailed-) k by class field theory. Let a = [s, M] with any s e S. Then a = id. on m ob n k, so that a can be extended uniquely to an element r of Gal (MBy the assumption (2), (u(y))t is meaningful for every V G Kla. We can therefore repeat the proof of Prop. 7.40, with p'(s) and N(il(s)) in place of p(x) and N(il(x)). Then we obtain an element a of Kx such that

aaP = N(il(s)) ,

a- pi(s)-1a= a,

(y e K/a) .

$(u(y))r = $(u(a - pi(s)- 'y)) Obviously a is unique for s.

Define sol : S

—

C" by

v2(s) = (a/ g(s))1

.

By the same reasoning as in the proof of Prop. 7.41, we can show that VA is trivial on M", and go2(s)=(p/(s) -') 2 for s e AL. Now define k"i' as in the proof of Prop. 7.41. By our assumption, we have k c Mao • k. Let U be the open subgroup of M; corresponding to k(' M at. Then UcS. Let s be an element of U such that so., =1 and p'(s)e W„, where Wa is as in (7.8.4). Then, by the same argument as in the proof of Prop. 7.41, we can show that v l(s)= 1,

216

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

which proves the continuity of so2 . By our definition of ¶02, we have 02 ...7 ¶02 ° NUM. Since S= N k/m (k;), the homomorphism so/ : S---C" is completely determined by 02. Therefore our problem is reduced to the possibility of extending i0 2 to M. This can be settled by the following lemma, on account of the equality [M;:S]=EM,o nk: M]. LEMMA

7.45. Let G be a commutative topological group, H an open sub-

group of G of finite index, and so a continuous homomorphism of H into C. Then there are exactly [G: H] continuous homomorphisms of G into C" which coincide with io on H.

Decompose G/H into the product of finite cyclic groups P„ •-- ,P,. of order m„ , m r , respectively, and take, for each i, an element a, of G which generates P, modulo H. Let ci be any m,-th root of io(ari), and put PROOF.

io'(haf l

(2, ) = P(h)*

(h

c:r

E

H, e, E Z).

It is now easy to verify that so' is a well-defined continuous homomorphism of G into em, and so' =io on H. It is also clear that the number of such extensions is [G: H], and any extension of io to G can be obtained in such a manner.

Let us now show that, for any given A, there is a model which is iso- and satisfies the conditions of Th. 7.44 with K* as M. morphic to A over 0, , t r of A of finite order For a given (A, C, 0), we can always find points t„ so that the structure

Q = (A, C , 0 ; t„

,

has no automorphism other than the identity map. With any such t„ let k' be the fi eld of moduli of Q (see p. 130). Then there exists a structure Q ' = (A' , C' , 0' ; t 1',..., 4)

which is isomorphic to Q and defined over k'; moreover, such a Q' is unique up to isomorphisms rational over k' (see [75, II, 1.5]). By Cor. 5.16, k' c K:b. Moreover, we have (7.8.8)

All the points of A' of finite order are rational over K b .

To see this, take an isomorphism : Gin /u(a)---. A' so that (A', C', 0') is of type (K, 0; a, () with respect to e'. Let CIE Aut (C/K: 6 ). Apply Th. 5.15 to Q' with s= 1. Then we find an isomorphism r of C'/u(a) to A' such that (A', C', 0') is of type (K, 0; a, C) with respect to eff, and Nu(y))° = eff(u(y)) for all y e K/a. Then we obtain an automorphism r of A' such that eff = r oe,. It can easily be seen that r is an automorphism of Q', so that r -,1. It

7.8

THE ZETA-FUNCTION OF AN ABELIAN VARIETY OF CM-TYPE

217

e, .r,

follows that hence e'(u(v)) is invariant under a for every y E K/a, which proves (7.8.8). Thus A' and k' satisfy the condition (2) of Th. 7.44 with K* as M. (We know even that k' c KL.) Under certain circumstances, we can take k' to be the fi eld of moduli of (A, C, 0). For example, assume the following set of conditions: (7.8.9) (i) End (A)n 0(K) = 0(o g) with the maximal order o f( in K; (ii) OK has no roots of unity other than ± - - 1; (iii) co ic has a prime ideal I) such that N(1))=3. Take an element b of K that generates t - 'a/a, and put t = e(u(b)). Let r be an automorphism of (A, C, ; t). Then r =OW with a root of unity e contained in oK, and t = rt, so that eb b mod a. Since e= -±-1 and b is of order 3, we have e = 1. Therefore (A, C, 0; t) has no automorphism other than the identity map. On the other hand, we have: (7.8.10) (A, C, ; t) and (A, C, 0) have the same field of moduli. To see this, let a be an element of Aut (C) which is the identity map on the field of moduli of (A, C, 0). Then there is an isomorphism ô of (A, C, 0) toe (Ac, Ce, 0°). We see that the point t has the property = {a

e OK I 0(a)t = 0} ,

and ± t are the only points of A satisfying this condition. Therefore (3 -1 t°' =± - t. It follows that either ô or --(3 gives an isomorphism of (A, C, 0; t) to (A°, C° , 0° ; t°), hence (7.8.10). Now, by the principle explained above, we obtain a structure (A', CI, 0 , ; t') which is isomorphic to (A, C, ; t) and defined over the field of moduli of (A, C, 0), and which satisfies the condition (2) of Th. 7.44 with K* as M. In particular, if A is an elliptic curve and j its invariant, then the field of moduli of (A, C, 0) is K(j). In this case, the number of characters of IC:1 as in (1) of Th. 7.44 is exactly Elf( j): K], the class number of K. For K=Q(V----cl) with d square-free, the condition (iii) of (7.8.9) is satisfied if and only if d * 1 mod (3). - -

-

Next let us give an example for which (2) of Th. 7.44 is not satisfied. We have just shown the existence of an elliptic curve E, defined over K(j), whose points of finite order are all rational over Ko . Take an elliptic curve E' defined over K(j) and isomorphic to E over O. Suppose that E' also satisfies (2) of Th. 7.44 with K as M, j. e., all the points of E' of finite order are rational over Kat,. Then we see easily that any isomorphism À of E to E' is rational over K. But this is not always the case, since the smallest

218

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

field of definition for A containing K(j) is not necessarily contained in Kab(For example take any p such that p 2 E K(jE) and p Œ Ktio , and define an isomorphism 2 as in Prop. 4.1.) Thus E' cannot satisfy (2) of Th. 7.44 with K as M, for such a choice of A. For an elliptic curve E with complex multiplications, Deuring [12, IV] determined the zeta-function of E over a field which does not contain the imaginary quadratic field in question. We shall now generalize this result in the following form: THEOREM

7.46. The notation being as in Th. 7.43, let k o be an algebraic

number field of finite degree, over which (A, C) is rational. Suppose that A is simple, 0(oF)c End (A), every element of O(OF) is rational over k o, and k o nK* is the maximal real subfield of K*. Define the characters 02 of (k oK*); as above with k oK* as k. Then C(s; Alk o, F) coincides, up to finitely many Eule? factors, with L(s,0,). More precisely, for almost all primes q of k o, the Euler q-factor of C(s; A/ k 0, F) is the product of the Euler p-factors of L(s, 0 1) for the prime factors p of q in koK*.

Note that every element of End (A) is rational over k oK*, see [81, § 8.5, Prop. 30]. A typical example is the case where A is an elliptic curve, and ko = Q(3) (see (ii) of Th. 5.7). One has K* = K and k,K* = K(j) in this case. The " bad Euler factors" will be discussed after the proof. PROOF. Put k = k oK*. Then [k : k o] =2. Let p denote the complex conjugation, and r an element of Gal (k ao/k o) which is non-trivial on k. Since 0(K)= End() (A), we can define an automorphism E of K by 0(a)r= 0(a`). We have r = p on K*, so that

tr 0(a`). tr 0(a)r= tr ø(a) P = tr ø(a°)

(a E K) .

Since A is simple, this implies that e = p on K, by virtue of [81, § 8.2, Prop. 26]. Therefore 0(0' = 0(aP). Put Eo = iou. Then r - ' induces an automorphism of the module eo(K /a), which is semi-linear with respect to the action of 0(a). Therefore, w EcT1 ($0(le ) ) is an isomorphism of K/aP to IC/a, which is linear with respect to the action of the elements of the order of a. (Note that a and aP have the same order, on account of the assumption 0(oF)c End (A).) Thus we obtain an element z of K; such that zaP = a, and $0(zw)r=$ 0(wP) for all w E K/aP, j. e., E o(v)= $ 0(zvP)r for all y E la. Now, for every x E k;, f is a meaningful element of k;, and r[x7 , k]=[x, k]r. Note also that p(xr)=/.1(x)P. Define a as in Prop. 7.40. Then

= eo (zve)rxr , ki= eo(zzeyx , k2r = sb(a P(x)-1 zv) r = eo(aP • p(f) -1 v)

(v

E

K la) .

7.8

THE ZETA-FUNCTION OF AN ABELIAN VARIETY OF CM-TYPE

219

Further aPa= N(il(x)), and aP - p(xT la =a. Therefore aP is the element of Ifx corresponding to x", hence (7.8.11)

Sba(x')

Let p, cp . A, pp, and I? be as in the proof of Th. 7.42 and Th. 7.43, assuming that A has good reduction modulo p, hence 02 is unramified at p. Let q be the restriction of p to kw and p, the Frobenius endomorphism of A of degree N(q). Suppose that p ;:)". We can take 4 as cor. Since pp =p i in this case, we have, by (7.8.6) and (7.8.11), (*)

det [1 — g(P4)X] =[1-0,(cp)X][1-0,(cp)X]

=[1-01(c1),(][1-01 (c;)X]. Next suppose that p =p and NW= N(q) 2. Put a = 0(cp). Then a = 0(cp) = 0(d). aP, so that a e F. Now we have pl =pp, so that, by (7.8.6), det [1— R(pi)X] = (1—aX)2 . commute k/k Let a = (--.). Then we have 0(01 = 0(a), so that pp does not ci scalar matrix, hence with 0(a) for a E K, e F. It follows that np,) is not a

(4,4)

det [1— g(pp)X] = 1—aX 2 =1-01(cp)X 2 .

Taking the product of (*) and (**) with X = N(i) t for all " good " q, we obtain our assertion. It remains to discuss the "bad Euler factors ", for which the last statement of our theorem does not hold. In view of Th. 7.42, it is sufficient to consider the primes q of /4 such that 0, is unramified at the prime factors of q in k. The above discussion shows that the bad factors may occur for the primes of ko ramified in k. Other primes are actually " good ". In fact, one has PROPOSITION 7.47. The notation and the assumptions being as in Th. 7.46, let q be a prime ideal in k o. Then A has good reduction modulo 0 if and only if q is unramified in k, and A has good reduction modulo the prime factors of q in k. The last statement of Th. 7.46 holds for such a prime q. k o (resp. k) by the PROOF. Let gt o(1) (resp. R(I)) be the field generated over positive integers m. We coordinates of the points of A of order lm for all By [81, §83, see easily that every element of End (A) is defined over R o(I). follow directly from Prop. 30], K* CR 0(1), hence R 0(1). R(1). Our assertions ] (see (7.8.5)). this fact and the result of Serre and Tate [66

220

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

7.9. Supplementary remarks A. Change of model and the field of definition In § 7.5, we have determined the zeta-function of a special model V s of T'VD* over Q. Actually there exist curves V defined over algebraic number fields k of finite degree birationally equivalent to V s over 0, but not necessarily over k. Therefore one can naturally ask about the determination of the zeta-function of any such V over k. The same question may be asked for the abelian varieties A s , or its factors A, A , considered in §§7.5-7.6. The complete solution of this question seems rather difficult. We shall discuss here only special cases. (I) Let S, V 3, and A s be as in §§ 7.3-7.5. Let k be a finite abelian extension of Q of conductor (r), and let m = [le : Q]. Then there are m characters X 1, , X„, of (Z/NZ)x such that (1- uf)kif =

(1-x 1(p)u)

for every rational prime p, not dividing r, which decomposes into m/ f prime ideals in k, where u is an indeterminate. If p denotes such a prime ideal in k, SOp (resp. rp) denotes the Frobenius endomorphism of A s of degree N(p) (resp. p), and Ri denotes the /-adic representation of End (As), then one has det [1— ufRi(sop)]

Therefore, if we put, for f(z)=

L(s, f,X)=

det [1—u X t(p)RK2r p)] .

a ne 2rint/I

E

sk

(rff) ,

an • X(n)n - '

as in § 3.6, and if {h„ , hi} is as in (7.5.4'), then the zeta-function of V s (or A s) over k coincides, up to a finite number of Euler factors, with the product rEt, H:= 1 L(s, h,„ X 1 ),

which is holomorphic on the whole s-plane, and satisfies a functional equation, on account of Remark 3.58, Prop. 3.64, and Th. 3.66. (II) In the next place, consider an arbitrary quadratic extension k of Q, of conductor (r). By virtue of a result of Weil [94], one can construct an abelian variety B s defined over Q and an isomorphism A of A s onto B s defined over k such that AC = —A for the generator a of Gal (k/Q). The couple (B 3,2) is unique up to isomorphisms over Q. If Op is the Frobenius endomorphism of gs of degree p, we have, for almost all p, op2=x(p)Irp, where X is the character of (Z/rZ)' corresponding to k. Therefore the zeta-function of B s over Q coincides, up to a finite number of Euler factors, with the product

SUPPLEMENTARY REMARKS

7.9

221

L(s, h„, X) ,

which is holomorphic on the whole s-plane, -and satisfies a functional equation. We can of course make a similar consideration by taking a factor A of As, considered in Th. 7.14, in place of A s .

B.

Rational points of an elliptic curve

The group of rational points of an elliptic curve defined over an algebraic number field, a function field, or a local field, has been a subject of extensive study. An excellent survey of this topic is given by Cassels [5], in which the reader can find references up to 1966. Here we content ourselves with mentioning only THE CONJECTURE OF BIRCH AND SWINNERTON-DYER [1] If the zetafunction C(s; E/Q) of an elliptic curve E defined over Q has a zero of order hO at s=1, then the group of rational points of E over Q has rank h. :

They verified the conjecture for many curves, especially for curves of type y2 = xs—Dx. If E is a curve V s of genus 1 isomorphic to 4)*/r 0 (N) for N belonging to the values of (7.5.6), then C(s ; E/Q) is, possibly up to bad factors, given by p

CO

E7.1 an = r(s) -1(2703 j o f(iY)Y - icly with an element f(z)= E,12.1 an e2'i of sxr,,(N)). The last integral is convergent for all s (see Proof of Th. 3.66). Since V s is of genus 1, we have div (f (z)dz) =0, so that div (f) can be obtained from the formula of Prop. 2.16. Checking the elliptic points of (N we see easily that f has no zero on the imaginary axis, except at oo. Since f(iy). Enc°,, an e -2"Y takes real values, it follows that C(s ; E/Q) does not vanish at s= 1. Birch has verified that this fact is in agreement with the above conjecture.

ro

),

C. The Euler factors for the primes where the variety has bad reduction To define the zeta-function of a curve or an abelian variety, we have considered only the primes for which the variety has .good reduction. It is of course natural to seek the Euler factors also for the " bad" primes. Néron [53] has shown that an abelian variety over a local (or global) field has a model which has the "best behavior" for the reduction process modulo the prime in question. By means of this result, one can define the "conductor"

222

ZETA-FUNCTIONS OF ALGEBRAIC CURVES

of an abelian variety over a number field, and its Euler factors for bad primes (at least for elliptic curves). For details, the reader is referred to Ogg [54], Serre and Tate [66], and Weil [98]. With such factors and the notion of Tamagawa number, the above conjecture of Birch and Swinnerton-Dyer can be formulated in a more precise form (see [1] and the article by SwinnertonDyer in [6]).

CHAPTER 8 THE COHOMOLOGY GROUP ASSOCIATED WITH CUSP FORMS 8.1. Cohomology groups of Fuchsian groups We shall now construct a certain cohomology group isomorphic to sk(r), which was first found by Eichler. Here k is any (odd or even) integer 2. To define it, we start with the usual definition of the cohomology group Hi(G, X) with an arbitrary group G and a left G-module X. We fix an associative ring R with an identity element, and denote by R[G] the group ring of G over R. In later applications, R will be Z or a field. We assume that X is an REGImodule, and denote by C i(G, X), for an integer i 0, the R-module of all maps of G i =GX xG (the product of i copies) into X; we understand that C°(G, X)= X. For u CAC, X), define an element au of C'i(G, X) by

if i=0,

au(a)= (a 1)u —

al • u(a2

au(cr i , a2 , ••• ,

,

ai+1)

•••

-FE‘j=1( - 1)izi(a1, 1 (

- -

It can easily be verified that

-

B i(G, X)=

if j > O.

1Y +1 u(a2, --• , ai )

33=0. fu

Z i(G,

••• at+i)

Put

C%(G, X) I au =0} , if i=0,

{0

x»

if i> 0,

Hi(G, X)=Z i(G, X)/13(G, X), X° = {xE X I ax=x for all a e G} We call IP(G, X) the i-th cohomology group of G with coefficients in X. Clearly H°(G, X) and Z°(G, X) can be identified with X a We observe that Zi(G, X) c onsists of all maps u of G into X such that

(8.1.1)

u(a P)= u(a) 1 au(S) - -

(a,

pe G) ,

and 13'(G, X) consists of all maps y of G into X such that (8.1.2)

v(a) = (a-1)x,

(a G G)

224

THE COHOMOLOGY GROUP ASSOCIATED WITH CUSP FORMS

From (8.1.1), we obtain u(1) = 0,

with an element x„, of X independent of a. and (8.1.3)

u(a -1).

—

a -1 u(a)

(a E G).

Now fix a subset Q of G, which may or may not be empty, and denote by cz(G, )0 the R-submodule of C 1 (G, X) consisting of the elements u with the following property : (8.1.4)

u(r)

E

(71. 1)X for every ir E Q. -

Then we put

ZZ G (

Bz(G,

,

x) = zi(c, x)nc4(G, X)

= a(cz(G, X)),

1-14(G, X) = 11%G, X), I-14(G, X) = ZZ(G, X)/13 1 (G, X), 1-1Z(G, X) =Z 2(G, X)/.13(G, X). Note that Bl(G, X)cZ(G, X). Now we shall consider the case where G is a Fuchsian group of the first kind. Here we understand that G is a subgroup of SUR)/(4-1), and not of SUR). We denote by P the set of all parabolic elements of G. We are going to establish an " isogeny " of 1-14(G, X), with a certain subset Q of P, to a certain cohomology group defined with respect to a simplicial complex on 0. If 0IG is compact, and G has no elliptic elements, such an isogeny is actually an isomorphism and a special case of a well-known isomorphism due to Hopf, Eilenberg, MacLane, and Eckmann. It is therefore our task to modify the standard argument so that the difficulty arising from parabolic and elliptic elements of G can be eliminated. Take a set {el, , Er } of representatives of elliptic elements of G, i. e., a minimal set such that every elliptic element of G is conjugate in G to a power of some ci. Let ei be the order of e) , and E the least common multiple of e„ ..• , e,.. We put E = 1 if { E} is empty. Let 0* be the union of 0 and the cusps of G. Let c„ , c„, be the points of 0*/G corresponding to the cusps of G. Take a small open disc Dk on 0*/G containing ck so that the closures of D I , ••• , D., are disjoint from each other. For example, if 00 is a cusp of G corresponding to ck , we can take D k to be the image of (z 0* I Im(z)> yl for a suitably large y, as described in § 1.3. Let 0„ be the inverse image of (0*/G)—(U7.1Dk) by the map 0* ---.0*/G. We make a simplicial complex K with the underlying space 0„ so that the following conditions (8.1.5-8) are satisfied :

8.1

COHOMOLOGY GROUPS OF FUCHSIAN GROUPS

225

(8.1.5) Every element of G induces a simplicial map of K onto itself.

(8.1.6) The fixed point of EJ on ef, is a 0-simplex of K; we denote it by d i. (8.1.7) There exists a 1-chain tk of K which is mapped onto the boundary of (8.1.8)

D k•

There exists a fundamental domain for tio/G whose closure consists of a finite number of simplexes of K.

One can construct such a K, for example, by taking a fundamental domain for Vc/G as considered in the proof of Th. 2.20, and removing the parts corresponding to the D k• Let (A„ a, a) be the chain complex, with coefficients in R, obtained from K, with the usual boundary operator a and the (unit) augmentation a defined by a(Ei s,p1).z,s, for sj E R and 0-simplexes p,. Since Oo is homeomorphic to a Euclidean plane, we have an exact sequence (8.1.9)

a

a

a

0 — A2 — A, — Ao R

—

In view of (8.1.5), A i becomes an REGImodule, and action of R[C]. By (8.1.7), we have

.0.

a

commutes with the

(k =1, •-• , m)

(8.1.10)

with a 0-simplex q, and an element ir k of P. Then every parabolic element of G is conjugate to a power of some ir k. Put Q= {Ito ••Let ils(X) denote the module of all R[G]-1inear maps of Ai into X, and let a: AI(X)_. A1(X) be defined by au = ua for u e MX). Further let AZ(X) be the submodule consisting of all u in MX) such that u(t k) r k —1)X for every k. Then we put

Z' (K , X) = fu E Al(X) I au =0} , Bi(K, X)=3A' - '(X), ZZ(K, X)= Z'(K, X) n AZ(X) , BZ,(K, X)=a,4 43(x), •

X)=Z°(K, X),

H (K, X)= Z4(K, X)/.13 1(K, X), •

X)= V(K, X)/B(K, X).

Note that I31 (K, X)c Z(K, X) in view of (8.1.10).

226

THE COHOMOLOGY GROUP ASSOCIATED WITH CUSP FORMS

PROPOSITION 8.1. There exists, for i=0, 1, 2, an R-homomorphism g' of H4(K, X) into I-14(G, X), and an R-homomorphism p of I-112(G, X) into H4(K, X) such that g' of' = E • (identity map of H4(G, X)) , ft g' = E • (identity map of I-14(K, X)). Especially, if R is a field whose characteristic is isomorphic to 1-4K, X).

0 or prime to E, I-112(G, X) is

(M 1, a, a)

First consider a well-known chain complex of the following: PROOF.

(8.1.11) A, for an integer ordered sets [ a o , a1 , (8.1.12)

a: A

0, is the free R-module generated by all the , ai] of i+1 elements of G.

is defined by

a[a.,

, a i ] = Et.0 ( - 1)[ao,

(8.1.13) a(E. b[aa= Et, b., for (8.1.14)

consisting

a„+,,

,

, ai].

a b„[a.] e M. with b i, e R.

G acts on Mi by the rule A[ao, -•- tri]= Cif3a0, •-• ,

cza

It is well-known that

(8.1.15)

a •-•

a M2

M1

a

a M0

R

0

is exact.

Denote by AP(X) the module of all REG] - linear maps of Mi into X, and define a: mi(x). mi+i(x) by au ua. For every u e C‘(G, X), put aTao,

, ai]) = ao u(aVa i, a 1 1a2 , ••• , aT2 lai).

Then we see that u .—,11 gives an R-isomorphism of C i(G, X) onto AP(X), and

au. au.

We are going to define an R-linear map f:

fa= af ,

(8.1.16)

a f = Ea ,

(8.1.17)

f(d;)= (El e;) -

(8.1.18)

f (t k) = E [1, r k]

fa = af

[4]

such that :

111

(a E G), ( j =1, (k =1,

, r), , m).

Such an f can be obtained by the standard argument by induction on i, with a little care about dj and t k . In fact, first define Ad )) by (8.1.17), and put f (a(di)) = af(d f) for all a e G. Then take a finite set S o of 0-simplexes so that every 0-simplex, other than the elliptic points of G, can be written as a(p) with a unique p E S. and a unique a e G. We include the points q k of (8.1.10) in So. Then we put f(a(P))= E • [a] for all a e G and all p e So.

COHOMOLOGY GROUPS OF FUCHSIAN GROUPS

8.1

22T

Similarly we fix a finite set S, of i-simplexes, for i =1, 2, so that the a(s) for all a E G and all SE Si form a free R-basis of A i. We include the t k in S1. Obviously af= Ea. Therefore af(as)=0 for every s E S,. By virtue of (8.1.15), we can define f(s) so that af(s)=f(as). In particular we can put f(tk)= E. [1, Irk] without contradiction. Then we put f (a(s))= af(s) for every a e G. Next, for s E S„ we have af(as). 0, hence we can define f(s) so that af(s)=f(as), in view of (8.1.15). Then we put f(a(s))=af(s). To an element u E C'(G, X), we assign an element w of A'(X) by w =170f. If u e GAG, X), w(t k)= E. u(ir k)e (rk -1)X, hence u e MX). Moreover it can easily be seen that the correspondence u w commutes with a, hence defines a homomorphism f of 1-14(G, X) into 1-14(K, X). By a similar argument, we can define an R-linear map g: A i satisfying the following two conditions :

(8.1.19)

ga=ag ,

ag = a,

(a e G) ,

ga = ag

(8.1.20) g([1, xkl= tk-F(irk - 1)bk with a 1-chain b k such that

ab,, =P0—q,,.

Here po is a fixed 0-simplex in S. To define such a g, first put g([a]) = a(p o) for all a e G. Then define g([1, a]) so that ag([1, a ]) =a(p.)—N, and put g(Cer, pl=g([1, a -1 13]). In particular we can define g([1,rk ]) as in (8.1.20). Since ag(a[1, a, p]). 0, we can define g([1, a, fi]) so that ag([1, a, IT =g(a[1, a, fi]) in view of the exactness of (8.1.9). Then we put g(Ecr, 19, r3) ag([1,a - ' 151, cr - 'r]). To an element x e Al(X), we assign an element y of C'(G, X) so that xog. If x e A4(X), Y(rk)=.9([1, ra=x(tk) -F(7rk - 1)x(bk) E (Xk - 1)X, hence ye X). Moreover, we can easily verify that the correspondence x 4 3? commutes with a, hence defines a homomorphism g' of 1-14(K, X) into 1-14(G, X). Let us now construct an R-linear map with the following properties : ,

c(c,

(a E G) ,

(8.1.21)

Ua= aU

(8.1.22)

fog

(8.1.23)

U([1, pr,,]) e

—

E • (identity map)= au+ua, (Irk —1 )M2

We first observe that f(g(x)). Ex for xe M.. Defining U=0 on M., we see that (8.1.22) is satisfied on M.. Let a be an element of G other than the ir k . Since au(g([1, a])) E[1, a]) = f(g(a[1, a])) Ea[1; a]=0 , —

—

we can define U([1, a]) so that &Jo,

= f(g([l, a]))—E[1, a],

228

THE COHOMOLOGY GROUP ASSOCIATED WITH CUSP FORMS

in view of (8.1.15). If a= rk , we have to choose U([1, a]) more specifically. Since 3f(b k)=0, we can find an element n k of M2 so that an k = f(b k ). Put U([1, xt ])=(lrk -1)nk. In view of (8.1.18) and (8.1.20), we have

OD — EEL rk]=(2rk— l)f(bk)=

du ([1 , 7r0)

Now we put U([a, fi ]) = aU([1, a- ',3]). Then (8.1.22) is true on A1 1 . we have to define U([1, a, fi]) so that

Further

&IV, a, 13 1=fCg([l, a, ,3]))—E [1, a, 43]— uoR a, /al This can actually be done, since the boundary of the right hand side is 0. Putting U([a, 13, r ]) = au([l, a'r]), we obtain the desired U. Let x E P(G, X). Then there exists an element y of Ci - '(G, X) such that y = o U. By (8.1.22), we obtain .t of og E2= (If i1, y =O.) If i=2, one has rk) = i(U(El, rO))= (2 rk -1).t(n e rk —1)X, —

ay.

hence y e CaG, X). This shows that g i of ( = E. (identity map) for i = 0, 1, 2. with the following Similarly we obtain an R-linear map V: properties :

(8.1.24)

Va =aV

(8.1.25)

go f

(8.1.26)

V(t k)= O.

—

(a E

E • (identity

map)= ay+ va ,

Since aogof =E - a on A o, we can define V(s) for se S o so that aV(s) =g(f(s))—Es. In particular we can put V(th)=Eb k . As for cli, we take a 1-chain h in A, so that ah; =po—d and put V(d .,)= (El e ;) • Ee„-Co' E(h). Then we can put V(a(p))=aV(p) for a E G and for an arbitrary 0-simplex p without contradiction. By the procedure similar to the construction of U, we define V on S, and S 2 so that (8.125) is satisfied, and put V(a(s))= a V(s) for a E G, s E Si. The choice (8.1.26) is possible in view of (8.1.18) and (8.1.20). Note that V =0 on AV Let u E Z i(K, X). Then u ogof—Eu=a(uoV). If i = 2, we have u(V(tk)) = 0, hence uoVe Ad(X). This proves that floe= E. (identity map), and completes the proof of Prop. 8.1. Actually the isomorphism of 1-43(G, X) and HaK, X) can be seen immediately. In fact, if w E r(K, X), then w(p) is independent of p. Therefore rw(P)= w(r(P))= w(p) for all r E G, hence w(p) E X G = MG, X). Conversely, any element of XG corresponds to an element of I-14(K, X). Thus I/4(K, X)

is always isomorphic to X° =114(G, X).

COHOMOLOGY GROUPS OF FUCHSIAN GROUPS

8.1

229

Let Y be the R-submodule of X generated by (a-1)X for all a e G. Then 11:22(K, X) is isomorphic to X/Y.

8.2.

PROPOSITION

Take a fundamental domain F for O. modulo G as described in <8.1.8). We may assume : PROOF.

<8.1.27) F is simply connected; , ag are the 2-simplexes contained in F, one has

<8.1.28) If a„

a(Ef-----1 at) — E :7 =1 a k(t k) -4- Et=,(8, with some a k,

g, e G and

1).5,

some s i e A,.

T hen G is generated by the 131 and akir kai', and A, is generated by the r(a i) for all i and all r e G. Therefore an element u of Z 2(K, X) is determined by the values u(ai). Let us put u(F)=D., u(a.). Suppose u(F) e Y. Then there exist elements y k and z1 of X such that u(F)=

( 8.1.29)

rki rkai i — 1)Y k-i-E14-1(191 - 1)zi .

We can find an element w of AZ(X) so that u = aw, wcto=(rk-1)ailYk, and w(s i)= z 1. In fact, we first define the values of w at t k and si as specified. Then we set the values of w at the 1-simplexes lying inside F, one by one, so that u(a j ). w(aa i). This is possible in view of (8.1.29). Then extend w to the whole A, by the property wr = rw for all r e G. Thus u E B(K, X), if u(F) E Y. Conversely, if u = aw with WE A,/X), we have u(F)=w(aF)=

w(ak(tO)+Ei=t (igt

-

1)w(st) E Y.

This completes the proof.

8.3. Suppose that R is a field, and X is a finite dimensional vector space over R. Let g be thegenus of G\0*, Y be as in Prop. 8.2, and let PROPOSITION

C= dim (X),

C' = dim (X/ Y) ,

e . =dim ({xE X I ex=x}) 72k

= dim ((mk-1)X)

j =1,

, r),

, m),

where dim( ) denotes the dimension over R, and the e i (resp. irk) are representatives for the elliptic (resp. parabolic) elements of G as in the above discussion. Then dim (114(K, X)).(2g-2) dim (X)+C -FC'

72k+ Eti..1 (dim (x)— Ç).

Let K be as above, and Ni the number of G-inequivalent i-simplexes in K. Then we see easily that PROOF.

230

THE

COHOMOLOGY

GROUP ASSOCIATED WITH CUSP FORMS

N 0 —N,-EN2 -Em=2-2g,

dim (A °(X))= N 0 • dim (X)—E 1 (dim (X) --.,), dim (i14,(X))= N, • dim (X)—Ez., (dim (X)—) k), dim (A 2(X))= N,- dim (X). Further we have E2.0 (-1) dim (H4(K, X)). dim (A°(X))—dim (A4(X))+ dim (A 2(X)) . Our assertion now follows immediately from these relations, Prop. 8.2, and the isomorphism of HZ(K, X) with X°. Let P be the s t of all parabolic elements of G. (8.1.30)

Then we have

HAG, X). 114(G, X).

To prove this, it is sufficient to show that ZAG, X)=Z4(G, X). Obviously ZAG, X)c Z(G, X). Let u e Zd(G, X), and re Q. Then u(r) = (r —1)x with xe X, so that, by (8.1.1), u(rn)=(1-0r+ -f-rnt - ')u(r)=(irni-1)x for any positive integer m, and by (8.1.3), u(r - ')= --r - mu(701)=(71- 7ft —1)x. Therefore, for every a E G and every p E Z, we have u(aIrPa - ')=(arPa - i-1)(ax—u(a)). Now every element of P is of the form ara' with r e Q, a e G, and pe Z. Therefore u e ZAG, X), so that ZAG, X)=Z4(G, X), q. e. d.

8.2.

The correspondence between cusp forms and cohomology classes

an (n+1)-dimene C2 and for every integer n 0, let us define v sional column vector [ ] by For

[

u

[

= '(un,

, un- kv k ,

, uvn -3 , vn).

Then we can define a representation p n : GL2(C)---GLn+I(C) by (8.2.1)

P.(a)[T— Cce [NY-

If n =0, we understand that [

]o

=1, and p o(a)= 1 for every a e G. There

exists a unique non-degenerate bilinear form on Cn+1 , represented by a real matrix e„ such that (8.2.2)

u

v

•

en •

Y

= det[ u x T.

v

y

231

CUSP FORMS AND COHOMOLOGY CLASSES

8.2

We see easily that 9.= 1, and

ten = (-1)ne„ ,

(8.2.3)

`p(a)9p(a)= det (a)

(8.2.4)

La(iz)f=i(a, zr` Pn(a)[ IT

(8.2.5)

GL,(R)n Ker (io,1).

1 {12 } {±1}

(a e GL,(R), ze

0),

if n is odd, if n is even.

Let r be a discrete subgroup of SL,(R) which is a Fuchsian group of the first kind, and r=r1(rn{-1-1}). Let P (resp. P) denote the set of all parabolic elements of r (resp. 1'). Let X be a r-module, which we consider a r-module in a natural way. (This means that if —1 e r, —1 acts as the identity map of X.) Consider the following condition on X:

(8.2.6)

If

XE

X and 2x= 0, then x=0.

Under this assumption, if u e zi(r, iy) and —1 e r, then 0 = u((-1)2)= u(-1)+u(-1), so that u(-1)=0. Then u can be considered as an element of zi(r, ,v) in a natural way. Therefore zi (r, iy) (resp. Bi(r, X)) can be identified with Z);(r, X) (resp. Btu', ,y» in a natural way, so that H,(r, iy) can be identified with H(P, x). Now we consider a representation 7 of r into GL,(R), with any r> 0, satisfying the following two conditions: (8.2.7) f maps (8.2.8)

r into

a compact subgroup of GLT(R);

The kernel of 7 is of finite index in

r if r has cusps.

Then we denote by sk(r, f) the vector space of all holomorphic maps f of 0 into Cr satisfying the following two conditions : (8.2.9)

f(a(z))j(a, z) - k =7(a)f(z) for all a e r;

(8.2.10)

The components of f belong to Sk(Ker (f)), if meaningful in view of (8.2.8).)

r has cusps. (This is

The vector space sk(no,o) of §3.5 is an example of sk (r, vr). In § 9.2, we shall give an example of 7 such that Ker(f) is not of finite index in r and r has no cusp. If qf is absolutely irreducible and —1 e r, then sk(r,yr)* {0} only when Vr( 1)=( 1)k. Further, if f is the direct sum of two representations f i and f2, then sk(r, 7) can be identified with the direct sum of sk(r,71) and sk(r, f2). Therefore, without losing much generality, we shall hereafter -

-

232

THE COHOMOLOGY GROUP ASSOCIATED WITH CUSP FORMS

assume

(8.2.11)

V" (

-

1) = ( 1)k if —le -

r.

By virtue of the assumption (8.2.7), we find a positive definite real symmetric matrix P such that tV(a)PYr(a)= P for all a e . Then we define a positive definite hermitian inner product on sk(r,lr) (depending on P), by (f, g) =

nto

sk(r, vr); z. x+ iy) .

(f,

ViDgr-y"dxdy

This is a generalization of the Petersson inner product of §3.4 ; the convergence of the integral can be shown in a similar way. Hereafter we fix r and Y., and consider .37,44r, IT) with a non-negative integer n. Our principal aim of this section is to find an isomorphism of S„...,(T , V) to the cohomology group HAT, X) with a suitable r-module X. First we define, for every fE s.„(r, V), a holomorphic vector differential form b(f) with values in CrOen1•1 by

(8.2.12)

b( f) = fO[ zi T dz

If n =0, we understand that t(f)=f(z)dz.

(8.2.13)

W= P® 6,,,

.

Put

geo=gr(a)OP.(a)

(a

En

In view of (8.2.4), (8.2.5), and (8.2.9), we obtain

(8.2.14)

tX(a)WX(a)=-- W

(8.2.15)

b(i) 0 = X(cr)b( f)

(a e

n

(a E I') ,

where ca means the transform of a differential form by a. Since X(a) is real, we have also Re (1( f )) 0 a = X(a)- Re (b(f))

(8.2.16) where Re(

(a E r) ,

) stands for the real part. Therefore we can define an R-valued

R-bilinear form A( f, g) on (8.2.17)

s.4.2(r,w)

A( f, g)=

by

'Re (b(f)) A W- Re (b(g))

In view of (8.2.2), we have t t-(f) A Wi.(g) =

—

(204+1 • tfPg • yndx A dy, so that

(8.2.18a)

A( f, g) = (2i)' '[(f, g)+(— 1)(g, f)] ,

(8.2.18 b)

A( f, g) =

(8.2.18c)

A( f , in - Jg)= 2' • Re ((f, g)).

Therefore, .4(

g) is non-degenerate.

, f) ,

8.2

233

CUSP FORMS AND COHOMOLOGY CLASSES

Now we consider Rr OR n+ 1 (resp. CrOC'') as an Kr] - module (resp. C[r]-module) through the representation X, and also as an R[P] - module on account of our assumption V(-1)=( 1) if —1 e r. (resp.C[Imodul), Hereafter we denote this R[r] - module (resp. C[r]-module) by X (resp. Xc). If it is necessary to specify n and V, we write X=Xf. Fix any point zo of 0. For Je .3 7, +2(T , 49.), put -

f 2 b(f) 1 v - -

with any fixed vector y of X. Since b(f) is holomorphic, F(z) is independent of the choice of the path of integral. For every a e r, we have, by (8.2.15),

a(2) F (a(z))= fb( f)+

a (zo)

b(f)+v = X(a)F(z)-1-t(a) ,

a (20)

where t(a)= f

b( f) -1- C1 — X(a)iv.

Therefore we see that

t(ap)= t(a)+X(a)t(P) , so that t e zi(r, e). We observe also that the change of y (and hence the change of zo) affects t only by an addition of an element of mr, e). Suppose that r has a cusp s. Take p e SL,(R) so that p(s)= co, and r I. hi JP with h > 0, generates fr e Ker (0") I r(s) =s} (see § 2.1). We can PAO 1 put kJ', z) - n -21(p - '(z)) = 0(q) with a holomorphic C'-valued function 0(q) in q = e-'2/ on account of (8.2.10). Then putting P(w) =AV', w) 1 I(w), we have f(w)w ndw =

- w)).i(P - ', P(2) f(pA

z) - 4-2P(w)dw

0(.0)

p(z)

=

0(ewih)P(w)dw

.

P(.0)

Since p(w) is a polynomial in w, and 0(0)=0, the integral has a limit when p(z) tends to oo, j. e., z tends to s (with respect to the topology of to*). Therefore we can meaningfully put F (s) =11m F(z). Then 2-8

F(s)= F (r(s))= X(r)F(s)-1-t(r) . This proves that t e zkr, Xe). Taking Re (b(f)) in place of b(f), put

(8.2.19) with any fixed a e X.

(8.2.20)

f(z)=

Re (b( f )) 1 a - -

Then

t(a(4) = X(a)f(z)+u(a)

(a e

r)

234

THE COHOMOLOGY GROUP ASSOCIATED WITH CUSP FORMS

with an element u of Z11(T, X). As is shown above, the cohomology class of u is uniquely determined by f, and independent of the choice of z.. Therefore we can define an R-linear map v of V) into by

s.+2(r,

H;.(r, x)

v(f)=the cohomology class of u. THEOREM 8.4. For every (even or odd) n._13 and every representation V of r satisfying (8.2.7, 8, 11), the map p is an R - linear isomorphism of sn+,(r, r) onto HAr, Xr). A result of this type, in a somewhat different form, was first given by Eichler [18] in the case where n is even and 45 is trivial. The theorem in the present form, under some restrictive conditions, was proved in the previous papers : I. [71, Th. 1] when n is even and V is trivial. II. [74, Th. 2] when n is even and r has no cusps. This method is applicable to the case of odd n. III. [48, Prop. 4.4] when r has no cusps. This includes also the case of the product of several copies of 0. A further generalization was given by Matsushima and Murakami [47] for discontinuous groups acting on a bounded symmetric domain with compact quotient. Here we shall prove the above theorem only in the case where Ker (V) is of finite index in T. This together with the previously known results will give a complete proof. Let f and g be elements of sn+2(r, V). Define f and u as in (8.2.19) and (8.2.20). Similarly, put g(

with any fixed b E X.

z) = f .

20

Re [b( fn+ b

Then

(8.2.21)

g(a(z))= X(a)9(z)-F v(a)

with an element y of have

zmr,,y).

(a E

r)

Since df= Re [b( f)] and dg= Re [t(g)], we

A( f, g) = f r \I:cif A Wdg .

r\D

Take a fundamental domain 17 for constructed in the proof of Th. 2.20. Here we do not take small circles around cusps and elliptic points as considered there. Since d(1147dg).'df A Wdg, we have

where

611 is the boundary of 17. As is observed in the proof of Th. 2.20,

285

CUSP FORMS AND COHOMOLOGY CLASSES

8.2

we have all =E2 [s2 --002)], with 1-simplexes S2 and elements a2 of that

A( f, g) = E.

f

sat

f Wdg fa f

02(8.1)

r, so

ti Wdg

By virtue of (8.2.20), (8.2.21), and (8.2.14), o a 2 ) W • d(g 0

fWdg =

=

a 2)

Wcig Ff 'u(o 2)WX(a 2)dg , -

S

32

hence, by (8.1.3) and (8.2.14),

(8.222)

A( f, g)=

E2 t u(ai')Wf dg . s

Now suppose that w(f)= 0. Then, choosing the constant vector a of (82.19) suitably, we may put u =0. Then (82.22) implies A( f, g) = 0, for every g Sii+Jr , V). Since A( f,g) is non degenerate, f must be 0. This proves that the map 5) is injective. In the next place, we compute the dimension of HMI', X), assuming that is trivial. In this case X= R', and X =p. (The condition "V( - 1)=(--1) if n is odd.) We are going to show if — 1E " then implies that —1 EE -

(8.2.23) The dimension of H (f' , X) over R is twice the dimension of Sn+2(11) over C. )2t, C, , s„ and r„ , r. be defined for f as in § 8.1. Let Let s,, and C' be as in Prop. 8.3. Since N ( -, x)=1-1(f, X), we have, by (8.1.30), Prop. 8.2, and Prop. 8.3, (8.2.24)

dim (H(r, 20)= (2g-2)(n+1)+C+Ci +Ell, 17 k +Erp:,,(n+1--e 1)

.

,v)) Suppose first that n = 0. Then v k = 0, and e, = C= C' --= 1, hence dim =2g, so that (8.2.23) is true. Next suppose that n > O. The Jordan canonical 1] according as 1 1 ] or [0 form of (the matrix representing) rk is [ 0 —1 1 the corresponding cusp is regular or irregular (see g 2.1). Therefore, looking 11\ +1 ([ 0 ± I) , we see easily that at the form of p.— in + 1 k

To determine roots w', oj n-2 ,

if n is odd and the cusp is irregular, otherwise.

let e1 be the order of ei. Then p(e) has n+1 characteristic (02-n , arn with a root of unity co, whose order is e, or 2e 1

236

THE COHOMOLOGY GROUP ASSOCIATED WITH CUSP FORMS

according as ei is odd or even. Therefore roots different from 1. We can show that

is the number of these

n+l—E1.2. [(n+2)(e 1 -1)/2e,3 , where Ex3 means the largest integer x. We omit the details of the verification of this formula, since it is quite elementary and rather tedious. Finally we have C (' O. To show this, let x e X", e., p n(a)x= x for all a e r. Put p(z). txen[ n n .

pEs_ n(r)

if

r has

By (8.2.5), p(a(z))j(a, z )" = p(z) for all a e no cusps.

SL,(R) so that p(s)=oo, and

If s is a cusp of

r, take

hjp generates

r.

Therefore

an element p of

fr e r i r(s) , s).

Put

z)". Then t is a polynomial in z, and t(z F2h). t(z). Therefore t must be a constant. It follows that pEc.. 7,(r). Since G...(r) 0) for n> 0 by Th. 2.23 and Th. 2.25, we have p= 0, so that x =O. This proves that Xr. {0}, hence C =0. To show that Ci. 0, let Y be as in Prop. 8.2. Let x be an element of X such that iyen x. 0 for all y e Y. Then for every w e X and every a e r, we have -

0= '[(19.(a - ') - 1)w3c9,,x= l wen(Pn(a) - 1)x, so that (p„(a) 1)x= 0, hence x e xr. Since Xr. {0), this proves that Y. X, hence C' = O. Thus we have determined ei, 72k, 4: and Putting these numbers into (8.2.24), and comparing the result with dim (s.+2(r)) given in Th. 2.24 and Th. 2.25, we obtain (8.2.23). Since we have already seen that so is injective, this completes the proof of Th. 8.4 for the trivial V. The case of non-trivial will be proved in the next section. -

8.3. Action of double cosets on the cohomology group Let r, and r, be commensurable Fuchsian groups of the first k ind, given as subgroups of SL 2(R), and 4 a semi-group contained in GLI(R), and containing and such that ar i a' is commensurable with r, for every a e 4. We assume that LI is stable under the main involution e (see p. 72) of /112(R). Let R be an arbitrary associative ring with identity, R[4] the semi-group-ring (monoid ring) of 4 over R, and X an REJImodule. We are going to define an R-linear map

r1

(8.3.1)

re

(r iar) x :

where P, is the set of all parabolic elements of r,. Let r 1 ar 2 = x), define a map y of r, into X be a disjoint union. For every u e as follows. Given r E 1-' 2, let a,r = ria; with some j and some r

ACTION OF DOUBLE COSETS ON THE COHOMOLOGY GROUP

8.3

Obviously aj , a, is a permutation of fa„

(8.3.2)

237

, ad j. Put

v(r).

It can be verified in a straightforward way that ye zi(r2, ,y); moreover, v Bi(r,, ,y) if u E mr„ x). Further, the cohomology class of y does not depend on the choice of a1. To see this, let is = 3,a, with cli e ri . Then pi r = (3a- 1 (3.TA, and

(8.3.3)

Ei

Z 1aA- 11/(31 11371) v(r)± (r —

n (41,1(3i- ') .

Thus we obtain the same cohomology class as before. We shall now show that v z). 2(r2, ,y). Let Ir e P, and aar = E,a, with Ei E r,. In view of (8.3.3), it is sufficient to show that E, alti(ej E (r-1)X with a special choice of ai . (Note that we may choose the a i depending even on 7r.) Therefore we take a subgroup il generated by ir, and consider disjoint coset decompositions

r,ar, = u r i CA , T,CA =

r i Ce ,

where m is the smallest positive integer such that 7r' e TIC; therefore m may depend on C. Then we take {7r1 to be fa i l. Since Cer= Cle+1 for v <m-1, and Cell - Jr =(ClenC - 9C, we have aiu(e i)= Ec C'u(CrmC - 0. We have C7r"'E P„ so that u(C7rmC - ') = (CzniC - i-1)y( with an element y c of X. Since CeCen= enC`C, we have

n

EL cri,i(L)= Ec (r— 1)y e (7r— 1)X,

q. e. d.

Thus we have shown that y determines an element of Hh(r 2, X) independent of the choice of {ad. Therefore we define (r iar,), to be the map which assigns the cohomology class of y to the cohomology class of u. The notation being as above, let be a multiplicative map of 4 into GL T (R) which maps r, and r, into compact subgroups of GL,r(R). Define X(a) for a e -1 by (8.2.13), and put k n+2. Suppose that ?P( 1)=( 1) if —1e 4. We can now define a C•linear map [riar2]k ,r of sk(r„Y ) to sk(r,, V) by -

(8.3.4)

f Criar23k,r = det (a) k- El=t W(cagaL( 2 ))i(aL, 2) -k

(JE

-

sk(l' ,,v)) .

It can be verified in a straightforward way that the right hand side belongs to sjr,,v), and is independent of the choice of {a d ; moreover we have

(8.3.5)

b(f I Eriar21,0= E=1 X(a)t(f)ocri •

PROPOSITION 8.5. The diagram

238

THE COHOMOLOGY GROUP ASSOCIATED WITH CUSP FORMS

Eriar,Jk,r

(

—s,c(r2,w)

1",arar

is commutative, where yo, and so, are the maps defined in § 8.2, and X=

X.

Let f E sk(r„lv), and g= f l[ri cer,I,r. Define and u by (8.2.19) and (8.2.20). Let r E 2 , and air =riai with n E r, as above. Then so z(g) is represented by a cocycle w which is given by PROOF.

w(r) =

r('°)

Re (b(g)).

By (8.3.5) and (8.2.20), we have

w(r) = Ei X(4) $ 20

Re (b( f))0 a,

= Et=3 X(cf)Ct(air(z0))—Rat(z0))3 = El.1 gaDEl(nai (z0))--4(a2(zo))3 = Ef=1 X(«)[u(ri)+X(Ti)fi(cri(zo))—

(zo))3

= Ef.I X(«)u(73+[X(T)-1]x , where x = Ef., X(al)f(a,(z.)). Thus w belongs to the same cohomology class as the cocycle y determined by (8.3.2). This proves our proposition. Let us now complete the proof of Th. 8.4 for non-trivial W. Let r = Ker (W), and let P. be the set of all parabolic elements of ro. Consider r e ar by taking a to be the identity element. We see that sk(r,,,w) is the direct sum of r copies of sk(ro), hence the map spo

: sk(ro, P.) — No(1 °, iy)

is surjective by what we have already proved. By virtue of this fact and Prop. 8.5, it is sufficient to show that (r • 1 r), is surjective. Therefore, let Let u be the restriction of t to r e. Define I o at, and t E zxr, y by (8.3.2) with r E and n E re. Then

v(T)= EfI x(cr)t(a traii)= d t(7) +(X(7) — 1) Et .1 t(aTi) . This implies that y belongs to the same cohomology class as d. t, hence cro• 1- r),,- is surjective. This completes the proof of Th. 8.4.

THE COMPLEX TORUS ASSOCIATED WITH CUSP FORMS

8.4

8.4.

239

The complex torus associated with the space of cusp forms

Let r be a discrete subgroup of SL,(R), which is a Fuchsian group of the first kind, and P the set of all parabolic elements of r. We consider a r-module D, which is a free Z-module of finite rank. Put D.= DØ R. Then the natural injection of z(r, D) into zAr, Do defines a Z-linear map

Regard

Hixr,

j

(8.4.1)

HAT',

D.) as a vector space over R.

PROPOSITION

8.6. The image of

subgroup of maximal rank) of

H ( I', D)

by j is a lattice (j. e., a discrete D.), and Ker(j) is finite.

HAI',

The group r has a finite set of generators, say {a 1 , •-• , (For example, the elements {Ti} of the formula (1) in the proof of Th. 2.20 form a set of generators of r, cf. Ex. 1.35.) Then every element u of zAr, with any r-module X, is completely determined by u(a,), ••• ,u(a,). This shows that zAr, D) (resp. zji(r, DR)) is finitely generated over Z (resp. R). Further we obtain an R-linear injective map PROOF.

u

(u(a,),

, u(a,,J)

of zxr, DR) into D. The conditions (8.1.1) and (8.1.4) can be written in the form 0

E

(1)

(h=1, 2,

---)

with R-linear endomorph isms Eht Of D. which are stable on D. Similarly, MT, D.) is characterized by the equations

Fmu(c ri) = 0

(2) with maps

Fhi

(h= 1, 2, .--)

of the same type. Put Z' =

(11 E

B' = Z'

zikr, D.)I u( T) E D

n Dv,

for all

DR).

From (1) and (2), we see that Z' (resp. B') is a lattice of z(r, D.) (resp. B' can be identified with a lattice of H(r, DR) B'(1-1 , DR)), and hence = zxr, DR)IR(r, D R). Let Q be a finite subset of P considered in § &1. We can find a positive integer t such that t [D

Or

—

1)D „jc(n. --1)D

for every r E Q. By the same argument as in the end of § 8.1, we can show that t c zxr, D). Therefore the image of HAr, D) contains t -(01e),

240

THE COHOMOLOGY GROUP ASSOCIATED WITH CUSP FORMS

and is contained in Z' /B', hence our first assertion. To prove the finiteness of Ker (j), define a map 2: DR- D'it by ((a,-1)x, • , (am -1)x)

(xe DR) .

Then we can find a positive integer r such that r • [A(D),-- Dim] c 2(D). Let u e B' nz , D). Then (u(a,), • , u(a„,)). 2(x) for some x e DR . Since 2(x) e 2(DR)nDm, we can find an element y of D so that r • 2(x). 2(y). Then r • u e Bi(r, D). Since z(r, D) is finitely generated over Z, this implies that Ker (j) is finite, which completes the proof. From the above proposition, we obtain especially (8.4.2)

HAT, DR)=

D) Oz R.

Now suppose that the above D satisfies the following condition (8.4.3)

The 1M -1-module DR is isomorphic to the direct sum of a finite number of modules X nri i, , X 77,3 of the type discussed in § 8.2.

Then, by Th. 8.4, there exists an R-linear isomorphism ,a of the direct sum

s= sn142(r, Yroe

HAr, D

onto

sns+2(r,

Put L = ,a( j (H ( T , D))). Then L is a lattice in S, so that we obtain a complex torus SI L. Let a be an element of GL(R) such that ara is commensurable with r. Then rar acts both on G and on Hil(r, DR). The action commutes with II by Prop. 8.5. Moreover, it is stable on H ( I', D), if a'D c D. Therefore the action of TaT defines an endomorphism of SIL. For example, let r be a subgroup of SL,(Z) of finite index, and let D= Zn' with n 0. Through the representation p„, we can regard D as a r-module. Then the R[r]-module DR is nothing but X,F with the trivial representation as IT, so that S. sn.2(r). Therefore we obtain a lattice L of s +2(r) which is stable under (rann+2 for every a e M 2(Z) p G (R). This proves the statement (3.520), which we needed for the proof of Th. 3.48. It was also shown in [71] that sn+z(r), L has a structure of abelian variety if n is even. In [74], this result was generalized to the case of S/L of a more general type. For further discussion of the cohomology of this type, the reader may be referred to the papers mentioned in p. 234, Verdier [88], Kuga [41], and Deligne [9]. One should also note the investigations in the higher dimensional case by Matsushima, Murakami, Raghunathan, and Garland.

CHAPTER 9 ARITHMETIC FUCHSIAN GROUPS 9.1. Unit groups of simple algebras So far, our number-theoretical investigation has been restricted to the Fuchsian groups of congruence type contained in GL 2(Q). We shall now show, without detailed proofs, that most of our results can be generalized to arithmetic Fuchsian groups obtained from quaternion algebras. In this section, we shall discuss the group of units of an order in an arbitrary simple algebra over an algebraic number field. Let B be a simple algebra over Q. Then we can define the adele ring B A and the idele group B; of B as follows (cf. [96], [99]). Put

B. = B R

= B OQR ,

B p = B OQ

(p: rational prime).

Qp

Take any Z-lattice g in B, and put gp = gO z Zp, and op(g)= {a e Bp I

gpa Cgp} •

Then B A is the subring of &xi-1p B p consisting of all the elements (a., — , a p, —) such that ap E op ) for all except a finite number of p. BA contains a subring (

o(g) = B. x il p 00,

which is a locally compact ring with respect to the usual product topology_ We introduce a topology into B A by taking o(g) to be an open subring of BA. One can also define BA to be simply BO Q A. Now BxA , as an abstract group, is just the group of all invertible elements of B A. In other words, B,x, consists of all the elements (a., — , a p, ---) such that a p E o p(g)x for all except a finite number of p. B; has a subgroup o(g)x = Bolxll p ope` ,

which is a locally compact group with respect to the usual product topology. We introduce a topology into B; by taking o(g)x to be an open subgroup of B. The definition of the topological ring BA and the topological group B; does not depend on the choice of g. It should also be noted that the topology of B; is not induced from that of BA.

242

ARITHMETIC FUCHSIAN GROUPS

Hereafter we write

GQ

for 134 and put

,

Gp = B; ,

GA =13:4'

.

One can regard GQ as the group of Q-rational points of an algebraic group G defined over Q, and GA as the adelization of G. If the reader is not familiar with the general theory of algebraic groups and their adelization, he may consider G Q and GA just new symbols for Bx and B. Denote by G o the nonarchimedean part of GA, and by G, the identity component of G. These symbols are in agreement with those of Ch. 6, if B= M,(Q). We identify B (resp. G Q) with a subset of BA (resp. GA) by means of the diagonal injection x x, x, ---). Let F denote the center of B, and v the reduced norm of B to F. The map v can be naturally extended to a map of BA to FA, which we denote again by v. (Note that Id = det, if B = A/12(Q)) Put GA I

14-0 = 1 } s

G41= {X e GQ v(x)= 1} . Then the following theorem is fundamental and well-known. THEOREM 9.1. (1) G Q is a discrete subgroup of GA. (2) GZ\G',‘, is compact if B is a division algebra. (3) For any open subgroup S of GA containing G., the orbit space GQ\GA IS is finite. (4) For any open subgroup T of G.u, containing G:, the orbit space GZ\G1 IT is finite. For the proof, see Weil [96], [99]. These facts can be generalized to reductive algebraic groups, see Borel [2], Borel and Harish-Chandra [3], Mostow and Tamagawa [52], and Godement [24]. Let g be the number of archimedean primes of F, and let FOQ R. Then we can put

(9.1.1)

B = /30111 ED

(9.1.2)

F.= F., ef)

•

ED

Bou g

,

ED

where Foot is the center of Ba,i ; F., is either R or C; B.,i belongs to the algebras of the following three types : M(R), M n (C), Mn(H), where H denotes the division ring of Hamilton quaternions. Put G.,= B. Then G.= G. i x •XG0, g , and Gœi belongs to the following three types : G 1,7,(R), GL 7,(C), G1, 7,(H). Put G (1; = G'An G: = G GY. 1 = nGuA . Then GG,x •-• xGrug. Now fix any open compact subgroup To of Gict, and put

T=To Gra, T r =Tn Gt4

9.2

FUCHSIAN GROUPS OBTAINED FROM QUATERNION ALGEBRAS

243

Let r denote the projection of T r to G oo. Then r is a discrete subgroup of G. Moreover, r\Gg. is compact if B is a division algebra. PROPOSITION

9.2.

PROOF. By (1) of Th. 9.1, GZ is discrete in OA, so that rr is a discrete subgroup of Tox at. Since To is compact, we see, by (3) of Prop. 1.10, that the projection r of rr to Gr. is discrete in Gr.. On account of (4) of Th. 9.1, G',4T is an open closed subset of G. Suppose that B is a division algebra. By (2) of Th. 9.1 and Prop. 1.3, one has Gz T= GK with a compact subset K of T. Since T=T oGX, we can take K in the form K=T oll with a compact subset H of G. Write every element of G1 in the form (x,y) with X E GI and y E Glt. Since G 0„c04T 0H, every element (1, y) with y e Gr., can be written as (1, y)= (a, a)(t, h) with a e GI, teT o, and h ŒH. Then a e Gr) rAT = TT . Since y = ah, this shows that Gro. rH. By Prop. 1.3, T\G: is compact.

9.2. Fuchsian groups obtained from quaternion algebras By a quaternion algebra over a field k, we understand a central simple algebra over k of rank 4. Let I denote the algebraic closure of k. Then, an algebra R over k is a quaternion algebra if and only if RO k ii is isomorphic to Al20-0 over I A quaternion algebra R over k is either isomorphic to M2(k), or a division algebra. Let tr and i) denote the reduced trace and the reduced norm of R to k. Then we can define an involution e of R over k (i. e., a k-linear one-to-one map of R to itself such that (xy)€ = yl x‘, x= (x90 by (x e R).

x+xl = tr (x)

In fact, if f is any i-linear isomorphism of R 0, -k to M,(i), and f(x)=[ 4: bi d J' then we see that tr (x) = tr ( f (x)), and hence `!

—ba ]=j.`f(x)j - ' ,

It follows that e defines an involution of R over k, id(x). xxl, which we call the main involution of R. We can easily verify that (y - lxy)e=y - 'xey for all x E R, y e R". we now Coming back to the simple algebra B and its center F of § 9.1, make the following assumptions :

r 0 —1 i where 3= I_ 1 cd-

(9.2.1) F is totally real;

(9.2.2) .8 is a quaternion algebra over F. so that the components Then, all the components Fooi of (9.1.2) must be R, H are the only quaternion Bee, of (9.1.1) is either M2(R) or H, since M2(R) and

244

ARITHMETIC FUCHSIAN GROUPS

algebras over R.

Changing the order of Bœ, if necessary, we may assume { A12(R)

(1 S.
B.,..1 =

(r
H

where g= [F: (2 ], and r is the number of the archimedean primes of F which are unramified in B (see the explanation below about P B). In the following discussion, we always assume

(9.2.3)

r> 0,

and fix, once for all, the identification of Bcei with M 2(R) or H. The groups G.„ G. +, and GI in the present case can be written as GL 2(R)rX(Hx) s- r G. + = G LI(R)T x (H x)g " 1* ,

SL 2(R)r X (H")g ,

where Hu = (x E H I v(x)= 11.

9.3. The notation being as in Prop. 9.2, let r, be the projection to the factor SL 2(R)' of G.+, under the assumption (92.1, 2, 3). Then r,

PROPOSITION

r

of is a discrete subgroup T'\SL 2(R)r is compact.

of SL 2(R) 7.

Moreover,

if B is a division algebra,

This follows immediately from Prop. 9.2, and Prop. 1.10, since Hu is compact. Now let GL(R) act on 0 as before. Then GL1 - (R)r acts on Or componentwise. Put GA+=GoGc.+ , GQ+ = GQn G A+ •

We define the action of an element a of GQ+ on Or to be the action of the projection of a to the factor GL(R)T of G. Observe that F x is contained in GQ+, and coincides with the set of all elements of GQ+ which act trivially on 0r. We denote by ri the injection of F into R obtained by identifying F., with R. Then GQ+

= {a E B f v(a)ri > 0

i r)} .

PROPOSITION 9.4. Let K be a totally imaginary quadratic extension of F, and q an F-linear isomorphism of K into B. Then q(K*) is contained in G Q+ , and every element of q(Km), not contained in F, has a unique fixed point w on Or, which is common to all such elements of q(Km). Moreover, q(Kx) Conversely, if an element a of G Q+ , not contained in = {r E GQ+ I r(W) = F, has a fixed point on Or, then F(a) is isomorphic to a totally imaginary

FUCHSIAN GROUPS OBTAINED FROM QUATERNION ALGEBRAS

9.2

245

quadratic extension of F. We call w the fixed point of q(Kx) on Or

.

Let a e K F, a = q(a), and let affi, , e) be the projections of a to B.„ ••• , Let ai be an isomorphism of K into C which coincides with r i on F. Since K is totally imaginary, we see that the eigen-values of a( ' ) are a't and eiP, where p denotes the complex conjugation. Therefore, a") gives an elliptic transformation on 0 (see § 1.2), and hence has a unique fixed point wi on 0. Put w= (w„ , w,.). Let p E q(Kx). Then fi(w)= fia(w) = a(i63(w)), so that p(w) is a fixed point of a on Or. Since w is the only fixed Observe that point of a, we have p(w)= w. Suppose r(w) = w with r e the isotropy subgroup PROOF.

E GL(R)

!

e(w)=w}

is isomorphic to Rx • SO(2) (see § 1.2), and hence commutative. It follows that 7 commutes with every element of q(K x). Since q(K) is its commutor in B, 7 must belong to q(K). Now, conversely let a e GQ+, a EE F, a(z)=z with z E Or. Let d i) denote the projection of a to 130,i. Observe that a") does not belong to the center R of B.,. Therefore aw, •-• ,a (r) are elliptic, so that none of the eigen-values of aw, ••• , cir) can be real, hence the first r archimedean primes of F corresponding to r„ ..• , r,. are ramified in F(a). The remaining g r archimedean primes of F are ramified in every quadratic subfield of B, since they correspond to the factors H of B.. Therefore we obtain the last assertion. —

One can naturally ask the following question : (i) How many quaternion algebras B, with a given r, over F can be obtained? (ii) What type of quadratic extension K of F is embeddable in B? To answer these questions, let F, denote the completion of F with respect to an archimedean or a non-archimedean prime y of F. Put B.= BOF F„. Let PB denote the set of all y such that Bo is a division algebra. A prime y contained (resp. not contained) in PB is said to be ramified (resp. unramified) in B. Then the following assertions hold : (9.2.4)

PB is a finite set consisting of an even number of primes.

(9.2.5)

For any finite set P with an even number of archimedean or non-archimedean primes of F, there exists a quaternion algebra B over F, unique up to F-linear isomorphisms, such that P= PB.

(9.2.6)

A quadratic extension K of F is F-linearly embeddable in B if and only if KOF F, is a field for every prime v e PB.

246

ARITHMETIC FUCHSIAN GROUPS

These results are special cases of Hasse's theorems on simple algebras over algebraic number fields (see for example [99]). Observe that PB contains exactly g r archimedean primes which correspond to the factors B.7.4.1= ..= Boe,= H. The set PB can be empty ; we have then B = M,(F). Therefore B is a division algebra if PB is not empty, especially if g> r. —

Let us now consider the case r= 1. Then we see that B is either a division algebra, or isomorphic to Al,(Q). Therefore the group r, of Prop. 9.3 is always a Fuchsian group of the first kind ; TAO is compact unless B is isomorphic to M,(Q). We consider F as a subfield of R, and assume that the projection of F to the first factor GIVR) of G.+ (i. e., r, in the above notation) is the identity map of F. This assumption is not absolutely necessary, but simplifies our discussion. Let K, q, and w be as in Prop. 9.4. In view of the assumption just made, we obtain q(P)[

for all p

E

K.

7 ]=

[

WI

],

or q(47 ]= 4 11)1 ]

(We are considering K as an algebraic number field in the sense

of 0.4, so that K is a subfield of C.) We call q normalized if q(p)[ wi j= p[ wi ] for all pE K. If q is not normalized, its "complex conjugate" q' defined by Cp)= CA) is normalized. Thus the non-trivial fixed points of GQ+ on cf,) are in one-to-one correspondence with the normalized embeddings of totally imaginary quadratic extensions of F into B. Our present discussion generalizes that of § 4.4, except that we have nothing here corresponding to the elliptic curves considered there. Anyway we shall be interested in the values of automorphic functions at these points. Before going further, let us insert an example of the representation V of § 8.2. Let p, denote the projection map of G Q into the i-th factor of G oe. Observe that P i is injective. Assume g> 1, and let r and 1-1 ' be as in Prop. 9.3. Then pi is an isomorphism of r to r,. If i> 1, p, maps r into Hu It is well-known that there is a homomorphism f of 11 14 onto .

SO(3)= {X e GL,(R)I 'X X= 13) , such that Ker (f)= {±1}. Therefore foP i oPi-1, for i >1, maps P into a compact subgroup of GL,(R). This gives an example of 7 considered in § 8.2. One can further obtain some interesting examples of r'-modules D satisfying (8.4.3), which are composed of these fo Pi oPT' by the operation of direct sum and tensor product. But we shall not go into details of such modules in this book.

9. 2

FUCHSIAN GROUPS OBTAINED FROM QUATERNION ALGEBRAS

247

Let us identify F with a subgroup of G A, and denote by F` the closure of FxF:a.f. in F. It can easily be verified that FcG, is the closure of Fx in GA. Now denote by 2 the set of all open subgroups S of GA+ containing PG. + and such that SIF`Goe,., is compact. Put, for each S E 2,

(9.2.7)

r s =Sr\G Q+

.

Observe that Fxc s . PROPOSITION

9.5. For any S E 2, the group r s/Fx, as a transformation

group on 0, is a Fuchsian group of the first kind. Since r s and r s , are commensurable for any two members S and S' of 2, it is sufficient to prove our assertion for one S. Take any maximal order o in B, and put R=G c.,+ xilp 0; where op = oOr Zp, and T =RnOLS=FxR, T R = RnG Q+ , 1-17 =TnGZ. Then SEZ, and r s .F ,T R . Let E denote the group of all units in F. Then v(x)E E if X E rR . Put Ec" = {e 2 I e E E}, and P=fr E r RI V(r) e E( 21. Then Y R :pj is finite, since [E: E (2)] is finite. If T E I", then (T)= e 2 with e E E, so that v(e - 'T). 1. Since eE R, e'T is contained in r r . This proves that r, cEr7. From our definition of r' and r i, we obtain Er 7 Cr,, so that rt =Err. Therefore PROOF.

,

Ers:Fxrrj.[ForR:Fxr'][r R :rij<00. As is seen above, by virtue of Prop. 9.3, kind. This proves our proposition.

rr is

a Fuchsian group of the first

Define a homomorphism a of GA to Gal (Fab/F) by

a(x)=[v(x) -1, F]

(x E G.

(For the notation [s, F] with se F, see § 5.2.) We see that Fx • v(S) is an open subgroup of F of finite index for every 5 E E. By class field theory, it corresponds to a sub field of Fab of finite degree over F, which we denote by k s. Then Lemmas 6.16 and 6.17 are true in the present case. It should also be noted that Lemma 6.15 is true with GI in place of SL,(A), by virtue of the approximation theorem due to Eichler [15] and Kneser [39]. We are ready to state the first main theorem of this section, which is a generalization of the discussion of § 6.7 and Th. 6.31. THEOREM 9.6. There exists a system Vs,

frs(4, (S, TEE; xEGA4.)},

formed by the objects satisfying the following conditions. (1) For each SEE, (V s, (p s) is a model of 0*/I's in the sense of § 6.7, where 0* is 0 or 0 ■.)QU {col, according as B is a division algebra or not.

248

ARITHMETIC FUCHSIAN GROUPS

(2) Vs is defined over k s. (3) Jr.s(x), defined if and only if xSx -1 C T, is a morphism of V s onto Vf.(z), rational over k s, and has the following properties:

(3.) J3(x) is the identity map if (3 b) Jr3(x)" )0 ./.3R(Y) = ./TR(Y),'

XE

S;

(3 c) frs(a)CWs(z)J=Wr(a(z))f or every aEG Q ,. and every zE0 (if aSa - lc T). (4) Let K be a totally imaginary quadratic extension of F, q a normalized F-linear isomorphism of K into B, and w the fixed point of q(Kx) on 0 (see Prop. 9.4). Then, for every S E 2, Vs(W) is rational over K a6 . Moreover, for every U E K:4 , one has

vr(w )(" ' =.frs (q( u)- ') Ev s(w )3, where T = q(u) -1 5q(u).

The system is unique in the following sense. THEOREM 9.7. If two systems {V s, y) s, Jrs(x)} and {VL ioLR•s(x)} satisfy all the conditions of the above theorem, then there exists, for each SE 2, a biregular isomorphism Ps of Vs to VL rational over ks, such that fi•s(x) 0 P s = Pri' 0 Jr s (x)

So. = PS 0 ¶°s , for all S, T of 2 and all X E

G A+

satisfying xSx - ' c T.

It is easy to give a generalization of Prop. 6.33, which may be left to the reader as an exercise. In the next place, to generalize Th. 6.23, put 3

--{

foso31fE ks(Vs)} ,

3=-- 1/4„J.s zs • -

Then (1) of Th. 9.6 implies that CF's is the field of all automorphic functions with respect to r s . Also we have ks= Faonas, and Fab=Crl. For every X E G4.1., we can define an automorphism r(x) of over F by

(9.2.8)

(f ovr)r's) = f'z' 0.1rs(x)c)

PROPOSITION

(f E kr( VT), S = X -1 TX) .

vs

9.8. The symbol r(x) has the following properties.

( i) r(xy)= r(x)r(y), i. e., r defines a homomorphism of G44. into Aut (W). (ii) (iii)

r(x)=a(x) on Fab. hr(a) (z)= h(a(z)) for every h E 15 a E ,

GQ4.,

and zE

0.

The equality (ii) follows directly from the definition (9.2.8); (i) from (3b) of Th. 9.6; (iii) from (3,) of Th. 9.6. PROOF.

Now Th. 6.23 and Th. 6.31 can be generalized as follows. THEOREM 9.9.

The sequence

FUCHSIAN GROUPS OBTAINED FROM QUATERNION ALGEBRAS 249

9.2

1— Fe a.+--■ G A+

r --. Aut (W) — 1

is exact. The map r is continuous, and induces a topological isomorphism of G A+ IF`G., onto Aut (W). Moreover, for every SE 2, one has (1) S= (xe G A+ 1 hecx) = h for all h E s}, i.e., r(S)= Gal (/as). (2) as = (h E 1 he(s) = h for all x e S).

THEOREM 9.10. Let K, q, and w be as in (4) of Th. 9.6. Then, for every he defined and finite at w, the value h(w) belongs to K a., and

a,

h(w )C ts.A1 = hr ( C ti)-1) (w)

for every uE K: 4.

Th. 9.10 follows immediately from (4) of Th. 9.6. PROOF of Th. 9.9. It is straightforward to see that hr(s) = h for he g and XE S. Conversely, suppose that r(x)= id. on as. By (ii) of Prop. 9.8, r(x)= id. on ks. By the generalization of Lemma 6.17 mentioned above, we have x= sa for some sE S and a E G Q .,.. Then, for every f Eks(Vs), we have

f ()Ws= (f owsr") = folsr(sa)owr = f V&A«) o W r , where T = a'Sa, so that Sos = isr(a) 0 VT, and hence v s(z)=so s(a(z)) for all z E O. Therefore a E r s, so that x E S. This proves the first equality of (1) of Th. 9.9. It follows from this result that

Ker (r)= rlsez S= PG. + . Now we can repeat the proof of Th. 6.23, and obtain the surjectivity and the continuity of r. If B # MA), we can dispense with the discussion about cusps. The equality (2) of Th. 9.9 follows from (1) and Prop. 6.11. PROPOSITION 9.11. (i)

Let Ge denote the closure of G Q+ Gœ + in GA.

Then

GC= PG Q,GuA = (xE G A+ k(x) E F e } . (ii) For every SE 2, Ge nS is the closure of rsao+ in GA. (iii) r(Ge ns)= (a e Aut (W) I a = id. on Fat, • as } = Gal (a/Fab -

s).

To prove this we need LEMMA 9.12. Let E + be the group of all totally positive units of F, E. the projection of E + to the non-archimedean part of F:„ and f o the closure of E. in F. For a positive integer n, put Er = {X II I X E E},

Fe(n) = {X n I XE Fe } .

Then Fe .E.F'F:o+, f o = E0E, and Fe = FAFe( n) for every positive integer n.

250

ARITHMETIC FUCHSIAN GROUPS

Let {U.}:-1 be a family of compact groups which form a basis of neighborhoods of the identity in the non-archimedean part of F. Let X E Ft. Then, for every m, there exists an element y„, of Fx such that y,T,Ix E U„,F .4. Put e= yT'y„,. Then ern e E+ and e,TilyTixE U„,Fex. +. Therefore the non-archimedean part of mix belongs to f o. This shows that Fe Cf 0F x F:+. Since the opposite inclusion is obvious, we obtain the first assertion. Next, since {xn 1 X E Eo} is of finite index in E9, we have [E„f6") : Eso]<00. We see also that is closed, since it is the image of the compact set E 0 under the continuous map x .---) x 3. Therefore E0f6") is closed, hence the second assertion. The last assertion follows easily from the first and second ones. PROOF.

Er

PROOF of Prop. 9.11. Since Fx cG c1+, we have Fe cGc. The strong approximation theorem, mentioned above, (of which Lemma 6.15 is a special case) implies that GI c G Q+ U for any open subgroup U of GA , so that GI C Ge. Therefore we obtain FeGe+GlcGec{xE G A+ I v(x) e Ft}. Let x e G A+ and v(x) e Fe . By Lemma 9.12, i(x)= ab' with a e Fx and b EFe. We see that a is totally positive. By virtue of the norm theorem of simple algebras (see, for example, [99, p. 206, Prop. 3]), we have a = v(a) for some a E B x = G Q. Then v(b - ' ce 1 x)= 1, so that x= ba - (a- ' b- ' x) e FeGQ.,G1, which proves (i). Next, let S e Z. For every open subgroup U of G„, we have Ge c G Q+ U. Therefore, if UcS, we have G` nSc(G Q ,nS)- U = rs U, so that Ge n S is contained in the closure of rsGe,,+. Since the opposite inclusion is obvious, we obtain (ii). BY CO, we have Ge = {x e G A+ I a(x)= 1} . This together with (1) of Th. 9.9 proves (iii). EXAMPLE 9.13. Let m= 7, 9, or 11, and let F=F„,=Q(C+C-i) with C=6.27 ". Then [F: Q] = 3, 3, 5, respectively. Since [F: Q] is odd, there exists, by virtue of (9.2.5), a unique quaternion algebra B over F which is unramified

at all non-archimedean primes of F, the archimedean prime of F corresponding to the identity map of F,

ramified at all the remaining archimedean primes of F. Take a maximal order o in B, by which we mean a maximal subring of B that is a free Z-module of rank [B: Q]. Put op = o(Dz Zp for every rational prime p, and U= Gos+ x Hp o;„ S = Fx(1. Since U is open in GA , we see that PG.+ c S. Moreover, U1G.+ is compact, so that S E Z. It can be shown that CA = G Q U and F ;4( = 1 • v(U), so that k 3 = F. (This follows from the fact that the class number of F in the narrow sense is one.) We see that r s = F mr(0),

9.2

where

FUCHSIAN GROUPS OBTAINED FROM QUATERNION ALGEBRAS

251

r(0)= (7. E D 1 r0 = 0, (r)> O}. can prove that rs IF x is a " triangle group" generated by three

Now one elliptic elements 7„ 7,, 7,„, of order 2, 3, m such that 7,7,T.=1; tVrs is of genus 0; every elliptic element of T sIFx is conjugate in r sIF* to a power of 7 2 , 7 3 , or T„,. Let z,, z„ and z , be the fixed points of 7 2 , 7„ and r on -0 respectively. Since 0/r5 is of genus 0, there is a Trautomorphic function on f.) which gives a biregular isomorphism of 0/r 3 onto the complex projective line V. One can normalize such a function T by the condition (9.2.9)

so(z,)= 1,

ço(z,)= 0,

T(z„,). co .

Then (V, T) can be taken as the member (V3, T s) of the system of Th. 9.6 for the present B. On account of (9.2.6), for every totally imaginary quadratic extension K of F, there exists a normalized F-linear isomorphism q of K into B. Moreover, one can take q so that q(o K)co, where o fc denotes the maximal order in IfIf q and q' are such F-linear isomorphisms of K into B, there is an element a of GQ+ such that cr - lq(p)a=q/(p) for all pE K. Then there exists a fractional ideal a in K such that q(a)o= ao. The ideal a is principal if and only In this way we if 7 - ' q(p)r= q'(p) for all p E K, with an element r of can show that, if h is the class number of K, there are exactly h points wi , --- , wh, modulo rs-equivalence, which represent the fixed points of q(Kg) for all such q satisfying q(ox)co. From (4) of Th. 9.6, we obtain

r8.

(9.2.10) The values T(w,), .-- , T(w h) form a complete set of conjugates of T(w 1 ) over K, and K(T(w,)) is the maximal unramified abelian extension of K. For q, q', a, and a as above, let w be the fixed point of q(K"), and (K(92(w))1K ). Then Th. 9.10, or (4) of Th. 9.6, implies a= (9.2.11)

So(10° = 40(a -1 (z)) .

We observe that (9.2.10, 11) are similar to Th. 5.5 and (5.4.2). Actually it can also be shown that {T(11) 1), ... , T(w h)} is a complete set of conjugates of ço(w,) not only over K, but also over F. Thus w is an analogue of the modular function j. The condition (9.2.9) corresponds to j(i) = 1, j (el' 3 ) = 0, j(co)= co. Finally we note that in the case m=7, r s/Fx is the Fuchsian group with the least measure of the fundamental domain, which was mentioned at the end of § 2.5. Unfortunately, the proof of Th. 9.6 is too long and intricate to include in this book. It needs a detailed analysis of certain families of abelian

252

ARITHMETIC FUCHSIAN GROUPS

varieties parametrized by the variable z on O. These abelian varieties play, to some extent, the role of elliptic curves in Chapter 6. The proof of Th. 9.7 is comparatively easy ; it may be a good exercise to give a proof at least in the simplest case B= NI,(Q). Actually we can generalize our theory to the case of algebraic groups whose arithmetic subgroups act on a product of Siegel upper half spaces. For details, the reader is referred to [77], [78], [80]. As for Ex. 9.11, see [77, 3.18]. We can of course propose a further generalization to the whole family of semi-simple or reductive algebraic groups whose arithmetic subgroups act on bounded symmetric domains. The case of unitary groups over algebras with involutions of the second kind has been treated by K. Miyake [49]. Therefore, roughly speaking, the theory has been established for one half of the family of all bounded symmetric domains of classical type. It seems quite likely that it can be extended to the remaining half. It is not clear, however, whether the semi-simple groups of exceptional type can be included in this framework. The theory of Hecke operators can be developed also for the groups r s of the above type. We can then construct Dirichlet series, similar to those of Ch. 3, which have Euler product and functional equation (see [74]). Further, these Dirichlet series, for cusp forms of weight 2, provide, the zetafunctions of the curves Vs of Th. 9.6, exactly in the same manner as in §§ 7.4, 7.5. For details, the reader is referred to [77], [80, 2.23], and [50]. Finally we mention that the curves V s, or rather the above theorems, are in close connection with Ihara's recent investigation [34].

APPENDIX The purpose of this part is to recall a few elementary facts on algebraic varieties, especially on algebraic curves and abelian varieties. We do not mean to present an introduction to algebraic geometry for the reader who is totally unfamiliar with the subject. Our intention is merely to remind a more experienced reader of some fundamental definitions, after Weirs Foundations [90], and to make sure what terminology we are using, and what results are referred to in the text. 1. We fix a universal domain D, which is an algebraically closed field of infinite transcendence degree over the prime field. If the characteristic is 0, we often take the complex number field C as D. By a field, we always mean, except when the contrary is stated, a subfield of D over which Q is of infinite transcendence degree. If k is a field and x=(x„ ••• , x n) is a set of elements x, of D, we denote by k(x)=k(x„ — , x n) the field generated by X1 , ••• , x„ over k, which is again a field in that sense. We say that k(x) is a regular extension of k, if k is algebraically closed in k(x), and k(x) is a separable algebraic extension of a purely transcendental extension of k, or equivalently, if k(x) is linearly disjoint from the algebraic closure of k, over k. Consider an affine space 91„ and a projective space q3„ over D, of dimension n, with a fixed coordinate system. Let a =(a1 , — , an) and b=(b 1, — , bn) be points of 9f„. We say that b is a specialization of a over a field k, if F(b„ ••• , b n)= 0 for every polynomial F(X„ — , X„) with coefficients in k such that F(a„ .-. , an)= O. Then we denote by [a—b; k] the ring of all elements of the form P(a)1Q(a) with polynomials P and Q with coefficients in k such that Q(b)* 0. For a point x.(x 0, x„ — , x„) of q3 n, let a2(x) denote the point (x0/x2, x 1/x 2 , — , x,/x) of W7i.,_„ whenever x1 = 0. For x E q3,, and y e q3„, we say that y is a specialization of x over k, if there is an index 2 such that x 2 # 0, Y2 *0, and al( y) is a specialization of ai(x) over k. More generally, put

x $nr x Win, x — x91,4 , and let x= (x"), ••• , x (r-")) and y = (y" ) , •-• ,y (r")) be points of X, where x"' and y") are points of q3ni or %,,, i , according as i ._ r or i>r. Then we say that , y"'"" )) y'r+i ), y is a specialization of x over k, if y = (a21(y")), •-• , is a specialization of x' = (a it (x ( "), — , x(r."), •.. , x (r+s )) over k, for some 2„ ••• , A. We then put —

[x—y ; k]=[x'—.y' ; k],

APPENDIX

254

since this ring does not depend on the choice of ,Z1 , •-• ,A r. We denote by k(x) the field k(a 4(x (1)), ••• , x(r -F", ••• , A set V of points of X is called a variety (or an algebraic variety) if there exists a field k and a point x of X such that (i) V is the set of all specializations of x over k; (ii) k(x) is a regular extension of k. (The condition (ii) implies that V is absolutely irreducible in the usual terminology.) If V, x, and k are in this situation, we say that : V is defined (or rational) over k; k is a field of definition (or of rationality) for V; x is a generic point of V over k; V is the locus of x over k. The transcendence degree of k(x) over k is uniquely determined by V, and called the dimension of V. A variety contained in V is called a subvariety of V. A point of V is a zero-dimensional subvariety of V, and vice versa. A subvariety of VI, (resp. 4.1 ) is called an affine (resp. a projective) variety. We say that a projective variety V is defined by equations Fi(X„, •-• , X ii). O (i=1, •-• , 0 if these , X, ] , the ideal of all the polynomials polynomials generate, over Q[X 0, vanishing on V. 2.

If two varieties V and W are given, we can find a common field k of rationality for V and W; further we can find a generic point x of V over k and a generic point y of W over k such that k(x) is linearly disjoint with k(y) over k. Then the set-theoretical product V x W is the locus of (x, y) over k, so that it is a variety. A subvariety T of V x W is called a rational map of V to W, defined over k, if, for a generic point (u, v) of T over k, one has k(u, v)= k(u), and u is a generic point of V over k. We say that T is de fined at a point a of V, if there exists a point b of W such that (a, b) E T, and [v b ; k]C [u a ; k] . 3.

The point b is uniquely determined by a under that condition, so that we put b = T(a). Especially we always have T(u). v. If S is a rational map of W to a variety X defined over k, and if S is defined at y, then we denote by So T the locus of (u, S(v)) over k, which is a rational map of V to X. We call T a morphism if T is everywhere defined on V. T is called birational if k(u)= k(v), and y is generic on W over k. If that is so, we denote by T - ' the locus of (y, u) over k, which is a rational map of W to V. We say that V is birationally equivalent to W over k, if there is a birational map of V to W defined over k. T is called a (biregular) isomorphism if it is birational, and both T and T - ' are morphisms. 4.

A rational map of V to the affine 1-space 91 1 is called a function (or

255

APPENDIX

rather a meromorphic function) on V. All the functions on V form a field, not contained in Q unless dim ( V) , 0, which is denoted by D(V). All the elements of D(V) rational over a field k of definition for V form a subfield of D(V), denoted by k(V). Then k(V) is linearly disjoint from D over k, and Q(V)= k(V). For a generic point x of V over k, the map »—■ f(x) gives an isomorphism of k(V) onto k(x).

If V is the locus of x over k, and a e V, we say that a is a simple point of V, or a is simple on V, if there exists a birational map T of V to a subvariety W of A n satisfying the following conditions: (i) T is defined at a, and T -1 is defined at T(a); (ii) If b=T(a), y =T(x), and r= dim (V), then there are n—r polynomials F,(X„••• , X„) (i=1,••• ,n—r), with coefficients in k, such that F4(y) = 0 (i= L •-• , n—r), and 5.

rank

(0] .1 rL ax

This definition does not depend on the choice of W and T. V is called nonsingular if every point of V is simple. If the universal domain is C, 9I, and q3„ are viewed as complex manifolds. Then every non-singular variety of dimension r has a natural structure of complex manifold of complex dimension r. Every projective variety is compact. 6. Let V be a variety defined over k, and o an isomorphism of k into Q. Take a generic point x of V over k. Then we can extend a to an isomorphism r of k(x) into Q. Put x'=xr. Then the locus V' of x' over k° is meaningful, and depends only on V and a, i. e., it does not depend on the choice of x and r. We put V'= V°, and call it the transform of V under a. If T is a rational map of V to W rational over h, we can define T and 1r, and observe that Ta is a rational map of V° to W°. Especially if f e k(V), then f° is a function on V. If T is defined at a point a of V rational over k, then T° is defined at a°, and T(a)° =T°(a°). The symbols, V, x, and k being as above, let W be another variety with a generic point y over k. Suppose that there is an isomorphism of k(W) to k(V), which induces an automorphism p of k. Then there exists a birational map h of V to Wo which is characterized by the property

e

(6.1)

= fP oh for every f e k(W).

To show this, define an isomorphism r of k(y) to k(x) by f(y)r= f 4(x) for J E k(W), so that the diagram

256

APPENDIX R. f ek(W)

T!

Ay) e k(Y)

k(V) a g

r

41

k(x) a g(x)

is commutative. Since f is generic on WP over k, and k(yr)= k(x), we obtain a birational map Je of V to WP, defined over k, such that J(x)=yr. Then we have fe(x) = f( y) 7 = fP( y 7) = fP(Je(x)), hence (6.1). If 72 is an isomorphism of k(X) to k(W) with another variety X defined over k, which induces an automorphism a of k, then Jr : W--.X° and Jre : V---X 0 " is meaningful, and (6.2)

.fre ---- A 0 .Iz .

7. Suppose that the characteristic of D is p > 0, and let q= pe with an integer e. Then a .--oaq is an automorphism of D. We denote by Vq the transform of a variety V under this automorphism. (In the usual circumstances, Vq will not be confused with the product of q copies of V.) If e> 0, and V is the locus of x over k, we can define a morphism F of V to Vq rational over k by F(x)= xq, which is called the q-th power morphism (or the Frobenius morphism of degree q) of V to Vq. Let T be a rational map of V to W, and F/ the q-th power morphism of W to W. Then we have PoT=TqoF,

(7.1)

(where Tq is of course the transform of T under the q-th power automorphism of D). In other words, the following diagram is commutative : T

Tq 8. If W is a variety of dimension n, there exist an n-dimensional vector space Dif (W) over D(W) and an D-linear map d: D(W)--.Dif (W) with the following properties : (8.1)

d(fg)= f - dg+ g . df

(82)

{di'', — , df} is a basis of Dif(W) over f2(W) if and only if f2(V) is

(f, g e D(W)),

separably algebraic over f2(f 1 , •-• , f.).

APPENDIX

27

The couple (Dif (W), d) is uniquely determined by W, up to isomorphisms. An element w of Dif (W), called a differential form on W of degree one, can be written as w = E t gidfi with gi and fi in 12(W). Let W be defined over a field k. The form w is called rational over k, if gi and fi are chosen so as to be contained in k(W). Let Dif (W; k) denote the elements of Dif (W) rational over k. Then Dif (W)=Dif (W; k)O kiw ,D(W). An isomorphism a of k into Q induces an isomorphism of Dif (W; k) to Dif (W' ; le) by co' = Ei gt • dff. We say that w is finite at a point a of W if w= Ei g, • dfi with functions gi and fi which are defined at a. Let T be a rational map of a variety V into W. If there is a point c of V such that T is defined at c and w is finite at T(c), then we can define an element w o T of Dif (V) by w 0 T = E i (g 0 T) d(f i oT). We denote woT also by 3T(w). If V, W, co, and T are rational over k, and is an isomorphism of k into K2, then (w oT)°= wet° T°. We call a differential form w on a projective variety W holomorphic, or of the first kind, if w is everywhere finite on W. We denote by 2(W) the set of all holomorphic elements of Dif (W), and put 2(W ; k)= D(W)n Dif (W ; k) for any field k of rationality for W. Then 2(W)=0(W; k)e k Q. 9. A variety V is called an algebraic curve, or simply a curve, if V is of dimension one. If a field k of definition for V is perfect, we can find a nonsingular projective curve which is birationally equivalent to V over k. Let V be a projective non-singular curve defined over k. Then all the notions and results of § 2.3 can be generalized to the present situation. In fact, we only have to replace W, K, and C by V, 12(V), and D. The divisors on V and the symbols div ( f), L(A), 1(A), etc., can be defined in the sam e manner, without any modification, except for the following point : The relation (2.3.1) should be (9.1) df= 0 if and only if D(V) is inseparable over D(f). This is of course a special case of (8.2). The genus of V is defined, for example, by Prop. 2.13, or (2.3.2). Then Prop. 2.11, Th. 2.12, and Prop. 2.14 are true. A divisor on V is also called a 0 cycle on V. -

10. A projective variety A is called an abelian variety if there exist morphisms f: AxA .44 and g: A .44 which define a group structure on A by f(x, y)= x+y, g(x)= x. Additive notation is used, since any such group structure on a projective variety can be shown to be commutative. The neutral element is accordingly denoted by O. If the variety A, and the morphisms f and g are defined over a field k, then we say that the abelian variety A is defined over k. Let A and B be two abelian varieties. By a homomorphism of A into B, —

—

—

258

APPENDIX

or an endomorphism when A= B, we understand a morphism 2 of A into B satisfying 2(x+y) = 2(x)--F 2( y). If 2 is birationai, we call it an isomorphism, or an automorphism when A= B. Suppose A and B have the same dimension. Then a homomorphism 2 of A into B is surjective if and only if Ker (2) is finite. Such a 2 is called an isogeny of A to B. If A, B, and A are rational over a field k, and x is a generic point of A over k, then we put deg (A) = [k(x): k(2(x))]

(=[k(A): k(B) 0 A]) .

The integer deg (A) does not depend on the choice of k and x. If deg (2) is prime to the characteristic of k, then Ker (2) is of order deg (2). If there exists an isogeny of A to B, A and B are said to be isogenous. We denote by End (A) the ring of all endomorphisms of A, and put End Q (A)=End(A)O z Q. 11. Let A be an abelian variety of dimension n with C as the universal domain. Then A, as a complex manifold, is isomorphic to a complex torus Cn/L, with a lattice L in C. Here, by a lattice in Cn, we understand a discrete subgroup of Cn which is a free Z-module of rank 2n. Let QL denote the Q-linear span of L. Then End (A) (resp. End Q (A)) can be identified with the ring of all C-linear transformations in Cn which send L into L (resp. QL into QL). Therefore we obtain two faithful representations of End Q (A): End,2 (A) — End (Cn, C)

(a:- Mn(C)),

R°: End Q (A) — End (QL, Q)

(,-- A n(Q)) .

R:

We call R (resp. R°) the complex (resp. rational) representation

of End() (A). It can easily be seen that R (resp. R°) is equivalent to the representation of End Q (A) on .0(A) (resp. on the first cohomology group of A). From Lemma 3.49, it follows that R° is equivalent to the direct sum of R and its complex conjugate. An arbitrary complex torus Cn IL has a structure of an abelian variety if and only if there exists an R-valued R-bilinear form E(x, y) on Cn satisfying the following three conditions: (11.1) E(x, y)= —E(y, x). (11.2) The value E(x, y) is an integer for every (x, y) e Lx L. -1 y) in (x, y) is symmetric (11.3) The R-bilinear form E(x, Nr-definite. We call such a form E a Riemann form on Cn IL.

and positive

APPENDIX

259

12. A divisor of an algebraic variety V is an element of the free Zmodule formally generated by all the subvarieties of V of codimension one. Let A be an abelian variety defined over a subfield of C, isomorphic to a complex torus Cn/ L. Take a basis {g1, — , g„ } of the vector space C" over R, and define real coordinate functions x l, — , x,„ on Cn by u = El:, xt(u)gi for u e C4. Then, for a Riemann form E on C"/L, there exists a divisor X of A whose cohomology class is represented by the differential 2-form Ei(j E(gt , gi)dx i A dx j . (Here we identify A with C" V L for simplicity.) Since E is unique for X, we say that X determines E (with respect to the fixed isomorphism of A onto C"/L), if X and E are in this situation. Let two divisors X and X' on A determine Riemann forms E and E'. Then the following three conditions are equivalent : ( i) X is algebraically equivalent to X'; (ii) (iii)

X is homologous to X'; E=E'.

13. Let A be an abelian variety defined over a field of any characteristic. A polarization of A is a set C of divisors of A satisfying the following three conditions: (13.1) C contains an ample divisor (in the sense of Weil [90, p. 286]). (13.2) If X and X' belong to C, there exist positive integers m and m' such that mX is algebraically equivalent to

m'X'.

(13.3) C is maximal under the conditions (13.1, 13.2). A polarized abelian variety is a structure (A, C) formed by an abelian variety A and its polarization C. If C is a polarization of A, there always exists a divisor X. in C such that every X in C is algebraically equivalent to m)(0 with a positive integer m. Such an X. is called a basic polar divisor of C. If the universal domain is C, and if A is identified with a complex torus C"/L, the condition (13.1) is equivalent to (13.1') Every X in C determines a Riemann form. Let E be the Riemann form determined by a divisor in C. Then we can define an involution (i. e., an anti-automorphism of order one or two) p of End Q (A) by E(tix, y)= E(x, 2Py) for 2 e EndQ (A). Here we identify End Q (A) with a subalgebra of End (C", C) as in No 11. We call p the involution of End Q (A) determined by C, since it is independent of the choice of X and C"/L. One can actually define such an involution also in the case of positive characteristics. For the detailed discussion of this and other topics concerning abelian varieties, the reader is referred to Weil [92], [95], and Lang [43].

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INDEX abelian variety, 257; — with complex multiplication, 126ff., 211ff. adelization : — of GL 2, 143-144; — of a simple algebra, 241 affine variety, 254 algebraic correspondence, 77, 169 algebraic curve, 257 algebraic number field, xi algebraic variety, 254 arithmetic Fuchsian group, 247 automorphic form, 28-29 automorphic function, 28, 30 automorphism: — of an abelian variety, 258; — of an elliptic curve, 106ff. _ basic polar divisor, 259 birational map, 254 birationally equivalent, 254 Birch-Swinnerton-Dyer conjecture, 221 biregular isomorphism, 254 canonical class, 36 class field, 116 CM-field, 124 CM-type, 125 cohomology group, 223 commensurable, 5 commensurator, 51 complex multiplication : — of an abelian variety, 1261f. ; — of an elliptic curve, 102 conductor of an order, 106 congruence subgroup, 20; principal—, 20 conjecture : — of Birch and SwinnertonDyer, 221; — of Hasse and Weil, 168; — of Ramanujan, 89 covering of a Riemann surface, 19 cusp, 8, 18 ; —s of the modular group, 14 degree : — of a covering, 19; — of a divisor, 35; — of a double coset, 51 ; — of a rational map, 112, 258 differential form, 36, 257 ; — of the first kind, 36, 257

Dirichlet series : formal —, 60-61 (see also zeta-function) discrete subgroup, 3 divisor : — of a Riemann surface, 35; — of an algebraic curve, 169 ; — of an algebraic variety, 259 eigen-function of Hecke operators, 77ff. eigen-values of Hecke operators, 77ff. Eisenstein series, 32-33, 78 elliptic curve, 96 elliptic elements of the modular group, 14-15 elliptic function, 98 elliptic matrix, 5 elliptic point, 8, 18; —s of the modular group, 14-15 elliptic transformation, 5 embedding : normalized —, 103404, 246 endomorphism : — of an abelian variety, 258; — of an elliptic curve, 102ff. equivalent : birationally—, 254; linearly —, 35; — under a transformation group, 1 Euler characteristic, 18 field of definition, 254 field of moduli :— of an abelian variety, 130-131; — of an elliptic curve, 98 field of rationality, 254 Fourier coefficients, 29 Fourier expansion, 29; — of J, 33; — of d, 33, 50 Frobenius correspondence, 177 Frobenius morphism, 256 Fuchsian group of the first kind, 19 function of an algebraic variety, 254-255 functional equation of a zeta-function, 93 fundamental domain, 15, 42 Gauss s um, 91 generic point :— of a variety, 254 ;— for meromorphic functions, 137 genus: —of a compact Riemann surface,

266

INDEX

r.

18; — of rye, a congruence subgroup, 23 good reduction modulo a prime 114, 213 Hasse-Weil conjecture, 168 Hecke operator, 76, 79 Hecke ring, 54; — of SL 7I (2), 55ff. ; — of a congruence subgroup, 65ff. homogeneous element of a Hecke ring, 60 homomorphism : — of an abelian variety, 257; — of an elliptic curve, 96 Hurwitz formula, 19 hyperbolic matrix or transformation, 5 inseparable morphism, 112 integral form, 30 invariant of an elliptic curve, 97, 99 involution : main —, 72, 243; — of the endomorphism ring of an abelian variety, 259 irregular cusp, 29 isogenous, isogeny, 96, 258 isotropy subgroup, 1 1-adic representation, 100, 189ff. lattice : — in a complex vector space, 98, 126, 258; — in a number field, 104; — in a rational vector space, 56 level, 20, 30 L-f unction, 213 linear fractional transformation, 5 local parameter, 17 locus of a point, 254 loxodromic matrix or transformation, 5 main involution, 72 243 maximal order of a number field, xii, 104 maximal ray class field, 116 measure: invariant — of the upper half plane, 41 ; — of a fundamental domain, 41, 42, 44 Mellin transformation, 94 model of 152 modular correspondence, 77, 172ff., 176 modular equation, 110 modular form, 30 modular function, 30; — of level N ra-

rwi,

tional over a cyclotomic field, 137 modular group, 14 morphism, 254; inseparable or separable —, 112 non-singular, 255 normalized embedding, 104, 246 normalized isomorphism into End Q (E), 113 1-cycle, 169 orbit, 1 order in a number field, 104 order of an elliptic point, 9 origin of an elliptic curve, 96 parabolic matrix or transformation, 5 Petersson inner product, 75 polarization, 259 polarized abelian variety, 259 primitive matrix, 108 principal congruence subgroup, 20 projective variety, 254 proper algebraic correspondence, 169 proper ideal, 104 purely inseparable morphism, 112 quaternion algebra, 243 quotient topology, 1 Ramanujan conjecture, 89 ramification index, 19 ramified : prime — in a quaternion algebra, 245 rational map, 254 reduction modulo a prime, 114 reflex of a CM-type, 126 regular cusp, 29 regular extension, 253 Riemann form, 258 Riemann surface, 17 Riemann-Roch theorem, 36 separable morphism, 112 specialization, 253 stability group, 1 subvariety, 254 theta-series, 95

267

INDEX transformation group, 1 triangle group, 251

Weierstrass function, 98 weight of an automorphic form, 28

universal domain, 253 unramified : character — at a prime, 213; prime — in a quaternion algebra, 245

zeta-function, 89ff. ; — of a curve, 167; — of an abelian variety, 167-168 0-cycle, 169, 257 2-lattice, 56, 104

ERRATA P. 6, line 12:

r az + b lcz

+dj

P. 19, line 5 from the bottom:

should read

1-•\‘),

z+q [prz + s I • should read

IN .Ç)*.

P. 22, line 11 from the bottom: Prop. 1.38 should read Prop. 1.37. P. 24, line 7 from the bottom: Lemma 1.39 should read Lemma 1.38. P. 25, Proof: The first paragraph should read as follows: PROOF. TO prove (4), put a [ N 0 and Too = fy E Fl y(00 )

01

with F = FM. Then Fo(N) = a - lFanT. By Prop. 1.37 we have a disjoint decomposition F = lieeR Fo(N) with a set R consisting of v. elements. Then Far = UeeR Fa6F„, which is also a disjoint union. (Indeed, if ya6F„ = F„ with y E F, then a - Iya6F„ = er„, and hence a -1 ya E F n a -1 1'a = Fo(N), so that = e'.) Thus v. is the number of cosets PriF„ in Far. Now Far consists of all the elements p in M2 (Z) with det(f3) > 0 whose elementary divisors are 1,

N. Therefore we see easily that FaF =

ur [ a 0

b d

with integers

a, b, d such that a > 0, ad = N, 0 b < d, and (a, b, d) = 1. (Cf. Prop.

3.36 below.) Thus our problem is to check when F[ oa db F„ = F [

la' blrl nil for some db , /10 holds. Suppose y ra bl LO di = LO d' ] l0 ii

1n y E F and m E Z. Then y must be of the form y = [ 0 , so that a = a', d = d', and b + nd = b' + ma. This holds if and only if b = h' (mod (a, d)). Since (a, b, d) = 1, there are exactly p((a, d)) choices for b with different F [ oa db F, and hence we obtain (4). P. 33, line 2 from the bottom: Insert bn after "coefficients."

270

ERRATA

P. 41, line 6 from the bottom: FP (dz ye

P. 46, last line: P. 53, line 20:

FA6e

27_, should read

should read

should read

Fg(dz)k.

TA6: 71.

P. 54, Proposition 3.6: One has to assume TA n {-±1} = F

{-± 1}

here.

P. 65, lines 21 and 22: "surjective ring-isomorphism" should read "injective ring-homomorphism." Pp. 69-70: The last seven lines of page 69 and the first six lines of page 70 should read as follows: To prove that the multiplicity of l'al"" in (F' F') • (P riP) is 1, we first take

1 0 4' so that F'F' = F'e ' and ‘ --=- LO r I (mod tNk), qJ

where k is

a positive integer such that mINk. This is possible by Prop. 3.31 and (2) of Lemma 3.29. Let P4'F' = U•P‘ei be a disjoint decomposition with

1 0 si E F'. We have P nr9 = U7:01 Pni with ni = [ 0 m

]

(see Prop.

3.33 below). Suppose P 4.77 = P e1 ii,, that is, yCn = ‘ei n; for some y E

* tb

F'. Put y = [* *

u tv ] and ei = [ * * ] . Taking the upper right

entry of y‘-ri modulo tNk, we obtain tbmq utj + mtv (mod tNk), and hence m kV. Since (u, m) = 1, we have/ = 0, 77 = ni , so that PC = F'4.61. This proves that the multiplicity is 1, that is, Pa l." = (F' F') - (P TIP). It follows that

deg(PaP) = deg(/' /-'') • deg(P nP). Therefore, the multiplicity of P aP in (PriP) • (PP) is 1, and hence P ar = (P •riP) • (P CP), which completes the proof of (3).

P. 72, line 1: Lemma 1.39 should read Lemma 1.38. P. 76, line 5: = f should read

= (cild)kf.

P. 79, line 10 from the bottom: After "proposition," insert "and by (1) of Prop. 3.32." P. 87, lines 2 and 13: = 1 shculd read

= (-1)k.

271

ERRATA

P. 90, line 9:

0(q)

should read 1 0(01-

P. 108, line 8 from the bottom: (m (m - 1)/2) • ans2 should read (m(m - 1)/2) • am ci + ma,nc2. P. 130, line 16 from the bottom: "units of K" should read "roots of unity ‘ in K such that ‘a = a ". Pp. 170-171: To deal with the situation in which some, but not all, groups contain -1, it is simpler to consider the groups and the elements in G14 (R)/R<. Pp. 183-185, Theorems 7.14 and 7.16: A better formulation and clarification of some points in the proof can be found in On the factors of the Jacobi an variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523-544. P. 190, line 5: "multiple of d" should read "multiple of h." P. 195, line 14: + pu 2 should read - pu 2. P. 196, Remark 7.27: (B): The statement ha = fh on line 11 from the bottom is erroneous. In fact, h Œ = fir= if (e 2 zilr) cr = e 2 irimir, and so the group generated by t is Q-rational only as a whole. However, some Qrational points of finite order on A s can be obtained as follows. Take N to be a prime for simplicity and put f (2) = [A(Nz)/3(z)rim with the greatest common divisor m of N - 1 and 12. Then f E Q(Vs) and div(f) = q(P. Po), where Pc is the point on Vs corresponding to a cusp c, and q = (n 1)1m. Thus P,. - Po corresponds to a Q-rational point t onAs annihilated by q. It is not difficult to show that t has order exactly q. For example, for N = 11, 17, 19, 23, 29, and 31, one obtains a Q-rational point on A s of order 5, 4, 3, 11, 7, and 5, respectively. P. 227, line 9:

CI, should read C.

P. 227, line 18: g([1, a -10]) should read ag([1, a -1 0J). P. 233, line 11 from the bottom: j(p -1, z)

should read i p-1,

P. 234, line 11 from the bottom: f should read g. P. 236, line 3: Both es should read ej . P. 252, line 7:

9.11 should read 9.13.

P. 255, line 9: A P. 259, line 4:

gn

read

21„.

should read g2n .

(

10).

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