Abelian Varieties with Complex Multiplication and Modular Functions: (PMS-46) 9781400883943

Table of contents :
Contents
Preface
Preface to Complex Multiplication of Abelian Varieties and Its Applications to Number Theory (1961)
Notation and Terminology
CHAPTER I. Preliminaries on Abelian Varieties
1. Homomorphisms and divisors
2. Differential forms
3. Analytic theory of abelian varieties
4. Fields of moduli and Kummer varieties
CHAPTER II. Abelian Varieties with Complex Multiplication
5. Structure of endomorphism algebras
6. Construction of abelian varieties with complex multiplication
7. Transformations and multiplications
8. The reflex of a CM-type
CHAPTER III. Reduction of Constant Fields
9. Reduction of varieties and cycles
10. Reduction of rational mappings and differential forms
11. Reduction of abelian varieties
12. The theory “for almost all p”
13. The prime ideal decomposition of an N(p)-th power homomorphism
CHAPTER IV. Construction of Class Fields
14. Polarized abelian varieties of type (K; {φi} )
15. The unramified class field obtained from the field of moduli
16. The class fields generated by ideal-section points
17. The field of moduli in a generalized setting
18. The main theorem of complex multiplication in the adelic language
CHAPTER V. The Zeta Function of an Abelian Variety with Complex Multiplication
19. The zeta function relative to a field over which some endomorphisms are defined
20. The zeta function over smaller fields
21. Models over the field of moduli and models with given Hecke characters
22. The case of elliptic curves
CHAPTER VI. Families of Abelian Varieties and Modular Functions
23. Symplectic and unitary groups
24. Families of polarized abelian varieties
25. Modular forms and functions
26. Canonical models
CHAPTER VII. Theta Functions and Periods on Abelian Varieties
27. Theta functions
28. Proof of Theorem 27.7 and Proposition 27.9
29. Theta functions with complex multiplication
30. The periods of differential forms on abelian varieties
31. Periods in the Hilbert modular case
32. Periods on abelian varieties with complex multiplication and their algebraic relations
33. Proof of Theorem 32.4
Bibliography
Supplementary References
Index
About the Author

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Abelian Varieties with Complex M ultiplication and M odular Functions

Princeton Mathematical Series E d it o r s : L u is A . C a f f a r e l l i , J o h n N . M a t h e r , 1.

and E l ia s M . S t e in

T he C la ssica l G roups b y H erm a n n W eyl

3. A n Introduction to D ifferen tia l G eo m etry b y L u th e r P fa h le r E ise n h a rt 4. D im e n sio n T h eory b y W. H u re w icz a n d H. W allm an 8. T h eory o f L ie G roups: I b y C. C h e v a lle y 9. M ath em atical M eth o d s o f S ta tistics b y H a r o ld C ra m e r 10. Several C o m p lex V ariables b y S. B o c h n e r a n d W. T. M a rtin 11. Introduction to T o p o lo g y b y S. L e fsc h e tz 12. A lg eb ra ic G eo m etry and T o p o lo g y e d ite d b y R. H. Fox, D . C. Spencer, a n d A. W. Tucker 14. T he T o p o lo g y o f Fibre B u n d les b y N o rm a n S te e n r o d 15. F ou n d ation s o f A lg eb ra ic T o p o lo g y b y S a m u el E ile n b e rg a n d N o rm a n S te e n r o d 16. F u n ctio n a ls o f Finite R iem an n S u rfaces b y M en a h e m S c h ijfe r a n d D o n a ld C. S p e n c e r 17. Introduction to M a th em a tica l L o g ic , V ol. I b y A lo n z o C hurch 19. H o m o lo g ic a l A lg eb ra b y H. C a rta n a n d S. E ile n b e rg 20 . T he C o n v o lu tio n T ransform b y I. /. H irsch m a n a n d D . V. W id d er 21 . G eo m etric Integration T h eory b y H. W hitney 22 . Q u alitative T h eory o f D ifferen tia l E quations b y V. V. N e m y tsk ii a n d V. V. S te p a n o v 23. T o p o lo g ic a l A n a ly sis b y G o rd o n T. W hyburn (rev ised 1964) 24. A n a ly tic F u n ctio n s b y A h lfo rs, B ehnke, B ers, G ra u e r t e t al. 25. C o n tin u o u s G eo m etry b y John von N eu m a n n 2 6 . R iem an n S u rfa ces b y L. A h lfo rs a n d L. S a rio 2 7 . D ifferen tia l and C om binatorial T o p o lo g y e d ite d b y S. S. C a irn s 28. C o n v ex A n a ly sis b y R. T. R o c k a fe lla r 2 9 . G lob al A n a ly sis e d ite d b y D . C. S p e n c e r a n d S. Iya n a g a 30. S in g u la r Integrals and D ifferen tia b ility P roperties o f F u n ction s b y E. M. S tein 3 1 . P ro b lem s in A n a ly sis e d ite d b y R. C. G u n n in g 3 2 . Introduction to Fourier A n a ly sis on E u clid ea n S p a ce s b y E. M . S tein a n d G. W eiss 33. E tale C o h o m o lo g y b y J. S. M iln e 34. P seu d o d ifferen tia l O perators b y M ic h a e l E. T a ylo r 36. R ep resen tation T h eory o f S e m isim p le G roups: A n O v e r v iew B a sed on E x a m p les b y A n th o n y W. K n a p p 3 7. F ou n d ation s o f A lg eb ra ic A n a ly sis b y M a sa k i K a sh iw a ra , T akahiro K a w a i, a n d Tatsuo K im u ra . T ra n sla te d b y G o ro K a to 3 8. Sp in G eo m etry b y H. B la in e L a w so n , Jr., a n d M a rie -L o u is e M ich elso h n 3 9. T o p o lo g y o f 4 -M a n ifo ld s b y M ic h a e l H. F reed m a n a n d F rank Q uinn 4 0 . H y p o -A n a ly tic Structures: L o ca l T h eory b y F ra n g ois T reves 4 1 . T he G lo b a l N o n lin ea r Sta b ility o f the M in k o w sk i S p a ce b y D e m e tr io s C h risto d o u lo u a n d S erg iu K la in e rm a n 4 2 . E ssa y s on Fourier A n a ly sis in H onor o f E lia s M . Stein e d ite d b y C. F efferm an, R. F efferm an, a n d S. W ain ger 4 3 . H arm onic A n a ly sis: R eal-V ariable M eth o d s, O rthogon ality, and O scilla to ry Integrals b y E lia s M. Stein 4 4 . T o p ics in E rgod ic T h eory b y Ya. G. S in a i 4 5 . C o h o m o lo g ic a l Induction and U nitary R ep resen tation s b y A n th o n y W. K n a p p a n d D a v id A. Vogan, Jr. 4 6 . A b elia n V arieties w ith C o m p lex M u ltip lica tio n and M odular F u n ction s b y G o ro S h im u ra

ABELIAN VARIETIES W ITH COMPLEX MULTIPLICATION AND MODULAR LUNCTIONS

Goro Shimura

P R IN C E T O N U N IV E R SIT Y PRESS P R I N C E T O N , N E W JE R S E Y

Copyright © 1 9 9 8 by Princeton University Press Published by Princeton University Press, 41 W illiam Street, Princeton, N ew Jersey 0 8 5 4 0 In the United Kingdom: Princeton U niversity Press, Chichester, W est Sussex A ll Rights Reserved

Library of Congress Cataloging-in-Publication Data Shimura, Goro, 1930— A belian varieties with com plex m ultiplication and modular functions / Goro Shimura. p.

cm. — (Princeton mathematical series ; 46)

Includes bibliographical references and index. ISB N 0 -6 9 1 -0 1 6 5 6 -9 (alk. paper) 1. A belian varieties.

2. M odular functions. II.

Q A 5 6 4 .S 4 5 8 5 1 4 .3 — dc21

I. Title.

Series. 1997

97 -8 6 7 3

CIP

Parts o f the present work appeared in an original version in C om plex M u ltiplication o f A belian Varieties a n d Its A p p lica tio n s to N um ber Theory, © 1 9 6 1 by The M athematical S ociety o f Japan This book has been com posed in Tim es Roman Princeton U niversity Press books are printed on acid-free paper and m eet the guidelines for permanence and durability o f the C om m ittee on Production G uidelines for B ook Longevity o f the Council on Library R esources http://pup.princeton.edu Printed in the United States o f Am erica 1 3 5 7 9

10 8 6 4 2

Contents Preface

vii

Preface to Complex Multiplication of Abelian Varieties and Its Applications to Number Theory (1961)

ix

Notation and Terminology I. Preliminaries on Abelian Varieties

C hapter

C hapter

C h apter

3

1. Homomorphisms and divisors

3

2.

Differential forms

7

3.

Analytic theory o f abelian varieties

19

4.

Fields o f moduli and Kummer varieties

25

II. Abelian Varieties with Complex Multiplication

35

5.

Structure o f endomorphism algebras

6.

Construction o f abelian varieties with com plex multiplication

40

7.

Transformations and multiplications

47

8.

The reflex o f a CM -type

58

III. Reduction of Constant Fields 9.

35

68

Reduction o f varieties and cycles

68

10.

Reduction o f rational mappings and differential forms

74

11.

Reduction o f abelian varieties

83

12.

The theory “for almost all p”

87

13.

The prime ideal decom position o f an V (p )-th power homomorphism

C h apter

xiii

IV. Construction of Class Fields

96

101

14.

Polarized abelian varieties o f type ( K\ { 1, both the cases K = K* and K ^ K* may occur. The abelian extensions of AT* thus obtained from A do not provide all the abelian extensions of K * unless n — 1; at any rate, the classical results of Kronecker and Hecke are included in our main theorems as particular cases. It is noteworthy that the prime ideal decomposition of the iV(p)-th power endomorphism itp of A(p) is fundamental in our whole theory, where A(p) denotes the reduction of the variety A modulo a prime ideal p of a field of definition k for A. The above result is in close connection with the investigation of the zeta function of the abelian variety A . In fact, a more precise analysis of n p shows that the correspondence p -> n p determines a Grossencharacter of the field k. We are then led to the expression of the zeta function of A by the product of several Hecke L -series attached to Grossen-characters (Main Theorem 4 of Chapter IV); this is a generalization of the results of Weil and Deuring mentioned above. We now give a summary of the contents. Chapter I is an exposition of more or less known results on abelian varieties, which are mostly given without proofs. The only exception is Section 2, where we have given a detailed (but elementary) treatment of invariant differential forms on abelian varieties. Section 3 deals with the analytic representation of abelian varieties, their homomorphisms and divisors by means of complex tori. In Section 4, first the notion of polarized varieties is introduced and then the definitions of field of moduli and Kummer variety are given. Chapter II is devoted to the algebraic part of the theory of complex multiplication. Sections 5 and 6 contain a necessary and sufficient condition that an algebraic number field K of degree In be realized as the endomorphism-algebra of an abelian variety of dimension n. Section 7 is the study of mutually isogenous abelian varieties in connection with the ideals of the endomorphism-rings; Section 8 concerns the phenomena which are essential

only in the case of dimension n > 1 and related to the definition of the number field K*. Chapter III contains the theory of reduction of algebraic varieties modulo a prime divisor of the basic field. We shall prove in Section 13 the fundamental theorem concerning the prime ideal-decomposition of A (p)-th power homomorphism. Our final aims are achieved in Chapter IV. The first step (Section 14) is the investigation of the relations between abelian varieties, of the same type of complex multiplication, whose polarizations are also of the “same type.” Then, in Section 15, we prove the first main theorem; an unramified class field is obtained by the field of moduli. A similar argument together with the analysis of the points of finite order gives us also class fields, whose characterization is the object of Section 16. These results are obtained assuming the endomorphism-ring to be the principal order of the number field. In Section 17, the case of nonprincipal order is completely investigated. The last section is devoted to the determination of the zeta-function of an abelian variety of the type described above. A large part of the contents was prepared in collaboration of both authors during 1955-56 and published in 1957 in Japanese as the first six chapters of the book with the title “Kindai-teki Seisu-ron” (Modem number theory). The English version was then planned; but owing to the sudden death of the second named author in the autumn of 1958, the work had to be completed by the person left behind. The present volume is not a mere translation, however; we have written afresh from beginning to end, revising at many points, and adding new results such as Section 17 and several proofs of propositions which were previously omitted. The present monograph owes much to the idea of Weil [54], though we have not necessarily indicated explicit references in the text. I take this op­ portunity to acknowledge my cordial gratitude to Professor Andre Weil for his constant advice, suggestions, and encouragement. I wish to acknowledge also my thanks to Mr. Taira Honda who read the manuscipt and contributed many useful suggestions. University of Tokyo February 1960

Goro Shimura

Notation and Terminology We denote by Z, Q, R, and C the ring of rational integers, the fields of rational numbers, real numbers, and complex numbers, respectively, and by Q the algebraic closure of Q in C. If i is a rational prime, Z t and denote the ring of f-adic integers and the field of f-adic numbers, respectively. Given an associative ring R with identity element and an /^-module S, we denote by R x the group of all invertible elements of R, and by S™ the module of all m x ^-matrices with entries in S. We shall often put Sm = S™. Thus Cn (resp. R'1) is the vector space of ^-dimensional complex (resp. real) column vectors. If V is a vector space over C of dimension n, a lattice of V means a discrete subgroup of V isomorhic to Z 2,\ We denote the identity element of the ring R” by 1„, or simply by 1, and the transpose of a matrix X by fX. We put G L n(R) = (R ")x , and S Ln(R) = { a e G L n(R) | det(a) = 1 } if R is commutative. For a complex number or more generally for a complex matrix a we denote by Re ( a ) , Im(aO, and a the real part, the imaginary part, and the complex conjugate of a. If o t \ , . . . , a r are square matrices, diag[oq, . . . , a r] denotes the matrix with a \ , . . . , a r in the diagonal blocks and 0 in all other blocks. Terminology and basic notation concerning algebraic geometry are essen­ tially the same as in W eil’s trilogy [44], [45], and [46]. In particular, a variety or a subvariety always means an absolutely irreducible one. If V is a variety, we view V as the set of all its points rational over the universal domain. For a finite algebraic extension k' of a field k we denote by [kf : k]j and [k' : &].v the inseparable and separable factors of the degree of k' over k. If a is an isomorphism of a field k\ onto a field &2, then for z £ k\ we denote by z° the image of z under o with the rule zaT = (zay for another isomorphism r of &2 onto a field. Furthermore, if Y is an algebro-geometric object defined with respect to k \, we denote by Y° the image of Y under a. If Y is defined by some polynomial equations, then Y° is defined by the transforms of the equations under a . If k is a field and x is a point of an affine (resp. a projective) space, we denote by k(x) the field generated over k by the coordinates (resp. the quotients of the coordinates) of the point x. We use also the notation k(x) for the points on an abstract variety (cf. [44]).

Method of citation. There are two lists of references: Bibliography and Supplementary References. The former is the same as in the 1961 book and its articles are numbered from 1 through 57. Each item in the latter list contains a roman capital with or without more letters and numerals. Therefore the reader should be able to determine in which list the article in question appears.

Abelian Varieties with Complex M ultiplication and M odular Functions

CHAPTER I

Preliminaries on Abelian Varieties 1 . H om om orphism s an d Divisors The purpose of this section is to recall briefly some of the basic concepts on abelian varieties defined over arbitrary ground fields. For the general theory of abelian varieties, we refer the reader to Weil [46] and Lang [26]. 1.1. By a group variety we understand an algebraic variety (affine, projec­ tive, or abstract) G with a group structure such that the map (x, y) b-> xy of G x G into G and also the map x i-> x -1 of G into G are both rational mappings defined everywhere. Such a G must be nonsingular. We say that a group variety G is defined over a field k if the variety G and these maps are defined over k. A group variety is called an abelian variety if it is complete in the sense of [44], which is the case if it is a projective variety. In fact, every (abstract) abelian variety defined over k is biregularly isomorphic to a projective variety over A, and therefore we practically lose nothing by assuming it to be projective. It is a wellknown fact that the group law of an abelian variety is commutative. Therefore we use the additive notation and denote the zero element by 0. Whenever we speak of an abelian variety A defined over a field k , we always assume that A as a group variety is defined over k in the above sense. If a subvariety of an abelian variety A is a subgroup, then it has a natural structure of abelian variety, and is called an abelian subvariety of A. An abelian variety is called simple if it has no abelian subvarieties other than {0} and itself. (Some authors call it absolutely simple.) Let A and B be two abelian varieties. By a homomorphism of A into B, or an endomorphism when A = B, we shall always understand a rational mapping X of A into B satisfying A(x + y) — A(x) + A(y) generically. If that is so, then A is defined everywhere and X(x + y) = X(x) + X(y) for every x, y e A. We denote by Ker(A) the kernel of X. We shall often write Ax for A(x). If such a A is birational, then it must be biregular, and we call it an isomorphism, or an automorphism when A = B. We denote by Hom(A, B) the set of all homomorphisms of A into B , defined over any extension of a given field of definition for A and B, and put End(A) = Hom(A, A). It is a basic fact

that Hom(A, B) is a free Z-module of finite rank; moreover if A and B are defined over k, then every element of Hom(A, B) is defined over a separably algebraic extension of k. We put also Hoitiq(A, B) = Hom(A, B) (g>z Q and EndQ(A) = End(A) 0 z Q. Clearly EndQ(A) has a structure of an algebra over Q, and End(A) is an order of the algebra. We denote the identity element of EndQ(A) by 1^. More generally, if X e HomQ(A, B) and ji e Hom Q(£, C), then we can define the product /xA naturally as an element of Hoitiq(A, C). For two abelian varieties A and B of the same dimension, there exists a ho­ momorphism of A onto B if and only if there exists a homomorphism of B onto A, in which case A and B are called isogenous, and any such homomorphism is called an isogeny. Given A e Hom(A, B) with A and B of the same dimension, take a field of definition k for A, B , and A and take also a generic point jc of A over k. If A is an isogeny, we put v(A) = [k(x) : k(Xx)], v s ( k ) = [ k ( x ) : k { Ajc)].y,

v#*(A) = [ k ( x ) : k ( X x ) ] j ,

and otherwise we put v(A) = vv(A) = v, (A) = 0. These numbers do not depend on the choice of k and x. If A is an isogeny, then v.v(A) is the order of Ker(A). For every isogeny A of A onto B there is a unique element A' of HomQ(Z?, A) such that A'A = 1^ and AA' = 1#. We write then X' = A-1 . 1.2. The €-adic representation of homomorphisms. For an abelian variety A and a rational prime i we put oo

Se(A) =

(J K e r e n s . 1

If A is of dimension n and i is different from the characteristic of a field of definition for A, then g^(A) is isomorphic to the direct sum 971 of 2n copies of the additive group Q^/Z^. We call any one of the isomorphisms of ^ ( A ) onto VJl an i-adic coordinate-system o f We consider every element of 971 a column vector of dimension 2n with components in Q^/Z^. Let B be another abelian variety of dimension m and A a homomorphism of A into B. Choose t - adic coordinate systems D of 0^(A) and ru of %t(B). Then there exists an element M of (Z ^ )^ such that ro(At) = M v(t) for every t e If we fix D and m, the correspondence A h* M can be uniquely extended to a Q-linear map of Hoitiq(A, B) into (Q ^ )^ , which we call the f-adic representation of HomQ(A, B) with respect to d and tv. In particular, if A = B and v = ro, this is a ring-homomorphism of EndQ(A) into (Q ^)^Now let Mt denote an £-adic representation of EndQ(A) with respect to a fixed v as above. Given £ e EndQ(A), let P (x) = X 2" + a \X 2"~] + • • • + Cl2 „

be the characteristic polynomial of the cij are rational numbers, and

Then the following facts are known:

P(^) = C n + a ^ 2n~] + - - - +

02 ,,

= 0;

moreover, the polynomial P is determined by £ independently of the choice of i and t-adic coordinate-system; furthermore, if% e End(A), then a; e Z and

(1)

v(£) = det(M ,(*)).

We call P the characteristic polynomial of £ and the roots of P the characteristic roots o f §; we also put

(2)

tr($) = tr(M*(£)).

1.3. The Picard variety of an abelian variety. Given an abelian variety A, let Qa(A) and Qi{A) denote, respectively, the set of divisors on A algebraically equivalent to 0 and the set of divisors on A linearly equivalent to 0. Then there exists an abelian variety A* canonically isomorphic to Qa{A )/Q i(A ), which is called the Picard variety of A. Every divisor Y contained in Qa{A) defines a point of A*, which we denote by C1(F). Let B be an abelian variety and B * the Picard variety of B. For every homomorphism A of A into B , we obtain a homomorphism A* of B * into A* such that (3)

A.*(Ci(y)) = c i( A .-'(r))

whenever A-1 (F) is defined. The mapping A -> A* is uniquely extended to an isomorphism of HomQ(A, B) onto HomQ(Z?*, A*); we denote by the image of a by this isomorphism and call it the transpose of a. If a e Hoitiq(A, B) and p e HomQ(Z?, C), we have t (f3a) = ta tft. Let X be a divisor on A; we shall denote by X u the transform of X by the translation x —> x + u on A. Now define the mapping (px of A into A* by the relation (4)

D f gives a linear mapping of V ( V ) into £2(V), and hence defines an element of 2)(V ), a differential form of degree one on V; we denote it by d f \ then we have d f • D = D f . We see that D (V ) is generated over Q( V) by the forms d f for / e £2(V). If V is of dimension n, then there exists a set of n functions {gi, . . . , gn) in k(V) such that k(V) is separably algebraic over k { g\ , . . . , gn). If {g \ , . . . , gn} is such a set, dg 1, . . . , dgn form a basis of D (V ) over £2(V). By our definition, every differential form 00 on V has an expression

where the / (?) are elements of Q ( V). We shall say that a differential form co on V is defined over k if co can be written in the form (/) with the (p{i) and the \j/j in k(V). The set {gi, . . . , gw} being as above, a differ­ ential form

is defined over k if and only if the / (/) are contained in k(V). Let V' be a simple subvariety of V. We shall say that a differential form co on V isfinite along (or at ) V' if co can be written in the form co = ^T(/) f a) dgi] • • -dgjr where the / ° B x B - A B, a ( x ) = x x x,

/3(x x y) = A(A) x /x(y),

Xo(x x y) = X(x) x 0,

y ( z x w) = z ~h w,

pio(x x y) = 0 x pu(y).

We have then A+/x = y/3a, yXo& = A, ypt^a = /x, sothat 0, k(x) is purely inseparable and separable over k( xq, u) \ so we have k(x) = k( xq, u). It follows that we have [k(x) : k( xq)] < q s, since the u\ are contained in k(xq)\ in particular, we have [k(x) : k( xp)] < p s. Suppose that we have [k(x) : = p r < p s\ then there exist r quantities Vj in k(x) such that k{x) — k( xp, v ) . By the first assertion of our lemma, the number of linearly independent derivations of k(x) over k( xp) is not greater than r. This is a contradiction since every derivation of k(x) over k gives a derivation of k(x) over k( xp). Hence we must have [k(x) : k( xp)] = p s. This completes the proof. THEOREM 1. Let A and B be two abelian varieties and X a homomorphism o f A into B; let k be afield o f definition fo r A, B and X, and x a generic point o f A over k. If the linear mapping 8 X ofT>o(B) into Do (A) is o f rank r, then the following assertions hold:

(i) k(x) is separably generated over k(Xx) if and only if dim^(Ax) = r. (ii) Assuming that A and B have the same dimension n, we have V/ (X) = 1 if and only if n — r. (iii) Forn as in (ii), ifk is o f characteristic p ^ 0 and ifk(Xx) D k( xq) fo r a pow er q = p e (e > 0) o f p, then we have v(A) = V/(A.) < q n~r. Proof. Let n and m be respectively the dimensions of A and B . Let F denote the subfield { f °X \ f e k{B)} of k(A). Let D be a derivation of k(A) over k. We shall prove that (8 Xco) • D = 0 for all co e D o(B) if and only if D F = 0. Take a basis {co\ , . . . , com] of D o (£ ; k) over k\ then, for every / e k (B ), there exist, by Proposition 3, m functions gi in k(B) such that d f = ^ giCOi. If (8 Xco) • D — 0 for all co e D o(B), we have

= d( f ° X) • D =

• D = 0; i

this shows D F = 0. Conversely, suppose that D F = 0. Every co in D o(B) can be expressed in the form co = J T f d h i with f e £L(B), hi e k(B). We have hence (5Xa>) • D = Y ^ ( f i ok)d(hi°X) ■D = i

= 0. i

Thus we have proved that (8 Xco) • D = 0 for all co e D o(5 ) if and only if D F = 0. Now, by our assumption, / ( x ) gives an isomorphism of k(A) onto A(x), and F corresponds to A(Ax) by this isomorphism; so there exist exactly n — r linearly independent derivations of A(x) over A(Ax). By Lemma 2, we can find n — r elements u \ , . . . , un- r in A(x) such that A(x) is separably algebraic over A(Ax, u \ , . . . , un- r). If dim^(Ax) = r, the u\ must be independent variables over A(Ax), so that A:(jc) is separably generated over A(Ax). Conversely, if k(x) is separably generated over A(Ax), then by Proposition 16 of [44, Chapter I], the dimension of A(x) over A(Ax) is n —r, so that we have dim/:(Ax) = r. This proves assertion (i). If m = n and v,(A) = 1, A(x) is separably algebraic over k (Ax); consequently, by what we have just proved, wehaverc = dim ^A x) = r. Conversely, if rank 0. Then, by Lemma 2, we have [A(x) : k(Ax)] = [(/c(x) : k(Ax, x q)] < q n~r \ this proves (iii) of our theorem. C OR O LLA R Y . Let B be an abelian variety and A an abelian subvariety of B. If a denotes the injection o f A into B, we have

M 2 ) 0( £ ) ) = 3)o(A).

This is an easy consequence of (i) of Theorem 1. PR O PO SIT IO N 6. Let A, B, A, k, x be the same as in Theorem 1. Suppose that the characteristic p o f k is not 0. Then:

(i) We have 8 X = 0 if and only ifk(Xx) C k( xp). (ii) Assume that A and B are o f the same dimension; ifVj(X) = 1, we have k(x) = k( xq, Ax) fo r every pow er q = p e with e > 0; conversely, if k{x) = k( xq, Ax) fo r some q = p e with e > 0, we have v/(A) = 1. PROOF. The proof of Theorem 1 implies that 0. Then there is no derivation of A(x) over A(Ax) other than 0, so that A(x) is separably algebraic over A(Ax), namely Vi (X) = 1.

PROPOSITION 7. Let A be an abelian variety of dimension n, defined over a field o f characteristic p 0. Then, V/(/?l/0 is a multiple o f p ” and the order o f Ker(p 1a ) is a divisor o f p n.

p r

^

a

P ro o f. By Proposition 5, we have 8 (p \ A) = 8 (\a + ••• + 1a) = p 8 \a =

0. Hence, by (i) of Proposition 6, we have k( px) C k( x p), where k is a field of definition for A and jc a generic point of A over k. It follows that Vi (pl A) = [k(x) : k(px]j > [k(x) : k( xp)] = p n. Since the order of Ker ( p \ A) is equal to vs { p \ A) and vs ( p \ A)Vi(p\a) = v (/?U ) = /?2", we obtain our proposition. Let A be an abelian variety of dimension n and k a field of definition for A. Denote by End(A; k) the set of all elements in End(A) defined over k and by EndQ(A; k) the subset End(A; k) 0 Q of EndQ(A). For every k e End(A; k), 8 k gives a linear transformation of Do (A; k). We have seen above that the relations 8 (k + pt) — 8 k + 8 pc, 8 ( kp) = 8 pc8 k hold, so that the mapping k ^ 8 k gives an anti-representation of End(A; k). As Do(A; k) is a vector space of dimension n over k, we obtain, with respect to a basis of Do (A ; k) over k, an anti-representation of End(A; k) by matrices of degree n with coefficients in k. If k is of characteristic p ^ 0, we have 8 ( p \ a ) = 0; so our representation is not faithful. If k is of characteristic 0, we get a faithful representation. In fact, if 8 k = 0, the rank of 8 k is 0, so that by (i) of Theorem 1, we have dim^(Ax) = 0; this implies k — 0. In case of characteristic 0, we can extend uniquely the representation to a representation of Endq(A; k). We shall call this anti­ representation a representation of End q ( A; k) by invariant differential form s. 2.9. Differential forms on a curve and its Jacobian variety. In the sequel, we denote by D o(E ) the set of all differential forms on an algebraic variety V of degree 1 of the first kind. Proposition 8. Let C be a complete curve without singular point and J a Jacobian variety o fC and cp a canonical mapping o fC into J. Then, co —> co xp gives an isomorphism of ®o( J) onto Do(C). P ro o f. Let g be the genus of C ; denote by Cg and Jg the product C x • • • x C of g copies of C and the product / x • • • x / of g copies of / , respectively. Let k be a field of definition for C, 7, ando(J) and S)o(C) are of the same dimension g. The symbols C, J and cp being as above, let C' be another complete non­ singular curve, J ' its Jacobian variety, and cp' a canonical mapping of C ’ into J ’\ and let k be a field of definition for C, / , cp, C \ J '', c p Let X be a positive divisor of C x C' rational over k. X determines a homomorphism of / into J' as follows (cf. Weil [45, 46]). Take a generic point x of C over k and put n

then, there exists a homomorphism A of J into J f, defined over k , and a point b on J ’, rational over A, such that n

A and b do not depend on the choice of k and x . PROPOSITION 9. The notation being as above, let Co be a complete non­ singular curve with a generic point z over the algebraic closure k\ o fk such that k{z) = k { x, yi, . . . , yn); let p and the q v be the rational mappings o f Co into C and into C ’ defined by p{z) = x, qv(z) — yy with respect to k\. Then, fo r every co € we have n

v=\

P ro o f. Define the rational mappings a, p, y as follows:

where the numbers of the factors in the products are both equal to n, and a(z) = y\ x • • • x yn,

p ( u { x • • • x un) =