Abelian Varieties, Second Edition (Tata Institute of Fundamental Research, Bombay Studies in Mathematics)

TATA INSTITUTE OF FUNDAMENTAL RESEARCH STUDIES IN MATHEMATICS General Editor : S. RAGHAVAN 1. M. Herve : 4EVERALCOMPLEX VARIABLES

2. M. F. Atiyah and others : DIFFERENTIAL ANALYSIS 3. B. Malgrange : IDEALS OFDiFFERENTiAaLE FUNCTIONS

4. S. S. Abhyankar and others : ALGEBRAIC GEOMETRY 5. D. Mumford : ABELIAN VARIETIES L. Schwartz : RADON MEASURES ON ARBITRARY TOPOLOGICAL SPACES AND CYLINDRICAL MEASURES

7. W.L. Baily, Jr., and others : DISCRETE SUBGROUPS OF LIE GROUPS AND APPLICATIONS TO MODULI

ff. C. P. RAMANUJAM : A TRIBUTE 9. C. L. Siegel : ADVANCED ANALYTIC NUMBER THEORY

10. S. Gelbart and others : AUTOMORPHIC FORMS, REPRESENTATION THEORY ANDARITHMETIC

ABELIAN VARIETIES DAVID MUM1'ORD With Appendices by C. P. RAMAIti[JJAM and. YURI MA'AM

Published for the

TATA INSTITUTE OF FUNDAMENTAL RESEARCH, BOM$AY

OXFORD UNIVERSITY PRESS

Oxford UniversiiyPress, Walton Street, Oxford OX2 6DP LONDON NEW YORK TORONTO DELHI BOMBAY CALCUTTA MADRAS KARACHI KUALA LUMPUR SINGAPORE HONG KONG TOKYO NAIROBI DAR ES SAL AAM CAPE TOWN

MELBOURNE AUCKLAND and associates in BEIRUT BERLIN IBADAN MEXICO CITY NICOSIA

® Tata Institute of Fundamental Research, 1970

First Published 1970 Second Edition 197'4 Reprinted 1985

Printed in India by R. N. Kothari, Konam Printers, Tardeo, Bombay 400 034 and published by R. Dayal, Oxford University Press Oxford House, Apollo Bunder, Bombay 400 039.

INTRODUCTION Tsis Boom is based on a series of lectures delivered in the winter of 1967-68 at the Tata Institute of Fundamental Research. These lectures were subsequently wri bten up, and improved in many

ways, by C. P. Ramanujam. The present text is the result of a joint effort.

To write a thorough treatise on abelian varieties would be a formidable job. This book covers roughly half of the material that

I think should be in a reasonably complete treatment. We have covered:

the basic material developed in the books of Weil [W 1] and Lang [L], (ii) the techniques from the theory of schemes, developed by Cartier and Grothendieck, which have given us a clear picture of the situation in characteristic p, (i)

(iii)

the basic analytic theory developed in the book of Conforto [Co].

Unfortunately, my treatment of these topics is not as elementary

as it could be, and quite possibly a student will find the subject more accessible if he reads the earlier treatments of the subject instead of or as well as mine. However, I have attempted to keep the discussion as simple as was compatible with the amount of material to be covered. In particular, I recommend Chapter 2 (which is independent of Chapter 1) as the easiest. Many of the techniques which are generalized in Chapter 3 to subtler scheme situations are treated here in a more transparent classical. setting. Were the book to continue, the topics which I would have liked to treat would be : 1.

Jacobians,

II. Abelian schemes: deformation theory and moduli, III. The ring of modular forms and the global structure of the moduli space,

vi

ABEISAN VARIETIES

IV. The Dieudonn6 theory of the "fine" characteristic p structure,

V. Arithmetic theory: abelian schemes over local, global fields.

I don't believe the word "Jacobian" is ever used in this book. Rather stubbornly I wanted to prove that the theory of abelian varieties could be developed without the crutch of " reduction to Jacobians ". One of the main reasons this is possible is that I have used systematically the higher cohomology groups: I am especially fond of the proof of the main theorem of § 8, which replaces

Theorem 4, p. 99 of Lang [L]. But I have to admit that some people might feel Lang's argument is more geometric. For a treatment of Jacobians, the reader should look at Weirs and Lang's books: especially the very important Theorem 31, p. 117 of Weil, which Lang strangely omits. For abelian schemes, some of the basic facts can be found in my book, Geometric Invariant Theory, Ch. 6, [Ml].. This area has been greatly clarified by recent work of Raynaud which should appear soon. The. connection of modular forms with moduli spaces of abelian varieties can be found in Baily [B] and Shimura [Sh], as well as in their talks at the Boulder Summer Institute [B-M]. A purely wloebro-geometric treatment of the "theta-null werte", which are special modular forms, is in my paper [M2]. It is interesting to ask whether further ties between the analytic and algebraic theories exist: e.g. an algebraic definition of the Eisenstein series as a section of a line bundle on the moduli

space. For the Dieudonn6 theory, see Manin [Ma], Oort [0], the S6minaire Heidelberg-Strasbourg [D-G], Tate [Ti], and the papers

of Barsotti [Bt]. Among the vast literature on the arithmetic theory, let me only mention the N6ron model [N] and the stable reduction theorem for this [Gl], Kodaira [K], the Mordell-Weil theorem [L-N], the report of Cassels' [C], and Tate [T2]. Some of the material in this book is new and has not been published elsewhere. This includes the results of §16 on the index of a nondegenerate line bundle, and the results of §23 on thetbeta-groups

INTRODUCTION

vii

T(L) in the case where 0c is not separable. Simplifications in §6, § 13

and §16, the very elegant appendix to §4 characterizing abelian varieties as complete varieties I with arbitrary composition morphisms X x I -% admitting a 2-sided identity, and the treatment in §21 of the local invariants of division algebras with involutions of the second kind are all due to C. P. Ramanujam. I want to thank C. P. Ramanujam for all his efforts and to thank the Tata Institute for the very pleasant and stimulating environment which encouraged these lectures. It is a pleasure to acknowledge the help

of the very able staff of the Tata Institute, of the Fulbright foundation, of Mrs. Laura Schlesinger, and of the National Science Foundation.

PREFACE TO SECOND EDITION AFTER THE publication of the first edition of this book, it was translated into Russian and published in Moscow together with an appendix on the Mordell-Weil Theorem by Yuri Manin. Moreover, C. P. Ramanuja.m, continuing my lectures at the Tata Institute, lectured on and wrote up notes on Tate's Theorem on homomorphisms between abelian varieties defined over finite fields. Reprinting the book at this time has given the opportunity to expand

it by including these two results as appendices. Also, a nice remark by M. Nori giving an alternate approach to Application I,

§ 6, has been included at the end of Chapter II. I would like to

thank Manin, Nori and Ramanujam for their collaboration, S. S. Rangachari for his translation of Manin's appendix, and the Tata Institute for overseeing the production of the second edition.

CONTENTS INTRODUCTION

V

CHAPTER ANALYTIC THEORY.

1.

1.

2. 3.

II.

. . . Complex Tori . Line bundles on a complex torus Algebraizability of tori

ALGEBRAIC THEORY VIA VARIETIES.

8.

Definition of abelian varieties . Cohomology and base change . The theorem of the cube: I Dividing varieties by finite groups The dual abelian variety: char 0

9.

The case k = C .

4. 6. 6. 7.

III. 1 0.

1 1. 1 2.

1 3. 1 4. 1 6. 1 6.

1 7.

.

1

13

24 .

24.

39

.

39

46 65

65 74 82

.

89

ALGEBRAIC THEORY VIA SCHEMES.

The theorem of the cube: 11 89 Basic theory of group schemes . 93 Quotients by finite group schemes 108 . . The dual abelian variety in any characteristic . 123 Duality the of finite commutative group schemes. 132 Applications to abelian varieties 143 . Cohomology of line bundles . 150 Very ample line bundles . . 163

IV. HOm(X, X) AND THE l-ADIC REPRESENTATION. 18. 19. 20. 21. 22. 23.

1

Etale coverings Structure of Hom(X,X) . Riemann forms . Positivity of the Rosati involution Examples . The group 3z(L). The case k = C .

.

. . .

.

167

167 172 183 192

210 .

221

.

235

CONTENTS

Appendix I : The Theorem of Tate by C. P. Rainanujam .

240

Appendix IT: Mordell-Weil Theorem by Yuri Manin .

261

Bibliography

276

Index

278

I ANALYTIC THEORY Complex Tori. We shall investigate in this chapter a compact connected complex Lie group X of dimension g, i.e. a compact 1.

connected complex manifold of dimension g with a group structure on the underlying set such that the maps XxX-+X, X-).X defined

by (x, y) ,--y x. y and x i---s x 1 are holomorphic. Let V be the tangent space to Xat the identity point e e X. V is a complex vector space. Recall that for every complex Lie group X, with tangent space V at e, for every v e V there is a unique holomorphic homomorphism

such that d¢, takes the unit tangent vector to C at 0 to v e V. (Cf. Hochschild, Structure of Lie Groupa, p. 79 and p. 195). Moreover the function 4,(t) in t and v is a holomorphic map C X VAX. The exponential map exp: V -;% is defined by exp(v) =#,(1). Because of the uniqueness property characterizing 4 4,(4), hence 0,(t) = exp(tv). Therefore, if we identify as usual the tangent

space to V at 0 with V itself, the differential of exp at 0 is the identity map of V onto V. Returning to a compact connected, now, we first prove: (1) X i8 a commutative group.

In fact, for x in X, define C. to be the conjugation map X X, C. (y) = xyx 1. The differential (dC.), is an automorphism of V and xi--i (dCY), is a holomorphic map of%into Aut(V) c End(V). Since End(V) is a finite-dimensional complex vector space and the PROOF.

only holomorphic functions on a compact connected complex manifold are constants, we deduce that (dW.), is independent of x e X, hence (dW.), = (dC,), =1 r. Now for any homomorphism T: Xl -s Xs of complex Lie groups, T (expa,y) = expx, ((dT ).y)

2

ABELIAN VARIETIES

This follows from the uniqueness property characterizing the homomorphisms t a r exp$,(ty) from C to X. It is easy to prove from this that CC(expy) = exp((dCC)ey)

Since (dC.). =1y, this shows that CC(expy) = expy, so exp(V) is in the center of X. Since d(exp) is the identity, it follows from the implicit function theorem that exp defines a homeomorphism of a neighborhood of 0 e V with a neighborhood of e in X. Since I is connected, this implies that exp(V) generates % as a group, and it follows that % is commutative. (2) The exponential map exp: V -+ % is a surjective homomorphism of complex Lie groups with kernel a lattice' U in V, and induces

an isomorphism V/U.4 S, i.e.i is a complex torus. Let x, y e V. Since X is commutative, the map C a % defined by tf-s (exp2x). (expty) is :a holomorphic homomorphism, and the image

of (at / u by the tangent map is easily seen to be x + y. Now for any ze V, the map t -a exp(tz) is characterized as the unique holomorphic

homomorphism, whose tangent map takes (at ) to z e V. Hence (exp tx).(exp ty) = exp t(x + y) and putting t = 1, we find that exp

is a homomorphism. It is surjective since on the one hand I is connected, while on the other hand exp(V) contains a neighborhood of e and hence an open and closed subgroup of X. The kernel U is a discrete subgroup of V, since there is a neighborhood N of 0 in V such that expI.V:N -> S is injective. The induced homomorphism

V/U -+ % is holomorphic by definition of structure of complex manifold on V/U, and is an algebraic isomorphism of groups. The

tangent map at the identity of this map is an isomorphism, and hence by the inverse function theorem, the inverse is holomorphic at e and hence holomorphic everywhere on % (translations being holomorphic isomorphisms on both V/U and S). Therefore % is isomorphic to V/U. Since lattices are the only discrete subgroups of vector spaces with compact quotient, V must be a lattice. tBy definition, a la# ee in a real veotor apace V is the subgroup generated by a basis of V.

ANALYTIC THEORY

3

From now on, we use additive notation for the group operation in X. We will fix the notation 7r: V -+ X for the exponential homomorphism for the rest of this chapter. (3) As an abstract group, X is divisible (i.e. nX = X for ne Z, n = 0) and if for n e Z, n 0 0, X. is the subgroup of elements anni-

hilated by n, X,, (Z/nZ) 29.

PxooF. By (2), we see that as a real Lie group, X is isomorphic to (R/Z) 29 = (S')w, where S' is the circle group. Hence we have (3). (4)

We have canonical isomorphisms

H'(X,

Z)

group of alternating r-forms

Ux...xU-- Z

PsooF. (V, 7) is clearly the universal covering space of X, hence U= Fr-1(0) is exactly ar1(X,0). Since for any good topological space X Hl (X, Z) = Hom(rrl (X), Z),

the assertion is correct for r =1. Then to prove it for all r, it will suffice to show that cup product induces an isomorphism

A'(H'(X, Z))

H(X, Z), all r.

(*)

But note that if (*) is correct for spaces X1 and X. with finitely generated cohomologies, then by the Kiinneth formula, (*) holds for

%1x%,: A'(H1(X1 X X2, Z))) H'(X,, X X2, Z) 11,

A'[H1(X1, Z) ® H1(X2, Z)]

2

2

E [APH1(Xl, Z) ® A R1(X2, Z))] -> I HP(X1, Z)®H°(g2, Z). P+a-r

P+e-r

ABELIAN VARIETIES

4

(Note here that (*) for X, X$ implies H'(Xj, Z) is torsion-free, hence the for term in Kiinneth disappears.) But our torus X is a product of S"'s, for which (*) is trivially valid.

Computation of the groups HQ(X, 0'), where SZ' = sheaf of holomorphic p forms on X. (5)

The cohomology groups Ha(X, la-) are one of the most significant invariants of any compact complex manifold X, and their computation for a torus will take up the rest of this section.

Let V = T0,g be the tangent space to I at 0 (regarded as a complex vector space), and let T = Homc(V, C) be the complex co-

tangent space to X at 0. By translation with respect to the group law on X, every complex p-covector a C AT extends to a translation invariant holomorphic p-form wQ on X. In fact, let Ty:X-+ X be the map TT(y) = x+y. Then define (wa)y = T*Z(a). Moreover, the map a i--r w, defines a homomorphism of sheaves: (*) ) f1p eX(&cA'T which is easily checked to be an isomorphism. In other words, L'

is a globally free sheaf of Ox-modules. Since the only global sections of Og are constants, the global sections of S' are exactly the trans-

lation-invariant p-forms wa. In fact, because of the isomorphism (*) we get:

HQ(X,i2')=HQ(X, OX® A'T)=H°(I,OX)0 AV T.

The main result that we want is Tnionxss.

If

Homc.aty r(V, c), then there are natural

isomorphisms HQ(X, O$)nAQT for all q, hence

HQ(X,£l )cA'T®AQT. Our proof of this (due to C. P. Ramanujam and related to that of Weil [W2]) depends on the well-known Dolbeault resolution: a

a

ANALYTIC THEORY

5

where'P'F is the sheaf of C ' complex-valued differential forms of type-(p, q) on X, and a is the component of the exterior derivative d For background on this, see Gunningmapping P>F to Rossi, Ch. 6. The '1',q are fine sheaves, hence the above resolution defines isomorphisms

$4(X

{a-closed (0, q)-forms on $} a{space of (0, q - 1)-forms on X} Moreover, if f = WOO is the sheaf of C° complex-valued functions on X, then just as with holomorphic forms, there is an isomorphism Ox)

01.9 : W ®C[APT ®A*]

2* WR4

taking Efi ® a{ to Efwad where co. is the translation invariant (p, q)-form with value a e APT ® AFT at 0. Note that these translation-invariant forms co. are all closed. In fact, since wAB = wq A ww, it is sufficient to check this for a of degree (1, 0) or (0, 1). Since s : V X is a local isomorphism, it is sufficient to check that

0. But considering a itself (which is in T (9 T) as a function on V, Fr*(wa) = da. Therefore d(.*wu) = dsa = 0.

Now let A' = ® MT be the exterior algebra on T. Let a = F

P(X, W). Then via ¢0,4, we get an isomorphism

a or AF

P(X, 5670').

If we define a differential a on the set of spaces on the left by a(f® a) =afA a, then (because the w,, are closed), the complexes a ®c A' and P(X,') are isomorphic. Therefore HH(X, Os) 9 HF(a Oc A').

Our aim is now to show that the inclusion is A' -+ a ®c A' defines

an isomorphism of cohomology, i.e. AF HH(a ®c A'). We will do this by Fourier series. Let fr. be the measure on X induced by. the Euclidean measure on V, and so normalized that the volume

µ(X) of X is 1. We define a C-linear map lc: a 3 C by putting µ(f) 5fµ. For any vector space W over C, we denote by Few the

s

ABELIAN VARIETIES

6

map A®13v: a® W a W: in particular, we get a map ILA:00CA - A' which is X -linear and such that pAoi = IdA. LEMMA 1. PROOF.

For w e a®cA', we have LA(8w) = 0.

Since pA is A-linear, it suffices to prove that j (df)

= 0 for f. e a. Choosing a basis w1.... , wn of T, we can expand c7f e a ®cTas Eh;® wi. The coefficients h; are all of the form D(f), where D is some invariant vector field on X. Therefore the lemma follows

from the elementary fact that if f is a C°-function on V, periodic with respect to the lattice U, and D is a translation-invariant vector field on V, then J

D(f)dx=0

vjU

(dx = some Euclidean volume element).

Let U* = Hom(U, Z). If A e U*, then A extends to an R-linear map A: V - R and we can then form the function x -> e2"ia(z> on V. This function is invariant under the action of U, hence it equals

exov, where ex is a C`°-function on I. Now define a C-linear map QA: a -+C byQa(f) = p(e_af) =J c_r,f.p. More generally, for S any vector space W, define Qa: a®cW -. W by Q, (f® w) = µ(e_ a f ).w.

The Qa(f) are the Fourier coefficients of f: for every f e a®cW we get the expansion

f=

e®®Qa(.f ) aeu.

The QA are compatible with C-linear maps W -->- W' just as a is in particular, Qa: a ®c A' - A is a A'-linear map.

For the remainder of this proof, we choose a Hermitian norm 11 on the complex vector space V. As usual, this induces a norm on T, hence on the whole exterior algebra X. II

Moreover, define the mapping C: U* -a T as follows:

U*_

Homa(V,R)cHomR(V,C)e[TED T]

projection

ANALYTIC THEORY

7

This makes U* into a lattice in 2', hence by restriction we get a norm 11

II on U* too.

LEMMA 2. (1) The, map f {QA(f)1,2a. is an isomorphism of a onto the vector space of all maps Q: U* -+C decreasing at ao

faster than II A 1I-", all n, i.e. IQ(A)I = 0(11 All-n), all n. (2)

For all w e tl ®,AP Qx(aw) _ (- 1)P2ni[Qx(w) A C(d)].

PnooF. (1) is standard Fourier analysis. To prove (2), note that *(ae_x) = a(e-2^ix) _; - 21rie-2iax.5A = rr*[- 2arie_x ®C(l)],

hence ae_x = - 2aie_x®C(d). ea®AP, 0= x))

Therefore, by Lemma 1, for all

= IeA(e_x . aw) + (- 1)9'1 21ri sA(We_x A C(A))

Qx(aw) + (-- 1)1-121riQx(w) A C(h). The following is well known. LEMMA 3.

Let W be a complex vector space, D e Homc(W, Q.

Then D extends to a map DJ : AMW -+AP'1W for all p, called interior multiplication by D, such that (1) P

(- 1)P-kD(gk).XiA ... A XkA...AXq;

DJ (X, n...ngP) _ k-1

(2)

in particular, if DXo =1, for all w e A' W,

DJ (w AXo)+(DJ w) A X0=w. We are now all set to prove that is A' -+0®cA is a homotopy equivalence for d-cohomology. For every A e U*, A 0 0 define an element A* E Homc(T,C) using the Hermitian inner product 0.

We wish to give a direct geometric description of every (holomorphic) line bundle L on a complex torusl. By the corollary, the line bundle n*(L) on V is trivial. If we choose an isomorphism

X: 7r*(L) ) C x V the canonical action of U on it*(L) (i.e. the action such that the quotient of ar*(L) by U is just: the original bundle L) carries over by

means of X into a linear action of U on the trivial bundle covering the action of U on the base V by translations. Let us denote by H* the multiplicative group H°(V, Op) of nowhere vanishing holomorphic functions on V. Since the only holomorphic automorphisms of a line bundle fixing the base are given by multiplication by non-vanishing holomorphic functions, we see that the action of U on C X V is given by

ABELIAN VARIETIES

14

((X, z) --s ¢ (a, z) = (e (z) . a, z + u), all u e U

(A)

where e e H*. Writing down the condition that = 0.+., (%,z), we see that u r--s e« is a 1-cocycle for U with coefficients in H* : e4(z + u'). eu.(z).

Further, if the trivialization X is altered by multiplication by a nowhere vanishing holomorphie function f on V, {ej is replaced by the cohomologous cocycle e;,(z) = e*(z)f (z + u)f (z)`'.

Therefore we have defined a map from H'($, 0x*) to H1(U, H*). But we can go in the other direction too. If we start with a 1-cocycle Y J with coefficients in H*, then define a line bundle L on I as the

quotient of C x V by the action of U given by (a,z) a -- (%(z). a, z + u). Therefore we have found an isomorphism

: H'(U, H*) =i 111(1, Os) More generally, for any sheaf F on S, there is a natural map 0: H1(U, r(1, ir5F)) --)- H'(%, J IT). The definition and properties of are recalled in an appendix to this section. Since.H'(V, 0*) = (1),

all i > 1, the . defined in the appendix is also an isomorphism. Let us check that the isomorphism just obtained and that of the appendix are the same. In fact, choose an open covering {V, } of $ by small enough connected open sets VV. Then

(a) it '(V{) =disjoint union of connected open sets u + WW,

allueU. (b) If ir{ = restriction of a to W n,: W; morphism.

(c) If V; n Vj i6 0, then 3 ui5 e V such that vri-1(Vi n V;) = lr; ' (V { n Vs) + u#l.

V{ is a homeo-

ANALYTIC THEORY

15

The map of the appendix by definition takes a group 1-cocycle {e.} to the Cech 1-cocycle {fii}, fii e T'(V1 n V1, Og) defined by `(z)). fs(z) = But {f{i} defines the line bundle L which is the union of trivial

line bundles C x Vi, modulo the patching

CXV,

Cx Vi U

Cx(Virr,VV)

tt

(a, x)

U

CX(V1nVi) (a f i(x), x).

But ri is an isomorphism of C X Wi with C X Vi. so L can also be described as the union of trivial line bundles C X Wi, modulo the patching

Cx Wi

C X Wi

U

U

Cxr,I(VinV,) (a, x)

F.

CXori- '(YinVi) (a.A(ri(x))' x +U..

Now the disjoint union of C x Wi is just the line bundle C X V pulled back to U Wi, and the set of above identifications is just the equivalence relation on this pull-back bundle induced by the equivalence relation on C x V given by the action of the group U. Therefore, L is exactly C x V modulo U. On any complex analytic space X the exact sequence

0ZOX cig0

t According to the conventions of the, appendix, we should take the action of U

on Ha as given by (u h) (z) = h(z - u), u e U, z c V. But then, if eu satisfies the condition above, fs = e_u is a 1-cocyclo for this action, and conversely. Thus, such associated I- eoeycle is given by f;i a r(Vi n vi, OX*),

fif(e) =f-nii(7ri 1(z)) = euii(?ri (z)), which is the formula we have above.

ABELIAN VARIETIES

16

defines a co-boundary S : H'(8, &*x) -* H2(X, Z). If a line bundle L corresponds to a cohomology class A a H'(I, &,*), then S(A) is calle 1. the first Chern class of L. In our case, suppose L is defined as above by a 1-cocycle {ej with values in H*. We want to calculate the first Chern class of the corresponding line bundle. First notice

that since H'(V, Z) = (0) for i > 0, it follows from the appendix

that the maps ¢: Hi(U, Z) = H'(U, H°(V, Z)) -; H'(X, Z) are isomorphisms. If H is the ring of holomorphic functions on V, we have an exact sequence es-tc

t! ))-40

0----+H°(V,u*(Z))--U

II

II

H H* Z (since V is simply connected), so that by, the compatibility of 4 with S (see Appendix) we get the diagram H'(U, H*)

H'(X, OS)

S

H5(U, Z)

S

H2(X, Z).

Hence identifying H2(U, Z) and H2(X, Z) by the above homomore2a« Z defined by AF(ul, us) = F(ul, u2) - F(us, a1) maps the group of 2-cocycles Z2(U, Z) into the space of alternating linear maps U x U 3 Z, and induces an isomorphism LEMMA.

A: H2(U, Z)

Hom (AF U, Z) = A2 Hom (U, Z).

Further, for z=, 77 e Hom(U, Z) = H'(U, Z), we have A( u 71) _ n 71.

ANALYTIC THEORY PROOF.

17

First we cheek that if F e Z2(U, Z), E = AF is bilinear.

We have

F(u2, us) - F(ul + u2, us) + F(u1, u2 + u3) - F(u1, u2) = 0, u, E U. (i)

In this equation, instead of ul, u2 and u3, substitute ua, ul and u=

(respectively u1, u3 and u2) and call the equation so obtained (ii) (resp. (iii)). Then (i) + (ii) - (iii) gives us that E(u3, U, +u2) =E(us, ui) +E(u3, u2)

Since E(u, u) = 0 and E(u, v) = - E(v, u), it follows that E is alternating bilinear. Now suppose F = SG is a coboundary. Then AF(u1, us) _ (SG)(u1, u2) - (80)(u2, u1)

_ [G(u2) - G(ul + u2) + G(uy)] - [G(uy) - G(ul + u2) + G(us)] = 0.

Hence A induces a homomorphism R2(U,Z) -a Hom(A2 U, Z) A2Hom(U, Z).

Now, since we have an isomorphism 0 of H*(U, Z) onto H*(X, Z)

where I is a torus, taking cup products to cup products (see Appendix), and we know that H*(X, Z) is the exterior algebra on H1(I, Z), it follows that H*(U, Z) is also the exterior algebra on H'(U, Z) = Hom(U, Z). Thus, to prove that A is an isomorphism, it suffices to prove the last statement of the lemma. But now, if (resp. rt) is given by the homomorphism f(resp. g) of U into Z, u ,l is given by the 2-cocycle (see Appendix) c(s, t) = f(s).g(t), so that A(e u q). is given by the map: A(¢ u ,t)(s, t) = f(s)g(t) -f(t)9(8) (f A 9)(s, t).

REMARK. We have thus an isomorphism H2(.$, Z) E A

H2(U, Z) -) A2Hom(U, Z). This coincides with the isomorphism H2(.$, Z)-+A2Hom(U, Z) defined in §1, using cup product in H*(X, Z) and the isomorphism H'(,%, Z)

Hom(U, Z.). In fact,

0 commutes with cup products and A has the property that it maps cup product into exterior product, by the lemma, and

18

A33ELLAN VARIETIES

0: H'(U, Z) = Hom(U, Z) - H'(X, Z) is easily checked to coincide with the inverse of the isomorphism of §1 using the naturality of vS. Thus, in future, we can unambiguously identify H'(X, Z) with A'Hom(U, Z). Returning to the line bundle L arising from an {ej a Z1(U, H*) we state formally our conclusions as a PRoposixiox. The Chern class of the line bundle corresponding to a Z1(U, H*) is the alternating 2 form on U with values in Z given by

E(ui,"..,=f,,,(z+u,)+f,,,(z)-.f,,,(z+&2)-f,,(z), (z arbitrary in V) (**) where c(z) = e2 u4c: COROLLARY.

If we extend E R-linearly to a map V X V i R, E

satisfies the identity E(ix, iy) = E(x, y) for x, y e V. PROOF.

In fact, since E represents an element of H2(X, Z)

in the image of H'(X, O$) H2(X, Z), its image by H2(X, Z) -r H2(X, O,x) must be zero (and conversely). Now, this last map

i

factorises as H2(X, Z) - H2(X, C)

If we put Homa(V, C) = Homc(V, C) Q) Homcanii(V, C) = Tp T, we H2(X, O8).

have established isomorphisms H2(X, C) c A2(TG) T) = (A2 T) Q (T(& F)@ (A2T), and H2(X, Ox) : A2T, and j goes over into the projection A2(T(D T) A2 T. Further, i(E) is nothing but the real linear extension of E (of. Remark 3, §1), which again we denote by E. Write E = E, + E2 + E3, where B, e A2 T, E2 e A2 T, and Ea e T(9 T.

The reality of E implies that El = E2, so that j(E) = 0 if and only if E = E3, and this holds if and only if E(x, y) = E(ix, iy). Our next aim is to give as explicitly as possible all line bundles on the complex torus X, or equivalently, to find the simplest kind of representing cocycles (eu) for all cohomology classes in H1(U, H*).

This is in turn equivalent to finding a. system of functions holomorphic in V and satisfying (*).

ANALYTIC THEORY

19

Thus we assume given to us an alternating form E: U x U Z, with E(ix, iy) = E(x, y) and we seek to find { fu} satisfying (*) and (**). Let us look for solutions f, which are linear in z (not necessarily vanishing at 0). We use the following elementary result. LEMMA. Let V be a complex vector space. There is a 1-1 correspondence between the Hermitian forms H on V and the real skewsymmetric forms E on V satisfying the identity E(ix, iy) = E(x, y),

which is given by

E(x, y) = Ira H(x, y) H(x, y) = E(ix, y) + i E(x, y). The proof is left to the reader.

Let H correspond to the given E; then one checks immediately that the functions fu(z) = 2i H(z, u)

P.

satisfy (**) for any constants Pu, and the reader can also check if he likes that these are the only linear solutions of (**), holomorphic in z modulo coboundaries. Substituting' in (*), we get a further condition: - ift,+s= E iZ for all u1, u2EU. Writing iftu = Y,. + JH(u, u), this reduces to 11 2H(-l, s&2) + i1,a1 + i&'

Yul + Yuq `. yul+u2 + j iE(ul) u2) a 2Z.

Now it is still permissible to modify if. by the coboundary of a C-linear form L on V, or what is the same, we may replace yu by y - L(u) with L: V -+ C being C-linear. The above equation shows that Rey. is additive in U, and hence extends to an R-linear map A: V -* R, and there is a unique C-linear form L on V with ReL = A (viz., the form defined by L(v) = A(v) - iA(iv)). Modifying y by this L, we may assume that y is pure imaginary. Writing a(u)= ellyu we see that a has to satisfy the conditions

Ia(u)l =1

20

ABEL[AN VARIETIES 41(61 + us) = e:.sca.,v.) 41(61)41(62)

We can check that given E, there always exists such an a, or equivalently, that there always exists a map 8: U -> R such that

8(u1 + u2) - S(ul) - S(u2)

E(u1, u2) (mod 1) for all u1, u2 E U.

This is left as an exercise to the reader.

We have thus proved the LEMMA. Let H be a hermitian form on V such that if E = Im H, E(U x U) c Z. Let a: U-+ C* = (z e C* I IzI = 1) be a map with

41(u1 + u2) = e: s(ex,ax),a(61)a(62), u, E U.

Such maps a exist for any given H as above.

If we put

a(u) a s(:,u>+} S(u,u

then u i---, eu is a 1-cocycle on U with coefficients in H°(V, 0v) = H$, the Chern class of the associated line bundle being E e H2(%, Z). DEFLNITION_

L(H,a) is the quotient of C x V for the action of U A, z + u).

given by 0u(A, z) =

Note that the map (H, a) m

i. {eu} satisfies the condition that if

{e( .0} corresponds to (H;, a;), {e( .l).eu2)} corresponds to (H1+H2,

Therefore we have an isomorphism of line bundles L(Hv al) ® L(H2, 412) e L(H1 + H2, ala2).

The main theorem of this section is THEOREM OF APPELL-HUMBERT. Any line bundle L on the complex torus I is isomorphic to an L(H, a) for a uniquely determined (H, a) satisfying the conditions of the above lemma. We have isomorphic exact sequences 0 -) Iiom(U, C:) ---> (Group of data (H. a)) ---> Group of hermitian

H:VxV --Cwith

I -r 0

(ImH)(Ux U)CZ M

A

0 -* Yic'X

*- Pie B - ---CRarfR.(X,

Z) -+ &°(R, Bz)) - 0

21

ANALYTIC THEORY

where Pic X is the group of line bundles on X, Pic°X the subgroup of those which are topologically trivial and the last vertical map is given by

H t---.ImH (with the usual identification of H2(X, Z) with alternating integral 2forms on U).

Paoor. We have already shown that an alternating integral 2-form E on U, considered as an element in H(X, Z), maps into 0 in H'(X, ©$) if and only if E(ix, iy).=E(x, y) when B is extended R-linearly to V x V; that is, if and only if it is Im H for H Hermitian. Thus v is an isomorphism. By definitions and the above lemma stating existence of a for given H, the first row is exact. Since the topological triviality of a line bundle L is equivalent to the vanishing

of its Chern class, and since v is an isomorphism, the second row is also exact. To prove the theorem, it suffices to show that A is an isomorphism. It a e Hom(U, C*) with A(a) = 1, we can find g e H* = H°(V, 0*y) with g(z + u)

a(u).

g(z)

If K is a compact set in V with K + U = V, it follows that for any z c- V, I g(z) ; < SupgI g(z) J, since ) a J = 1. Hence g can only be a constant, so a = 1, which shows that A is injective. Consider the commutative diagram ezai( )

H'(U, C) -- H' (U, H) ---) Ker [H'(U, H*) -9. H$(U, Z)]

H'(X, C) --> H'(X,ox) -- Ker [H'(X, tg) --*- H2(X, Z)] = = Pic°X where the vertical maps are isomorphisms and the maps denoted by ez"tc) are surjective. But we proved in §1 that H'(X, Oa) is surjective. It follows therefore that every line bundle L ePic°(X)

is presentable in the form C x V modulo an action of U of form &.(X, z) = (A.a(u), z + u), where a: U--> C* is a homomorphism.

ABELIAN VARIETIES

22

But as we saw on p. 20, by an automorphism of C x V, we can always normalize such actions so that Image(a) c C*. Therefore A is surjective.

APPENDIX TO §2

We want to study cohomology of sheaves in the situation: Y=.$/G, where G is a discrete group, acting freely and discont;^uously on a good topological space X (i.e. V x eN, x has a neighborhood U. such that U. n a(U2) _ 0, a 0 e). Let 1r: X > Y be the projection. First recall the definitions of the cohomology of abstract groups.

Let M be a G-module and let CP(G, M) _ (group of functions f : G"-+ M}; 8: CP

* C"}1 the map p-1

Sf(a°,...,o,) =ao(.f(al,...,a,)) + I I (-". 1)ti+1f(o J1 ,...Iai.ati+1,...'a,) i=0

+ (- 17+1 f(Q°,..., ap-1)i

ZP(G,M) =Ker(S); B"(G,M) =Im(S);

J

HP(G,M) = ZP(G,M)/B"(G,M) =derived functors of Al i-> H°(G, M) , where H°(G, M) = {m e M I a(m) = m, all a e G} (also written MG).

Given a G-linear pairing M x N * P of C-modules, get u : HP(G, M) x H4(G, N)

HP+Q(G, P)

via

f u 9(a1, ... , ap+4) = A01".., a,)* (al..... aa)9(u,+1, ... , ap+4)

all feCP(G,M),9eC4(G,N).

We want the result: V sheaves F on Y, there is a natural map ¢:HP(G,P(X,ir It has the properties;

23

ANALYTIC THEORY (a)

If

0 -+.°F' -} .1---) .°$"

)0

is an exact sequence of sheaves on Y, and

0-, r(x, 7r*.5F')

) r(x, 7r* !F)

is exact, then we get a homomorphism from the cohomology sequence of HP(G, ) to that of HP(Y, -). (b)

The natural maps 0 are compatible with cup product.

(c)

If i > 1,

(0), then

¢: HP(a, r(x, a*,)) ------ HP(Y,-W) is an isomorphism. To define 0, choose a covering { Vj

of Y such that for each i,

(1) 7r 1(V;) = U -(U{), U; c X open such that res it : U. .-

V7,

.60

(2) V i, j, there exists at most one a e O such that U, n oUj

0;

call it o;, if it exists. Define a map from group co-chains to Cech fco-tchains:

P .VP(a, r(7r*.5F)) --> by

o(17*){ol[f(v;o;.....,osP_1;P)l

where (7r*); 1: r(x, 7r*- q7) .---- r(V;,.F) is the map res

r(x, Tr*' )> r(U{, 7r*.`F) F

Pd

7r*

r(Vti,.').

It is easy to check that 8¢P = 0P+i8, hence the 4), induce a map 0: HP(a, r(x, 7rx*F)) i HP(Y, JIF). Properties (a) and (b) follow immediately by computation. To prove (c), we use induction on p: for p 0, it is obvious. In general, embed S in an injective

Op sheaf F and let .V _ '

Then we find

24

ABELIAN VARIETIES

")-*H(X,n*8r)

(o)

hence tl'- IIG,

H'-'(a,

Hn(0, P("*jr)) -- s. H'(a. P(**.e

))

I

to

to prove that r(7r*-,F') is an injective G-module, and H'(X, 7r*.Ir') = (0), i > 1, because then it follows that (0), i > 1, hence 01, ¢2 are isomorphisms by the H`(X, uduction hypothesis, hence 02 is an isomorphism. We need

that

LEMMA. If .' is an injective Op-sheaf, then 77*F is a flasque OZ-sheaf and r(7r*.F) an injective (1-module. Pxooxr.

For all G-modules M, let M be the constant sheaf on X

with value M. There is an obvious action of 0 on M compatible with its action on X. Then C1 acts also on 7r* (M), so we can form 7r*(M)o. It is easy to check that Homo(M, r(7r*,,-,)) m HomoY(7r*(M)a, F). So if M1 c M2, then 7T* (M1)o c 7r* (M2)a, hence

Hom,,(7r* (M2)o,-fl _. HornsY(7r* (M1)G, F) r(7r* '5r)) is is surjective, hence Homo(M2, surjective. This shows that r(7r*.F) is injective. Secondly, .°F' injective implies JF flasque, and since 77 is a local homeomorphism, then 7r*.°V' is flasque too. (Cf. Grothendieck, Sur quelques points d'algebre honwiogique, Tohoku Math. J. (1957), esp. Ch. V, p. 195.) 3.

Algebraizability of tori. We have seen that any line bundle

L on the complex torus X = V/U is isomorphic to a unique line bundle of the form L(H, a) where H: V x V --> C is hermitian with

E = Im H integral on U X U, and a is a map U --,- Cl, satisfying L(H, a) is the quotient of Cx V for a(u1 +u2) the action of U given by

ANALYTIC THEORY

25

0.(A, z) eu(z)

a(u) a*&(:,u)+i.aiu,u)

We now investigate the sections of L(H, a). These sections are in a natural one-one correspondence with sections 0 of the trivial bundle C X V over V (i.e. holomorphic functions 0 on V) which are

invariant under the above action of U, that is, which satisfy the functional equation 0(z + u) = eu(z) 0(z) = a(u). e°E(z u)+}ea(u,u) 0(z), z e V, U E U.

Such a function is called a theta function for the hermitian form H and the multiplicator a.

First consider the case when H is degenerate. Since E = Im H and H(x, y) = E(ix, y) + iE(x, y), we have

N={xeVIH(x,y)=0,yyeV}={xeVIE(x,y)=0,yyeV}. It follows from the first expression for N that N is a complex subspace of V. And since B is integral on U X U, it follows from the

second expression for N that N n U is a lattice in N. If 0 is an associated theta-function, we must have 0(z + u) = a(u) O(z), y u e N n U.

Thus, if K is a compact subset of N with N=K+(N n U), we must have I B(zo-F-z')I< Sup 10(zo-I-C) I =c(zo), (e$

for all z' c- N. Therefore, by the maximum principle for holomorphic

functions, 0(zo +z') = 9(zo) for z' e N and 0 is constant on cosets mod N. It follows from the earlier equality that if 000, then a(u) =1

for u e N n U. Thus, if ,l: V -* V/N is the natural map, we see that any theta-function for (H, a) is of the form 8o,7, where B is a theta-function on V/N for the lattice 77(U), the hermitian form H induced by H, and the multiplicator a obtained from a by passage to quotient from U to U/N n U. Now H is non-degenerate on V =VIN. Thus the study of the theta-functions for (H, a) is reduced to the study of theta-functions for (H, a) on the quotient V =V/N,

ABELIAN VARIETIES

26

and we may restrict ourselves to the case when His non-degenerate. In particular, we see that if His degenerate with null space N, if 0 vanishes at z e V, it vanishes on the coset z+Y, so that any section a of L(H, a) which vanishes at an x e X =V/U also vanishes on the

coset x+X' where X' is the subtorus N/U n N c X. In particular, we see that if the sections of L(H, a) define a morphism of X into projective space at all, this morphism has to factor through the

quotient torus X/X', X' = N/U n N. Thus L(H, a) cannot be ample if His degenerate. Next, suppose there is a complex subspace W c V of positive dimension such that H(w, w) < 0 for w e W, w rh 0. Let K be a compact subset of V with V = U + K. Let zo e V and w e W, and

write w =d+u, deK,ue U. We have I O(zo + w) I = 9(zo + d + u) I = 9(zo + d) I e

and since ReH(zo+d,u)+ JH(u,u) =ReH(zo+d,w) - ReH(zo+d,d)+ JH(w,w) + JH(d, d) - ReH(w, d) _ JyH(w, w)+Re H(zo, w) + c(d, zo).

Of the terms on the right, for fixed zo, the first is a real negative definite quadratic form in w, the second linear in w and the third is bounded (since d stays in a compact set K), so that the expression

tends to - co as w-w in W, and applying the maximum principle to 9(zo w) as a function of w, we conclude that 9(zo + w) . = 0, hence 0 0. Thus L(H, a) has no non-zero sections in this case. Therefore, if H is not positive definite, L(H, a) cannot be ample.

From now on, we work under the assumption that H is positive definite (and E = Im H integral on Ux U, as always). We shall prove the following PROPOSITION.

When H is positive definite and E = Im H is

expressed as a matrix using a basis of U over Z, we have

dim H°(X, L(H, a)) = dim [space of theta functions with respect to (H, a)1

= + 1/det E.

ANALYTIC THEORY

27

PRooF. The idea of the proof is as follows. Since in eu(z), z occurs

in the exponential linearly, one might hope that by multiplying 0 by

where Q is a suitable quadratic function one will be able

to obtain periodicity for the new function with respect to a big sublattice U' of U. We can then expand this periodic function as a Fourier series, and the behavior of 0 with respect to lattice points not in U' can be expressed in terms of the Fourier coefficients. This enables one to compute the number of linearly independent solutions. Let then eu(z) = a(u).e"a(I,u>+;,.acu.u)

as usual, and let 0 be a

holomorphic function on V satisfying 9(z + u) = eu(z) 0(z). If B: V x V i C is any complex symmetric bilinear form, and if we put 0*(z) =

9(z), B*(z) satisfies the modified equation

9*(z + u) = a(u)

a sublattice U' of U of rank g e (= dim V) such that (1) E(U' X U') = 0, and (2) if W = R. U', W n U = U'. Then W n iW is a complex subspace of V on which E and hence His identically 0. SinceHisnon-degenerate, W n iW=(0),

and so V=WQiW=CA U'=C®a W. SinceE(Wx W)=0,H has a real symmetric restriction to W, and by the above, there is a unique symmetric complex bilinear B on V such that B I W x W = H I W X W. By C-linearity in the first variable, H(z, w) = B(z, w)

for wE W,zE V. Since EI U'x U'=0,aI U':U'-iC; isahomomorphism, and we can find a C-linear form A on V with A real on W and a(u) = e2nta(u> for u e U'. The functional equation for 0* shows

then that e'2

0*(z) is periodic with respect to the lattice U'.

Let us write

= Homz(U', Z) c Homc(V, C), and expanding

e-2"ia(E).0*(z) in a Fourier series, we obtain the expression 0*(z) _ 7 cz. e2,,1(a(=>+a(:))

(1)

X=3'

Now, for any u e U and u' E U', (H - B) (u', u) = H(u, u') - B(u, u') = - 2i Im H(u, u') = 2i E(u', u) and if u e U' is defined by u(u') = E(u', u) and extended C-linearly to V we deduce that (H- B)(z, u)

ABELIAN VARIETIES

28

= 2i u(z). Substituting the Fourier series (1) in the functional equation we get for any u e U, ex. e2"1 Uc)+1(u)1 p2-4X(z)+1(z)1 = a(u)eiAU(u)

CX. e2, 1X(z)+1(z)+a(z)1

tea'

Va'

and comparing coefficients, Cz = a(u). e:1u(u)-2virL(u)+A(u)1 . CX-u _

(2)

Thus, if M is the image of U under the homomorphism U -->. U' given by u F--+ u, we see that the cX are uniquely determined once they are specified for X running through a system of representatives

of U' IM. (Note that if u, u2 e U with u1 = u2, then E(U', u1- u2) _ 0 so u1 - u2 e U' and one checks that the relations (2) obtained with

u1 and u2 for u are the same.) We shall check conversely that given any system {cj . of constants satisfying (2), there exists a corresponding function, i.e. the series (1) is the Fourier series of a holomorphic function. It suffices to check the uniform absolute convergence of (1) on compact subsets of V. Fixing a Xo a U' it suffices

to prove this for the solution c,, such that ex = 0 if X - Xo M and cXO =1. Writing X = Xo + u for X e Xo + M, when z lies in a compact

set K c V, the series (1) is majorized in absolute value by e2.f.(z)J I CXpt u .

const. / E!V

hence by coast.

e"TMQu)+AAQ! uEM

where 1j uj; denotes a suitable norm on M, and A a positive constant determined by Xo, K, a and H. Since the sum V= W (D iW is direct, we can find R-linear maps V -+W such that z =¢(z) i

Since u is real on W and E(W x W) = 0, we get Im u(u) = Im[u(¢(u)) + iuu(O(u))]

ANALYTIC THEORY

29

Further, 0(u)= 0 4n. u=#(u) c> ueW Cu u= 0, so that Im u(u) is a negative definite quadratic form on M. Thus, the above series converges very rapidly.

We deduce that the dimension of the space of theta-functions for (H, a) is the index U'/1Nf.

Thus we have only to show that if U is a free abelian group of order 2g, E a skew symmetric bilinear form on U into Z non-dege-

nerate over Q, U' a direct summand of U of rank g on which E = 0 and n the order of the cokernel of the map U -->- Homz(U', Z) the defined by u --E diagram 0

ii) l" My

0

---) U I#

0

) .U/U'

0

ly

)0 with exact rows and columns, the definitions of a, P, and y being U'

via E, using the fact that a and y are transposes of each other up to sign,. hence have cokernels of same orders, and the fact that r3 has a cokernel of order I det E 1. We can now prove the main theorem of this section. TrrEO nM OF LEFSCHETZ. Let $ be a complex torus V/U, H a hermitian form on V such that E = Im H is integral on U X U, at a and L = L(H, a) map U -+ C, , with a(ul + u2) = a(ui)

ABELIAN VARIETIES

30

the associated line bundle on X. Then the following statements are equivalent.

Given any complex subtorus Y of X, there is an integer N > 0, and two points x1, x2 a X, xx - x2 e Y such that a section a of a(xi) = 0, a(x2) 0. (1)

(2)

The hermitian form H is positive definite.

(3) The space of holomorphic sections of L®" give an imbedding of X as a closed complex submanifold in a projective space, for each n > 3.

PROOF. We have already shown earlier that (1)

(2), and

(3) u (1) is clear. It remains to assume (2), and deduce (3). We use the expressions "theta-functions for (H, a)" and "section of L(H, a)" interchangeably, making the identifications indicated earlier.. We shall prove (3) for n = 3 (the cases n> 3 being proved quite similarly).

Firstly, if 0 is a section of L(H, a) and a, b e V, then 0(z - a). 0(z - b).B(z + a + b) is a section of L(3H, as). In fact, on making the substitution z + u for z in this function, it acquires a factor a(u)3 exp {nnH(z - a, u) + orE (z - b, u) -F 77H(z + a + b, u) en.sa(z, u)+},.SH(u,u) 32 + H(u,u)} = a(u)2

which proves the assertion. Hence, if zo e V, there is a section of Ls not vanishing at z0. In fact, one has only to take a non-zero section 0 of L(H, a), which exists by the Proposition, and then to choose a, b e V such that 0(z0 - a) 0 0, O(zo - b) 0 0 and 0(z0 + a + b)

# 0, and put 0 to be the product 0(z -. a).B(z - b).B(z + a +b). Thus, if 00,..., Bd is a basis of the sections of L', we get a welldefined holomorphic map

0:8-+Pd given, in terms of homogeneous coordinates, by O(ir(z)) _ (00(z), B,(z), ... , Od(z)) a Pd, z e V.

Next, we prove that O is an injective map. If not, there exist Z11 z2 a V, zz - z= 0 U, and a non-zero constant y e C* such that for

ANALYTIC THEORY

31

all theta-functions q for (3H, a3), we have c(xs) = yO(xl). In particular, for any a, b e V and any theta-function 0 for (H, a) we have b)6(z2 + a + b). 9(z, - a)9(z, - b)e(z, + a + b) = yO(z2 a)9(zsWe now consider both sides as functions of a (fixing b), and take

logarithmic derivatives, so as to eliminate y. Writing w for the (meromorphic) differential i- , we obtain the relation

-w(z,-a)+w(z,+a+b)= -w(z2-a)+w(z2+a+b),ab eV which means that the differential w(z2 + z) - w(z, + z) is translation

invariant in z, hence of the form dl(z), where l is a C-linear form on V. But then, this is also the differential of log

6(z + z2)

d(z + z,)

so that

we obtain an identity 9(z+z2) = A,.eu(t).9(z+z,)

for some A, a C*. Writing a = z2 - z, this may also be written as 0(z + a) = Ae=(z)B(z)

with a fixed A e C*. Making the substitution z r--} z + u (u a U), using the functional equation for B and comparing the multiplicators

on both sides, we get that euu) U E U,

or irB(a, u) - 1(u) a 2zriZ, u e U.

This implies that irH(a, u) - 1(u) = TrH(u, a) -1(u) + vr(H(a, u) H(u, a)) = irH(u, a) -1(u) + 2aiE(a, u) takes only pure imaginary values for all u e V, hence the same holds for 7rH(u, a) - 1(u), and this

being complex linear in u, we must have rH(u, a) = l(u) for all u e V. But then it follows that 21ri.E(a, u) a 2iriZ for u e U, hence

a E U'- = {x e V I E(x, u) a Z, V u e U), which is a lattice in V

containing U as a sublattice of finite index. Since a U by assumption, U + Za # U, and the equation 8(z + a) = A. el('). 8(z) = A'. swat:,o)+}va(.,o). B(z)

shows that 9 is actually a B-function for the lattice U + a Z, the hermitian form H and a suitable multiplicator a' on U + Za extend-

ABELIAN VARIETIES

32

ing a.

Now, this must hold for any section 0 of L(H, a), and the

dimension of the space of such 0's is i/(detuE), the root of the determinant of E for the lattice U. On the other hand, if U' U + Z a # U, the dimension of the space of theta-functions for the

lattice U' and H and any multiplicator a' is V(detaE), the root of the determinant of E on the lattice U'. But since we have evidently

Xa detoE> dota.E,

and since there are only finitely many possible a"s extending a, it follows that almost all theta-functions for H, a and U are not theta-functions for H, a', and U' for any a'. This is a contradiction. Hence @: X -)..pd is injective.

To complete the proof of the theorem, we have only to establish that 0 induces an injective map of tangent spaces at all points of

. If not, there is a zo e V and a tangent vector Z a, - at zo with i

azi

not all a; = 0 mapped into the 0 vector at O(a(zo)) in V. There is then an ao e C such that for all q Er(X, L(3H, as)), ao ON) +

D(log

i.e.

ai 0- (ZO) = 0,

_ - ao

a for all 0 as above, where D => a;- - Take

O(z) = B(z - a)B(z - b)B(z + a + b) as before, with a, b e V and e e r(L(H, a)).

If we put f(z) = D(log 0)(z), we obtain

f(zo - a) +f(zo - b) +f(zo -i- a + b) = - ao for all a, b e V. One concludes easily that f is a linear (not necessarily homogeneous) function of z. Integrating the equation f(z)

D(log 8)(z), we obtain that there is an a e V, a 0 0 such that for all d e C, we have

ANALYTIC THEORY

33

O(z+A) =e"+VO).9(z) for some constant c. One concludes as in the earlier step (by writing down the transformation formulae for both sides, for the ) z + u, u e U) that for all A e C, Aa belongs to the lattice Ul = {z e V I B (u, z) a Z, V u e U1. This is a contradiction. substitution z 1

We next recall some definitions.

Let X be an algebraic variety over C. There is a canonically associated analytic space structure on the underlying set of X. We denote this analytic space by .Xhoa, and its structure sheaf by Or hod. Often, we will not be so explicit, and talk of holomorphic functions

on X, holomorphic maps from or into %, etc. Also, we shall say that an analytic space I is algebraic or algebraisable if there is an algebraic variety Y such that Yho1 = X. Note further that an

algebraic variety X is complete if and only if

is compact.

(This is an easy consequence of Chow's lemma.)

We now recall the THEOREM of Cxow.

Let %be a complete algebraic variety and Y a

closed analytic subset of'ho1 Then Y is Zariski closed in X.

Chow proved the theorem for % = P , but the above version follows immediately from this and Chow's lemma.

An easy consequence is that if I and Y are complete algebraic varieties and f:'ho! -a Yho, is a holomorphic map, then f considered as a map from X to Y is an algebraic morphism. To prove this, let r in Iho, x Yo1 = (X x Y)ho1 be the graph off; it is a closed analytic subset, hence a closed algebraic subset of X x Y. For every (x, f(x))

C- r, the projection r -s $ induces a local homomorphism of the algebraic local rings 0z.x -s 0(z,f( )),r Let 0i = 64, 02 = Oc=.1c=»,r

Then I claim that 0, that the projection. r

0, is an isomorphism. First, use the fact % is propel and bijective : by Zariski's

Main Theorem, this means that 0, is a finite Ol-modnle. Let IF. = d(zf(=n.r.ho1 and O., = C x ho1 Since r r.1 is an analytic isomorphism, we get a diagram:

ABELIAN VARIETIES

34

I

f O1

30

02.

In particular, 01-a 02 is injective. If rn; = maximal ideal in 0;, then dividing by mj2, we get o11m2 4

A Ql}ml l

192lm2

22I

J2Im2 02

hence m2=m102 +m2. Therefore the G. module m2/m102 becomes

(0) after ®p2 02/m2: hence by Nakayama's lemma, m2 = m102. Therefore the 01-modulb 02/01 becomes (0) after ®ni 011m1: hence by Nakayama's lemma, 02 =01. This shows that r X is an algebraic isomorphism, hence that f is an algebraic morphism.

In particular, we see that a compact, complex space has at most One further fact that we will need is that if I is a complete algebraic variety, every meromorphic function f one algebraic structure?

t For non-compact complex spaces, this is quite false. For instance, Serre [Si] p. 108, has given the following example: for every 1-dimensional abelian variety X over C, there is a unique algebraic group G which is a non-trivial extension (as alg. group):

0 ---) C -) G

) % -) 0.

It is easily checked that if 0G isits (algebraic) structure sheaf, then r(Oo) C. But taking the universal covering of G, one checks that analytically, 0 = VJU, V=a 2-dimensional complex vector space, U = {n,w,+n,w2pnieZ} where w1,w2 are

a C-basis of V. If G. denotes the 1-dimonsional affine algebraic group, given by C- {0} under multiplication, it follows easily that G and Gm x Gm are two different algebraizations of the same analytic group!

35

ANALYTIC THEORY

on Xhal is a rational function on X, i.e. in the function field C(X); this can be proved similarly by considering the "graph" of f, and applying Chow's theorem.

Getting back to complex tori, we have COROLLARY. Let X = V/U be a 9-dimensional complex torus. The following are equivalent.

(1) X is the complex space associated to a projective algebraic variety,

(2) X is the complex space associated to any algebraic variety, (3)

there exist g algebraically independent meromorphic functions

on X, (4) .

there is a positive definite hermitian form H on V such that Im (H) is integral on U x U.

PROOF.

(1) n (2)

. (3) are obvious, (4) .(1) has been proved

in the theorem. It remains to prove (3) n (4). Let fl,... , fg be the independent meromorphic functions. Let D{ be the polar divisor of f{, D = ED{, and L the line bundle associated to D. Then L admits g + 1 sections °b'..., va such that whenever f, is regular, vi = f; ao. By the theorem of Appell-Humbert, L

L(H, a)

for some hermitian form H on V with ImH(Ux U) c Z, some a. Since L has sections. at all, by the discussion preceding the theorem, we know that H is positive semi-definite. Let V. be its degenerate

subspace, and Xp =V0/Vo n U the corresponding subtorus of X. Then the quotient torus X/Xo equals (V/ Vo)/image (U), and H is induced by a positive definite H on V IV, such that Im(H) is integral on the lattice U/U n V0. Therefore, by the theorem, X/Xc is a pro-

jective algebraic variety. But also, by the discussion earlier, we know that if a is any section of L, the zeroes of a are unions of cosets of X. Applying this to the sections E a,o{, it follows that all the analytic sets f{ = constant are unions of cosets of Yo, hence each f{ is induced by a meromorphic function Jt on X/Xo. Let C(X/Xc) be the function field of X/Xo; then

ABELIAN VARIETIES

as

g=tr. degcC(7,...,7,) < tr. degcC(X/Xo)=dim(X/Xo) 0, then let H(z, w) = 1 z.w. Im (w) One checks immediately that ImH = E has values E(1,1) _ E(w, w) = 0, E(w, 1) = - E(1, w) = 1, hence H satisfies the hypotheses of the theorem. Moreover, several projective embeddings of X are very well-known in the classical theory (cf. Hurwitz-Courant): for example, if P(z)

1a

2

+

I

(z

-n-mw)2

(n+ mw)21

1

n,m'(o,o) LL

is the Weierstrass p-function, then p is a meromorphic function, periodic with respect to 1, w, with double poles at the points n'+ mw. The map: z'--0. (1, $)(z), .4'(z)) (projective coordinates) induces an isomorphism of X with a plane cubic curve of the form Xo X2 = 4X1 + aX02X1 + bXo (for suitable a, b depending or w).

On the other hand, in dimensions > 2, it is easy to see that almost all tori are non-algebraic. Ti fact, we can check that on almost all

tori X, Pic(X) = Pico(X) or equivalently (by the theorem of Appell-Humbert) that there is no skew-symmetric E = V x V --)" R which is (a) integral on U x U, and (b) satisfies E(ix, iy) = E(x, y).

Let 8 = V/U, and put T = Homc(V, C), T = Homc at1(V, C) as before. Consider the map A2(Hom(U, Z)) I

C) 11

ASHomR( V, C) II

Ac(T©T) = (4T) 0 (T®T) ®(AZT)

ANALYTIC THEORY

37

We want to show that for almost all lattices U c V no element of

AZ(Hom(U, Z)) has image entirely in the middle factor TOY on the right. It will suffice to show that AZ (Hom (U, Z)) -a A2 T is injective. Bat Hom(U, Z) projects into a lattice in T, and for suitable choice of the lattice U in V, Im (Hom(U, Z)) is an arbitrary lattice in T. So the conclusions follow from LEMMA. Let V be a g-dimensional complex vector space. Then for almost all lattices U c V, the map AlU -+AZV m C is injective (hence all the maps AzU --AQV are injective, k < g).

If coordinates z1, ... , zo are introduced in V, and U is described by giving a basis (wi...... wig), 1 < i < 2g, then almost all can be inter-

preted to mean all g x 2g-tuples (cu0) not lying on a countable union of (g(2g) -- 1)-dimensional analytic subsets. We leave the proof of this lemma to the reader.

II ALGEBRAIC THEORY VIA VARIETIES Definition of abelian varieties. We now turn to the study of abelian varieties over an arbitrary algebraically closed field k. 4.

DEFINITION. An abelian variety X is a complete algebraic varietyt

over k with a group law m: X x X + X such that m and the inverse map are both morphisms of varieties.

Note that if k = C, then the underlying complex analytic space of an abelian variety is a compact complex analytic group, hence by the results of §1, it is a complex torus. When k =t- C, the first aim of the theory of abelian varieties is to show that an abelian variety has properties analogous to those enjoyed by a complex torus. Of

course, when char k = 0, many of these results can be proven by

reduction to the case k = C (Lefschetz's principle), but when char k 0 0, this is by no means possible. We shall want to answer the following basic questions.

Structure of X as an abstract group. We will show that X is a commutative and divisible group. We will also show that if nx denotes multiplication by n (n an integer > 0) on X, the kernel X. of nx, or what is the same, the group of elements x e X such that nx = 0, has the following structure QUESTION 1.

X,,

X,

(Z/n Z)'g if char k,f' n, (Z/pmZ)* if p = char k, m > 0,

where i can take every value in the range 0 < i < g = dim X. This integer i will be called the p-rank of X. QUESTION 2. Calculate the cohomology group H4(X, 011) (Up being the sheaf of p-forms on X). As in the classical cases we have canonical isomorphisms p

HQ(X,

Q

d') e A [H°(X,i1)1 ®k A[HI(X, Ox)],

tThis means, in particular, that it is irreducible.

ABELIAN VARIETIES

40

and

dim H1(X, Oz) =dim H°(X, 01) = g.

We also show that ir4(X) (in the algebraic sense, i.e. the projective limit of finite groups of unramified Galois coverings) is isomorphic to H(ZI)E' in char 0 and to fl(Z,)S x Zi, in char p.

f top

t

More precisely, we shall show that if Y

+ X is any morphism

such that a finite group G acts freely on Y in such a way that X becomes the quotient of Y for this action, there is an integer n > 0 and a commutative diagram

Further, Y carries a structure of abelian variety such that f and 9 are homomorphisms. QuESTrox 3,

The structure of Pic X.

We will show that there is an exact sequence

0 -+ Pic°X -) Pic X --) NS(X) -+ 0 where Pie°X has a natural structure of an abelian variety, and NS(X) is a finitely generated free abelian group, whose rank p is called the base number of X.

We will also try to find analogues of the classical description of NS(X) by Riemann forms E. Related questions are: (a) for a pair of abelian varieties X, Y, show that Hom(I, Y) is a free abelian group on a finite number of generators, (here Hom means the set of maps which are both morphisms of varieties and homomorphisms of groups), and (b) give a matricial representation of this group of homomorphisms (in the

ALGEBRAIC THEORY VIA VARIETIES

41

classical case, we have the representation induced in Hom(H1(X), H1(Y))). QUESTIOx 4. Characterize ample line bundles. More generally, compute the cohomology groups of arbitrary line bundles, and in

particular the dimension of the space of sections-this is the Riemann-Roch problem.

(i) We start with the observation that an abelian variety is everywhere non-singular. In fact, there has to exist a non-singular

point x0 a X, and for x e X, the translation morphism T(,,, X -r X, given by T(Z_,)(y) = x.xo l.y, is an automorphism of X carrying xo to x, so that x is again a non-singular point of X. (ii) As a group, X is commutative. We give two proofs, one here

and the second a little later. The first proof generalizes the proof we gave in the classical case. We consider not only the adjoint representation of X in the tangent space at the identity e, or in the space of differentials at e, but in each of the spaces (OS,/2g,) where Og,, is the local ring of X at e and 97tg,, its maximal ideal. For x e X, let CC: X -+ X be defined by CZ(y) = x y x-1, so that CZ(e) = e. Then C. induces an automorphism O',, of the vector space Ox,,/9 , deduced by passage to quotient from the automorphism Cz : Ox,, -> t!1 of the local ring. This induces a set theoretic map y: X a Aut(OS,,/TZ4 X's , x Q* and if we put on the latter group the natural structure

of an algebraic variety (viz. that induced from the inclusion Aut(Ox,,/lUl s) c End(Oz.,f mg,), the last being a finite-dimensional

vector space over k), one checks easily that this is a morphism of varieties. Since the latter is an affine variety and X is complete and

connected, y must be a constant map! Since y(e) is the identity, we see that Cy is the identity for all x e X and n > 0. But since n I11%,, = (0) this means that Cx : OZ, -+ Oz,, is the identity, so that

C. reduces to the identity in a neighbourhood of a in X. Since X is

irreducible, CZ is the identity on X for every z, that is, X is commutative. From now on, we write the group law in X additively. Moreover,

we will use the following notations: for x e X, we denote by

42

ABELIAN VARIETIES

T.: X -* X the translation morphism T.(y) = x + y; and the map z' > n.x will be denoted by na. If T = Tx,0 is the tangent space at 0 to X, 00 is the dual space

(iii)

T,*,, of differentials, then there is a natural isomorphism

n®®k O$

12x,

where £I$ is the sheaf of regular 1 farnss on I.. One defines this mapping as follows. To each 6 e Q., consider the 1-form o w on X defined

by (cue). = T'-x(9), that is, the unique translation invariant 1-form on X whose value at 0 is 6. It is checked easily that we is a regular 1-form on X. Thus, we have a natural homomorphism as above. Since at any point x, T*_s induces an isomorphism of the space of differentials at x onto the space L20, it follows that the above homomorphisms of sheaves induces an isomorphism of fibers at x reduced

modulo the maximal ideal Tlx,y at x. It follows by Nakayama's lemma that it is an isomorphism of sheaves.

Since X is complete and connected and H°(I, OX) = k, it follows from the above isomorphism that the everywhere regular forms on X are precisely the invariant forms. For every n not divisible by the characteristic p of k, the endomorphism n$ is surjective. (iv)

For the proof, we make the following observation. The tangent

space TX a. (0, 0) to IXX at (0, 0) splits canonically into a direct sum Tl O T2, where Ti (resp. T2) is the isomorphic image of

i T%,0 under the immersion 1 -+X x I given by x A-°r (x, 0)

(resp. X r (0, x) ). Identifying T; with T%0 = T by these T of the addition map m: I x X -> X, m(x, y) = x + y, is nothing isomorphisms, we note that the differential d(m): T ® T

but addition of components: d(m) (t1, t2) = tl + t2. In fact, it is sufficient to check this (by linearity) on the two summands T of T ® T, and for these, it follows from the fact that the composites

X

o) X x X -+ X and X i) I x I -+ X are the identity.

ALGEBRAIC THEORY VIA VARIETIES

43

It follows by induction on n that for any n > 0 (and hence also for n < 0), (dnx)o is multiplication by n. Thus, if p Pa, dnx is an isomorphism. If nx were not surjective, by the dimension theorem,

dime n%' (0) > 0, and we can therefore find a non-zero-9 e T

tangent to nx'(0) at 0. But then, we would have dn$(t) = 0 (since nx'(0) is mapped into the single point 0 by n1), which is a contradiction.

The following lemma, besides having other important applications, gives a second proof of the commutativity of X. RIGIDITY LEMMA. (Form I.)

Let X be a complete variety, Yand Z

any varieties, and f: $ x Y -} Z a morphism, such that for some yo a Y, f (X x {yq}) is a single point zq of Z. Then there is a morphism

g: Y a Z such that if p2: X X Y -e Yis the projection, f = gop2.

Choose any point xq a X, and define g : Y -- Z by g(y) = f(xq, y). Since X x Y is a variety, to show that f = gop2, PROOF.

it is sufficient to show that these morphisms coincide on some open subset of X x Y. Let U be an affine open neighbourhood of zq in Z,

F = Z - U, and G = p2 (f-'(F)); then G is closed in Y since X is complete and hence p2 is a closed map. Further yo 0 G since f (X x {yq}) = {zq}. Therefore Y- G = V is a non-empty open subset

of Y. For each y e V, the complete variety X x {y} gets mapped by f into the affine variety U, and hence to a single point of U. But this means that for any x e X, y e V, f(x, y) = f(x0, it) = gop2(x, y)1

and this proves our assertion. COROLLARY 1.

If X and Y are abelian varieties and f: X .). Y

is any morphism, f (x) = h(x) + a where h is a homomorphism of X into Y and a e Y.

Replacing f by f - f(0), we may assume f(0) = 0 and we have to show under this assumption that f is a homomorphism. PROOF.

--

Y defined by O(x, y) =f(x + y) Consider the morphism X x X -1(y) - f(x). Then q(X x {0}) =0([O} x I) = 0, so that it follows

by the above lemma that 0 - 0 on X x X, or what is the same, that f is a homomorphism.

44

ABELIAN VARIETIES

Note that in the proof of the above corollary, despite the additive notation used, no use of the commutativity of X was made. Thus, we may use it to give a second proof of the commutativity of X. COROLLARY 2. X is a commutative group.

In fact, the morphism of X into itself mapping each element onto its inverse is a homomorphism by Corollary 1. Hence, for (xy)- 1 = x ly-1= y Ix-1, and X is commutative.

x, ye X,

COROLLARY 3.

Let X be an abelian variety (with base point 0).

Then on the category of complete varieties with base point, the funetor

) Hom(S, X) (where Hom denotes morphisms preserving base point) is linear; that is, for S, T in this category, the natural map S&

Hom (S, X) x Hom (T, X) ---) Horn (S x T, X) given by (f, g) . -r Is, h(s,t) =f(s) + g(t) is a bijection. PROOF. That this map is injective follows by fixing s and tin turn to be the base points so and to of S and T, respectively, in the equation h(s, t) =f(8) + g(t). Next, take any Is e Hom (S x T, X), and put f(s) = h(s, to), g(t) = h(so, t), k(s, t) = h(s, t) - f(s) - g(t). Then k(S X {to}) = k({so} x T) = 0, and it follows from the rigidity lemma that Is = 0.

APPENDIX TO §4

We want to prove the following result. TxEoREM.

Let X be a complete variety, e e X a point, and

m:XXX

)X

a morphism such that m(x, e) = m(e, x) = x for all x e X. Then X is an abelian variety with group law m and identity e.

PROOF. We shall denote m(x, y) simply by xy. Introduce the morphism

#: I X X ---> X X X 0: (x, y) = (xy, y)

ALGEBRAIC THEORY VIA VARIETIES

45

Then 0-1 (e, e) = {(e, e)}, so that by the dimension theorem, dim (Image ) = dim (X X X). Since X x X is complete, this implies that 0 is surjective. In particular, given x e X, there is an z' E X with x' x = e. Thus, if r, = ((x, y) E S x X I xy = e}, and p;(i =1, 2) is the iW` projection of X x X, p5(I") = X. Choose an irreducible component I' of r' with p2(P) = X. Note that dim r > dim X.

If p; =pi I P, pi 1(e) = {(e, e)}, so that again by the dimension theorem, dim (Image pi) =dimX. Since r is complete, this implies that p' is surjective.

Let ¢: P x X X be defined by 0((x', x), y) = x'(xy). Then ¢(I' x {e}) ={e}, so by the rigidity lemma, 0((x', x), y) =0((e, e), y)) =y, that is,

(4) x'(xy)=y, y(x',x)EP, yEX. In particular, if (x', x) e P, then x'(x.x') = x'. Choose an (x", x') E P, and multiply the last equation on the left by x", to obtain x"(x'(xx')) = x"z But x"x' =e, and by (1), x"(x'(xx'))=xx'. Therefore if (x',x) EP, then not only is x'x = e, but also xx' = e.

Let x: P x X x X -- X be the map x((x', x), y, z) = x((x'. y)z). Since X(1' x {e) x {e}) = e, by the rigidity lemma, x((x' y)z) = e((ey)z) = yz.

Multiplying on the left by x' and using (1), we get (x'y) z = x (x((x'y)z)) = x'(yz).

Since x' is arbitrary in X (p,' being surjective), this shows that multiplication is associative. Thus X is a group with group law M. In particular, for any xo a X, the translation x i -- s xo x is an automor-

phism of X as a variety, and we deduce that X is non-singular. This also shows that +G is bijective. But also the tangent map of at (e, e) is an isomorphism since (x, e) = (x, e) and #(e, x) = (x, x). Thus y cannot be inseparable, so that by Zariski's Main Theorem, i is an isomorphism. The inverse of+J Y is a proper morphism of locally noetherian preschemes and.F a coherent sheaf of 0.y-modules on X, for all p > 0 the direct image sheaves RP f k (.y) are coherent sheaves of Dr-modules. THEOREM.

We recall the following definition. If f: X--Y is a morphism of preschemes and -F a quasicoherent sheaf on X, F is said to be flat

over Y or f -flat if for each x e X, 9f(for its natural structure of D),nzYmodule) is D Y,fc=j flat. It is easily shown that this condition is equivalent to requiring that for U cX, V c Y with Uand V affine open, and f(U) c V, F(U) is a flat & (V)-module.

The main result of this section is the following THEOREM. Let f: X -* Y be a proper morphism of noetherian schemes with Y = Spec A affine, and F a coherent sheaf on X, flat over Y. There is a finite complex K': 0 -+ K° i Kl ... -> K" -> 0 of finitely. generated projective A-modules and an isomorphism of

functore

H'(X x y Spec B, lw' (gA B) y H'(K'(&d B), (p > 0) on the category of A-algebras B. PROOF.

Choose a finite affine covering stl = {U;}tE1 of X by affine

open subsets. Then the Cech complex C' = C' (W, F) = O C'(8f, .F) of alternating Cech cochains on $C with coefficients in .°F is a finite complex of A-flat modules, whose cohomologies are isomorphic to the cohomology groups B'(X, .F).

47

ALGEBRAIC THEORY VIA VARIETIES

Moreover, for all A-algebras B, {U; x r Spec B) is an affine cover-

ing of X xY Spec (B), and C1(%, .f) ®d B is the module of Cech p-cochains of ®A B for this covering. Therefore HP(X x y Spec B, . " (&A B) Z HP(C' (&A B)

for all B, and, in fact, functorially in B. We need the following basic lemma LEMMA 1. Let C' be a complex of A-modules (A any noetherian ring) such that the Hi(C) are finitely generated A-modules and such

(0) only if 0 < p < n. Then there exists a complex K' of finitely generated A-modules such that K. 0 (0) only if 0 w p < n and KP is free if 1 < p < n and a homomorphism of complexes that CD

P(C), all i. K ' --. C' such that 0 induces isomorphisms H`(K') Moreover if the CP are A -flat, then K° will be A flat too. PROOF. We define, by descending induction on m, diagrams: am+l

am

} Km+1

Km

> Km+2 --) ...

j,#m+1

`Ym

j,cm+s

W Cm-1

Cm

am

Cm+1

am+1

s Cm+2

Put K9 = 0 for p > n. Suppose we have defined (KP, p> m + 1 such that the following conditions hold:

aP) for

0)

a'0P =0P+1aP, (p > m + 1).

(iii)

The LP induces isomorphisms in cohomology BQ(K') HQ(C') for q>m + 2, and a surjection ker a0+1 -* Hm+1(C*)_

(iv)

The KP are A-free and finitely generated, (p > m + 1).

(ii) aP+1oaP=0, (p>m+1).

We then construct K", a"` and 0,,, so as to satisfy (i)-(iii) with m + 1 replaced by m.

Suppose first that m > 0. Let Bin}1 be the kernel of the homoSince Bm+1 is finitely generated

morphism ker a"'+1

48

ABELIAN VARIETIES

over A (A being noetherian), we can find a finitely generated free module K'" and asurjection a': K'm -> R"I+1. Further, since H'"(C')

is a finitely generated A-module, we can find a surjection K'O A

--*- H'(C') with K'Infinitely generated and free. Let µ: K"">Z'(C')

be any lift of A, and 4 K""' > C"` the composite of µ with the inclusion Z"'(C') > C"`. We then put K"' = K'" Q+ K"'", and define a"': Km > K'"}1 by putting it equal to zero on K'O and equal to a'

on K'"". Since ¢m+l"c aCm, we can find 0m: K'"' >C"' such that =¢m+loa'. We then define 0m: K"' -> C"' as being equal to 0;,, on K''" and ¢m on K"'". The conditions (i)-(iii) are evidently fulfilled with m instead of m + 1.

Suppose then that m = - 1, that is, that {KP, aP} have been defined for p > 0 satisfying (i)-(iii). We then replace K° by K°/ker a° n Ker 00, and we take 00: K° - CO and a°: K° -> K1 to be the induced mappings. Putting KP = 0 for p < 0, we get a complex

0>K°>Kl>K2->K3->... K"-+ 0 and a homomorphism 0: K' > C' which by construction induces isomorphisins in cohomology. We have only to check that K° is A-flat when all the OP are A-flat. Consider the 'mapping cylinder' complex L defined by L" = K" p 0-1 for p e Z, and a: LP > LP+1 defined by a(x, 0) = (ax, ,(x)), a(0, y) = (0, - ay). If C" is the complex obtained from C' by shifting degrees by one (and making a sign change in a), C'2' = Ca-1, we have an exact sequence..of complexes 0 > C" -a V > K' -+ 0, and hence an exact cohomology sequence

and one sees from the definition that the cohomology maps H"(K') > Hp}1(C") e HP(C') are the ones induced by 4: K' > C'. Since these are all isomorphisms, HP(L') = (0) for all p e Z. But

ALGEBRAIC THEORY VIA VARIETIES

49

then 0 -> K° = L° - Ll -a L2 -+ ... -> L"+x 3 0 is exact and the modules V are flat for i > 1, hence K° is A-flat.

Applying the lemma to our case, we have a complex K', and a homomorphism K' -+ C' such that

HP(K') . HP(C') = R'(X,,), all p. Note that K° is A-projective, since it is A-fiat and finitely generated

over a noetherian A. It remains to check that for all A-algebras B, HP(K'(& AB) -+ HP(C'® AB) is an isomorphism too. This is a consequence of Let C', K' be any finite complexes of flat A-modules, and let C' -* K' be a homomorphism of complexes inducing isomorphisms HP(C') HP(K') for all p. Then for every A-algebra B, LE1,nSA 2.

the maps HP(C' ®A B) --) HP(K'(gA B) are isomorphisms. PROOF_ Construct the `mapping cylinder' L' exactly as in the proof of Lemma 1. As before, we see that L' is an exact finite complex of flat A-modules. Then it is easy to see that all the

modules ZP = Ker(LP

aP

> LP+') are flat too, hence

0) Z'.-+LP-*ZP+i- -0 is a short exact sequence of flat A-modules. Therefore

0-+ZP(&AB--) LP®A B

-ZP+i®AB-+0

is exact, from which it follows that L' ®A B is exact. But now L' ®A B is the mapping cylinder of the map K' ®A B -+ C' ®A B.

So using the cohomology sequence in reverse, it follows that HP(K' (gA B) -a HP(C' ®A B) are isomorphisms.

For any morphism f: X Y and y c- Y, we denote by X, the fiber over y off (i. e., the fiber product K X Y Spec k(y), considered as a scheme over k(y)), and for, quasi-coherent on X, we denote by

the sheaf,®o k(y) on X, Y We have then the following important corollary.

ABELIAN VARIETIES

50

Let X, Y, f and .° be as in the theorem (except that Y need not be affine). Then we have: COROLLARY.

(a)

For each p > 0, the function Y --)- Z defined by ya

(b)

9 dimk(v) H1'(Xv, Fv) is upper semicontinuous on Y.

The function Y-+ Z defined by (- 1)" dimk(V) H'(X5, Fv)

y - X(Fv) = p-0

is locally constant on Y.

PRooF. The problem being local on Y, we may assume Y affine. Let K' be a complex as in the proposition; by further localization, we may assume K'to be a free complex. Denote by d": K" K"+l the coboundary of K. We then have p(Kv, ` v) = dimkcv>[ker (d5 ®a k(y))]- dimk(Y)[Im(d"-1 ®A k(y))] = dimk(5)[K"®k(y)] - dimk(v)[Im(dp(& k(y))] -dimk(v)[Im(d"-1® Ic(y))]. (*)

The first term being constant on Y, (b) follows on taking alternating

sum of (5) over all p. We assert that for any p > 0, the function p, (y) = dimk(v) [Im(d"(& k(y))] is lower semi-continuous on Y. In

fact, if r is any integer > 0, and dp : ArK5 i AAK"+1 is the map induced by d",

{yeYI pp(y) HP(X,

,,)

is an isomorphism. If these conditions are fulfilled, we have further that HP-1(Xy, Fr,)

RP-' f* (F) ®aYk(y) is an isomorphism for all y e Y.

PROOF. Again assume Y affine, S' as in the proposition. (ii)

(i) is obvious. To prove (i)

LEMMA 1.

If Y is reduced and..

.

(ii), we need two lemmas.

a coherent sheaf on Y such that

dims[ ° ®0Yk(y)] = r, all y e Y, then JF is locally free of rank

ronY. PROOF. For any y e Y, let al,..., a, e. y lift generators of JFy® k(y). Since o,..., a, are extendable to sections in a neighborhood of y, we have a homomorphism a: OYr 1 v >F 1Y defined in a neighborhood V of y. Then a is surjective on the stalks at y, by Nakayama's lemma, so coker(a) is zero at y and hence in a neighbor-

hood of y. Thus, we may assume a to be surjective. Then by assumption, for every y' a V, the map a® k( y'): k(y )`-.FP. ®p v,k(y')

is an isomorphism. Thus, if 0 is the kernel of a, we have Dy c M., 0y for each y' a V. Since Y is reduced, this means that 0 = (0). Thus a is an isomorphism.

We apply this in the following LEMMA 2.

Let Y be a reduced, noetherian affine scheme, and let .°F

0

)D

ABELIAN VARIETIES

52

a homomorphism of coherent locally free Cy-sheaves. dim,,>[Im(0® k(y))] is locally constant, then there are splittings: be

If

.F=`IF 1®F2 C = D I1®02 E) (0), Im(o) c 01, and 0:.x'2 i £1 is an isomor-

such that phism, i.e.

0

[0

isom. 1 0

JI

PROOF. By Lemma 1, is locally free. If Y = Spec (A), M=11(Y, F), N =P(Y, ID), then this means that NJ/(M) is A-projective. Therefore N splits into the direct sum of ¢(M) and a second submodule isomorphic to N/#(M). Or, in sheaves, ;J = 0 1®02, where

D1= Im(#). Moreover, this shows that ¢(M) is A-projective, too, so M splits into the direct sum of Ker(#) and a second submodule isomorphic to ¢(M). Or, in sheaves, . = J® .°F2, where (0), 0: -'W" Z-)- k),

Now assume (i) holds. Let X be the complex given by the theorem. As in the proof of Corollary 1, dim[Im(dp-1 (& k(y))] and dim[Im(dp (& k(y))] are locally constant. By Lemma 2, applied

first to dp: Kr -s Xp+l, and second to dp_1: K"-1 -+ Ker(d,), we get splittings into projective modules:

Z,-1®K_1 BP®H,®K, Bp+1®K,+ K,

Kp+1

B, is an isomorphism, B, ® H, = Ker(d,), and d,: 1, --). Bp+1 is an isomorphism. It where Z,_1 = Ker(dp_1), dp_1: IL9_i follows immediately that

H'(K*0,B)=Hp®AB=H'(K) ®A B, all B and H'-'(K ®A B) = Zs-1 ®A B/Im(d,_5 (& B) = H"-'(K' ) ®A B, all B. This proves (ii).

ALGEBRAIC THEORY VIA VARIETIES COROLLARY 3.

53

Let X, Y, f and .' be as above (unlike Corollary

2, Y need not be reduced). Assume for some p that H'(X5, Ft,) _ (0),

all y e Y. Then the natural map RP-' f* (JF) ®ppk(y) - HP-1(X1 F,,) is an isomorphism for all y e Y.

PROOF. Again assume Y = Spec (A), K' as in the theorem. For ally a Y, we know that dP-1

KP-1 (D k(y)

) K' (9 k(y)

d9

) K'+1 ® k(y)

is exact. Split the vector space K' ® k(y) into W1 $ W2, where W1 = Image of K2'-1 ® k(y), and Wti is mapped injectively to K'+ 1® k(y). To prove the corollary at y, we can replace A by any localization Af, (f e A, f (y) 0 0). If we do this for a suitable f, we may assume that KP itself splits into a direct sum of free modules W1® W2 such that (a) W, = W; ® k(y), and (b) W1 c Im(dP-1). To do this, just lift a basis of W1 to any elements in the image of d'-1, and lift a basis of W2 arbitrarily. But then since W1® k(y) - K't 10 k(y) is injective, it follows that Ws -> KPt 1 is also injective if A is replaced again by a suitable localization Af. But

then Im(dP-1) n W2 = (0), hence W1 = Im(d'-1). Since W1 is a projective module the surjection K'-1 -a W1 --> 0 splits, and KP-1 =

Ker(d'-1)® W1. It follows that we have exact sequences

;P-2

Ker(d'-1) > H'-1(X,.5F)

K'-2 ® k(y) -+ Ker(d'-1) ® k(y) --- * HP-1(X,,,

0 0.

Therefore H'-1(Xv, F,) = H'-1(X, SP) ® k(y) as required. COROLLARY 4. Let X, Y, and °.F be as above. If .R f*(F) = (0) for k > ko, then Hk(X,,, F,,) = (0) for all y e Y, and for k> ko.

PROOF.

Use Corollary 3 and decreasing induction on k0.

COROLLARY 5.

A-algebra,

Let I, Y, f and.Fbe as above. Then if B is a flat

ABELIAR VARIETIES

54

HP(.X X . Spec B, F (&AB) a HP(X, F) ®4B. PROOF. This follows immediately, from the fact that for B flat over A, and any complex K',

HP(K" ®4B) = HP(KC) ®AB. COROLLARY 6. (Seesaw Theorem -provisional form). Let g be a complete variety, T any variety and L a line bundle on X x T. Then the set

T1 = {t c- T I L I x ), fl is trivial on .X x (t))

is closed in T, and if on % X T1, pa: I X T1 -+ T1 is the projection, then L I a x Tl = p,M for some line bundle M on T1.

PROOF. We first make the remark that a line bundle M on a complete variety I is trivial if and only if dim H°(g, _M) > 0 and dim H°(%, M- 1) > 0 where M denotes the sheaf of sections of

M. In fact, the necessity of these conditions is clear. Suppose conversely that they hold. The first implies the existence of a a

non-zero homomorphism Og-± M, and the second implies a non-zero homomorphism Ox

hence on dualizing, a non-zero

T

homomorphism M -) Ox. Hence r(a(l)) is a non-zero section of OX, and since % is complete and connected, r(a(1)) is a non-zero scalar. This implies that r o a is an isomorphism, hence a and r are isomorphisms. It follows that T1 is the set of points tof T such that dim H°(X X {t}, L I .X x {t}) > 0 and dim H°(X x {t}, L-1 I % x {t}) > 0, and it follows

from Corollary 1 that T1 is closed. Replacing T by Tl (so T is now merely a reduced scheme of finite type over k) and L by its restriction

to %x T1, we may assume that L I I x {t} is trivial for each t e T. Hence dim H°(.X x {t}, L I % x {t}) = 1 for all t e T, so that by Corollary 2, p2,, (L) = M is an invertible sheaf on T and 900T(kt)- L is an isomorphism. Since M is the sheaf of sections of M, then p2M = L. 6. The theorem of the cube: I THEOREM. Let X, Y be complete varieties, Z any variety and x0, yo and zo base points on X, Y, and Z, respectively. If L is any line bundle on X x Y x Z whose restrictions to each of {zo} x Y x Z, X x {y0} x Z and X x Y X (zo} are trivial, L is trivial.

RErrARS. Let T be a contravariant funetor on the category of complete varieties into the category Ab of abelian groups. Let X0,..., Xn be any system of complete varieties, x° a base point of Xi, and let 7q: 10 x ... x X. -+ X. x... x X x ... x XI (Xi indicating the omission of the i-th factor Xi) be the projection map, and

a;: X0x...xXix...xIn-+X0 X...xX,,the `inclusion'definedby 0'&0,..., xi-i> xi+i,...,

(xo,..., xi-l, 4, xi+a,..., xn).

Consider the homomorphisms

aT:flT(XOX...xI,X...xX.)-+T(X0x...xX.), i-0

$:T(Xox...XXX)

n

T(XXX...xjiX...XX.)

defined by c

n

C

tt

all(eo,... , Sn) _

n

{ (Si), MT(h) _ (a (+]), o (rl), ... , Qn 0

One then proves by an easy induction on n that we have a natural splitting T (X0 x ... x Xn) = Im a ® Ker P. The functor T is said to be of order n (linear if n=1, quadratic if n=2, etc.) if a is surjective,

or equivalently )3 is injective. (Note that the definition of a is independent of base points.)

Thus, the above theorem (when Z is also assumed complete) may be paraphrased as saying that the functor Pie X is a quadratic functor on the category of complete varieties.

ABELTAX VARTETTES

56

Now, if Ti (1 < i < 3) are contravariant funetors on complete

-± T. and T2 9 *. T3 are natural transformations such that Tl -* T2 g > T3 is an exact sequvarieties into A.b and Ti

ence, and if Tl and T3 are of order is, so is T2, as follows from the exactness of

O=Kerp4,(X0,..., .K) -+Kerf'(XO,..., Thus we get a proof of the theorem of the cube when the base field is C by observing that we have an exact sequence

H'(X, 0)-.H'(X, 0*)_*H2(X,Z),

functorial in X, and H'(X, 0) is linear (hence quadratic) and H2(X, Z) is quadratic in X, by Kenneth formulasPROOF OF THE T1EOREM (following Weil and Murre). By the `Seesaw theorem', it is sufficient to prove that for every (x, z)eX x Z,

the restriction of L to {x} x Y x {z} is trivial, since it is already given that L restricted to X x {yo} x Z is trivial. The following enables us to reduce the proof of the theorem to the case when X is a complete non-singular curve. LEMMA.

Let X be any variety and x0, x, a X.

Then there is

an irreducible curve C on X containiny xo and x1.

PROOF. We assume that dim X > 1. By the lemma of Chow, we may assume X projective. Moreover, by induction on dim X, it is sufficient to find a subvariety Y of codimension one in X containing

X birational, with X projective and dim f-'(x;) > 1. (In fact, if h is any meromorphic function on X with indeterminacies at xa and x, X can be taken to be the closure of the graph of h in X X P'). If X' c Pi'' is an imbedding, there is a hyperplane H of PN such that H n X'= Y' is irreducible, xo and x1. We can find an X'

by a theorem of Bertini, and H n f-'(x;) 0 0 since dim f-'(xi) 1. Then Y = f(Y') is irreducible in X and contains xo and x1, and the lemma is proved.

ALGEBRAIC THEORY VIA VARIETIES

57

Resuming the proof of the theorem, we can find for any x e X an irreducible complete curve C, in X joining x0 to x. Let x: C --> Cl be the normalization of C1 and it': C x Y x Z --> X x Y x Z the induced

map. The hypotheses of the theorem are clearly fulfilled for the bundle a'*(L) on C X Y x Z (with X replaced by C and x° by any point of C lying over x0), and it is sufficient to prove the triviality

of this bundle, since it would then follow that L restricted to {x} x Y x {z} is trivial for any x e X and z e Z.

Thus, we assume X to be a complete non-singular curve, and it is

even sufficient to show the existence of a non-void open subset Z' of Z such that L restricted to X X Y x 2' is trivial, since we would then have proved the triviality of L I X x YX{z} for z e Z', and it would follow by continuity that this holds for all z e Z.

Let l1 be the sheaf of regular 1-forms on X and let g-= dine H°(X, S2') be the genus of X. We can clearly find g points P1,..., P1 on X such that if D = E P;, dim H°(X, 01® Ox(- D))= 0. 1

Denoting by pl the first projection X x Y x Z -3 X, let L' be the line bundle L' = L ®pi (Lx(D)) (where Lx(D) is the line bundle associated to Ox(D)) on X x Y x Z, and for any (y, z) e Y x Z, let L'(5 z) be the restriction of L' to X x {y} x {z}. Since L'(1,,) = Lx(D),

we have dim HP(X, L'(v xo)) = dim H°(X, Ell ® Ox(- D)) = 0 by Riemann-Roch so that the closed set F = {(y, z) e Y X Z I dim HI (X, L'(1,,)) > 1} of Y x Z does not encounter Y X {z°}. But Y

being complete, we can find Z' open in Z and containing z° such

that Y x Z' n F = 0, so that by restricting ourselves to Z', we may assume H'(X, L'(1,z)) = 0 for all (y,z) e Y X Z. But this means that for all (y,z) e Y x Z,

dimes°(X,L'(,,z))=XL'(,,z))=X( In view of Corollary 2 to the proposition ifp23 : X X Y X Za Y X Z

is the projection, p23*L') is an invertible sheaf on Y x Z of rank

one and for any (y, z), the natural map p28* (L') ®k(y,z) -* H°(X,L'(V.,)) is an isomorphism. Let U be any open subset of Y X Z on which p23* (L') is trivial, and ac a r(U,pR3* (L')) = r(p231(U),.U) a

ABELIAN VARIETIES

58

generating section. Let Da be the divisor of zeros of au in p23 -1(U).

Since for U,U' open in Y x Z, a and vu. differ on U n U' by a nowhere vanishing function, we have b, n p 23(U n U') = DU' n p -23'(U n U'), so that we have an effective divisor D on X x Y x Z

-1(U) = DU. For each (y,z) e Y x Z, the restriction such that D ; p23 of ,5 to X x {y} x {z} is the divisor of zeros of a non-zero section of In particular, D restricted to X x {y} x {z°} and X x {y°} x {z} for any y e Y, z E Z must coincide with D = Y- Pi_ Hence, if

P EX, P P. (i = 1, ... , g), the restriction of b to {P} x Y x Z has a support S not meeting {P} x Y x {z°} or {P} x {y°} x Z. The projection T of S on Z is therefore a proper closed subset of Z, and since S is pure of codimension one in {P} x Y x Z, S must be of the form U {P} x Y x T;, Ts closed and of codimension 1 in Z.

But since S n {P} x {y0} x Z=O, it follows that S= 0, that is, the

support of D does not meet {P} x Y x Z for P e X, P # Pi. Hence D must be of the form

Zani({Pi}

i

x Y x Z), and restricting

0

to X X {y0} x {z0}, we see that b =E ({Pi} x Y x Z). Hence for any

(y,z) E Y x Z, L(v.z) is the lint bundle L1(D), and therefore L restricted to Ix {y} x {z} is trivial.

With X, Y, and Z as in the proposition, any

COROLLARY 1.

line bundle on X x Y x Z is isomorphic to p12(L) ® px3(M) ® pQ3(P) where pii is the projection of X x Y x Z onto the product of the i

and ja' factors, and L,M,P are line bundles on X x Y, X x Z, and Y x Z, respectively. PROOF. This is a consequence of the remark preceding the proof of the theorem. COROLLARY 2.

Let X be any variety, Y an abelian variety, and

f,g,h: X - . Y morphisms. Then for all L e Pic (Y), we have

ALGEBRAIC THEORY VIA VARIETIES

59

(f+g+h)*L= (f+9)*L® (f+h)*L® (g+h)*L® ®g*L-1 ®h*L-1

f*L-1

PROOF. Let pi : Yx Y x Y --> Y be the projection onto the ia' factor, put m,, = pi + pl : Y x Y x Y -> Y and m..- p1 + p2 + p3:

YxYxY -+Y. Consider the line bundle M= m*L ® m1zL-1® m13L-1® mzsL-1®p' L ® p*2L ® psL on Y x Yx Y. If q: Y x Y-r Y x Y x Y is the map q (y,y') = (O,y,y'), we have

q*M =n* L® qiL-1® qzL-1® n*L-1® O*L® qiL® qEL where 0, g1,g2,n: Y x Y -> Y are the 0 map, the projections, and addition. Therefore q*M is trivial. By symmetry, M is trivial on Y x (0) x Y and Y x Y x (0) too. By the theorem, M must be trivial on Y x Y x Y. Pulling back M by the map (f,g,h): X Y x Y x Y, the result follows. COROLLARY 3.

If X is an abelian variety, and n e Z, then for all

line bundles L,

olm g 1®(-1x)*L 42-

%)

n4L=L

11

PROOF. By Corollary 2 with f = (n+l)a, g=1g and h = (-1)a, it follows that the "second difference",

(n+2)*L® (n+1) *L-2® ngL = 1*a(L)®(-11)*L, hence for some line bundles M1, M2, we must have n(n-1)

n,*L= IL®(-ls)*L]

2 ®Mi®M2.

Putting n = 0 shows that M2 is trivial, and putting n=1 shows M1=L. COROLLARY 4.

(Theorem of the square.) For all line bundles L

x,y a %,

Tx+b L0L=T.L®T'L.

60

ABELIAN VARIETIES

Therefore if ,¢L(x) = isom. class of T, L® L-1 in Pic(X), ,, is a homomorphism from X to Pic(X).

PROOF. Apply Corollary 2 with X = Y, f and g constant maps with images x, y respectively, and h = identity.

In terms of divisors, Corollary 4 asserts that for any divisor D

on X, and x,yaX, (where - means linear equivalence).

In the rest of this book, we will always keep the notation #L for this very important map. Note that oL,®L' = 'Ll + cL} by(& in Pic(X)), (a)

(b)

OT4

(+ standing for the group law induced

#L

DEFrNITION.

K(L) = Ker(¢L) _ {xe X I TzL : L}.

PROPOSITION_ K(L) is a Zariski-closed subgroup of X. PROOF.

Apply the Seesaw Theorem to the line bundle m*L®p2L-1 on XxX (m: X xX iX being addition). It follows

that the set of x e X such that m*L ® p2*L- 1 is trivial on {x} x X is Zariski closed. But m*L®p,L-1 I{z}X$ = T.L® L-1, so this

set is K(L). APPLICATION 1. Let D he an effective divisor on an abelian variety X and L = L(D) the associated line bundle. The following conditions are equivalent.

(i)

The subgroup H = {xeX jTz*(D) = D} of X is finite (equality of divisors, not divisor classes).

(ii)

K(L) is finite.

(iii)

The linear system 12D I has no base points, and defines a finite morphism X

(iv)

L is ample on X.

ALGEBRAIC THEORY VIA VARIETIES

61

Paoov. The implication (iii) .(iv) is a general fact (EGA Ch. III,

(2.6.1) or (4.4.2)). We show next that (iv) * (ii). If K(L) is not finite, let Y be the connected component of 0 of K(L), so that Y is an abelian variety of positive dimension, and the restriction Lr of L to Y is ample on Y. Further, T*,(L,) = L,. for ally e Y. Hence, by the Seesaw theorem ifm: Y x Y-+ Y is the addition and A: Y x Y Y the projections, the line bundle m*(L7) ®pi(Lpl)®pa(LY1) is trivial on Y x Y. Pulling back by the morphism Y -a Y X Y, y

i--p (y, -y) gives us that Lr®( - lp)*(Ly) is trivial on Y. But Lf is ample, and so is (- lr)*(Ly) since - lp is an automorphism of Y, so that Lr® ( - lr)*(Ly) is again ample. This is a contradiction since dim Y > 0, which proves that (iv) n- (ii). The implication (ii) n- (i) is trivial, since K(L) D H.

We now show that (i) _ (iii). The linear system 12D I contains the divisors Tx (D) + T*Z (D), by Corollary 4. For any u e X, we can find an x eX such that u ± x 0 Sup p D, and this means that u $ Supp (T*(D) + T*_Z(D)). Thus, the linear system 12D I has no base

points, and defines a morphism c: X -.>. P. If 0 is not a finite morphism, we can find an irreducible curve C such that ¢(C) =one

point. It follows that for all B e 12D 1, either E contains C or is disjoint from C. In particular, for almost all x e X, C and T *(D) + T%(D) are disjoint. Now note the general fact. LE .a &.

If C is a curve on X and B is an irreducible divisor on X

such that C n E= 0, then E is invariant under translation by x, - x all x;cC.

If L = L(E), then L is trivial on C since C and E are disjoint. Therefore, Ts L, restricted to C, has degree 0 for all x e X. But then TZ(C) and E can never intersect in a non-empty PROOF.

finite set of points, since this would imply that Ti (-L) 1a had positive degree; i.e. for all x, either TZ(C) and .E are disjoint, or TZ(C) C E. Let x,, xE a C, y e B. Then T5_Z,(C) and E meet at y. Therefore

T,,_xi(C) c E, hence y - xy + x, a E. This proves the lemma.

62

ABELIAN VARIETIES

If D = En;D;, D; irreducible, then by the lemma, each Di is invariant under translation by all points xl - x2, x; a C. This contradicts (i), hence we have proved that (i) u (iii).

This enables us to show trivially that an abelian variety X is projective. In fact, let U be any affine open subset of X, D1,..., Dl C

the components of X - U and D the divisor D = E D;. We will 1

show that D verifies (i) above. We may assume after a translation that 0 e U. Then H = {x e X I Tz (D) = D} is a closed subgroup, and for x e H, U is stable for T. Since 0 e U, it follows that H c U, and H being complete and U affine, H is finite. APPLICATION 2.

An abelian variety X is a divisible group, and

for all n > 1, Xn is finite.

Considering the homomorphism nx: X-> X, it is clear that dim (ker nx) > 0 if and only if dim (Im(nx)) < dim X. Hence to

prove nx surjective, it suffices to check that X. = ker(nx) is finite. But let L be an ample line bundle on X. Then n(n-1)

n(n+1

nx* L=L 2

®(--lx)*L

2

Since (- lx) is an automorphism of X, (- lx)*L is also ample, and since Jn(n + 1) > 0, . n(n we see that nx*L is also ample. But then n$*.L cannot be trivial on any positive dimensional subvariety. Since is trivial, ker(nx) must be finite. APPLICATION 3. We can go even further and compute the order of X,,, when the characteristic p of k does not divide n. We first recall some general facts. Let X and Y be complete varieties both of dimension n, and let f: X -r Y be a surjcctive morphism. Then via f*, k(X) is a finite algebraic extension of k(Y), and we define the degree d (resp. separable degree, inseparable degree) of f to be the degree [k(X): k(Y)] of this extension (resp. [k(X):k(Y)]+, [k(X): k(Y)];). If f is separable, i.e. k(X) is separable over k(Y),

then d is the cardinality of f-1(y) for almost all yeY. If f is inseparable, the separable degree of k(X) over k(Y) instead is the

ALGEBRAIC THEORY VIA VARIETIES

63

cardinality of f-1(y) for almost all y. Moreover, a basic fact is that if Dl, ..., D. are Cartier divisors on Y, then we get the relation between intersection numbers; (f*DI..... f*D,.)a =d(D1..... D,.)y. Now suppose I and Y are abelian varieties. A homomorphism f: x-. Y is called an isogeny if it is surjective, with finite kernel. We have just seen that nx: X-- % is an isogeny. Then every isogenyf has a degree d, and since the cardinality of the kernel of f, # [ker fj, is the cardinality of f`'(y) for ally a Y, we see that

# [ker f j = separable degree (f ).

(*)

Now take the case f = nx. Let D be an ample, symmetric divisor

(i.e. (- l )* D = D) on I: we have seen that ample D's exist, and then D + (- 1%)*D is both ample and symmetric. Then by Corollary 3, nx*D is linearly equivalent to n2D. Therefore, if g = dim X, ex

ax

degree (nx) . (D..... D)g = (n%D..... n%D)j ax

=nsa(D..... D)1, hence degree (n8) =n7D. When is nx separable? If p,rn, then by the result above, p,4'degree

(nb) so ng must be separable. On the other hand, if p In, then we saw in § 4 that the differential d(ng) mapping Tg 0 to Tao is 0. Therefore, if w is any invariant differential form on %, nx**(w) is (a) still translation invariant, and (b) has value 0 in the cotangent space to I at 0, hence it is zero. Since the invariant differentials on % generate the sheaf S2} over Ox, they generate the k(%)-module

of k(X)/k-differentials. Therefore we find that the induced map nx on rational differentials SLk(x)11., is 0 if p I n. This implies that

the induced map on k(I) maps k(X) into k(X)P, and hence the inseparable degree of pr is at least pa. na'. Thus X. is a finite abelian It follows that if p14'n, # group killed by n such that for all m In, X. contains exactly mw

ABELIAN VARIETIES

64

elements of order dividing m. It is elementary group theory that the only such group is (Z/nZ) 20.On the other hand, Xa is annihilated

by p and is of order equal to the separable degree of pr, which is p{ for some i with 0 < i < g, so X, = (Z/p Z)'. Since X is divisible, it follows by induction on m that for any m > 1, X9m = (Z/pmZ)'. Summarizing, we have proved

PnoPosiTlox. (1) degree (n$) = n,4.

p r n.

(2)

ng separable .

(3)

If p,f' n, X ®(Z/nZ)29.

(4)

There is an integer i with 0 < i < g such that for all m > 1,

g, = (Z/p"Z)'. APPENDIX TO §6

We give an alternative proof of the fact that deg nr = n2' for n > 1, avoiding intersection theory.

For an invertible sheaf L on a complete variety X of dimension g, and any coherent sheaf v on X, P,.(n) =X(F(&L") is a polynomial

in n of degree < g. We shall denote the coefficient of n° in this polynomial by dL(-5F)/g!, so that dy(-59') is an integer > 0. We call dL(Ox) the degree of L, and denote it by deg L. The basic result is the following PBOrosrrsox. (1) Let X be a complete variety with an invertible sheaf L. For any coherent sheaf .° on X, let rank (.°) be the dimension over the function field uj X of the generic stalk of ., or equivalently, of the space of rational sections of ."". Then we have

dL(F) =rank (.F). deg L. (2) Let f: Y -. X be a surjective morphism of complete varieties of the same dimension g, and L an invertible sheaf on X. Then

deg f*(L) = (deg f).(deg L). PROOF.

(1)

It is a standard fact that we can find a non-zero

coherent sheaf of ideals 5 and an exact sequence

ALGEBRAIC THEORY VIA VARIETIES 0

O,.otv

65

--4.JF3---4.0

w ith ,°I a torsion sheaf. Since

has support of dimension

< dim %, X(9r (9L") is a polynomial of degree < dim %, and the

additivity of gives us that dL

(J°"')

= (rank ,9r). d. (J),

and using the exact sequence 0 ->.5 -s Ox -> Oz J.f 3 0, we see that dL(.f) = dL(O) =deg L. (2) For each p > 0, we have a canonical isomorphism R'f* (f *(L")) = R1'f* (Or) ® P. Taking alternating sums in the

Leray spectral sequence H'(X, RQf*(f*(L"))) zo. H"(Y, f*(L")), we

obtain that

X(f*(L")) =

P-0

(-1)'X(R'f*(Or)®L").

Since there is a non-void open subset U of I such that a finite morphism, R"f*(Or) have supports f1f-1(U):

in a proper closed subset of 8 for p > 0. Further, R°f*(Oy) is clearly a coherent sheaf of rank = degf. The assertion now follows on comparing coefficients of no on both sides. Now let % be an abelian variety, is a positive integer. Let L be an

ample symmetric line bundle on I; for any ample L, L.®(-l$)*L is both ample and symmetric. Then by Corollary 3, n*L is isomorphic to L"'. Therefore, if g = dim %, we get by the above proposition that deg nz. deg L = deg na(L) = deg L-" = nom. deg L, and since deg L > 0, we get that deg ng = n2g. 7. Dividing varieties by finite groups. Let f : % - Y be a morphism of algebraic varieties (over an algebraically closed field 1c). f is said to be etale if

66

ABELIAN VARIETIES (i)

f is flat,

(ii)

for all x eX, y = f(x) e Y, if ms and my are the maximal ideals in 0= and Oy, then f*(my)Oz = m2.

This is equivalent to assuming that f is a "formal isomorphism" in the sense: (i)' for all x e X, y =f(x) e Y, if Ox, Oy are the completions

of 0a, Oy, then the natural map f*: Oy-+Ox .

is an isomorphism.

(Cf. Mumford, Intro. Alg. Geom., p.353). When k =C, it is also equivalent to assuming that f is a local isomorphism of analytic spaces. Our main result is Let X be an algebraic variety, and 0 a finite group of automorphisms of X. Suppose that for any x e X, the orbit 0z of x TaEOREM.

is contained in an affine open subset of X. Then there is a pair (Y, er), where Y is a variety and ir: X -+ Y a morphism, satisfying the following conditions: (i)

as aSopologicalspace, (Y, rr) is the quotient of X for the 0-action,

(ii)

if Ir*(0x)o denotes the subsheaf of G-invariants of -*(61X)

for the action of 0 on a*(t9) deduced from (i), the natural homomorphism Op i rr* (O$)° is an isomorphism.

The pair (Y, 1r) is determined up to an isomorphism by these conditions. The morphism 47 is finite, surjective and separable. Y is afne if X is affine.

If further 0 acts freely on X (that is, if gx ,-£ x for any x eX and any g e 0 with gte), it is an 4tale morphism. Since the conditions (i) and (ii) determine the topology and structure sheaf of Y, the uniqueness assertion is trivial. Also, the problem of existence reduces to proving that if Y is the quotient I/O as a topological space, and is given the structure sheaf Oy = a;(Ox)°, it is an algebraic variety. Suppose we knew the theorem PRoov.

67

ALGEBRAIC THEORY VIA VARIETIES

(in its entirety) to be valid when X is affine. For any xaX, let U' be an affine open subset of X containing Gx. Then u = fl gU' is an pea

affine open subset of X containing z and stable for G. Thus X is covered by G-stable affine open sets U. Then each a(U) is open in Y, and 7f ' (rr(U)) = U, so by the affine case of the theorem a(U), with

the restriction of Oy, is an affine variety. But the open subsets ir(U) cover Y, so the theorem would follow for X.

We may therefore assume X = Spec(A). Let A = k[xl, ... , x j. Then G acts on A by the law g(f)(x) = f(g-'x), g e G, f e A, z e X.

Let v = order of G. For f e A and 1 < k < v, denote by a(f) the elementary symmetric function of degree k in fg(f )),0, and put B' =k[af(x9)]1 Y, where two isogenies f1:X -- Yl, f2:X . YR are considered equal if there is an isomorphism h: Yt > Ya

such that f2 = h o fl, which is set up by K = ker (f), and Y = X/K. FaooF. First start with a finite subgroup K c X. Then K acts freely on X by translations, so we can form the quotient (X/K, f),

and f is an etale surjective finite morphism X a X/K. On the other hand, X/K as a set is the quotient of the abstract group X by the subgroup K, hence it has a group structure. The group law is in fact a morphism. This follows by considering the diagram:

XxX

X

fxf

if X/K

X/K x X/K n

where m is the group law of X (a morphism) and n is the group

law of X/K (so far, just a map). But it is easy to check that X/K x X/K = X x X/K x K, and since the morphism fom: X x X -> X/K collapses the action of K x K, it factors through

X x X/K x K i.e. n is also a morphism. Similarly, it can be checked that the inverse map on X/K is a morphism. Therefore X/K is an algebraic group. Finally, X JK is the image. of a complete variety and therefore is complete. Thus X/K is an abelian variety, and f :X -* X /K is a separable isogeny. Clearly the

kernel off is K. Second, start with a separable isogeny f : X -a Y. Let K be its kernel, and as above form a new separable isogeny g:X >X/K. A morphism h in the diagram:

ABELL4N VARIETIES

74

X

l\f exists, since f collapses the action of K, hence it factors through the quotient $/K. But h is obviously bijective, and the reparability off implies the separability of h. Therefore h is birational too. Therefore by Zariski's Main Theorem, h is an isomorphism. COROLLARY 1. A separable isogeny f: X -3 Y is an Stale morphism.

Let f: X -a Y be an isogeny of order prime to p. Then the kernel off and the kernel of f*: Pic(Y) -+ Pic (X) are dual COROLLARY 2.

finite abelian groups.

PRooF. Apply Proposition 3 and Theorem 4. The dual abelian variety : char 0. We will use the hypothesis of char 0 only towards the end of this section.

8.

Dasi ITION. Pic° (X) is the subgroup of Pic(X) consisting of line bundles L such that the homomorphism 01, is identically zero.

By the theorem of the square, the image of each ¢L is contained in Pico (X), so we get an exact sequence:

0 -+ Pic° (X) -3 Pic(I) -. Hom(X, Pico (X)).

L r- L. The main purpose of this section is to show (in char 0) that Pico (X)

is naturally isomorphic to another abelian variety $, called the dual of X. We make some general observations about Pic° (X). (i)

LcPico(X) -4-- => Tx L= L,allxeX

cam*L_p;L®p2LonXX X.

ALGEBRAIC THEORY VIA VARIETIES

7b

PROOF. By the See-saw theorem, m*L®p*L-1®p2L-1 is trivial if and only if it is trivial on X x {a} and on {0} x X. But it always is trivial on {0} x X and its restriction to X x {a} is isomorphic to T*L® L-1.

If L e Pic°(X), then for all schemes S and all morphisms f, g: S-+ X, (f + 9)*L e f*L®9*L.

(ii)

PROOF.

Consider the last isomorphism in (i) and pull it back to S

by(f,9):S-aX xX. (iii)

If L e Pic°(X), nx* L r L".

PROOF. Apply induction to (ii). (iv)

For all L e Pic(X), nx*L : L"'(9 (something in Pic°(X)).

so In fact, by §6, nx* L= L" ' ®[L ®(- lx)* it suffices to prove that L®(- lx)* L-1 a Pic°(X). By translating L-1](n-"=)n

PROOF.

by x, we get

T,*(L(&(- lx)*L-1) e TL (g (- lx)* T*.L-1 Ts L®(- lx)*[L® T*xL-']® (-

lx)*L-1

in Pic°(X)

_ Tx L®L-1® T*zL®(- Ix)*L-1, by (iii) = L® (- 1x)*L--1 by theorem of the square. (v)

If L e Pic(X) has finite order, then L e Pic°(X).

#y(nx), all x c -X. If L" is trivial, 0 = ¢in(x) = Since X is divisible, this shows that 0L = 0. (vi) For all varieties S, and all line bundles L on X x S, if PROOF.

L.=Llxx(,), then L,1®L,o' ePic°(X), (s°, s, c- S). PROOF. Replacing S by open sets belonging to a covering of S, we can assume that L l(o)x8 is trivial. Further, replacing L by

L ®p*(L '), we can assume that L,. is trivial, and we must then prove that L, 6 Pic° (X), all s e S. We shall show that

ABELIAN VARIETIES

76

to*(L,) ® p*(L,-1) ®p2 (L, 1) is trivial for all s. In fact, construct a line bundle M on X x X x S µ(x, y, s) _ (x + y, 8) P1s(x, y, 8) = (x, a) P28 (XI y, 8) = (y, 8)

Then M is trivial on X x {0} x S, {0} x X x S and X x X x {s°}. Therefore by the theorem of the cube, M is trivial. But M, restricted to X x X x {s} is m*(L,)®pz(L 1) ®p*2(L,-1). (vii)

If L ePic°(X) and L is not trivial, then Hi(X, L)=(0), all i.

PROOF. If H°(L) # 0, then L = O1(D) for some non-negative divisor D. Then L-1=(-1g)*L = Og((-lx)*D), hence O® = Ox(D + (- 1$)*D). Therefore P + (- 1g)*D = 0, hence D = 0 and L: Ox which contradicts our assumption. This proves that H°(L) = (0). Let k be the smallest integer such that Hk(L) (0). Let a1: X-X x X be the map si(x) _ (x, 0). Using the fact that L(2)L-1

m*L = p*L(& p, *L and the Kiinneth formula, we get the diagram:

X 1 ) XxX

?X

s*

Hk(X, L) +- Hk(X x X, m*L) 4

m*

Hk(X, L)

2II

Hk(X x X,piL®p2L) 211

Hi(X, L) ® HH(X, L).

it)_k

Since moss = 1g, the dotted arrow is the identity. But if i +j = k > 1, then either i < k or j < k, hence in all cases

ALGEBRAIC THEORY VIA VARIETIES

77

H'(X, L) ® ''X, L) = (0). Therefore, the identity from Hk(X, L) to H5(X, L) factors through a (0)-group. So Hk(X, L) = (0) too.

We now come to the really key point in the theory of Pie°. THEOREM 1.

xex,

Let L be ample and M e Pie°(X). Then for some

McTZL®L'1, i.e. the map q, : X - Pic°(X) is surjective. PROOF. The whole idea is to look at the cohomology on X x X of the line bundle

K=m*L®p'L-1®pi(L-'(9 M-'). This cohomology is the abutment of two Leray spectral sequences associated to the two projections of X.x X onto X:

Hk+'(X x X, K),

(1)

H=(X, Rkpl,*(K))

(2)

Hz(X, R11p2,*(K)) - Hk+t(X x X, K).

Notice that on the fibres {x} x X of p1 and X x {x} of p2, K restricts to the line bundles K I(s)xx =

T:L (9 L-1® M-',

K 1, x t.)

Tx L ® L-1.

Therefore, if M = T. *L ® L-1 for any x, it follows that K I{s} x x is

a non-trivial line bundle in Pic° for every x. But by (vii), this means that all the cohomology groups of K I ,x} x a are (0). Therefore Rkp1, * (K) = (0}for all k by Corollary 2, §5. Therefore Hk(X X X,

_ (0) by spectral sequence (1): Now use the other spectral sequence. Since Tz L® L- 1 is nontrivial and in Pic° if x ik K(L), it follows that suPP (Rkp2,* (K)) c K(L).

Since K(L) is a finite set., spectral sequence (2) degenerates to ® Rkp2,* (K): = Hk(X x X, K). XEs(a)

ABELIAN VARIETIES

78

But Hk(X X X, X) _ (0), so .Rrp2,, (K) = (0) too. Therefore Hk(X, Klgx (x}) = (0) for all a, by Cor. 4, § 5. But KjxX{o} is the

trivial line bundle, hence has a non-zero H° ! Thus we have a contradiction, so that the theorem must be true. For a second proof of a slight weakening of the theorem, see Lang, p. 99. This important theorem shows that as an abstract group, Pic° (X) is isomorphic to the abelian variety X/K(L).. If Y is an abelian variety isomorphic as abstract group to Pic° (X), what properties would we expect, which would characterize this "extra structure" on Pic°(X)? (a) We want a line bundle P on X x X, the Poincare bundle such that for all a e X, the restriction P. of P to X x {a) represents the element of Pic° (X) given by at under the isomorphism Pic° (X) =k

Moreover, we require that PJ{01X1 is trivial. (These properties characterize P by the See-saw theorem.). (b)

For every normal variety S, and every line bundle K on

X xS such that (i) K,=KIxX{,} is in Pic°(X), for one and hence all

e e S, and (ii) KI{o}XS is trivial, the unique set-theoretic map f:S -+ X such that K, = Pf(,), is to be a morphism, and K is to be isomorphic

to (1g x f)*P.

It is easy to check that (a) and (b) uniquely characterize both X and P up to canonical isomorphisms. The problem is to construct such an X and P. So fix an ample L on X, and as suggested by the

theorem. take X to be the quotient X/K(L), constructed in §7. Let ir: X X be the given morphism. To construct P, we shall

use Prop. 2, §7. We want (1g xa)*P to be the line bundle K = m* L® pl L-1®p2 L-1 on X x X. [This clearly should be the case: apply (b) with S = X, K = m*L®p1 L-1®p,*L-1. Then f = IT so K should be isomorphic to (la x7r)*P]. According to Prop. 2, we

must lift the translation action of Ker(1g x a) = (0) x K(L) on

ALGEBRAIC THEORY VIA VARIETIES

79

X x X, to an action of the same group on the line bundle K. But for any a e K(L), compute the pull-back T(,a)K: ®T%)PiL-1®T(*

T(o a)K o T(o a)m*L

P *L-1

m*T;L®p,*L-1®pz TTL-1 L-1

° m*L ®Pi L-1®Pz since a a K(L). Therefore, there is an automorphism Y'a: K -a K covering the automorphism T(o a): X X X X x X on the base space. However, each 0. could be changed by a scalar, so there is no reason why Y'a°Y'6 = Y'a+a should hold if the O .'s are chosen

arbitrarily. However, if L-1(0) is the fibre of the line bundle L-1 over 0, then notice that there is a canonical isomorphism: K I (o) x x= m*L 1((,) x.r®PiL-11(o) x I®p*L-1 I(°} x x

L®

trivial bundle

L-1(0) x X

L_ 1

L-1(0) x X.

Suppose we require that the automorphism ¢d of K should restrict on {0} x X to the product automorphism: (a, x)

(a,x+a)

L-1(0) x X.-> L-1(0) x X.

Clearly, there is a unique a which has this restriction to {0} x .8. Since the restrictions then obey 0a0 Ob = Y'a+e> so do the Oa's themselves. With this action of Ker(lx x a) on K, we construct a P on X x X such that (1x x 2r)*P = K.

Notice first that for all a a X, if a = Tr(x), then

Pa-PIIx(a} der ir*(P) Idx(x}

L

®L-1'

Tx

i.e. P. represents the element #L(x) a Pic° (X). Therefore, if X is identified with Pic° (X) so as to make the diagram

ABELIAN VARIETIES

80

'L X

Pic°(X) 211

commute, the first part of (a) holds. Moreover, P 1 l0lx4 is the quotient of K I{ol,Xx by Ker(v), i.e. of L-1(0) x X by the product action of K(L). Therefore, P I(o)xj = L-1(0) x X, a trivial bundle. Thus the second part of (a) holds. To check that (b) holds, given S and K, consider the line bundle E = Pli (K) ®pxs(P`1) pn X x S x X. Then E I x x {«.,t n K, ® Pa 1, and the subset bf S X X: F = { (s, a) I E I x x 1, 5} trivial}

is Zariski-closed in S xX. But since E Ixxt,,al is trivial if and only if

K, s Pa, r is nothing but the graph of the set-theoretic map f. In particular, the projection r > S is a bijection. Now since the characteristic is Ot, this shows that r and S are birationally equivalent varieties, and since S is normal, r -->S is an isomorphism of varieties by Zariski's Main Theorem. Therefore r is the graph of a morphism,'i. e. f is a morphism. The last assertion in (b) follows from the See-saw theorem. REMARKS.

(1)

For every line bundle L on X, the map

i : X >X is a morphism. This comes out of applying the universal mapping property (b) to the line bundle m*(L) ®pl (L-'1)®pz(L-1)

onXXX.

IfX f

Y is a homomorphism of abelianvarieties, the induced map Pie Y->Pic K maps Pic°X into Pic°X, and thus we get (2)

a natural map f : Y> X, and this is a morphismn. In fact, if Q is the tThis is the only place whore we use char. _- 0! However, it. is quite essential. The X we have constructed would definitely be "`wrong" in char 1,.

ALGEBRAIC THEORY VIA VARIETIES

81

Poincar6 bundle on Y x Y, (f x 1)*(Q) is a line bundle on X x Y such that for y e Y, (f x VY)*Q I a x {v} represents f *(y) a Pic°X, and

by the universal mapping property, f : Y -> X is a morphism. (3)

If f : X--> Y is an isogeny, so is f : Y -* X, and there is a

canonical duality (of finite abelian groups) between Kerf and Ker f.

PxooF. We have seen in §7 that Ker(f) and Ker(f*: Pic (Y) Pie(X)) are dual. If y e Pic(Y) is such that f*(y) = 0, then this shows that y has finite order, hence y e Pic°(Y). Therefore, Ker(f *: Pic Y-. Pic X) = Ker f. Finally, since dim X = dim X = dim Y = dim Y, it follows that f is an isogeny too. The final point we want to make is that the relationship between X and l is in reality symmetric like the relationship between two

vector spaces set up by a bilinear pairing. We can see this as follows. DEFINITION.

Let X and Y be abelian varieties. A divisortal

correspondence between X and Y is a line,bundle Q on I x Y whose restrictions to {0} x Y and X x {0} are trivial. Let X and Y be two abelian varieties of the same dimension, and Q a divisorial correspondence between X and Y. The following are equivalent PROFOSxTIoN 2.

(1) (2)

is trivial, then x = 0, If Q lax{y} is trivial, then y = 0. If Q I{x} x Y

If these hold, then X = Y with Q isomorphic to the Poincar6 bundle PY of Y, and Y P X with Q isomorphic to the Poincare bundle Px of X.

PROOF. By symmetry, it suffices to deduce (2) from (1). If (1)

holds, there is

. an

injective morphism ¢:X - Y such that

Q = (0 x 1Y)*PY, PY being the Poincar6 bundle on Y x Y. Since dim X = dim Y and 0 is injective, ¢ is also surjective; since the characteristic is 0, this implies that is an isomorphism, i.e. X n Y.

ABELIAN VARIETIES

82

Now, let & : Y-- X be the morphism such that if Px is the Poineare bundle on X x X, (1x x &) *Pg = Q. To prove (2), we have to show that 0 is injective. If not, we can find a finite subgroup K c

K

(0), and#factorizes as

Y-71

) Y/K

>.$ where rl is the

natural homomorphism. Thus, if L is the line bundle (1g x#)*Px on X x Y/K, we have that Q v (1g xrl)*(L). Now, L induces a bomo-

morphism a: X -* (Y/K), and the isomorphism Q ; (1$ xn)*(L)

means precisely that the composite X. - (Y%K)

'7 + Y is the

homomorphism defined by Q. Thus this composite is an isomor-

phism. Thus a is injective, and since dim X = dim (Y/K), a and i are both isomorphisms. But we know that rl has a non-trivial kernel, viz. the dual abelian group of K. This is a contradiction, proving that is injective. The case k = C. We want to link up the methods of Chapters 1 and 2 in this section. Therefore we assume that the ground field k = C, and that X = V/ U (V a complex vector space, U a lattice) is an abelian variety over C. Recall that every line bundle on X is isomorphic to L(H, a) for a unique Hermitian form H on V such that E = Im H is integral on U x U, and a unique map a: U -a C i

9.

satisfying

a(u1 + u2) = es, E(u1. *h ui, ua a U. a(ul) a (u2)

L(H,a) is, by definition, the quotient (C X V)/U for the action

We shall do two things: (A) compute T,*(Z)[L(H, a)], (x a V), explicitly and hence interpret oz(H.a) in the analytic case. (B) Describe divisorial correspondences as .(H, a)'s and hence compute

the dual 1 analytically.

ALGEBRAIC THEORY VIA VARIETIES (A)

83

To keep the picture clear, it is convenient to generalize a

little. Let a discrete group U act freely and discretely on V1 and V2

and let T: V1 _+ V2 be a U-morphism. Let Xi = V;/U, and let T: Xl -X2 be induced by T. Suppose U acts linearly on C x V2 by 6u(A, z) _ (A. eu.(z), u(z)), z E V2, A e C, u e U,

where e,(z) is a multiplicative co-cycle eu1+uy(z) = eu,(u2(z)).e22(z).

Let L2 = (C X V2)/U.

Suppose we now want to describe the

line bundle L1 = T*(L2). Then Ll = X1 xx2 L2

-

[V1 X v ,(C x V2)]/U, where U acts on both factors.

Therefore, Ll = (C x V1)/U with the action 0a(A, z) = (,l-ee(Tz), u(z)), z e V1, A e C, u e U.

Now the co-cycle {ea} might be normalized to have a special form which {e.-T} might not have. Then we might use an automorphism of C x V1: 3 (A.g(z), z) (A, Z) I--

CxV,

CxV1

and carrying over the action of U, obtain a new description of L1 as (C x V1)/U with action 0.(A, z) =

u(z)).

Apply all this to the case V1 =V2 = V, X1=X2=X, T. translation translation by 17(a), and L2=L(H, a). It follows bya e V, T = that T (,)L(H, a) is (C x V)/U with action 0u(A, z) _ (A.a(u).eAII(2+a.u)+Ira(u,u), z + u)

_

eaH(a,u>] a E(z,u)+i-E(,,u)

z + u).

To simplify this co-cycle, take g(z) to be the non-zero holomorphic

function a "E(-). We get the new action a(A, z) = (A.a(u).e"CB(a,

e.+E(z,u)+J.H(u.u) z + ..I, u)

and since H(a, u) - H(u, a) = 2iE(a, u), we conclude:

ABELIAN VARIETIES

84

PaOPOSITION.

a)] a L(H, a. ya) where ya(u) = e2,0(11,u)

We get immediately lots of nice consequences. (i)

L(H a) (,r(a)) is the point of Pic° (X) represented by L(0, y, ).

(ii)

In particular, since yal .yay =Va,+a2, it follows that 0l,(H a) is a homomorphism; this is the theorem of the square.

(iii)

We find K(L(H, a)) = U-I-/U c V /U = X, where Ul = {afE(a, u) E Z, all u e U}.

(iv)

Therefore L(H, a) is in Pic°(X), as defined algebraically,

K(L(II, a))=XC=> U-L=V-n E=0 H = 0, i.e. L(H, a)is in Pie°(X), as defined analytically. (v)

Moreover,

K(L(H, a)) is finite

Ul/U is finite U J- is a lattice E is non-degenerate . u H is non-degenerate.

(vi) Notice that if H is non-degenerate, e.g. if L(H, a) is ample, then every homomorphism U> R is given by u --> E(a, u), for some a c- V. Therefore every homomorphism a : U > Cl is given by

u > e2"Wa,u) for some a c- V. Thus every element of Pic°(X) is equal to L(0, ya), some a e V; this proves the main theorem of §8 when k = C. (B)

Let X = = Vj/U;, i =1, 2, be two abelian varieties. Then Q =

L(H, a) on Xx x X. is a divisorial correspondence if Q is trivial on (0) x X2 and on Xl x {0). This means that (a) the Hermitian form H is 0 on {0} x V2 and on Vl x {0), and (b) a = 1 on {0) x U2 and on Ul x {0}. Define

ALGEBRAIC THEORY VIA VARIETIES

85

B(x1, x2) = H((x1, 0), (0, x2)).

Then B is an R-bilinear form on V1 X V2, complex linear on V1. and

anti-linear on V2. Let Im (B) _ . Then fi is integral on U1 x U2, and we get H((x1, x2), (y, ye)) = H((xl, 0), (y1, 0)) +H((xl, 0), (0, y2))

+H((0, x2), (y, 0)) +H((0, x2), (0, y2)) = B(x1, y2) + B(yv x2) a((u'1, n2)) = a((u1, 0)).a((0,

Thus the divisorial correspondence Q is determined entirely by B.

In order to find the map from XZ to Xl induced by Q, we next calculate the restriction of Q to X1 x {1r2(a2)}, a2 a V2. Let Q' = (C X V1 X V2)1 U1' If 1 x 'r2: X1 X V2 -+X1 x X2 is the natural map, thenQ' e (1 xir2)*Q, and Q' I81xca,}= Q I81xn,(a. The action of U1 on C x V1x V2 is given by (A.ena(al.xs)

x1, x2) =

x2 +u1, x2).

Restricting to V1 x{a2}, it follows that

equals (C x V1)JU1

with action Y'u }

(A, x1) _

en8('i.a'

xl + ui)

Modifying this group action by the automorphism of C X V1, scalar multiplication by e-11(11102), we get the action (A,x1) _ (A.e 2'

x1 +u1)

Thus we have PEGPOSITION. Q I% x

L (0, S42) where

Sax (ul) = e-2 ti e('

.

In particular, we see that in order that Q on XX X X2 satisfy the equivalent conditions of Proposition 2, §8, it is necessary and

sufficient that dim X1 = dim X2 and that for all

x2 a V2,

e U1. This implies that B is a - fl(u1,x2) a Z, all is non-degenerate pairing of V1 and V2. Hence, x2 a U2 -

ABELIAN VARIETIES

86

COROLLARY.

(i)

(ii)

Via Q, Xl = X2 and X2 m Xx if and only if

B is non-degenerate;

under f, U1 and U2 are dual lattices, i.e.

U2 = {x2 a V2 Au1, x2) eZ, all u1 e U1) and vice versa.

Explicitly, therefore, if X = V JU, then the dual abelian variety

X can be constructed as follows. Let C),

U' _ {l e T IIm l(u) a Z, all u e U}.

Then X=TIU'. The form B: V xT -;C is simply B(x, 1) = l(x), and the Poincar6 bundle P1 on XxX is simply L(H,a) with H((x1, 11), (x2, 12))

12(x1) + 11(x2)

a((u. l)) = e- trm!(u>

Moreover the line bundle on X corresponding to a point a(l) e 1 (1 e T) is then just L(0, a1) where a1(u) = e2nti rm 1(u) n e U.

There arises a small question of compatibility : we just constructed X = P-/U' and showed that X e Pic° (X). But in §2, via the exact sequence 0Z -± a$ we constructed an isomorphism

-* tx ---, 0,

Pic°(X) m H1(X, 6x)/H1(X, Z)

a'1ld in §1, we found an isomorphism H1(X,OX) o T. We would like to be sure that we have really found essentially the same description

of Pic°(X) twice. As we have just seen, our second description of Pic°(X) rests on the map

T -T Pic° (X)

l ; L(0, a1).

ALGEBRAIC THEORY VIA VARIETIES

87 eS,n

Let us compute the composite map T . Hl(01)---) Pic(X) which gave us the first description. This map is given by e2":

T c T®T f * H1(IC) H'(X,Ox) --+ Pic(X). Using group cohomology of U, we get a diagram by def.

HOm(U, C)

ofgp. coho.

H1(U C) ------} H'(U, H*)

T c T ©T f F HA (X, C)

where H*=multiplicatioe group of non-zero holomorphio functions on V, and where the square on the left commutes according to the compactibilities verified in §1. Therefore, ifl e T, the first description associates to l the U-co-cycle u -*

But e2, U(u)

g(z + u)

esoiRelqu)]

g(z)

where g(z) = e-2" z) is holomorphic in z. In other words, the first

description rests on the map Pic°(X)

1)--- L(O, a,*) as (u) = et.iRe[!(u)1.

So the two maps differ only by multiplication by 2i (experimental error!).

REMARKS ON EFFECTIVE DIVISORS by M. V. NORI PROPOSITION. Let f : X -* Y be a morphism of varieties, with X an abelian variety. For each x a X, let FF be the connected component off-'f(x) to which x belongs. Then there is a closed connected subgroup

F of X such that FF = x + F, for all x e X. PROOF.

Fix x e X. Let ¢: X x F. - Y by O(z, u) = f(z + u).

Since FF is complete and connected and 0({e} x FF) = f(x), by the rigidity lemma q(z, u) = O(z, x) for all z e X, it e P. In particular, f(z - x + FF) = f(z) for all z e X. Since z - x + FF is connected, we have FF contains z - x + F. Reversing the positions of x, z e X, we

actually have F= = z - z + F. for all z, x e X. In particular, if F=Fe,F,=z+F forallzeX. So it only remains to show that F is a subgroup of X.

Let u e F. Then F_u = - it + F, so e e F_,s. It follows that F_ F. = F. Therefore - u + F = F, for each it c- F, so F is a subgroup of X (by the very definition of F, F is closed and connected). From now on, F will be called the kernel off.

One checks easily the following THEOREM.

(i)

Let D be an effective divisor on X.

Let H(D) = {x eX I TF(D) = D} (equality of divisors, not equivalence). Clearly H(D) is Zariski closed.

(ii)

Let L = L(D), since L2 has no base points, let f : X -+P(H°(X,.L2)) = P11

be the corresponding morphism. Let F(D) = ker f, as in the above proposition. Then H(D)° = K(L)°'= F(D). Note that the above theorem implies Application 1 on p. 60.

III

ALGEBRAIC THEORY VIA SCHEMES The theorem of the Cube (II). In this chapter, we shall always mean by a scheme a scheme of finite type over an 10.

algebraically closed field k, and a point will always mean a closed point of the scheme. We begin with the following seemingly innocuous generalization of Corollary 6 to the semicontinuity theorem. PROPOSITION. Let X be a complete variety, Y any scheme and L a line bundle on X x Y. Then there exits a unique closed subscheme Y. c

Y having the following properties:

if Ll is the restriction of L to X x Y1, there is a line bundle Ml on Y1 and an isomorphism ppM1 = Ll on X X Yl; Y is any morphism such that there exists a line (b) if f : Z bundle K on Z and an isomorphism p2 (K) m (lx xf)*(L) on X x Z. . Y. f can be factored as Z --> Yl (a)

The uniqueness is immediate, since if Yl c .Y Y are two closed subschemes of Y satisfying and Yi ' (a) and (b), each of these closed immersions can be faetored through the other, and they must coincide. Paoozr.

Note next that if p2M1 o L, then M, m pa,*(L1) (by the Kunneth formula), hence to show that there is a line bundle M1 such that pzi1= LI, it is equivalent to showing that pR *(L1)=R1 is an invertible sheaf and the natural homomorphism p2 (Dlt1)

is an isomorphism. In view of this, we are reduced to proving that there is an open covering {Vi) of Y such that the proposition holds for X x V,--> Vi and the restriction of L to X x Vi. In fact, if we have done this, we Vi such that (a) and (b) are. obtain a closed subscheme IV, r

valid with Y replaced by V. Then clearly Wi n (Vi n V,) and W, n (Vi n V,) are two closed subschemes of Vi n VV such that (a) and (b) hold with Y replaced by Vi n V,, hence they are equal. Thus we obtain a closed subscheme Yl of Y such that Yl n VV-Wi,

ABELL4.2 VARIETIES

90

and (a) (because of our local reformulation) is clearly valid. As for

(b), the local version of (b) implies that for each i, f-'(V;)-+ V{ factorizes through W, hence f- 1(V{) -> Y factorizes through Y1, and

so Z

f)

Y factorizes through F1.

Thus we may assume Y = Spec A, and it suffices to find an open neighborhood of each point of Y in which the proposition is valid.

By shrinking Y if necessary, we may assume that we have a free complex 0

0) A'1 -- ... giving the direct images of

AN

L universally, as in §5. Let M be the cokernel of the transpose '¢ of 4: A'1

to,

B'1

A' - M

0. Then for .any A-algebra B,

B'" -* M ® LB -a 0, is exact, and hence so is 0 ->

HomB(M (DAB, B) -+ B'

B'1. This shows that for all f : Spec B

Spec A = Y, p%* ((lx x f)*(L)) n Homd(M, B). Now let F be the set of points y e Y such that the restriction Li, = L I % x {y) is trivial, so that P is a closed subset of Y by our earlier result (applied to Y and the restriction of L to % x Y,,). If Y' =

Y - F, the empty subscheme Yl = 0 of Y' satisfies (a) and (b) with respect to Y'. Hence, it suffices to show that for any y e F, y has an open neighborhood in which the proposition holds. If y e F, 1 = dim Ha(Y x {y}, Ll) = dim HomA(M, AJAR.) = dim,.

so

K vMI that by Nakayama's lemma, there is an element of M which generates M in an open neighborhood of y. Restricting ourselves to this neighborhood we may assume that M = A/QC, where $I is an ideal of A. Let Yi be the closed subscheme defined by I. L,' the restriction

of L to % x Yx and !,the associated sheaf. Then pp,*(L,') is the sheaf associated to Homd(A/8(, A/$() n A/$( on Yl, and is hence free of rank one. Consider the natural homomorphism pz*(ps*(L1')) A

--. L3 on % x Y. Since both sides are locally free of rank one, this is an isomorphism, at a point z eX x Y; if and only if the induced homomorphism of `fibers'

ALGEBRAIC THEORY VIA SCHEMES

91

[PS(P2.*(1z))z] ®psk- [L's]®0 k is surjective. Now, since Homd(A/91, A/9S)-?Homd(A/4X, AITZ,,) = H°(X x {y}, LAX x{y}) is surjective and LI.Z x {y} is trivial, A is

an isomorphism at all points of X x {y}. On the other hand, the set Z of points on X x Yi where ,l is not an isomorphism, being the union of the supports of ker A and coker A, is closed and does not meet X x {y}. Hence its projection into Y is a closed subset not containing y. By restricting ourselves to an affine open neighborhood of y not meeting this projection, we may assume that M c A/21, and that if Y1 is the closed subscheme defined by 91, condition (a) is fulfilled on Y1. We claim that (b) follows.

In fact, since the condition that f : Z -# Y factorize

through Y1 is local on Z, we may assume Z = Spec B affine, and B becomes an A-algebra via f. Further, we mayassume K trivial on Z, so that (l= x .f )*(L) = cVaxz and 2°s,*(lz x f)*(L) = ps,*(Oaxz) = IVs (since X is a complete variety). Hence we have an isomorphism of

B-modules B a Homl(A/%, B), so that $1.B= 0 and A B factors through .A/2(. Thus f factors through Y1, proving the proposition. Under the assumptions of the proposition, we shall refer to the closed subscheme Y1 of Y given by the proposition as the maximal

closed subscheme of Y over which L is trivial. We get the following strengthened version and direct proof of

the theorem of the cube. Let X and Y be complete varieties, Z a connected scheme, and L a line bundle on X x Y x Z whose restrictions to {x°} x Y x Z, X x {y0} x Z and X x Y x {z0} are trivial for some x0 a X, yc a Y, and z0 a Z. Then L is trivial. THEOREM.

PROOF. Let Z' be the maximal closed subscheme of Z over which L is trivial, so that Z' ; 0 since zo E V. We have to show that Z'=Z, and since Z is connected, it suffices to show 'that if a

point belongs to Z', Z' contains an open neighborhood (considered as an open subscheme of Z) of that point. Let us denote this point again by z0, by Tl the maximal ideal of Oz,,. and by I = I. the

ABEL kW VARMTTs

92

ideal defining Z' at z0, so that I C JJ1. We have to show that I=(0).

If not, since fl 9R' = (0) by Krull's Theorem, we can find an n>0

I, so that

integer n > 0 such that W j I, J7t"+1

1] C 10"ISRs+11

[MR+1 + I/ D

is a non-zero k-vector space.. Hence, if QC1 = 2R +1-} I, we can find an ideal Its with 211:) 212 i 90+1 and dim,

-

= 1. Hence $(1 = It,

+ k.a for some a e 411 and 21, n I but 41, 5 I. Let WO = R. Let z, be the closed subschemes of Z consisting of the single point z0, with structure sheaf so that: Z'

z

(z0 = Z, C Z1

z$

Let _.4 (i 0,1,2) be the restriction to % x Y x Z; of the sheaf Ti _of sections of L. Note that L1 are trivial on F x Y x Z0 , 8 X Y X Z1 respectively, since Z0, Z1 c Z', so that we have isomorphisms Iq=OYxrx; (i=0, 1). Further, since the structure sheaves of Z,,. Z1 and ZQ are related by the exact sequence 0 ---+ Ozo

mult. by a

Ozi restr. ---) Ozl

)0,

we also have an exact sequence of sheaves on the topological

space Z0:

0)La) Ls mult. by a

restr.

}L1 )0.

Consider the sections a P(% x Y X Z1, L1) equal to A(1) under the

isomorphism a: @XxPxty -} L1. The necessary and sufficient

ALGEBRAIC THEORY VIA SCHEMES

93

condition that L' be trivial is that, a can be lifted to a section s' of L,. In fact, if we can do this, multiplication by a' is a homomorphism A': Oa x r x z, -+ L, which reduced modulo the maximal

ideal of any point { of X x Y x {z°} is an isomorphism k _.} L= ®ot k, hence A' is an isomorphism (the sheaves being locally free). Conversely, if L, is trivial, using the induced trivialization of L1, the map P(L,) -+ r(Ll) becomes the map F(Oz,) -* 1(49z,),

which is surjective. Now, fix an isomorphism L = Oaxr. The obstruction to lifting a to I' (L!) is then an element e C- 1'(X x Y, OX, y). Since the restrictions of L to %x (y0) x Z and {x°} x Y x Z,

hence also to X x {y0} x Z, and {x°} x Y x Z are trivial, the restrictions of a to X X {yo} x Zl and {x°} X Y X Zl can be lifted to

L I X x {y0} x Z. and L I {x°} x Y x Z, respectively. This means that the image of by the maps B'(X x Y, OxxY) Bz(X, Oz) and H'(X x Y, OZxz) a B'(Y, OY) induced by x r-+ (x, yo) and y i--+ (x°, y) are zero. But by the Kiinneth formula, these maps induce an isomorphism $'(X x Y, OxxY) z B'(X, OX) ® H'(Y, 0Y). Therefore 1 = 0, and a can be lifted to X x Y X Z$, and L' is trivial. This is a contradiction, so Z' contains an open neighborhood of z°. Basic Theory of Group Schemes. We continue to work over a fixed algebraically closed field k, with schemes always of finite

11.

type over k, and with closed points only. Sob will denote the category of schemes of finite type over k. One of the most basic tools in the theory of schemes is the concept of S-valued points: if X and S are schemes, an S-valued

point of X is a morphism from S to X. The set of all such is denoted Homy (S, X) or %(S).

If X is fixed, the map S i--t X(S) is a contravariant functor: X: Sch°

) Sets.

The importance of this functor is this: ifX and Y are two schemes, then (a) a morphism f: X -> Y defines a morphism from the functor

94

ABELIAN VARIETIES

X to the functor Y (i.e. a. map X(S) that for every g: S 3 T, the diagram

S) AS

0 Y(S) for every S such

X(S) E----- 1(T) f(S) I

11(T)

Y(S) ( ---- Y(T) commutes), and (b) conversely, a morphism from the functor X to the functor _Y is defined by a unique morphism of schemes f: X-+ Y. (a) and (b) are really tautologies holding for any category: the reader should prove them for himself if he has not seen them. This

remark will turn out to be an excellent tool for constructing morphisms as we will see. Formally, (a) and (b) say that

XHX is itself a fully faithful functor from the category Sch to the category Fun (Sch°, Sets) of all contravariant functors from Sch to Sets, hence Sch is equivalent to a full subcategory of Fun (Sob°. Sets). Cf. Mumford (M 3), Cb. 2.

If Alg denotes the category of k-algebras of finite type, and R e Obj Alg, let us put X(R) = X(Spec R). Then X defines a covariant functor from Alg to Sets, which functor again we denote by the same symbol X. The elements of X(R) are also referred to as Bvalued points of X. If ' = Fun (Alg, Sets), it is once again an easy matter to show that Homse,(X, Y)

Hom, (g, '') is bijective,

so that Sch can be identified with a full subcategory of W. DEFINITION. A group scheme is a scheme a together with (a) a

multiplication morphism m: 0 x 0 -; 0, (b) an identity point, i.e. a morphism e: Spec k -* 0, and (c) an inverse morphism is 0 -gyp, such that the following axioms hold.

ALGEBRAIC THEORY VIA SCHEMES

(1)

95

(Associativity). The diagram m x 10

GxG t m

m

G

is commutative. (2)

The diagram 1a x e

1a

ex1Q is commutative. (3)

The diagram

GxG

0

Speck

GxG is commutative.

G

ABELIAN VARIETIES

96

Now let 0 be a scheme, and let us interpret the conditions for giving a structure of group scheme to G in terms of the functor which it represents (either on Sch or on Alg). In view of our earlier remarks and the fact that products in Sch correspond to products

of functors, m, i, and e can be interpreted respectively as maps G(S) x G(S) - G(S), g(S) _G(S), and as giving a distinguished element of G(S), functorially in S in the obvious sense. The conditions (1)-(3) then simply say that G(S) is a group for each S with in, i, and e defining the, group law, inverse and identity element,

and that for all morphisms S' -* S of schemes, the induced map g(S) -r G(S') is a group homomorpbism. We thus see that to give a group scheme structure on a scheme G is equivalent to making the set of S-valued points of G into a group for every S, functorially in S. It would also be enough to make the set of R-valued points of G into a group, for every k-algebra B, funetorially in R.

If x is an ordinary point of G, then the morphism m

R=: G _=L), G x {x} c G x G -) G is an automorphism of G, called right-multiplication by x. More

generally, if x: S -# G is an S-valued point of G, x induces an automorphism B. of G x S over S, which we can call rightmultiplication by x. We use mo(1a x x): G x S -+ G and let B. (mo(lo x x), p$) so as to get a commutative diagram.

axS

R'

=

axB

a

It is easy to check that with the group structure on G(S), we get R,.,, = R o R. Interchanging the factors in G x G, we can define left-multiplication L. similarly. LIB ALGEBBes. Let % be a scheme, and Ll, the sheaf of Kabler

differentials on % over k. By a vector field on I we shall mean

ALGEBRAIC THEORY VIA SCHEMES

97

a k-linear map of sheaves D: Ox -# Ox such that the induced map D: O%(U) --> O1(U) is a derivation over k, for every open set U. This is equivalent to saying that D factors: d

where f is an Ox-linear homomorphism of sheaves.

By a tangent vector at x e $, we mean a derivation 08,, -r k, or equivalently an element of Hom

Oyy

(QSx, k). Now, it is well known

that if V. is the maximal ideal of O,r. the natural map

-T S2g x ®

k,

tQ

is an isomorphism. Thus, a tangent vector is uniquely determined

by giving a linear form on T /V . Clearly, a tangent field in a neighborhood of x determines a tangent vector at x (by composition

of the derivation Oa,.--r ex, and the evaluation map Og,x which we shall call the value of the vector field at x. Let $ and Y be two schemes, and D a vector field on I. If pp (i = 1,2) denotes the i'1° projection of X x Y, we have a canonical isomorphism p*(S2Z) ®pQ (Sbp) - S2rxY, so that there is a

unique vector field D ® 1 on % x Y such that when factored through S2zxr, D® 1 agrees on p(f1) with D and it is zero on pz*(f.), i.e., D® 1(.f(&g) =Df®g. Now let 0 be a group scheme. A vector field D on 0 is said to be left invariant if the following diagram commutes:

D

1®D

ABELIAN VARIETIES

98

the vertical maps being the natural ones induced by the multiplim

cation morphism 0 x 0 -r G. PnorosiTIorr. For any tangent vector t at e to 0, there is a unique left invariant vector field on C having the value t at e.

Paoor. First we interpret tangent vectors and vector fields in a slightly different way. Let A be the k-algebra 1C[e]/(ea) and ri: k[e]/(e$)-+ k the homomorphism with q(e) = 0. If A and B are k-algebras and Bis an A-algebra, the k-derivations D of A in B are in one-one correspondence with algebra homomorphisms 0: A --r

[ ] = B ®r A such that ¢(a) = a.1 + multiple of c. This n'

correspondence is given by D -(- 0, q(a) = a.l + (Da).e. Thus, we deduce that if X is a scheme and x a point of X, a tangent vector at x is a morphism Spec A -+ X such that Spec (k)

Spec'A -+ X is the point x; and a vector field D gn X is an automorphism over A: D $ x Spec A

S x 8peo A

which restricts to the Identity la: X a X when you look at the fibres over Spec (k) I

I

Spec (A).

One then sees easily that a vector field D on a group scheme 0 is left invariant if and only if for the associated automorphism D, the diagram G x G x Sp ec

A-

la x D

Ox0

jinx1

Imx1 O x Spec A

-

x S pec A

D O

x S pec A

ALGEBRAIC THEORY VIA SCHEMES

99

is commutative. If D'= p1oD: G x Spec A - G, then, in terms of S-valued points x, y of f and l of Spec A, this diagram says: D'(z. y, 1) = x.D'(y, 1).

This clearly holds if and only if D'(x,l) = x.D'(e,l) for all x and 1. In other words, if 7 equals ploDo(e,l): Spec A-+ G then we want D to be right-multiplication by the A-valued point t of G. Therefore given any A-valued point t of U, we get a unique automorphism D of G x Spec (A) such that (*) commutes and p1oDo(e,l)= t. But

this means exactly that D is left invariant and has value t at e. Thus we get a canonical isomorphism of the k-vector space of left invariant vector fields on U and the tangent space at e to G. Now, given any two vector fields D1, Dx on a scheme S, considering

them as endomorphisms of C., we see that D=[D1, DQ]=D5D2 D2D1 is again a vector field on S, called the Poisson bracket of Dl and Dz. Furthermore, if char k= p> 0, D-1 is again a vector field.

If U is a group scheme, one verifies trivially that the Poisson bracket and pw' power operation take left invariant vector fields to left invariant vector fields. DEPixrrrox. The Lie algebra of a group scheme is the k-vector space of left invariant vector fields, together with the operation of Poisson bracket defined on it, as well as the pt" power operation if

char k=p> 0. We shall agree to denote the Lie algebra of U, H, G'1, ... etc. by g, Ij, g1, ... etc. We have then: (1) The map g x g -+ g, (%, Y) i- [S, Y], is bilinear over k and the map % i--. %p satisfies (AX)' = A'X'. (2)

For %eg,[%,$]=0.

(3)

For %, Y and Z e g, we have

[I, [Y, Z]1 + [Y, [Z, S]] + [Z, [I, Y17 = 0. (4) If char k = p > 0, there is a certain universal non-com-

mutative polynomial F. (depending only on p) in two variables such that

100

ABELIAN VARIETIES

ad (XP) _ (ad X)',

(X+ Y)'=X'+YP+FF(ad X, ad Y) Y, where ad X is the endomorphism of g defined by ad X(Y) _ [X, Y].

For the actual expression of FP, which is unimportant for our purposeg, we refer to [J]. It is however uniquely determined by the condition that if A is an associative algebra over k and we define [X, Y] = XY - YX and XP to be the pth power in A, A satisfies (4). Thus, P. may be calculated by taking A to be a free associative algebra on two generators X and Y over k, and showing that (X ± Y)P can be expanded as in (4) in A. Since we are mainly interested in the case when g is "abelian", that is, when [X, Y] = 0

for X, Y eg, it suffices for us to know that FP has 'no constant term', that is, FP(0, 0) = 0. Thus, in the abelian case, X -* X' is just a p-linear map (i.e. an additive homomorphism of g into itself with (AX)1' = APXP). In the case of the Lie algebra of an abelian variety, the pth power map is called the Hasse-Witt map.

We remark that if G is commutative, that is, if the diagram

is commutative, where a is the `switch map' a= (P21 PI) then g is abelian, 'hat is, [X, Y] = 0 for all X, Y e g. To prove this, we start with the following observation. Let D; (i = 1, 2) be vector fields on any scheme X, D3 = [D2, Dl] and .D;: X x Spec A -+X x Spec A be the associated automorphisms, where A = k[ej/(E2). Put A' = k[E, E']/(E2, e'2), and let of : A - A' be the 7o-algebra homomor-

phisms defined by al(e) = E, a2(E) = E' and aa(E) = EE. Then a, induces a morphism ¢i = Spec a,: Spec A' -* Spec A (1 < i < 3), and we get autollnorphisms

ALGEBRAIC THEORY VIA SCHEMES X x Spec A'

X`

101

X X Spec A°

Spec A'

by taking fibre products with Spec A' over Spec A. via i. One then checks easily (by taking S affine) that Xs is the commutator [Xv Xz] = X1 Xz XT I Xi 1 Now suppose G is a group scheme, let ti :

Spec A a 0 (i = 1, 2) tangent vectors at 0, and let Di be corresponding left invariant vector fields. Then Di is just right-translation with respect to ti, and if Ti : Spec A'-+ G, then Xi is right translation of G x Spec A' by the point Ti e G(A'). Hence X1 Xs Xi 1 Xz

1 is right translation by [T1i T21 e G(A'). Since G(A')

is a commutative group, it follows that [Ti, T2] = 0, hence X1 Xs X1 _3L Xz 1 = Xs = Identity, and [D1, D2] = 0.

THEOREM. Any group scheme over a field k of characteristic 0 is smooth (and in particular reduced). PROOF. We show in fact that if % is any scheme over lc of charac-

teristic zero and x a point of % such that there exist vector fields

D1,..., D. in a neighborhood of x with is = dimk

inducing X

independent tangent vectors at z, then % is smooth at x. We can choose xi (1 < i < n) in llQ such that they form a basis modulo I1% of

and (Di x,)(x) = Sip. Since clearly Di(9J7ix) c 9)2x'"1, Di

extend to derivations of the completion Ox of &,, Now, we have a unique continuous k-homomorphism a : k[[t1,... , t ]] -s Ox with a(ti) = xi. On the other hand, define fl : OZ k[[t1, ... , 4J]

by putting Df

R(f) _

(x).t

°i30

where D° f = Dop ... D, "nf, v! = v1! ... vn! and t° = t1'i ...

By

Leibniz's formula and induction on is, one checks trivially that P is again a continuous k-homomorphism of local rings. Now,

102

ABELIAN VARIETIES

a is surjective since its image contains a set of generators of M . and k[[t ... , t"]] is complete. On the other hand, P(x;) = t{ (mod(t ... , t")2) so that (3(xi) generate the maximal ideal of k[[t ..., t"]], and P is also surjective. Hence so is ,Boa, -so that flow

is an automorphism of k fft ..., tj]. Hence a is also injective, hence an isomorphism. Hence 0, and Oa are regular, and X is smooth at x. In positive characteristic, we will soon have plenty of examples of non-reduced group schemes. SUBGROUP SCHEMES, KERNELS, QUOTIENTS.

Let G be a group scheme, and H a closed subscheme. We say that H is a subgroup scheme if, denoting by i: H c . 0 the closed immersion, mo(i x i): H x H -a 0 factors through B: HxH

G

This is equivalent to saying that for every Se Obj Sch, H(S) is a subgroup of 0(S). ClW,rly H then becomes a group scheme in its own right, and is H -> G a homomorphism of group schemes (defined in an obvious way).

To any group scheme C over k of characteristic p> 0, one can associate in a natural way an increasing sequence 0" of closed subschemes (n> 0), all having support at the identity element of C, as follows. Let 0 = 0a be the local ring at the identity to C, M its maximal ideal, and X!P'> the ideal generated by {x "ix EM). Put G.= Spec so that G. is a closed subscheme of 0 with support at e. Let 0'=Ocas>,ax be the local ring of (e,e) on C x 0, so

that 0' is the localization of 0®r 0 with respect to the maximal ideal X12® 0 + 0(& M. The multiplication map m induces a local homomorphism of rings m*: 0 -* 0', so that for f c- 9, m*(f) = g/h,

ALGEBRAIC THEORY VIA SCHEMES

103

g e TZ 0 0 + 0® IJ1, b a unit in 0'; hence m*(fp") e [9JR('") ® 0 + 0 ® DlP">]. 0'. This proves that the composite Q. x G. 3 m

G x G -+ G factorizes through the subscheme G,,, proving that Gn is a subgroup cheme. One has evidently Lie (G") = Lie 0. for n >1. In any characteristic, define G", the additive group over k, as Spec k[T] = Al, the group law G. X G. -+ G. being defined by addition, or equivalently, by the homomorphiem m*: k[T] -- k[T] ® k[T], m*(T) = T ® 1 + 1® T. The group of EXAMPLES.

(1)

S-valued points Gg(S) is just the group r(S, o.). If char k = p > 0, we also write otpn for the subgroup scheme (G,)", that is IXpn = Spec [ kIT I]. We have Lie (Gn) = Lie (ayn)

La/OT, n > 1, and ocpn(S) = (f er(S, es) ; f'" = 0}.

In any characteristic, define G,,, to be the multiplicative group over k, coinciding as a scheme with Al - {0}, the group law being multiplication. Writing G. = Spec k [T, 1/T], the homomor(2)

phism m* : k[T, 11T] -> k[T, 11T] ®k k[T, 1/T] is given by m*(T) =

TO T. The group of S-valued points of G. is the group of units

r(s,e

.

If characteristic k = p > 0 we shall put a n = (G.),, = Spec k[T, T-1]/((T - 1)e) = Spec k[T]/(TP' - 1). We have Lie Q. = Lie Elpn = k.T a/aT (n > 1), and lsl,n(S) _ {f e r(S, op l fp"= 1}. Now, let G and H be group schemes and f : G -> H a homomorphism of group schemes. The fiber f-1(e$) = G x${eH} over the closed point e$ of H which is the identity element of H is a closed subscheme K of G. By definition, for any S-valued point of 0, ¢ : S -> G,

factorizes through K if and only if fo factorizes

through e$ : Spec k->H, or in other words, K(S) is the kernel of G(S) -NH(S). Therefore K is a subgroup scheme of G. As an example, consider the homomorphism G.

G. defined

by % P. The kernel is denoted by p,,. For (n, p)= 1, w is just

ABELIAN VARIETIES

104

a discrete group (i.e. reduced and finite group) isomorphic to the nt" roots of unity in k*. On the other hand, it follows from definitions that µ9 is the same group defined earlier.

Next, suppose 0 and H are group schemes, and 0: H -> G a homomorphism. A pair (0/H, al), where G/H is a scheme and .1 : G

0/H is a morphism, is said to be a quotient of G by

H, if it is universal for all pairs (8j) where S is a scheme, and f : G S a morphism such that the following diagram commutes: M. (¢x 1) Hx 0 I P1

f

It can be shown that quotients exist, that f is always flat and surjective, and further that when H is a normal subgroup scheme

in 0 (i.e. H(S) is a normal subgroup of a(S) for all S), 0/11 inherits a unique structure of group scheme such that

: 0-+ 0/H

is a homomorphism and has kernel precisely H. We will not need the result in this generality, but only in the special case when H is a finite group scheme. We will consider this in §12. LEI+T-xNvauLANT DIFFERENTIAL OPERATORS.

We want to study the algebra of maps OQ a OQ generated by left-invariant derivations. We first introduce the hyperalgebra

Hof0: H. = Ho+n..t(0, (;, k)

H=®% MG

where Hom o, means the maps L : 0, -). k which are continuous in the sense that L(U?N+') = (0) for some N. Alternately,

H=

lim O.dim-BIIb9eheu, ,

ZCQ

r(dz)*

ALGEBRAIC THEORY VIA SCHEMES

105

where W* means the k-vector space dual to a vector space W. If

b e H lies in the subspace r(O,)*, we will say that L is supported by Z. This definition makes it clear that H is an algebro-geometric analog of the space of distributions on a Lie group supported on a finite set. H has a lot of structure. (1) We get an associative and distributive convolution product

*:H®H-+H by defining (§)

Li*L2(f)=L1®L2(m*f)

More precisely, if L; is supported by Z;, i = 1, 2, and if Z3 c G is a finite subscheme such that the group law m factors : Z, x Zx ---n

_* X. n

then L1 * L2 is to be supported by Z,, and the equation (§) is to be interpreted with f e r(OZ,) and with in as the restriction of in to Z, x Z. (2)

Evaluation at any point x e G is a continuous linear map

Sx : Os -+ k hence an element Sx e H supported by the point x with reduced structure. S, is a two-sided identity for convolution:

8,*L=L*8,=L. Moreover, evaluation of elements L E H at the function 1 e r(Oa) induces an augmentation

6:H -.- k. Note that if 0 is finite and reduced, the Sx's are a basis of H over k, and since S= * 8, = 5,w, H is nothing but the group algebra k[G] of G.

ABELIAN VARIETIES-

106

The ordinary product of functions 0x ®05 -> 0x induces a co-product (3)

8:H-+H®,H. This satisfies many identities, which we will consider later in § 14 for finite commutative group schemes 0.

The interesting thing is that the elements L e H extend uniquely to left-invariant operators DL : 0a Oo such that, roughly, (DL f) (e) = L(f). These operators DL are combinations of differen-

tial operators (which for every open set U c 0, map 0o(U) to O&(U)) and translation operators f --> f', f'(x) = f(xa), (which map e10(U) to 0a(U.a 1)). We need some definitions.

DxarxITION. A differential operator D on a scheme X is a klinear endomorphism of the structure sheaf 0g such that there is an integer N > 0 having the property that if f e q'1is +1, some x e S, then Df(x) = 0. The least such N is called the order of D. For example, the differential operators of order 0 are multiplications by functions, and those of order 1 are the sums of derivations plus multiplications by f.'s. Now say L e H is supported by

Z= r

U Spec (0.'1D;j).

c-1

We then define an operator DL : 0a -> Oo which consists of a set of maps

DL: 0g(U)) 0a(V) whenever V.aa c U, 1 < i < n, compatible with restrictions. We define DL to be the composition :

10L

res

m* 0,

Oax0

) 19axz

'

It is easy to see that DL is a sum of operators DL { which are given by differential operators of order < d;, followed by right translation

by ai. In particular, if L e H,,, then DL is a differential operator. Moreover, DL is left-invariant, i.e. the diagram

ALGEBRAIC THEORY VIA SCHEMES

107

DL

1®DL

commutes. In fact, substituting the definition of DL, this diagram becomes the outer rectangle of 1® L

aG

aGXZ

1®1® L

1G®m* aGxG

0 aGxGxZ

The left-square commutes by associativity of the group law m, while the right square obviously commutes. It follows immediately from the definition that if f e aG(U), and a, e U, 1 C i < n, then DL f is defined at e and DLf (e) = L(f ). The correspondence L a--r DL generalizes the isomorphism T,,G {left-invariant derivations) used in defining the Lie algebra of G. In fact, the tangent space T, G can be naturally identified with a subspace of H: Tee

(L: a,,G- kIL(1)=0,I(l6)

(0))CH,

and if L e Tee, then DL is the unique left-invariant derivation with value L at e. An important fact is that convolution of L's goes over to composition of operators : (*)

DL l_, = DLi c DLL.

The proof is straightforward and is left to the reader. In particular, (*) shows that the Poisson bracket of left-invariant derivations can be interpreted by a commutator with respect to convolution product in H and that the p°D power on left-invariant

ABELIA1 VARIETIES

108

derivations can be interpreted as P power in H. Note that if 0 is commutative, then m*: C0 - apxp is co-commutative, i.e. COxQ

"switch factors"

commutes, and therefore convolution product in H will be commutative too.

This gives us a second proof that if 0 is commutative, then the bracket on its Lie algebra is zero. To illustrate hyperalgebras, look at the simplest cases 0 = p, and U = z . Writing lrp = Spec k[X]/(%p - 1) ap = Spec k[%]/(%p),

then in the first case all left-invariant differential operators are given by

f 0r ao f -h a1i df -1- ... +a,-,. (x d.X 1lp

f

and in the second case by

f r- p a0f+ a1 dX df -F...

ap_1

dp-1

dg"f

Thus H = k[D]/(D' - D) or = k[D]/(D'), where D = X dg or

12.

Quotients by Finite Group Schemes.

scheme U on a scheme S is a morphism:

p,:Gx%%

An action. of a group

ALGEBRAIC THEORY VIA SCHEMES

109

such that

(i)

the composite

X e Spec (k) x X is the identity ; (ii) the diagram

ex1

GxX P ) X

in x 1$

GxGxX

µ

GxX is commutative. (Here, as usual, e is the identity and m is multiplication.) This is equivalent to saying that G(S) acts on X(S) for every scheme S (or even every affine S), functorially in S. It is also equivalent to saying that for every S-valued point x e G(S), we are given an automorphism over S:

`XxS

XxS

such that (i)

(ii)

if x, y E G(S), then Tzo T = T0.3,;

if f: Si

S2 is any morphism, x: S2-.G is an S2-valued

point, so x of: SI - G is an S1-valued point, then Tr,f X XS'

lrxf XxS2-T.

commutes.

ABELIAR VARIETTh

110

Explicitly, the Tx's are induced by p via the formula T. = .(pooo(la x x), p2) where o: % x G -+ G x % is the switch morphism; and conversely, if we let S = 0 and take x to be the Gi-valued point la : 0 -s 0 of G, then p, o T.: X x G - % is just (except for the factors being reversed).

A morphism f : % -* Y is said to be 0-invariant if the diagram P

f

is commutative, i.e. in terms of S-valued points g and x of 0 and %, f(p(g, x)) = f(x). In particular, taking Y = A', we get the notion of a 0-invariant function. We say that the action of 0 on S is free if the morphism

(1k,p2):GxI*%x% is a closed immersion.

The action of a group 0 on a scheme % has a differentialgeometric as well as a set-theoretic aspect. Let H, be the part of the hyperalgebra of 0 with supports at e, and let L e H. Then L defines a differential operator Dy on % via

p" OX

)

L®1

Oax I ---) Ow. s - 09x-

It is easy to cheek that (a) DL,.L. = DL. o DL,, (b) Da, is the

identity, and (c) if L E T.,o c H then DL is a differential operator of order 1, i.e. a derivation from Oa to 0a. In particular, the Lie algebra of G is represented by derivations of Ox. Let °r be a coherent sheaf on X. A lift of the action Is to F' is by definition an isomorphism d: p*2(. ) -=* p'(J'F) of sheaves on

O x % such that the diagram of sheaves on 0 x 0 x S,

ALGEBRAIC THEORY VIA SCHEMES

111

r f*(,Ir)

19009

where 1 = p o (pz, pa), +] = P o (m x lg) = µ o (1° x /A), and pi is the ith projection of G x 0 x X, is commutative. A more manipulable way of defining a lift of an action µ to is to require, for every S-valued point f of 0, an automorphisml1 of the sheaf .IF ®Os on X x S covering the automorphism 1L1 of X x S -5F(& 49s

hJ

IxS

.IF ®Os

XxS P1

such that (1) the Vs are functorial in f, and (2)

o A,.

Having stated these definitions, let us generalize the principal results of §7 on quotients by finite groups to quotients by finite group schemes. The following theorem is proved analogously to the first proposition of §7, so that we content ourselves with stating the necessary modifications. THEOREM 1. (A) Let Gi be a finite group scheme acting on a seheme $ such that the orbit of any point is contained in an affine open subset of I. Then there is a pair (Y, a), where Y is a scheme and a: I -s Y a morphism, satisfying the following conditions: (i)

as a topological space, (Y, a) is the quotient of % for the

action of the underlying finite group; (ii)

the morphism a: % --) Y is 0-invariant, and if a*(Ox)°

denotes the subsheaf of a,(Ox) of p-invariant functions, the natural homomorphism OY -+ a,(Os)° is an isomorphism.

ABELIAN VARIETIES

112

The pair (Y, a) is uniquely determined up to isomorphism by these conditions. The morphism 7r is finite and eurjective. Y will be denoted

X/G, and it has the functorial property: V G-invariant morphisms f: X - Z, 3 a unique morphism g: Y a Z such that f = govt. (B)

Suppose further that the action of G is free and G = Spec (B),

n = dim2R. Then yr is a flat morphism of degree n, i.e. ir, (Ox) is a locally free Oy-module of rank n, and the subseheme of X x X defined by the closed immersion

(,A,p2):0xI- X xX is equal to the subscheme X x y X c X x X. Finally, if F is a coherent Oy-module, n*-5F has a naturally defined G-action lifting that on X, and .

F i.

1r*.5F

is an equivalence of the category of coherent 0y-modules (resp. locally free 0y-modules of finite rank) and the category of coherent Ox-modules with G-action (resp. locally free 0x-modules of finite rank with G-action). PROOF OF (A). As before we are reduced to the case when X = Spec A is affine. Let R be the ring of G, e: B -+ k the evaluation

map at e, m*: R-a R®k R and lc* : A -+ R®k A the homomorphisms o f k-algebras induced by m and s. LetB=AG={aeA11s*(a)=1(D a) be the algebra of G-invariants in A. Let Nmd : R ®kA -s A be the norm mapping (defined since R (&kA is free of finite rank over A), so that Nm is a homogeneous polynomial function over A of degree

n = dimk(B) which is multiplicative. Define N: A -+ A by putting N(a) = Nm(,e*(a)), so that N is again multiplicative and k-homogeneous of degree n. We assert that N(A) c B.

To prove this, we have to show that for any at e A, µ*(N(a)) = 1 ® Na. For any k-algebra B, denote by NmB the norm mapping R ®k B -a B. Define A - . R (9,A and : B ®kR ®kA -+ R ®kR ®kA by setting ¢(a) = 1® a,

( ®n0 a)=(m*(6)0 1)(101®a).

ALGEBRAIC THEORY VIA SCHEMES

113

Note that if f : B -* C is a homomorphism of k-algebras, we have Nmc o (1R ®f) = f a NmB. We thus have

14*oN=p*oNmA cp* = NmR®kA c (1R ®p*) 0 p NmR®kA o (m* 0 1A) o p* = NmR®kA ° # ° OR ®Sti) ° p*.

Now, if we consider R ®,kR ®kA as an B ®kA algebra via the homomorphism R ®kA -> R ®,t R ®kA given by 77 ® a E--r

1 ®77 ® a, 0 is an automorphism of the B ®kA algebra R®kR(%A, so that NmR®kA U 0 = NmR®.LkA. Thus, we obtain

*oN=NmR®A °(1R®0)

=#0NmA0 N.

This proves our assertion. Now, 0 also acts on % x Al, (by acting trivially on A'), so that N; A[T] -SA[T] is also defined. For any aGA, put X"(T)=N(T-a). Then X"(T) = T" + s, T"-' + + s" is 0-invariant, and is the characteristic polynomial of the endomorphism of the free A-module

R ®k A defined by the, element p*(a). Since e ® 1: B ®k A A is surjective and (E® l)(p*(a))=a, p*(a)- a defines the zero map on the quotient A of R®kA(via E(& 1), hence det(p*(a)-a)=X.(a) =0. The equation Xa(a) = 0 shows that a is integrally dependent on B. Thus A is integral over B. Since A is finitely generated over k, A is also integral over a finitely generated subalgebra of B. Thus A

and hence B is a finite module over this subalgebra, and so B is also finitely generated over k. Let Y= Spec B and let ar : X -s. Y be the induced morphism. Then it is finite and surjective. Using the map N, one sees as before that ar separates orbits, so that (i) holds. Further iris clearly 0-invariant, so that we have an inclusion Or c a5 (O1)°, inducing isomorphism of global sections. On the other hand, sr* (O1)° is a coherent OY module, being the kernel

114

A33ELLA-W VARIETIES

of the Op-homomorphism A : vr* (O$) -> a* (Og) % R, h(f) = µ *(f) - f ® 1, and this shows that Op = zr* (Or)e. This proves (ii).

f

PaooF of (B). For the construction of the G-action on x*(. -),

the basic fact is the following. If X

9

o Z are morphisms *Y and Pr a sheaf of Oz-modules, there is a natural isomorphism , J,(J""): (ge f)*(. ) ...

dition: if X

f ) Y 9)

f *g*h*(-,)

f*(g*(.F)), satisfying the following con-

Z -+ T are morphisms, the square f *(a,,A)

f *(hog)*(.f)

kh. (gof)*h*(. ) -4

j6h

(hog °f) *(F)

is commutative. Returning to our special case, the equality of the two composites

µ if Q x X: % --- Y enables us (by means of A) to define an isoPz

morphism A: P'`(g)- µ*(g), g = a*(F), and one checks that this is an action, using the above remark. On the other hand, let g be a sheaf on $ with a 0-action covering the action on X. A section a e g(X) is said to be ii-invariant if A(02*(o)) = µ*(o). Localizing, we have the notion of the subsheaf

A*(g)° of G-invariants of a*(g). This is clearly an Op-module which is coherent, being the subsheaf of it* (g) where two Oy-homomorphisms ir* (g) ir* (g ®Y B) coincide. As before, we have natural transformations S(°r): , ,r* (a*(. ))o and evident T(g): ar*(a* (g)a) -r g, and it is again sufficient to show that (when

the action is free) n is flat, G x

X x r X, and T(g) is an

isomorphism for every 0-sheaf g on % (as in the proof of Proposition 2, §7). In view of these remarks, since all the defini-

ALGEBRAIC THEORY VIA SCHEMES

115

tions localize, we may assume X = Spec A affine. Recall that the freeness of the action means that the morphism

(µ, p2): a x X ---) X X X is a closed immersion. Since it is U-invariant, (,z, p2) factors through a closed immersion of U x X into X x r X. On the ring level, this means that the homomorphism

A: A®BA-0 R®xA A(ai 0 a2)

(9 a2)

is surjective. We have to show (a) that A is flat over B = A° and A is injective, (b) if g is a G-sheaf on X with g(X)=M, A®BM°_.M is an isomorphism, and (c) if M is projective over A, M° is B-projective. Note however that (c) is an immediate consequence-of (a)

and (b). In fact, it suffices to show that M° is B-flat, i.e. the functor N.--i N®A° M° is exact on the category of A°-modules,

and since A is faithfully flat over A°, it

suffices

to show

that N r- p (N®AG M°)® A° A = (N® A° A) ®A, (A (DAa M°) (N ®A° A) ®A M is exact, and this holds in view of (a). To prove (a), we may pass to the ring of quotients of A and B with respect to S = B - 9, where 9R is a maximal ideal of B, so that we may assume B local and A semilocal.

If we consider A ®B A and R ®x A as A-modules through their second factors, A is a surjective A-homomorphism, so that R®k A is generated by s*(A). Since A is semilocal and µ*(A) generates the free module B®k A, one shows easily that there are {a;} (1 < i < n = dimr B) such that fc.*(a;) form a basis of B ®k A over A. Now suppose a, Al, ... , k eA. I claim:

aA, a, andA1..... keBl.

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ABELIAN VARIETIES

is obvious by applying 1s* and using the fact that Ec*(A;) = (1A-) if k. e B. To prove . , use the fact that The implication

(1®Ec*)(µ-*a) = (m*® 1)(,u*a) in R®k R®,, A, hence substituting the expansion of &* a, we deduce that A

A

Since the is*(af) are linearly independent over A in R ®k A, (1®p.*)(fc*a;) are linearly independent over R®k A in R®k R ®k A,

the latter being considered as an algebra over the former via the last two factors. Hence the above equation yields that 1®p*A; = 10 10A, i.e. A; a AO '= B. Applying the homomorphism a®1 to the equation µ*(a) = E(1®AM®BA #(m) =m®l -T(1®m). Since A is flat over B, it follows that NOB A is the kernel of the homomorphism +G=

#(m®a)=m®1®a...T(1®m)®a.

In other words, N®BA={2mi®aiaM®BAI

m:®1®ai=1 (10 i)®ai}.

But the associative law (a), applied to the element 1®1® m e A® A® M, says exactly that 7(10 m) a M®B A has the property described in this equation. Therefore regarding NOB A and T(1®M) as two subsets of M®BA, we find N®BAD T(1(gM). Now both of them are modules over the subring 1®A c A®BA. Moreover, N®BA is generated over this ring by the elements n®1, n e N, and since 7(1®n) = n®1, these elements are in T(1®M). Therefore NOBA =T(1®M). But finally the maps

118

ABELIAN VARIETIES

x

lox

T

),r(lox)

m i- 1®m are isomorphisms, since A is faithfully flat over B, hence we obtain

an isomorphism N ®sA = M. This is clearly the canonical map too, so (b) is proven. DEFZNrrION. A homomorphism f : X --> Y of group schemes is an epimorphism if f is surjective and f* : Or ->f* Ox is injective.

Suppose X itself is a group scheme and G is a finite normal t subgroup scheme, acting on X by right-translation. COROLLARY 1.

Then %/G is again a group scheme, Tr : X -> 1/G is an epimorphism, and G= ker (or). Conversely, if f : X --> Y is an epimorphism of group

schemes, and if G = ker (f) is finite, then Y = I/O. In other words, for every group scheme X we get a Galois-type correspondence between (a) normal finite subgroup schemes 0, and (b) finite epimorphisms zr : X --> Y. In fact, if the word "finite"

is dropped from (a) and (b), the correspondence is still correct, but we will not prove this. PRooF.

First, say X is a group scheme and G c X a finite normal

subgroup. Let m : X X X - X be the group law and consider the solid arrows in the diagram:

XX X rr X 4r

m

7r

X/G. X/G x X/G ---Then X/G x X/G is the quotient of X x X by G X G, and I claim that Tr o m is a G x G-invariant morphism. In fact, if

x1, xg, g1, g2 are S-valued points of X and G respectively, then 7r(-(x191 x xs92)) =r(x191'x292) =ir(xlx29s) =ir(x1x2)=7r(m(xj X X2))

where gs = (xz 191x2) g2 a G(S). It is easy to check that the dotted t Normal means G(S) is a normal subgroup of X(S) for every S.

ALGEBRAIC THEORY VIA SCHEMES

119

arrow defines a group law m' on %/0 in terms of which Tr is a

homomorphism. Now since G x 8 v I x y I, it follows that if xl, xz a %(S), then 1r(x1) =vr(x.) if and only if

some g e 0(S).

In particular, if K = kernel (n), xl a K(S) . a(xl) = rr(e) r xl = g, some g e CA(S). Thus 0 = kernel (Tr).

The second half of the corollary is harder. Let Tr: %-+ S/0 be

the canonical map. In view of the 0-invariance of f, there is a unique g: I/O -> Y such that n

f

I/O ------ Y g

commutes. By the first half of the corollary, I/O is a group scheme, and one checks easily that g is a homomorphism. In fact, g is also an epimorphism with trivial kernel and we are reduced to proving the special case that an epimorphism f: S -> Y with trivial kernel is an isomorphism. We recall that if g: S --> T is any morphism such that

g-1(t) is a finite set for every t c- T, then there is an open dense

set To c T such that if So = g1(To) gis, : So -> To is a finite morphism. In our case, say fix.: So --> Yo is finite. Then for every

x e %, if R= is right translation by %, we get a diagram: B.

1o ra

Pd

so f is a finite morpbism from X x to Y0. f(x) too. Since the open sets Yo. f(x) cover Y, f itself is a finite morphism. Therefore, to show that f is an isomorphism, it suffices to prove that the homo-

120

ABELIA,PT VARIETIES

morphism f*: Cr -*f* Px is an isomorphism. But we know f* is injective, and by Nakayama's lemma, f* is surjective if for all y e Y, the map

fv:krO.M., is surjective. But Spec(f*Og(& O,/D?y) is the fibre f''(y), and all the fibres of f are isomorphic by translation to the kernel of f. This kernel is trivial, so we deduce that for all y r= Y, f-1(y) is one point with reduced structure. Therefore fy is surjective, and j is an isomorphism. COROLLARY 2.

Let Y = I/O as in the theorem. Let g be any

coherent sheaf on X acted on by G. Then there is a natural isomorphism 7r*(-*g) = g ®x R.

P$ooF. Given the situation:

X'

g'

f

X

If Y

Y, 9

and a sheaf g onX, the natural homomorphism f *(f* (g)) -g induces g'*f*(f* (g)).+g'*(g), that is, f'*g*(f* (g)) or what is the same,

g*(f*(g))-f(g'*(g)). If f is an affine morphism and X'=Y' xp X, this is an isomorphism. In fact, the problem being local on Y and Y', we may assume both (and hence all four of X, Y, X', Y') affine, and the assertion is then obvious. Apply this remark with X =Y', f = g=2r, to deduce that ar*(a* (g)) =p2*p; (g) where p{: X xr X i X are the projections. Denoting the ith projection of G x X by q;, and using the isomorphism (µ, q2): O x X -* X X r X and the G-action on g, we get that n*(1r*(g)) = gz,i,µ*(g) = q2*qs(9)=g ®5R, which proves the corollary.

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121

We now want to prove a theorem on the Euler characteristic of inverse images of coherent sheaves for a class of morphisms. For this, we introduce some definitions. Let G be a finite group scheme acting freely on a scheme X such that the quotient X/G exists. Let F be a finite scheme on which G acts. Then G acts on X x F in an obvious way. It is easy to check

that this action is again free, that the quotient U = (X x F)/G V

exists, and that we have a natural morphism U -) V = X/G. The morphism a obtained in this way will be called the fibration with fiber F associated to the principal G-bundle X --> X/G. zr is finite and flat, so that it*(Oa) is locally free over Ov of constant rank. We shall call this rank the degree of v. THEOREM 2. Let nr : U -> V be a fibration with finite fibers associated to a principal G-bundle over V, where G is a finite group scheme. For any coherent sheaf .sue on V, we have X(-*( -Iv)) = (deg ir)

PRooF.

It suffices to prove the theorem when G acts freely on

U and V = U/G since in the general case, we have X x F-* X x FIG -+ X/G and the theorem would be true for the composite and for the first morphism, so that it is also true for the second.

Thus we assume V = U/G for a free action of G on U. For a coherent sheaf F on V, let ./ be the sheaf of ideals annihilating Jr, and call the closed subschemes of V defined by J the support of.°f.

Then F is a coherent sheaf on this subscheme. If the theorem is not true, since V is a noetherian space, we can find an F with support V' c V for which the theorem is not valid, whereas for any

coherent g with Supp g c Supp F, the theorem is valid. Set U' = n '(V'). Then, using (B) of the proposition, one sees that U' is a principal G-space over V'. Replacing U and V by U' and V' respectively, we see that we may assume the theorem to be valid V, that is, whenever Ann F 0 (0). It is further whenever Supp clear that if in a short exact sequence of coherent sheaves on V, the theorem holds for two of the sheaves, it holds for the third (remember it is flat).

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ABELIAN VARIETIES

Now, if V were reducible, we can find a short exact sequence 0--s. 91-+.Fa 92-3`0 with g; having proper support c; V, and the theorem would hold for .97. Hence we may suppose V irreducible. If J is the subsheaf of nilpotent elements of V and 5 0 0 we have

the exact sequence 0 -+ 55 -a.17-. J.O.. a 0 and both 5. " and shave supports proper closed subschemes of V. Thus we may assume V reduced and irreducible. Let r be the rank of the generic

fiber of ,W. Then there is a sheaf of ideals f on V and an injective homomorphism Jr > F with cokernel having proper support. Thus, the theorem holds for .F if and only if it holds for 5', and 0, we see that the theorem holds for 5 if and only if it holds for 0,,. We see therefore that it suffices to prove the theorem for one coherent sheaf on V of non-zero rank. But then, a* (Ou) is such a sheaf, since by Corollary 2 to the proposition, again by the exact sequence 0 --)-f -+ 01, --> OI,/S

X(-*(-*(Ou))) =X(On(&k B) = (deg 7r) X(Ou)

=(deg 'a) X('r*(Ou))

REneess. The reason we have insisted on an associated fiber space, rather than confine ourselves to a space on which G acts freely is the following. Let f: U-* V be a finite etale morphism everywhere of degree n. Then f can always be realized as the associated fiber space with finite fibers of a principal E(n)-space, where E(n) is the symmetric group of order n. In fact, in the n-fold fiber product U x y U x .... x y U, consider the set P of points (x1, ..., such that xx xj for i O j. Then P is both open and plosed in

U x vU x y... x yU and is stable for the natural action of E(n). Further, P is gtale over V, and V is the set-theoretic quotient of

P by E(n). It clearly follows that V is the scheme-theoretic quotient of P by E(n) and P is a principal fiber space with structure group E(n) over V. Let F be the reduced scheme with n points [1, 2, ..., n] on which E(n) acts naturally. Define a map P x F-s V by ((x1..... xx),i) a - x;. This map is invariant for the

action of E(n) on P x F and V is the set-theoretic quotient. Since P x F V is again 6tale, V is the scheme quotient of P X F by 1(n).

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123

Thus, the theorem is applicable to any finite etale morphism of constant degree n. 13.

The Dual Abelian Variety in any characteristic. For a line

bundle L on an abelian variety X, we had earlier defined a closed subset K(L) of X as consisting of the points x eX for which Ts (L) = L, and we had shown that it is a subgroup. We shall now

define a structure of subscheme on K(L). Namely, consider the standard line bundle M= m*(L) ® pl (L)-l 0 p2 (L)-1 on X x X, and define K(L) to be the maximal subscheme of X such that MIK(L) x X is trivial. (See § 10). We can interpret the S-valued points of K(L) roughly as the set of S-valued points f : S-# X such that L is invariant under translation by f. Namely, Xs = X x S, and let T,: Tj

Xs

Xs

be the automorphism of Xs induced by f (i.e.T1(x, s)=(x+f(s), 8) in terms of T-valued points x, s of X, 8).. Let Ls be the induced line bundle pl L on Xs. Now when S is a big space, the condition

27(L3) = Ls is too strong; for example, Ls and 27(L3) can be isomorphic on the sets X x U; for {U;} an open cover of S, without being isomorphic. The correct condition to look at is: (*) N a line bundle on S. Then I claim that (*) holds if and only if f is an S-valued point

T7 (Ls) = L® ® ps* N,

of K(L). P1

X PRooF. Note that the composite X x S -* X X S 1gxf m is just the composite X x S ----) X x X -* X, so Tf (Ls)v

(lgxf)*m*L. Thus Lsl(o>xs is trivial, while Tf(LB)I(o)xs=f*L-

124

ABELIAN VARIETIES

Now whenever (*) holds, the N which occurs can be identified by restricting both sides to (0) x S:

f*(L) c Tf(Ls) I(o)ks=Ls® p2N (o)xs=N. Thus (*) holds if and only if (IX X f)*m*L m p1 L®pz (f*L).

But (Ix xf)*-*L(g)p*L-'(&p,*(f*L)-'=(Ixxf)*M, so (*)holds if and only if (1X x f)*M is trivial, which means, by definition that f factors through K(L). An immediate consequence is that K(L) (S) is a subgroup of X(S), hence K(L) is a subgroup scheme of X.

Our next aim is to construct the dual abelian variety of X, imitating the procedure in characteristic 0. Choose an L which is ample. Then K(L) is a finite group scheme- We define X to be the

quotient abelian variety X/K(L). Let 7r: X > X be the natural homomorphism. As before, we wish to define the Poineare line bundle P on X x X by defining it as the quotient of the line bundle M on X x X by a suitable action of K(L) x {0} lifting the translation action on X x X (it is easily checked that the natural homomorphism X x X/K(L) x {0} -. X/K(L) x X is an isomorphism).

Recall that an action of a subgroup H c X on any coherent sheaf .i7 on X can be described as the giving, for each S e Obj Sch, an Os lifting the action of the (abstract) group H(S) on Fs = action on Xs, this action varying functorially in S in an obvious sense.

Let us agree to denote all objects obtained by base extension to S by a subscript S. Now, K(L)(S) consists of the subgroup of x e X(S) such that if T5: X, Xs denotes translation by x, V. (Ls) = L® ® Lo, where L. is the lift to Xs of a line bundle on S. On Xs Xs Xs = (X x X)s, Ms = ms(Ls) ® pi (Ls)-1® pi (Ls)', so that T(*o)(MS) = mmT.(LS) ® pi T, (Ls)"'' ® p2 (Ls)-' = MS (D MS* (LO)

® pf(Lo 1) = Ms. Thus, TT.o)(Ms) = Ms, and to define a lift of T(Z,o) to Ms, or equivalently an isomorphism of T (*.,o)(Ms) with Ms,

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125

it suffices to give an isomorphism of these line bundles on the subscheme XS x8 Os. This is because any two isomorphisms, either on Xs xsXs or on X. xs Os, differ by multiplication by a unit, and the groups of global units on these schemes, H°((X xX)s, 0*(axx)S) and H°(Xs xs Os, Ox*sxs0s), are both isomorphic to H°(8, 01),

hence the restriction map'H°((X x X)s, O*(%xI)3) .-H°(Xs xs0s,

sss°s) is an isomorphism. Next, let V be the 1-dimensional vector space dual to the fiber L/llJ2°L of L at 0 on X, and let V x X. be the trivial line bundle over Is with fibre V. Then, if i s Xs = Xs xs Os -* Xs xsXs is the closed immersion, i*(Ms)mi*m*(Ls) ®i*pi(L3)-1®i*pz (Ls)-1=Ls®LS 1®kV_V xXs, where all the isomorphisms are canonical. Therefore, we can choose

a unique lifting of the translation T(10) to Ms by requiring that this lifting when restricted to Xs xs Os, becomes the map 1y x T.

on V x X. It is then easy to check that the action of K(L)(S) on M8 defined like this is a group action, as required.

We conclude that there is a unique line bundle P on X x X = X/K(L) x X such that its pull back is isomorphic, as a line bundle acted on by K(L), to .M on X x X. The restrictions of P to {0} x X and X x {0} are trivial. Further, since L: X-** Pic°X is surjective with kernel K(L) we see that there is an induced isomorphism of

abstract groups X _:.)- Pic°X, and if a e X, the restriction of P to {a} x X, considered as a line bundle on X, is nothing but the element of Pic°X corresponding to a. Thus, to check that X is a

dual variety and' P a Poincard bundle on X x X, we have to prove the following THEOREM.

Let S be any scheme, and L a line bundle on S x X

such that L I S x {0} is trivial and L ; {s} x X e Pic°X for every e e S.

Then there is a unique morphism 0: SAX such that L: (¢ x 1$)*(P). PROOF.

Consider the line bundle M = pzS(P) (&.p ,*3(L)-1 on

S X X x X, and let rs be the maximal subscheme of 8 x 1

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ABELIAN VARIETIES

over which this line bundle is trivial. The main point is to show that if n: rs -a S is the restriction to rs of the projection S x X -+ S, it is an isomorphism. For this, it is clearly sufficient to show that for any closed subscheme S° of S having support at one point of S, (So xsrs)-*S° is an isomorphism. On the other hand, by definition of rs, if rso denotes the corresponding closed subscheme of 4 ° x X formed with respect to the line bundle L I S° x X on S° x X, we have So xs Ps=rso. Thus for the proof of the main point, we may assume S=Spec B, where Bis a finite-dimensional local k-algebra. Further, if s is the unique point of S, one sees easily that the statement to be proved remains unaltered if we replace L by L®p2 (L!{s} x X)-1,

(since 3 x e X such that L I {s} x X P I {x} x X) so that we may further assume that L I {s} x X is trivial. Now, for all points (s, x) e S x X, the restriction of M to {s} x X x

{x} belongs to Pic°X (since this is so for (s, 0)); and there are at most finitely many points (s, x) such that M restricted to {s} x X X (x} is trivial, since there are at most finitely many x e X such that m*(L)® pi (L)-1®p2 (L)- X x {x)=T,z (L)®L-1 is trivial. Hence, all the direct images RPp13 * (M) on S x X have discrete support,

so that by the Leray spectral sequence, IIP(S x X X X, M) On the other hand, R°p13 * (M) , RPp13,*(p (P))®L-1, so that we have isomorphisms of B-modules 23

H° (S x X,BPp13 * (M)).

HP(SxIxX,M)=RP(Sx.iXX,pQ3(P))=B(DkHP(XxX,P),p

0.

Therefore these cohomology groups are free B-modules. On the

other hand, consider the direct images

RPp12 * (M).

Since

M I {s} x {x} x X e Pic °X for all x and is trivial only for x = 0, all

these sheaves RPp1E * (M) are concentrated at the point (s, 0) e S x X. Let 0 -3 Ko --> K, -+ ... Ko -* 0 be a complex of free

modules of finite type over the local ring A= B ®k Oo,$ of (s,0) e S x K given by the base change theorem for direct images. Then Hi(K.) _ [R' p12,* (M)](,,o) are modules of finite length over A and hence also over Oo, x. Now we have the

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127

Let 0 be a regular local ring of dimension g, and.

LEMMA.

0 -+ K, a Kl a ... a K, -a 0 be a complex of finitely generated free modules over 0. If the H'(K.) are artinian modules, we have

H2(K.)=0for0 0 and that the result holds in smaller dimensions. Choose an x in the maximal ideal R of 0 but not in Re, so that PRooF.

0/0x is regular of dimension g - 1. Putting K. = 0 ®o K., we

have the exact sequence of complexes 0 -+ K. ---> 0, from which we get the exact sequence x

K. - K

x

HP(K.) _ * HP(K.) --+ HP(K.)---. This shows that the HP(F) are artinian.

x

HP+1(K.).

By induction hypothesis, x

N

HP(K.) = 0 for p A/Si1 is also surjective, hence an isomorphism. Thus, -,r: I's -+ S is an isomorphism. Now, for any scheme S and line bundle L on S X I satisfying the

hypothesis of the theorem, and any morphism ¢: S

I,

denoting by Pj:S -+ S x % the graph morphism, we have the t In fact, for every a c S, show that the endomorphism of (K,) given by multiplication by a is null-homotopic; hence so is the same endomorphism of (Ks).

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129

equivalence (0 x la)*(P) W L ra P4 factors through F3. So the theorem clearly follows from the fact that Py is already the graph of a unique morphism from S to X. COROLLARY 1.

We have

R'i(X x %, P) _

if i 0-

k(0)if i=g.g

PROOF. Using the notations of the proof of the theorem, with B = k, L trivial, we have established that K = k, i.e.

is a free resolution of k. On the other hand, any two free resolutions

of one module are homotopically equivalent, and the residue field k of any regular local ring such as 0,,2 has a well-known standard resolution, the Koszul complex. Namely, let xl,...,x, eT1,

lift a basis {J of Ol0/31110. Let L, be the free &0 module with the formal symbols eh A ... A eik (I< i1 < i, < ... < it < g) as a. basis. Then d

d0............ k

d,(eh A ... Aek) _

(-1)txtteji A ... Aeil A ... Ae{k

is the Koszul complex. Then the dual complex

0-ego-- K1) ......-)

is homotopic to the dual of the Koszul complex, which it is easy to see is still isomorphic to the original Koszul complex. This gives Corollary 1 by calculating cohomologies. COROII. aY 2.

For an abelian variety X of dimension g, dim,. HP(X,0) = (PI,

.

PROOF. In fact, HH(X, 0) is isomorphic c to the pth cohomology of the complex

ABELIAN VARIETIES

130

k-i-Kx® k---)....)Kg0 k-..0, which is homotopic to the Koszul complex tensored with k. Now, the differential operators of the Koszul complex tensored with k are trivial, so that we have dimkHP(X, 0) = rank of module of

p-cochains of Koszul complex = lf/ . CoRoLLA&Y 3.

There is a canonical isomorphism of the tangent

space to (0) on I and H'(I, Or). so that the tangent space to I at 0

PaooF. Let S = Spec

is canonically isomorphic to Homo(S, I), where Homo denotes the

set of morphisms of S into I mapping the unique point so of S onto 0. On the other hand, we have by the theorem, Homo(S,

{Line bundles on S x % trivial on {so) x I}

= ker{H'(S x I, 0,,*,,x) -+ Hl({so} xS, 0,*)). But now, we have an exact sequence of multiplicative sheaves 1, and as sheaves of abelian groups,-1 + c O1 Or . Therefore the

11+01- Osx101*- )

cohomology sequence gives

0 -# H'(O1)

Hl(S x %, 0s. z) ---4 H'(%, Ox*)

and this gives a natural isomorphism of the tangent space to 0 at I with H'($, O1), at least as abelian groups. It can be checked that this is actually an isomorphism of k-vector spaces.

Now, let f : I -+ Y be an isogeny of abelian varieties. By the theorem, we get a unique homomorphism f :. Y % of abelian varieties such that if P%, P y are the Poincar8 bundles on S x Z ana Y x Y respectively, we have

(1 xf)*(P1)= (f X 1)*(PY) Hence, if we denote this line bundle by Q, on applying the proposition of § 12, we get that

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131

X(Q) = deg f . X(P1) = deg f. X(Pr),

and since X(Px) = X(Pr) = (- 1)0 by Corollary 1, we get COROLLARY 4.

For an isogeny f : S -* Y of abelian varieties, deg f = deg. f.

For every line bundle L on S, the set-theoretic homomorphism Y'': S - Pico I rs $ is a morphism, and K(L), with COROLLARY 5.

its scheme structure as above, is its kernel.

PRooF.

In fact, L is the unique morphism from 8 to X

such that x 1s)*P= m*L®p,*L-10 p,L-1. Now by definition K(L) is the largest subscheme S of .1 such that

the bundle on the right is trivial on S x I; and in view of the Universal Mapping Property of S, ker(#L) is the largest subscheme

S of .I such that the bundle on the left is trivial on S x S. SYMMETRIC DEFn'rnoN OF X.

We wish to set up the relations between I and S in an obviously symmetric way. Define a divisorial correspondence between two abelian varieties %, Y of the same dimension to be a line bundle on S x Y whose restrictions to {0} x Y and % x {0} are trivial. PRorosrrroN. For a divisorial correspondence Q between I and Y, the following are equivalent. (a) There is no subscheme Z of I different from. (0) such that the restriction of Q to Z x Y is trivial. (b)

There is to 8ubscheme Z' of Y different from (0) such that Q

restricted to I x Z' is trivial. (c)

The absolute value of X(Q) is 1.

When these conditions hold, Y is canonically isomorphic to the dual of I, and % is canonically isomorphic to the dual of Y.

PROOF. By symmetry, it suffices to show that (b) Ga (c). Now,

we get a morphism f : Y-*. I such that (1g xf)*(P) =Q. Now, f

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132

has to be a homomorphism since f(O) = 0. Clearly, (b) is equivalent to saying that f has trivial kernel. If this holds, then since dim Y =

dim X = dim X, f is an isogeny, hence by Corollary 1, § 12, f is an

igoinorphism- Thus we have to show that f is an isomorphism if and only if I X(Q) I = 1. By Theorem 2 of § 12, if f is an isogeny, I X(Q) I= (deg f)- I X(P) I= degf

and the result follows. On the other hand, if f has positive dimen-

sional kernel we can choose a finite subgroup F c ker (f) of arbitrarily large order d. The map 1g x f: X x Y-- 0 X x X factors as

X x Y ---+ X x Y/F --) X x X, so that Q is the pull-back of a line bundle on X x Y/F. Therefore dI I X(Q) I by Theorem 2. §12, and since this holds for arbitrarily large d, x(Q) = 0. COROLLARY.

(The duality hypothesis.) For any abelian variety

X, the canonical morphism is X > X defined by the Poincare bundle P on X x X (regarded as a family of line bundles on X parametrized by X) is an isomorphism.

PaooF. In fact, the divisorial correspondence P on X x X fulfils (b) (or (c)), hence also (a). 14.

Duality Theory of Finite Commutative Group Schemes. Throu-

ghout this section, the ground field k is assumed algebraically closed, of positive characteristic p > 0. Let (2 be a finite commutative group scheme, so that Q' is ailine,

hence G = Spec (R), where R is a finite-dimensional k-algebra. The group law gives us a map µ: B -)- R ®k R, the inverse gives us a map i : R --).B, and evaluation at the identity e gives us an augmentation S: R . k. In the present case, the hyperalgebra H of (2 is simply the dual vector space R* of B, and we will use the notation R* instead of H. As in §11, the group law p dualizes to an associative multiplication

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133

p*: R* (Dk R* - -)- R*.

Since G is commutative, p is co-commutative and so p* is commutative i.e. R* is also a finite-dimensional commutative k-algebra. As in §11, the linear functional 8 is the identity element of R* On the other hand, if m: R ®, R -+ B is the multiplication of B, then m*: R* R* ®, B* will be a co-associative, co-commutative map. Together with the dual i*: R* --). R* of i, we get a group law and

an inverse making the scheme G = Spec R* into a second finite commutative group scheme. The identity point of G corresponds to the homomorphism R* - k gotten by evaluating linear functionals at leR. Thus, to every finite commutative group scheme O we have associated in a canonical fashion another finite commutative group

scheme G, which we shall call the dual of G. This construction is due to Cartier. We shall now give a more `geometric' definition (cf. Oort [OJ).

For group schemes Gi and H and any scheme S, let Homs(G, H)

denote the set of morphisms f : S x G -- S x H such that the diagrams

.SxGS

I

) SxH

(SxG)xs(SxG)

if xsf (SxH)xs(SxH)

m,'OTSxO If

ms'$ TSxH

are commutative, where ma Q and ms u are the multiplications of G and H lifted to S x G and S xH respectively. Such morphisms f will be called B-homomorphisms from G to H. One checks easily that if H is commutative, Homs(G, H) can be made into a commutative group by defining. f + g = ms $o(f, g). Given a morphism 7' --)-S, we have an associated homomorphism Homs(G, H) -* HomT(G, H)

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ABELIAN VARIETIES

given by f F-s f xs T, so that for given G, H, we get a funetor Sch Sets given by S + Homs(G, H), and this is in fact a commutative group-valued functor when H is, commutative.

Now suppose G = Spec A is a finite commutative group scheme and H the multiplicative group scheme Gm. We then assert that the funetor Sch -a Ab, S -+ Homs(G, Gm) is represented by G, that is, there is for each S an isomorphism 0(S) = Homs(G, Gm),

which is functorial in S. Now, if we restrict S to vary in the open subsets of a fixed scheme, it is clear that both sides define sheaves on this scheme. Hence by standard arguments, it suffices to establish the above isomorphism as S varies through affine schemes,

S = Spec B. Let us denote as usual the objects, morphisms, etc., obtained after base extension to R by a subscript R. Then the coor-

dinate rings of GR and Gm, become bi-algebras over B, that is, they are algebras with 1 over B, with co-multiplication maps AR -*

AR OR AR, etc., satisfying the usual identities, and co-identity maps AR R, etc. The notion of homomorphism of bialgebras is then clear. We then have G(R)=Hom(Spec R,G) = Homk.,g(A*, R) = HomR_a]E(R®kA*, R) HomR.a]B((AR)*, R),

where (AR)* is the R-algebra HomR(AR, R). Further, AR, HomR(G, Gm) = HomR.b ,S(R[T, T`'], AR) ` ¢(T). Clearly, an element where i is the inclusion defined by

a e AR is in the image of i if and only if (i) at is a unit in AR, and (ii) 1CR(a) =a®a. But in the presence of assumption (ii), (i) is equivalent to (i)' ER(a)= 1, where ER: AR R is the homomorphism given by the identity. In fact, (i) implies that ER(a) is a unit, and on the other hand (ER® ER)(,f(a)) = ER(a), that is, ER(a)2 = ER(a),

so that ER(a) = 1; on the other hand, if (i)' and (ii) hold and i*: AR AR is got from the inverse, we have commutativity of the diagram

ALGEBRAIC THEORY VIA SCHEMES

135

aft x GR

Spec R

and taking the pull-back of the function at by these maps, we find a.i*(a) = 1, so that a is a unit. Now, since AR is R-free of finite

rank we have a natural isomorphism AR -* HomR((AR)*, R) and under this isomorphism, elements a satisfying (i)' and (ii) go

over into element f e HomR((AR)*, B) such that f(l) = 1 and f(XY) = f(X). f(Y) for X, Y e (AR)* (since XY = µ*R(X (9 Y)). Thus we have set up a set-theoretic bijection between G(R) and

HomR(G, G.) natural in R. We leave it to the reader to check that this is an isomorphism of groups. In particular, taking S = G in the above isomorphism, we get a morphism G x G G. corresponding to the identity of G(G), defined by the homomorphism of k-algebras k[T,T-1) A*®k A, T i-a 3, 3 = the `diagonal element' of A*®k A(which corresponds

to lde liomk(A, A) under the natural isomorphism A*®kA -=* Homk(A, A)). One checks easily from this that this 'universal character' G x G -+ G. is in fact a bilinear map of group schemes .in the obvious sense.

Suppose G is a discrete (i.e. reduced) finite group of order n prime to p. By the above, the geometric points EXAMPLES.

(1)

of a form a group isomorphic to Hom(G, k*), which is of order n.

136

ABELIAN VARIETIES

On the other hand, if A is the ring of functions of 0, dim A = dim A*

= n, which shows that a is again reduced, and is isomorphic as a discrete group to Hom(G, k*). (2)

Next suppose G is reduced and isomorphic to Z/p"Z. For any

reduced G let k[G] be the group algebra of G. If we identify any g EG with the linear form A -* k which is evaluation at g, we get an isomorphism of vector spaces k[G]

A*, which is trivially seen

to be an isomorphism of algebras. Hence G has coordinate ring A* isomorphic to the group algebra of Zf p"Z, i.e. A* z k(%)/(%2'" - 1),

where I corresponds to evaluation at the generator 1 e Z/p"Z.

On the other hand, for f,g eA, 1(f.g)=(fg)(1) =f(1).g(1)= (1®1) (f ® g), which shows that the co-multiplication on A* is given by X i-s % ® X. Thus we have an isomorphism. (Z/p"Z) hence also

(I4

) a Z/p"Z.

Now, let 0 be any finite commutative group scheme. We shall say that 0 is of type l(local) or r(reduced) if the underlying space of G is a single point or if G is reduced respectively. We shall say that G is of type (1, 1) (resp. (1, r), (r, 1), (r, r)) if G is of type l and O is of type l(resp. G of type 1, Cr of type r; etc.). We shall show that any group admits a unique decomposition as a product

G=G,.,xG,,tx0,,.x

,,r

of groups of the indicated types. In fact, if GO is the connected component of identity in G, considered as an open and closed subscheme, and if Gred is the reduced group, the closed immersions G induce a homomorphism G°x Gred G° G and Gred'

- G which is clearly an isomorphism. Further, this decomposi-

tion of G into the product of a reduced and a local group is clearly unique. Thus, it suffices to show that each local (resp. reduced) group is uniquely expressible as a product G,, X O, (resp. G x G,j). Now, if G is reduced, 0 is uniquely expressible

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137

as a product Gl x Ga where Gl is of order prime to p and Ga is a p-group. By the above, Gl is again reduced and G. is local, which proves the assertion for reduced groups. When G is local, split G into its local and reduced parts. Dualizing back, this implies a

unique decomposition of a local G into groups of type

(1, r)

and (1, 1) respectively. This proves the assertion for local groups.

It follows from our discussion that the only groups of type (r, r) are those reduced groups of orders prime to p, hence direct products of cyclic prime power groups, the primes being distinct from p; that the only groups of type (r, 1) are p-groups, hence direct products of

Z/p"'Z's; and that the only groups of type (1, r) are duals of p-groups, hence direct products of There are plenty of examples of local-local groups. For instance,

the groups occ" are local-local. In fact, since Spec '[X] cannot be decomposed as a product (the tangent space being one-dimensional), it suffices to see that it is not isomorphic to µP". Since a9 is a quotient of oc n , it even suffices to see that oc, is not isomorphic to 1L,,. But now, if is the linear form on A = k[X]/(XP) defined by

0ifi'p--1 (x`)

1 if i =p -

1

in A*,

we have f2(r)_(e(& e)((l®X+X®1)')=0 if i < p - 1, which shows that A* has nilpotent elements. So far we have developed the circle of ideas involving homomor-

phisms from G to Gm. Next we turn to homomorphisms from G to G. and related results. We fix the notations G = Spec R, G = Spec R* as above and we let roman letters x, y, ... be elements

of R; greek letters a, P, ... be elements of B*. Let e and s* be the co-multiplications in R and B* respectively. We recall that in §11, we saw how R* operated naturally on the sheaf Oo. In our affine case, this means that there is a natural inclusion

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138

R* - Homk(R, B). Explicitly, if a: B -a k is an element of R*, we get a map Da: R -* R by the composition s

1(& a

B --) R®kR --} R. In particular, if a(1) = 0, a(91R;) = (0), then D. is derivation of R over k. The operators D. are all translation-invariant in the usual sense. Moreover the transposed maps Da: R* -> R* are just the compositions mult.

i.e. the maps

a.g, multiplication by a.

As an application, we can interpret Hom(G, Ga) in a new way: Homap.seh.(G, Ga) = Homb1-algeb=a(k[X], B*)

{acR*18*a=a® 1+l®a} (since a homomorphism from k[X] to R* is determined by the image a = ¢(X), and q is compatible with co-multiplication if and only if s*a = a ® 1 + 1(& a). Such a's are called primitive elements

of B*. If we associate to any a e R* the map D,, which is the transpose of multiplication by a, we see that primitive elements correspond precisely to invariant derivations of R: j aER *Is*a =a®1

1®a} e { D: R---+-R D an invariant derivation} = Lie(G).

fill

The conclusion is that Hom(( . Ga) = Lie(G).

Moreover, the pth power operation in Lie(G) is obtaihed by taking

p

D to Do ... o D, which corresponds to raising a to its pph power, G. with the which corresponds to composing a map Frobenius F: Ga-3Ga (since F*(X) =XP). Now look at the following type of group.

ALGEBRAIC THEORY VIA SCHEMES

139

DEFINITION. A group scheme G is of height one if it consists of a

single point e, and if xv = 0, for all x e 9)Ia. (G need not be commutative.)

It is easy to see that, as schemes, such groups G are isomorphic to Spec(k[X1, ..., Xn]/(X9, ..., Xc)). In fact, choosing X1, ..., Xn e9 , which induce a basis of and letting R =r(l? ), we find that R is a quotient of k[X1,..., Xn). On the other

hand, if we use the fact that B admits derivations Di: B -R such that Di(X;) = 8,1 (mod T1,), then it is easy to see that there

can be no relation of linear dependence over k among the monomials

F1 X°i, 0 0 of the classical equivalence of categories between Lie algebras and germs of Lie groups is the following THEOREM. The funcfor G6 i Lie G sets up an equivalence of the categories of finite group schemes of height one and finite-dimensional

p-Lie algebras over k (i.e. a finite-dimensional vector space with bracket and p'' power map, as in §11). We shall prove this only for commutative G, which correspond to p-Lie algebras with trivial bracket. PROOF.

For any Ic-vector space g with a p-linear map a -+ a(F),

let U(g) be the k-algebra S(g)/I, where I is the ideal generated in the symmetric algebra S(g) of g by elements of the form a(n -n

0, a e g. Note that if al, ... , an area basis of g, then R ai, 0 0, r > 0, m > 1, (p, m) = 1. We want to analyze the structure of the finite group scheme X = ker n1. Now, X. and X , are subgroup schemes of X,,, and we have a homomorphism X. x X - X,,. This is, in fact, an isomorphism since X. is the (r, r)-part of X,,, and X is the product of the (r, 1), (1, r) and (1, I)-parts of X,,. As we saw in §6, X. is a discrete reduced group isomorphic to (Z/mZ)'4. Thus, it suffices

to study the structure of X ,,, which we rename G,,. Suppose now that (Ge)=ed = (Z/pZ)'. Since X is divisible, for any n > 1, we have an exact sequence 0 -+ Gyred -+ Gn+l,red

G,'.d -; 0,

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147

and one deduces by induction that for any n> 1, (O, )md = (ZJp"Z)'.

Now, by Theorem 1 of this section, G" is the kernel of (e)g, so

it follows that there is an integer s such that for any n > 0, (C,")nd = (Z/p"Z)s. Thus, the decomposition of G. into its pieces is as follows: G. = (Z/p" Z)' x (Z/p"Z)` x GO

_ (Z/p" Z)' x where G0 is local-local.

x G.0'

Since G. is of order p2" a, we deduce that

there is an integer t > 0 such that r + s + t = 2g, and Gn is of

order 0. We shall show that the integers r = r1, 8 = s$ and t = to are the same for isogenous abelian varieties. It suffices to prove this for r

and s, since r+s+t=2g. Further, since sa=rg, it even suffices to verify this for r. Let f: X Y be an isogeny, with kernel of order k. p", we have that the order of (X-),.d is at most Since f(X ") c k times that of (Yp"),"d, that is, p'K < k.pj'ry for all n, hence rd < r7. Now, ker f is a finite group scheme, hence is annihilated by an N > 0, which shows that ker Nx ker f. Therefore, Nr

f

a* Y -i. X. Thus, Y X is an isogeny, factorizes as X and ry < r1. This proves that r, 8 and t are isogeny invariant. In particular, since for any abelian variety X, X and X are isogenous, we deduce that rx = rg = ex. Thus, we have that

G. = (Z/p"Z)' x (p.,pY x Gr

with G.0 local-local of order p"`, 2r + t = 2g. In particular, we see

that r < g. The integer r is called the p-rank of X, and is an isogeny invariant.

Now, since px induces the 0-map on Lie algebras, we see that Lie X = Lie (ker pa) = Lie 0, = Lie (pa)' p Lie G. Since GO, is local-local, the pth power map on Lie G; is nilpotent, whereas Lie(pp)' admits a basis e1, ... , e, such that e? = e;. We thus see

ABELIAN VARIETIES

148

that the p-rank of X equals the dimension of the semi-simple part of Lie X with respect to the pth power map. The same result holds for Lie X, since X and X have the same p-rank. On the other hand, we have established a canonical isomorphism Lie X c H1(X, Ox). Let F: Ox -a Ox be the Frobenius homomorphism F(a) = a1', and denote the induced p-linear mapH1(X,0x)>H1(X, Ox) again by F. We shall establish that under the isomorphism Lie X ^_- H1(X, e'), the pth power map in Lie X goes over into F. It follows that the

p-rank of X is also the dimension of the semi-simple part of H1(X, 0.) with respect to the Frobenius map F. Thus, we need to prove THEOREM 3.

Under the natural isomorphism Lie X H1(X, Ox),

the p=n power operation in Lie X goes over into the Frobenius map in H1(X, OX).

PnooF. First we give a description of the pth power operation on vector fields on a scheme X, using the functor X, analogous to the one given for the Poisson bracket in §11. Let D be a vector

field, interpreted now as an automorphism of X x Spec A over Spec A, where A = ([2) , which is the identity on the closed fibre

X'

-XxSpecA.

Let M = k[E1,... , e9]/(E2,, ..., P2), and let 77j: A -s M be the k-algebra homomorphisms defined by -%(E) = E, and let O, _ Spec 71,: Spec M> Spec A. By base change using 0,, D induces an

automorphism D, of X x Spec M over Spec M, and hence D' = D1 oD2o... o Dy is an automorphism of X x Spec M over Spec M. Let s, (1 0. We have the exact sequence

Hi(L'a'®LkH'(L' from which we have

dimH'(L'a'(& L4+') - dimH'(L'° (& L,-),< dimH'(H, L'a'® Lt+' In)

Y is an isogeny of abelian varieties of degree prime to p and L a line bundle on Y with X(L) 0 0, i(f*(L)) =1(L), In fact, we know by Cor. to Prop. 3, §7 that L is a direct summand off*(f*(L)), andH1(X,f*(L))=H'(Y,f*(f*(L))), so that H'(L) =A (0) H'(f*(f*L)) 0 (0) . H'(f (L)) 0 (0). Also if Lo EPi.c°X, then

i(L) = i(L®L°). In fact, for some x e X, L®L° = TT L, hence Hi(L) 96 (0) . H{(T? L) 0 (0) u. H(L® L°) 0 (0). In particular, =L°'® L° with L° a Pic°X, we see that i(L°') =i(L) since

for any n > 0, with p ,f' n.

Suppose then that N is a large square prime to p. If i(L) 0 i(L(&H), then i(Y) = i(LN(&HN), so there is a least integer a in 0 dimH'1(Li(g L2)

if

=IX(14®L's)I=NO (N, By the first lemma, we therefore deduce that there is a constant c > 0 such that for all large N (prime to the denominator of r if r is rational), NoI If (N, N) or

If r

a

c

If (N'N IA N':-'1' Now, f(t, 1 - t) = (t - r)1g(t) where g is non-vanishing at r. Thus

for all large N as above, there is a c' > 0 such that

ALGEBRAIC THEORY VIA SCHEMES

r 1V

-

TI

# 0).

Since we are making these assertions for any ample L, it suffices to prove (1) and (2) for any ample L with x0 = 0, since the general case follows by applying the result to a translate of L.

Thus, if (1) were not true (with xo = 0), we would have that for any D as above and any x, y e X,

ABELLAN VARIETIES

164

06SuppD-x n-

a(SuppD - x) u (SuppD - y)

u (SuppD+x+y). Since we may clearly choose y such that x1 does not belong to the last two members, we deduce that x E Supp D implies x E Supp D

- x1, that is, Supp D = Supp D -xl. Since the divisor D has no multiple components, this means that TA(D) = D. In particular, x,, e S(L), hence x, has finite order. Let x, generate the finite group F. We then have an 6tale morphism ir: X -* X/F, and Dl = rr(Supp D) is a closed subset pure of codimension one in X/F,

which we may consider as a divisor with all components of multiplicity one. Since a is etale, 7r*(D1) is again a divisor with all components of multiplicity one and has the same support as D, so that D = sr*(D1) and L = a*(Ox/z,(D,)). But note that dim H°(X, L) = X(L) _ (Order F).X(0x1F(Di))

= (Order F).dim H°(X/F, Ox/F(D1)) > dim H°(X/F, Ox,F(Di)).

Since the set of all divisors D, such that L = a*(0x1F(D1)) fall into a finite set of linear equivalence classes, this proves that all sections 8 E T(L) either define multiple divisors, or lie in one of a finite number of lower-dimensional subspaces 7r*r(OS1F(D,)). This is a contradiction, so (1) holds.

Similarly, suppose (2) is not true for a non-zero tangent vector t at 0, and let T be the invariant vector field defined by t. If (2) is false for all the divisors Ts (D) + T, *(D) + T*z_ (D), it follows

immediately that for all x e Supp D, the vector T. is tangent to D at x. Since D has no multiple components, this is equivalent

to the property: V U c X open, V local equations = 0 for D on U, (*)

T(#) = a.4, some a e 01(U).

In terms of the k[cl/(e2)-valued automorphism of X defined by T, (*)

just says that the divisor D - a subscheme of X - is inv-

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165

ariant. This implies that the k[e]J(e2)-valued point of X defined by t is in the subgroup K(L) of points leaving L invariant. Now

in characteristic 0, all group schemes are reduced, so K(L) is finite and discrete and this cannot hold unless t = 0. On the other hand, in characteristic p, let H be the smallest subgroup of K(L) containing t; then H c X(-13 and will be determined by its Lie algebra which will be the span of t and its pth powers. It

is easy to see that D will be invariant under translations by all points of H. [In fact, if H = Spec(R), the action of H gives a homomorphism of R* into the ring of differential operators on .X, mapping elements of 4 into the corresponding invariant derivations. Since b generates R*, and the sheaf of ideals

Og(- D) is stable under 1) by (*), it is also stable under R*, hence we get a homomorphism R* -s Diff(OD), i.e. an action of H on D.] Let X' = X/H, D' = DJH. From the results of §12, we find that ir: X -,- X' is flat and surjective, that D' is a closed sub-

scheme of X' and D = if xg.X. Therefore if 5' is the sheaf of ideals of D',

Ox(- D) = ,?' ®Oz,Or. Since D is a divisor, Oa( - D) is a locally free sheaf, so by Part (B), Theorem 1, § 12, 5' is a locally free sheaf, i.e. D' is a divisor too. Now D = lr*(D'), so we compute, as before :

dim H°(X, L) = deg Tr.dimH°(X/H, Cxl,(D'))

> dimH°(X/H, Osta(D )).

Exactly as before, this implies that all sections a e P(L) either define multiple divisors, or lie in one of a finite set of proper subspaces --a contradiction.

CHAPTER IV Hom (X, X) AND THE l-ADIC REPRESENTATION 18.

Etale coverings. The main result is the following

THEOREM. (Serre-Lang.) If X is an abelian variety, Y a variety and f : Y -+ X is an !stale covering, then Y has a structure of abelian variety such that f becomes a separable isogeny.

PROOF.

Let P,,, be the graph of the multiplication m: X x X--)-

X in X x S x X, and r, the inverse image in Y x Y x Y of r_ by f x f x f. Since (1)1" P,,, is an Bale covering, (2) p12: Pm-+X X X is an isomorphism, (3) we have the commutative diagram r,

YxY

fxf

and (4) f x f is an. stale covering, p12: P' --> Y x Y is an stale covering too. Choose a point yo e Y such that f(yo) = 0, and let P be the connected component of P' containing (yo, yo, yo) (which belongs to P' since f(yo) = 0 and (0, 0, 0) a Pm). Then the restric-

tion p: r -> Y x Y of p12 is again an 6ta1e covering, so that the degree of p equals the number of points of any fibre of p. We want to show that p is an isomorphism, or equivalently that there is one

point of Y x Y whose inverse image in r is again a single point. Let ol, a2: Y -* P be defined by al(y) = (ye, y, y), a2(y) = (y, ym y) (Since a;(Y) c P' and (yo, yo, yo) a o,(Y), it follows that o (Y) C P.)

Then the restriction of p to a2(Y) is a bijection of a2(Y) onto Y x {yo}. It therefore suffices to establish that p-1(Y x {yo}) = a2(Y), or equivalently that if q: P - Y is the restriction to r of p2: Y x Y x Y -r Y, q-1(yo) = a2(Y). Since a2JY) is an irreducible component of q-'(yo), it suffices to show that q-'(y0) is irreducible. Now, r is-non-singular, being Male over Y x Y and hence X x X,

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ABELI.9N VARIETIES

and since it is also connected, it is irreducible. Further, the morphism q: P i Y is smooth, being the composite of the etale mor-

phism P -> Y x Y and the projection Y X Y 2 Y. Finally a,: YAP is a section for q. Now the assertion that q- I (yo) is irreducible follows from the LEMMA.

Let f : X -3 Y be a proper smooth morphism of

irreducible varieties such that there is a section a: Y -X, f o a = ly. Then all fibres off are irreducible.

PaooF. We may assume Y = Spec A affine. Let B = P(X,Og), so that B is an A-algebra which is a domain since X is irreducible and a finite A-module since f is proper. The morphism f factorises h

as X g Spec(B) -± Y where Spec(B) is again an irreducible variety. But go a is a section of h, and since dim(Spec B) = dim Y, g o a is surjective, hence h is an isomorphism, and A =B.

Since f is smooth, its fibres are non-singular, and it suffices to show that they are connected. Let 0 - KO K, ... be a complex of free finitely generated A-modules giving the- direct images of Ox

universally, so that by the above, we have an exact sequence 0 A--)- KOOK,. Let y be any point of Y, 9Jl its maximal ideal in A. Since completion with respect to the P -adic topology is an exact functor, we have an exact sequence 0 -> A KO - K1, so

that!

ll m Ker [ K.

-' L l . But now, J

Ker [2"KO,1t"K,J = H°(f-'(y), Oaf°Oa)> so the natural map A

lim H°(f-'(y), oa' n Ox) is a ring

isomorphism. If f-'(y) were not connected, let f-'(y) = Z, u Z2, Zi closed,

Z, n Z2 = 0 and Zi 0 f-'(y). We can find a unique

f E H°(f -'(y), d X1Ft' Os) which reduces to 1 on Z, and 0 on ZS,

Hom(X,X) AND THE i-ADIC REPRESENTATION

169

and (f") defines an element f e lim Ho(01/ Q*Ox) = A with f8=f,

4-

and f 0 0 or 1. This is impossible since A is a local ring. The lemma is proved.

Returning to the proof of the theorem, we have shown that p1s: P -+ Y x Y is an isomorphism, so that v = ps o p121: Y X Y -}Y

is a morphism. Since, as we saw, r D a1(Y) and a2(Y), it follows that v(y, yo) = y = v(yo, y). Therefore, from the theorem proved in the Appendix to § 4, Y is an abelian variety with composition law v and zero element yo. Since f(yo) = 0, f is a homomorphism of abelian varieties. REnfAax.

If I is an abelian variety and f : Y -+ X is an

isogeny, then we can find an isogeny g: X -3- Y with fog=nz for

some n > 0. In fact, since kerf is a finite group scheme, it is killed by some integer n > 0, hence ker f g ker(ny). Since Y/ker f, it follows that n . factorizes as np = go f for a homomorphism g: X -i-- Y. But then fog = nr too, since for all x EX; x = f (y) for some y e Y, and therefore X

f-9(x) =f(9(f(y))) =f(ny) =nf(y) =nx: We can interpret our results in terms of the fundamental group 771(X). Recall, that if X is any non-singular variety, and xo eX is a base point, the group 771(X, xo) is constructed as followst: consider the set of all morphisms

YO

'--0

X0

together with a base point yo e Y lying over xo such that (1)

a finite group GF acts freely on Y and 1 YJG, and

(2)

Y is connected, hence Y is again a non-singular variety.

Given

two such

tFor details, cf. (021 pp. 60-61.

ABELIAN VARIETIES

170

f n

-A

X Xnxo

I

A

-

Yo C- Y .

recall that there is at most one morphism f : Y'--> Y' such that

A,

y{!

(i)

A"-

(ii)

f(Yo) =Y Y.

When f exists, there is a unique surjective homomorphism

p:(ir.(3r

such that f(u y) = p(a) f (y), all

e Qr., y e Y'. We order the

triples (Y, yo, or) by saying (Y', yo', ir')> (Y", yo, 7r) if such an f exists. Then the set of (Y, yo, ir)'s forms an inverse system, and we define

v,(X, xo) = lE lm Gr.

Now suppose X is an abelian variety and xo = 0. Then all such Y's are abelian varieties, and 0r is just the kernel of 17 acting on Y by translations. In particular, we see that irl(X) is abelian. To describe it more explicitly, it is convenient to break it up into the product of its l-primary piece for different primes 1. First suppose l = p. By the remark following the theorem, the set of etale coverings

X

l" 4- X

is cofinal in the set of all tale coverings Y -* X, #(Ker 7r) = l"', some m. Therefore, the l-adic component of r1(X) is the inverse limit of ker(la), or X.. This is called the l-adic Pate group of X. DEFRGTTION.

Tr(X) = lim Xr,,, where the inverse system is

Hom(X,X) AND THE l-ADIC REPRESENTATION

---*.xn+1

IX

171

4 Sin -...- 9Y%.

As an inverse limit of finite abelian 1-torsion groups, T1(X) has the structure of a module over the l-adic integers Z. Since X,, (ZJl"Z)10, it is easy to see that T=(X) m ZZ, as a Z, -module. Secondly suppose l =p. In this case, break up Ker(pg) = X°Pn x Mo

where XP°" is local and X' P. is reduced. Then by the remark

following the theorem, the set of etale coverings ir" in the diagrams

x

K

-

\1,

/.

Y"=SJ2;

-

is cofinal in the set of all btale coverings of X whose degree is a power of p. But Ker(7r") = Q7(X ,n) = %'p", so the p-adic component of vl(X) is again the p-adic discrete Tate group.

DErmmoN. T,(X) = lim XD" (the inverse system as before). TP(X) is a Z,-module and if r=p-rank of X, then clearly TP(X) (Z,)'. The full fundamental group is then given by vi(X) =

fJ Ti(X). Q primes i

Now suppose k = C, and X= VJU where, as usual, V is a complex vector space and U is a lattice. Then, in addition to the algebraic fundamental group as just defined, we have the usual topological fundamental group aitOP(X), which, as we saw in §1, is

canonically isomorphic to U. On the other hand, since

Xl".ln.UJUCVJU=X,

ABELIAN VARIETIES

172

it follows that T=(X)

lim ZK U/ U

with maps

U/U -L T. U/U,

T1(X) = lim U/l"U 1

with maps U/l" IU--> U/l"U. In other words, TT(X) is the l-adic completion of U = , (X), and

zr (X) _. jJ Tda') I

= lim U/n!U n

= full pro-finite completion of U

Structure of Hom(X, X). For two abelian varieties X and Y, we denote by Hom(X, Y) the group of homomorphisms of X into Y, and by End X the ring Hom(X, X). Further we shall put Hom°(X, Y) = Q ®ZHom(X, Y) and End°(X) _ Q®ZEndX(End°X is classically called the algebra of complex multiplications of X). Composition of homomorphisms extends to a unique Q-bilinear map Hom°(X, Y) x Hom°(Y, Z) -a Hom°(X, Z), so that we can 19.

form a category whose objects are abelian varieties, and morphisms from X to Y are elements of Hom°(X, Y), the so-called category

of "abelian varieties up to isogeny". We have seen that given any isogeny f : Y X, there is another isogeny g: X -+ Y such

that fg = n1, and this proves that in the new category, isogenies are isomorphisms. Thus in future, whenever we have an isogeny f : Y -* X, we shall denote by f-x its inverse in

Hom(%,X) AND THE 1-ADIC REPRESENTATION

173

Hom°(X, Y). It is also clear that we can give the following more fancy definition of Hom°(X, Y):

Hom°(X, Y) = lim Hom(X', Y).

(Iga) THEOREM 1. (Poinoare's complete reducibility theorem.) If X is an abelian variety and Y an abelian subvariety there is an abelian subvariety Z such that Y n Z is finite and Y + Z = S.

In other words, X is isogenous to Y x Z.

Let i: Y X be the inclusion, and i s X -+ Y its dual X - X is homomorphism. Let L be ample on X, so that an isogeny. We take Z to be the connected component of 0 of .E'(ker i). We then have dim Z = dim ker i > dim X - dim Y = dim X - dim Y. Further, by definition of i and ¢L, if z e Y, then Pxoor.

z e ly '(ker i) n Y

Ts L® L-1 l y is trivial z a K(L j.).

Since L l Y is ample, K(L I y) and hence Z n Y is finite.

This

means that the natural homomorphism Z x Y - X has finite kernel, and since dim(Z x Y) = dim Z + dim Y> dim X, it is also surjective.

REMAE$. Over the complex field, the complete reducibility theorem is very simple to prove. In fact, let X = V/U with V a

complex vector space and U a lattice, and H a positive finite hermitian form on V which is non-degenerate with E = Im H integral on U x U. Then any abelian subvariety Y of X is of the form V1/U n Vl where Vl is a complex subspace of V with V 1 n U a lattice in V1.

If V2 is the orthogonal complement of V

for H, then (a) V. is also the orthogonal complement of V for E, hence the. lattice U n V2 is of maximal rank in V2; and (b) Vl n V2 = (0) since H is positive definite. Thus, if Z = V2/V2 n U, Z is a complex subtorus of I such that Y n Z is finite. The restriction of H to V2 gives a Riemann form on V2, which shows that Z is an abelian subvariety.

ABELIAN VARIETIES

174

In fancy language, the theorem shows that the category of abelian varieties up to isogeny is a "semi-simple abelian category, all of whose objects have finite length". More concretely, we get the following corollaries by standard arguments. DEFINITION. An abelian variety is simple if it does not contain an abelian subvariety distinct from itself and zero. CoaoLLAnr 1. Any abelian variety X is isogenous to a product

X, x ... x Xkk where the X; are simple and not isogenous to each other. The isogeny type of the Xi and the integers n; are uniquely determined.

(Proof standard.) For X simple, the ring End°X is a division ring. For any abelian variety X, if X = X, x ... x Xkk, with X; simple and not isogenous, and A = End°X;, then ConoLLAnY 2.

End°(X) = Mnl(DI) ® ... E) Mnk(D,E).

(Here Mk(R) = ring of k x k matrices over R.) PnooF. For X simple, any non-zero endomorphism of X

is an isogeny, hence an invertible element in End°X, which proves the first assertion. As for the second, Hom(I 1, .i) = (0)

j, so End°X = p IEnd°(X;i). And End°(X,i) is clearly i-i the algebra of matrices of order n; on the division algebra D. We shall say that a function # defined on a vector space V is a polynomial function of degree n if restricted to any for i

finite-dimensional subspace, it is a polynomial function of degree n or, equivalently, if for any v°, v, a V, #(x°v° + xlvl) is a

polynomial in x° and xl of degree. n. Thus for instance, we have seen that X(L) extends to a homogeneous polynomial function of degree g on the vector space NS(X)®zQ. THEonnm 2.

The function 0 i -- deg 0 on End X extends to a

homogeneous polynomial function of degree 2g on End°$.

Hom(.,X) AND TB i-ADIC REPRESENTATION

175

Since for ¢ e End X and n e Z, deg no = deg n1. deg 0 = n'deg it suffices to show that for 0, 0 a End X, the function P(n) = deg (no + 0) is a polynomial function. If L is an ample PROOF.

line bundle, we have that deg (no + 0) = X ((n$ + )*(L) ) X(L)

Therefore it suffices to show that X((ns6+#)*(L)) is polynomial

in n. Putting L(") = (no + #)*(L) and applying Corollary 2 of the theorem of the cube to the three morphisms no + 0, 0, 0 respectively we get that L("+,>® L(.+2 1)0 L(")® (20) *L-1® #*L®'Q*L = 1,

from which it follows by induction on n that for suitable line bundles L1, L2, and L, on X, L("1 = L1'("-1)12®L2®L3.

Since x(L) is a polynomial function of L, X(L(")) is a polynomial in n. To go further and prove, in particular, that dimQHom°(X, Y)

it seems to be essential to use some entirely new method. If k = C, we can compute Hom(X, Y) very quickly like this.

is finite,

Let X1 = V1/U1, gX = dim X1, X2 = V1/U21 g, =dim X2, Vs complex vector spaces, U; lattices.

Then every algebraic homomorphism f : Xl -+ X2 lifts complex-analytic homomorphism f : V1

to a

V2. As is well known,

such f's are simply the complex linear maps from V1 to V2. Conversely a complex linear map L: Vl

V2 induces an analytic

homomorphism f : $1 i X2 if and only if L(U1) c U2, and by Chow's theorem (cf. §1) all analytic homomorphisms from $1 to X2 are algebraic.

ABELIAN VARIETIES

176

This proves :

Homtu (X1,X2)

L:V1- V2I L complex-linear, L(U1) c U2 }.

In particular, L is determined by its restriction to U1 so we get an injection T: Hom abelian (Xi, X2) -+ Homz(U1, Us). varie6iea

Since Ui is the topological fundamental group 1r(X;) (or the homology group H1(Xi)), the map T is just the functorial repre-

sentatio._ of maps between spaces via maps between al's (or Hr's). If we introduce bases, we have a faithful representation of Hom(X1, X2) by 2g1 x 2g2-integral matrices. In particular, Hom(X1, X2) is a free abelian group on at most 4g1g2 generators.

Even when the group field k is not C, an analog of the above method works. This consists in using the free Z1 module Ti(X) instead of the free Z-module U = vrl°j'(X). In fact T1(X) is just the i-primary component of the algebraic fundamental group irl(X), and when k = C, T1(X) is nothing but the l-adic completion of U. If X1 and X2 are two abelian varieties, every homomorphism f:X1 X 2 restricts to maps f : (X1)1n (X2)1" and hence it induces a map

T1(f): T1(Xi) - - T1(X2) The map f 1-- s'1'1(f) itself is a canonical homomorphism: T1: Homsbellan(X1, X2) a Homz1(T1(X1)1 Ts(X2)), va 1etea

known as the 1-adic representation. In fact, in terms of bases of T1(Xi) over Z1, this represents homomorphisms f by 2g1 x 2g2 matrices with coefficients in Z1. We now prove THEOREM 3.

For any pair of abelian varieties X and Y,

Hom(X, Y) is a finitely generated free abelian group, and the natural map Z1®zHom(X, Y) -+ Homzi(T,(X), T1(Y))

(*)

IIom(I,I) AND THE l-ADIC REPRESENTATION

177

induced by Tt: Hom(X, Y) -* Homyi(T1(X), T,(Y)) (l any prime 3-4 char k) is injective.

PROOF. Note that since Hom(X, Y) is torsion free, we have an inclusion Hom(X, Y) c Hom°(X, Y).

Step I. For any finitely generated subgroup M of Hom(X, Y),

Q M n Hom(X, Y) = {0 e Hom(X, Y) I no e M, some n : 0) is again finitely generated.

To prove this, choose isogenies f XX" -)- X and Y -> f Y1"J where Xi, Y; are simple abelian varieties. Then Hom(X, Y) gets

mapped injectively into n]om(Xi, Yf), so that it suffices to id

prove this result for X and Y simple. If X and Y are not isogenous, Hom(X, Y) = (0), so that we may assume that they

are; in this case using the injection Hom(X, Y) - End X induced by an isogeny Y a X, we are reduced to the case X = Y and X simple. By the earlier theorem, there is a homogeneous polynomial function P on End°X such that for 0 e End X, P(¢) = deg 0 e Z. Since any ¢ # 0 is an isogeny, P(o) > 1 if o e End X and o :A0. Now Q.M is a finite-dimensional space, and I P(¢) I < 1 is a neighborhood U of 0 in this space. Therefore U n End (X) = (0), so End X n Q.M is discrete in Q.M and hence is finitely generated.

Step II. For any l 9i p, the map (*) is injective.

In fact, it suffices to show, in view of Step I, that for any finitely generated (hence free) submodule M of Hom(X, Y) such that M = Q.M n Hom(X, Y), Zt ®zM

> Homz'(TT(X), TI(Y))

is injective. Let fl, ..., f, be a Z-base of M. If this map is not

injective, since the right side is Zt-free, we can find ai a Zl with at least one °c{ a unit such that E a; fi.i--+ 0. Hence we can find P

integers ni (1 < i < p) not all = 0 (mod 1) such that Tj(E -if) I

maps Ti(X) into lTT(Y):

By the very definition of T1(f), this

ABELIAN VARIETIES

178 9

means that

n; f. = f maps X1 into 0. But then, f factorizes as

X -+ X ) Y, g

9

m; f;.

and since g e Q.M n Hom(X, Y) = M,

Thus Enn f; = Mm, f;, and f1, ... , f, being a basis

of M, Z I n; for all i, a contradiction.

The theorem now follows. In fact, because of the injectivity of (*), Hom°(X, Y) is finite-dimensional over Q, and because of Step I, Hom(X, Y) is finitely generated, and being torsion free, it is also free. COROLLARY 1.

PRoop.

Hom(X, Y)

Z' with p < 4dim X.dim Y.

In fact, the rank of Hom(X, Y) is at most that of

Homzl(Tt(X), P (Y)) which is 4dim X.dim Y. COROLLARY 2.

For any abelian variety X, the group NS(X) _

PicX/Pic°X is free of finite rank (called the base number of X).

PRoo . In fact, the homomorphism L i-- 4 induces an injection of NS(X) into Hom(X, X). COROLLARY 3.

End°X is a finite-dimensional eemi8imple algebra

over Q.

Let A be a'finite-dimensional associative algebra over a field r, which, for simplicity, we assume to be infinite. By a trace form on A over P, we mean a r-linear form

S:A ->r such that S(XY) = S(YX) for X, Y e A. A norm form on A over r is a non-zero polynomial function

N: A--) r (i.e. in terms of a basis of A over r, N(a) can be written as a polynomial over r in the components of a) such that N(XY)== N(X).N(Y) for X,Y a A. The following lemma is well known, but we include a proof for the sake of completeness.

Hom(S,X) AND THE I.ADIC REPRESENTATION

179

Let A be a finite-dimensional associative simple algebra

LEMMA.

over a field r (assumed infinite) with center A, separable over r. There is a canonical norm form NO and a canonical trace form TO of A over A such that any norm form (resp, trace form) of A over r° is of the type (NmairoN°)k with k an integer > 0 (resp. oTr° where 0: A -r r is a r-linear form). If [A: A] = d2, NO is homogeneous of degree d.

Pnoor. When r = A is separably closed, A can be taken to be a matrix algebra MM(r). In this case, the elements %Y- YX span the vector subspace of matrices of zero trace, and any norm form gives rise to a rational homomorphism of algebraic groups GL(d) -> G.. This shows the validity of the lemma with Tr = matrix trace, N = matrix determinant.

In the general case, let f be the separable closure of r, o;: A P (1 < i < [A: r]) the various imbeddings of A in 1' over r, and P,;) the field r considered as a A-algebra through a,. We have an isomorphism of T' algebras

A®rr = A ®A(A(& rr) cf F1 A®A r;;r Denote the image of a e A® rP under this isomorphism by If N is any norm form on A over r, it extends to a norm form of

Aq rP over P, and defines a norm form N; on A® r1'(;) by the equation .N (e) =N(1, 1, ..., ¢,1, ..., 1). By what we have seen, Ns = (N°-)"; , where N° is the reduced norm of A®AF(o over r(r, so that we get

N(a) = H x(ma)) . We shall show that the norm a -+ II N°(#;(a))" of A ®rr over P comes from a norm of A over r by base extension if and only if all the n, are equal. Since I' is the separable closure of r, N comes from r if and only if for any automorphism a of r over r, we have N((1(9 a)a) = u(Na), that is to say,

180

ABELIAN VARIETIES

FT N (4i ((1®a)a))"i = o rT N0j(0i(a))"i

.

i Now, there is a permutation 7r of the integers from 1 to [A: P] such that a 0 of =a (i), and we have the commutative diagram ci

so that (since the second. vertical arrow is an isomorphism of simple algebras over the isomorphism a of separably closed base

fields) we get N>,,,()((10 a) (a))] = a N°(oi(a)), and on substitution, we see that we must have n (i) = n, for all i. Now, the jalois group of P over r acts transitively on the imbeddings over A in P, so that we must have all the ni equal.

Thus we see that we may take N°(a) =

in the

i lemma, and then N(e) = Nm, .(NN,A(a))" . The assertion about the trace is even simpler.

DEFINITION. NmArr o N° will be called the reduced norm of A over P and TrAfr o TO will be called the reduced trace of A over P.

We can now prove the following important TsEoREM 4. Let f be an endomorphism of an abelian variety, and T1(f) the induced endomorphiem of T7(X) (l:A characteristic). Then

deg f = det Ti(f ), hence

deg (n.1z -f) = P(n), where P(t) is the characteristic polynomial, det (t - TI(f)), Of TIM.

The polynomial P is manic of degree 2g, has rational integral coefficients, and P(f) = 0.

HOm(X,X) AND THE I-ADIC REPRESENTATION

181

PBooF. The functions f i o dog f and f det TI(f) both extend uniquely to norm forms N, and N2 respectively of degree 2g on the semi-simple QI-algebra QI®z End X, where QI is the quotient field of ZI. If j 1 denotes the l-adic absolute value, we assert that ! NIa I = ! N2a ! for all a e QI ®z End X. In fact, it

suffices to verify this by homogeneity for a eZI ®z End X, and by continuity for a e End X. Thus we have to show that the power of I dividing deg f equals the power of l dividing det T, (f). Now, the

power of I dividing deg f is the order of the kernel of XI - f- - XI for n large, or what is the same the order of the cokernel of this

map for n large. On passing to the limit, it is also the order of the cokernel of TI(f), which is 1, where v is the power of I occurring in det TI(f).

Now let QI®Z End X,= HAS

be the

decomposition

;m1

of

QI®z End X into a product of simple algebras. The norms N1 and N. go over into norms on 11A;, i.e. into power products Ni(al, ... ,

r

_;f N°(a5)"j (i = 1, 2),

where N° are. the norm forms on A, over QI of lowest degree, by the lemma. On taking ay =1 for j =A j 0, we deduce that = 1 for all ado a Ago. Since Nje is homogeneous !N;o(s ) of positive degree, we see (by multiplying oeo by 1) that vl;® = v,,,, and since this holds for all j0, N, = N2.

This proves the first statement of the theorem. The second follows on substituting n.11 - f for f and using TI(n.Ix- f) = n.1TI(x) - TI(f). Now P has to be monic of degree 2g and, since P(n) is an integer for all is, its coefficients are all rational.

Further, since End X is a finite Z-module, f is integral over Z, so j f and hence TI(f) satisfies a monic equation over Z. Hence all the eigenvalues of the matrix TI(f ). are algebraic integers, and

its characteristic polynomial has coefficients

which are algebraic integers. Since the coefficients are also

182

ABELXAN VARIETIES

rational, they are rational integers. Hence P(f) is a well-defined element of End X, and we have finally T1(P(f)) = P(T, f) = 0,

so that P(f) =0. The above polynomial P(t) (which belongs to Z[t] and is independent of 1) is called the characteristic polynomial of f. Its constant term and minus the coefficient of tf-1 are called the norm and trace respectively of f. DEFINITION.

By the lemma proved earlier, we see that if End°X = A, x ... X Ak where A. are simple algebras over Q, and we denote the components of an f e End°X in As by f, and the reduced norm of A; over Q and the reduced trace over Q by Nm° and Tr° respectively, we have k

Nmf=fl(Nrn°f)"i, 5

i=1 k

Trf = where n; are integers > 0.

:-1

n,Tr°f

COROLLARY. Let X be a simple abelian variety of dimension g, K the center of the algebra End°X, [K: Q] = e, [End°X: K] = d2. Then de divides 2g.

PROOF. We have Nm f = (Nm°f)" for some n. But Nm is a polynomial function of degree 2g, and Nm° is a polynomial function of degree de.

RErQARK. When the characteristic of k is zero, with assumptions as in the above corollary, one can say even that d2e divides 2g. In fact, we may assume (by the Lefschetz principle) that k is the complex field. Let X = VI U as usual. Then the division ring

End°X admits a faithful representation in the rational vector space iT®Q, so U®Q becomes a vector space over End°Y. Hence dimQ U® Q = 2g must be divisible by dimQ End°X = d2e.

This is definitely false in positive characteristic. In fact, for any characteristic p > 0, we shall see in §22 that there exists an

Hom(X,X) AND THE i-ADIC REPRESENTATION

183

elliptic curve X with p-rank 0 and that for such a curve, End°X is non-commutative of rank 4 with center Q. Thus, in this case, de = 2 = 2g. DEFINITION. A simple abe,lian variety X is of (CM)-type if de = 2g where d2 is the rank of End°X over its center If, e is the degree of K over Q and g the dimension of X.

Now, in a division algebra A of rank d2 over its center K it is well known that all maximal commutative subfields have degree d over K. Thus, a simple abelian variety X is of (CM)-type if and only if End°X admits a subfield of degree 2g (since in any case, de < 2g). Let l be a prime different from the characteristic of k, and let i" be the group of l"-th roots of unity in k*. We have homomorphisms µri}1 pi" given by e --) 1, and 20.

Riemann forms.

this makes {µi"} a projective system. Let us put Ml = lim Kr,. M1 has the structure of Zt module. Since evidently we can choose isomorphisms pr, = Z/l"Z such that the maps µI"+1 µr. go over

into the natural maps Z/l"*1Z -Z/l"Z the projective limit is (non canonically) isomorphic to the Z, itself.

Now, let n be any integer prime to the characteristic. We have set up a canonical isomorphism of ker ng with the dual of ker nx,

that is to say, we have defined a pairing which we will call Z.: X. x (X). -+ N, where p. is the group of n-th roots of unity in

P.

Recall the definition:

Take a e X. and A e (X)", and let A correspond to the line bundle L. Then L" is trivial, so n4L is trivial too and

L =All x X

( action of X. O"(a,z) = (X(u).Ct, z + u)

for a character X: X. - k*. Then e"(a,A) = X(a)-

ABELIAN VARIETIES

184

In other words, we take the canonical action of X,, on nx*L and carry it over to an action of g" on the trivial bundle, where it is

given by a character X. It is useful to have an alternate definition of i using divisors instead of line bundles. Let D be a divisor such that OX(D)=L. Since L" and nnL are trivial, there are rational functions f and g

on I such that

(f) = nD, (9) = n$1(D) Then

(naf) = n.ng1D = (9"), so for some constant a,

g"(x) = a.f(n.x), all x EX. It follows that [g(x)/g(x + a)]" = 1 for all x e %, i.e. g(x)/g(x + a) is a constant n-th root of unity, and we can prove g(x)

LEMMA.

a"(a,h) =

PnooF.

Let g(xg+)

9(x + a)

a)

= 77(a)- Consider the diagram of maps of

sheaves:

n* Ox(n$1D) 0. It suffices therefore to prove the first statement. Choose and fix bases for T1(X) and M. Applying Theorem 3 §19, we get

[El Y', = c - (RI), [E+L]AV-1 A V*(L)

= c . (Hn-1. 0*(H)). for some l-adic constant c. Therefore, A [E,6]"0 [EL]AO-1

= (H4-1.0*(H)) (E')

Since we have EO'(L) = ELo(o X 0), we are reduced to proving the equation

Hom(X,X) AND THE 1-ADIC REPRESENTATION

193

[BU]AI-1 A (V -(o Y' x #)) Lnu

where ¢' is the transpose of 0 with respect to E". This is purely a problem on linear algebra. We may utilize a basis el,e2, ... ,e20 of Qi® TT(X) such that EL(e2ti_1, e2:) = 1 and EL(ez:-1, ej) _ EL(e2j, ej) = 0 if j 2i or 2i - 1. Then the left side becomes (by definition of exterior multiplication)

i iv...,ip odd

r

;1

ELMeip), 0(eip+1))' ,,...,i,odd

(9-1)f ` i odd

=

1

00'(ej+j)) +EL(Y''Y'(ei), ei+1)

29 t odd

= 29 APPLICATION I.

STRItCTt7RE of End°,X FOR SDNPLE X.

We have seen that for a simple abelian variety 1, D = End°X

is a division algebra of finite rank over Q with an involution x i-a x' such that if x 0, Tr(xx') > 0 where the trace is the reduced trace over Q (or any positive multiple of it). We shall now give the classification, due to Albert, of all pairs (D,') where D is a division algebra of finite rank n over Q and

x, -r x' is an involution such that TrD,Q(xx) > 0 for x e D, x

0. We shall consistently use the following notations. The

center of D will be denoted by K, and K° _ {x e K I x' x} is the set of elements of K fixed by the involution. We put [D: K] = d1 [K: Q] = e and [K°:Q] = eo (so that n = e32 and e = e° ore = 2e°

according as the involution is trivial on K or not). Without further mention, we make use of the fact that the restriction of TrD/Q to any simple subalgebra of R®Q D is a positive multiple of the reduced trace over R of this subalgebra.

194

ABELIAN VARIETIES

i < r1) be the set of distinct STEP I. Now, let a,: K0 real imbeddings of Ko, and a,l+i : Ko -> C (1 <j< r2) a set of complex imbeddings such that any non-real complex imbedding of KO in C is either a certain a,;+s or a complex conjugate of some ax}j. Thus, r, + 2r2 = ea. We then have an isomorphism of R-algebras a; R ®QK0

R'= X C'=,

all (9 x) = (a1(x).... , a,1(x), &,+1W, ... , a,l t,. (x) )

Since the involution is the identity on Ko, for x e K,*, we must

have Tr x2 > 0, and the same must hold in R®QKo also (in fact, this quadratic form has to be positive semi-definite on R (&QKo

by continuity, and its null space, being the orthogonal comple-

ment of the whole space for this quadratic form, must be a rational subspace. But then, Tr(x.x') > 0 for x e Ko, x 0 0, so that it has no null space). But this implies that r$ = 0, as is trivially seen, so that KO is totally real. If now K 0 KO, K = Ko(i/a) for some a e KO, Va f Ko, and Now, P. ®QK = (R ®QKo)(D8o K ^ fl

(&K. K

ri-1

where R,ri) is R considered as a Ko-algebra via o . Now, K is isomorphic as an R-algebra to either R x R or to C according as

ari(a) > 0 or ari(a) < 0, and the restriction of the involution interchanges the factors in the first case and is complex conjugation in the second case. Again from the positive definiteness of Tr(x.x )

on R®QK, one deduces easily that R x R cannot occur, as a factor, i.e., K is totally imaginary, and ari(a) < 0 for all i. We shall say that the involution is of the first kind if K = Ko, and otherwise we say it is of the second kind. STEP II. If the involution is of the first kind, the involution defines an isomorphism of D and its opposite algebra over the center K, so that its class in the Brauer group Br(K) of K is of order 1 or 2.

If this order is one, we must have D = K. Next assume that the order is 2. Since by a theorem of 13asse-Brauer-Noether, the rank of a central division algebra over a number field is the square of its order in the Brauer group, D must be of rank 4 over K (i.e. a

Hom(X,S) AND THE I.ADIC REPRESENTATION

195

so-called quaternion division algebra over K). In this case, there

is a canonical involution x -+ x* of D over K given by x* = TrD1gx - x where TrO is the reduced trace. (To check this is an involution, extend D to the algebraic closure of K, so that

we are reduced to the case of a 2-by-2 matrix algebra over a field, when this is trivial to check.) By the theorem of Skolem-

Noether, there is an a e D - {O) such that x' = ax*a-' and the condition that x' = x gives us that a* = e. a with e e K*. But now, a = a** (e.a)* = c2a, so that e = f 1.

Now, ifs=1, a*=Tr.11 a - a=a, so that aeK and x'=x*. We have an isomorphism

R®QD ~ (R®QK)®$D-2

(R(j)0ED) x ... x (R(.)(&ED) (*)

where R 0, that is, (Tr A)2 > TrA2, and this is false for A =110

1).

Hence all the factors in

the above decomposition of R®QD are isomorphic to K and the involution restricts on each factor to the canonical involution.

Since for the standard involution on K, we have Tr(x.x*) > 0 if x # 0, the conditions derived (when e = 1) are necessary and sufficient.

Next consider the case when e = - 1. In the decomposition (*), let ai be the image of a in R(i) ®gD = D{, so that on this factor, the involution takes the form x r--? a`(TrD x - x)a, i, and we have

also a* = TrDI5 a; - a; _ - a{, so TrD,,R a, = 0. Suppose now that D; is R-isomorphic to K. Since aa* is real and positive, a; satisfies an equation x2 + A2 = 0, A e R*, and hence by SkolemNoether, we can choose an isomorphism of D({) with K such that

a; goes to Ai e K = R + Ri + Rj +Rk. But then, if x = xo + x,i + x2 j + xsk, we have Tr (z. z) = 2(xo + xi - xs - x$) which

ABELIAN VAEtIETIES

196

is not positive definite. Hence, each D, is isomorphic to M2(R). Further, K[a] is a subfield of D stable for the involution such that a' = - a, which shows that a2 e K and a2 is negative in every real imbedding of K. Thus, each a, satisfies a minimal equation a; = hs e R in M2(R) with A,. < 0. Again by Skolem-Noether (or trivial checking) we can choose an isomorphism R(, ®ED M2(R) such that a; goes to pi( 1

'1 with

, > 0, fr, a R. But one checks

that in this case, `the involution on this factor is nothing but the

transpose (in the sense of matrices), and we certainly have 0. Thus the conditions Tr(A.'A) > 0 for A e M2(R), A derived are necessary and sufficient for the positive definiteness of Tr(x.x').

STEP III. We come now to the case of an involution of the second kind. We summarize the results of class field theory concerning the Brauer groups of an algebraic number field and p-adic fields in the following theorem. The Brauer group of a p-adic field is canonically isomorphic to Q/Z. If L D K are two p-adic fields with [L: K] = n, the induced map Br(K) >Br(L) goes over by means of the above isomorphisms into multiplication by n in Q/Z. TJ EOEEM. (1)

The Brauer group of R is cyclic of order two, and we identify it with the unique cyclic subgroup of order 2 of Q/Z. The Brauer group of C is trivial. (2)

For any central simple algebra 'D over an algebraic number

field K, and any finite or infinite, place v of K, if K, denotes the completion of K at v, let Inv, (K) denote the element of Q/Z corresponding to 'the class of DO, K, in Br (K,). Then we have an exact sequence

fBr(K,)-}Q/Z--0 where the second map is gotten by forming the sum of the elements

in QJZ.

Hom(X,X) AND THE l-ADIC REPRESENTATION

197

Let us now look at the involutorial division algebras of the second kind over Q. Let a be the restriction of the involution to K, so that a induces an automorphism of Br(K). The existence of an

involution of the second kind implies that a(cl(D)) _ - cl(D), or equivalently, using the above theorem, that for any place v of K, Inv,(D) + Inv (D) = 0. (A) Since we have shown that K is totally imaginary, this condition is always fulfilled for infinite v. Suppose then that (A) holds, so that D is isomorphic to the opposite algebra to the conjugate algebra D(,). This means that we can find a map D D, x i --+ x* such that for A EK, (Ax)* =a(A)x*, (x +,y)* =x* + y* and (xy)* = y*x*. By Skolem-Noether, any involution inducing a on K must be of the

form x' = ax*a- 1 for some a c- D, a = 0. Since x i.-- x** is a K-automorphism of D, we must have x** = axa' 1 for some a E D, and since ax*a 1 = (x*)** = (x**)* _ (axa 1)* = a*- 1x*a*, x E D,

we deduce that a*a e K, and since (a*a)* = a*a, a*a E K0. In order that xa- +x' = ax*a 1 be an involution, we must have that as*- laxa la*a- 1 = z for all x e D, i.e., a.a*- la e K, or equivalently, a'1a* =pa for some s EK. If we put q(x) =a1x* for xED, then 0 is a-linear, and the solvability of #(a) =,ua with a 0 implies that

((x*a)'la

=a(a*a)-1 =a 1aaa la*'1 =a1(ala*)* =02(a) p.al&.a =Nmxlx, lr..a,

so that a*a c- Nmxis K*. Conversely, if this holds, let (a*a)-1 = NmxlxoA, so that for any x E D, if a = Ax + 4,(x), we have #(a) = a(A) O(x) +

((x*a)-1

x = a(A) (Ax + O(x)) = a(A).a.

Thus, under assumption (A), with * and a defined as above, the necessary and sufficient condition for the existence of an involution is that a*a e NmEJX,K*. Since K/Ka is a quadratic (hence cyclic)

extension, this holds if and only if a*a is a norm in each (K0),, from K,,, vo being any place of K0, and K, being the direct product

of the completions of K at all places of K lying over vo. If vo is

198

ABELIAN VARIETIES

infinite and a; : K, --)- R is the corresponding imbedding, DO. ,R

= D®$(K®g R) D®, C=M,(C) and * extends to a map of Md(C) onto itself of the form X*=AX* A-1, AeG.L(d, C). Hence X**=A.A`-1.X.At.A-1, so that theimage of a in D®$OR is AA.11-1 for some A e C*, and a*a has for image J.12 which is a norm from C.

Thus, it suffices to look at the Archimedean v0. Again, if there are two extensions of vo to K, K,o is the direct product of two copies of (K.),, as a (K0),,-algebra, so that the norm condition is vacuous.

Thus, we are left with the case of a v of K such that av = v. In

this case, Inv,(D) = 0 or I by (A). If Inv,(D) = 0, D®$ K, is a matrix algebra over K, and A& - a(A) ` is an involution of D ®g K inducing a on K,, so that, by the previous reasoning applied in the

local case, a*a is a norm. Suppose now that Inv,(D) = J, so that D, = D®x K, is a matrix algebra over the quaternion division algebra Q on K,. Since a induces the identity on Br(K,) (see the theorem above), condition (A) gives us a a-linear map Q ->Q,

X i-a X, such that (X Y) = YX. If we put X = P X R-1 for some fl e Q and X* = A X` A-1 for all X e D. and some A a D we see that upto a factor which is an element of the center, a equals AAt-1fl, and a*a differs from A. (A A`-lY)"t . A-1(A A`-1Y) = A(R A-1A`)

At-10

p=11 by a factor in Nm1,,0(K, ). Hence a*a is a norm in Ka, if and only if

is, hence, if and only if Q admits an involution inducing

a on K,. Suppose ' is such an involution. Then we have (by the functoriality of trace) that Tr z' = a(Tr x), so that if is Q ->Q

H

i(x') is an is the canonical involution of Q, i(z') = i(x)'. Thus x automorphism of order two of Q inducing a on K. If we put Qa = {x e Q 19S(z) = zJ, Q0 is a K,-subalgebra of Q and K, ®B,, Qc -->

Q is an isomorphism. But now, Qc is of rank four over Kp hence a quaternion algebra, and since Br(K00,) -+ Br(K,) is, by

Hom(X,X) AND THE I:ADIC REPRESENTATION

199

means of the canonical isomorphisms with Q JZ, nothing but multiplication by 2, Q = Q0 ®ga Kp is a matrix algebra over K which is a contradiction. Thus, if K0 is a totally real field, K a purely imaginary quadratic extension of K and D a central division algebra on K, the necessary and sufficient condition for the existence of an involution of D inducing the non-trivial automorphism o of K over Ko is that besides (A), we also have

Inv,(D) = 0 if ov = v.

(B)

Suppose then that (A) and (B) hold and let x i.-> x*

STEP IV.

be an involution. We shall then show that there are positive involutions too and we will classify them. For this, choose an isomorphism eo X

D®QR

Md(C) x Md(C) x ... x Mg(C).

Then the given involution has an extension to the right side given by (Il, X2, ... , I,a) --* (A1S°4A1 1, ... , A,.1% AA1) with .A{ =-1tAl, 91; e C*, A; a GL(d, Q. We must have I rJ; I = 1, and on replacing A; by a scalar multiple, we may assume n. = 1, so that .A{ = A;. Hence, if A =(A1,..., AO), we have A* = A. The set of A e D ®QR with A* = A is of the form V ®Q R where V is a Q-subspace of D, so that we can find an cc e V such that a® 1 is arbitrarily close to

The map x H x' = a lx*a is again an involution of D whose extension to Md(C) x ... x Md(C) is arbitrarily close to (I..... , I,,,) -r (,X .... , 8;a). Hence for q. a good A E D ®Q R.

enough approximation to A, TrDJQ(x. x') > 0 if x

0, x e D.

On Md(C) x ... x M4(C), this involution is of the form

(I1,...,5,,) H(A1 1A11,...,A,o

,,Ae,1),

with A hermitian and close to I, so that the A{ are positive definite. Let B, be a positive definite square root of Al. Modifying the chosen isomorphism D®Q R Md(C) x ... x Md(C) by the

inner automorphism given by B = (BI, ..., B,a), we see that we

ABELIAN VARIETIES

200

may assume that the extension of the involution to Md(C) x ... x M4(C) is the standard one

(Xi,..., X..)t--r (.X ...... which is certainly positive. Thus, when (A) and (B) hold, we have found one positive involution on D and an isomorphism D®0R

MM(C) x... x Md(C)

such that the involution goes over into the standard one written

above. Suppose * is any other positive involution, so that x* =ax'a'-x, a' = Aa for some A E K. Since A.k =Nm$1 A = 1, we can write A =

Qµ

for some 1` a K, and when a is replaced by

fU

p,a, the involution is unchanged whereas the new a satisfies a' = a. Hence a goes over into (Ax, ... , A..) a Md(C) x ... x MM(C) with

A; hermitian. Positivity of Tr(x x*) gives us the condition that Tr(X A. XI A, x) > 0 for X e M4(C), or equivalently, for some unitary U and any X e Ma(C),

Tr(UXU-IA,UX'U-'Ai 1)=Tr(XU-'A,UX'U-1A; IU) > 0. Choose U so that U- 'A; U is real diagonal :

=D,.

U- 'A; U = 0

as

We must then have Tr(X D X` D' 1) > 0 for all X e Ma(C). But if X = (rU), a

Tr(XD$tiD-x) = i.tax

lxjxjaa , r

so the condition is that D is positive definite or negative definite. Since we may replace a by - a, this proves that all positive involu-

tions are of the form x r- a x' a I where the hermitian matrices A. are positive definite. Summarizing, we have

Hom(%,X) AND THE l-ADIC REPRESENTATION

201

Let D be a division algebra of finite rank over Q with an involution ' such that TrDnQ(xx') > 0 for x e D, x 0. Let K be the center of D and KO the subfceld of elements of K fixed by ' . Then (D,') is one of the following types. THEOREM 2.

TYPE I. D = K = KO is a totally real algebraic number field and the involution is the identity.

TYPE II. K = K, is a totally real algebraic number field and D a quaternion division algebra over K (i.e. a central division algebra of rank 4 on K) such that for any imbedding a: K ->R, R(.)®KD = M2(R).

Let x* = Tr x - x be the standard involution of D and a e D such that a2 e K and a2 is totally negative. Then the involution is of the form x' = a x*a 1, and conversely, any such map is a positive involution. For any such involution, we can choose an isomorphism

R®QD--->M2(R) x ... x M2(R) (e=[K: Q] factors) such that the involution extended to the right side by R-linearity is given by (.I, ... , S,) - (%i, ... , Xi). TYPE III_ K = KO is a totally real algebraic number field and

D a quaternion division algebra over K such that for any imbedding a : K -*R, R(,)®K D N K,

where K is the standard algebra of quaternions on R. In this case the involution ' is the standard one, x' =TrDIKx- x, and there is an isomorphism

R®QD-*Kx...xK, carrying the involution

into the product of the

standard

involutions in each factor K.

TYPE IV. K, is a totally real algebraic number field, K a totally imaginary quadratic extension of K. with conjugation a over K0. Then D is a division algebra with center K such that (i) if v is a finite place fixed by a, Inv,(D) =0, and (ii) for any finite 0. place v of K, Inv,(D) +

ABELIAN VARIETIES

202

In this case, there exist totally positive involutions x t s x' and isomorphisms

R®QD * Md(C) x ... x Md(C) which carry the involution into the standard involution (X1,, ...,Xee ) 2l,, ). Given one such ', any other positive involution

of D is of the form x* = a x' a-' with a e D, a' = a and such that the image of 1® a by the above isomorphism is of the form (A,,..., Aee) with A; hermitian positive definite.

The following table gives the numerical invariants in all four types, and also indicates the restrictions on these invariants

when D=End°X where X is a simple g-dimensional abelian variety. The symbols e, eo, and d have the same significance as before, and S = {x e D I x' = x} and 71 = dimQ S

dimQ D

Type e

I IV

when D= End°X, dim X = g

Restriction in char p > 0 whenD=End°X dim X = g

1

e1g

elg

2} eo 2 }

2elg

2eJg

2elg

elg

e°d$Ig

eodlg

eo

II III

dI

Restriction in char 0

1

eo

2eo d

Excepting the indicated restrictions, all the assertions contained

in the table have been proved. As for the restrictions, they are immediate consequences of the three divisibility results established earlier, viz. (i) in char 0, dim D12 dim X, (ii) in char p > 0, ed12 dim X, and (iii) if L is a subfield of D whose elements are fixed by the involution, [L: Q]Jg.

One might ask to what extent the conditions derived above on the endomorphism rings of a simple abelian variety are complete, that is, given a division algebra of one of the four types and an integer g fulfilling the restrictions imposed

Rom(X,X) AND THE I-ADIC REPRESENTATION

203

above, whether there exists a simple abelian variety of dimension

g having the given algebra as endomorphism algebra. In characteristic zero at least, the answer

is known and is

due to Albert. The result is that there always exists such an X, excepting when D is of type III or IV and the quotient g/2e in the first case and g/e° d2 in the second case is 1 or 2. Even in these exceptional cases, it is known what further restrictions ensure the existence of an X (of. Shimura, [Sh], esp. §4).

On the other hand, not much seems to be known in positive characteristics. APPLICATXON II.

THE RIEMAx

HYPOTHESIS.

We first prove the

Let X be an abelian variety,' the Rosati involution on End°X defined by some ample line bundle and a e End X such that a'a == a e Z. Let wl,..., co, be the roots (in C) of the characteristic polynomial P of a. Then the subalgebra Q[a]c End X generated by a is semi-simple, and PRoPOsITIoN.

(i)

wi 12 = a for all i;

(ii)

the map wi r a is a permutation of the roots w;. w;

PROOF.

a

wi

Note that (ii) is an immediate consequence of (i) since

and P is an integral polynomial. Next, let Q(X) be the

minimal polynomial over Q of a (as an element in End %). I claim

that P and Q have the same complex roots. In fact, since P has integral coefficients, and P(a) = 0, QIP. But also P is the characteristic polynomial of TT(a) in the matrix representation

Tt: End(X) -+ End(TIX). If w e QI(algebraic closure of QI) is a root of P, then w is an eigenvaluc of TI(a), hence Q(w) is an eigenvalue of Ti(Q(a)). But TI(Q(a)) = 0, so Q(w) = 0, i.e. all roots of P in QI are roots of Q. Therefore PjQ° for some n, and P and Q have the same complex roots too.

204

ABELIAN VARIETIES

The restriction S of the trace on End°X to Q[a] is a trace form on Q[a] satisfying S(X.X') > 0 if X e Q[a], X t 0. Further, since a is invertible in End°X and Q [a] is finite-dimensional, it follows that a-'1 a Q[a]. Hence a' = a/a a Q[a], so Q[a] is stable for the involution. If 21 cQ[a] is any ideal in Q[a], and b is its orthogonal complement in Q[a] for the quadratic form S(X.X'), b is again an ideal and 2I n b = (0), 92 O+ b = Q[a]. Thus Q[a] is semisimple, hence isomorphic to K, x K. x ... x K, where K; are algebraic number fields. The involution, being an automorphism of Q[a], permutes the factors K. But since S(X.X') > 0 for every X e 0, the involution must take each K. onto itself, and therefore S is a trace form on each E over Q with S(X.X') > 0 if X = 0. Hence each Ki is either totally real with identity involution or is a totally imaginary quadratic extension of a totally real subfield with complex conjugation for involution. Now, the roots w of the minimal polynomial of at are precisely the images of a under

the various imbeddings 0, of the K; in C. Since 4j(x') =#;(x) for all x e Q[a], it follows that

We shall apply this proposition to obtain a proof of the Riemann hypothesis on abelian varieties over finite fields. Let F = FQ be a finite field with q = pf elements, and X°

a scheme of finite type over F. (We do not consider X° as a variety whose points are geometric points with values in an

algebraically closed field, but as a scheme in Grothendieck's sense.) We define the Frobenius morphism on Xo, ao: Xo -->Xo,

to be the identity on the underlying space together with the homomorphism Ox. -> Gx, of structure sheaves given by f i-* fQ.

Note that this is a homomorphism of sheaves of F-algebras since AQ=A for A E F, so 7r° is a morphism over Spec F. Now

let k be the algebraic closure of F5, and let X be the k-scheme X = k OF X. The morphism iT: X -; X obtained from a° by base extension is called the Probenius morphism on X, relative to F and to I. Let us see what this looks like on the geometric

(or closed) points of X. Suppose that X. = Spec A, where

205

Hom(X,X) AND THE i.ADIC REPRESENTATION

A =F[11, ..., Im](5511, so that Xe is embedded as a closed subseheme

in AF, and the closed points of X can be considered as elements of the set km. The morphism Tr is defined by the homomorphism of

F-algebras A ->A sending Xi into iso that if (x,,

, x,,,) is a

geometric point of X, 17 maps it into the point (xi, ... , x.11). In particular, a point (x,,... , xm) is fixed by e if and only if x;" = x;, i.e. if and only if x; is a rational point over the field FQ" with q" elements. Further, the Frobenius morphism has the functorial property that if f : Xo -> Yc is a morphism of F-schemes and iro,x, and mo.r, are the Frobenius maps of X0 and Yo, respectively, vo y0o f

= f o vo x,. Finally, it is clear that the map induced on tangent spaces by w at any point of X is 0, since D(f)=0 for any derivation D of a ring A of characteristic p and f e A. ThEOREM 3.

(Lang.)

Let Xp be a scheme over F. such that

X = k OF Xo is an abelian variety. Then Xo has at least one point rational over F. PRoor. If a is the Frobenius morphism, then a must have the form v(x) = xo + f (x) for some closed point xo e X and some endomorphism f of X. Then 1-f is an endomorphism of X. Since 7r and

hence f induce the zero map on the tangent space at 0, 1 - f induces the identity on this tangent space. Therefore ker(I-f) is 0-dimensional and 1-f is surjective. Then if(1-f)(x,)=xo, it follows that x, =xo+f(xl) =v(x,), hence xi is rational over F5. Therefore, if X is an abelian variety, by choosing an appropriate origin 0 e X, we can always assume that 0 is F-rational. Then each v' fixes 0 and is therefore an endomorphism of X. Moreover, 1 - v" induces the identity on the tangent space at 0, so it is also a separable endomorphism. Hence we obtain: N. = Number of FQ"-rational points of X = #(Ker(1 - r)) 4°f

= deg(1 - 7').

But if w ... , w2, are the roots of the characteristic polynomial of

ir, then the characteristic polynomial P"(t) of 7r", for all n, is 21

11(t- w ). Since deg(1 - e) = P"(1), it follows that i-i

ABELIAN VARIETIES

206

20

i=i

We now wish to show 1wil = Vq: this is the Riemann hypothesis. Since it suffices to prove that Jw;' I = %/q'" for some m, we may replace F. by F,,,, Xo by F m ® F Xo and sr by rr'" if necessary. By

doing this, we can assume that there is a line bundle L0 on Xa such that L = k®FLO is ample on X (since any line bundle on X is of this form for suitably large m). Denoting by ' the Rosati involution with respect to L, we shall prove that (i) 7r' O7=q, so that the proposition applies. But by the definition of 'this means that (ii)

(OL(r(x))) = qcS (x),

all x e X.

But ira acts on Og, by f k--+ fe, so it follows that i L0 4 Lo. Therefore yr*L = LQ and

(ill)

*(T *L® L-1) a T*7r*L® (,r*L)-

(TxL®L- s)®0, Since the line bundle on the left represents and the (ii) and hence (i) are line bundle on the right represents correct.

We summarize our conclusions in THEOEEM 4. (Weil.) Let Xo be a scheme over FQ such that X= k®FX0 is an abelian variety. Let N,, = the number of points of X rational over F,,. Then sa

1) i-1

where wi a C and they satisfy (i)

Iw,!=v'q,

(ii)

wni = q1-: for some permutation 7T.

Rom(X;X) AND THE Z-ADIC REPRESENTATION

207

COROLLARY. For some constant C, I A'" - q" o I < C. q"v-D for

all n. Another application of the proposition is THEOREM 5. (Serre.)

For any n > 3, and any L ample on an

abelian variety X, the restriction homomorphism

a e Aut X a*L a L®(in something) Pico,X)

1

l

J

-* Aut(X")

is injective (here X" = scheme-theoretic kernel of n1). PROOF.

If a*L

L

PicoX (something1

l in

then

hence

o a = 0y. This means that for the Rosati involution defined by L, a'a=1. Hence by the proposition the roots of the characteristic polynomial of a are algebraic integers all of whose conjugates have absolute value 1, and hence are all roots of unity. Suppose now that a restricts to the identity on some 8"(n> 3). Then the restriction of a-1 to X,, is 0, so that (a-1)=np for some fl e End X. We deduce that if w is any characteristic root of at, ao

w -1 = nn where rt is an algebraic integer. We now have the LEMMA. If w is a root of unity such that to = 1 + nrl where n is a rational integer > 3 and 7) an algebraic integer, then w = 1. PROOF.

If not, by raising w to a suitable power, we may assume

that w is a primitive p-th root of unity for a prime p. Taking norms over Q in the equation co - 1 = nrl, we obtain p-1

F1 (1-w{) =np-'.N,

t-z

where N = (- 1)p-'Nm rl is a rational integer. But the left side is p-1

the derivative at X = 1 of Xp- 1 = 1I (X - co'), that is, p. i-0 Hence np-' divides p, which is impossible if n > 3.

Applying the lemma, we deduce that the characteristic roots of

a are all 1, so that 1 - a is nilpotent. But by the proposition, Q[a] is semi-simple, so it has no nilpotent elements. Thus a =1.

ABELIAN VARIETIES

208

ArPLro&TION III. STRVOTUBE OF NS°(I).

Let X be an abelian variety and let NS°(X) = NS(X) ®Q. As in § 20, if we fix an ample .1 on X, then we can identify NS°(X) -* {a a End°X I m' = cc}. P

In particular, NS°(X) has a natural structure of Jordan algebra over Q if we define a ° P =jP 1(P(a)P(p) + p(p)P(a)), at, fi e NS°(X),

using composition in End°X. What can we say about this Jordan algebra? First of all, the fact that Tr(p((X)2) > 0, all a e NS°(X), a # 0, implies immediately that NS°(X) is formally real, i.e. n

a;0

iv l

(of. Braun, Koecher, [B-K] Ch. 11, § 3). Now the formally real Jordan algebras over R have been classified: of. Braun, Koecher, Ch. 11, § 5. In our case, we do not get all possible such algebras by forming NS°(X) OR. In fact, we have THEOREM 6. NS°(X)® R is isomorphic to a product of Jordan algebras of the types: Af,(R) = r x r symmetric real matrices ,a£°,(C) r x r Hermitian complex matrices af,(K) = r x r Hermitian quaternionic matrices, i.e. IX = X, where z -> x is the standard involution on K.

Psoor. Decompose End°(X) ® R into a product of copies of M (R), M (C) and ,°.K). Then NS°(X)®R is isomorphic to the set of fixed points here under a positive involution. But it is easy to check that every such involution (a) fixes each of the factors K = R, C or K, and (b) by inner automorphism of each [Of. the proof of factor, can be put in the standard form X Theorem 2, Step IV, for the case K = C; the other cases are analogous. One checks first that every positive involution of where L = A. One then MA(K) is of-the form X F-)

Hom(S,%) AND THE l.ADIC REPRESENTATION

209

checks that A = U.D. U-1 where D is diagonal with real entries either all positive or all negative and "U= U-1. Finally, solve =ED = E2 and use the inner automorphism defined by UE U-1 to put the involution in standard form.] Fix isomorphisms 0 and qi: (I) End° (X)®R

fJM,,(R) x ]JAf,1(C) x f M,(K) U

jP NS°(X)®R

f.r,,(R) x fA"(C) xf Y,(K).

What happens to the polynomial function X: NS°(X)®R-+R? For any x e End°(X) OR, let #,,1(x), 01,2(x), op (x) denote the components of ¢(x) in the above decomposition. Then we know that the function "degree" can be written (II)

deg(x) = I1det(0,.1x)a' IUldet(q,,2x)12b' I1Nm(¢:,,x)' where Nm : Mt(K) -s R is the reduced norm (the multiplicative polynomial of degree 2t). Now on NS(X), deg(px) = a. x(x)2 for some constant a. It follows that the a, are even and that the function X can be written (III) X(x) =cast. IIHNm(#;,sx)ci all x E NS°(X)®R. Note that ,y, 2(x) is Hermitian, so its complex determinant is real. Here "HNm" is the "Haupt norm" of BraunKoecher (Ch. 2. §4), a. polynomial function of degree t from .e' (K)

to R. If a° e NS°(X)®R is the point defined by the ample L on I used to set up p, then p(.1°) = 1, so 0,,; (A0) = I, so the constant in the above formula is X(A°). Using the results of §16, it follows finally that: (IV) If X(x) 0 0, then (o ieg.e igenvalues) + ' b{ (# neg. igenvalues) i(x)

2

f

# neg. eigenvalues L,`of+:.sx }.

210

ABELIAN VARIETIES

(The eigenvalues of a quaternionic Hermitian matrix H are defined

as the entries of a diagonal matrix D such that H = U.D. U-', and =U = U-'.) Since ample line bundles L are characterized by X(L) 0 0, i(L) = 0, it follows from (III) and (IV) that the images

of the ample line bundles in NS(X) are exactly the totally positive elements of the formally real Jordan algebra NS°(X) ® R. 22. Examples. FIRST EXAMPLE: ABELIAN VARIETIES OF CM-TYPE OVER C.

Let X be a simple g-dimensional abelian variety, D = End°(X), K = center of D, d' = [D: K] and e [K: Q]. Recall that ed 12g

and that we have called X of C.M-type if ed = 2g. We wish to classify these when k = C. A glance at the table in §20 giving the types of division algebras D tells us that we must have K = D, K a totally imaginary quadratic extension of a totally real field K° of degree g over Q.

We pose the problem a little differently. Suppose we are given a totally real number field K° of degree g over Q and a totally imaginary quadratic extension K of K°. We consider all pairs (i, X) where X is an abelian variety over C of dimension g and is K

-+ End°X is an imbedding of the field K in the ring End°X. We define two such pairs (i, X) and (j, Y) to be equivalent if there

is an isogeny a: X -+ Y such that if a : End°X-+ End°Y is the induced isomorphism, we have a o i =j. It is easily checked that this is an equivalence relation. Our object is to exhibit a complete set of representatives for the equivalence classes.

Let (i,X) be any such pair, V the tangent space of X at 0 and U the kernel of the exponential map from V to X, so that we

have a natural isomorphism V f U -- - X. Then End X acts faithfully as a ring of C-endomorphisms of the vector space V, leaving U stable. Thus, if we put i '(End X) = A c K, A is an order (i.e. a finitely generated subring of maximal rank) in K and U becomes an A-module. Thus, Q. U C V becomes a vector space over Q®ZA = K, and since both K and Q. U are of dimen-

Hom(X,X) AND THE l-ADIC REPRESENTATIO14

211

sion 2g over Q, Q. U is a one-dimensional K-vector space. Hence,

if we choose a non-zero element uo a U, the map 0: A --> U defined by a i - + a. uo is an injection of A into U, such that the index [U: ¢(A)] < + oo. Changing X by an isogeny, we can first shrink U so that U= ¢(A), and then increase U so that U = ¢(Ao), Ao=ring of integers in K. Next, the map c extends to an R-linear map which we still denote by 4:

R®QK=R0zA0

0) R(gzU=V.

It follows that 0 defines an isomorphism between the real tori:

(R®QK)/Ao ) V/U = X. Note that if a e A0, then this isomorphism has been set up exactly so that the endomorphism i(a): X -* X corresponds to multiplication by 1®a in R ®QK.

Next, let CD denote the complex structure on the real vector space R®QK obtained by pulling back the complex structure on V via 0. Since multiplication by 1® a in R®QK(ae Ao) goes over via 0 to a complex-linear map from V to V, it follows that in the complex structure 'D, multiplication by 1® a is complexlinear too. In other words, CD actually makes the R-algebra R ®QK

into a C-algebra as well as a C-vector space. We now invert this whole construction. DEFINITION.

If K is as above, A0 =integers in K, and CD is a

structure of C-algebra on R(&QK, then let X(K,CD)

and let im :

the complex torus R®QK/Ao,

(X(K, (D),.I(K,C))begivenbyim(a)=map

induced by molt. by 1® a.

We have shown that for given K as above, and any pair (i, X), there is a structureC of complex algebra on the real algebra R®QK such that (i, X) is equivalent to (io, X(K, C)). Our next aim is to show that (i) for any structure CD of complex algebra on ROQK, X(K, 0) = R ®QK/1 ® A 0 is an abelian variety, and (ii) for

212

ABELIAN VARIETIES

different complex structures d>x and ID, on ROQK, (i 1, %(K, dbi,)) and (i,2,1(K, D2)) are not equivalent. To prove (i), let us look more closely at a structure QI of complex

algebra on R ®QK. Giving such a 0 is equivalent to giving a homomorphism D of R-algebras, (D: C -a R®QK. Now, if 0;(1 ` i < g) are the distinct embeddings of KO in R, we have an isomorphism of R-algebras

R®QK- Z. (R(,)(Dx.K) x (R(s>®a.K) x ... x(R(o) ®s.K),

®aA r(A®a, A®a,---,A®a), where R(,) is R considered as a K0-algebra through oi. Thus, giving an R-algebra homomorphism d>: C -+R®QK is in turn equivalent to giving P.-algebra isomorphisms C;: C --Z. R(;)(DKOK

for 1 < i < g. For each i, there are clearly two such possible Risomorphisms C-*R(;) ®g0K. Thus, we see that there are exactly 2° possible 4) on R ®QK, and each 'D is uniquely determined by giving the corresponding R-isomorphisms (Di: C R,>(D%K (1 < i < g). Let T;: R(j)®,,0K -+ C be the inverse of Oi, so that r; restricted to K ;: 1® K cR(;) ®KOK is an imbedding of K in C extending the imbedding o; of Kp in R. We can choose an element

a e K such that a2 a Ko and ,(a) = ip, fl; a R, A> 0. In fact, if K =K0(VS) then T,(V) =iy; with yi a R*, and we can find 71 e K. such that si(n) has the same sign as y;, and we can take a = 7A/S.

We may further assume a to be an algebraic integer. If r;(a) i f; (1 c i < g), we define a Hermitian form H on (R ®QK, (D) by putting a

H(x, y) = 2

1i(x) T,(y), x, y e R®QK.

This form is clearly positive definite, and we shall show that Im H is integral on the lattice A0. In fact, for x, y e A0, we have a

Im H(x, y)

2Re 7

i-t

n(y)

Hom(S,%) AND THE I.ADIC REPRESENTATION

213

2> Re Ti(axy) i-)1

TrKjQ(axy) eZ,

where for any y e K, y denotes its conjugate over K. Thus, for any complex algebra structure iA on R (&QK, and the lattice A° in it, H defined above is a Riemann form and X(K, End°X an embedding, with the equivalence relation defined above. Then there are exactly 29 equivalence classes, and as (D runs through

complex structures on R®QK which make R®QK a C-algebra, the pairs (X(K,d1), i.) give a complete system of representatives in the distinct equivalence classes. REMAR.as. (1) It is not true that X(K,1) is always simple. It can be shown that in order for X(K,4D) to be simple, it is necessary and sufficient that there does not exist a proper subfield L of K satisfying the following conditions: (i)

L is a quadratic extension of L n K°,

214 (ii)

ABELIAN VARIETIES

if I is given by the set of imbeddings rl, ..., .r, of K in C,

and ifr;ILnKO =riI Lr) K.,then r;IL=r;IL. If such an L exists, X(K, (D) is isogenous to a power of X(L,'Y), where `Y is given by {r;

IbnE0}.

(2) Let us specialize to the case of dimension one, that is, the case of elliptic curves over C. If X is an elliptic curve over C, either End°X =Q or End°X = Q (-,/ - d) for some square free d e Z, d > 0. Moreover, given any imaginary quadratic field Q(-%/ - d), there is an elliptic curve X with End°Xr Q(-,/ - d), and upto an isogeny, X = C/{n + m V - dln,m a Z}.

SECOND EXAMPLE: ELLZiric CtTEvES IN CH'ARACTERI8TIC p > 0.

We begin with recalling some basic facts concerning abelian varieties of dimension one (or elliptic curves). These facts are immediate consequences of our general theory, as the reader may verify for himself.

Let X be an abelian variety of dimension one. We shall denote the divisor corresponding to a point P by [P]. Then, for any divisor

D on X, we have x(O$(D)) = degD, and if further degD > 0, x(Oz(D)) = dimH°(O$(D)) and H'(O%(D)) = (0). A divisor D belongs to Pic°X if and only if deg D=0. A divisor D=Bn;[Pj of degree 0 is linearly equivalent to zero if and only if EnP; = 0 on X. Any divisor D of degree > 3 is very ample.

Suppose now that the characteristic is either 0 or greater than 2. Let 0 be the identity element of the group X and let P1, P2, and P. be the points of order two on X. Since dimH°(0x(2[0]) = 2, we can choose a non-constant function x having a double pole at 0 and regular elsewhere. Subtracting a constant from x, we may assume

x(Pi) = 0, and since the sum of the zeros with multiplicity is 0 and there are exactly two zeros, we deduce that PI is a double zero ofxandthere are no other zeros. Thus, by dividing by a constant, we may assume that x(P2)= 1 and x(P3) = A e k*. By applying the above argument to x-1, we deduce that A,0, 1. SineeH° (0x(3[0])) is of dimension 3, we can find a function y having a triple pole at 0 and regular elsewhere. Bysubtractinga suitable linear combination

Hom(X,X) AND THE

REPRESENTATION

215

ax+b from y, we may assume that y(P1) = y(P2)= 0, and since the number of, zeros is 3 (taking multiplicity into account) and the sum of the zeros is 0, we deduce that y has simple zeros at P1, Pa and

P2=-P1- P2. Both the functions y2 and x(x-1)(x-A) have poles of order 6 at 0 and double zeros at Pl, P2, and P$ and no other zeros or poles anywhere, so that they differ by a non-zero scalar factor. Replacing y by a non-zero scalar multiple, we arrive at an equation

Xa:y2=x(x- 1)(x-A)

(N,,)

for X- {0} in A2(k). Conversely, the projective curve yet = x(x- t).

(z-At) has no singularities and is of genus 1 for A 0 0 or 1, and hence defines an elliptic curve Xa inP2(k).

We wish to find all possible values of A for which X is of p-rank 0, for characteristics p > 2. We know that the p-rank is 0

if and only if the F`robenius map in HI(X0) is trivial. The meromorphic form dx on X is regular in X --- {0} and vanishes at P1 = (0, 0), P2 = (1, 0) and Pa = (A, 0) to the first order, and no-

where else, since dx(P) = 0 and x(P) = a implies that x - a vanishes to the second order at P, hence 2P must be 0. Thus, the form w = dx/y is regular and nowhere vanishing in X - {0}. It follows that to must be regular and non-vanishing at 0 also. If we put Ua = X - {0}, U1 = X - {P1}, 21 = (U4, U,) is an afrze covering of X. A 1-cocycle for this covering is a regular. function

fin Uo n U1 = X - {0} - {P1}, and this is a coboundary if and

only if f = g - h with g regular on U. and h regular on U1. Consider the linear form on C'($Y, 0) = P(UO n U1, 0) defined by ) Res,,(fw). Since the residue at any point of a meromorphic form with a pole (of any order) at a single point of X and no other

fa

poles is zero by the residue theorem, we see that Resp;(fw) =0 if f is a coboundary. On the other hand, the function y/x is regular on Uo n U1 and has a simple pole at P1, so that Respl (y/x. w) 0 0.

Since dim H1(X, 0) = 1, we deduce that the above linear form k, and also that y/x induces an isomorphism H1(X, 0) e P(U0 n U1, 0) defines a non-zero cohomology class inHl(1,0). Hence, the Frobenius map on H'(X, 0) is trivial if and only if

216

ABELIAN VARIETIES

yple r= I'(Uo n` U3, 0) is a coboundary, hence if and only if Res.,

(f

.

0. 1,1.,,

dx Y

)

dx) Resp, (e dx) = Res , r (yaP1 X%1_1

Xy

`

X

coefficient of xP-1 in

= 2.

-1) (y2)

(P-1)

(P_ 1)

(P-1)

coefficient of x = 2.

(P-1)

=x T (x-1) 2 (x-A)-2 2

In

(P-1) (P-1) (x - 1) s (x - A) s

(P- 1) 2

2

P-1 R

(v)

=f 2(D(A),

where y-1

2

Now, 4)(0)=coeff. ofx(P-1)12 in x(P-1)12(x- 1)(P-1)12, and 4'(1)=coeff.

of

x(P-1)12 in (x- 1)P-1=

x - 1 = 1 + x +... +

xP-1, so that (D(0)

0. Thus, every root of D defines an elliptic curve 0, (D(1) Xx of p-rank 0, and these are the only elliptic curves of p-rank 0 upto isomorphism.

Let us call an elliptic curve in characteristic p> 0 supersingular if its p-rank is 0. We have then shown that in characteristic p > 2, any supersingular curve is-isomorphic to one of the Sx where A is a root of (D(A). If p = 2, it is not hard to see that there is exactly one supersingular elliptic curve, namely

y2 + y = x3. We omit this. Therefore, in any positive characteristic, there is one and there are only finitely many supersingular curves upto isomorphism.

Hom(I,I) AND THE I-ADIC REPRESENTATION

217

We now study the algebra End°1 of an elliptic curve over a field of characteristic p > 0. Let k be the algebraically closed field over which we work. We shall say that an abelian variety X over k is defined over a subfield k° of k if there is a scheme S° over k° such that I = k ®F,%. We can then easily establish that there is a finite algebraic extension k,. of k°i a rational point 0 in kl®F, g°= X and a morphism m1: S1 xk1ll --11 over k1 such that on base extension, we get (upto an isomorphism) the triplet (I, 0, m). In future, when we speak of abelian varieties defined over various fields, we will assume that 0 is rational and m defined over this field. Another remark in this connection is that if P. % Y is a separable isogeny and if either $ or Y is defined over an algebraically closed subfield k° of k then f, I and Y are all defined over k°. This is clear if we assume S defined over k°, since Y is the quotient of I by its kernel which is a reduced finite subgroup of I, and all points of finite order in I are k°-rational. Suppose on the other hand that Y is defined over k°. By induction, we may assume f of prime degree 1. If 10 p, then there is a separable isogenyg: Y-> %, defined over k, such that fog: Y-->Y

is lp, so that we are reduced to the first case. Suppose then that l = p, and let G be the infinitesimal part of pp, and Y' = Y/G. Then, G and Y' are defined over k°. Then there is a separable

isogeny g: Y' -.I, defined over k, such that the composite Y --> Y-> I -> Y is pp, so that we are reduced to the first case again. TH OREM. (Deuring.) Let I be an elliptic curve in characteristic p > 0. We have the following equivalences.

(a) I cannot be defined over a finite field .uu End°X = Q. (b) Suppose $ is defined over a finite field k°. Then, (i)

End°I is imaginary quadratic over Q - p-rank of I is

1 e 77-" 0 p% for suitable integers n, m, where it is the Frobenius morphism over k°. (ii) If, however, the p-rank is 0, then End°S is the (upto isomorphism, unique) quaternion division algebra S0, over Q which satisfies

InvK,) = 0 if t is finite and

p and Inv

) = Inv,,S(,,) _4. -

218

ABELIAN VARIETIES

Finally, there exists at least one and upto isomorphisms, at most finitely many .1 in each characteristic for which (ii) holds. PxooF. First consider the following three statements.

(A) X is of p-rank 0. (B) End°X is non-commutative. (C) X is defined over a finite field, and if it is the Frobenius morphism over this field, ii" = p$ for some n and m > 0. We shall establish that (A) (B) and (A) e: (C) in that order.

Suppose then that (B) holds. A look at the table of §21 tells us that End°(X) is a central simple quaternion algebra over Q, hence End°(X) is also a central simple algebra over Q. If the .p-rank of X were one, Q9®.T,(X) would be a one-dimensional Q,-vector space in which Q,, ®QEnd°X admits a representation, which is impossible. Hence X has p-rank 0, proving (A).

Next, suppose (A) holds and suppose End I were commutative,

so that End°X = K is an algebraic number field. Since every elliptic curve isogenous with X is again of p-rank zero, and there are only finitely many isomorphism classes of curves of p-rank 0, if R is the ring of integers of K, we can find an integer N> 0 such that for every X isogenous to I, we have N.B c End X. Choose it prime l not dividing pN such that Rl is a prime ideal in B. (It is known that such l exist.) Let a be a non-zero element of TI(X) not

divisible by l and let K. be the cyclic subgroup generated by the image of a under the natural homomorphism TI(X) -+ X. Then K. C K,,,,. Again by the finiteness of the number of isomorphism classes of curves of p-rank 0, we can find integers m> n such that K, CE,, and there is an isomorphism J: XI K, * g/K». Thus, if 71: X/Kn a X/Km is the natural homomorphism induced by the inclusion K. c Xs we get an endomorphism a = 71 o e

End S', where I' = X/Km, such that a has cyclic kernel of order 111, k > 0. Since degree a and hence NmxiQa is a power of l and Rl is a prime ideal in B, we must have a = l'. u where u is a unit in B

Hom(X,X) AND THE I-ADIC REPRESENTATION

219

and r > 0. So Na = l'.Nu, and Nu a End X'. Now, the degree of

Nu is N2 since u is a unit in B, so that the i-primary part of ker(I'. Nu) is (Z/l'Z)5. On the other hand, the l-primary part of ker(Na) is isomorphic to kera, which is cyclic. This contradiction proves (B).

We now prove that (A) (C). We have already shown that (A). implies that X is defined over a finite field. Let a be the Frobenius morphism over this field. Since EndI is finitely generated, we can find a finite extension of degree n, say, such that

every element of EndI is defined over this extension. Hence ir" commutes with End X, so a" belongs to the center of End X. Since (A) holds, so does (B), so that End°X is a quaternion algebra with center Q. Hence, ]r" is an integer, and by consideration of degree, lr" =±pm for some m > 0, so 1r2" = ps". Conversely, if a," = px" for some n and m > 0, then since 17 is bijective, pg is also bijective and X has p-rank 0.

. (C). Next if End°X We have thus shown that (A) (B) is non-commutative, it is a quaternion division algebra over Q, and since for 1 0 p, Qt®QEnd°X -+ EndQ,(Ql (9 z, T, (X)) is

injective and both sides have dimension 4 it is an isomorphism. Therefore Invt(End°X) = 0 if l is finite and l # p. Since EInvn(End°X) = 0, the sum being over.the finite and infinite places of Q, and Inv,,(End°X) = 0 or J, we deduce that Invr,(End°X) =Inv. (End°%) = J. This establishes (b) (ii). Next, we show that if X is defined over a finite field, End°X : Q. We have proved this for X of p-rank 0. Suppose then that p-rank of X is 1. As before, it is an endomorphism of X which is bijective and of degree a power of p, so that it cannot equal ma for any integer m. Thus, 17 e End X, 7r 0 Z. It follows from the table of possibilities of §21 that then End°X is an imaginary quadratic extension of Q. This proves (b) (i).

We have also established therefore that if End°% -= Q, I cannot be defined over a finite field. Suppose finally that X is not defined over a finite field. We may assume X is the normal form

220

ABELIA11 VARIETIES

(Np) in characteristic p > 2, with A transcendental. (If p = 2, the argument still works if a somewhat different normal form is used.) Let us call this curve Xa. Since any two transcendental elements A and A' over the prime field are conjugate over the prime field, we

see that if EndXA = A, then End X,, A for any other transcendental µ over the prime field. Now, End°X must be either Q or an imaginary quadratic extension of Q, since the only other possibility is that of a non-commutative division algebra, in which case, by the implication (B) (A) above, X. has p-rank 0 and A must be algebraic. Suppose then.that End°X is an imaginary quadratic extension of Q. Then we can find an element a in A such that a2

= N e Z, N < 0. Hence for any transcendental µ over the prime field, there is an a a End X, with a,2, = N. Suppose l is a prime

not dividing pN and e TI(X,). Let $, be the cyclic group generated by the image of f under the map TI(XA) -, Xim, and let p: X1 -# Xa/K be the natural map. By our remarks preceding the theorem, X,1/K is also not defined over a finite field,

and is therefore of the form Xfor some µ transcendental over the prime field. With aF, as above, since the map End° X,, -->. End° X, given by a i - + p-1 o a o p is an isomorphism, we deduce

that (p1 o ar, o p)2 = N1, and since as = Ng and End° XA is a commutative field, p 1 o a o p= f ax and a o p=± p o aA. Thus, aa(kerp)c kerp, that is, aA(K )c K,,. Since this holds for every n, we deduce that ax(e) = a¢ for some a e ZI. Thus, for ax acting as an endomorphism of TI(XA), every vector is an eigenvector, so that as acts as a scalar on TI(XA). Since the characteristic polynomial of ax has integer coefficients, this scalar must be rational, and its square

cannot be negative. This contradiction shows that if X is not defined over a finite field, End° X = Q, thereby proving (a). A far-reaching generalization of part of this theorem to higher

dimensions has been proven by Tate and Grothendieck. Suppose

k has char p, and X is a simple abelian variety defined over k.

TasoaEM. X is isogenous to an X' defined over a finite field if and only if X is of CM-type.

Eom(X,S) AND THE i.ADIC REPRESENTATION

221

( was proven by Tate:[T2]; n was proven by Grothendieck:

Tate shows further :

[G1]).

THEOREM.

If X is defined over afnite field *o, ar is the .Frobenius

morphism over ko, and End (X, ke) is the ring of k5-rational endomorphism8, then

End(X, ko) ® Q1 = centralizer of T,(ir) in Uoni 1(T1(X), T1(X)).

The group 97(L). There is a second approach to the Riemann form of a line bundle, via a technique which is important in other 23.

contexts such as the theory of moduli, and the closer study of linear systems on an abelian variety. We first make a grouptheoretic digression to explain the class of group schemes that we will need. DEFnrrrjoN. A theta-group will be a system of group schemes and homomorphisms i

IT

1 - + G. ---) G - + K -+ 1 such that

(a) K is commutative (but G need not be); (b)

3 an open covering {U;} of K and sections o, of ir:

(c)

i

is a closed immersion, making G. into the kernel of rr;

(d) G. ccenter of G. When K is a finite group scheme, there is a global section a : K-)- 0 for n, and then as a scheme, G = G. x K (i.e. define ¢ : G. x K-* G by ¢(a, k) = i(a).o(k)), Having made this splitting, the group law on G can be carried over to a "twisted" group law

on G. x K. There will be a morphism

222

ABELIAN. VARIETIES

f:KxK -+Gm such that the twisted group law is

(a, k). (a', k') = (a. a'. f(k, k'), k + k'), (*) where a, a' are S-valued points of Gm, k, k' are S-valued points of K, and K is written additively. f must be a 2-co-cycle : f(k + k', k"). f(k, k') = f(k, k' + k'). f(k', k')

and changing the section v has the effect of altering f by a coboundary:

f *(k, k') =f(k, k'). 9(k + k'). g(k)`i 9(k')-'. Conversely, given any such f, (*) makes G. x K into a theta-group.

In other words, the set of all theta-groups over a fixed finite K is isomorphic to the cohomology group H2(K, G.), computed via morphism cochains.

The deviation of G from commutativity is easily measured by taking the commutator. For any two S-valued points x, y of 0, (1) xyx 'y-' is an S-valued point of G. and (2) it depends only on ir(x) ar(y) and not on x, y. Therefore there is a morphism

e:KxK - ) G. such that

xyx 'y-' = e(7r x, 17Y),

all x, y E 0(S), all S.

It is easily checked that e is a skew-symmetric bihomomorphism : (a)

e(k + k', k") = e(k, k"). e(k', k")

(b) (o)

e(k, k) = 1.

If 0 admits a global section of K and so is described by a 2-co-cycle

f, normalised by the condition f(0, 0) = 1, then e(k, k') =f(k, k')If(k', k). In case K is finite, the bi-homomorphism e also can be expressed asymmetrically as a homomorphism : (d)

y: K-+ K.

Rom(X,X) AND THE I-ADIC REPRESENTATION

223

In fact, if we regard K x K and G. x K as group-schemes over K via ps then (e, p2): K X K -> G,,, x K is a K-homomorphism, i.e. a K-valued character of K, or a morphism y : K --> K. If

< , > : K x K -+ G. is the universal pairing, then in terms of S-valued points k, k' of K, y is given by e(k, k') = ((L) and j : W(L)

- K(L) by the homomorphisms of functors:

Gm(S) = r(S, on -+ Aut(L/X) (S) d 6(L) (S) mult. by e 91 (L) (S) ; Aut (L/X) (S) ---4- K(L)(S) ei

P

a HI the S-valued point f of I such that a covers Tf

Jt

Then i is clearly injective, j is just the projection p, and Im(i)

Ker(j). Since there are sections locally to p: L-+ I, there are also sections locally to j : f(L) -. K(L). Finally, if e er(S, ), then the automorphism, mult. by e, clearly commutes with all other automorphisms a e Ant (L/%)(S), so i(G,,,) is in the center of

9(L). This proves that i(L) with i ad j is a theta-group. el: K(L) x K(L) 3 G. is the skew-symmetric bi-

DEFINITION.

homomorphism associated to the commutator in the theta-group er(L).

Look at the case L e Pic°$. Then K(L) _ l and the morphism

ez takes the complete variety I x I to the affine variety G.. Therefore eL - 1 and 9(L) is a commutative group scheme, which is an extension ofl by G.. It. can in fact be shown that this map: Pic°(S)

Ext'

(S, G,,,)

Comm.group

L r-s V(L) is an isomorphism (Theorem of Serre and Rosenlicht).

Suppose next that the line bundle L arises from a divisor D: L = G ,(D). We leave it to the reader to check that the discrete group W(L)t can be described as follows.

91(L)t={(x,f)Ixe1,fek(X),Ts1D=D+(f)} This works since

A

228

Lrax VARIETIES

T.-+',D-D.- (Tz g.f)=T;'(Ty 1D-D)+(TE 1D- D) -Td 1(9)-(f) =Tz 1(T;'1D-D-(9))

+(2 1D-D-(f))=0. The subgroup k* _ (G.),,, c ((L)x corresponds to the pairs x = 0, f = a e k*; the projection 97(L)r -s S(L)Y corresponds to the map (x, f) i ---s X. FUNOTOBIAL PROPERTIES OF eL.

In the following formulas, the symbols x, y etc. are to be understood as B-valued points for any k-algebra R. One could equivalently interpret the formulas as commutativity of certain diagrams of morphisms. With this specific understanding, we shall often omit B from the statements and proofs and speak as though

we were just dealing with ordinary points, but all the assertions are to be understood in the stronger sense mentioned.

If f : X -a Y is a homomorphism of abelian varieties and L a line bundle on Y, we have (1)

e(',)(x, y) = eL(f(x), f(y)), x, y e f 1(K(L)). (2)

For any line bundles L1, L2 on X, eLiaL'(x, y) = eLi(x, y). eZ'(x, y), x, y e K(L1) n K(L2)

(3) For eL3 = ex's. (4)

algebraically equivalent line bundles Ll, L2 on X,

For x e %(L) and y e ng' (S(L)), eL" (x, y) = eL(x, ny).

(5)

For x e .., y e nal(-K(L)) _ 0.z 1($), (n any integer with

PT -n) e.(x, OL(y)) =CL4(x, Y).

.Moo s. (1) We may assume f(x) = j(e), f(y) =j(-1), where j: W(L) -* K(L) is the natural homomorphism. (When x, y are B-valued points, we. can find f, -I after localizing on Spec R.) We

Hom(X.X) AND THE I-,9DIC REPRESENTATION

can then lift

and 71 to automorphisms 0,

and T respectively, and then

229

of f*(L) covering P. lifts e q t"1 1, which

means precisely (1). (2)

Again we may assume that there are automorphisms

respectively of L;, i =1, 2, covering T. and Ti,. Then 41® 0 and 1 ®' z are automorphisms of L® ® La covering T,,, T, respectively, and the commutator of these two automorphisms is the tensor product of the commutators of ¢4 and q; (i =1, 2), which proves (2).

(3) Write Lg = Ll®L2 1, so that L8 a Pic0X and Ll = L2®L3, K(L3)= I. Apply (2) to the line bundles LQ and Lz.

(4) By replacing the ring R by a ring B' D B if necessary, we may assume x = nz for some z e ng1(K(L)). (In fact, we have only to take Spec R' = Spec B Xx(L)n81(K(L)) which is finite and

flat over Spec R, so that it is affine and B' z) B.) We then have z, y e n$1(K(L)) = K(L°), so that by (1) applied .to ng(L) and (2) applied repeatedly, and making use of the algebraic equivalence of nx(L) and L"', we obtain eL(nz, ny)

= ex

(r,

y)

= e ns(z, y)

=

eL"(nz, y),

which is formula (4).

As in (4), we may assume that y = as for some z, and the equation to be proved assumes the form (5)

en (x, 4L(y)) = eL" (x, nz) = epo (x, z) = e°x

(x, z)

since L°' is algebraically equivalent to nx(L) and z e K(L") = K(ng(L)). The fact that z e K(nn (L)) means that we have an automorphism a of nx*(L) covering the translation T. (localize Spec R if need be).

230

ABELTAN VARIETIES

Let us agree to denote (temporarily), for line bundles M, N on I, the line bundle associated to the locally free sheaf of germs of homomorphisms of M into N by Hom(M, N), so that we have

a natural isomorphism Hom(M, N) f M-'®N. Note that there is a natural action of X. on any pull back ns(M) covering translations, hence also natural actions on tensor products, Horns and translates of pull-backs (this last, since any translation commutes with any dther). With this understanding, we have natural isomorphisms of the following line bundles on X commuting with this X. action: na(T,*,(L))®n*x(L-1) - T*(nx*(L))®ns(L)-1 na(T*VL®L-1) - Hom (nx* (L), T z(na(L))].

ng(Tr*,L®L-') is isomorphic to the trivial But what is bundle, and Z.(-, L(y)) is given in the usual way by the natural

action of X. carried over to the trivial bundle. Equivalently, nx* (T,*, LO L-1) has a nowhere vanishing section, unique up to scalars, and a"(., OL(y)) is given by the action on X on this section. Now make this computation on the bundle on the right instead of the one on the left. A nowhere vanishing section of the line bundle

0)

nx*(L) covering on the right is just an isomorphism nx*(L) Ta, and the natural action of x e X on the right maps this section into the section 0': nx(L) - nx*(L) defined by

where j : nx*(L) - ng(L) is the natural isomorphism covering = e"(x, OL(y)).#. Applying T.. We must therefore have this in particular to the automorphism a covering T. chosen earlier,

we get that %o ao fix' o a ' = e"(x, L(y)). But this means by h(y)). definition that z) = As a corollary, we get a second proof of the skew-symmetry of the Riemann form of L: e"(x, 9L(y)) = eL"(x, y) = eL"(y, x)-1 = e"(y, 4.L(x))-'.

if x, y e X. The formula (5), coupled with (4), shows how the Riemann forms can be computed from the es's and conversely,

Hom(I,I) AND THE

REPRESENTATION

231

how all the eL's, on points of order l", can be computed from the Riemann forms. THEOREM 2. Suppose 7r: X-.Y is an isogeny of abelianvarieties, and L is a line bundle on X. Then there is a natural one-one corres-

pondence between (a)

isomorphism classes of l ine bundles M on Y such that 7r 'M = L,

(b)

homomorphisms a: ker 7r i ((L)

ker7rr PROOF.

%.

lifting

the

inclusion

This is just a restatement, in a special case of the

general descent theorem in §12 for coherent sheaves with respect to quotients by finite group schemes. Use the fact that Homa(ker 7r, 1Y(L)) a

actions of ker iron L covering its translation action on X

COROLLARY.

Given 7r: X-* Y and L as above. Then a line bundle

M on Y such that 7r*M ; L exists if and only if ker(ir) c %(L) and eL Ika

. x ker,. =1.

PROOF.

Let p: 9P(L) 3 S(L) be the projection and let G

p-1(ker 7r). Then G is a theta-group over ker a, and by part (i) of the lemma at the beginning of this §, eL I ker ker . = 1 r- G is commutative e=:,- G = G. x ker 7r as group-scheme . . 3 a M exists. homomorphism a: ker 7r-> G such that poa = 1 We now prove the following theorem, promised in §20. TBEOREM 3.

If L is a line bundle on an abelian variety I and

n eZ, L = M" for some line bundle M if and only if K(L) X,,.

PROOF. The `only if' part follows from the equation jSL =no,,. Assume conversely that $(L) X,,. To show that L = M" for some line bundle M, it guffiees to show that Il' = nx*(N) for some N.

Indeed, we would then have that (L®N-")"e Pic°S, hence also L®N-" ePie°S, and if we choose P e Pic°S such that L®N-" P", we would obtain L = (N® P)".

ABETXLN VARIETIES

232

To show that L° . n8(N), we merely compute, for any B-valued points x, y of X,,, eL°(x;y) = eL(x>ny) = 1.

The desired conclusion then follows from the corollary to Theorem 2.

Our last result concerns the non-degeneracy of W(L). We need some preliminary results. Suppose e: K x K -p- G. is a skew-symmetric bi-homomorphism on a finite group K. Let y: K-# K be the associated homomorphism. Then if H cK is a subgroup, I claim that there is a second subgroup Ha characterized by the property:

if k is an S-valued point of K, keHl(S) 4==;. {for all S'/S, all S'-valued points k' of H, e(k,k') =1}.

In fact, restricting characters of K to H defines a morphism q: K-> H, and Hl is clearly the kernel of q o y: K-. H. Now suppose 7r: %-; Y is an isogeny of abelian varieties, M is a line bundle on

Y and L = a*M. Let H = ker(a): then as we have seen H is a subgroup of K(L) such that e'Jaxa 1. In other words,HcHl. The result we require is

Lie 2.

K(M) = Hl/H.

PsooF. Let x: S-->% bean S-valued point of X. We must show that x is a point of Hl if and only if Irx is a point of K(M). It will suffice to prove this when S = Spec(.), B a local ring, in which case S carries only trivial line bundles. Then, using the descent theorem of §12, 7rx a K(M)(S)

(S x M) = S x M

n-

3 an isomorphism 2 (S x L) : S x L .

commuting with the action of ker it on these two line bundles

.

Eom(X,X) AND THE L-ADC REPRESENTATION

233

Now suppose a : H - T(L) is the homomorphism giving the action of H on L for which M is the quotient. Then we continue our equivalences: 3 an S-valued point w of 2F(L) such that

p(w) = x and w commutes with a(H) {eL(x, h) = 1, all S'-valued points h of H} e

x e H1 (S).

The same argument shows in fact that W(M)

{

centralizer of a(H) a(H) in 9(L) I

but we do not need this fact. We now apply the lemma to THEOREM 4. Let L be a non-degenerate line bundle. on an abelian variety X. If H c K(L) is a maximal subgroup such that eL la x s = 1,

then H = Hl and order (H)2 = order (K(L)).

Let H be a maximal subgroup scheme of K(L) such that eL'H X H . 1. Let Y = X /H, and let 7r: X -a Y be the natural homomorphism. By the Cor. to Th. 2 there exists a line bundle M on Y such that rr*M a L. The fact that H is maximal means that there are no further isogenies Tr': Y-, Y' for which M = ir'*(M') except the identity. PRoo1r.

LEMMA 3. Let L be a non-degenerate line bundle on an abelian variety X. If there are no isogenies rr: X -aY of degree > l such that L = 7r*M, some line bundle M on Y, then I X (L) I = 1. PROOF. If I X(L) I > 1, then K(L) is non-trivial. Then there exists a subgroup E c K(L) of prime order, i.e., E e Z/lZ, Z/pZ, ,uy or ac.,. Look at the inverse image E in 9F(L): this is a thetagroup scheme G over E. By the lemma at the beginning of this section, G is commutative, hence G a G. x E, hence there exists a

homomorphism:

ABELIAN VARIETIES

234

E Therefore by Theorem 2, L descends to X/E, contradicting the assumption. So J y (L) I= 1.

Returning to the proof of the theorem, we deduce that 1(M) = (0) and I X(M) I = 1. Therefore by Lemma 2, 1 =HJ-, and by the results of § 16, I X.(L) I = deg(as) = order (H),

X(L) I = deg(OL) = order (S(L));

so order (H)2 = order S(L). Every abelian variety X is isogenous to a principally polarized abelian variety Y, i.e. one which carries an ample line bundle L with X(L) = 1. ConoLLAR.Y 1.

.

Pxoos Apply the theorem to any ample L on X, and let Y = X/H, H maximal in S(L) with eL Isx$ = 1. Then L = ir*(M) and M is ample with X(M) = 1. CoBoLLAEY 2. If L is a non-degenerate line bundle on X, then Or(L) is a non-degenerate theta-group.

Puoos. Let y: S(L) a%(L) be the homomorphism associated to eL and suppose D is its kernel. Choose an H c1(L) with H =

Hi and order(H)5 = orderl(L), as in the theorem. Now since y(D) = (0), we find eLI DXE(L) = 1 and eLls(L)XD = 1

Therefore D c Hl, so D cH and also all characters y(x) annihilate

D, all x e X(L)(S). Now by definition, Hl is the kernel of the homomorphism %(L)

4- S(L)

q

*. H. It follows that

Im(go y) c (H/D) and that we have an exact sequence

Hom(X,%) AND THE l-ADIC REPRESENTATION

235

0 - H -- K(L) y , HID. Therefore

order K(L) < order H. order HID = (order H)2/order D. This proves that order D = 1.

The next step in the development of the theory of thetagroups is to show that (1) all representations of non-degenerate theta-groups which restrict to the identity character on the center G. are completely reducible, (2) that there is only one irreducible representation with this property,, and (3) that when L is nondegenerate, i =i(L), then T(L) acts naturally on H`(%, L) and that this is the irreducible representation in (2). For these facts and their application, of. [M2]. 24.

The case k = C. The purpose of this last section is to tie

together the algebraic approach of this chapter with the analytic

methods of Chapter I. In particular, I want to relate the analytic and algebraic Riemann forms of a line bundle L, and I want to show how the positivity of the Rosati involution follows immediately from the positivity of the analytic Riemann form in its guise as a Hermitian form when L is ample.

As always, let X =V f U, V a complex vector space and U

a lattice. Let L = L(H,a) be a line bundle on I, where H is a Hermitian form on V such that E =Im H is integral on U, and a: U -+ Cl is a function such that a(ul + u2) = a(ni) a(zc2).e.:sru,.u,

Consider the group ii of analytic automorphisms ifr,,u of C x V given by o.u (a>z)

=

aEC*,wcV. Then

z + w)

236

ABELIAN VASIETIES

P = a. z. a

8(e.ea)

so 4 is an extension analogous to the theta-groups of

§ 23:

1 -* C' -* c -'" ) V -+ 1 Yatp H to

The data at defines a lifting i of U into U

n T i(u)

p

-V

v = a(u).

eg ECu.u)

so thatL(H;a) is, by definition, the quotient C x V/i(U). The commutator in 9, as in the theta-groups of §23, is given by a V x V ..I-..* C* as follows: I e (v,w) = e2a;B(vm)

Therefore, if Ul = {u e V E(u,u') a Z, all u' e U} as before, it follows that the group To def = p-'(U-1) = {V,,. I v e Ul} is the centralizer of i(U)in 4. Therefore, all the automorphisms ve Ul, descend to automorphisms 4ra, of L(H,a):

Cx V

t L(H,a)

0a'°

-0...

Cx V

I --->- L(H,a).

Hom(X,X) AND THE I-ADIC REPRESENTATION

237

This gives us a natural homomorphism a - 9a(L(H,a)). But we saw in § 9 that K(L(H,a)) = Ul/U, so we get in fact isomorphic extensions: i

) C*

is/i(U)

Ul/U

1

1 -) C*-) T(L(H, a)) -,- K(L(H, a)) -+ 1. It follows that the commutators in these two groups are equal, hence we have proven

If L = L(H,a) is a line bundle on X = V/U; B = Im H, and a : V - X is the natural map, then for all x, y e Ul: THEOREM 1.

e Z,r:E(zv) = eL(irx,

y).

Since our ground field is C, there is a canonical primitive nth root of 1 for all n, namely = e2 1". Therefore the module

M,=lunwm

has a canonical basis element C too, given by the sequence e2"ly' epm. We can now relate the Riemann forms E: U x U-> Z, where E = Im H, EL:TIX x TX -3 MI, defined in § 20. Let srI denote the natural map from U to TIX, i.e. irl(u) is given by the sequence un = ir(u/l") a Iln, with luA+l = v,W. Then if u, v e U, EL(7r:u, arty) = the sequence eIn(un, ¢LVv)

= the sequence = the sequence = the sequence

vn) (§ 23, property (6)) -2,ri1"E(,/Zn, °1I°)

(Theorem 1)

-E(u, v). C.

Thus, except for sign, EL is the ZI-linear extension of B from U to TIM

238

ABELIAN VARIETIES

Applying this to the case where X is Y x Y and L is the Poincar8 bundle P, we see that the canonical non-degenerate

integral pairing of the lattices U and U corresponding to Y and Y, of the analytic theory (cf. §9, part (B)) is, up to sign, the same as the canonical non-degenerate l-adic pairing ej of TTY and, T,Y (cf. §20).

Next, consider the Rosati involution of End°(X). Analytically, we use the interpretation: End°(X) = jt

set of complex-linear endomorphisms T: V -+ V 1 J}

such that T (Q. U) c Q. U

Then the natural involution is the adjoint with respect to H: H(T*x, y) = H(x, Ty), all x, y e V. Since if x e Q. U, then for all y e Q. U, E(T*x, y) = E(x, Ty) a Q, it follows that T*x must be in Q. U too, i.e. T* e End°(X). If T'

is the image of T under the algebraic Rosati involution, then for all x, y e U, E((T*- T')x, y) = Im H(T*x, y) +EL(T'Trx, rrty) = Im H(x, TO + EL(rrtx, Tray)

= E(x, Ty) - E(x, TO = 0 .

Thus T* = T'. Now for any complex-linear operator T: V - V, if T* is its

adjoint, then T*T is a positive self-adjoint operator on the Hermitian vector space V, hence all its eigenvalues are positive, hence the complex trace, Tr(T*T), is positive. If T e End°(X), so that T*T maps the rational vector space Q. U into itself, its trace here is just twice its complex trace; and its l-adic trace in TTX = U® Zl is equal to its rational trace. Therefore for any of these traces,

Tr(T*oT) > 0, all T e End°(X), T = 0.

Hom(X,X) AND THE

REPRESENTATION

239

Thus the positivity of the Rosati involution is obvious from the existence of the positive definite H with Im H = E. In a sense, we have shown that over any ground field one can reverse this argument: namely, using the positivity of the Rosati involution, we have realised NS°(X) as a formally real Jordan algebra in which the ample L's are the positive elements.

APPENDIX I THE THEOREM OF TATE

By C. P. RAMANUJAM

WE HAVE seen that if X and Y are abelian varieties over a field k (algebraically closed) and l a prime different from the characteristic of k, the natural homomorphism Z1®zHom (X, Y)

T

--

Homzi(Tz(X), Tt(Y))

(1)

is injective. It is obviously of importance to know the image. By a field of definition of X, we shall mean a subfield'k0 of k over

which there is a group scheme X0 and an isomorphism of group schemes X0 ®k,, k . X. We also say that X is an abelian variety defined

over ko. Now choose a common field of definition k for X and Y; we may and shall assume ko to be of finite type over the prime field. Let ko be the algebraic closure of ko ink. Since nx and EZ) are morphisms defined over ko, their respective kernels X. and Y consist entirely of k0-rational points, or in other words, the points of

finite order in X and Y are Z. -rational (i.e., they come from korational points of Xo®ku ko and Ya(&i., ko resp.) Thus, if G =

is the Galois group of ko over ka, G acts on the groups X. and Y. M

m

compatible with the homomorphisms X.,, --* X. and Ym -+ Y,,. Hence G acts continuously on the Zi-modules TT(X) and T1(Y), where we give 0 the Krull topology and T1(X) and Ti(Y) their l-adic topologies.

Furthermore, let y be any homomorphism of X into Y. Since the points of finite order in X are ko rational and k-dense, and p maps points of finite order into points of finite order, p is defined over ku and hence over a finite extension k, of ko in k0 (i.e., comes by base extension from a homomorphism of group schemes Xo (&ka kl Yu®kokl). It then follows that if H=G(ko/k) c G is the subgroup of G fixing k1, for any k6-rational point x of X, we have hp(x) = p(hx)

APPENDIX I : THE THEOREM OF TATE

241

for all h e H. Thus, if we make G act on Homz1(T1(X), Tt(Y)) by putting (gl)(x) = g(X(g-lx)) for g E 0, AE Homz (TT(X), T1(Y)) and x c- T1(X), we see that h Tt(p) = T1(p) for h e H. In other words, for any (p E Hom (X, Y), there is a neighborhood H of the identity in G fixing T1(p). We see therefore that if for any G-module M we denote by M« the subgroup of elements fixed by some neighborhood of e,

the image of (1) is contained in Homz,(T1(X), T1(Y))t°>, and we obtain a homomorphism Zt (&z Hom(X, Y)

HomZ1(T1(X ), TAY))

.

(2)

Tate conjectures that (2) is an isomorphism (or equivalently (2) is surjective) for any field of definition k° of finite type over the prime field. He proves this for abelian varieties defined over a finite field. It has also been proved when k° is an algebraic number field and X = Y is of dimension one (Serre). We shall now reproduce Tate's proof in the case of finite fields. THEoREMM 1. Let X and Y be abelian'varieties defined over a finite field k°. Then the homomorphism (2) is an isomorphism.

The rest of this section will be devoted to the proof, which we give in several steps. STEP I. It Suffices to show that the homomorphism

Qe®z Ho-(X, Y) = Qe®Q Hom°(X, Y) --+ HomQ1(V1(X), V1(Y))tvi

(3)

where V1(X) = Q1®zt T1(X), is bijective.

In fact, if this were so, the image of the homomorphism (1) would be a Z1-submodule of maximal rank in Homz1(Tt(X), Tt(X))(4). On the other hand, if this image is M, Homz,(T1(X), TT(X))JM has no torsion; for, suppose p e Homz,(Tt(X), TT(X)) and 1,p e M. We can 3 19, so that for all large find i(r E Hom(X, Y) such that n, T1(W n) =l p,,, pn E Homzt(Ti(X), T1(Y) ). Thus,

(TT(X)) c l TI(Y)

and f,, vanishes on X. Thus 0. admits a factorisation #,,= X. o d r = I Xn and the X. converge to a certain. X in Z1(gz Hom(X, Y) with T1(X) = (p, so that p E M. This proves our assertion, and establishes that .111 = Hoinz1(T,(X), T1(Y))(").

ABELIAN VARIETIES

242

STEP II.

It suffices to show that for any abelian variety X defined over a finite field x

QI®Q Ends(X)

]SndQ)(I',(:v))("')

(4)

is an isomorphism.

In fact, if (4) is an isomorphism with X x Y instead of X, since we have the direct sum decompositions as G-modules Q,®Yad'(XxY)a--(Q,®o End'X)®(Q,6vEnd' fl LQ/8oHom9(X.Y))®(Q,EaHom'(Y.X))

I

1

1

End(V,(XxY)) .-- End )V,(X))®End )VdY))ED Horn

I

i

Y,(Y))®Hom(r,(Y),1',(X)),

the above diagram is commutative and the first vertical arrow is an isomorphism, (3) is an isomorphism. Thus, henceforward we shall restrict ourselves to a single abelian variety X defined over a finite field, and prove that (4) is an isomorphism.

STEP III. It suffices to show that A is an isomorphism for one 1 and th ii dimQ) End (VI(X))(o) is independent of I T char. k.

This follows from the fact that the dimension of the left member in (4) is independent of l and A is always injective. STEP IV. To establish (4) for an 1, it suffices to show the following. Let E be the image of d, and F the intersection over all neighborhoods of e in G of the subalgebras generated in EndQ)(V1) by these neighborhoods. Then F is the commutant of E in End(V1).

In fact, since E is semi-simple, if the above were true, it would follow from von Neumann's density theorem that E is the commutant of F. Further, since EndQ1( VI) is of finite dimension over Q1, F

actually equals the subalgebra generated by sonic neighborhood, and all smaller neighborhoods generate the same subalgebra. Hence the commutant of F is precisely EndQ1(V))u . Now, Steps I-IV are valid for any field of definition k,,, and do not make use of the fact that ko is finite. The next step makes use of a

APPENDIX 1: THE THEOREM OF TATE

243

certain hypothesis which is easily seen to be true for finite fields, and probably holds for any field of finite type over its prime field. We therefore state it as a hypothesis, verify it for a finite field and deduce its consequences.

Let X be an abelian variety defined over a field k0, 1 a prime. We

then state the HYPOTHESIS (k0, X, 1): Let d be any integer > 1. Then there exist upto isomorphism only a finite number of abelian varieties Y defined over ko, such that: (a)

there is an ample line bundle L on Y defined over ko with

X(L) = d; (b) there exists a ko isogeny Y- X of 1-power degree.

(If X - X0 ®Aak and Y = Yo ®,ok, a line bundle L on Y is said to be defined over ko if there is a line bundle Lo on Yo such that L - Lo (&A.. k, and an isogeny Y ->-X is said to be defined over ka if it arises from a homomorphism Yo-- X0 by base extension to k.) For finite fields ko, we have in fact the following stronger LFMMs 1.

If ko is a finite field, d> 0 and g> 0, there are upto

isomorphism only finitely many abelian varieties Y defined over ko of dimension g and carrying an ample line bundle L. defined over ko with X(L) = d.

Pxoox. The line bundle L$ on Y is defined over ko and gives a projective embedding of Y as a k. -closed sub.variety of projective space of dimension 30X(L) -1=30d-1. The degree of this subvariety is the g-fold self intersection number of L3, that is, 39.(L)o.d = 3r.g!d. Thus, there corresponds to it a k6 -rational point of the Chow variety of cycles of dimension g and degree 39.g! din P'Od-'. Since ko is a

finite field, there are only finitely many k, -rational points on this Chow variety, which proves the lemma. STEP V. Suppose X is an abelian vareity defined over ko and L an

ample line bundle on X defined over k0. Suppose for a prime

244

A13ELIAN VARI I' TLES

1 - char. k5,Hyp(ko,X,1) holds. Let JVcV,(X) be a subspace of V,(X) which is G-stable and is maximal isotropic for the skew-symmetric form EL. Then there is an element u nE with u(V1(X)) = IV. PROOF.

Set T = T,(X), V = V1(X) and for each integer n > 0, T. = (T n 1V) + 1"T.

If +/r": T"(X) -r X," is the natural homomorphism, set K. Y. = X 1K, and let v": X ->Y" be the natural homomorphism. Then (1"),t factors as X

=r Y. -+ X.

Since ,r" o A. o 7" = l"rr", we

obtain that a" o A. = (1%.x so that a"((Y")1") = ker 9r" = K..

Furthermore, for any m>n, )1,.((I'")1",)z) (a"ovr") (X,-) =X:m-., so that T!(d") (T1(Y,j) D P T1(X), T,(A") (T,(Y")) = T"

We now verify that Y. is an abelian variety defined over ko and that 1 is a ko morphism. First note that if X = Xo ®5,k where Xo is a group scheme over ko, the points of X1", are separably algebraiq

over ko. In fact, since (1")t: X - X is etale. so is (1")r0: Xo

Xo,

so that (1").ta (0) consists of points whose residue fields are separably

algebraic over ko. Further, since W and hence (T n W) + l"T are G-stable by assumption, K. is also G-stable. Thus the fact that Y. as a- variety is defined over ko and a": X -* Y. is defined over ka follow from the following general LEMMA 2. Let X be a quasi-projective variety defined over ko and H a finite group of automorphi8m8 of X such that

(i) there is a separably algebraic extension of ko over which the automorphisms of 11 are defined; (ii) for any automorphism a of the algebraic closure ko of ka over

kc, 11° = H.

Then X/II is defined over ko and the natural map X - X/II is defined over ko.

APPENDIX I : THE THEOREM OF TATE

245

Choose a Galois extension k,(ko over which the automorphisms of Il are defined. If U is a ko-afhne open subset of X, f7 AU is PROOF.

Aen

again ko-affine open and 12-stable, by assumption (ii) of the lemma.

Thus, we may assume X affine, X = Spec A, A = A0 ®ko k, A0 being a k0 algebra. If Al = A0 ®ko k1, ll operates on A,, and An (9k;k = An. Further, G = Gal (kllko) operates on A1, and since by assumption A 11 A-1 = H for any A e G, it follows that All

is d-stable. Let B be the ko-algebra of G-invariants, and A) a basis of k1/ko. If E a; ® O1 a An, ai a A0, then A (E a1 ® 0,) = Lai ®A (Bi) a An, and since det (A (Bi))a,o i 0, a; ®1 e An n A. c B, which proves that An = B ®., k,.

This proves the lemma.

Thus, as a variety, Y. is defined over ko and or,: X -s Y. is defined over lco, so that or,(0) is k,; rational. Since the addition map

m: Y,, x Y. - Y and the inverse 1: Y.

Y, are defined over a separably algebraic extension of ko and are invariant under the action of Gal(ko/k ), they are again defined over ko. Hence Y, is defined over ko as an abelian variety. Now, (l")x = A. o or, and or, are defined over k hence A is defined over ko, since for a ko regular function p on an open subset of X, p o A. is k1-regular for a Galois extension k, of k, and is invariant under Gal (kt Jk,,) since p o A, o n,= po(d")X is.

Let d be the degree of the ample line bundle L on X. We shall produce an ample line bundle of degree 20.d defined over ko on each Y,. We have e1(x, pa;,(L) (y)) = B

(L" (x, y)

= BL(A,(x),

eet(T,,, T,)

eL(111 T, T n W +1" T)c1"M1

for any or, y e Z 1(Y,), since W is isotropic for J. Since et: T,(Y,) x T1(Y,)->Mtisnon-degenerate, it follows that pA.(,)(Tt(X))cl"TI(X), 901Li=1"4r for some Y,, >Y..

ABELIAN VARIETIES

246

It follows from the theorem of §23 that 41= 9Ln for some line bundle L,, on Y,, defined over the algebraic closure, hence over some normal extension of k0. We may assume that L" is symmetric. Hence,

if p denotes the characteristic of k° if this is positive and p = 1 if the characteristic is zero, for a suitable integer N > 0, LPN is defined over a Galois extension k1 of ko. We now have LE1c iA 3.

Let Y be an abelian variety defined over ko and La Pic°Y

a line bundle defined over the algebraic closure 1° of k° in k. Let and denote by a(L) the line bundle on Y defined over ka ae Gal obtained by pulling buck by the morphisml l y. x Spec a: Yo ®,.. ko -a Yo (&a A0. Then a(L) e Pic" Y. PROOF.

Let M be an ample line bundle on Y defined over k°, and

consider the line bundle Y = m*(M) ® p*(M)-' (& p2(M)-1 on Y x Y. We can find an algebraic point y e Y such that N I {y} x Y= L. It is then easy to see that N j {ay} x Y = a(L), so that a(L) a Pic° Y.

We shall make use of the notation a(L) introduced in the lemma

in future. Further, if L1 and L2 are two line bundles on Y with L1® L2 ' e Pic° Y, we shall write L1 = Ls.

Resuming the earlier discussion, if a e Gal (k /ko), we see that a(LO) is also symmetric, and a(L"N)t" a*(L"), so that since is torsion-free a L,"nNfor every a e Gal (k,/k°). Hence, and L",N differ by an element of order 2 in PicO Y. Thus, if we put M. = Ln"N, we have M,, for every ac-Gal (k1/ko) and M;," We now have

Let Y be a complete variety defined over ko with a k, rational point, k1 a Galois extension of k° and L a line bundle on Y defined over k, such that for every a e Gal(k,/k°), a(L) - L. Then L LEMMA 4.

can be defined over k°.

Paoov. Put X = X°®4k, X1 = X, (9k. k1 and .let vr:X1 -* X° the natural morphism. Since projective modules of constant rank over semi-local rings are free, we can find an affine covering 11= (U,)1 , of X0 such that L/rr-'(U;) is free. Let {act;) be the 1-cocyale

APPENDIX 1: THE THEOREM OF TATE

247

with respect to the covering {ir '(Ui)} with values in Oal defining L.

Our assumption implies that for any a e G = Gal (k/k°); we have fi., a r(v-'(U;), ,) such that

a l'

rj.o

If yo is a ko rational point of Xo with yoe Uio and y, 7r-1(yo), by dividing the fl;., by P; .,(y1), we may assume that gi.,,(yt) = 1. Now,

- f;."

a?-(a;;) a,i

i.o>

aT ((X;j)

a (a;j)

a(a;i)

0,i;

a (9;,.) g;.o a(fli,.) flj,a

so that for any a, r e G, i, j e 1, -` P;,' = Nj,e, a(Rj,.)-' f3;,o' in .U; n Ui.

Since X is complete, P;,,> a P71 = C,,> for all i, and taking i = io and evaluating at y1Q, weppsee that

Grant for the moment that if A is any local ring. of X° and B = A®k.k,, and if B* is the group of units of B, H' (0, B*) = (1). We deduce that there is a covering {U;,z}aE,t, of U; and y;,Q a IF(- -' (U;.,),

& .) such that a(Y;a) via

in i '(U,, ).

The cooYcle a;i yin with respect to the covering {U+=} aEA is cohomoiEJ

logous to { aii ),and we have

/

Yj'

\\\

Yia

aij

Yjd

r Yia = Yin = w-aii-Q---a;i p;e

Nio Y.

Yia

e r (Uia r' Uin, Fo).

Yia

It only remains to show that if A is the local ring of a point on X° and B = k, ®ta A, H' (G, B*) = {1}. Let B be the quotient field of B; we then have the exact sequence H°(G, R*) -+H°(G, R"`/B*) -- H'(G, B*) -+ H'(G, B*) = {1}.

248

AI3ELIAN7 VARIETIES

An element of H°(G, R* f B-*) is represented by an element f c R*

such that f e B* for all veG, and we shall. show that we can write f= gu where g e H°(G, R*) and u e B*. Writing f=

with fD e B,

F e A, we see that we may assume that f e B. Now, since f e B* for all a e 0, the ideal Bf is G-invariant, hence of the form B S?t, R( being an ideal in A. But now, since Bf = kl®kn 4(, 2( is A-projective, hence

principal. Thus, Bf = Bg for some g e A and f = gu, g e A, u e B*. This completes the proof of Lemma 4.

It follows that the line bundle M. on Y. is defined over k°. Now, the g.c.d. of 2p" and 1" is 1 or 2 according as I is 2 or not. Hence, we can find integers a, b such that 2apN + bl" = 2. Define N. - M;"® A*(L)b, so that N" is defined over k°. Further, N = L2aPN+61" = L2 so that N. is ample with X(N") = 21 X(L") = 29. l-"'

(,1n(L}}

= 29.1-n9 deg. A. X(L) = 29X(L) = 29d.

An alternative way of producing an ample line bundle of degree 29d on Y is to observe that (i) if Y is an abelian variety defined over k°, and we construct Y as Y/ti(L) for a line bundle L defined over k°, Y is defined over k° and the Poincare bundle P on Y x Y is defined over k°; (ii) if since A,,(L) is defined over and finally; (iii) if P" is the Poineare k°, so is (pa,-,,L) and hence also bundle on Y. x Y and X = (l, #): Y. -+ Y. x Y,,, then X*(P,,) = N. is

line bundle defined over k° with 9,.v-.-: 2 q, (see § 3:3). so that

X(Nn)2 = deg TNn = 229 deg > = 2291-2n0 deg

229 1-2n9X

(deg An)2 X (L)2 = 221 X (L)2.

Anyhow, we deduce from Hyp(k°, X, 1) that there is an infinite set I of natural integers with smallest integer n and isomorphisms vi: Y,, Z. Yi for all i e I. Consider the elements u;, = Avian 1 e E nd° X and their images u; a Endo (V1). We have u;(T") = T; c 2'n for i e I,

APPENDIX I : THE THEOREM OF TATE

249

and since Endz (T,,) is compact, we can select a subsequence (u;)XI

which converges to a limit u'. Since E is closed in End (VI) and u3 a E, u' also belongs to I,'. Since T is compact, u'(TA) consists of elements of the form x = Jim x; where x, a T3, and since

the sets T, are decreasing, it follows that u(T,,) = je fl fl T; = T n W. 1E

Hence u(V) = W. This completes the proof of Step V.

Step VI. Suppose that for any finite algebraic extension ka of k(, Hyp(kl, X, 1) holds, and that F is isomorphic as a Qi-algebra to a direct product of copies of Qt. Then (4) is an isomorphism. PROOF. Replacing ka by a finite algebraic extension kl over which all elements of End X are defined and which is such that Gal (k,Jk,)

generates F in End(V1), we may assume that ko itself has these properties. Let D be the commutant of E in End(Vi), so that D D F. We first show that any isotropic subspace W for EL which is F-stable is also D-stable. We proceed by downward induction on dim W. If W is maximal isotropic, i.e. if dim W= g, we can by Step V find a u e E such that u(V) = W, and hence DW = DuV = uDV =uV = W, which proves the assertion. Suppose then that dim W = r < g and the assertion holds for F-stable isotropic subspaces of dimension r + 1. The orthogonal complement W -L of W for EL is also F-stable, since EL

is invariant under the action of Gal (k/ko). Further, since any simple F-module is one-dimensional and dim W1-dim W = 2g-2 dim W = 2(g - r) > 2(g - g + 1) 2, we can find F-stable one-dimensional subspaces Ll and L2 of W' such that the sum W ± L1+ L2 is direct. By induction hypothesis, W + Ll and W + LZ are D-stable, hence so is their intersection W. This completes the induction. We deduce

that any eigen-vector for F in V is also an eigen-vector for D. It follows that D c F. (The decomposition of V into factors V; corresponding to the simple factors of F.reduces this assertion to the evident statement that an endomorphism of V, for which every element of

V; is an eigen-vector is a scalar multiplication). Hence F = D, completing the proof of Step VT.

250

ABELIAN VARIETIES

STEP VII. End of proof of theorem. We assume from now on that k° is a finite field. By replacing it by

a finite extension if necessary, we may assume that every element of End X is defined over k°. Let N be the Frobenius morphism over k°.

Then IT belongs to the center of End° X, and hence Q[ir] is a comimitative semi-simple subalgebra of End" X. We shall first show that there are an infinity of primes l for which Q,®Q Q[7r] is isomorphic as a Q,-algebra to a direct product of copies of Q. In fact, writing Q[ir] = K, x ... x K, where Ki are finite extensions of Q, it suffices to show that for an infinity of primes 1, each Q1® Ki

is isomorphic to a product of copies of Q. Let K be a finite Galois extension of Q in which all the Ki are embeddable. Then it suffices to show that for an infinity of 1, K®QQ, splits as a product of copies of Q, as a Qi algebra. It suffices for this that there is one simple factor of K ®QQ1 isomorphic to Q1. In fact, if K ®Q Q1 = L1 x ... x L, the Galois group IT of K,1Q permutes the factors L. It also acts transi-

tively on the simple factors. For, if not, suppose L1 x ... x Lr is ir-stable; then the element (1, 1, .... 1, 0, 0, 0) e L1 x ... x Lk is TtimC5

77-stable- On the other hand, since 7r fixes only the elements of Q in

K, it fixes only the elements of Q1 in Q,®Q K, that is, elements of the form (a, a, ... , a) E L1 x ... x Lk with a e Q1. This proves the assertion.

Now, choose an algebraic integer a of K generating K over Q, and let F(X)eZ[XJ be its irreducible monic polynomial over Q. Since K®QQ1 Q,[X]((F(X)), it is enough to find an infinity of l for which F(X) has a zero in Q1. Let A be the discriminant of F(X),

and l any prime not dividing A such that F(X) = 0 (mod 1) has a solution n in Z. Then F'(n) # 0 (mod 1), so that by Hensel's lemma, n can be refined to a root of F in Z1. Thus, we are reduced to proving the following LEMMA 5- Let F(X) e Z[X] be a non-constant polynomial. Then there are an infinity of primes l for which F(X) = 0 (mod 1) has a

solution in Z.

APPENDIX I : THE THEOREM OF TATE

251

PRoor. Let F(X) = a°X° + a1X"-1 + ... + a,,. The lemma being trivial when a" = 0, since X is a factor of F(X) in this case, we may assume a 0 0. Further, by substituting a"X for X in F and removing the common factor a,,, we may assume a = 1. Let S be a

finite set of primes p. If N = II P, then Pcs

F(vN) = a° v" N" + ... +a"_ 1 vN + 1 = 1 (mod N) so that no prime of S divides F(vN). On the other hand, F(vN) o f 1 for v large, hence has a prime factor l not belonging to S.

Next, we show that for all 1 : ehar.k°, the dimension of Endo1 V1() is the same. Again, assume that every element of End X is defined over k°, so that the Frobenius it belongs to the center of End°X, hence to the centre of Qt ®Q End°X. Then Q1[ar] is semi-simple

and V1 is a Q1[ir]-module, so that the image 17 of it in EndQ1Vt is semi-simple. The characteristic polynomial- P(t) of i in EndQ1V1 has coefficients in Z independent of 1. Further, the closed subgroups of Gal (k°/k°) generated by Tr" form a fundamental system of neigh-

borhoods of e in this group, so that EndQ, VI') is the commutant of ir'"! in EndQ1V1 for n large. The characteristic polynomial of ir'41 has for roots 6' where Or, ..., 8r, are the roots of P(t) repeated with multiplicity. For all n large, the number of distinct elements of

8',', ..., 0% as well as their multiplicities is the same. Thus, our assertion is a consequence of the following lemma, applied to an algebraic closure of Q1. Let A and B be absolutely semi-simple endomorphisms of two vector'spaces V and W of finite dimensions over afield k respectively, with characteristic polynomials PA and PB. Let LEMMA 6.

PA = fl

PM(P)

P PB=IIp"(P)

P

be the decompositions of PA and PB as products of powers of distinct irreducible monic polynomials p. Then the vector space

E={pEHomk(V, W) fpA=Brp}

ABELIAN VARIETIES

252

has dimension

r(Pa, Ps) = E m(p)n(p) deg p, v

and this integer is invariant under any extension of the base field k.

PROOF. Make V (reap. TV) into a k [X]-module by making X act

through A (reap. B). Denote the k[X]-module k[X]/(p(X))by M5. Because of our assumption of semi-simplicitity, we have isomorphisms of k[X] ,modules V

HM.-O)

Ti' . H Mp (c2) is a special case of (b,) . (b:,) when

dim X = dim Y. Further, we have seen during the proof of the Riemann hypothesis in § 20 that if w,(i e I) and wj(j eJ) are the roots of P. and P1. respectively and N. and N,, are the number of rational points of X and Y respectively in the extension of degree n of k0, we have

N = 11(1w;°)7 j thus, we have to show that P.r = P1. rI (1 - w `) = R (I

every n > 0. The implication implication, note that

is obvious.

J(1-w")=

-W,11.) for

To prove the other

(-1)ISIw; SCI

ie!

')IT, 47",

jeJ

2, C.1

where I S I, I T I denote the respective cardinalities and w5 = rj w;, ;e8

w'T = rl w,. Multiplying the given equation by t", where t is a ;Er

variable, and summing as formal power series, we obtain (_1)ISI

Sc!

= 1 l-s tw

(-1)ITI !'C.1

1

l - ho.

q1I', comparing the poles on both sides on the Since I w; I = I wj t I = q'12, we obtain that there is a bijection e: I iJ such circle

that w; =

Hence, Px = I'x

Before we come to the next theorem, we need some preliminaries. Let X be an abelian variety defined over ko, so that X = Xo®1 k for

some group-scheme X, over ko. Let Y be an abelian subvariety of X, which is a ko-closed subset, so that if A: \ -> X, is the natural

APPENDIX I : THE THEOREM OF ?ATE

255

morphism, there is a closed subset Yo of X. with A-1(Yo) = Y in the' set-theoretic sense. We give Yo the structure of a reduced subscheme of X0. If mo : X. xA.0 Xo --). Xo is the multiplication morphism, our hypothesis implies the set-theoretic inclusion mo (Yo x A:o Yo) c Y. Hence

mo restricts to a morphism md: (Yo x ko Yo)red. > Y0 If we can assert that Y = Yo®Aok and Yo x A..Yo are reduced, it would follow that Y is

an abelian variety defined over ko. Both of these are consequences of the assertion that the function field RA.o(Y0) is a regular,extension of ko, or equivalently that RA.,(Y0) is a separable extension of kd. This

is always true (vide S. Lang, Abelian varieties, Chap. I), but we shall not prove this, since we shall need it only when ko is a finite (hence perfect) field, so that this is trivially satisfied. Next, suppose X is an abelian variety defined over ko, and Y an abelian subvariety which is k, -closed. We want to show hat there is an abelian subvariety Z of X defined over ko such that Y + Z = X and Y n Z is finite. We know (vide §18, proof of Theorem 1) that if L is an ample line bundle -on X, we can take Z to be the connected component of 0 of the group Z' = {zEX; TZ(L)® L-1I, is trivial}, so that if we can ensure that Z is defined over ko for a suitable choice of L, we are through. Now if we choose L to be a line bundle defined over ko, Z' is defined over the algebraic closure ko of ko and is stable for all automorphisms of T. over ko. Hence Z' is k, -closed. Further, the conjugations of ko over ko permute the components of Z', and since Z is a component of Z' containing the k-, -rational point 0, Z is also stable under these conjugations. Hence Z is k,-closed, and it

follows from the comments of the earlier paragraph that Z is an ahelian variety defined over k0. It follows from this by repeating the arguments of § 18 that if we call an abelian variety X defined over ko to be k, ,-simple if it does not contain an abelian subvariety Y defined over ko with YT{ll}, Y=X,

then (i) any abelian variety defined over ko is ko isogenous to a product of 1., -simple abelian varieties, and (ii) if X is k,; isogenous to a product X't x ... x X,", where X; are k. -simple and X; and'XX are not ko isogenous if i

j, then End'o X c JI (D,) x ... N

where DA = End', (Xi) are division algebras of finite rank over Q.

ABELIAN VARIETIES

256

We now have THEOREM 3.

Let X be an abelian variety of dimension g defined

over a finite field k0. Let 7r be the Frobenius endomorphism of X relative

to ko and P its characteristic polynomial. We then have the following statements: (a) The algebra F = Q[7T] is the center of the semi-simple algebra E = Enc1Ao(X); (b) End' (X) contains a semi-simple Q-subalgebra A of rank 2g which is maximal commutative; (c)

the following statements are equivalent:

(c1) [E: QI = 2g, (e2) P has no multiple root,

(ca) E=F, (ci) E is commutative; (d) the following statements are equivalent:

(d1) [E: QI = (29)2,

(d2) P is a power of a linear polynomial,

(d3) F=Q, (d4) E is isomorphic to the algebra of g by g matrices over the quaterniondivision algebra Dj,over Q(P==char. k0) which

splits at all primes l r p, co, (d5) X is ko isogenous to the g-th power of a super-singular curve, all of whose endomorphisms are defined over k0; (e)

X is ko isogenous to a power of a k0-simple abelian variety if and

only if P is a power of a Q-irreducible polynomial. When this is the case, E is a central simple algebra over F which splits at all finite primes v of F not dividing p, but does not split at any real prime of F. PROOF

center of EI

It follows from the main theorem that FI =QI®QF is the QI ®Q E, which proves that F is the center of E.

APPENDIX I : THE TREORM OF TATE

257

Suppose E = Al x ... x A, is the expression of E as a product of simple algebras Ai with centers Ki. Let [Ki : Q] =ai, and [Ai : Ki]= b;. We can choose subrings Li of Ai containing Ki with Li semi-simple

and maximal commutative, [Li : Ki] = bi. Then L = L1 x .... X L,. is a semi-sii ple Q-subalgebra of B which is maximal commutative,

and [L: Q]

E ai bi. Now, for any 1, we have 1

r

E®QQi= fl (Ai®QQi1=

(Ai®xi(Ki®QQI))

i-1

Ai®xiKi

'i

with K;, fields. On the other hand, if

where K® ®Q Qt = H i-1 S

P = II Pi" is the decomposition of P over Qi into a product of powers of irreducible polynomials over Q:, and if we consider Ti(X)

as a Q1[T]-module by making T act via rr, we have an isomorphism of Q,[T)-modules Ti(X)

T

-`

T] Qp,)

Q(!l'.)

so that E®QQ1, being the commutant of it in EndT1(X), is isomorphic to

.-1

Comparing the two factorisations of E ®Q Q1 and keeping in mind that K;, is the center of A® ®xi K;i, we deduce that (i) for any prime 10 p and any prime v of Ki lying over 1, A i splits at v and (ii) there

is a partition of [1, s] into r disjoint subsets I1,..., I, such that "i

Ai®QQ: ^' fT Ai®. K;i t-1

rT J

1

.E/i

A13ELIAN VARIETIES

258

It follows that m, = b, for v eIi and E [S, : Q,]

[K;i : Q,]

[K; : Q] = a,, so that r

a` b` - 7 i=l .eli

a

-'A: Q1]

m,,[S, : Qt] v=1

m,deg P,= deg P= 2g. This proves (b).

Since F is the center of E and E contains a maximal commutative subring of rank 2g, (cl), (c3) and (c4) are equivalent, and since E commutative o El commutative a m, = 1 with the above notations, these are also equivalent to (c2). This proves (c).

Now, [E: Q] = (2g)' if and only if QI®QE = M2p(Q1), hence if and only if s = 1, S, = Q, or equivalently, P is a power of a linear polynomial. In this case, Q1 is the center of Qi®QE, so that Q is the center of E, and conversely, if this holds, Ql ®Q E is the commutant of Q7 in End V7, so that it is the whole of End V1. Thus M2p(Q,), E is a (d,), (d2) and (d3) are equivalent. If Q1®QE central simple algebra over Q whose invariants at all finite primes l -f= p are 0. Since its invariant at the infinite prime is 0 or I and the sum of invariants at all primes is 0, B is either M2p(Q) or M1(D,) where D, is the quaternion algebra over Q splitting at all finite primes l p. The first possibility is ruled out, since X cannot be a product of 2g abelian varieties. This proves that (dl) o (d4). In view of our

remarks preceding the theorem, (d4) is equivalent to saying that X at CO, where C is an elliptic curve with End1.o C D5. We have then shown that C is supersingular (§22). This proves (d).

Let Q be the product of the distinct irreducible factors of P. Since F = Q [ar] is semi-simple, and P(ir) = 0, we have Q(1r) = 0. Further, rr acts as an endomorphism of V,, any irreducible factor over Q, of the characteristic polynomial*P divides the minimal polynomial of rr, so that Q is the minimal polynomial ofiroverQ. Now, X is k,-isogenous

APPENDIX I. THE THEOREM OF TATE

259

to a power of a 1-0 -simple abelian variety if and only if E is simple,

hence if and only if the center F = Q[zr] of E is a field. Since F ^_- Q[X]/(Q(-Y)). F is a field if and only if Q is irreducible, or equivalently, P is the power of an irreducible polynomial Q. If F is

the center of E, we have shown earlier that E splits at any finite prime v of F not dividing 1. Suppose v is a real imbedding of F, so that v(or) is a real number. Since v(rr) satisfies P(v(,r)) = 0 and the roots of P have absolute value -%/q, we must have v(7r) = ± Vq. If q is a square, v(a) e Q and F = Q, so that the equivalent-condition of (ds) and (d4) implies that E does not split at co. If q is not a square,

F = Q(. /p). Let ki be the quadratic extension of ko and ir' = ir 2 the Frobenius over ku. Then ,a2 a Q, so that the center F' of E' = Ends. X is Q. Appealing to (d), we conclude that E' = M,(D,). On the other hand, we have F' cFcEcE', and E is the commutant of F in E'. By a known result on central simple algebras, we see that E and F O ..E' define the same element of the Brauer group over F, that is, E is the image of E' under the natural map Br(F') -+ Br(F). Since both the real primes of Q(-%/p) lie over the real prime 00 of Q

and E' has invariant h at co with respect to F= Q, E has invariant I at either of the real primes of F = Q(,/p). This completes the proof of (e). COROLLARY. Any two elliptic curves defined over finite fields with isomorphic algebras of complex multiplications are isogenous (over the algebraically closed field k).

In particular, any two supersingular elliptic curves are isogenous.

Suppose X, Y are supersingular elliptic curves. We can choose a common finite field of definition k such that Endk X and PROOF.

Endk,Y are quaternion algebras over Q, so that they have Q for center. Thus, their Frobenius morphisms rrx and w y lie in Q. Since they must both have absolute value -Vq where q = card (k0), we see that vr' = 7r2, = q. Thus, if ku is the quadratic extension of k,,, there is an isomorphism T1(X) - T1(Y) carrying the action of 7r.r into -a , , and w ', = 7-, 11 are the Frobenius morphisms over "k1. By Theorem 2, X and Y are isogenous over ku. where y r r

260

ABELIAN VARIETIES

Next suppose End°X

K, End° Y n K for some imaginary

quadratic extension K of Q. Choose a common finite field of definition of X and Y over which all their endomorphisms are defined and

all the points of order p are rational. Now, Qp®Ql4,nd°X admits a one-dimensional representation in TT(X). Hence, p splits into a product of two distinct primesp and.' in K which are conjugate, and Qp®Q End°X K, x Ks,,. Suppose for instance that Qp®o End°X

acts on TT(X) via Ky. By what we have said, it follows that irx = 1(i), and since Nm orx is a power of p, (1r x) has to be a power of ,p' in the ring of integers of K. A similar assertion (possibly with p replacing ti') holds for,ry. By altering the isomorphism End° Y^-K

by the conjugation of K if necessary, we may assume that (v,.) is also a power of p'. Since Nmor,r= Nmiry = q = pf, we see that in the ring of integers of K, (7rx) _ (irl.) = P'f so thatTrr and ,r, differ by a

unit, i.e., a root of unity since K is imaginary quadratic. Thus 'rr = ar'y in K for suitable n, and they have the same minimal equa-

tion over Q, of degree 2. Since this has to be their characteristic polynomial, X and Y are isogenous over an extension of degree n of k°.

APPENDIX II MORDELL-WEIL THEOREM

By Yu. I. MANIN 1.

Statement and sketch of the proof.

Let K be any

finite

extension of the field of rational numbers Q, X _ an abelian variety defined over K. We shall prove the following result. THEOREM.

The group X(K) of rational points of the variety X is

finitely generated.

Poincare had already conjectured this result for elliptic curves. It was Mordell who proved this conjecture of Poincare, over Q, and Weil generalised this proof to the higher dimensional case, by introducing a series of new ideas and technical tools. Both the theorem and the method of its proof play a central role in modern "Diophantine Geometry". The generalisation of Mordell-Weil theorem to the case of the base

field being a field of finite type (over the prime field) is given by ,S. Lang'. No essentially new ideas are required for the same.

The proof of the theorem consists of three steps of completely different kinds. We shall sum up these results in the form of three assertions. PROPOSITION 1.

Let n > 1 be any integer. Then the group

X(K)/nX (K) is finite. This is known as "Weak Mordell-Weil Theorem" and -its proof is given in §2. PROPOSITION 2.

There exists on X(K), a symmetric bilinear scalar

product X(K) x X(K) -a R (x, y)

--> <x, y> with the following properties:

(a) <x, x> > 0 for all x e X(K),

(b) the set {x e X(K) I <x, x> < C} is. finite for all C > 0.

j S. Lang: Diophantine Geometry, In+,erscience, New York (1962).

ABELIAN VARIETIES

262

The construction of this scalar product is based on the theory of heights of Weil-Tate. This is contained in §3 and §4.

The arithmetic of the field K and the geometry of the variety X will be used in these two steps. The third-a purely formal deduction of the finite generation of the group X(K) from the above properties, we shall see now. ,

Let r be an abelian group with the following

PROPOSITION 3.

properties

(I) r/nr is finite for all n > 1; (2) there exists on r a symmetric bilinear scalar product

r. x r -). R: (x, y)-w--4<x, y) such that properties (a) and (b) of Proposition 2 are satisfied. Then r isfnitely generated. PROOF. First, we choose a system of representatives x, ... x, of all

the classes of r/nr. We observe further that there exists a constant C with the following properties:

<x, x> > C - <x-x;,x-x;> foralli=1....,8. (1) In fact,

<x - xi, x -x;) = <x, x> - 2 <x, x;) + <xi, x;) and

<x, x> < <x', xi>'tQ <x

x)112

(2)

and hence <x -- x;, x - x;) increases asymptotically as <x, x> for <x, x)

oo.

[The Schwarz inequality (2) follows from the fact that the quadra-

tic form <mx + nx;, mx + nx;> has for its discriminant <x, x;)2 <x;, x;> <x, x> and is non-negative for all integers m, n.]

We set now M = (x,, ..., x) u {x a r/<x, x> < C} where C is the constant above and prove that the finite set M is a set of generators for the group r. In fact, if it were not the case, there exists an element x e r which

does not belong to the subgroup ro generated by M with <x, x>

APPENDIX II: MORDELL-WEIL THEOREM

263

minimum (for these values form a discrete subset of R in view of the inequality (2)). Obviously <x, x> > C. Besides x - x; = ny for some x; e M and y e P. By using inequality (2) we obtain

= ns <x - xi, x - xi> < n2 <x, x> < (x, x). Therefore y e I'o and then x = ay + x; e r0, in contradiction to the choice of x. The proposition is proved. This part of the proof of Mordell-Weil theorem is sometimes called

"method of descent". In fact, the proof by contradiction gives. a method of representing any point x e r as a linear combination of elements of M, by successively "descending"' from x to x' = x

x;

n

and from x' to z" = x -' .2 and so on, until the "heights" <x°, jo> n of the new points no longer decrease. Weak Mordell-Well Theorem. The symbols X, Khave the same meaning ini this paragraph as in the previous one. Let K z) K be. the

2.

algebraic closure of K. We consider geometrical points of X over the field k and its subfields. We fix an integer n > 1, and denote by the group of geometrical points of order n in X and by µ the group of nth roots of unity. We shall prove below Proposition 1, under the additional assump-

tion that X. c X(K) and µ c K. This can always be achieved by passing from K to a finite extension and it is obvious that the strong form of Mordell-Weil theorem for the extension yields the theorem for the field K.

Let L = K(nsx X(K)) -- the composite of all fields ink obtained by adjoining n-Ix, x eX(K). Then L is a finite abelian extension of K. FUNDAMENTAL LEMMA.

DEDUCTION OF PROPOSITION 1 FROM THE FUNDAMENTAL LEMMA.

Let G be the Galois group of L(K. Consider the maps 1(K) 3 Hom (G,

x

fx

f.,(s) = s(n-lx) -n-lx, xeX(K), 8eG.

ABELIAN VARIETIES

264

It is clear that f, is independent of the choice of n -lx, since

It is easily verified that fl is a homomorphism and that fr r: Further

fr = 0 ' n x e X(K) Q x e nX(K). Therefore the map x'W -* fr defines an imbedding X(K)/nX(K) c --i Horn (G, By virtue of the fundamental lemma, the latter group is finite and that gives the required.

We shall divide the proof of the fundamental lemma into several

steps. All the necessary algebraic geometric considerations are concentrated in the proof of Lemma 1, the rest is purely arithmetic. LEMMA 1.

Let A be any ring of integers in the field K. There

exists an open subset Y = Spec As c Spec A, a scheme X -' Y, projective over Y, Y-morphism m= X x X -> X and a section e: Y -a X with the following properties: (a) the fibre X over a generic point of the scheme Y (together with

the restrictions m and e of the morphisms m. and a to this fibre) is isomorphic to the abelian variety X over K; (b) X is a group scheme over Y and all fibre (schemes) over closed points y e Y (together with the restrictions of the morphisms m and e to these) are abelian varieties; (c)

the mapping in

(a)

induces an isomorphism of groups

X(Y) =a X(K). OUTLINE OF PROOF. Suppose more generally, we are given an algebraic variety defined over K. By taking a finite cover by means

of affine sets, we select a finite number of generators of the coordinate rings of each of these sets and a finite number of generators of the corresponding ideals. We obtain thus a finite presentation of the variety by means of a finite number of polynomials and rational functions-th a finite number of variables. The coefficients of these

polynomials lie in a subring A,y, c K of finite type over K. Thus our variety can be considered as a generic fibre over Spec As, defined

by the "same equations". An analogous construction "model over

APPENDIX II: MORDELL.1VEII. THEOREM

265

a ring" is possible for any diagram consisting of a finite number of varieties and morphisms over the field K. By applying this argument to the given defining abelian variety X over K, we can meet the requirement in (a). In order to establish assertion (b), it can be shown that one has necessarily to reduce still Spec A, by puncturing at a finite number of points. The most important requirement consists in that, for y e Spec As. the fibre scheme Xy be a smooth variety. We omit the proof of the fact that this holds for all but a finite number of points if the generic fibre is a smooth variety. The other requirement in (b) can be secured in an analogous manner, if one applies this result also to the diagrams of relevant morphisms. Finally, the condition (c) is automatically fulfilled if we choose Spec AS so small, that A 1 be a principal ideal ring (this is always possible, for the group of ideal classes of A is finite). In fact, the unique invertible sheaf over Spec A. will be the structure sheaf. Since X -> Y is a projective morphism, X is a closed subscheme in Py and it is sufficient to check that PY(y) P`(K). This follows, for example, from Proposition 3 of Lecture 5 in Mumford's book*. Indeed, according to this Proposition P} (yl = ((so,

.

, s.), si E As, si coprime}j R

where R is the equivalence relation (s,_., (uso, ... , us.), U E AS* (invertible elements). But since AS is a principal ideal ring, it is

clear that all points of P'(K) lie in the above described set. This concludes the proof. We observe that now considerably more subtle and precise facts on the models of abelian varieties over rings and in particular on the behavior of "degenerate fibres" (N6ron's theory) are known. From Lemma i follows immediately COROLLARY I. Let x a !(Y). Consider x as a closed subscheme of ! and denote by n-'(x) the closed subscheme in X, the inverse image of x under the

morphism n 7y, multiplication by n. Then the natural projection n-'(x)-+ Spec (AS) is etale over all points y E Spec As, whenever char k(y) ,r'n. t 1). Mumford:

Lectures on curves on an algebraic surface, Annals of Math.

Studies 59, Princeton University Press (1966).

ABELIAN VARIETIES

266

This follows from the fact that every component of n-'(x) maps onto y and the number of different geometrical points in the fibre PROOF.

Xy over the pointy is equal to the order ker (Xy ) X;,) i.e. n°, g = dim X, if char k(y) n. Since this number coincides with the degree of the blojective morphism, the definition of the staleness gives the required. Reformulating this result in the classical terminology, we obtain

There exists a finite set E of prime ideals of the ring A, such that for any point xeX(K) and y e n-'(x)eX(K), the COROLLARY 2.

extension K(y) D K is unramified outside E. (In E, it is necessary to include Spec A \ Spec A. and divisors of n.)

We return now to the fundamental lemma. The field K(n-' x) is normal over K, because the conjugates of the point n-I x are of the form n-'x + y, y c X c X(K) and K(n` Ix) = K(n- Ix + y). This also gives the abelian nature of the Galois group and its action on n-'x defines an isomorphism with a subgroup of X,,. Hence the composite L = K(n-I X(K)) is abelian of exponent n over K and is ramified over only a finite number of prime ideals of the field. Consequently the fundamental lemma will be proved, if we verify the following fact. LEMMA 2.

The maximal abelian extension of exponent n of an

algebraic number field K, which is unramified outside a fixed finite set E of prime ideals of the field, is finite. PRooF. Let F K be the maximal abelian extension of exponent

n, H, its Galois group. Consider the exact sequence I -a µ } K*,4o1. The exactness of the sequence of cohomology groups of IC* Hom (H, A.). Gal(K/K) gives K,/(K-)" -,, Hom (Gal (K/K), (We use here that H'(Gal (K/K), K*) = 1 and H'(Gal (K/K), p.) = since µ cK*). In other, words, there is a wellHom(Gal(K/K),

defined nondegenerate bilinear pairing:

HxK*/K*"-*p,, (s, a(K*)*)

Va

I

I

(3)

APPENDIX II:

THEOREM

267

(Here the group H has Krull topology and K*IK*°, discrete topology).

To any subfield K c L c F we obtain from Galois theory a subgroup HL c H which leaves its elements fixed. Let B c K* be the inverse image in K* of the subgroup, orthogonal to HL with respect to the pairing (3).-By making use of this point of view, it is not difficult to conclude that

L=

K(... , Vbi, ...), bi a B. This is clearly the algebraic part, the so-called Kummer theory. We now make use of the arithmetic of the field K. It is necessary

to explain when the extension K(,/a) is unramified. outside E (divisors of n are contained in E). Local considerations show immediately that the necessary and sufficient condition for this is v,,(a) 0(mod n), for all y f E. The group of elements with this property in K*, splits into finitely many classes mod K*". This follows easily from the theorems on the finiteness of the number of ideal classes of E-units (Dirichlet - Minkowski - Hasse).

Thus, we see that to obtain the extension K(n-1%(K)) it is sufficient to extract nth roots of a finite number of elements of the field K. This concludes the proof of the fundamental lemma and the weak Mordell-Weil theorem. Height of points of projective spaces. As before, K denotes an algebraic number field of finite degree. We recollect some elementary facts from the arithmetic of these fields. By paints of the

3.

field K, we mean objects of one of the following three kinds: (a) prime ideals of the ring of integers of the field K (non-archimedean points)

(b) imbeddings K -r R, (c) pairs of complex conjugate imbeddings K C. Let v be one such point of the field. This uniquely defines normalised I ... 1, of the field K satisfying the following properties. Let x e K; then x 1, = I z' J where x' is the image of x under the in bedding of K in R or C if v is an archimedean point.

I p I,, = p if v is non-archimedean and the ideal divides the ideal (p), p e Z, a prime.

268

ABELIAN VARIETIES

In the sequel, we will often make use of the following simple formulas. For all x, y e K Ion 1 xy 1, =log I x I,, + log j y j,

log I x + y Ins max (log I x 1 log I iL) + C, c'

(4)

log 2 if v is archimedean, 0

if v is non-archimedean.

We denote by K, the- completion of the field K under the topology

induced by the normalised ... Iz and set n - [K,.:Q,.]. For any I

element x e K, x

0

nn log 1 X!, = 0

(5)

(the sum makes sense, for, j x I for all except finitely many v). For the sake of ^,onvenience, we set log 0 m, less than any arbitrary real number. Let (x0, ... , x") E K"+' be any system of elements not all of which are zero. DEFINITION - LEMMA 1.

h(xa, .... r.) -

The number

?:6t

fK

n, max (log ! xj

(6)

l

is well defined and has the following properties: (1)

h a function on the K-points P" (K) of the projective space P" provided with a distinguished system of co-ordinates. This function is called the height.

(2) h(x) > 0 for all x e P"(K); (3) h(x) does not depend on the choice of the field where the projective coordinates of the point x he.

Pnoor. The first assertion follows from the fact that max log I dxi I, = max log I x, ;,, + log I Ax I i

and "product formula" (4) applied to A.

APP1 NDIX II: MORDELL-WEIL THEOREM

269

For the proof of the second, we remark that, dividing all the coordinates of the point by a suitable non-zero number, we may assume that one of the coordinates is a unit; then max log I x; I. > 0 i

for v non-archimedean so that h(x) > 0.

For the third assertion, it is sufficient to prove for pairs of fields L D K, one contained in the other. Let w run through all points of the field L which are extensions of v. Then K,O L = Qi L. from

where [L: K] n. - I n so that, f'or all x e K, [K Q)

(L: Q) wlo n Since the valuation I ... I,., on K coincides with I ... lo, this proves

the third assertion. EXAMPLE.

Let K

Q, xi e Z with g.c.d.-(x,,.-X1) = 1. Then

h(x0... , x") = max log I xi I.

In particular, the number of points of bounded height in P"(Q) is finite.

More generally; PROPOSITION 4.

Let C > 0, d > 0, de Z, P" projective space

with a fixed system of coordinates. Then the set

{x a P"(K) /h(x) < C, deg K(x) < d}

is finite.

PROOF. We reduce this assertion to the case K = Q with the help of the following transformation. Consider points x = (.To, ... , x")

of degree < d and denote the corresponding point (X0,... , X,") of the projective space over Q of forms of degree deg k(x) in n + I variables. The coordinates of this point would be coefficients of the norm form Nm (E xi Ti), the norm taken over Q. The mapping

(xo, ... , x") -w.--j (Xo, ... , Xm) defined on the set of points of fixed degree, has finite fibre, since one and the saine image is possible only for conjugate points (the norm foim splits into linear factors, the factorisation being unique upto ordering). Thus it is sufficient

270

ABELIAN VARIIiTI S

to prove that points of bounded height go over to points of bounded height. This is almost obvious. Making use of the inequality (4), we find that, for some constant do (where v is any point of the Galois extension over Q generated by k(x)). max log I x; !, < d max, log I .r; I r + d,. i.e

i

where dr may be assumed to be zero for all non-archimedean points of the field and v' runs though the conjugates of v over Q. Hence

h( (...... X,,) < d'h (xo, ... , a;,,) + d" where d' and d" can be evaluated in terms of n and d.

The assertion is proved. We shall analyse now the dependence of height on the system of coordinates. Let f1, f2 be two real-valued functions on some set. We all them equivalent (fl'-, f2) if I fl - f2 I is bounded. Let h h2 - heights in P"(K) defined with respect to two systems of coordinates. Then hl ~ h2. PROPOSITION 5.

PROOF. Let A = (a;f) be the matrix describing the passage from one coordinate system to the other (xu (4,

(xo

..... ,

A.

By means of formula (6), we find that

maxlog; xi"I,<maxlogIx;!r+maxlogIa;jl,+log(n+1) if v is archimedean, and max log ; x, !r < max log I xi i

+ max log I ai t.)

if v is non-archimedean. Hence h2(x) < hl(x) + h( ... a;; ...) + c' for all x E P"(K). Making use of the invertibility of A, we can show analogously hi < h2 + C. This proves the assertion. Taking into account this result, in future we will consider heights.

only upto equivalence and calculate them for some convenient coordinate system.

APPENDIX 11: MORDELL.WEIL THEOREM

271

Height associated with an invertible sheaf. Let now X be any variety defined over a field K, q : X --> P" a morphism in the projective space over K. The height h* is ,'efined on the geometric points X(K) by-the formula, h,,(x) = h(g1(x)) where his as before the height in the projective space P".

4.

PRoposrrsox 6.

Let X be any complete variety over K, and

p: X y Pk, : X -> P' two K-morphisms. If p*(Op;(1)) _ #*(19pz(1)) over X, then hm- h& in X(K). PROOF.

Let L = p*(Opk(l)) = *(Opt(1)). Then from the defini-

tion of the morphism X: X Proj (S(F(X, L))) we have X*(O(1)) =L. It suffices to show hx '-- ho; then by symmetry hx -hy.

We may suppose that p(x) is not contained in any proper linear subspace of Pk. Choose a basis (To, ..., TT) c p*(I'(Pk, Opk(1))cr(X,L) a n d extend it to a basis (To, .. , of the whole space I'(X, L). The

height hx can be computed in the system (To, . ., and h* in the system (T0, ... , TA.). Let x e X(K), (xo, ... , x.,,) - coordinates of the point X(x); (xo, ... , ;rk)-coordinates of ?(x). Firstly

max log Iav, 1 < max I xi 1o =.> h9 < hX. i6k

n

The estimate in the other direction is somewhat less trivial.

Since X is a complete variety, the image X(X) c P" is closed. Let I c K[T., ... , Ti,] be the defining ideal in the homogeneous coordinate ring B - K [To,..., Since the sections To__ , Tk of the defining

morphism q1 do not all have a common zero on X, their images in R generate an ideal, the radical of which is called "irrelevant" ideal R+, generated by forms of degree > 1. In particular, there exists an integer q > 0 and forms F,, a K[ To, ... , of degree q - 1 such that k

Tk+1 -

F;,{To, ... ,

T; E r, i =1, ..., n - k.

;_o

Consequently, for any point (xo, ... , k-I -'R+1=

X(x), i=..

imu

1, ... , n- k.

(7)

ABELIAN VARIETIES

272

By applying the estimate (6) to the right side, we obtain q log ; xk+1 !,, 5 (q - 1) max log I xi I + Inax log I xi ,, + C,, j:;n iR, a function on it

satisfying the condition s

x; - L h(xi + xj)

h -1

0;

(8)

k-1

(the left aide can be considered as a function on r x r x r). Then there exists a symmetric bilinear pairing b = r x r --> R and a homomorphism 1: l' -->. R such that

h(x) ' h(x) Rf J b(x, x) + 1(x)These conditions define b and d uniquely,

PROOF. We set firstly

f(zI, xs) = h(xi + x3) - h(x1) - k(x3)Then f is symmetric and is "approximately bilinear". N(xl + x1, x3) - N(x1, X3) - N(x1, x3) ti0

(9)

ABELIAN VARIETIES

274

and analogously in the second argument. Since the proof follows from the relation (9) in terms of h, it is enough to verify the equivalence of this with the corresponding equation (8). We set now b(x1, x2)

= lim 4-" P(2"x1, 2"x2). N..

One obtains the existence of the limit and the equivalence b simultaneously, if one notes that 4-(n+i)p(2n+1xl 2"+I x2) = 4 "f(2"xi, 2"x2) + 4-("+1) 0,,,

where 0" = 0,,(x,1, x2) --.0 so that 4_h

v

f(2Nx1, 2Nx2) = f(xI, x2) + 2 4

+i) 9,,

N=(1

From the formula (9) follows immediately the bilinearity of b.

We set now A(x) = h(x) - I b(x, x). Then

)'(x1 + x2) - A(XI) - A(x2) = PI1 x2) - b(xl, x2) ^' 0. Therefore, by applying the same averaging process to A, we obtain a homomorphism

l: P-iR, l'..A: l(x) = lim 2-" A(2"x). 11

Summing up,

h=ib+A- b+1. The uniqueness of b and l is obvious. DEFINITION -LEMMA 2.

Let X be an abelian variety over an

algebraic number field K, L E Pie X. Then the height AL on X(IC) satisfies the condition of Lemma 3. Consequently the functions bL, If, and hr are uniquely defined and the latter among them is called Tate height of the point o;I X (associ4-ted to the line bundle L).

Pnoor. Apply the theorem of the cube (to be exact, Corollary(2)

of it) to the projections Pi X x X x X - X. Then we obtain a

s

l pi, *LcS(pl+p2)*L-'®(p1--p3)*L-1cg(p2+pa)*L-I ®pi*Lrl.

APPENDIX II:

THEOREM

275

By calculating the height of the point (xl, x2, x3) e X X X X X with respect to the line bundle on the left side with the help of the theorems on properties of hL we obtain 3

3

s=1

/

t£i<j53

i=1

This concludes the proof. REMARK. Let q,: X->Y, be a morphism of abelian varieties, L any line bundle on Y. Then km.(L) - AL o q) whence h@.(L) = AL o R and in particular be.(L)-= bLa p, 1*.(L) = ZL . qi . By applying the latter

relation to the mapping pp(x) = - z, and setting p*(L) =L-, we obtain bL- = bL,1L- -1L. Let now L be a symmetric line bundle, i.e. L- = L. Then IL = 0 and hL(x) _ I bL(x, x)6.

Proof of Proposition 3. We select a symmetric very ample line

bundle L on X and choose < x, y > = bL (x, y). Since L is very ample, <x, x > > 0 for all x e X (K). In fact, otherwise, hL(nx) ,jn2 < x, x - co as n --.>. oo and this would contradict the fact k.L(y) hL(y)

and in the class of functions equivalent to kL(y) there exists a non-negative one (second property of height-compare DefinitionLemma 1).

Further, from Proposition 4 it follows that the set {x e (X (K) Jh,(x) 5 c} is finite and therefore the same is

true of the set (xeX(K)/<x, a> cC). This completes the proof of Proposition 3 and that of the theorem of Mordell-Weil.

BIBLIOGRAPHY [A-G]

[B]

A. ANDREOTTI and H. Oa&ux T : Thborbmes de finitude pour Is cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193. WALTER BAILY : On the theory of 0-functions, the moduli

of abelian varieties, and the moduli of curves, Annals of Math. 75 (1962), M. [B-K]

H. BRAUN and M. KOEORRR : Jordan Algebren, SpringerVerlag, 1966.

[B-M]

ARmANII BoREL and GEORGE MosTow : editors, Algebraic

groups and discontinuous subgroups, American Math. Soo. Providence, 1966. [Br]

L. BREEN : On a non-trivial higher extension of representable

abellan sheaves, Bull. American Math. Soc. 75 (1969), 1249. [Bt]

Metodi analitici per varietb, abeliane in caratteristica positiva, Annali della Sc. Norm. Pisa,

IAOOPO BARSOTTI :

appearing in several parts, 1964-1966. [C]

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[Co]

[D-G]

Abelsche Functionen and algebraische Geometric, Springer, Berlin, 1956. MIOEEL DEMAZURE and PIERRE GARRIEL : SEminaire FABIO CONFORTO :

,lteidelberg-Strasbourg. [G1]

ALEXANDRE

GRoTRENDIEox :

S¢minaire

de

geometric

algebrique. 1968. [G2]

A. GROTEENDIEag.: Seminaire 1960-61.

de

geometrie

algebrique,

[G3]

A. GROTEENDIEOE : Seminaire

de

gdometrie

algebrique,

1963-64 (Schemas en groupes). [Go]

ROGER GDDEMENT :

Topologie algebrique

et thdorie

des

faisceaux, Hermann, Paris, 1964. [G-R]

[H]

ROBERT GUNNING and Huao Rossi : Analytic functions of several complex variables, Prentice-Hall, 1965. G. HOOESOHILD : The structure of Lie groups, Holden-Day, San Francisco, 1965.

BIBLIOGRAPHY

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(J]

N. JAcoBsox : Lie algebras, Wiley-Interscienee, New York,

[K]

Kuwnmo KODAmA . On compact analytic surfaces, Analytic functions, Princeton Univ. Press, 1960. SERGE LA ea : Abelian varieties, Interscience-Wiley, New

[L]

1962.

York, 1959. [L-N]

SEEuE LANG and A&DRE NvEox : Rational points of abelian varieties over function fields, American J. Math. 81 (1959), 95.

[Ml]

DAVID MumFORD :

Geometric invariant theory,

Springer,

Berlin, 1965. [M2]

D. Muam'oxn : On the equations defining abelian varieties, Inv. Math. 1 (1966), 287.

[M3]

P. Mumtsoai : Introduction to algebraic geometry, forthcoming. Yusr MAxrx : The theory of commutative formal groups

[Ma]

over fields of finite characteristic, U8pekhi Mat. Nauk. 18 (1963), No. 6, p. 1; transl. in 2?ussian Math. Surveys, Macmillan. [N]

ANDRE Naaox : Modbles minimaux des varietes abeliennes sur les corps locaux et globaux, Publ. I.H.E.S. No. 21, 1964.

[0]

FSAxs OoBT : Commutative group schemes, Springer Lecture Notes, Vol. 15, 1966.

[Si]

J.-P. SERRE :

[Sh]

Goao SEIMIIRA:

troupes algebrique et corps de classes, Hermann et cie., Paris, 1959.

On analytic families of polarized abelian varieties and automorphic functions, Annals of Math. 78 (1963), 149.

[TI]

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[T2]

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AxDRE WEm :

Varietes abeliennea at courbe8 algebrique,

Herman, Paris, 1948. [W2]

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INDEX Abelian variety ................... Action of a group scheme on

39

Group scheme .....................

94

a scheme ........................ Appell-Humbert; theorem of.

108 20

Hasse-W itt map .................. Height-l-group scheme........ Hodge-decomposition .........

100 139 9

Base number .....................

40,178

Hyperalgebra (of a groupscheme) ........................

104

C.M.-type ..........................

183 211 133

Index of line bundle............

150

construction of...

Cartier dual ........................ Characteristic polynomial of endomorphism of abelian variety ........................... Chem class (1st) .................. Chow, theorem of ............... Complex homomorphisms....._ Complex multiplications;

algebra of ....................... Cube, theorem of ...............

182 16 33

kind .............................. Tsogeny ........................... ;of height 1............

194 63 145

Sacobian-(never used) ...... .Tordan.algebra structure on

vi

2

Neron-Severi-group...........

208

172

1-adic representation............

176

55,91

Degree of a finite morphism;

also separableinseparable-,

62, 121 9

De

classification of endomorphism algebras of elliptic curves .................. Differential operator(on a scheme) ..................

Dearing;

order of .......................... Divisorial correspondence.... Dolbeault-resolution............

Dual abolian varietyarbitary characteristic.,.,,. Dual abelian varietycharacteristic 0 ............... Duality hypothesis .............. eu-pairing ........................... Epimorphism

of

Lang; theorem of-on existence of rational points.....

Lattice .............................. Lcfschetz; theorem of......... Lie-algebra........................

217 106 106 81

205 2 29 99

Neron-Severi-group .............. 40, 178 Nilpotent part of ry -linear map

Non-degenerate line bundle...

Norm form ........................ Norm; of enlomorphism of abelian variety ............... Norm; reduced ..................

143 155 178 182 ISO

4

Pfaffian of skew-symmetric

matrix ........................... Picanl-group .....................

154

123

Po i ncare-bu,u l I c .................. Poineare complete reduci-

78

bility theorem .................. Poisson.bracket ..................

173

p-rank of Abelian variety......

147

55 69

74 132

183

group-

schemes ..........................

Involution, of first and second

Etale ..............................

118 65

Quadratic function ............

Exponential map ................

1

Quotient of a group scheme

Vibration associated to principalbundle .....................

121

Flat Sheaf ........................

46

Fundamental group of variety

169

Quotient by a finite group......

21

99

by sub-group scheme.........

104

Riomann-form of a divisor... lticmonn hypothesis............ ltiomann-Roch theorem........

186 206 150

INDEX

Rigidity lemma .................. Rosati involution ............... See-saw theorem

...............

Semi-continuity theorem...... Scini-simple part of p-linear map ............................ Serve and Roscnlicht; theorem of .....................

43 189 54 50

143 227

Serre; theorem of- on automorphierns

of

abelian

varieties ........................

207

Simple abelian variety......... Square.; theorem of............ Sub-group scheme ...............

174

; normal-

59 102

Supersingular elliptic curve...

118 216

S-valued point ..................

93

279

Tate group; l-aclic ............... Tate group; p-adie discrete...

170 171

Theta-function .................. 25 Theta group ....................... 221,225 non-degenerate

Trace form ........................ Trace of endomorphism of abelian variety ............... Trace; reduced ..................

223 178 182 180

Type of a finite commutative group scheme: local,

reduced . ........................

136

Vanishing theorem ............. Vector field ........................ , left-invariant......

15096 97

Weierstrass-function ............

36