Abelian Varieties, Lecture Notes - Mathematics , Study notes of Mathematical Methods

Download Abelian Varieties, Lecture Notes - Mathematics and more Study notes Mathematical Methods in PDF only on Docsity!

Abelian Varieties

J.S. Milne

Version 2. March 16, 2008

These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course.

 Alas, the notes are still in very rough form.

BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} }

v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages.

Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page.

The photograph shows the Tasman Glacier, New Zealand.

Copyright c^ 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder.

• Introduction
• I Abelian Varieties: Geometry
  • 1 Definitions; Basic Properties.
  • 2 Abelian Varieties over the Complex Numbers.
  • 3 Rational Maps Into Abelian Varieties
  • 4 Review of cohomology
  • 5 The Theorem of the Cube.
  • 6 Abelian Varieties are Projective
  • 7 Isogenies
  • 8 The Dual Abelian Variety.
  • 9 The Dual Exact Sequence.
  • 10 Endomorphisms
  • 11 Polarizations and Invertible Sheaves
  • 12 The Etale Cohomology of an Abelian Variety
  • 13 Weil Pairings
  • 14 The Rosati Involution
  • 15 Geometric Finiteness Theorems
  • 16 Families of Abelian Varieties
  • 17 N´eron models; Semistable Reduction
  • 18 Abel and Jacobi
• II Abelian Varieties: Arithmetic
  • 1 The Zeta Function of an Abelian Variety
  • 2 Abelian Varieties over Finite Fields
  • 3 Abelian varieties with complex multiplication
• III Jacobian Varieties
  • 1 Overview and definitions
  • 2 The canonical maps from C to its Jacobian variety
  • 3 The symmetric powers of a curve
  • 4 The construction of the Jacobian variety
  • 5 The canonical maps from the symmetric powers of C to its Jacobian variety
  • 6 The Jacobian variety as Albanese variety; autoduality
  • 7 Weil’s construction of the Jacobian variety
  • 8 Generalizations
  • 9 Obtaining coverings of a curve from its Jacobian
  • 10 Abelian varieties are quotients of Jacobian varieties
  • 11 The zeta function of a curve
  • 12 Torelli’s theorem: statement and applications
  • 13 Torelli’s theorem: the proof
  • 14 Bibliographic notes
• IV Finiteness Theorems
  • 1 Introduction
  • 2 The Tate Conjecture; Semisimplicity.
  • 3 Finiteness I implies Finiteness II.
  • 4 Finiteness II implies the Shafarevich Conjecture.
  • 5 Shafarevich’s Conjecture implies Mordell’s Conjecture.
  • 6 The Faltings Height.
  • 7 The Modular Height.
  • 8 The Completion of the Proof of Finiteness I.
• Bibliography
• Index

Notations

We use the standard (Bourbaki) notations: N D f0; 1; 2; : : :g, Z D ring of integers, Q D field of rational numbers, R D field of real numbers, C D field of complex numbers, Fp D Z=pZ D field of p elements, p a prime number. Given an equivalence relation, Œ denotes the equivalence class containing . A family of elements of a set A indexed by a second set I , denoted .ai /i 2 I , is a function i 7! ai W I! A. A field k is said to be separably closed if it has no finite separable extensions of degree

1. We use ksep^ and kal^ to denote separable and algebraic closures of k respectively. For a vector space N over a field k, N _^ denotes the dual vector space Homk .N; k/. All rings will be commutative with 1 unless it is stated otherwise, and homomorphisms of rings are required to map 1 to 1. A k-algebra is a ring A together with a homomorphism k! A. For a ring A, A^ is the group of units in A:

A^ D fa 2 A j there exists a b 2 A such that ab D 1 g:

X Ddf Y X is defined to be Y , or equals Y by definition; X  Y X is a subset of Y (not necessarily proper, i.e., X may equal Y ); X  Y X and Y are isomorphic; X ' Y X and Y are canonically isomorphic (or there is a given or unique isomorphism).

Conventions concerning algebraic geometry

In an attempt to make the notes as accessible as possible, and in order to emphasize the geometry over the commutative algebra, I have based them as far as possible on my notes Algebraic Geometry (AG). Experts on schemes need only note the following. An algebraic variety over a field k is a geometrically reduced separated scheme of finite type over k except that we omit the nonclosed points from the base space. It need not be connected. Similarly, an algebraic space over a field k is a scheme of finite type over k, except that again we omit the nonclosed points. In more detail, an affine algebra over a field k is a finitely generated k-algebra R such that R ˝k kal^ has no nonzero nilpotents for one (hence every) algebraic closure kal^ of k. With such a k-algebra, we associate a ring space Specm.R/ (topological space endowed with a sheaf of k-algebras), and an affine variety over k is a ringed space isomorphic to one of this form. An algebraic variety over k is a ringed space .V; OV / admitting a finite open covering V D

S

Ui such that .Ui ; OV jUi / is an affine variety for each i and which satisfies the separation axiom. If V is a variety over k and K  k, then V .K/ is the set of points of V with coordinates in K and VK or V=K is the variety over K obtained from V by extension of scalars. An algebraic space is similar, except that Specm.R/ is an algebraic space for any finitely generated k-algebra and we drop the separatedness condition. We often describe regular maps by their actions on points. Recall that a regular map W V! W of k-varieties is determined by the map of points V .kal/! W .kal/ that it defines. Moreover, to give a regular map V! W of k-varieties is the same as to give natural maps V .R/! W .R/ for R running over the affine k-algebras (AG 4.37). Throughout k is a field.

v

Introduction

The easiest way to understand abelian varieties is as higher-dimensional analogues of ellip- tic curves. Thus we first look at the various definitions of an elliptic curve. Fix a ground field k which, for simplicity, we take to be algebraically closed. An elliptic curve over k can be defined, according to taste, as:

(a) (char.k/ ¤ 2; 3) a projective plane curve over k of the form

Y 2 Z D X^3 C aXZ C bZ^3 ; 4a^3 C 27b^2 ¤ 0 I (1)

(b) a nonsingular projective curve of genus one together with a distinguished point; (c) a nonsingular projective curve together with a group structure defined by regular maps, or (d) (k D C/ an algebraic curve E such that E.C/  C= (as a complex manifold) for some lattice  in C.

We briefly sketch the proof of the equivalence of these definitions (see also Milne 2006, Chapter II). (a) !(b). The condition 4a^3 C 27b^2 ¤ 0 implies that the curve is nonsingular. Since it is defined by an equation of degree 3 , it has genus 1. Take the distinguished point to be .0 W 1 W 0/. (b) !(a). Let 1 be the distinguished point on the curve E of genus 1. The Riemann- Roch theorem says that

dim L.D/ D deg.D/ C 1 g D deg.D/

where L.D/ D ff 2 k.E/ j div.f / C D  0 g:

On taking D D 21 and D D 31 successively, we find that there exists a rational function x on E with a pole of exact order 2 at 1 and no other poles, and a rational function y on E with a pole of exact order 3 at 1 and no other poles. The map

P 7! .x.P / W y.P / W 1/, P ¤ 1; 1 7! .0 W 1 W 0/

defines an embedding E ,! P^2 :

On applying the Riemann-Roch theorem to 61 , we find that there is relation (1) between x and y, and therefore the image is a curve defined by an equation (1).

2 INTRODUCTION

(a,b) !(c): Let Div^0 .E/ be the group of divisors of degree zero on E, and let Pic^0 .E/ be its quotient by the group of principal divisors; thus Pic^0 .E/ is the group of divisor classes of degree zero on E. The Riemann-Roch theorem shows that the map

P 7! ŒP  Œ 1 W E.k/! Pic^0 .E/

is a bijection, from which E.k/ acquires a canonical group structure. It agrees with the structure defined by chords and tangents, and hence is defined by polynomials, i.e., it is defined by regular maps. (c) !(b): We have to show that the existence of the group structure implies that the genus is 1. Our first argument applies only in the case k D C. The Lefschetz trace formula states that for a compact oriented manifold X and a continuous map ˛W X! X with only finitely many fixed points, each of multiplicity 1 ,

number of fixed points D Tr.˛jH 0 .X; Q// Tr.˛jH 1 .X; Q// C


:

If X has a group structure, then, for any nonzero point a 2 X, the translation map taW x 7! x C a has no fixed points, and so

Tr.ta/ Ddf

X

i .1/i^ Tr.tajH i^ .X; Q// D 0:

The map a 7! Tr.ta/W X! Z is continuous, and so Tr.ta/ D 0 also for a D 0. But t 0 is the identity map, and so

Tr.t 0 / D

X

.1/i^ dim H i^ .X; Q/ D .X/ (Euler-Poincar´e characteristic).

Since the Euler-Poincar´e characteristic of a complete nonsingular curve of genus g is 2 2g, we see that if X has a group structure then g D 1. The above argument works over any field when one replaces singular cohomology with ´etale cohomology. Alternatively, one can use that if V is an algebraic variety with a group structure, then the sheaf of differentials is free. For a curve, this means that the canonical divisor class has degree zero. But this class has degree 2g 2 , and so again we see that g D 1. (d) !(b). The Weierstrass }-function and its derivative define an embedding

z 7! .}.z/ W }^0 .z/ W 1/ W C= ,! P^2 ;

whose image is a nonsingular projective curve of genus 1 (in fact, with equation of the form (1)). (b) !(d). A Riemann surface of genus 1 is of the form C=.

Abelian varieties.

Definition (a) doesn’t generalize — there is no simple description of the equations defining an abelian variety of dimension^1 g > 1. In general, it is not possible to write down explicit

(^1) The case g D 2 is something of an exception to this statement. Every abelian variety of dimension 2 is the Jacobian variety of a curve of genus 2 , and every curve of genus 2 has an equation of the form

Y 2 Z^4 D f 0 X^6 C f 1 X^5 Z C


C f 6 Z^6 :

Flynn (1990) has found the equations of the Jacobian variety of such a curve in characteristic ¤ 2; 3; 5 — they form a set 72 homogeneous equations of degree 2 in 16 variables (they take 6 pages to write out). See Cassels and Flynn 1996.

4 INTRODUCTION

to one about abelian varieties, replace 1 by g (the dimension of A), and half the copies of E by A and half by A_. I give some examples. Let E be an elliptic curve over an algebraically closed field k. For any integer n not divisible by the characteristic, the set of n-torsion points on E, E.k/n, is isomorphic to .Z=nZ/^2 ;and there is a canonical nondegenerate (Weil) pairing

E.k/n  E.k/n! n.k/

where n.k/ is the group of nth roots of 1 in k. Let A be an abelian variety of dimension g over an algebraically closed field k. For any integer n not divisible by the characteristic, the set of n-torsion points on A, A.k/n, is isomorphic to .Z=nZ/2g^ , and there is a canonical nondegenerate (Weil) pairing

A.k/n  A_.k/n! n.k/.

Let E be an elliptic curve over a number field k. Then E.k/ is finitely generated (Mordell-Weil theorem), and there is a canonical height pairing

E.k/  E.k/! R

which is nondegenerate module torsion. Let A be an abelian variety over a number field k. Then A.k/ is finitely generated (Mordell-Weil theorem), and there is a canonical height pairing A.k/  A_.k/! R

which is nondegenerate modulo torsion. For an elliptic curve E over a number field k, the conjecture of Birch and Swinnerton- Dyer states that

L.E; s/ 

jTS.E/j jDiscj jE.k/torsj^2

.s 1/r^ as s! 1;

where  is a minor term (fudge factor), T S.E/ is the Tate-Shafarevich group of E, Disc is the discriminant of the height pairing, and r is the rank of E.k/. For an abelian variety A, Tate generalized the conjecture to the statement

L.A; s/ 

jTS.A/j jDiscj jA.k/torsj jA_.k/torsj

.s 1/r^ as s! 1:

We have L.A; s/ D L.A_; s/, and Tate proved that jT S.A/j D jT S.A_/j (in fact the two groups, if finite, are canonically dual), and so the formula is invariant under the interchange of A and A_.^4

REMARK 0.1. We noted above that the Betti numbers of an abelian variety of dimension g are 1;

2g 1

2g 2

; :::; 1. Therefore the Lefschetz trace formula implies that

P

r .1/ rC 1 2g r

D

1. This can also be proved by using the binomial theorem to expand .1 1/2g^.

EXERCISE 0.2. Assume A.k/ and A_.k/ are finitely generated, of rank r say, and that the height pairing h
;
iW A.k/  A_.k/! R (^4) The unscrupulous need read no further: they already know enough to fake a knowledge of abelian vari- eties.

is nondegenerate modulo torsion. Let e 1 ; :::; er be elements of A.k/ that are linearly inde- pendent over Z, and let f 1 ; :::; fr be similar elements of A_.k/; show that

j det.hei ; fj i/j .A.k/ W

P

Zei /.A_.k/ W

P

Zfj /

is independent of the choice of the ei and fj. [This is an exercise in linear algebra.]

The first chapter of these notes covers the basic (geometric) theory of abelian varieties over arbitrary fields, the second chapter discusses some of the arithmetic of abelian vari- eties, especially over finite fields, the third chapter is concerned with jacobian varieties, and the final chapter is an introduction to Faltings’s proof of the Mordell Conjecture.

NOTES. Weil’s books (1948a, 1948b) contain the original account of abelian varieties over fields other than C, but are written in a language which makes them difficult to read. Mumford’s book (1970) is the only modern account of the subject, but as an introduction it is rather difficult. It treats only abelian varieties over algebraically closed fields; in particular, it does not cover the arithmetic of abelian varieties. Serre’s notes (1989) give an excellent treatment of some of the arithmetic of abelian varieties (heights, Mordell-Weil theorem, work on Mordell’s conjecture before Faltings — the original title “Autour du th´eor`eme de Mordell-Weil” is more descriptive than the English title.). Murty’s notes (1993) concentrate on the analytic theory of abelian varieties over C except for the final 18 pages. The book by Birkenhake and Lange (2004) is a very thorough and complete treatment of the theory of abelian varieties over C.

Chapter I

Abelian Varieties: Geometry

1 Definitions; Basic Properties.

A group variety over k is an algebraic variety V over k together with regular maps

mW V k V! V (multiplication) invW V! V (inverse)

and an element e 2 V .k/ such that the structure on V .kal/ defined by m and inv is a group with identity element e. Such a quadruple .V; m; inv; e/ is a group in the category of varieties over k. This means that

G

.id;e/ ! G k G m ! G; G

.e;id/ ! G k G m ! G

are both the identity map (so e is the identity element), the maps

G

! G k G

id  inv ! ! inv  id

G k G m ! G

are both equal to the composite

G! Specm.k/

e ! G

(so inv is the map taking an element to its inverse), and the following diagram commutes

G k G k G

1 m ! G k G ? ?ym 1

?ym

G k G

m ! G

(associativity holds). To prove that a group variety satisfies these conditions, recall that the set where two morphisms of varieties disagree is open (because the target variety is separated, AG 4.8), and if it is nonempty, then the Nullenstellensatz (AG 2.6) shows that it will have a point with coordinates in kal.

8 CHAPTER I. ABELIAN VARIETIES: GEOMETRY

It follows that for every k-algebra R, V .R/ acquires a group structure, and these group structures depend functorially on R (AG 4.42). Let V be a group variety over k. For a point a of V with coordinates in k, we define taW V! V (right translation by a) to be the composite

V! V  V

m ! V: x 7! .x; a/ 7! xa

Thus, on points ta is x 7! xa. It is an isomorphism V! V with inverse tinv.a/. A group variety is automatically nonsingular: it suffices to prove this after k has been replaced by its algebraic closure (AG, Chapter 11); as does any variety, it contains a non- singular dense open subvariety U (AG, 5.18), and the translates of U cover V. By definition, only one irreducible component of a variety can pass through a nonsin- gular point of the variety (AG 5.16). Thus a connected group variety is irreducible. A connected group variety is geometrically connected, i.e., remains connected when we extend scalars to the algebraic closure. To see this, we have to show that k is algebraically closed in k.V / (AG 11.7). Let U be any open affine neighbourhood of e, and let R D .U; OV /. Then R is a k-algebra with field of fractions k.V /, and e is a homomorphism R! k. If k were not algebraically closed in k.V /, then there would be a field k^0  k, k^0 ¤ k, contained in R. But for such a field, there is no homomorphism k^0! k, which contradicts the existence of eW R! k. A complete connected group variety is called an abelian variety. As we shall see, they are projective, and (fortunately) commutative. Their group laws will be written additively. Thus ta is now denoted x 7! x C a and e is usually denoted 0.

Rigidity

The paucity of maps between projective varieties has some interesting consequences.

THEOREM 1.1 (RIGIDITY THEOREM). Consider a regular map ˛W V  W! U , and as- sume that V is complete and that V  W is geometrically irreducible. If there are points u 0 2 U.k/, v 0 2 V .k/, and w 0 2 W .k/ such that

˛.V  fw 0 g/ D fu 0 g D ˛.fv 0 g  W /

then ˛.V  W / D fu 0 g.

In other words, if the two “coordinate axes” collapse to a point, then this forces the whole space to collapse to the point.

PROOF. Since the hypotheses continue to hold after extending scalars from k to kal, we can assume k is algebraically closed. Note that V is connected, because otherwise V k W wouldn’t be connected, much less irreducible. We need to use the following facts:

(i) If V is complete, then the projection map qW V k W! W is closed (this is the definition of being complete AG 7.1). (ii) If V is complete and connected, and 'W V! U is a regular map from V into an affine variety, then '.V / D fpointg (AG 7.5). Let U 0 be an open affine neighbourhood of u 0.

10 CHAPTER I. ABELIAN VARIETIES: GEOMETRY

and identify V  fw 0 g and fv 0 g  W with V and W. On points, f .v/ D h.v; w 0 / and

g.w/ D h.v 0 ; w/, and so
Ddf h .f ı p C g ı q/ is the map that sends

.v; w/ 7! h.v; w/ h.v; w 0 / h.v 0 ; w/:

Thus
.V  fw 0 g/ D 0 D
.fv 0 g  W /

and so the theorem shows that
D 0. (^2)

2 Abelian Varieties over the Complex Numbers.

Let A be an abelian variety over C, and assume that A is projective (this will be proved in 6). Then A.C/ inherits a complex structure as a submanifold of Pn.C/ (see AG, Chapter 15). It is a complex manifold (because A is nonsingular), compact (because it is closed in the compact space Pn.C/), connected (because it is for the Zariski topology), and has a commutative group structure. It turns out that these facts are sufficient to allow us to give an elementary description of A.C:/

A.C/ is a complex torus.

Let G be a differentiable manifold with a group structure defined by differentiable^1 maps (i.e., a real Lie group). A one-parameter subgroup of G is a differentiable homomorphism 'W R! G. In elementary differential geometry one proves that for every tangent vector v to G at e, there is a unique one-parameter subgroup 'v W R! G such that 'v .0/ D e and .d'v /.1/ D v (e.g., Boothby 1975, 5.14). Moreover, there is a unique differentiable map

expW Tgte .G/! G

such that t 7! exp.tv/W R! Tgte .G/! G

is 'v for all v; thus exp.v/ D 'v .1/ (ibid. 6.9). When we identify the tangent space at 0 of Tgte .G/ with itself, then the differential of exp at 0 becomes the identity map

Tgte .G/! Tgte .G/:

For example, if G D R, then exp is just the usual exponential map R! R. If G D SLn.R/, then exp is given by the usual formula:

exp.A/ D I C A C A^2 =2Š C A^3 =3Š C


, A 2 SLn.R/:

When G is commutative, the exponential map is a homomorphism. These results extend to complex manifolds, and give the first part of the following proposition.

PROPOSITION 2.1. Let A be an abelian variety of dimension g over C:

(^1) By differentiable I always mean C 1.

2. ABELIAN VARIETIES OVER THE COMPLEX NUMBERS. 11

(a) There is a unique homomorphism

expW Tgt 0 .A.C//! A.C/

of complex manifolds such that, for each v 2 Tgt 0 .A.C/, z 7! exp.zv/ is the one- parameter subgroup 'v W C! A.C/ corresponding to v. The differential of exp at 0 is the identity map Tgt 0 .A.C//! Tgt 0 .A.C//: (b) The map exp is surjective, and its kernel is a full lattice in the complex vector space Tgt 0 .A.C//:

PROOF. It remains to prove (b). The image H of exp is a subgroup of A.C/. Because d.exp/ is an isomorphism on the tangent spaces at 0 , the inverse function theorem shows that exp is a local isomorphism at 0. In particular, its image contains an open neighbourhood U of 0 in H. But then, for any a 2 H , a C U is an open neighbourhood of a in H , and so H is open in A.C/. Because the complement of H is a union of translates of H (its cosets), H is also closed. But A.C/ is connected, and so any nonempty open and closed subset is the whole space. We have shown that exp is surjective. Denote Tgt 0 .A.C// by V , and regard it as a real vector space of dimension 2g. Recall that a lattice in V is a subgroup of the form L D Ze 1 C


C Zer

with e 1 ; :::; er linearly independent over R; moreover, that a subgroup L of V is a lattice if and only if it is discrete for the induced topology (ANT 4.14, 4.15), and that it is discrete if and only if 0 has a neighbourhood U in V such that U \ L D f 0 g. As we noted above, exp is a local isomorphism at 0. In particular, there is an open neighbourhood U of 0 such that exp jU is injective, i.e., such that U \ Ker.exp/ D 0. Therefore Ker.exp/ is a lattice in V. It must be a full lattice (i.e., r D 2g/ because otherwise V =L  A.C/ wouldn’t be compact. (^2)

We have shown that, if A is an abelian variety, then A.C/  Cg^ =L for some full lattice L in Cg^. However, unlike the one-dimensional case, not every quotient Cg^ =L arises from an abelian variety. Before stating a necessary and sufficient condition for a quotient to arise in this way, we compute the cohomology of a torus.

The cohomology of a torus.

Let X be the smooth manifold V =L where V is real vector space of dimension n and L is a full lattice in Rn. Note that V D Tgt 0 .X/ and L is the kernel of expW V! X, and so X and its point 0 determine both V and L. We wish to compute the cohomology groups of X. Recall the following statements from algebraic topology (e.g., Greenberg, Lectures on Algebraic Topology, Benjamin, 1967).

2.2. (a) Let X be a topological space, and let H .X; Z/ D

L

r H^ r (^) .X; Z/; then cup- product defines on H .X; Z/ a ring structure; moreover

ar^ [ bs^ D .1/rs^ bs^ [ ar^ , ar^2 H r^ .X; Z/, bs^2 H s^ .X; Z/

(ibid. 24.8).

2. ABELIAN VARIETIES OVER THE COMPLEX NUMBERS. 13

THEOREM 2.3. Let X be the torus V =L. There are canonical isomorphisms ^r H 1 .X; Z/! H r^ .X; Z/! Hom.

^r L; Z/:

PROOF. For any manifold X, cup-product (2.2a) defines a map ^r H 1 .X; Z/! H r^ .X; Z/, a 1 ^^ : : :^^ ar 7! a 1 [ : : : [ ar.

Moreover, the K¨unneth formula (2.2b) shows that, if this map is an isomorphism for X and Y and all r, then it is an isomorphism for X  Y and all r. Since this is obviously true for S^1 , it is true for X  .S^1 /n. This defines the first map and proves that it is an isomorphism. The space V  Rn^ is simply connected, and expW V! X is a covering map — therefore it realizes V as the universal covering space of X, and so  1 .X; x/ is its group of covering transformations, which is L. Hence (2.2c)

H 1 .X; Z/ ' Hom.L; Z/:

The pairing ^r L_^ 

^r L! Z, .f 1 ^^ : : :^^ fr ; e 1 ^^ : : :^^ er / 7! det .fi .ej //

realizes each group as the Z-linear dual of the other, and L_^ D H 1 .X; Z/, and so ^r H 1 .X; Z/

' ! Hom.

^r L; Z/: (^2)

Riemann forms.

By a complex torus, I mean a quotient X D V =L where V is a complex vector space and L is a full lattice in V.

LEMMA 2.4. Let V be a complex vector space. There is a one-to-one correspondence between the Hermitian forms H on V and the real-valued skew-symmetric forms E on V satisfying the identity E.iv; iw/ D E.v; w/, namely,

E.v; w/ D Im.H.v; w//I H.v; w/ D E.iv; w/ C iE.v; w/:

PROOF. Easy exercise. (^2)

EXAMPLE 2.5. Consider the torus C=ZCZi. Then

E.x C iy; x^0 C iy^0 / D x^0 y xy^0 , H.z; z^0 / D z zN^0

are a pair as in the lemma.

Let X D V =L be a complex torus of dimension g, and let E be a skew-symmetric form L  L! Z. Since L ˝ R D V , we can extend E to a skew-symmetric R-bilinear form ERW V  V! R. We call E a Riemann form if

14 CHAPTER I. ABELIAN VARIETIES: GEOMETRY

(a) ER.iv; iw/ D ER.v; w/I (b) the associated Hermitian form is positive definite. Note that (b) implies that E is nondegenerate, but it is says more.

EXERCISE 2.6. If X has dimension 1 , then

V 2

L  Z, and so there is a skew-symmetric form EW L  L! Z such that every other such form is an integral multiple of it. The form E is uniquely determined up to sign, and exactly one of ˙E is a Riemann form.

We shall say that X is polarizable if it admits a Riemann form.

REMARK 2.7. Most complex tori are not polarizable. For an example of a 2 -dimensional torus C^2 =L with no nonconstant meromorphic functions, see p104 of Siegel 1948.

THEOREM 2.8. A complex torus X is of the form A.C/ if and only if it is polarizable.

PROOF (BRIEF SKETCH) H)W Choose an embedding A ,! Pn^ with n minimal. There exists a hyperplane H in Pn^ that doesn’t contain the tangent space to any point on A.C/. Then A \ H is a smooth variety of (complex) dimension g 1 (easy exercise). It can be “triangulated” by .2g 2/-simplices, and so defines a class in

H2g 2 .A; Z/ ' H 2 .A; Z/ ' Hom.

^^2

L; Z/;

and hence a skew-symmetric form on L — this can be shown to be a Riemann form. (HW Given E, it is possible to construct enough functions (in fact quotients of theta functions) on V to give an embedding of X into some projective space. (^2)

We define the category of polarizable complex tori as follows: the objects are polariz- able complex tori; if X D V =L and X^0 D V 0 =L^0 are complex tori, then Hom.X; X^0 / is the set of maps X! X^0 defined by a C-linear map ˛W V! V 0 mapping L into L^0. (These are in fact all the complex-analytic homomorphisms X! X^0 .)

THEOREM 2.9. The functor A 7! A.C/ is an equivalence from the category of abelian varieties over C to the category of polarizable tori.

In more detail this says that A 7! A.C/ is a functor, every polarizable complex torus is isomorphic to the torus defined by an abelian variety, and

Hom.A; B/ D Hom.A.C/; B.C//:

Thus the category of abelian varieties over C is essentially the same as that of polarizable complex tori, which can be studied using only (multi-)linear algebra. An isogeny of polarizable tori is a surjective homomorphism with finite kernel. The degree of the isogeny is the order of the kernel. Polarizable tori X and Y are said to be isogenous if there exists an isogeny X! Y.

EXERCISE 2.10. Show that “isogeny” is an equivalence relation.