Analytic Theory of Abelian Varieties 9780521205269, 0521205263

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London Mathematical Society

Lecture Note Series

—s.

Analytic Theory of Abelian Varieties

H. P. F. SWINNERTON-DYER

CAMBRIDGE

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London Mathematical

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14

Analytic Theory of Abelian Varieties H.PRFESWINNERTON-DYER

Cambridge - At the University Press 1974

Published by the Syndics of the Cambridge University Press Bentley House,

200 Euston Road,

American Branch: (c)

London NW1

32 East 57th Street, New York,

Cambridge University Press

Library of Congress

Catalogue Card Number:

First published 1974

Printed in Great Britain at the University Printing House,

N. Y. 10022

1974

ISBNi0052152052653

(Brooke Crutchley,

2DB

Cambridge

University Printer)

74-77835

Contents

page Introduction Chapter I - Background 1.

Compact Riemann curves

surfaces and algebraic

2.

Doubly periodic functions

3.

Functions

of several complex variables

Chapter II - Theta functions and Riemann forms

4.

Reduction to theta functions

5.

Consistency conditions and Riemann forms

6.

Construction of theta functions

Chapter III - Morphisms, 7.

Morphisms

polarizations and duality

of Abelian manifolds

8.

Duality of Abelian manifolds

9.

Representations of End (A)

10.

The structure of End (A)

Appendix - Geometric theory

References Index of definitions

and of tori

14 21 29 30 37 44 53 53 60 67 73 80 87 89

ili

Digitized by the Internet Archive in 2020 with funding from Kahle/Austin Foundation

https://archive.org/details/analytictheoryofOO00swin

Introduction

For fixed

2 R“”

n>

0 let

c"

denote n-dimensional

the underlying 2n-dimensional real space.

Rae - that is, a free abelian group on thus with the induced topology T

A

2n

Let

torus.

T=

c"/A,

A

bea lattice in

generators which spans

is discrete and

has a natural structure as a complex manifold,

writing itas

complex space and

T = RA

Rol

is compact.

which we recognize by

and with this structure it is called a complex

The most interesting case is when there are sufficiently many

meromorphic Chapter II;

functions on in this case

denoted by A.

T

T, in a sense that will be made precise in is called an abelian manifold,

and is usually

Thus from one point of view the study of abelian mani-

folds is essentially the

variables having

2n

study of meromorphic

independent periods.

functions

of

n

complex

Thus it forms a natural

generalization of the theory of elliptic functions - that is, of doubly periodic functions of one complex variable,

Historically,

of the subject was the study of compact Riemann

term

‘abelian manifolds'

comes

A compact Riemann non-singular algebraic

from the connection with Abel's theorem.

and this already gives a connection between

algebraic geometry and abelian manifolds.

that abelian manifolds

non-singular

varieties.

indeed the

surface is just another way of describing a

curve;

strengthened by Lefschetz

surfaces;

the other parent

This connection was much

and the great Italian geometers,

who showed

are an important tool in the problem of classifying

Conversely any abelian manifold can be embedded

in projective space as a variety in the sense of algebraic geometry;

we wish to emphasize this point of view, instead of an abelian manifold.

geometric methods,

initiated by Weil;

when

we speak of an abelian variety

The study of abelian varieties by purely

valid over fields of arbitrary characteristic,

was

see [14], [15] and Lang [5]. (Numbers in square brackets

refer to the list of references

at the end of this volume. ) An up-to-date

account may be found in Mumford [8].

More recently Shimura and others

have shown that the theory of abelian manifolds has important application

to algebraic number theory;

for Shimura's work see [9], [10] anda

series of papers published during the 1960s,

largely in Annals of

Mathematics. The main object of this book is to give an account of the standard theory of abelian manifolds which presupposes basic complex variable course.

not much more than a

It contains all the material

on abelian

manifolds which is needed for the applications to algebraic geometry and to number

theory;

indeed it is based on a course

of lectures delivered

in Cambridge which was designed to lead on to an exposition of some

Shimura's work. applications.

But it does not contain an exposition of either of these

I have included some geometrical results,

suppose a knowledge of algebraic geometry;

may omit them without inconvenience. ring of endomorphisms

and they do pre-

the reader who lacks this

Also §10, on the structure of the

of an abelian variety,

requires some knowledge

of the theory of algebras and of algebraic number

necessary

of

theory;

however

the

results from the theory of algebras are stated without proof

at the beginning of §10. The book starts with two sections closely connected with the main

theme though not essential to it.

In §1 there is a survey of the theory of

compact Riemann surfaces - without proofs of the key theorems because to include proofs would take so much space that it would unbalance the

book.

Proofs may be found for example in Gunning [4].

brief account,

with proofs,

the special case

n= 1

In §2 there is a

of the theory of elliptic functions.

of the theory,

This is

but it is untypical for two reasons.

Because one can give a simple description of divisors when

n=

1, there

are many explicit formulae in that case which cannot be generalized; the Riemann form,

which plays a major part in the general theory,

in so trivial a form when

n=1

that its presence goes unnoticed.

also appears

The

rest of Chapter I contains some general results on functions of several complex variables which are needed later.

necessary and sufficient conditions on

constant meromorphic functions, the construction of those functions. sequences

vi

of these results,

A

Chapter II is concerned with

for

T= c"/A

to admit non-

or to be an abelian manifold,

and with

It also contains the immediate con-

in particular theorems about the projective

embedding of abelian manifolds.

Chapter III contains the standard theory

of abelian manifolds themselves,

in particular the study of the ring of

endomorphisms

of an abelian manifold and the matrix representations

these endomorphisms, the group

and the duality theory which

of divisors on an abelian manifold.

of

essentially describes

Many of the results in this

Chapter can be stated in purely algebro-geometric terms,

the proofs which have been given for them are analytic.

even though

I have therefore

given in an Appendix a brief account without proofs of the geometric theory

of abelian varieties over a field of arbitrary characteristic, far the analytic theory remains

to show how

valid.

The most elegant account of the analytic theory is that given in

Chapter VI of Weil [17]; but that account was written to show how Hodge theory can be applied to a particularly and it therefore assumes

simple kind of complex manifold

a great deal of background knowledge.

same is true of the account in Mumford [8], Chapter I.

The

Two books which

assume no more knowledge than this one are Conforto [2] and Lang [6]. Conforto's treatment is very old-fashioned;

in particular a substantial

part of his book is taken up by Poincaré's original proof of the key existence theorem,

theta-function on

that to every positive divisor on

T

corresponds a

c". whereas the proof given here, based on Weil [16],

only takes a few pages. though perhaps more

Lang's book is very similar in spirit to this one,

modern;

but I believe there is enough difference in

the material covered to justify publishing this also.

Much

of the theory

can also be found in Siegel [11]. Nothing in this book is original,

except perhaps the errors.

have therefore only ascribed a result to someone

I

if it is generally known

by his name, I would like to express my gratitude to André Weil,

Serge Lang and

S. J. Patterson for the help which they have given me at various levels; without them this

book would never

have been written.

NOTATION As is usual,

numbers, sometimes

C, Q, R

the rationals,

and

Z

denote respectively the complex

the reals and the rational integers;

also used to denote a quadratic form.

but

Q

is

From the beginning of

vii

Chapter II onwards,

and

A

V = Cc" is a complex vector space of dimension

isa lattice in V;

general,

and by A

Appendix,

A

Zine, ae

V;

Similarly base for

is denoted by

it is known to be an abelian manifold. n

z)

and

T

in

In the

defined over an

k, not necessarily of characteristic zero.

usually denote points of V base for

V/A

is an abelian variety of dimension

arbitrary field

Zi

when

the complex torus

n

Both

Sas en, boii w,)

and their coordinates with respect to a fixed

they may also denote the corresponding points on

2X denotes an element of A;

but note that

A, rather than a set of coordinates for

A.

AL

baal

Finally

T

or

A,

Aon is a E

and

H

denote respectively the alternating and the Hermitian version of a Rie-

mann form on

V X V

with respect to A.

By convention,

a Hermitian

form will be C-linear in its first argument and C-antilinear in its second;

the reader is warned that Weil [17] adopts the opposite convention. Since the distinction between Theorems subjective,

they share a common

system of numbering;

principle has not been extended to Corollaries.

Vili

and Lemmas

is purely

however this

|-Background

1.

Compact Riemann surfaces and algebraic curves A Riemann

surface is a topological space which in a neighbourhood

of any point looks like the complex plane. Riemann

(Some writers also require a

surface to be connected. ) More precisely,

is a Hausdorff space

N

of P

For each point

P

@¢ from

c>

0.

satisfy the following consistency condition.

any two points of

N,

of S we are given an

andahomeomorphism

lz - o(P)| < c inthe complex z-plane for some morphisms

overlap;

S such that their corresponding

then

surface

S which is endowed with a complex one-dimensional

structure in the following way.

open neighbourhood

a Riemann

9, ° o

is holomorphic on

N

toa disc

These homeoLet

P., Pe be

neighbourhoods

o,(N, qN,).

N.;

Tre t(zjeis

a function which is holomorphic and has non-zero derivative at z= 9(P), then we can replace

@ by

f° @ (with corresponding changes in N

c) without changing the complex structure of S. variable at P;

Wecall

and

¢ a local

clearly it is then a local variable at each point of N.

We can now transfer all the standard terminology of the theory of

functions of a complex faces.

For example,

holomorphic

at

P

w

z=

is holomorphic

n

at P

¢ is

¢(P), where

wd®9;

w

n

at

defined ina neigh-

it is said to be holomor-

at P, or to have a zero or pole of order

Ww has the corresponding property.

Again,

P,

has a zero of order

A differential

can be written inthe form

phic or meromorphic

z=

in a neighbourhood of

if po g

¢(P), and similarly for poles. P

at

the consistency condition is just what is needed

wy is then holomorphic

has a zero of order

bourhood of

if

¢?

is

yw defined in a neighbourhood of P

afunction

if po

a local variable at P; to ensure that

variable from the complex plane to Riemann sur-

n

at P,

It is easy to see that none of these

properties depends on the choice of the local variable

¢.

It must be

remembered that a residue is associated with a differential rather than If w= wWd@ has at worst an isolated singularity at P, with a function.

then its residue at P

is

(27)? Jw

which lies in N, goes once round induced in

N

by

except perhaps

taken round a closed contour

P

anticlockwise under the orientation

9, and contains no singularity of P

itself.

[

w

inside or on it

It follows that the residue is equal to the co-

efficient of 1/(@ - ¢(P))

in the Laurent

series expansion of w in terms

of ¢ - ¢(P), and also that it is independent of the choice of 9. In general one is primarily interested in those functions and differentials on a Riemann

surface which are everywhere

and in what follows we shall confine ourselves

to them.

is said to be of the first kind on a Riemann surface phic everywhere

on

§ if it is holomoron

S and

and of the third kind if it is meromorphic

S; differentials of any given kind form a C-vector space.

ential is called exact if it is of the form

on

A differential

S, of the second kind if it is meromorphic

has residue zero at every pole, on

meromorphic;

dy

where

A differ-

WY is meromorphic

§; clearly an exact differential is of the second kind, but not every

differential of the second kind is exact. We can now translate Cauchy's theorem into the language of Riemann

surfaces:

Theorem 1. and let

w

Let

I be acontour on

be a meromorphic

continuously,

S.

Suppose that

with its endpoints if any remaining fixed.

remains constant as long as and it changes by moves

differential on

S, not necessarily closed,

27i

TI does not move

Then

IT varies [pe

across a singularity of w,

times the residue at a Singularity whenever

IT

across that singularity. In particular,

if

second kind, the value of

IT is a closed curve and Jpe

w

is of the first or

depends only on the homology class of

I’; this value is called the period of w to see that

w

with respect to

IT.

It is easy

is exact if and only if all its periods are equal to zero.

Henceforth we shall also assume that that it is connected.

S is compact,

and usually

(A compact Riemann surface is the disjoint union

of finitely many compact connected Riemann surfaces,

so that results in

the general case follow at once from those in the connected case; they are often more

but

complicated to state in the general case. ) At each

point of S there is a canonical local orientation,

obtained by means of

@

-1

y , ; from the canonical orientation of the complex plane.

tions are compatible,

so

S itself is oriented.

Since it is compact it

can be triangulated and the triangulation is finite; just a sphere with

g handles,

for some

g=

group H,(S, Z) is afree abelian group on Theorem

pact Riemann the sum

2.

(i) If w

surface

§ then

of the residues

(ii)

If

pact Riemann

of

w

0, and its first homology

2g generators. differential on a com-

at these poles is

S then

function on a com-

wy takes every value the same number

In particular

of

wy has as many poles

zeros.

Proof.

If w

of accumulation on phic at that point.

had infinitely many poles these would have a point

S, by compactness, Now triangulate

and sum the results;

and

w

would not be meromor-

S, integrate

w

Applying these results to the special case as many poles as zeros, the value

c

valence

of

and writing

w=

~-c

w.

The number

has been

we find that

Ww wefindthat

Lemma

3.

Let

6, w

surface

there is a polynomial

Wy takes

wy takes each value is called the

be non-constant meromorphic

S, of valences

F(X, Y)

wy has

This completes the proof

Constant functions are deemed to have valence

a compact Riemann

such that

of times

w

once in each direction.

dwW/y)

for

as many times as it has poles.

of the Theorem.

round each triangle

(i) follows at once since in the sum

integrated twice along each side of each triangle,

Y

S and

0.

WY is a non-constant meromorphic

surface

S is

has only finitely many poles on

times (allowing for multiplicities). as

topologically

is a meromorphic

w

These orienta-

m, n

functions on

respectively.

of degrees at most

n in X

S is connected then

F(@, ¢)= 0. If moreover

0.

and

Then

m

in

F can be

chosen to be irreducible.

Proof. by

w

c which is not a value taken

Choose a complex number

at any of the zeros

points of S atwhich

or poles of

y=c.

For any

6, and let r=0

@ and

ers Po be the

the differential

w = 6 dy/(W - c) has a simple pole with residue its other poles are at the poles of

P.,

J)

at each

Ee

wy, and at each of these the

residue is a rational function of that

2 e'(P,)

c.

It now follows from Theorem

is a rational function of

c

for each

r, so that the same

is true for the elementary symmetric functions of the satisfies an equation of degree

tions of w;

whose

o(P ).

n

in

Thus

6

coefficients are rational func-

and hence the minimal equation connecting

degree at most in

n

2(i)

6 and

wW has

6 and hence by symmetry has degree at most

m

w.

Now suppose that minimal

equation.

S is connected and that

If F

for some polynomials

F(6, y) = 0

is not irreducible then we can write

F, and

be the canonical neighbourhood

F,. of

Choose a point

P

P

on

is the F = FF,

S andlet

associated with some

N

local variable

at P; then F (6,

WF, (8, 7) = 0

in N,

and so (by the analogous result for the complex plane) either

F (6, Wy) = 0 tim iy oe

F (6, w)= 0

by analytic continuation

4.

Riemann surface degree 1 over

F.

S, which contra-

This completes the proof of the Lemma.

The meromorphic

functions on a compact connected

Given any two points

meromorphic function P

ee and P,

y on S_ such that

on S, there isa

V(P,) #V(P,).

on S there is a meromorphic

is a local variable at

function

Moreover, @ on

are enough meromorphic

functions on

equipped with plenty of functions.

phic

W

on

n>

0; then by Lemma

is algebraic of degree at most is zero,

For suppose that

3 any meromorphic function on

C(wW).

Since the characteristic

it follows from standard results in field theory that the field of

all meromorphic

C(y).

n over

S it is likely to

Given a non-constant meromor-

S, the proof of the first sentence is easy.

has valence

that there

§; in many practical cases this is

since if one is presented with an explicit

come

Y

S which

P.

The difficult part of the proof is the second sentence,

irrelevant,

then

S form a finitely generated field of transcendence

C.

given any point

Assume the former;

F,(6, Ww) = 0 everywhere on

dicts the minimality of

Theorem

in N.

functions on

S is algebraic of degree at most

n

over

§S

Corollary.

Any compact connected Riemann surface can be

regarded as an irreducible non-singular

algebraic curve over

C, and

vice versa.

Because the functions on

S separate points,

S can be recon-

structed from a knowledge of the field of meromorphic and this field is of the sort that corresponds

functions on it;

to an irreducible curve.

The

curve is non-singular because the field contains a local variable at each point.

For the converse,

we need only check that the Riemann

corresponding to an irreducible curve is connected;

the connected components

surface

if this were not so,

of the Riemann surface would give rise to com-

ponents of the curve. The additive group of divisors

on

whose generators are the points of S. on

§ is the free abelian group

Let

Ww be a meromorphic function

§S which is not identically zero on any connected component of

and let

Py

eens

a be the poles and

Q.> Rist

repeated according to their multiplicities;

S,

Q, be the zeros of y,

then the divisor of

WY is de-

fined to be

(y)=Q, glee se

sor neXs = eae 1

The divisor of a differential

as

w

is defined in a similar way and is written

(w), though of course a differential need not have as many poles as

zeros.

Taking divisors is a homomorphism

of non-zero meromorphic

zeros;

from the multiplicative group

§ to the additive group of divisors.

functions on

The kernel of this homomorphism

is just the functions with no poles or

these have valence 0 and are therefore constant on each connected

component

on

n

of

S.

In particular,

if S is connected the divisor of a function

§ determines the function up to multiplication by a non-zero constant,

The image of the homomorphism,

that is the group of divisors of functions,

is called the group of principal divisors;

and two divisors are said to be

linearly equivalent if their difference is a principal divisor. of adivisor

2 n Pp, is defined to be

sors have the same 2D np,

degree,

Z n>

thus linearly equivalent divi-

but not necessarily vice versa.

is said to be positive if every

The degree

A divisor

n, = 0; this induces on the group

of divisors a partial ordering which is compatible with addition.

Since

the quotient of two differentials is a function,

tion and a differential is a differential,

and the product of a func-

it is easy to see that the divisors

of differentials precisely fill a linear equivalence class;

this is called

the canonical class and every divisor in it is called a canonical divisor.

There are two key theorems which tell one something about the structure of the group of divisors,

the Riemann-Roch

theorem which gives

information about principal divisors and Abel's theorem which describes the group of divisor classes modulo linear equivalence.

Riemann-Roch theorem we need some notation.

Let

compact

Riemann

S.

C-vector

space consisting of those meromorphic

which

(WY) +a

a4.

Since

L(a) = {0}

and

a

adivisor on

S be a connected Denote by

functions

= 0, together with the zero function,

l(a) = dim L(a);

of

surface,

thus

Z(a)

Theorem

whenever

the

on §S for

and write

wy on

S,

deg(a) < 0.

5 (Riemann-Roch).

Riemann surface.

w

L(a)

depends only on the linear equivalence class

deg((w)) = 0 for any non-zero function

and 2(a)=0

To state the

Let

S be a compact connected

Then there is an integer

g = 0, depending only on

§S,

such that

U(a) = deg(a) + k= g Fi(k-a) for any divisor

(1)

a and any canonical divisor k.

The condition that

S should be connected may be dispensed with,

if we suitably modify the definition of L(a) However this is not a real generalization,

and allow

g to be negative.

for the equation (1) for general

S can be obtained by addition of the corresponding equations for the

connected components

Corollary 1. Proof.

of

S.

Deg(k) = 2g - 2 and

Applying Theorem

5to

L(k- a) = deg(k) - deg(a) +1-

and comparing this with (1) gives note that

2(0)=1

because

2(k) =¢, if S is connected.

k - a

instead of a

¢ + l(a),

deg(k) = 2g - 2.

L(0)=C;

gives

thus

Now take a =

Z(k) =g.

k and

The Riemann-Roch theorem is a duality theorem, and

Z(k-a);

doing so.

relating

/(a)

it can be written in self-dual form but nothing is gained by

In the special case when

deg(k-a) 2g - 2 we have

2(k-a)=

Cay = des(a) +1 eit

0; so (1) takes the form

-des(a) > 2¢-2,

(2)

which is the form in which it is most frequently used.

Corollary 2.

The differentials of the first kind form a C-vector

space of dimension

Proof.

g, provided

S is connected.

This is just the equation

bea given non-zero

7(k)=g

For

let

w

the

Ww

ww

is a differential of the first kind if and only if w is in L((w)), which

where

has dimension

differential;

inanewform.

then the differentials are just

wW runs through all meromorphic

functions on

S, and

g.

The statements

S is not connected;

of both these Corollaries need to be modified if

in that case both

2(k)

and the dimension of the space

of differentials of the first kind are equal to (g - 1) plus the number of connected components The integer

of

S.

g in Theorem

connected its genus is equal to the

5 is called the genus of

g which we have already defined

topologically by the condition that H,(S, 2g generators.

Let Ss) and

is holomorphic

of order and

r>

r for

S, be compact connected Riemann sur-

f: Ss,=> S, is said to be holomorphic if for any point

then a map

on Ss, and any local variable

La

Z) is afree abelian group on

To prove this we consider maps from one Riemann

surface to another. faces;

S; if S is

ar

at

f if

The point

e, at {(P,) P.

on S,

the function

o. of

is said to be a ramification point

1o, o f - 9, ° Ps

has a zero of order

r

at EG

1; it follows from the compactness of S, that if f is non-

constant it has only finitely many points of ramification.

P, is a point of

S, then the degree of foie)

Moreover

if

does not depend on the

choice of P, provided that points of ramification in fei cy) are taken with multiplicities equal to their orders

n

then we say that

f is an n-to-1 map.

of ramification;

if this degree is

Theorem 6.

Let f:

S, ad S, be a non-constant holomorphic

n-to-1 map between two compact connected Riemann surfaces, i qr

a

eee?

be the orders of the ramification points of f.

28, ee where g,

n(2g,, - 2) +

Let

w,

(3)

g, the genus of

be a differential on

a differential

w,

on

w,.

By means

of f we can pull

Ss, which is not a ramification point of f then

,

if et

oF ef

w,

back to

is a point of

is a local variable

has a zero (pole) at Pe

if and only if

has a zero (pole) at {(P,), and both zeros (poles) have the same

multiplicity. r

w,

for con-

of points of ramification

S.: With the notation above,

at Eye and it follows that

S,-

S, and assume

venience of description that none of the images of f are zeros or poles of

Then

ty - 1)

is the genus of S, and

Proof.

and let

then

Ww)

On the other hand,

d(9., of)

has a zero of order

Thus the divisor of

divisor of

o

w,

and

then

n

r-1

at PB.

is equal to the sum

and hence so has

of the pull-back of the

r-1.

r

To prove (3) we now evaluate the

(,) both directly and through this decomposition.

In particular let infinity;

is a ramification point of order

and all the ramification points of f, a point of order

being taken with multiplicity degree of

if P

S,

be the complex plane including the point at

f can be any non-constant meromorphic

is the valence of f.

has a double pole at infinity;

28, Seer ena 2

Moreover

function on Ss,

e = 0 since the differential

dz

so (3) reduces to

- 1).

(4)

If one now constructs a triangulation of

S, which has all the images of

ramification points among its vertices,

and lifts it to a triangulation of

S.

a straightforward calculation shows that the rank of H(S,,

Z) is

28:

In the classification of connected Riemann surfaces, the only discrete-valued invariant that occurs; surfaces

the genus is

all connected Riemann

of the same genus form a single continuous

system.

How can

one most easily calculate the genus of a given connected Riemann surface? If the surface is defined topologically then one should use the topological

description of the genus; triangulation,

in particular,

if the surface is given with a

as in modular function theory for example,

one can use

the formula

2g - 2 = e-v-f

where the triangulation has

v

vertices,

e

edges and

surface is defined by means

of its function field one should use either

deg((w)) = 2g - 2 for some suitably chosen differential of the covering formulae (3) and (4); the various

f faces.

w

If the

or else one

formulae that occur in

algebraic geometry can be deduced without difficulty from these. Now let

&

denote the C-vector

kind on a compact connected Riemann

space of differentials of the first 0.

By

which is C-linear in the first argument and additive in the second;

so

Theorem

surface

S of genus

g>

1 the pairing

(w, T) > fyw induces a map

2 xX HIS,

this map induces

H(S,

Z)>C

a canonical homomorphism

Z) > 2* = Hom(Q,

Cail the image of this map most

2g generators. S; to an arc

point of

If we change the arc Q* of

A;

C).

it is obviously a free abelian group on at

an arbitrary g Pp corresponds an element of Q* given by wr Joe. be afixed point and

Now let O OP

OP,

and

O

leaving

is changed by an element of A; 2*/A,

(5)

P

the same,

thus to P

P

this element of

corresponds an element

and by additivity this correspondence can be extended to a

homomorphism

{Divisors on S$} 7 */A. If we restrict this homomorphism

depends on the choice of the point

(6) to divisors of degree zero,

O;

it no longer

and it is just what we need to pick

me

out the principal divisors. Theorem

7

(Abel).

With the notation above,

Q* - that is, a free abelian group on

2g

considered as a real vector space.

is a lattice in

generators which spans

Moreover

restricted to divisors of degree zero,

A

{2*

the homomorphism

(6),

is onto and its kernel is just the

group of principal divisors.

Another way of stating the first sentence is to say that (5) induces a homeomorphism between H,(S, Z) with the discrete topology and its image

A

with the topology induced on it as a subset of

*.

Abel's theorem identifies the group of divisor classes of degree zero with the complex torus the Jacobian of S.

2*/A

of dimension

Not surprisingly,

g, which is also called

this complex torus is actually an

abelian manifold - that is, it has enough meromorphic functions on it to separate points.

This will be proved in §9; for the proof we shall need

further information about the periods associated with

A

or, which comes S.

to the same thing,

about

This information is contained in Riemann's

relations;

to state them it is convenient to choose a normalized base for

H(S,

So let

Z).

Ve Sone

Ep ne a

Ls be closed curves on

S

such that

(i)

the corresponding classes generate

(ii)

no two of these curves have a point in common,

except that for each

v the curves

Yr, and

i

H,(Ss, Z); and

Ie)

have one

common point at which they cross with the orientations shown in the diagram.

Such curves can be found, for if we consider handles we can choose

S as a sphere with

Yr, to go once round the pth handle and

g ,

to

run along the ye handle and back along the surface of the sphere.

By considering intersection numbers we see first that the homology classes of these 2g curves are linearly independent and then that they form a

base for LS or

H,(s, Z).

Moreover any closed curve which does not meet any Di must be homologically trivial.

Theorem 8 on on § and and letle

10

Cu

(i) Let w Cu'

1?

w, be differentials of the first kind

beth e the periods i of oF with i respect to ro r,i

respectively.

Then

z( CAP lv CLs2v (ii)

and let

=vcl

Let

2c ey)

w

)=0.

(7)

be a non-zero

differential of the first kind on

C), Cy, be its periods with respect to

(i) Cut the Riemann surface S*.

The boundary of

a typical one of which consists of

r, and

S*

r, once in each direction.

is zero,

an indefinite integral

w*

each taken twice,

as in

Y Jy yj

The

Yj

jj

f, on

w,

S*

S*

has

l

which

Mf}

is a single-valued holomorphic function. Moreover

of

since such a curve is

homologically trivial in S; so

of

iy and g pieces,

consists

each of

integral of tr round any closed curve in

I'

S along all the S*

as one goes round this piece

of the boundary one traverses

by and

Then

(8)

ie and call the result

the figure;

ES respectively.

0s

ImZce ¢)

Proof.

Pr

S

if w*

is any holomorphic

Ll

IJ

differential on

S*

then the integral

round the boundary of S*, each piece of the boundary being tra-

versed so as to leave

S*

particular this holds for

on the left, is zero by Cauchy's theorem. w* = f,w..

But to any point of

In

r, correspond

two points of the boundary of S*, and the values of f, at these two points differ by in

cay because any path in S*

S to Tr.

that connects them is homologous

Taking account of directions,

the two parts of the boundary of S*

the integral of fs

which correspond to

round

rs is

“Cy Jp, meta (ce), = cc!) /2ia

wii UCoat crfone :

so we need only prove that the integral is strictly positive. =u+iv;

Write

then

fw/2i = d(u’ + v’)/4i + 4(udv - vdu), By Green's theorem we obtain

J fw/2i = $ J(udv - vdu) = Jf fdudv where the simple integrals are taken over the boundary

double integral is over

S*.

By means

S*

where

= é+in

This proves (8).

of the Riemann-Roch

theorem we can put into canonical

form the function field of a Riemann surface of small genus.

first the case

g=0.

Let

P

be any point of S; then

and hence there is a non-constant meromorphic

only singularity is a simple pole at P.

morphic functions on

and the

The double integral is positive since the

integrand at any point P is equal to |df/dd|*dédn is a local variable at P.

of

§ is just C(z), and

z-plane including the point at infinity.

2(P)=2

function

By Lemma

Consider

z

on

by (2) S whose

3 the field of mero-

S can be identified with the

We have already seen in the proof

of (4) that the z-plane has genus 0; what we have now shown is that it is essentially the only connected compact Riemann surface of genus 0.

Consider now the case

Now (2) implies

7(P)=1,

meromorphic functions

g=1

u, v

on

with respectively a double anda

ai

and again let

2(2P) = 2

and

relation must involve both

be any point of S.

S with no singularity other than triple pole at P.

v’, uv, u’, v, u and 1 lie in L(6P)

must be linearly dependent over

P

1(3P) = 3; so there exist

C.

which has dimension 6; so they

Moreover the linear dependence

u°> and

in uv

v by v + au+b

and

term in u’,

v;

Vv, for otherwise one of its terms

for suitable

and then by replacing Multiplying

u

or

v

u

By

a, b we can get rid of the terms by

u+ec

we can get rid of the

by a suitable constant now reduces the

linear dependence relation to the form

se

and

The seven functions

would have a pole of strictly greater order than any of the others.

replacing

P

2

for some

3

agensuercre sum Ee constants

(9)

g, and

g,-

(The notation,

and the factor 4, are

chosen for historical reasons which will be explained in §2.) freedom left to us is to replace for some

c #0;

u

and

this would replace

v by

g, by

c’u

and

ev

and

g, by

Gige

The function field of S is at most quadratic over

argument of Theorem 4; so it is precisely

C(u, v).

4(u - a)? (u - 8) we could write

=a Sets w’

and

w

Gye.

C(u), by the for if it were of

v/2(u - @) whence

would have a simple pole at P

contrary to 2(P)=1.

and no other pole,

The condition that the right hand side of (9) splits

into three distinct factors is

=

w=

respectively,

Moreover the right

hand side of (9) splits into three distinct linear factors,

the form

The only

a3

g - 278, #0;

we therefore write

2

j = 1728g, /(g, - 27¢,) so that P.

j is finite and depends

(We shall see below

P.)

Conversely,

only on

S and perhaps on the choice of

that in fact it does not depend on the choice of

consider the function field C(u, v) where

connected by (9) with

g - 278, #0,

It can be verified that

differential which has no poles or zeros; in the next paragraph. Moreover

du/v

Hence

g=1

By Theorem

isa

indeed this is implicitly proved

by Corollary 1 of Theorem

is the unique differential of the first kind,

plication by a constant.

u, v are du/v

5, to any divisor

a

5.

up to multiof degree 0 on

S there is a function

f, unique up to multiplication by a constant,

that

(f) +a

+ P=0;

and hence there is a unique point

Q

on S such

that

(f) + a+Pz=+Q.

This is the same as saying that

a

is linearly

equivalent to Q - P; identification

and in this way we identify

S=C/A

such

S with its Jacobian,

the

being induced by

Q du/v Q-> Ip which is a holomorphic map.

On the one hand this means

that

S has a

natural structure of abelian group once one has chosen a base point

P;

on the other hand it means that a change of base point merely corresponds to a translation on

of P.

C/A.

Thus any Riemann

In particular,

j does

not depend on the choice

surface of genus 1 corresponds to an equation he

of the form (9), and such surfaces are classified by the points of the affine line. Now consider those Riemann admit a meromorphic

function

such a surface cannot be

C(u).

u

C(u)

some polynomial u

which

f(u) =

of valence 2.

it to be

0, where

infinity.

If n

there is

u (n-1)/2

local variable

differential

v

eee

Meroe degree

v;

du/v

if n

n points at

and the one or two points at

is even there are two points at infinity anda is

1/u.

It is now

has no poles or zeros

order

is even.

if n

2) if n

easy to check that the

except perhaps at infinity;

n - 3 at infinity if n

3(n-

with no repeated root.

is odd there is one point at infinity and a local variable

has a zero of order

odd and

n

y= f(u) for

S except for the

is a local variable,

at each of them

3n- 2

0 which

The function field of

C(u, v) where

is a local variable at any point of v=

g>

so it must be a quadratic extension of

We can therefore suppose

Now

surfaces of genus

is odd,

Sothe genus of S is

is even.

For

g >

1

it

and two zeros each of

$(n- 1) if n

is

such a surface is called

hyperelliptic.

Every Riemann surface of genus 2 is hyperelliptic.

k be a canonical divisor;

then

deg(k) = 2 and

is linearly equivalent to a positive divisor.

we may therefore assume u

on

§S with

(u)+k

=0.

&

positive. This

u

Indeed,

let

Z(k) = 2, so that k

Without loss of generality

There is a non-constant function

has at most two poles and therefore

has valence 2; valence 1 is impossible since a surface with a function of valence 1 must have genus 0.

Using the results of the last paragraph,

we see that the function field of S has the form where

C(u, v) with

eS

f(u),

f is a polynomial of degree 5 or 6 with no repeated root. For any

g which are

g>

2 there exist connected Riemann

not hyperelliptic;

surfaces

of genus

but I do not know an easy proof of this

fact.

2.

Doubly periodic functions

In this section we consider the complex torus dimension 1, where

A

is a lattice in C;

genus 1, but we shall develop its theory results of §1.

14

Let

AL

A,

C/A

of complex

this is a Riemann surface of directly rather than quote the

be a pair of generators of A, ordered so

that

Im(a, /r,) > 0, and let

Zo be a point of C;

of the parallelogram whose vertices are

ZO au A, and by

I

denote by II the interior

z a Zo te Ao

Z

tf AL tr Ao»

the boundary of II taken anticlockwise, Zim

as in the

N eat A

Cone?

figure.

In what follows we shall always suppose

Z, $0 chosen that none

of the functions we are concerned with has a pole or zero on

from this,

ZO is of no importance.

elliptic function with respect to

f(z)

on

C

A

IT; apart

By a doubly periodic function or we shall mean

a meromorphic

function

such that

f(Zer NW)

7)

for.all-A)

this is the same as saying

ind:

f(z + A,) = i(704 A,) = f(z).

Elliptic functions

are so called because they were first obtained by inverting the integrals

involved in finding the length of an arc of an ellipse.

Theorem 9. zero,

and let

Let

eo aes

f(z) be a doubly periodic function not identically

aA

be its poles and

II, account being taken of multiplicity. Peer on Proof.

= Qa

FP We

sje

Then

i

Since

one £'(z) 0

qa

and

+ Q, mod A.

i, a

In each integral we compare the contributions

Z

m=n

Q, its zeros in

have

ree dz = 27i(n - m),

parallelogram.

Q; Biers

f'(z)/f(z) a ZOD,

dz = 2mi(2Q, - 2P). (10) of opposite sides of the

is doubly periodic,

ieit, ta,

we have

FZ)

i

owen) ale

£'(z+A,)

“Say” H@FX,y

}dz = 0;

adding the similar equation for the other two sides and comparing with

the first equation (10), we obtain 7 aler

Que a

2a zil(z) f(z) era

m=n.

ZN

Olea I +a ta,

Similarly zf'(z) f(z) uz

Zita, -r, £"(z) i, Sa CZ

= for some integer

n,-

Zot, [-A, log fe = 2min,>,

Adding the similar equation for the other two sides

and comparing with the second equation (10), we obtain A) a FID

v

U

=i

Nar tn PN

1

WD

= (0)seavoyel AN,

This completes the proof of the Theorem. be stated and proved as Theorem

theorem for

C/A,

10;

It has a converse,

since the differentials of the first kind on

respond to the constant multiples of

We cannot have the one zero of f(z)

m=n=1

which will

the two together are just Abel's

C/A

cor-

dz.

in Theorem

9, since the one pole and

in II would have to coincide;

so the simplest non-

constant doubly periodic function that we can hope to construct will have

double poles at the points of A such a function;

then

and no other singularities.

f(z) - f(-z)

Let

f(z) be

is doubly periodic with at most simple

poles at the points of A, and therefore must be constant - and since it is

an odd function of z this means it must be zero, by considering

f(z)

af +b

f(z);

f(z)

is even,

and

instead of f we can normalize it so that

zo + O(z?) near the origin.

to construct

Thus

This is in fact enough to show us how

but there is a slight difficulty over convergence which

leads us to construct its derivative first.

This satisfies

f'(z)= -2z,°+0(z);

so write

P1(z) = @'Z; A) = @'(z; 2, A,) = -22(z +d) where the sum is over all

convergence;

X in A.

Here there are no problems

(11) about

indeed for this and the sums that occur in (12) and (14) one

can easily prove the following.

16

*

Let

U

be any compact subset of C

and

delete from the sum the finitely many terms that have a pole in

the sum that is left is uniformly absolutely convergent in U.

U;

then

This result

is enough to justify all the formal manipulations that follow. Clearly

@'(z)

is odd and doubly periodic.

single-valued because

@'(z)

has residue 0 at every pole,

@(z)=2°° + i {ue 20 dus where the prime denotes that the sum

Clearly

@(z)

is even.

proceed as follows.

because its derivative vanishes;

So

is

is over all non-zero

@(z)

2X in A,

is doubly periodic

but it is tedious to justify this;

The function

g(z) = @(z + A,) - ®(z) and

@(z + r) = @(z), and similarly

which is

oh tig) ~=a-*| 212)

One way to prove that

is by rearranging the series,

Its integral,

g(-2,)

= 0

is constant

because

@(z + A.) = @(z).

instead we

@(z)

is even.

This function

@(z), which is the simplest possible doubly periodic function,

is called

the Weierstrass ®-function. Now

@'2 - 4@?

is an even doubly periodic function which has at

most poles of order 2 at the points of

tracting a suitable multiple of ®

function

f(z) say;

and

A

and no other singularities.

we obtain a holomorphic

Sub-

doubly periodic

f(z) - f{(0) must vanish identically by Theorem

9.

So there is an equation of the form

@'* — 4@° - g,° rae where

g, and g,

depend only on

corresponding to @(z)

(13) A.

Moreover

the function on

C/A

has valence 2, so that as in the proof of Theorem

4 the field of doubly periodic functions is at most a quadratic extension

of C(@).

Hence this field is precisely

C(®, ®') with

®

and

®'

alge-

braically related by (13). Since integral which

@(z)

has residue 0 at every pole,

it has a single-valued

we write after a change of sign as the Weierstrass

zeta-

function

(lee

E {®(u) - uw? }du=z

+2" {(2t+n)*-a7* +2077}; (14)

here as in (12) the prime denotes that the sum is over all non-zero A in A. The function €(z + A) - €(z) is constant for any fixed » because its

ike

derivative vanishes;

should vanish.

but there is no longer any reason why this constant

Define

CZ)

Nae,

by

eizy= 40, 1?

SZ

r,) —{(2) = 15;

(15)

a trivial induction argument shows that

Nest + A

CZ

for any integers

n,

and

the proof of Theorem

ie

ae) = te

n,-

Moreover

+

atl

an argument similar to that in

9 shows that

Jpb(z)dz = 211 = ALN, - ASN)

(16)

the right-hand equality here is known as Legendre's The singularities

of

¢(z)

point of A; sothe integral of at each of these points.

relation.

are simple poles of residue 1 at each

€(z) will have a logarithmic branch point

We obtain a single-valued function by taking the

exponential of this integral;

the result is the Weierstrass

sigma-function

o(z) = exp {log z + f7(€(u) - wu )du} = zi" pee exp 0

Xr

Xr

9?

where the product is taken over all non-zero

2X in A.

an odd holomorphic

are simple ones at the

points of A.

function whose only zeros

Clearly

2 ee

v=1,

ing that

By

2

o(z)

and some

constants

(17) B.

Setting

is odd and does not vanish at

z=

=2y

and remember-

2X, we obtain the value of

substituting this back into (17) we obtain

o(z + r,) = -o(z)exp { n (2 + 2A,,)} for

is

Integrating (15) and exponentiating gives

o(z + r,) = B,a(z)exp(zn,,) for

o(z)

v=1,

2.

(18)

From this we can obtain the general functional equation

o(z + 2) = (-1)"o(z)exp {m(z + 3a)] where

18

A= nr,

fe nr,

7

F 1 eee

nn tn +n,

and

n

n

ie)

are integers.

In many ways the sigma-function is the most important of the Weierstrass functions, the extra complications of its functional equation being counter-balanced by the simplicity of its divisor.

This is

illustrated by the proof that follows.

Theorem 10. be points of Suppose

For some

II, not necessarily

n>

1 let Pisses

Pe and Q.; rey Q,

all distinct but such that no

P. ai wens te PL = Q, t--.

and whose zeros

Proof. and

in

II

are just the

Choose points

R.> Ee;

II are just the

Q

R, such that R, = Pe mod A

2R, — 2Q 05 then the function

sa mecte

o(z-Q, ) oo

o(z-Q )

Fal; e-B)

has the correct poles and zeros,

C2) and it is doubly periodic by (18).

An important generalization of the sigma-functions theta-functions;

these are the holomorphic functions

any

A in A

say

F(z, ), such that

there corresponds

O(z. + 2) = O(z)exp-F(z,.

an inhomogeneous

x

and

zeros in II with the correct multiplicities;

W(z) = O(z)/lo(z- Q,)... oz-Q

Lemmal1l.

where

Q(z)

such that to

linear function of z,

(20)

is a theta-function not identically zero,

is a trivial theta-function,

6(z)

is given by

2).

Clearly it is enough to check this for

@(z)

Q,

+Q, mod A.

Then there is a doubly periodic function whose poles in Re

Pi, isa

also that

n

d,-

Now assume that

and let Q,> are;

Qe be its

thus

)}

that is, a theta-function with no zeros.

Any trivial theta-function has the form

is a quadratic polynomial;

conversely any such

exp Q(z)

exp Q(z)

is a trivial theta-function.

Ihe)

Proof. log 6(z)

Let

be a trivial theta-function;

6(z)

then any branch of

is holomorphic in the z-plane and satisfies

log 0(z + 2,) - log 6(z) = F(z, 2,) = O(1 + |z|) for

2.

v=1,

II and it takes only

to gofrom any

=.)

or

=n

steps each

|z|)

O(F=+

is bounded on

log 6(z)

Since

(21)

z toapoint of II,

it follows from (21) that

log 6(z) = O(1 + |z|*); and this implies that

log 6(z)

is a quadratic polynomial.

The converse

is trivial. Lemma

11 and the argument above it shows that any theta-function

can be simply expressed in terms

ways

of writing them down.

zeros of

not.

Let

6(z)

of sigma-functions;

but there are other

First of all, we consider the number

in Il - the zeros are periodic even though

F(z, r,) =a

z+ b, LOrs) =

6(z)

of

itzelf is

eta then

6'(z + A,)/8(z + A») = 6'(z) /6(z) + a,

and so the number of zeros of 6(z) ae

; OZ)

2mi “I

O(z)

A.a enti =

in II is

-A_a

Dial

271

by an argument like that in the proof of Theorem

4, - roa, = 27min for some integer which is the number of zeros of

6(z)

9.

In particular

n= 0

(22)

in II.

We now show how to construct all theta-functions with given

b, satisfying (22).

It is convenient to construct not

6(z)

a,

itself but

1 Soe : -1 w(z) = 6(z)exp{ 24,2, z(z r,) br, 2 } where the multiplier has been chosen to ensure

yz + A,) = ¥@). 20

(23)

The other functional equation for

yw has the form

w(z + A,) = ¥(z)exp(az + b) for some important.

a, b; here

a= -2mind,

1

(24) by (22), and the value of b is not

By (23) we can write 2

ViZ=an),

cy, exp(27ivz/X,)

(25)

-%O

for some constants

C3

and (24) now becomes

Chin= Sy exp {2mivr, /r, -b}. A simple induction argument shows that

Cham = Cp exp {tim(m-1)na, /A, + 2mivma, /r, - bm} for every integer

(provided

n>

Hence we can choose

0) and the other

Im(Q, /,) > 0 the series

m.

c), are uniquely

C, die away rapidly as

€

arbitrarily

open iors Since

|v| > oo: in particular the

(25) is absolutely convergent. It follows that for

n>

0 the theta-functions with given

form an n-dimensional C-vector

of Theorem tions.

Cor seen

space.

a» b,

We could use this to give a proof

10 which does not involve constructing the Weierstrass

Again,

we can express

o(z)

multiplication by a suitable constant.

as an infinite sum

Unfortunately

func-

in this way,

up to

the constant involved

can itself only be written as an infinite sum or product,

and it is not

possible to give a concise account of the calculations involved.

‘Further

information may be found in Weber [13] or Tannery and Molk [12]. 3,

Functions of several complex variables Much of the theory of functions of several complex variables is a

routine generalization of the corresponding definitions and results in the

theory of functions of one complex variable.

open subset of co morphic in U

point

then a function

f(z,,

For example,

be an

let U

atone z) is said to be holo-

if it is differentiable at each point of U

(o> bore. Cy of U there correspond constants

- that is, to any

B,, ...,

B

n

such that

Zl

(Ze, vey ZV t(Oey iene Gite If we write

Dae

function of the Riemann

2n

oF ly

this is equivalent to

real variables

ee

meme

f being a differentiable

and satisfying the

n

Cauchy-

equations

of/dy, = i of/ ox, It follows that power

(ie Geeto

(Vie

ees

f can be expanded about

ea):

(o> noe

on as a multiple

series S

co

m

f(ziye pez 5) =e oe) a(n emenee are)Um) Tee (ee)

m

a

0

absolutely convergent in some neighbourhood of over the coefficients of this power

(o> Byeusns

oh

More-

series are given by a multiple

Cauchy formula

f(z Prey z)dz.. Bes dz 1

(277i) where

1

Sp

n (z,-6

ye

aeae

i*. is a closed contour in the Zz,-plane surrounding

W 2 D, Neo

boundary is gence;

n Sp

dD, where

D.

ae

o

and

D,, is the closed disc in the Z)-plane whose

However there is no analogue of a circle of conver-

indeed the domain of absolute convergence

of a multiple power

series can be quite complicated.

Let

U

bea connected open subset of Cc".

A function

g(z,, siete

is said to be meromorphic on

U

of U

and if for each point

P

of U there exist a neighbourhood

of P

and two functions

and

a

eel

at every point of N(P)

both meromorphic

on

U

z.)

if it is defined on a dense open subset f . holomorphic in N(P) at which

g is defined.

N(P)

such that Two

functions

are identified if they are equal at every point at

which they are both defined.

By abuse of language one can speak of

having the value infinity at certain points, the situation for meromorphic

¢g

but even then (in contrast with

functions of one complex variable)

have no definable value at certain points - consider for example

g may

g = Zz,/%,

at the origin.

In the theory of functions of one complex variable,

each zero of a

holomorphic function (and each zero or pole of a meromorphic function)

22

is isolated and can be considered separately. which vanish at a given point functions holomorphic

at

P

Moreover,

the functions

form a principal ideal in the ring of

P, and that ring is a local ring.

So one can

speak of a function having a zero of given order at P, and that one integer gives the most important information about the behaviour at

P.

With functions of

plicated,

n>

1

complex variables

of the function

matters

are more

com-

primarily because the set of zeros of a holomorphic function

will have complex dimension

n - 1, at least in an intuitive sense.

a set does not naturally break up into local pieces;

give any but local descriptions,

Such

yet we cannot easily

and indeed the only easy description of

the set of zeros of f will be by means

of f itself.

thinks of the divisor of a meromorphic

function as 'zeros minus poles’,

Intuitively one still

and a general divisor as an element of the free group generated by

‘respectable’ point sets of complex dimension necessary to proceed rather differently. divisors on

c”. the definitions

of dimension Let let

i

n by means {u,}

that the system

can be transferred to complex manifolds

of local coordinates,

function on

{Uys ie

1; but formally it is

For convenience we consider

be a covering of c

be a meromorphic

n-

much as in $1.

by open sets, Cy

andfor each

not identically zero.

@

We say

is a description of a divisor if the following

consistency condition holds:

if ue n Uz

is not empty then

holomorphic functions on

In such a case,

of course,

a

ae

and

La

NU B"

the holomorphic functions will be inverses of

each other and will therefore have no zeros in Uy q Us this means

that

oe

So intuitively

and fg have the same poles and zeros in Uy nN Ue

To any refinement of the covering

Lv

} there corresponds

refinement of the description of a divisor. have a common

are equal to

refinement,

Since any two open

an obvious

coverings

in defining two descriptions of divisors as

equivalent we can confine ourselves to the case when they are both based Uo fi We shall say that the two descriptions on the same covering.

and

uO

ea

are equivalent if and only if f,/g,

equal to holomorphic functions on

UF

for each

a.

and Ey/tg are An alternative

which works even if the two descriptions are not based on the open covering, is to say that two descriptions are equivalent if

definition, same

22

their union is a description.

Now a divisor on

of ,descriptions of divisors.

phic function

f on

C”

c"

is an equivalence class

The divisor of a holomorphic

is the equivalence class of the description

A divisor is said to be positive if it has a description that

or meromor-

ie is holomorphic

in

Oe

for each

a;

if so,

LU

ifi

ico £)3 such

every description of

it has this property. It is now a straightforward matter to develop the elementary

theory of divisors and to show that they have all the properties which one expects by analogy with divisors on Riemann surfaces or in algebraic geometry - with two closely connected exceptions.

It is not obvious how

to prove that every divisor is the difference of two positive divisors, how to prove that two positive divisors have a highest common For both these proofs,

we need the Weierstrass

which tells us how far we can normalize

neighbourhood of a point

P

nor

factor.

Preparation Theorem,

a holomorphic

function in a

by multiplying it by a function which is holo-

morphic and non-zero at P.

To simplify the notation,

we take

P

to be

the origin.

Theorem 12

(Weierstrass).

in a neighbourhood of the origin,

Be

het

Be

0 and

and has a zero of order g(z,,

ron

in Zs

a2,

f(z, Cee Moreover

g=f

Iz, | =c zeros

24

9(z,) does not vanish identically

m= 0

at z, = 0.

Then there exist a function

and non-zero at the origin and a polynomial

Spee

z) =n z)

aay

a,(z., eee

are holomorphic

z) es g(z., Ree

We canassume that

g and

m>

is both forced and acceptable.

inthe Z,-plane, where

of 9(z,) inside or on

Zz)t... ta

at the origin,

z_ plz, ; Zap veey

these conditions determine

Proof. p=1,

suppose that

m,

p(Z, ; Zap veey where the

Zz) be holomorphic

9(z, ) be the restriction of f to

Zz) holomorphic

of degree

Let f(z, , aes

let

c>

(Z,,.. »Z,) |

such that

Zz).

p uniquely for given

0 since if m= 0 Let

I

f.

the choice

denote the circle

0 is chosen so that there are no

I’ except for the m-fold zero at the origin;

and let

N

(z.,. Ste

be a connected open neighbourhood of the origin in the

z_)-space such that f has no zeroon

IX N.

For any

(z,, +++, 2) in N, I(z., stash 2 is the number

ay pf

-1 of

wz,

of zeros of f inside

function of Z, alone. valued;

ey.

IT, where

So this is equal to

m

at the origin and is integer-

on the other hand it is continuous in

all (Zz, ayer

z) in N.

Sponge

these being complex numbers

For any integer

r=0

Cte

Denote the

m

f is considered as a

N.

So itis equalto

corresponding zeros of f by depending on

holomorphic

p

and

functions of the

roots are the

the z,-plane.

Zip ceey

hence so are the ele-

Cu _Now let p(z,) be the poly-

oy and let

D

denote the interior of

By writing it as an integral over

IT we see that

in D X N, and by construction it is non-zero

g=f/p

oy rai st

p must vanish at

some suitable neighbourhood of the origin,

oo

IT in

f/p

in DX

is

N;

so

Moreover,

have the properties stated inthe Theorem.

any candidate for

Ze

Z

oz

and this is clearly a holomorphic function in N;

mentary symmetric

for

we have

J C= Tal Ip

nomial whose

m

as long as these lie in

and this is enough to prove

uniqueness.

Corollary. hood of the origin.

and two functions

Let

u

u, v holomorphic in N

¢ holomorphic

U of N

f= u, iS

and such that

in U

of the origin

and such that f= u/v in N,

For any open subset

eA holomorphic in U

there is a function

N

Then there is an open neighbourhood

with the following property. functions

z) be meromorphic in a neighbour-

f(z,» eae

t=

such that

and any

Le,

gu and

v=

in U. Intuitively,

this says that the fraction

sible terms everywhere in N.

Of course,

u

u/v=f and

v

is in lowest posare not uniquely

determined.

25

ov

Proof.

We may assume that

otherwise we can take

u=0

phic functions in some

neighbourhood of the origin such that

Neither

u, nor

Z pote

z

Ail

WW

v, vanishes,

if necessary,

n

WO BZ

HBooo

both u, andv,

(Zz.

oe

v

= ZS

D

N.; the roots of u,

0 donot vanish.

f= u,/V,.

Applying Theorem

is a disc in the z,-plane and

and

be holomor-

we can assume that the restrictions of u,

and non-zero in N

K

and

for

so after a linear transformation on

and

which are polynomials in Zs

Let

Let u,

z_)-space) and a decomposition

holomorphic

in N

v=1.

we find that there is a neighbourhood

the origin (where

the

and

f does not vanish identically,

N,

N = D X N,

of

is an open set in

f = gu,/V,

UV,

12 to

where

g is

are holomorphic functions

Moreover,

for

(z.,, atenes Z)

es regarded as polynomials in Z,

be the field of functions of Zax ceey

in

all lie in D.

Ze meromorphic

in

N5 then u, and ve regarded as elements of K[z, | have a highest common factor efficient 1.

p(z,)

For

and we can require

p(z,) to have leading co-

(z., ens z.) in N, the roots of p(z,)

and so the coefficients of p(z,) are bqunded; they must be holomorphic

in N,.

being also meromorphic

Thus we can divide

u, and

without altering the holomorphicity of their coefficients;

we can without loss of generality take u,

elements of K[z, |. such that

and

We now set u= gu, and

be the resultant of u, and

Vee then provided

SILOM

v, by p

in other words

ve to be coprime as

v= Me

Let Riz., aces z)

(z.,. Passes z) in N, is

R # 0, the zeros of u, and v, are respectively the zeros and

poles of f regarded as a function of Z, in D. position

all lie in D

f= u/v

does not depend on the choice of the original decompo-

te u, Ne; and u, oe

v5 /v is holomorphic in N.

Suppose we repeat the process

coordinates,

obtaining this time

last sentence, the origin,

It follows that the decom-

both

u/u, = VAN

after a linear transformation

f = u,/V,3

and

by the final remark

uu

of

in the

2.) /v are holomorphic at

so the only effect is possibly to decrease

N

and to multiply

both numerator and denominator by some invertible holomorphic function.

Now let f= us ive in U decomposition

f = u/v

andlet

P

be any point of U.

we may suppose the coordinates so chosen that satisfy the conditions of Theorem

26

Since the

is essentially independent of choice of coordinates,

12 at P.

u, v, uy and

fa all

Now the polynomial parts of

u

and

v

at P

are coprime,

since they are factors of u,

pectively;

and so the canonical decomposition

from

by multiplying numerator

u/v

morphic

and non-zero

of this proof, point of U

at

uw nu

P.

Ms /v

of f at P

and v, resis obtained

and denominator by a function holo-

By the final remark

in the second paragraph

is holomorphic at P;

and since

P

is any

this completes the proof of the Corollary.

Theorem

13.

Any divisor on a complex n-dimensional manifold

can be written as the difference of two positive divisors. Proof.

Suppose that

To each point

P

of each

of P

contained in Lb

Nop

such that

Corollary in divisor,

and

Neg

Ly,

1

and functions

f jae? WV yp

Nop:

Clearly

aP’ where

Up and

Vop

Nop

holomorphic

in

satisfies the conditions of the last iN yp:

so to prove the Theorem

{N

i } is a description of the divisor.

we can construct a neighbourhood

i

is a description of our original

it is enough to show that

iN op

Up }

} are descriptions of divisors. Suppose Nop meets ‘oP In the intersame. the be may Q and P a and 6 or

section we have

UEP’ ap a

§p ~ "8Q’ "QO

and hence by the last Corollary and invertible in Nop A No:

Uyp MaQ =D

/VeQ

is holomorphic

This is just the consistency condition we

need, A similar argument proves that two positive divisors have a highest common

factor;

but we shall not need this result in what follows.

It is convenient to include one more Lemma

Even

in this section.

if we are initially given a covering with lots of good properties, these are likely to be lost during the processes described above; so it is

desirable to know that we can refine any covering to a 'good' covering. Lemma

14 below is stated for a complex torus

c"/A,

where

A

isa

because this is the case which we shall need; there is an analogous result for Cc” with ‘finite’ replaced by ‘locally finite’ which is We shall say that a set U in c"/A is an open proved inthe same way. lattice in

a

hypersphere if its inverse image in c"

is the disjoint union of an open

27

hypersphere

Iz, - A at

F Iz.

Ch

and its translations by elements of A.

r In other words,

it is the image of

an open hypersphere in Cc” which is small enough for its translations not to overlap itself, Lemma cl /A;

then it can Proof.

taining this c"/A

14.

P

Uo:

Let

Up

be a covering of the complex torus

be refined to a finite covering by open hyperspheres.

To each point

P

of cUy/A

we can choose a

and an open hypersphere containing

P

Uy

con-

which is contained in

The set of all these open hyperspheres forms a covering of

which refines the original covering;

and because

cA

is com-

pact we can select from this a finite covering.

It is essential for the application that the new covering should be

locally finite, that the open sets involved and all their intersections should be 'nice’ sets, and that their inverse images should be disjoint unions of open balls.

That the sets

are open hyperspheres

is just a

device to simplify the construction of partitions of unity' in §4.

28

ll- Theta functions and Riemann forms

This Chapter is mainly concerned with the question:

complex torus

T = cA

when does a

have non-constant meromorphic functions on

it, and how does one construct them when they exist?

It is not possible

to mimic the method used in §2 to construct the Weierstrass ®-function, essentially because when

n>

1 the set of poles of a multiply periodic

meromorphic

C”

does not break up naturally as a union of

function on

local objects;

so a series expansion analogous to (11) or (12) can no

longer be expected to converge,

and indeed in general any neighbourhood

of a given point will contain poles of infinitely many terms.

Moreover,

it is not likely that there will even be a description of the set of poles simpler than that given by the meromorphic

function itself.

We therefore

fall back on the other method outlined in §2, by showing that any multiply periodic function can be written as the quotient of two theta-functions

by then constructing all theta-functions for The first part of this program case

n=1

and

A.

is carried through in $4,

we could give a very simple proof of this result,

In the

because we

had a good description of the possible divisors of multiply periodic functions and because we had a large enough supply of theta-functions;

so

from our stock of theta-functions we could construct a meromorphic thetafunction with the same poles and zeros as our given doubly periodic function. For

This reduced the problem to the study of trivial theta-functions. n>

1 this method does not work;

the given multiply

instead we have to construct from

The

periodic function the theta-functions that we need.

ideas involved are really those of Hodge theory and of cohomology, but because one has so explicit a description of T the proof can be made to appear totally elementary;

but this does make the motives for some of the

steps obscure.

The second part of the program is complicated by the fact that

A

only admits non-trivial theta-functions if it satisfies a further condition;

29

this is the existence of a non-trivial Riemann form which is at least positive semi-definite.

(This condition played no part in 82 because it

is automatically satisfied when

n= 1.) In $5 we prove that this condition

is necessary;

that a Riemann form for

in 86 we assume

construct all theta-functions associated with it. tain other

related results;

in particular

function field of an abelian manifold,

A

exists and

Both these sections con-

§6 contains a discussion of the

and the theorem of the projective

embedding of an abelian manifold. In Chapters II and III it will be convenient to denote by V particular space

Cc” inwhich

lying real space

Row

be denoted by

T;

A

The complex torus

or sometimes

V/A = c"/A

but it will be denoted by A

abelian manifold - that is, if A 4.

is embedded,

the

the under-

will in general

if it is known

to be an

admits a positive definite Riemann form.

Reduction to theta functions

In most of this section,

space spanned by the lattice

we shall consider

A, and

otherwise stated functions on

V

or

T= T

V/A

= x ah ly

as a real vector

asareal torus;

and unless

will merely be complex-valued

infinitely differentiable functions of the vos where

V

2n

real local variables

x, and

What forces us to this is the fact that Lemma

16 below has no holomorphic analogue,

It turns out also to be convenient

to make the complex change of variables from

Xx

V, to

Zi, 23

the fact ~

that the new variables are complex-valued does not invalidate any of the standard formalism. domain

U

Moreover,

a function

f is holomorphic

if and only if it satisfies the Cauchy-Riemann

and in terms

in an open

equations in

U;

of the new variables these take the convenient form

af/ dz, = OeefOraV\=5) 0 2skee

vans

In view of this, one says that a differential r-form is holomorphic if it can be written

A

ie

2 L Lay trers creme i ce Lee ‘ii 1

with all the

30

f, i Wee ote

;

Be

mea

acaa

holomorphic.

oe

Again,

an (r+s)-form is said to

be of type

2

(r, s) if it can be written

“a « reeset

DREAM

duane de 2

«jez, ossig7 025 “dz, A... Adz,

Tea

Sie.

:

1

where now there is no restriction on the functions involved,

type

(r, s) then

of type

(r, s+l).

it is of type If A

n=1.

is the sum of a form of type

Moreover

(r, 0) and

dw

w

is of type

is of

is holomorphic

if and only if

(r+l, 0).

is the obvious generalization of that given in §2 for the case

Itisafunction

r in A

an r-form

If w

(r+l, s) andaform

isa lattice in C” then the definition of a theta-function with

respect to A

z

dw

s

0@(z)

there corresponds

holomorphic

aform

in

Cc" and such that to each

F(z, A) inhomogeneous C-linear in

and satisfying

6(z + A) = O(z)exp F(z, A). The divisor of

@ is positive and periodic;

function if its divisor is null - that is, if

case

log 6(z)

Lemma where

is holomorphic,

Q(z)

w

on U

6(z)

is never

zero,

is inhomogeneous quadratic in Z,,...,

a l-form

such that Proof.

(Poincaré).

15 on

such that

U

In this

and an argument like that in the proof of

11 shows that the trivial theta-functions are just the

Lemma and

@ is called a trivial theta-

Let

U

dw=0.

exp Q(z)

Z Z,.

be a convex open subset of c" Then there is a function

f

df= w. For some fixed point

O

in U

write

f(P)=

Ge where

the integral is taken along any polygonal arc lying entirely in U; then to prove the Lemma we have only to show that f is well-defined - that is, independent of the path of integration.

that

For this it is enough to show

Jw = 0 when the integral is taken over any closed polygon,

and an

obvious dissection reduces this to the case of the boundary of a triangle A.

But by Stokes' theorem ee

Jf de = 0.

The result holds if U

is any open ball, but the proof in this more general

Bi

case involves topological complications.

Lemma 16, hyperspheres subscripts Oe on

Let

and let

r= 0

i, j such that

U; NU.;

suchithat)U. ae

a

YL

1 U, 1 U j

1)

whenever

Suppose that for each pair of

U; q uF is not empty we are given an r-form

k

OM.

1

Oi

ees

i, j, k

is not empty these forms satisfy

+ oe + Os = 0 in

(..=@,6¢

U; n WF nU,..

w, on U; for each W, mig,

Pal

(26)

i such that (27)

J

U; nN uF is not empty.

Proof. {U,,

be an integer.

and suppose that for each triple of subscripts

Then there is an r-form

We now

iU,, ++, Uy } be a covering of T by open

From (26) we deduce first

a = 0 and then

Or = ere

construct a partition of unity corresponding to the covering

lene Uy! - that is, a set of infinitely differentiable functions

fi Ristes fy on

(i)

T

(ii)

For this purpose,

gpot

such that

f =0 in U. and f =0 outside U_ for each v v v v f. ae eels fy = 1 at each point of T. associate with

fy LP 2

at See)

2

for some constant

c;

2

2

< © Gat Se

?, on

T

is the set of points satisfying

2

that this is possible follows at once from the defi-

nition of an open hypersphere in function

UF real local coordinates

such that U,

ctoaeueve aah;

v, and

T

given at the end of §3.

Define the

by

$,=exp{-1e°- -ni-...- 8 -92)) in Uy, and

oS

0 outside

U,.

differentiable everywhere

f = ¢,/¢ then the

82

Clearly on

T.

$7

0 in U, and

Now write

“ite ale + ow) LOU RU 31

ty have the required properties.

ace Let

NS

oF is infinitely

w;—

fo) eel fe e e

i in

fy? wn

U;

where any term which is undefined at a point is to be taken to be zero at that point.

It is easy to check that

w, is an r-form (that is to say, the

functions that occur in it are infinitely differentiable) and (27) holds.

Lemmal7. let w*

Let

w

bea 2-formon

be the induced form on

nates on

V.

Thenthere

on V, andal-form such that

and let

isal-form

¢= ec es,

suchthat

Se -

dw=0,

be real coordi-

Came ensZ1) T, with inducedform

wy on

on V, where the

Ci

y*

are constants,

w* = dy* + dd.

Proof. coordinates,

that

is not affected by a change of

A

consists of all points with integer

We have

2

_

where the

The truth of the Lemma so we may assume

coordinates.

w

V

T=V/A

“A oe

. are periodic and infinitely differentiable;

so each of them

can be expanded as a multiple Fourier series

+ ,(m) exp{2mi(m,é, : f=— 2a, +... +m, £,,)} where the sum is over all integer vectors factors are not altered by differentiation, tials cannot interact;

m.

Since the exponential

terms with different exponen-

so for each integer vector

d[multiple of exp{2miZm,é}

(28)

m

we have

in w*] = 0.

To prove the Lemma we must therefore prove first that it holds for the individual components

w* =

of

w*, that is in the special case

(29)

{exp(27izm_£ ) Jzza, dé, “A des,

and second that in the general case the resulting formal Fourier series

for

w*

is a 1-form - that is, the functions that occur in it are infinitely

differentiable. In the case

m=0

henceforth we may assume

wecantake

m #0

y*=0

and

¢= aE

so

ta

and so without loss of generality

33

m, #0.

Moreover

in (29) we may assume that

=a = 0 unléss.

i
LA, oT .

-l

nt+v

if itholdsfor

n

by A Ae = -E( n+v’ L(A, by (43

the arguments at the beginning of

of these cases have been dealt with already.

LH, v between 1 and

+ m2) }+

A,ap

The functional equation (42) with .

.

:

now takes the form

A(z +d.) = O(zexp {2mi[d 2, + JA, .))]) which is equivalent to the set of equations

C( em:

aera

1

Vv

- d,, +++, Vv

m,)/e(m,,

= exp {27i[> m buaL

Tia

0x m, , V/A,

A, CA,

The identity map on

V

is an isogeny of degree

m,

and

where the underlying map

and

[A ave

A] =m,

whence

induces

V/mA, > V/mA , =e V/A, oe V/A, where the two outer arrows isogeny

%, : tT, > Ty

are

Ue

and the inner arrow defines an

by considering the composition of two successive

arrows we obtain the results required.

For the converse,

we note that

m6

so the equation ve ° o = m6

shows that

kernel.

similarly,

Using

%, ° ee = m6

each onto and have finite kernel,

is onto and has finite kernel; eS is onto and

we find that

pe has finite

g. and

~

are

and this is enough to show that they are

both isogenies. We shall say that geny

Tt = T..

A is isogenous to

By Lemma

33 this relation is symmetric,

easily that it is an equivalence relation. abelian manifold and manifold.

now if H

a

DS if there exists an iso-

Now suppose that

is isogenous to

T,3 then 1M

Co = Ve

and T, = V/A,

For we can take

and it follows iy

is also an abelian with

is a positive definite Hermitian Riemann form on

respect to A.

is an

V

A, =) Ay

with

it is one also with respect to A.

There is an alternative way of defining an isogeny.

Denote by

55

Hom(T, , a) and by

the additive group of holomorphic homomorphisms

End(T)

the ring of holomorphic

Hom (T,; Ts) = Hom(T, , It follows from Lemma

33 that

endomorphisms

of

T) SQ, End (T) = End)

@ in

Hom(T,,

T)

case the two inverses are the same.

Inparticular

geny if and only if it is invertible in End |(T).

and write

&Q.

) and in this

¢:T-~T

is an iso-

It will follow from Theorem

is an abelian manifold this condition is equivalent to

being a divisor of zero;

but if T

(47)

is an isogeny if and

only if it has both a left and a right inverse in Hom |(T,,,

34 that if T

T;

T, => T,

@ not

is merely a complex torus this need not

be so. One major reason for the importance of isogenies is Poincaré's

Complete Reducibility Theorem for abelian manifolds, which are stated below as Theorem

various forms of

34 and its Corollaries;

this too does

not necessarily hold for complex tori.

Theorem 34. and let A, CA,

Let

be the image of @.

3

XB

A

1

zontal arrows

A, is an abelian manifold

A

3

XB

3

2

diagram as shown,

in which

shall extend this

For

v=1,

2 write

notation to The map

which we also denote by

¢.

v= 3

A, = Va

and

are

@¢ extends to a C-linear

Now

A, = ove /OM,

pA,

spans

is a complex torus.

Hermitian Riemann form on

Ve

dim ves = ly5 We

as soon as we have shown that

ov,

and it is discrete because it is a subgroup of A,

map

A,

Ve —- ve

as an R-vector

space,

so it is a lattice in

If H,

is a positive definite

with respect to A.) its restriction to

ve has the same properties with respect to A;

56

B,

the vertical arrows are isogenies and the lower hori-

is a complex torus.

and

B, and

are the obvious maps.

Proof.

ov,

A,)

es

and there is a commutative

abelian manifolds,

be an element of Hom(A,, Then

d

a

A.

¢: A, age

so

A, is an abelian

manifold. Now choose a positive definite Hermitian Riemann form

Ys with respect to let wv

H,.

AL

let

We

be the kernel of @ acting on

be the orthogonal complement of We

Since

A, n Ww,

A, and pA,

have ranks

must have rank

en, - 2n,

since it is discrete. plement of we

on

E,; the alternating form associated with

which spans

Ww,

as an R-vector

fore a lattice in W.projection

We q AL

Let

B,

space,

We q A, which has rank

so

A

spans

be the eer

ioe

Qo: he >: ee

identified by means

is now clear that Finally, for

WA,

ae then the

and we have

with kernel

Ve ae

and

Woe

so they can be

of a suitable isomorphism between

WwW, and Vee

It

A, => A, x B, is onto, and therefore it is an isogeny.

the proof that

B, is an abelian manifold is analogous to that

A,; it suffices to consider the restriction to Ww

multiple of

and is there-

Certainly

the projections

are both epimorphisms

We n AY

Wo

A, =? B,

constructed the left-hand part of our diagram.

Moreover

en,

AL , be the projection of AL

Ve = WwW, induces a homomorphism

dim A, = Chim A, x B,-

E,

since

has rank

én, - én,; on the other hand it contains

TS has rank

2n.,,

Let

oe

of We q AG

that

Since the kernel of this projection is

W,>

with respect to

is the orthogonal com-

we

A, x AD this means

and therefore it is a lattice in

Vis and

én, respectively,

H,, and hence even the orthogonal complement is integer-valued on

on

and is therefore a lattice in We

It is easy to see that

with respect to

in Le

én, and

H,

of a suitable integral

H.

The argument for the right-hand part of the diagram is similar but simpler.

We have identified

Vig = ov,

as a subspace of

vis let We

be the orthogonal complement of Ney in VE with respect to H,. argument that was used above for WW,

shows that

in aN

We, Ws

let

B,

be the complex torus

manifold for the same

reason that

induces a homomorphism

is constructed; being onto,

and

A,

is.

We n A,

N A,

The

is a lattice

which is an abelian

The inclusion map

WwW. SG ve

B, > A,; so the right-hand part of the diagram

dim A, = dim A, x B,» so that the right-hand arrow,

is an isogeny.

This completes the proof of the Theorem.

Bi

Corollary 1.

Any subtorus or quotient torus of an abelian mani-

fold is an abelian manifold.

Proof.

The argument which was used to prove

A, an abelian

manifold works for any subtorus of an abelian manifold.

A-~T

is ahomcmorphism

atorus.

A

is an abelian manifold and

T

The argument used to construct the left-hand side of the diagram

in the Theorem

torus

onto, where

Now suppose that

shows that

T,; thus

T X T,

A

is isogenous to

T X T,

is an abelian manifold and,

Corollary already proved,

sois

A

complex

T.

To state the next Corollary, that an abelian manifold

for some

by the part of the

we need a further definition.

We say

is simple if it is not isogenous to the product

of two non-trivial complex tori (or of two non-trivial abelian manifolds, which comes

as saying that

to the same

A

thing in view of Corollary

has no non-trivial subtorus,

trivial quotient torus. no divisor of zero,

Moreover,

A

for an element of

1).

33 that

A

is the same

is simple if and only if End(A) End(A)

has

is a divisor of zero if and

only if its image has dimension lower than that of A.

from Lemma

This

or that it has no non-

It now follows

is simple if and only if End (A) isa division

algebra.

Corollary 2.

Any abelian manifold is isogenous to a product of

simple abelian manifolds;

and the factors are uniquely determined up to

isogeny.

Proof.

Induction on the dimension

shows that any abelian mani-

fold is isogenous to a product of simple abelian manifolds, only to prove the uniqueness

g: : A, x saci be an isogeny,

where the

After re-ordering the

A

Let

clause.

x 210

a

Be

and B,

are simple abelian manifolds.

B, if necessary,

we can assume that

is the smallest partial product that contains cannot contain

so

B,

58

GA,.

nS B.. for otherwise

@ would be a dimension-reducing

we may assume that

oA,

e

so we have

map.

AW

Now

9(A,

it would contain

B, IAS neeos B. arena Ay A,

and

After further re- ordering

does not contain

B;

and there-

fore its intersection with

B,

is finite because

B,

is Simple.

Thus the

map

: x os yrA,?

x OE

which is the restriction of Again,

the projection of

a

Br

@ followed by projection,

oA,

on

cannot be just the identity element because then in

B, Seto

as Bu so it must be the whole of

The composite map

A, is simple, dimension, So

A, > B,

has finite kernel.

BL has image a subtorus of

is onto,

so it is an isogeny.

A,

B.; this

would be contained

B, since

B,

is simple.

and it has finite kernel because

Thus

A,

and

B, have the same

whence the source and target of y have the same dimension.

wW is an isogeny too, and the Corollary now follows by induction.

This Corollary enables us to describe for any abelian manifolds

A

and

simple and not isogenous then

m, A=A A,

where

First note thatif

Hom (A, B) = 0.

m.,. KEXeAe Es r

Clearly

Hom (A; : A, +6 and this consists of the ing fixed isogenies to A However

End (A).

and

B;

and

B

are

n, n,. Ale XC eA 1 r and some of the

m,;

is the direct sum of the

Hom (A, B)

in the division algebra End (A,).

A

Next suppose that

Bietedy A, are simple and non-isogenous,

may vanish.

n.

and

1

B.

Hom (A, B) and End (A)

n x m,

matrices with elements

The general case now follows by apply-

and a similar treatment works for Hom

there is no such simple decomposition for

or

End.

Corollary 3.

submanifold of A. that

BNC

Let

A

be an abelian manifold and

Then there is an abelian submanifold

Proof.

so

denote the image of B, in A.

BX

C-—A

C >A

Apply the Theorem to the inclusion map

that in the notation of the Theorem

C

BX

isa finite set and the natural map

is anisogeny;

the kernel of this isogeny.

and

A, =A

The map

P

isin

and

an abelian

C of A

such

is an isogeny.

B~ A, so

A, = A, = B;

and let

has finite kernel,

B, +A

BNC

B

only if (P, -P)

is in

This proves the Corollary.

3)

8.

Duality of Abelian manifolds Let

T=V/A

be acomplex torus;

later in this section we shall

need to assume that it is an abelian manifold. divisor on

T

divisors on

corresponds

T

a Riemann form.

We saw in §5 that to any We shall say that two

are algebraically equivalent if the corresponding Riemann

forms are the same;

clearly this is an equivalence relation which is com-

patible with addition and subtraction of divisors, linear equivalence.

and it is weaker than

The terminology here comes

from algebraic geometry,

and the definition is an ad hoc one chosen to simplify the development

the theory;

of

it would be more natural though less convenient to start from

a different definition and prove the present one as a theorem. The underlying idea is to consider two divisors as equivalent if one can be continuously transformed into the other within the set of all divisors. A Riemannform E with respect to A is determined by a matrix with integer elements; so the set of all possible Riemann forms with respect to A

naturally has the discrete topology,

and therefore two

divisors which are equivalent in the sense of this paragraph must have the same Riemann form.

Conversely

if two divisors

linearly equivalent then there are a divisor

tion

f on

T

a te a, by

such that f=

©.

a

aa a,

and

a.

and a meromorphic

func-

and

- a

system as

c

varies,

and the system contains both

This result can be extended to algebraic equivalence by 35 below, at least in the case when T is an abelian

and the general case follows from this by means

23 or the remarks

Lemma

35.

after Theorem

Let

A

Then there is a point

x

of Theorem

28,

be an abelian manifold,

algebraically equivalent to zero,

and

on

a

adivisor

on

A

b a positive non-degenerate divisor

A

suchthat

a

is linearly equivalent

to (b- b.), where the suffix denotes translation by x.

60

are

of Lemma

manifold;

on A.

a,

Thus the divisor

describes a continuous

means

and

is the divisor defined by f=0

{divisor defined by f=c}

aq

a ;

a :

Proof.

By the argument leading up to Lemma

meromorphic theta-function

6(z)

associated with

a

20, there is a

(that is, whose

to V) with a functional equation of the form

divisor is the pull-back of a

6(z + r) = O(z)exp {27if(A)} f:A—>C.

for some homomorphism associated with

Let

8,(2) be any theta-function

b, and suppose it has functional equation

6 (z + 2) = 8, (z)exp {2mi[L(z, A) + J(a)]} where the conventions and notation are those of §5.

Write

W(z) = @, (z)exp {27iR(Z) }/6,(z - x) is C-linear;

R(z)

where

then

~(z)

is a theta-function associated with

(b - b,); and it satisfies the functional equation W(z + A) = W(z)exp {27i[L(x, A) + R()] ie So to prove the Lemma

we need only show that we can choose

x

and

R

so that

£(A) = L(x, A) + R(A)

The argument above Lemma and

H

for all

(48)

A.

20 shows that we can take

is non-singular by hypothesis.

In this case

L(z, w)=-2iH(z, w),

L(x, w)

is aC-

and can be made equal to any assigned C-antion to linear function by suitable choice of x. Again, f is the restricti A of an R-linear function on V, which can be written

antilinear function of w,

f(z) = 4 {f(z) + if(iz) } + 2 1f(z) - if(iz) i second one is here the first term on the right is C-antilinear and the R, and Thus (48) can be satisfied by suitable choice of x and C-linear.

this proves the Lemma. x

is unique,

extension

for

f(A)

It is not claimed (nor even true in general) that is really only determined mod 27i and hence its

f(z) is not unique.

61

For any abelian manifold tively the groups of all divisors,

zero,

A, let

G, G, and

denote respec-

of divisors algebraically equivalent to

and of divisors linearly equivalent to zero;

The group

G,

thus

G5

G, = G).

G/G,, which is called the Néron-Severi group of A, is

naturally isomorphic to the group of those Hermitian Riemann forms which can be written as the difference of two positive semi-definite Riemann forms;

this is a free abelian group on at most

precise number

depending on number-theoretical

group will not be further studied in this book. G,/G).

Toadivisor

a

in

of @(z)

properties

of A.

the This

We now turn to the group

er there corresponds

normalized meromorphic theta-function

né generators,

an essentially unique

6(z), and the functional equation

has the form

6(z + AX) = O(z)exp {27if(A)} where

f: A *R/Z

a character only if a character

on

is ahomomorphism.

A, uniquely determined by 4;

is in G). x

can be obtained from

and

a meromorphic

x(A)

Char A = (R/z)22

is trivial if and

theta-function

and

We have therefore constructed

of commutative groups

G_/G, = Char A. But

is

- this is the step in the argument that would

were merely a complex torus.

a natural isomorphism

x(d) = exp{2m7if(A)}

Conversely the construction of §6 shows that every

therefore from a divisor a fail if A

Thus

(49) is areal torus.

positive non-degenerate divisor;

Moreover,

then by Lemma

let b

bea fixed

35 the map which takes

x in A tothe class of (b- b.) is an epimorphism A > G,/G); and the proof of Lemma 35 implies that the composition of this map with (49) is continuous. It is therefore natural to hope that we can give CharA a complex structure in such a way as to make it an abelian manifold and to make the map A-~Char A just described an isogeny. For this purpose denote by

V->C

V the space of C-antilinear maps

- that is, the space of R-linear maps g such that g(iz) = -ig(z); thus Vv is a C-vector space of dimension n, as V is. Write e (z) = Img(z); then g, is anR-linear map V~>R and

62

g(z) = -g, (iz) a ig, (2). Conversely if g, is any R-linear map this formula defines a C-antilinear map of R-vector

so there is an isomorphism

spaces

V = {R-linear maps Denote by

g;

V7?R

A

V>R}.

the lattice in

to the dual of A;

thus

A

(50)

V which corresponds under this isomorphism

is a lattice in V and consists of those

e (A) C Z.

which takes

g in V to Xr where

X (A) = exp { 2mig, (X) }; and the

Thus we have defined

A.

kernel of this is precisely

V—>Char iN

(50) induces a map

The isomorphism

Vv such that

g in

of

an isomorphism

real tori

V/A = Char A.

(51) V/A =A , which is at any rate a com-

Henceforth we shall write V is a C-vector

plex torus since

positive non-degenerate divisor on

space.

A;

b, as above,

Let

be a fixed

and write

¢:A> A

(52)

A> G,/G, > Char A >A, where the first arrow takes x in A tothe class of (b- b.) and the other two arrows are the isomorphisms (49) and (51). From now on, ¢ will always denote this map;

for the map

depends on

b

but it is not necessary to make this dependence clear in

the notation since we shall only consider one

Theorem

Proof.

Let

and

H

be the Hermitian and alternating Riemann

E

an isogeny,

@ is onto,

and since

(50) involves

This makes the map

and shows that it is C-linear; know that

and (48) then requires

The isomorphism

g(z) = H(x, z).

and

A

35 we

In the notation of the proof of Lemma

b .

L(z, w) = -ZiH(z, w);

f(z) = E(x, z).

finally

¢ is an

With the notation and hypotheses above,

36.

forms associated with

and

ata time.

A is an abelian manifold.

isogeny and

can take

b

A

V-—V

R(z) = ZiH(z, x)

ee

f and so

underlying

a priori it was only R-linear. and

We already

A have the same dimension;

is an abelian manifold so is A.

¢ explicit so

¢ is

This proves

63

the Theorem. H

or

It also shows that

@ really depends not on

b

E - that is, on the algebraic equivalence class of b.

reason,

it is often written

but only on For this

Pry in the literature.

We shall call A the dual of A.

To make this terminology accep-

table, we have to show that there is a natural identification of A with For this purpose we identify

defined by

z

z(g) = g(z), where

in

V

A.

with the C-antilinear map on Vv

g runs through the elements of

a

It is

easy to check that this definition is compatible with the structure of V

as a C-vector space;

and since the imaginary part of z(g)

in the notation above,

the dual of A

is -g, (2)

is A.

Let us temporarily denote the underlying pairing

V x vr-c

by

zx 8 8(z) = (2778), where the subscript will be omitted whenever possible.

of Hom(A,

B) or

End(A),

we shall use the same symbol also for the

underlying map of vector spaces. ponds a C-linear map

Given any element

Thus to any

@

in

Hom(A,

B)

corres-

a: Va => Vp: and thus also its transpose

ty : Veat vie defined as usual by

(az, Us = (7,

og)

for all z in V, and g in Vo manda oniaieetincst that

Me maps

A

into

ae

so it induces a homomorphism

which is called the transpose of works as one would hope,

A with

melee easy to check -

a.

Moreover,

ty -BoA

all the standard formalism

including the fact that

Ny = a

when we identify

A, Let

H

be a given positive definite Hermitian Riemann form for A,

and let ¢ be the isogeny (52) determined by H. anti-automorphism on

We define the Rosati

End (A) by

a ent om tae: Lemma other words

a'

37.

(53)

H(az, w) = H(z,

is the adjoint of

for any

a@ with respect to

a" = a, so that (53) is an involution,

64

a'w)

and

= @.

z H.

and

w

in V; in

Moreover

Proof.

This is a straightforward calculation.

It was shown in

the proof of Theorem 36 that (z, ¢w) = H(w, z); so

H(a'w,.z) = H(¢"} ‘agw, z) = (z, 'agw) = (az, ow) = Hlw, az) and on taking complex conjugates the first assertion follows.

a"=a.

Again H(z, w) =(w, $z) = (tow, z) whence

{z, gw) = H(w, z) = {z, tow) by taking complex conjugates; It would be nice if we could obtain from on

with respect to A, in such a way that

V

and this

which completes the proof of the Lemma.

gives ‘¢ = H

Hence also

Zs

H@, g)=H(d

-1

f, @

-1

a dual Riemann form

i is the same as

H;

The obvious candidate would be

to be impossible.

but this seems

H

(54)

8g),

put this is in general not a Riemann form because its imaginary part is not integer-valued on A x A. However, a suitable multiple of H isa

Riemann form, polarization.

and we can maintain duality by introducing the idea of a A polarization P on an abelian manifold is a set P of

positive non-degenerate divisors such that

(i) integers n, a A

ny

given any divisors and

n,

4, and na :

such that

a. in P, there are non-zero

is algebraically

equivalent to

and

(ii)

P

is maximal

subject to (i).

a polarization is just a positive definite Hermitian together with those of its multiples which are Riemann

In analytic language, Riemann form

H

The meaning of a map of polarized abelian manifolds is obvious, is a map of abelian manifolds and B is polarized, there and if AB

forms.

is an induced polarization on

For practical purposes

A.

one can afford to think of a polarization of

29 as equivalent to a projective embedding of A. Indeed by Theorem and of ASpelt a a polarization induces a set of projective embeddings the projective embeda, are algebraically equivalent, then by Lemma 35 So the various projective dings they generate differ only by a translation.

A

one another induced by a given polarization are all related to Veronese mappings by means of translations on the abelian manifold and Conversely any projective embedding of A deteron the ambient space. embeddings

65

mines a polarization.

It is easy to verify that the Rosati anti-automorphism enly on the polarization defined by

if A

H, and not on

H

itself.

(53) depends Moreover,

is polarized then (54) determines a dual polarization on

the standard properties of duals;

and

varieties,

A> A is a multiple of

and the analogous map

A, with all

@ is a map of polarized abelian g.

Again,

if a@ isin End (A) then straightforward calculation gives t

t (a') = gag” = (‘a)’,

If we use

Theorem

@ to identify

36 that

V

and

A is the dual of A

ve it follows from the proof of

with respect to the alternating

Riemann form

E; thus the degree of @ is equal to the square of the

Pfaffian of

We shall say that

choose

H

E.

A

is principally polarized if we can

in the polarization so that

remarks

above,

base for

A

¢ is an isomorphism;

by the

this is the same as saying that by choosing a suitable

we can put the matrix of E

in the form

abelian manifold admits a principal polarization; matrix representation

a

a) for

E

(: A

Not every

but by considering the

we easily see that any polarized

abelian manifold is isogenous to a principally polarized one.

In the more

advanced theory,

principally polarized abelian manifolds have a number of important properties which have not been proved and may not even be true for arbitrary abelian manifolds; see for example Maass [7] passim. We defined the Jacobian of a compact connected Riemann surface

S - or, which comes to the same thing,

of an algebraic curve defined

over C - in §1, just after the statement of Theorem 7, One reason why principal polarizations are important is the following result.

Theorem 38,

Let S be a compact connected Riemann surface;

then its Jacobian is an abelian manifold which has a principal polarization.

Proof.

8, so that

We use the notation of §1, and in particular of Theorem g is the genus of S and Pr, ener D2 r', wees Is isa

normalized base for H,(S, Z). The map that takes w to (c,, Foe, BOM): in the notation of Theorem 8(ii), has trivial kernel by (8). So it is an isomorphism,

66

by comparing dimensions,

and we can choose a base

Wir rees

w®_

for the differentials of the first kind on

the period of

Denote by

Sa

matrix of the metric.

a

the period of

oye

Again,

and write

w=

with respect to

Ey, is 1 if w#=v

ory with respect to

then it follows from Theorem

let

z_

Zivceey

2 a

S such that

Civ

and 0 otherwise.

YP and let

8(i) that

@

®

be the

is sym-

be any complex numbers not all zero,

in Theorem

8(ii); then it follows from (8) that

Imy2 22576). vp pv > 20, whence

Im @

is positive definite.

Now take the elements

r

tr: Beales

the C-vector Let

H

as abasefor space

V

of A

corresponding to

A, andthefirst

- which was called

r;

nets a

g of them as a base for &*

in the notation of

be the positive definite Hermitian form on

V X V

§1.

whose matrix

Direct calculation

representation with respect to this base is

(Im 6) +,

shows that the matrix representation of E

with respect to the given base

oLe A» is

¢ . so

polarization.

9.

H

is a Riemann form and corresponds to a principal

This proves the Theorem.

Representations of End (A)

with elements in

m>

F, where

0 is aninteger and

There are two representations of End(A) tant in the analytic theory,

space

and

F

matrices

is any ring.

End (A) which are impor-

and a third which can be described in the ana-

lytic theory but only becomes

element of End (A)

m X m

will denote the ring of

In what follows, M,,{F)

important in the geometric theory.

induces an endomorphism

Any

of the underlying vector

V= ct: so there is a representation

End (A) > End(C") ~ M,(C) Again, any element of End(A) which is called the complex representation. induces an endomorphism of A, so that there is a representation

End(A) + End(A) * End(Z“") » M,, n (Z); and by tensoring with

Q this induces

67

End(A) + End(A ® Q) = End(Q*") = M,,Q). Either of these is called the rational representation.

Obviously both the

complex and the rational representation are faithful.

Now let

Z bea fixed rational prime;

the letter

by analogy with the geometric theory,

in which

characteristic

For each

End(A)

of the underlying field.

induces a homomorphism

p

2 is used here

is reserved for the r>

0, any element of

on the group of 1*- division points of A

(that is, on the kernel of 1”6); and this gives a representation

End(A) > End((Z/Z The sequence

compatible,

a

“M, (Z Tay

of representations

corresponding to increasing

r

are

and taking the limit we obtain a representation

Zn , End(A) + End(Z)) ~M,, (Z,);

and once again tensoring with

Q

(55)

induces

End 0 (A) + End(Q*") l = M, Zl (Q,) Hither of these is called the Z-adic representation.

division points can be identified with

The group of lee

2~"A /A, and the compatibility

statement above corresponds to the homomorphism

(ina ere given by multiplication by representation

(56) Z.

Thus the Z),-module which determines the

(55) is just the limit of the groups

under the homomorphisms

(56),

a free Z),-module of rank

2n;

of all infinite vectors of A

and

la, eee

(a

This is the Tate module

- each

r->

a, is an 1" -division point

0, and addition and multiplication

The group of 1”- division points is also

isomorphic to A/l ait. and the homomorphism the natural homomorphism

A/l

module isomorphic to A ® Z) theory is concerned,

68

T)> which is

explicitly it can be described as the set

sy see ) where

are defined component-wise.

of 1" - division points

Te

(56) is then replaced by

ALT AS thistivaken theorate

and it follows that, so far as the analytic

the /-adic representation is equivalent to the rational

representation and need not concern us further.

Once a coordinate system for can be written as an

V

is chosen,

n X 1 column matrix with complex elements.

we also choosea base for

A, the complex torus

described by means

n X 2n

of the

V/A

correspond to

A; we shall denote this matrix by corresponds to multiplying

Varying the coordinate system in V

on the left by an element of GL(n, C);

changing the base of A

ponds to multiplying it on the right by an element of GL(2n, can use these operations to put V/A =A

is an abelian manifold,

U

corres-

Z).

We

but when

into canonical form;

which is the only case that interests

it is convenient at the same time to make use of the existence of a

us,

polarization of A. A

U

If

is completely

matrix whose columns

the elements of the chosen base for U.

any element of A

So, as at the end of §5, we assume that the base for

is so chosen that the alternating Riemann form

E

associated with

the polarization has the form

0 D0)

(Cp and the

last

D=

diag(d,, ae

qd, are positive integers.

elements

n

ag

where

qd)

As in the proof of Theorem

of the chosen base for

A

24, the

are linearly independent over

so we can choose any convenient multiples of them as a base for V. We therefore suppose that the base for V is chosen so that U can be

C;

written

U=(S

here

D) where

S is in M_(C)

and

S=L+

L, M

iM;

are in M_®).

The condition that

H

defined by (37) should be Hermitian and positive definite is just that E(ix, y) should be symmetric and positive definite; to express this, write

E

x=x, tix, =(L+iM D) (2) where

gy 5, are real

real vector space

n X 1 column matrices and

generated by

AHL? Bae oe

ron

x, X, are in the

Thus

69

ai

we already know that

M

and

(x, - LM

a

x);

is non-singular because the columns of U

linearly independent over

1,

-1

§, =D

R.

With a similar notation for

y in terms of

n,, we find

E(ix, y) =

x, My, tt

+ "My,

f 5

il mL - L)M"’y,.

t

This is symmetric and positive definite if and only if L symmetric

and

M

is positive definite.

and

M

The set of matrices

are both

S=L+

satisfying these conditions is called the Siegel upper half-space; of the natural generalizations

Given the rational

End(A)

of the classical upper

representations

and let

y and

then

in End(A).

For

Proof.

and

For let

@

bein

p be its complex and rational representations Conversely if y in M_(C)

yU = Up

and

p

in

then each of them defines an element

M,, (2) by M,,,,Q).

The rational representation of End (A) is equiva-

of the complex representation and its complex conjugate. The matrix

@) is non-singular

above its determinant is equal to det(2iM)det(D)

@

half-plane.

and End (A).

End (A) we need only replace

Lemma 39. lent to the sum

of End(A)

yU=Up.

M,, (2) are such that

let

iM

it is one

U, we can easily recover the images of the complex and

respectively;

@

are

bein End (A)

y respectively;

because

in the notation

which is non-zero,

and have rational and complex representations thus

yU=

Up

andso

yU=Up.

Now

p

These can be put

together to give

CG and since

U

OU

qq) = Ge

@) is non-singular this proves the Lemma.

Lemma 40.

Suppose that

a #0

is in End (A) and

image under the Rosati anti-automorphism;

then

a'

is its

Tr(aa') > 0 where

the trace is that derived from the rational representation.

Proof.

By Lemma

37, @'

positive definite Hermitian form

70

is adjoint to H;

a

with respect to the

so by a standard theorem

of elemen-

:

tary algebra the trace of the complex representation of

strictly positive.

(Indeed,

if we choose a base for

represented by the unit matrix,

of a

by the matrix

V

is real and

suchthat

H

is

and denote the complex representation

(a;,), then the complex representation of a@' is

(a..) and the trace of the complex representation of

DD, las,| at

aa@'

) Lemma

aa@'

40 now follows from Lemma

is

39.

The characteristic polynomial of the rational representation of an

element of

a;

a

of End (A)

is called the rational characteristic polynomial

if we denote it by

os then it has degree

cause the rational representation is faithful. alternative way of describing

from

End(A)

i

2n

and

f (a) = 0

There is an important

We can extend the definition of degree

to End (A) by means of

(57)

deg(ra@) = r"deg(a) for any

in End(A)

@

be-

r

and

in Q.

Now we have

f i (r) = deg(r5 - @) for any r6- a

r

in Q

isin

and

End(A)

@

in End (A), where

6 is the identity;

for when

this merely states that the degree of an endomor-

phism is the determinant

of its rational

which is obvious,

representation,

and the general case follows from (57).

Using our results on the structure of U, we can now construct a ionclassifying space for the set of abelian manifolds with a given polarizat that is, with a given matrix

the Siegel half-space

D.

We have seen that to every point

corresponds

S of

an abelian manifold with the given

and that every such abelian manifold can be obtained in this one However, a given abelian manifold corresponds to more than

polarization, way.

matrix S because there is more than one base for A which takes the So if 2 denotes the Siegel representation of E into canonical form. upper half space and

G

the stabilizer of E

space we are looking for is

=/G

in

Z), the classifying

GL(2n,

where the action of

G

on

2

is still

in order to remain compatible with the classiLet o differently. cal notation in the case n= 1 we proceed somewhat therefore on U from be in GL(2n, Z); we wish o to operate on 2 and

to be described.

However,

Wh

the left, so we define its operation as

U > uto, and we then have to mul-

tiply this on the left by a suitably chosen element of GL(n, C) so as to renormalize it.

c= then

Write

Oar,oe.)

(58)8

uto = (sta ot p's ay oFp's)

and to renormalize we have to multiply

on the left by D(S*y + D's)"'. The resulting point of the Siegel space is therefore

t p(s‘y + D's)"(sta + D‘g); but we know this to be a symmetric

matrix,

so we can replace it by its

transpose and write the transformation as

S > (aS + BD)(yS + 6D)" *D. Of course it would have been much more sensible to write

2n X n

U

asa

matrix ab initio, but in this matter we are the prisoners

of

history. It remains to consider the action of

o

the conventions we have adopted this is clearly

G, the symplectic group associated with

on

E = ( 0

Dy and with 0 So we define

-D E > oEto,

D, to be the

o in GL(2n,

Z)

such that

dy = oF ‘o,

(59)

This is equivalent to- tone oe ET}. and in the special case corresponds

to the classical symplectic

condition becomes

‘oR = E.

group,

Eee

-E

D=I,

which

and so this

Using the partitioning (58) the condition

(59) reduces to ti ait aD6-f~Dy=D

and

eal aD,

t : yD 6 symmetric.

For this classification to be useful we need to show that G acts discentinuously on 2 - that is, that if N is a compact subset of = there are only finitely many o in G such that NN ON is not empty. But if S isin

U2

NON

the corresponding Hermitian

form

H(z)

%z)is

positive definite and lies in a compact set which depends only on

its value at any

ae

where

>

is bounded independently of o.

rors

ron is the given base for

This means that the

finite set, and-hence the same is true of

N;

o,

ones

so

A,

lie in a fixed

In particular it follows that

a polarized abelian manifold only admits a finite group of automorphisms.

In contrast with the case

n=

1, it is not known whether

2/G

naturally has the structure of a complex manifold.

It seems

trouble can occur at fixed points of elements of G.

Already in the case

n=1,

=/G

is not compact;

but Baily and Borel [1] have shown for all though with much

n how it can be compactified in a satisfactory way,

greater difficulty than in the case

10.

likely that

n=

1.

The structure of End (A)

We saw at the end of §7 that to determine the structure of End (A) for an arbitrary abelian manifold A it suffices to be able to do so when A is simple; moreover, if A is simple then End (A) is a division algebra. In this section we investigate the possibilities for End (A) For the reader's convenience we begin by sum-

A.

n=dim

for given

marizing without proofs some

more

standard results on division algebras and

We know that End (A) is finite

Deuring [3] or Weil [18], Chapter IX. dimensional

over

Q

because the rational representation is faithful.

Until further notice, centre is

that

{0} FL L.

for proofs see

generally on finite dimensional simple algebras,

F

K

K

be afield and

and which is finite dimensional

is simple over

and

let

F itself.

K

- that is, F

over

F

L

we also assume

K;

has no two-sided ideal. other than

For any field L> K wewrite

is a simple algebra with centre

an algebra whose

F,; = F ®, L; then

and it is finite dimensional over

The first major structure theorem is that any such

F

is isomorphic

is a division algebra;

to a complete matrix ring M_(D) where D F determines n uniquely and D upto isomorphism.

and

Moreover, with

of the obvious identification of K with the centre of D, the dimension subfield of D D over K is a perfect square m’; and every maximal splits F if Fy is has degree m over K. We say that a field L>K the notation isomorphic to M_(L) for some r, which must be mn in of L which is above; then L splits F if and only if there is a subfield 713

K-isomorphic

over

K

to a maximal

is inner - that is, is of theform

element

c

and let

Every automorphism

x > ce'xc

D

q@ be any element of D.

m*

over its centre

The characteristic polynomial of

of the K-vector

plication on the left, is a perfect mt power; m,

F

for some invertible

bea division algebra of dimension

regarded as an endomorphism

degree

of

in F.

Now let

K

subfield of D.

space

D

a,

acting by multi-

its mA root, which has

is called the reduced characteristic polynomial of

way to define it is to take the minimal polynomial for

@

@.

over

K

choose that power of the minimal polynomial whose degree is

m.

Another

and The

second and last coefficients of the reduced characteristic polynomial, with the standard sign changes,

reduced norm of @ over

are called the reduced trace and the

K.

If k is a subfield of K with

we can define the reduced trace and the reduced norm of means

of the usual tower rules;

a

[K:k] over

finite, k by

and they enjoy all the standard properties

of traces and norms.

Suppose that

F

and

G

being finite dimensional over centre

are simple algebras with centre K;

then

F ®& G

is a simple algebra with

K, and the division algebra underlying it depends only on the

division algebras underlying

F

and

G.

In this way we obtain a law of

composition on the division algebras with centre

over group,

K, both

K

and finite dimensional

K.

Under this law, these division algebras form a commutative called the Brauer group of K. The identity element of this group

is K itself; elements as

We write

the inverse of D D

Br(K)

If K

is the division algebra which has the same

but whose law of composition is defined by

qd, x qd,7dod..

for the Brauer group of K.

is an algebraic number field or a local field we can describe

explicitly its Brauer group and the homomorphism

Br(K) > Br(L)

by mapping the class of

K, to the class of

F, a simple algebra over

We start with local fields;

braic extensions of C, and

Br(C)

Br(R)

FL.

is trivial because there are no alge-

has order 2, the non-trivial element

corresponding to the classical quaternions.

Br(R)

obtained

Thus each of

Br(C)

and

has a unique embedding into Q/Z, which we shall use shortly. If K is an algebraic number field and @ a finite prime of K, then there is a canonical identification Br(Ke) =Q/Z. Now let K be an algebraic

74

number field and

F

a simple algebra with centre

finite set of primes

Fe a

®

in

K, the algebra

Be Ke: where the prime

that takes

F

to

@

F

K.

For all buta

splits over

Ke;

so writing

may be finite or infinite, the map

IIF@ induces a homomorphism

Br(K) > ® Br(K@).

This can be embedded in an exact sequence

0 > Br(K) > ©@ Br(Ko) >Q/Z

> 0,

where the penultimate map is given by addition in Q/Z canonical identifications

number field and

q

above.

in L

isaprime

It follows that if F

can determine whether

L

if the image of F

hence if D of D

over

r

have degree

in K, then with

K, we

has order

in Br(K)

(This result need not be true if K

r then there are

F, but none of lower degree; F, the maximal subfields

has dimension

D

andso

In

F.

in K, the set depending only on

is the division algebra underlying K

®

is just multiplication by

by examining the factorization in L

F

r which split

of degree

of K

extensions

lying above

is another algebraic

is a simple algebra with centre

splits

of a certain finite set of primes

particular,

if L > K

Br(Ke) = Br(Lq)

these identifications the map

[Ly : Ko!

Moreover

together with the

r’ over

K. The

is not an algebraic number field.)

particular case of this which we need is

Let K be an algebraic number field and let D be

Lemma 41,

a division algebra with centre

K

and finite dimensional over

K

admits an anti-automorphism which leaves either

phism between

D

has order 1 or 2.

in

D

has dimension 1 or 4 over

K.

can be regarded as an isomor-

and its inverse in the Brauer

D

If D

elementwise fixed, then

isa quaternion algebra over

The anti-automorphism

Proof.

Br(K)

D

or

D=K

K.

group;

By the remarks above,

so the image of

this means that

K, and this proves the Lemma.

We now return to considering the structure of End (A), where is a simple abelian manifold.

algebra,

and denote by K

Write

D=

End (A) so that

the centre of D;

D

A

is a division

and choose once for all a

and its associated Rosati anti-automorphism, which an involution on K; is an involution by Lemma 37. Clearly this induces

polarization of A

VS

we shall say that

A

is of the first kind if this involution on

and of the second kind otherwise,

field of the involution. in the second

K

is trivial,

K, for the fixed

K= KY in the first case and

[K: K | =2

case.

Lemma then

Clearly

and we shall write

K

42.

K

is totally real;

andif

A

is of the second kind

is totally complex and the involution induced on

K

is complex

conjugacy.

Proof.

Let

no divisor of zero,

a be any element of D= End (A). the minimal polynomial for

and both its reduced characteristic polynomial

a@

Since

over

Q

D

contains

is irreducible

(in the sense of this sec-

tion) and its characteristic polynomial with respect to the rational repre-

sentation described in 89 are powers

of this minimal polynomial.

Thus

the corresponding traces are positive multiples of one another and Lemma 40 implies

Trp if

@

isin

2) K

or

>

O

ity a0:

(60)

K) this also applies to the trace for

K/Q

or

K /

respectively. Suppose first that infinite prime of K

find

a

in K

K)

is not totally real and let

- that is, a complex embedding

such that

oa

all its conjugates except o,a

are small. 2Re(o, a)?
C.

is large and has argument near

because its dominant term is So

Oo. be a complex

So

and since complex conjugacy induces K aE it must be the Rosati involution

This completes the proof of the Lemma.

0

Recall that any quaternion algebra

D

with centre

K

has a canoni-

cal involution given by

Kt

Tr

(x) -x

D/K

where the trace is the reduced one;

the elements

involution are precisely those in K. of this and a K-automorphism xe? ce xke

for some

automorphism;

of D

fixed by this

Any K-involution of D

is compounded

of D, and therefore has the form

non-zero

c

in D.

This is certainly an anti-

it is an involution if and only if its square is the identity,

and a little algebra shows that this happens if and only if c ‘c* with every

x

in

D.

Thus

c*=Ac

with

c

is in

in

have! e* isin

possibilities when

A

is of the first kind.

A

is of the first kind,

so that

K,

K=

then one of the three following cases holds:

is totally real;

(I)

D==K. is a quaternion algebra over

D

(II)

ponent of D EO R_ is isomorphic to M,(R); c* =-c,

such that

3

2

anti-automorphism

(III)

in the second case

we can give a complete account of the

Suppose that

43.

Theorem

In the first case

K.

With the notation above,

in D

K, and applying the

or -1l.

and the involution is the canonical one;

K

abel (@)) 20)

A

A= 1

canonical involution to this we see that

commutes

;

c’ is in K

is given by

@'=c

ponent of D ®.R

such that every com-

c

and there is an element

and totally negative, =I

and the Rosati

a*c,

is a quaternion algebra over

D

Q

3

K

K

such that every com-

is isomorphic to the classical quaternions;

and

C—O

41 applied to the Rosati anti-automorphism shows or D is a quaternion algebra over K; and in the that either D—K must latter case the discussion above shows that the anti- automorphism To prove the Theorem we need only show that be of one of two kinds. III. Note these two kinds of automorphism correspond to cases TI and corresponding that D ®R is the direct sum of r= [K: Q] components, a simple algebra to the r embeddings of K into R; each component is Proof.

Lemma

77

of dimension 4 over M,(R) on

or to the classical quaternions.

D

Moreover

where

our two involutions

Tr(aa') = 0 for each component by replacing

b isin

K

a

and we can

by

ba,

and is large for one particular embedding of K

into

and small for all the other embeddings. Suppose first that

@'=

a*,

It is easy to check that

non-negative for the classical quaternions,

but not for D ®R

when it is equal to

M,(R), when it is equal to 2 Det(x);

aq! = c late,

Choose a base

all anti-commute

OS

Tr(xx*)

is

2 Norm(x),

so every component of

is isomorphic to the classical quaternions and we are in case III. Now suppose that there exists

k

either to

induce corresponding involutions on each component;

show that

R

R, and must therefore be isomorphic

in

1, c, j, k for

and their squares

a, oF ca, ae ja, stika,

c

with the

are in

D

such that

D K;

over

c* = -c

K

and

such that

c¢, j,

then if

a, in K, calculation shows that

2024 ex O ne 2D 2 Trp pg(@a"')‘y= 2(a, . ca + tja, 7 k a). Each component tive and

of this is non-negative if and only if ce

i? and

k?

are totally positive;

is totally nega-

and the latter cannot happen if

D ®R _ has a component isomorphic to the classical quaternions.

So we

are in case II and this completes the proof of the Theorem.

For completeness

of the second kind.

we list, as case IV, what happens when

A

is

One can obtain further information about the structure

of D

in this case (see for example [8], pp. 196-200) but this involves using much deeper results about the structure of algebras. (IV)

D

is a division algebra whose centre

plex quadratic extension of a totally real field

of the Rosati anti-automorphism

to

K

K 3

K

isa totally com-

and the restriction

is complex conjugacy.

Albert has shown that any division algebra of type I, II or III can

be isomorphic to End (A) for a suitable

A, and for type IV he has given

necessary and sufficient conditions for this to be possible. However, the only easily answered question that one can ask is what constraints are imposed on n= dim A by fixing End (A). To consider this, write

e=[K: Q], ean

78

: Q] and d° =[D:K].

Lemma

divides

n.

44.

Incase

I, e

In case IV, ed

Proof.

. Since

as a vector space over

D

divides

actson D

divides

n.

n.

A ®Q#=

andtherefore

this deals with cases II, I] andIV.

In cases II and III, 2e

ou 2n

the latter can be regarded

is divisible by

[D : Q|= ed’;

Now consider case I, and let E

denote the alternating Riemann form associated with the given polariza-

tion of A.

For any endomorphism

a

of A

wehave

we are in case I, and it follows from Lemma defines an alternating Riemann form on times that of

E, where

©

A.

37 that

a@'=a

z X w>E(z,

Its determinant is

morphism;

Apply this result to m5-

a, where

aw)

det(®(q))

denotes the rational representation;

the determinant of any alternating Riemann form is a square,

det(@(a@)).

because

and since

so is

6 is the identity endo-

it follows that

det(@(m6 - a)) = det(mI - #(a)) is a square for every integer

degree

m.

Since this is a polynomial in

2n, it must be the square of some polynomial

coefficients;

and since

representation of

f2

f(a) = 0.

Moreover

also

f is a power of the minimal polynomial for

that

[Q(@) :Q]

we find that

e

divides

n;

of

is the characteristic polynomial of a faithful

a, this implies

divides

m

with integer

f(m)

choosing

@

a

over

f° and hence This implies

Q.

so that it generates

K

over

n, and this completes the proof of the Lemma.

Wy

Q

Appendix: Geometric theory

Many of the results in Chapter III can be stated in purely geometric language,

even though the proofs given there are analytic.

The

object of this appendix is to state how far they can be regarded as geometric theorems,

by outlining without proofs some

theory of abelian varieties. algebraic geometry;

of the geometric

The language used will be that of classical

for an account in scheme-theoretic

language see

Mumford [8]. Let

V

irreducible.

bea variety, V

not necessarily complete,

is called a group variety if there is given a group law

on the points of V

such that the structure maps

which respectively take

u X v

to uv

in the sense of algebraic geometry. definition for

V

non-singular or

and

V X V>V

v to vy

and

We say that afield

k is a field of

considered as-a group variety if V, the two structure

maps and the identity element of the group are all defined over this case

V

must be non-singular,

by a group variety defined over

V

and there-

for this reason we could without much

to be irreducible,

An abelian variety over

In

k, and the other irreducible components

are cosets of this subgroup;

loss have required

k.

the irreducible component of V

which contains the identity element is a normal subgroup of V of V

V7>V

are regular maps

C

and some writers do so.

should be so defined that it is just the

geometric realization of an abelian manifold;

in particular it should be

complete and irreducible,

and the group law should be commutative. It turns out that the first of these properties implies the last, so an abelian variety is defined to be any complete irreducible group variety.

Theorem

45.

(i) The group law on an abelian variety is com-

mutative.

(ii) manifold;

Any abelian variety defined over

fold which is non-singular is an abelian variety.

80

C

is also an abelian

and conversely any projective embedding of an abelian mani-

The completeness

map from any variety Theorem

V

46.

(i) Let

to an abelian variety;

(ii)

such that VX

then

Let

abelian variety;

of an abelian variety

to A

A

also implies that any

must be well-behaved in various ways: f: VA

bea rational map of a variety

f is regular at every simple point of V.

f{£: VX W-A

bea

rational map of a product to an

then there are rational maps

f(v X w) = f, (v) Se f,(w)

i :V-—>A

and

for every simple point

f. :WrA

v X w

on

Wz.

(iii)

Let

G bea group variety and

which takes the identity element of G

f:G—~A

a rational map

to the identity element of A; then

f is a homomorphism.

The last part is a generalized analogue of Theorem portant consequence

32,

One im-

is that any map from the projective line to an abelian

variety must be constant;

for the projective line can be given a group

structure corresponding to the additive group of the underlying field after removing one point,

and a different group structure corresponding to the

two points,

multiplicative group of the field after removing constant map can be a homomorphism

Again,

it follows that the group law on

knowledge

and let

for both these group structures.

A

is uniquely determined by a

of the underlying variety and the identity element.

Let A bean abelian variety defined over

Theorem 47 (Chow). k

and no non-

B

be an abelian subvariety of A;

some finite separable algebraic

The main importance

no continuous families

then

B

is defined over

extension of k.

of this result is that it shows that there are

of abelian subvarieties

of A

- a result which is

trivial in the analytic theory. We can no longer use the definition of isogeny in §7; instead we say that a homomorphism f: A, => A, is an isogeny if its kernel is

finite, it is onto, and properties

dim A, = dim A,.

imply the third.

As before,

any two of these

The degree of f as an isogeny is its degree

this need not be the same as the order of the kernel because With these conventions Lemma 33 remains true, and of inseparability. . / Moreover the Poincare so isogeny is still an equivalence relation.

as a map;

81

Complete Irreducibility Theorem

34 and Corollaries

true (with 'variety' for 'manifold' throughout, map

mo

istic

p with

still has degree

(Z /mz)°™,

However

ding only on v

satisfies

m°",

p prime to

A

m,

of course).

and in characteristic

Again, the

0 or character-

its kernel is still isomorphic to

in characteristic

p there is an integer

such that the kernel of p'6

0 =v=n

2 and 3 of it remain

v

is isomorphic to

depen-

(Z/p'Z)”:

but can take any value in this range.

The standard definition of linear equivalence in algebraic geometry corresponds hand,

to the analytic definition given in §6;

on the other

the definition of algebraic equivalence of two divisors on a torus

given in §8 was purely ad hoc. sors on

However

an abelian variety defined over

in the standard geometric

it can be shown that two diviC

are algebraically

equivalent

sense if and only if the corresponding divisors

on the associated abelian manifold are algebraically equivalent in the

sense of §8 - indeed a proof of this is outlined in $8. on

A

A positive divisor

is said to be non-degenerate if it is ample - that is, if some

tiple of it induces a non-singular projective embedding responds to the definition in §6 in view of Theorem

which follow its proof. moreover

48.

Let

xX, y any points on A.

this cor-

29 and the remarks

With these new definitions Lemma

we have the so-called Theorem

Theorem

of A;

35 still holds;

of the Square:

b be any divisor on an abelian variety

Then

mul-

(b- b. - . = bee

A

and

is linearly equivalent

to zero.

In the analytic theory this was too trivial to be worth stating; but in the geometric theory it is a difficult and important result. With the help of it, we can geen give G,/G, the structure of an abelian variety

(which we again call

A);

and for any non-degenerate divisor

we can again define an isogeny

¢:A—> A by mapping

point on A corresponding to the class of (b - b,). defined over a field

k then we can take

A

x

on

b A

on

A

tothe

Moreover if A

to be defined over

k.

is

All

that part of the formalism

remains valid.

of §8 which does not involve the Riemann form The Néron-Severi group G/G, is finitely generated, as

indeed it is for any variety.

a:A-B

82

A homomorphism

induces ahomomorphism

of abelian varieties

a* : G(B) > G(A)

on the groups of

divisors;

and since this preserves both algebraic and linear equivalence

it induces way that

ty : B> A. we = aq and

Moreover

in sucha defined

The definition of a polarization is still valid in

context;

and we now say that a polarization of A

cipal if we can choose a divisor ated

A

s = 9; and the Rosati anti-automorphism

by (53) is an involution. the geometric

A can be identified with

is prin-

} in the polarization so that the associ-

@ is an isomorphism. The Jacobian

abelian variety.

of a curve

I

is the group

G,/G,

This realization is obtained by means

realized as an

of a geometric

construction which gives the Jacobian a natural projective embedding,

or

more precisely a natural polarization induced by the unique polarization

on

I.

Theorem

is principal.

38 can now be strengthened to say that this polarization

Not every abelian variety is isogenous to a Jacobian,

but

the following result is almost as useful. Theorem variety

B

49.

Given an abelian variety

(depending on

A

A

there is an abelian

and not unique) such that

A X B

is iso-

genous to a Jacobian.

We now turn to the representations of End (A).

In characteristic

zero we can obtain a complex and a rational representation by embedding

and regar-

k, the least field of definition of A, into the complex numbers ding

A

as an abelian manifold.

But even in this case the l-adic repre-

sentations have certain advantages over the rational representation, although they are equivalent to it; for if k denotes the algebraic closure of k then the Galois group module

T)

Gal(k/k)

acts non-trivially on the Tate The extra structure

and thus on the l-adic representation.

thus obtained is useful,

particularly in number-theoretic

In characteristic

applications.

p it is known that neither the complex nor the

rational representation can exist in general.

Indeed for any

p there is

of genus 1, defined over the field of p- elements, for which End (T) is that quaternion algebra over Q which splits at all primes tion in except p and infinity; and this algebra has no faithful representa

a curve

either

I

M_(C)

or

M,(Q).

However for

(55) is defined as in §9 and is faithful. statement that there are

1 #p

the l-adic representation

Moreover,

p- p-division points on

if we define A

(so that

v by the 0 =v =n)

83

then a similar construction gives a p-adic representation This however

v=n,

orif

is not necessarily faithful,

v>

0 and

A

End(A)>M, (2,,):

though it certainly is so if

is simple.

Although in characteristic

p we do not have a rational represen-

tation, we can still define the rational characteristic polynomial of an element of End(A). an endomorphism

In accordance with standard conventions,

of A

Lemma 50. identity element. coefficients

in

which is not an isogeny we write

Z

f(X)

of degree

is

deg(a) = 0.

Let a be any fixed element of End(A) There is a polynomial

if a

and 6 the

2n

and with

such that

f(m) = deg(m5d - a)

for every integer

m.

Moreover

f(@)=0

in End(A).

This can easily be extended to elements teristic 0, f(X)

In charac-

is clearly the characteristic polynomial of the rational

representation of

@;

so in characteristic

rational characteristic polynomial of

the trace of

of End (A).

a.

a

p we define it to be the

and its second coefficient to be

(Its constant term is just

deg(@)

and needs no new

name. ) It turns out that this is also the characteristic polynomial of the l-adic representation of

40 by Tr(aa@')>

a, for

1#p;

moreover

we can replace Lemma

0 if a #0, with the new definition of trace.

To what extend can we describe the alternating Riemann form or at any rate the invariants

qd; ACCES

by purely geometric

The first step is to mimic the pairing

AXA

Z

means?

induced by (50),

division point on A, and let

thus

ma

Let 4

OF of its matrix representation,

0 be aninteger and

be adivisor

on

A

¥ an m-

corresponding to

V;

is linearly equivalent to zero and is therefore the divisor of

some function

f.

Moreover

f(mx) = {g(x)}™

for all

on

and

A; then

g(x)

x

there is afunction

on

g(x +u)

quotient is an mit root of unity.

on u and V, so we write

gx +u) =e (u, Hg(x) 84

m>

E,

A.

Now let

u

g on

A

such that

be an m-division point

have the same mA power and so their It is easy to check that this depends only

_

as a definition of the m

Lemma

th

root of unity

eis

51. _ In the notation above,

in characteristic

v).

Cen is a bilinear form;

0, or in characteristic p with

m

and

prime to p, it has

trivial kernel in each argument. Now fix a prime

For each

r>

2 #p

and let

m

run through the powers

0 choose an isomorphism

of l.

{2*-th roots of unity} = Z/l*

in such a way that the diagrams

{2 See

eosccanity

a

Zi

'

{2"-th roots of unity} + Z/1* commute,

where the left-hand arrow is the iv power map.

limit of the functions

Then the above is

compounded with the isomorphisms

er

a bilinear form

erat (A) x T)(A) >Z I" In characteristic

0 and with the natural choice of the isomorphisms

this can be identified with the form A x A-~Z Let

b

beadivisor

A

of (50).

@ the associated map

and

T, (A)=o T, (A) which we also denote by

this induces a map —ExX n> e(é, 7)

on

In characteristic

of E, bearing in mind that

A >A;

9.

Then

T) (A) Xx T,(A) = Z) which is easily

is a bilinear form

shown to be alternating.

above,

0 it is just the l-adicization

T,(A) can then be identified with

thus from it we can read off the powers

of

JZ in the

A ® Zy;

qd, and by letting

1

run through all primes we can even recover the d. Moreover in characalterteristic p these alternating forms give a partial substitute for the nating Riemann form

E.,

Finally we consider the structure of End (A).

Once again we can

use the confine ourselves to the case when A is simple, so that we can we still have notation of §10. Since, with the revised definition of trace, Tr(aa') > 0 for

a #0,

Theorem

43 and the classification of the possible

that lead up End (A) into four types both remain valid and the arguments to them only require minor changes.

However the proof of Lemma

44

85

depends essentially on the existence of the rational representation and

the modification of the argument that is needed in characteristic

p leads

only to the following weaker result.

Lemma 52. p, and let n;

e, ey and

incase II, 2e For

occur.

86

For

Let A be a simple abelian variety in characteristic

n= 1

n>

d be as in $10.

divides

n;

Then in cases I andIII,

and in case IV, e,d divides

this is best possible;

1 very little is known.

e divides

n.

in particular case III does

References

Baily,

W.

L.

and Borel,

A.

Compactification of arithmetic

quotients of bounded symmetric domains.

Ann.

Math.,

84 (1966),

442-528. Conforto,

F.

Abelsche

Funktionen und algebraische Geometrie.

Springer (1956). Deuring,

M.

Algebren.

Gunning,

R. C.

Berlin (1935).

Lectures on Riemann surfaces.

Princeton (1966).

Lang,

S.

Abelian varieties.

Interscience (1958).

Lang,

S.

Introduction to algebraic and abelian functions,

Addison-

Wesley (1972). Maass,

H.

Siegel's modular forms and Dirichlet series.

Springer lecture notes, Vol. Mumford,

D.

216 (1971).

Abelian varieties. Y.

G. and Taniyama,

Shimura,

Oxford (1970). Complex multiplication

varieties and its applications to number theory.

Japan 6.

Tokyo (1961).

Shimura,

G.

Automorphic

lecture notes, Siegel,

C. L.

Vol.

Publ.

functions and number theory.

of abelian Soc.

Math.

Springer

54 (1968).

Topics in complex function theory,

Vols. I to II.

Wiley-Interscience (1972). J. and Molk,

Tannery,

J.

Fonctions elliptiques.

Paris (1893-

1902). H.

Weber, Weil,

A.

déduisent.

Weil, A.

Lehrbuch der Algebra,

Vol.

III.

Brunswick (1908).

Sur les courbes algébriques et les varietés qui s'en

Paris (1948).

Variétés abéliennes et courbes algébriques.

Paris

(1948). Weil,

théta.

A.

Théoremes

fondamentaux de la théorie des fonctions

Sém. Bourbaki, 16 (1949). 87

[17]

Weil, A. Introduction a l'étude des variétés kahléeriennes.

[18]

Weil, A.

Paris (1958).

88

Basic number theory.

Springer (1967).

Index of definitions

abelian manifold, abelian variety,

isogeny,

v, 51

isogenous,

54, 55, 81

v, 80

algebraic equivalence of divisors,

l-adic representation of End(A), 68 linear equivalence of divisors,

60, 82

Sod Brauer group of afield, canonical divisor, class,

local variable,

74

1

normalized theta-function,

canonical

40

6 period of a differential,

complex representation of

End(A),

Pfaffian,

67

complex torus,

2

44

polarization of an abelian manifold,

v

65 degenerate divisor,

positive divisor,

47, 82

degenerate theta-function,

degree of an isogeny,

5, 24

principal divisor,

42

5

principal polarization,

54, 81

66, 83

differential of the first (second, third) kind, divisor,

ramification point,

2

5, 24

doubly periodic function, dual abelian manifold,

elliptic function,

holomorphic,

Riemann form,

64

Riemann

38

surface,

1

Rosati anti-automorphism,

2

genus of a Riemann group variety,

68

15

15

exact differential,

7

rational representation of End(A),

surface,

/

70

simple abelian manifold,

58

group,

Tate module,

21, 30

hyperelliptic Riemann

Siegel upper half-space,

symplectic

80

surface,

14

64

theta-function,

72

68 19, 31

89

transpose of a map,

trivial theta-function,

64

19, 31

type of a differential form,

valence of a function,

3

Weierstrass ®-function,

90

17

31

—

—_— '

.

.

~

ee

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Editor: PROFESSOR G, C. SHEPHARD,

University of East Anglia

This series publishes the records of lectures and seminars on advanced topics of mathematics held at universities throughout the world. For the most part, these are at postgraduate level, either presenting new material or describing older miterial in a new way. Exceptionally, topics at the undergraduate level may be published if the treatment is sufficiently original.

Analytic Theory of Abelian Varieties H. P. F. SWINNERTON-DYER, F.R:S. Professor of Mathematics, Cambridge University and Master of St Catharine’s College

The study of abelian manifolds forms a natural generalisation of the theory of elliptic functions, that is, of doubly periodic functions of one complex variable. When an abelian manifold is embedded in a projective space it is termed an abelian variety in an algebraic geometrical sense. This introduction presupposes little more than a basic course in complex variables. The notes contain all the material on abelian manifolds needed for application to geometry and number theory, although they do not contain an exposition of either application. Some geometrical results are included however.

O 521 20526 3