The Schottky problem and second order theta functions

Bert van Geemen

December

Introduction

The Schottky problem arose in the work of Riemann To a of g one can

asso ciate a p erio d matrix which is an element of a space H of dimension g g Since the

g

Riemann surfaces themselves dep end on only g parameters if g  the question arises

as to how one can characterize the set of p erio d matrices of Riemann surfaces This is the

Schottky problem

There have b een many approaches and a few of them have b een succesfull All of them

exploit a complex variety a ppav and a subvariety the theta divisor which one can asso ciate

to a p oint in H When the p oint is the p erio d matrix of a Riemann surface this variety is

g

known as the Jacobian of the Riemann surface A careful study of the geometry and the

functions on these varieties reveals that Jacobians and their theta divisors have various curious

prop erties Now one attempts to show that such a prop erty characterizes Jacobians We refer

to M Lectures III and IV for a nice exp osition of four such metho ds to vdG B D for

overviews of later results and V for a newer approach

In these notes we discuss a particular approach to the Schottky problem which has its

origin the work of Schottky and Jung and unpublished work of Riemann It uses the fact

that to a genus g curve one can asso ciate certain ab elian varieties of dimension g the Prym

varieties In our presentation we emphasize an intrinsic line bundle on a ppav principally

p olarized ab elian variety and the action of a Heisenberg group on this bundle A systematic

study of the geometry asso ciated to these leads in a natural way to Prym varieties Moreover

one nds several other remarkable prop erties of Jacobians which suggest geometrical solutions

of the Schottky problem

Acknowledgements I thank the organizers of the conference Variedas ab elianas y func

tiones theta for the opp ertunity to present these lectures and for providing the pleasent working

conditions in Morelia

The Schottky problem

Introduction We recall the basic results on p erio d matrices of Riemann surfaces References

are ACGH C GH and CGV Then we briey discuss mo dular forms a reference is Ig

December

Period matrices

Let C b e a Riemann surface of genus g we consider only compact Riemann surfaces in

2g

these lectures On the group H C Z Z there is an alternating nondegenerate

1

intersection form A symplectic basis of H C Z is a basis f g satisfying

1 1 g 1 g

i j i j i j ij

with Kroneckers delta so if i j

ij ii ij

0

The complex vector space of holomorphic one forms on C is denoted as usual by H C

C

0

it has dimension g Given a path in C and an H C one can compute the integral

C

R R

0 0 

as a map H C C thus it is an element of H C the dual We will view

C C

0

vector space of H C If is a closed path the integral only dep ends on the homology

C

class of which we denote by the same symbol H C Z Thus we get a map

1

Z

0 

H C Z H C

1 C

This map is injective in fact much more is true

Theorem Let f g b e a symplectic basis of H C Z Then there is a unique

i j 1

0

basis f g of H C such that

1 g C

Z

j ij

i

R

0  0

H C form the dual basis of the basis f g of H C Thus the elements

C j C

i

1

A symplectic basis of H C Z thus determines a basis of H C For this we only

1 C

use the We now use the to dene a complex g g matrix

i j

Denition The p erio d matrix of C with resp ect to the symplectic basis of

i j

H C Z is the matrix

1

Z

M C with

ij g ij j

i

0

and H C as in Theorem

j C

0 

Remark The p erio d matrix determines the image of H C Z in H C In fact

1 C

R

0 

of H C we have using the basis

C

i

Z Z Z Z

i1 i2 ig

g

1 2

i

0

since b oth sides give the same result when applied to the basis elements H C

j C

December

Torellis theorem Torellis theorem asserts that one can recover the Riemann surface

from its p erio d matrix There are many pro ofs of this theorem all of them use the Jacobian

its theta divisor which are asso ciated to a p erio d matrix see also

The Siegel upp erhalf space

The Schottky problem The Schottky problem basically asks for equations which

determine the p erio d matrices of Riemann surfaces among all g g matrices There are two

well known prop erties of p erio d matrices so p erio d matrices are symmetric and

ij j i

I m the imaginary part of which is a symmetric real g g matrix denes a p ositive

g t g

denite quadratic form on R xI m x for all x R We write I m

This leads to the following denition and theorem

Denition The Siegel upp erhalf space H is

g

t

H f M C I m g

g g

Theorem Let b e the p erio d matrix of a Riemann surface Then H

g

The subset H of M C is actually complex manifold with the complex structure

g g

induced from that on M C In fact H is an op en subset of the vector space of symmetric

g g

0 0

g g matrices if I m then also I m for any symmetric with suciently

1

small co ecients The dimension of H is g g

g

2

We investigate H in more detail to see what kind of equations for p erio d matrices one

g

should exp ect

The symplectic group To dene the p erio d matrix of a Riemann surface we had

to choose a symplectic basis Any two symplectic bases are related by an element of

 

I

t

S pg Z fA M Z AE A E g with E

g 2g 0 0 0

I

One can show that acts on H as follows

g g

 

a b

1

H H A A a bc d A

g g g

c d

where a d are g g blo cks of A The p erio d matrices of a Riemann surface X are a orbit

g

in H Thus rather than study the p erio d matrices of Riemann surfaces in H one could study

g g

their images under the quotient map

H A nH

g g g g

The action of on H is prop erly discontinuous but not xed p oints free and A is complex

g g g

C using the j invariant for elliptic curves variety with singularieties if g If g A

g

Riemann surfaces of genus

December

Mo duli spaces Let M b e the mo duli space of Riemann surfaces of genus g It is a

g

variety whose p oints corresp ond to isomorphism classes of Riemann surfaces Then we have a

well dened holomorphic map

j M A X

g g g

where is a p erio d matrix of X Torellis theorem implies that j is injective

The Schottky problem can now b e reformulated as the problem of nding equations for the

image of j

0

Denition Let J H b e the set of p erio d matrices of Riemann surfaces Its image

g

g

in A is

g

0

j M I mag eJ A nH

g g g g

g

0

The subvarieties J and j M are not closed see We dene the Jacobi lo cus J to b e

g g

g

0

the closure of J in H

g

g

0

J J H

g g

g

Decomp osable matrices A H will b e called decomp osable if lies in the

g g

orbit of matrices in diagonal blo ck form the diagonal blo cks b eing matrices in upp er half planes

0

of lower dimension The set J J in H consists of decomp osable matrices the diagonal

g g

g

blo cks b eing p erio d matrices of Riemann surfaces of lower genus This follows from a result of

Hoyt

Mo dular forms From Teichmuller theory one knows that the subset J is actually

g

0

an irreducible subvariety of H of dimension g for g if g one has H J J

g 1 1

1

Since J is a complex subvariety of H it is natural to ask for holomorphic functions f on

g g i

H such that f for all i implies J The fact that J is invariant under suggests

g i g g g

that we could try to nd such f which are invariant It is known that a variant of this idea

i g

will work

A Siegel mo dular form of weight k is a holomorphic function on H which transforms in the

g

following way under

g

 

a b

k

f H C f A det c d f A

g g

c d

in case g one has to add a certain growth condition on f for i The mo dular

forms of weight k form a complex vector space which has nite dimension For suitable large

k a basis f f of this vector space gives an everywhere dened map

0 N

N

H P f f

k g 0 N

k

Since f A det c d f for each i we have A and thus the map

i i k k k

factors over A nH In this way we get a map

g g g

N

A nH P

k g g g

December

A fundamental result is that for suitable k the map embeds A so A A The

k g g k g

image of and thus A itself is a quasi pro jective variety that is a Zariski op en subset of a

k g

pro jective variety For this pro jective variety one can take the Satake compactication of A

g

which has the following nice settheoretic description as a disjoint union

A A A A

g g g 1 0

A and for suitable k here A is dened to b e a p oint Actually the maps extend to maps

g 0 k

one has A A A

k k g g g

N

The closure of j M is a pro jective subvariety of P Thus it is dened by a nite set

k g

of homogeneous p olynomials Determining these p olynomials as well as the mo dular forms of

weight k gives a solution to the Schottky problem The b est type of equations for the Jacobi

lo cus J are thus homogeneous p olynomials in mo dular forms

g

Note that if F CX X is a homogeneous p olynomial of degree d and f f is

0 N 0 N

a basis of M k then F f f is a mo dular form of weight k d Thus the

g 0 N

equations for J we lo ok for will b e mo dular forms

g

We just explained that the nicest equations for the p erio d

matrices are mo dular forms These mo dular forms are obtained from homogeneous p olynomials

N

which dene the closure in P of the image of the comp osition

j

N

M A nH P

g g g g

The map is given by mo dular forms on H To nd the image of p oint C M one

g g

has determine a p erio d matrix H of C and then evaluate the mo dular forms at In

g

practice these two transcendental steps cannot b e made explicit except for very sp ecial cases

for instance when the Riemann surface C has many automorphisms

However any Riemann surface is an and can thus b e dened by a p olynomial

F CX Y in two variables Using the algebraic geometrical approach one nds that the

co ordinates of j C are given by p olynomials in the co ecients of F For example if the

Q

2

curve is hyperelliptic and F Y X a Tomaes formulas essentially compute these

i

co ordinates cf M In the classical literature one nds several other partial results for more

general curves

Table We conclude this section with a table It shows that the Schottky problem is

trivial for g and that for g one equation might b e sucient

J g dim A dim J co dim

g g g A

g

1 1

g g g g g g

2 2

December

Ab elian Varieties

Complex tori and p olarizations

We return to the problem of nding the mo dular forms which solve the Schottky prob

lem It turns out that nature has already done most of the hard work To explain this we

introduce ab elian varieties and show how they are related to H and its quotient A nH

g g g g

In a sense we take the longest p ossible route However it is an interesting one where we

see various imp ortant geometrical ob jects and constructions At the end we get for free an

explicit map

g

2 1

A nH P

g g g

which we will introduce and study in detail in the next chapter since it app ears to b e of great

imp ortance for the Schottky problem The variety A is a nite covering of A

g g

The intrinsic approach which we sketch here is due to Mumford but is implicit in the

classical literature on theta functions Readers already familiar with theta functions will nd

the classical results in the corresp onding notation in the next chapter and might want to skip

this chapter

We study line bundles on complex tori and we recall that A is the mo duli space of principally

g

p olarized ab elian varieties Then we introduce the Heisenberg group and give some applications

Complex tori As a rst step we will asso ciate a geometric ob ject a

to a H Then we characterize the tori obtained in this way in Theorem

g

2g g

Since I m the matrix I m is invertible The image of Z in C under the g g

g

matrix I is then a lattice in C

g g g

Z Z C

g

The quotient of C by this lattice is denoted by

g

X C

0

it is a compact complex variety a torus In case J is the p erio d matrix of a Riemann

g

surface C this torus is J C the Jacobian of C Using notation from the previous lecture we

have

0 

J C H C H C Z

C 1

The p olarization Let V b e a complex g dimensional vector space and let X V

b e a complex torus Then we can choose a basis B fa a b b g for the lattice

1 g 1 g

in such a way that fa a g is a Cbasis of V The matrix M C dened by

1 g B X B g

P

a then has an invertible imaginary part but I m need not b e p ositive denite b

B ij j B i

j

nor is necessarily symmetric

B

We should recall however that the p erio d matrix of a Riemann surface was dened with

resp ect to a symplectic basis wrt the intersection form The formulation of the following

December

theorem is not so elegant since various conventions are not compatible Note that if B is a

symplectic basis for an alternating form E on and we dene

0

B fa a b b g with B fa a b b g

1 g 1 g 1 g 1 g

0

then B is a symplectic basis for the form E

Theorem A complex torus X V has a basis B for which H i there is an

B g

alternating form E Z such that

B is symplectic basis for E

g

The Rlinear extension of E to C R satises the two conditions

Z

g

E iv iw E v w E v iv v w C

and v in the second condition

In case H the basis B fe e e e g of with e the ith standard basis

g 1 g 1 g i

g

vector of C and the alternating form E dened by the matrix E satisfy these two conditions

0

0 0

Pro of Let B and E satisfy the conditions of the theorem and let B b e as ab ove Then B is a

0

symplectic basis of and E is given by the matrix E on the basis B Let i V V x ix

0

2 0

b e multipication by i i and let J b e the g g matrix of i wrt the Rbasis B of

V The conditions in translate in the following matrix identities

 

t

J E J E E J with E

0 0 0 0

g

To determine J we dene W Ra Ra R and

1 g

R W W V u v u iv

2 2

Since iu v i v ui we have iR u v RJ u v where J W W is given by the

matrix

 

1

J R iR

By denition of the map

B

2

I W V

B

maps u v to u a u a v b v b Thus

1 1 g g 1 1 g g

1 1 1 1

J i R R iR R

Let X iY and X Y M R then v v v v v X v iY v

g 1 2 1 2 1 2 2

R v X v Y v so we have

1 2 2

     

1 1 1

X XY Y XY X XY

1 1 1

R J R

1 1 1

Y Y Y Y X

December

t

The matrix J E J is then

0

 

t 1 t 1 t 1 t 1 t 1

Y X X Y Y Y Y X X Y X

t 1 t t 1 t 1 t t 1 t 1 t t 1 t 1

Y Y X Y X X Y X Y Y Y Y X X Y X X Y X

t t t t

The condition J E J E is equivalent to X X Y Y since Y is invertible X X

0 0

t 1

i we have a zero in the upp er left corner then we have a I upp er right i Y Y I ie

t

Y Y the rest follows

The condition EJ is

       

1 1 1

Y Y X Y X

equivalently

1 1 1

XY Y XY X X Y

t

Since a matrix A is p ositive denite i S AS is p ositive denite where S is invertible i

t 1 t 1 1

A A AA is p ositive denite this condition is equivalent to Y 2

Line bundles

To understand the signicance of an alternating form with prop erties as in Theorem

we recall the basic facts on line bundles Line bundles on a complex variety X are

N

imp ortant in the study of maps from X to a pro jective space P There is a natural line

N N

bundle usually denoted by O on P Given a holomorphic map X P the pullback



of O to X is a line bundle L O on X Moreover the map is given by global

sections of L Thus a knowledge of line bundles and their sections allows one to determine all

embeddings if any of X in a pro jective space We restrict ourselves to a discussion of line

bundles on a torus References for line bundles on tori are M Chapter and LB Chapter

Co cycles To construct a map

N

X V P

one could use holomorphic functions f V C i N satisfying the transformation

i

rules

f z c z f z z V

i i

with c indep endent of i and c z for all z V Then f z f z

i i

N

and the map V P z f z factors over X V The transformation

i

law implies that

c z c z c z

+

replace z by z in the transformation rule This is the co cycle rule and fc g is called

2

a co cycle for

December

Line bundles Given a co cycle fc g one can construct a complex variety L of dimen

sion dim X as follows Let act on V C by the rule

z t z c z t

that z t z t follows from the co cycle condition The orbit space L

V C is a complex manifold and has a holomorphic map L X induced by z t z

L

The triple L X is a line bundle Any line bundle on a torus can b e dened by a co cycle

Global sections and theta functions A holomorphic function f V C satisfying

the rule f z c z f z denes a map

s V V C z z f z and sz sz

hence s gives a welldened holomorphic map s X L which obviously satises sz z

L

for all z V Such maps s X L are called holomorphic sections of the line bundle L and

such functions are called theta functions Any section is obtained from a The

global sections form a nite dimensional Cvector space denoted by X L

Classication of Line bundles

Isomorphism of bundles Given a torus X V we now want to determine all

line bundles on X as well as the vector spaces X L

0

Two line bundles L L on X are isomorphic if there is a biholomorphic bre preserving

map

0

L L

0

y y

L L

id

X

0

X X

L L

0 0

which is linear on the b ers Let c c b e the co cycles dening L and L Then corresp onds

to a biholomorphic map

V C V C z t z z t

which intertwines the actions of given by the two co cycles so

0

z c z z t z z c z t

for all z t

The Picard group The set of line bundles on the torus X mo dulo isomorphism is

0

an ab elian group with tensor pro duct as group law equivalently pro duct fz c z c z g

of co cycles This group is called the Picard group of X P icX It can b e identied with the

1 

sheaf cohomology group H X O From the exp onential sequence

X

e

 2 if

O ef e Z O

X X

December

one obtains an exact sequence

c

0

P ic X P icX NS X

2 11

here NS X is the NeronSeveri group of X which is the subgroup H X Z H X of

1 1 g (2g 1) 0 2

H X O H X Z is a complex torus The map c is Z and P ic X H X Z

the rst Chern class of a line bundle For X V the group NS X is canonically isomorphic

to the group of Zvalued alternating bilinear forms E on with E ix iy E x y

NS X fE H om Z E x y E y x E ix iy E x y g

0

The group P ic X is canonically isomorphic to

0

P ic X H om U with U fz C jz j g

0

we do not discuss the complex structure on P ic X

Given H om U the corresp onding line bundle L is dened by the co cycle

fc z g thus the co cycle do es not dep end on z Actually any homomorphism



C denes a co cycle in this way but the bundle L is isomorphic to a unique

L

App ellHumbert data Given a H om U and an E NS X it is not

p ossible in general to dene canonically a line bundle on X However one can write down

co cycles which exactly parametrize P icX These co cycles are determined by App ellHumbert

data which are dened as follows For E NS X dene a Hermitian form

H H V V C H v w E v iw iE v w

E

that H is Hermitian follows from E ix iy E x y E y x Next one considers maps

not homomorphisms in general

E ()

U

0 0 1

Note that if are such maps then is a homomorphism U A pair H is

called App ellHumbert data for the torus X and it denes a co cycle by



H ()+ H (z )

2

c z e

Let L b e the line bundle on X dened by this co cycle

(H )

Theorem App ellHumbert Let X b e a complex torus Then each line bundle on X

is isomorphic to a L for uniquely determined App ellHumbert data H

(H )

December

Very ample line bundles It is now an interesting problem to determine the vector

spaces X L and to see for which bundles the global sections dene an embedding of X in

(H )

a pro jective space Such line bundles are called very ample The precise results are not so easy

to state but the following result due to Lefschetz is classical We recall that an Hermitian form

H on a complex vector space V is called p ositive denite and one writes H if H v v

for all v V fg In terms of E I mH the condition H is obviously

E v iv v V fg

Theorem Lefschetz Let H b e App ellHumbert data on a torus X

n

If H then the line bundle L is very ample for any n the bundle L is

( nH ) (H )

then called ample Conversely if L is very ample then H

(H )

Denition A complex torus is called an ab elian variety if it has a very ample line

N

bundle equivalently if it has an embedding X P In that case Chows theorem implies

that the image X is a pro jective variety that is is dened by homogeneous p olynomials

RiemannRo ch The reader now recognises the two conditions on the p olarization

we met in The rst garantees that E NS X the second that there exist very ample

line bundles on X

One can determine the dimension of X L in terms of and H To state a weak

(H )

version of this result we recall that if E Z is an alternating form then there is a

generalized symplectic basis fa a b b g of such that

1 g 1 g

E a a E b b E a b e

j k j k j k j j k

with if j k and e Z with e dividing e we adopt the convention

j k j j j 0 j j +1

that any e divides These e are uniquely determined by E and are called the elementary

i j

divisors of E In case E v iv for all v we have of course e for all j

j

Theorem Let L b e an ample line bundle on X so H and let e e b e

1 g

(H )

the elementary divisors of E I mH Then

dim X L e e e

(H ) 1 2 g

Principally p olarized ab elian varieties

The tori which interest us particularly are the Jacobians of curves and more generally

those dened by a H Theorem and the general results ab ove show that such a torus

g

X X comes with a given element E NS X which denes ample line bundles L with

(H )

H H and E has elementary divisors e e so dim X L Note

E 1 g (H )

that H implies all e and thus dim X L i e for all j

j (H ) j

December

Translates of ample bundles For any a X we have an isomorphism

T X X x x a

a

the translation by a One can pullback line bundles along a translation This gives a map

 

L cL The map P icX P icX which preserves the Chern class cT T

a a

0  1

X P ic X H om U a T L L

L

a

is a homomorphism which is determined by E cL in the following way

2 iE (a ~ )

a e

L

where a V maps to a X V since E Z this do es not dep end on the choice of

a In case detE this map is surjective with kernel

2 2

ker Ze Z Ze Z

L 1 g

0 0

In particular if two ample line bundles L L have the same Chern class cL cL then

0 1 0  1 0

L L P ic X X is isomorphic to T L L for some a X and thus L and L

L

a

 0

L T are translates of each other L

a

If L is ample detE so the map induces an isomorphism on the tangent spaces at

L

0 1 1 1

the origins Since P ic X H X O H X Z its tangent space at is H X O and

X X

1

H X O we get T X

X 0

Denition A principally p olarized ab elian variety ppav for short is a pair X E

with X a complex torus and E NS X satisfying

H dim X L

E

(H )

E

Equivalently a ppav is a pair X L with X a complex torus and L an ample line bundle with

0 0

dim X L but ppavs X L X L will b e identied if cL cL that is if L and

0

L are translates The dimension of a ppav X E is dened to b e dim X

0 0 0 0

Two ppavs X E X E are isomorphic we write X E X E if there is an

0  0  2 0 2

isomorphism X X with E E where H X Z H X Z is the map

induced by

Mo duli of ppavs With these denitions one can verify that A nH is the

g g g

mo duli space of g dimensional ppavs roughly sp eaking

A fX E g

g

December

The theta divisor of a ppav Given a ppav X E of dimension g any two line

0 0

bundles L L with cL cL E are translates and they have upto scalar multiple a

unique nonzero global section Thus the zero lo cus of such a section is a variety

(X E )

of dimension g which dep ends only on X E and is called the theta divisor of X E

In case X E JC with E the intersection form Riemann proved that the divisor is

JC

isomorphic to the image of an Ab elJacobi map

(g 1)

I mag eC JC

JC

This divisor thus carries interesting geometrical information on the Riemann surface

It is known that the dimension of the singular lo cus of is at least g This prop erty

JC

has b een used to study the Schottky problem

Symmetric line bundles Given a ppav X V E there is no canonical way to

nd a line bundle L on X with cL E However one can consider App ellHumbert data

H with fg U The corresp onding bundles have the prop erty that

E



L where for n Z they are symmetric L

n X X x nx

2g 2g

There are such and thus such bundles in fact if denes a symmetric bundle then

so do es where fg is a homomorphism

To see that such indeed exist let B fa a b b g b e a symplectic basis of

1 g 1 g

We dene

t

g g mn

Z Z U m n

B B

g g

where m n Z Z corresp onds to m a m a n b n b and

1 1 g g 1 1 g g

t

m n m n m n It is easy to check that H are App ellHumbert data and

1 1 g g B E

thus dene a symmetric line bundle M on X and X M is a ppav

B B

An intrinsic line bundle Although it is imp ossible to dene intrisically a line bundle

L on a ppav X E with cL E one can dene such a line bundle with cL E In fact

2E ()

since we have the line bundle

L L I mH E

(H )

2

on X This bundle is isomorphic to M for any symmetric line bundle with Chern class

cM E Since the elementary divisors of E are e we get

i

0 g

dim H X L

This bundle will b e very imp ortant in the remainder of these notes

December

Heisenberg groups

Introduction We now recall an interesting asp ect of line bundles on an ab elian

0

variety It p ermits one to nd an intrinsic basis of the vector spaces H X M for any line

bundle M on X More precisely one nds a nite set of such basis and each such basis is

dened up to multiplication by a constant For the sake of simplicity we restrict ourselves to

the case of the intrinsic bundle L on a ppav References for this section are Ig LB Chapter

and K

Let X E b e a ppav and let L b e the intrinsic line bundle on X Since cL E we

get from that



L a a X ker X X L T

L

a



Given a X there is no intrinsic isomorphism T L L If L is given by a co cycle fc z g

a



then T L is given by the co cycle fc z ag but since c z c z a one has to choose a

a

map as in to get an isomorphism This forces us to consider the Heisenberg group of

L Its elements are couples of a torsion p oint and an isomorphism of bundles



=



H L f a a X T L Lg

a

With the natural group law the Heisenberg group turns out to b e nonab elian in we

give a concrete description of H L There is an exact sequence



C H L X

The rst non trivial map is t t the second is a In fact the map H L X

is obviously surjective and the only isomorphisms of any line bundle on a compact complex



variety with itself are scalar multiples of the identity The subgroup C is the center of H L

g

2

C as follows The group H L acts on the vector space of global sections X L



as T s

a

A basic fact is that this action is irreducible the only invariant subspaces are fg and the space

itself

0

A concrete description of the Heisenberg group H L and its action on H X L is

obtained as follows We dene a group

 g g 

H H C ZZ H om ZZ C

g Z

with pro duct the term mu makes it nonab elian

t u l s v m tsmu u v l m



must b e torsion so f ZZ by a homomorphism f to C Note that the image of

n 

for a unique n ZZ The map f n gives an isomorphism H omZZ C ZZ

December

The Heisenberg group H L is isomorphic to the group H An isomorphism which is the

identity on the subgroups C is called a theta structure

=





id H L H

C

jC

There are only a nite number of theta structures the elements u l H have order at

most and together with C generate H moreover there are only a nite number of elements

of order at most in H and thus in H L so a theta structure is determined by the map it

induces from the nite set of order at most in H L to the corresp onding nite set in H

The group H has a natural representation the Schrodinger representation on the vector

g

2 g

as follows C space V F unctionsZZ C

g

t u l f u tl u f u u t u l H u ZZ

g

with The vector space V has a natural basis of delta functions f g

(Z2Z)

if u

g

ZZ C u

if u

The main result on Heisenberg groups asserts that this representation of H on V coincides with

0

the action of H L on H X L The irreducibility of the representations and Schurs lemma

imply that given a theta structure H L H there is an essentially unique isomorphism

0

V which intertwines the representations The basis of delta functions in V then H X L

0

gives a canonical basis of the vector space H X L The consequences of this remarkable fact

are discussed in the next chapter We summarize the results in the theorem b elow

Theorem Given a theta structure H L H there is a unique up to scalar

multiple isomorphism

0

T H X L V

which satises

T ()

0

H X L PV

y a y T as a T s a

T ()

0

H X L PV

1 0

In particular the elements T are a basis of H X L and in these sections give a map

g

2 1

X P PV

g

where the co ordinates of PV are parametrized by ZZ

December

We need one more fact Since t H acts by scalar multiplication on V it acts



trivially on PV and thus the Schrodinger representation induces a representation of H C

2g 

ZZ on PV Let a t u l H map to a H C then we write

U a PV PV

for the pro jective linear map induced by the action of a on V

0 0 

For x X any two elements x x H L are related by t for some t C

and thus a theta structure induces an isomorphism a level two structure which we also denote

by with some abuse of notation



=

2g  

ZZ H C H LC X

With this notation the following diagram is commutative



PV X

1

T yU a y

(a)



PV X

so the translation by twotorsion p oints on X is given by pro jective transformations on PV

Geometry of second order theta functions

Introduction We work out the consequences of the theory of the previous chapter

and show that we recover some classical results from theta function theory In particular we

nd intrinsically dened maps X PV and H PV

g

In section we consider the intersection of the images of these maps Somewhat surpris

ingly we nd that if X is a Jacobian the intersection is rather large This is the rst hint that

second order theta functions are rather ecient at detecting Jacobians

Next we consider the intersection of a tangent space of H with X again the

g

Jacobians b ehave in a p eculiar manner This led Izadi to a geometrical solution of the Schottky

problem in the case g

Classical theta functions

Classical notation We return to the upp er half plane and the comp ex tori X

g

C

2g g

The standard basis of Z gives a symplectic basis B of Z Z the corresp onding

symmetric line bundle M will b e denoted by M cf Thus X M is a ppav This

B

bundle can thus b e dened by a co cycle fc g as in but Riemanns original co cycle fe g

is more convenient for various purp oses for example Riemanns co cycle is holomorphic as

function of but the c are not since H is not holomorphic in The explicit formula for

Riemanns co cycle is

t t

i( k k +2 k z )

e z e l k

December

The global section of M is called Riemanns theta function In general the global sections of

a line bundle M on X with Chern class cM k cM for k Z are called theta functions

0

of level k

The intrinsic bundle L cf over X will b e denoted by L Its dening co cycle is

2 2

fe g since

B

Canonical basis For applications of Theorem which gives bases of X L

it is imp ortant to know that given H the intrinsic line bundle L on X has a natural

g

theta structure

H L H

 g

It has the prop erty a b t a b for some t C and b ZZ

g  0

H omZZ C The corresp onding canonical basis of H X L is given by the second

order theta functions

X

t t

2 i( (n+ ) (n+ )+2 (n+ )z )

2 2 2

z e

g

n2Z

g

where ZZ and one may take representatives with comp onents f g That

i

is the isomorphism T dened by the theta structure satises



=

0

T H X L V

Maps As a consequence of the previous results we now have for any H the

g

natural map

X PV z z

We see that the maps for various glue together in fact the theta functions are holomor

g

phic in b oth z and Thus we get a map from H C to the pro jective space PV These theta

g

functions and these maps were well known classically but the approach with the Heisenberg

group emphasizes that the construction is a canonical one

Theta constants Since an ab elian variety X V has a canonical p oint the

origin any theta structure denes a canonical p oint PV Thus we get a map

from the mo duli space of pairs X E of ppavs with theta structure to PV dened by

X E

In the classical picture this gives the map

H PV

g

the co ordinates are called theta constants but they are not constant in Since there is

more than one theta structure one should not exp ect that this map factors over A nH

g g g

However this is almost true

For an even p ositive integer k we dene normal subgroups of nite index of

g

S pg Z by

k fA A I mo d k g

g g

December

0 0

a b

0 0

and diagb diagc mo d k k k A k A I k

g g

0 0

c d



We denote by AH the subgroup of automorsms of H which are the identity on C H

0 0 1

If H L H are theta structures the map H H is in AH and for

AH the map H L H is a theta structure In this way the set of theta structures

is a principal homogeneous space under AH There is an isomorphism of nite groups

AH

g g

and the space A nH is the mo duli space of ppavs with a theta structure The

g g g

group AH is the Galois group of the covering A A

g g

Theorem Igusa The map factors over the quasipro jective variety A

g

nH The induced map denoted by the same symbol

g g

A PV

g

has degree on its image and its dierential is injective at any p oint of A It is not

g

known if is an embedding The closure H of the image of is a pro jective variety of

g

dimension g g in PV

Mo dular forms Thus we have a very explicit map of a nite cover of A to a

g

pro jective space The co ordinate functions of the map are mo dular forms of weight

q

so basically transform with deta b but the sign of the ro ot has to b e sp ecied

Taking the second Veronese image of this map given by all pro ducts one gets a map

whose co ordinate functions are mo dular forms of weight The theory of automorphisms of

the Heisenberg group or equivalently the classical transformation laws for theta constants

imply that the nite group acts on PV in such a way that is an equivariant map

g g

A A A

g

Thus one can obtain mo dular forms for by taking all homogeneous p olynomials on PV which

g

are invariant under the action and substituting the for the variables

g g

Kummer varieties We consider now the map X PV According to

3

Lefschetz Theorem for n the map given by global sections of the bundle M on the the

2

ppav X M embed X in a pro jective space We are interested in the bundle L M

however

The map given by the global sections of L is not injective In case is indecomp osable

the variety X is isomorphic to the quotient of X by the involution x x second order

December

theta functions are even z z as one veries from the Fourier series in

This quotient variety is called the Kummer variety of X and will b e denoted by

K X

2g

The Kummer variety has singular p oints corresp onding to the twotorsion p oints X

these are the xed p oints of the involution

K X fg S ing K X X

In case is decomp osable the ppav X is isomorphic to a pro duct of lower dimensional ppavs

and the X is isomorphic to the pro duct of their Kummer varieties K Chapter

2

Intersections

We face the obvious question what is the intersection

H X PV

g

b etween the mo duli space and the Kummer variety This question was raised in GG in view

of the dimensions of the spaces involved one wouldnt exp ect any intersection at all for g

From the denition of it is however obvious that at least the origin of K lies in the

intersection

We consider the preimage of this intersection in X and call it Y so

1

H X Y

g

In case is the p erio d matrix of a Riemann surface a relation b etween theta functions discov

ered by Fay implies that Y has dimension at least two

0 

Recall that J C H C H C Z For any p q C we can dene an element

C 1

in this space simply denoted by p q as follows Take any path in C starting in q and

R R

0

will change H C C If we choose another path ending in p then we get a map

C

R R

where is a closed path so gives an element of H C Z Therefore the class of to

1

R

in J C dep ends only on p and q and this is the desired p q J C

For g one then obtains a surface in J C

C C fp q J C p q C g

With this notation Fays result is

Theorem Let b e the p erio d matrix of a Riemann surface C Then

fx J C x C C g Y

December

Remarks In case g one has Y C C It would b e interseting to know

for which one has dim Y but I dont know of any results b eyond g

We will consider an innitesimal version of this condition in where we replace the

mo duli space by its tangent space

One should note that the p oints of order of J C X are contained in Y since

p p C C A further study of these p oints which we carry out in section do es lead

to rather explicit Siegel mo dular forms related to the classical SchottkyJung relations which

give nontrivial equations for the p erio d matrices of Riemann surfaces

Sketch of pro of of Theorem Let x J C with x p q let D x

0

then D q p q linear equivalence of divisors Interpreting J C as P ic C the

2

O p q Let s variety of line bundles of degree on C this implies that O D q

C

b e the global section of O p q which is zero in p and q Inside the global space of the line

bundle O D q we can now consider the subvariety C of p oints satisfying s The

C D

subvariety C is an irreducible curve provided p q and the bundle pro jection induces a

D

map C C which ramies only over p and q Thus the genus of C is g The pullback

D D

0 0

map H C H C induces a Norm map N m J C J C The kernel

C D C D

D

of N m is an ab elian variety of dimension g and has a principal p olarization induced by the one

on J C This ppav is called the of the cover

D

X for some H Fay Fa proves that for a suitable pair of H Thus kerN m

g g

X one has X and kerN m with J C

x H

g

Therefore x Y The case p q follows by taking the limit p q In that case the cover will

b ecome singular and in fact the Prym variety in that case is b est seen as a ppav of dimension

g

Lo cal intersections

Tangent spaces Let H b e indecomp osable Then the image of X PV

g

is isomorphic to the Kummer variety K X The p oint is

singular on K Let

2

g

PV t t

f01g 0 k l

z z

k l

The second order theta functions z z are all even thus the rst order deriviatives

vanish The embedded tangent space of X at is the span of these p oints

1

T ht t i dim T g g

K 0 0 k l 1k l g K 0

2

The classical Heat equations for second order theta functions

2

z i z

k l

z z

k l k l

December

g 2

for all H z C and i these are easy to verify from the series denition of the

g

in Therefore we also have

t i

k l k l

k l

The vectors on the righthand side of the equation ab ove and t together obviously span the

0

embedded tangent space T to H at t Thus this space coincides with

g 0

(H )( )

g

T we denote it by T

K 0

T T T

K 0

(H )( )

g

Remark This may b e seen as a geometric version of an intrinsic isomorphism The

2 

Kummer variety K is singular but its tangent space is still dened by mm where m

O is the maximal ideal in the lo cal ring at K This lo cal ring is the ring of invariants

K 0

g

O Thus O may b e identied with the ring of under z z of the lo cal ring O

C 0 K 0 X 0

even convergent p ower series in g variables Any f O has a Taylor series expansion

K 0

X

a z z H O T f z f

ij i j

ij

We obviously have

2

f m f and f m f a i j

ij

2  

Thus mm is spanned by the monials z z Each z is an element of T thus we nd

i j i

X 0

2 1 2

S H X O S T T

X 0 K 0

the last isomorphism comes from the principal p olarization on X as in

On the other hand the rst order deformations of an algebraic variety X are parametrized

1

and thus T O by H X T Since the tangent bundle of X is trivial T

X 0 X X X

1 1 1

H X O H X O The deformations of X parametrized by H pre H X T

g X

serve the p olarization This gives an identication

2 1

T T S H X O and so T

K 0 H ) H )

g g

1

g g is dened by The linear pro jective subvariety T PV of dimension

00

2

1

g

g g linear equations We identify these equations in the following way

2

P

a X b e the equation of a hyperplane in PV Then we can pullback H to X Let H

P

a z along the map X PV This pullback is the theta function

H

 0 0

H PV O H X L H

H

2

Then we nd that t lies in H i and similarly t H i z z

0 H ij i j H

Therefore the dening equations for T are the elements of the vector space

n o

0

H X L m

00 0

where m stands for multiplicty at zero since these theta functions are even the multplicity

0

is also even That is

T H

H 00

December

Explicit equations We will need the following observation later Let P

H C X b e a homogeneous p olynomial which is zero on the pro jective variety

g

PV Then

X

P

aX with a H

P

X

is a linear form on PV which is zero on the tangent space T at a to H Thus its restriction

g

to X gives a section in In case a is a smo oth p oint of H these linear forms cut

00 g

out T PV and the corresp onding sections span

00

The lo cal intersection We will now investigate an innitesimal version of the

H see That is we replace H by T its embedded intersection X

g g

tangent space at

We dene

F fx X x g

00

Then by the previous discussion F is the preimage of this intersection

F X H X T

2

00

H

For p erio d matrices of Riemann surfaces this intersection is again large and it is related in a

somewhat surprising way with the intersection X H

g

The incusion in the following theorem was observed by Fay Gunning van Geemen and

van der Geer the inclusion is due to Welters

Theorem Let X J C the Jacobian of a Riemann surface C Then

F C C except if

0

we are in the case that g and that C has two distinct line bundles of degee with dim H

3

this is the case for the generic curve of genus In that case we write the p oints in P ic C

1 1 1 1

corresp onding to these bundles by g and h and we have F C C fg h g

3 3 3 3

The surface C C J C is singular at the origin J C Its tangent cone there

viewed as subvariety of PT is the canonically embedded embedded curve Thus if C is

J (C )0

non hyperelliptic Theorem gives a pro of of Torellis theorem it can also b e used to prove

Torelli for HE curves with a little bit of extra work

For dimension reasons one do es not exp ect that dim F for a H This leads

g

to the following conjecture which was proved in case g by Izadi Iz For g we thus

have a geometrical solution to the Schottky problem More rened versions of the conjecture

and variants are discussed in GG BD and D

?

Conjecture Let H b e indecomp osable then dim F J

g g

December

Example The case g is easy to understand Fix an indecomp osable H and

3

let T T Then dim T so T is a hyperplane in PV The corresp onding section spans

T

and its zero lo cus must b e C C One can also verify directly that C C jj the

00

2

linear system dened by L We have m and if z F z H O T is the Taylor

0 T T 4

series of then the quartic p olynomial F is the dening equation for the canonical curve C

T 4

2

if X J C with C non hyperelliptic In case C is hyperelliptic one has F F and C C

4

2

(2) 1

is the surface C g counted with multiplicity two its tangent cone is the rational normal

2

curve of degree g counted with multiplicity two

SchottkyJung relations

Introduction We consider the eigenspaces of the linear maps U a PV PV for a

2g 

ZZ H C and of course the intersection of these eigenspaces with b oth the mo duli

space H in and with the Kummer varieties X in PV in

g

This leads us to a geometrical picture of the classical SchottkyJung relations We show

how these relations can b e used to construct to mo dular forms which are zero on the lo cus of

p erio d matrices of Riemann surfaces Next we recall the known results

Eigenspaces of Heisenberg group elements



Let a H C b e a p oint of order two let and let U a PV PV b e the pro jective

transformation dened in We denote by a X the p oint for which the following

diagram commutes





PV X

1

T a a y U a y

a







PV X

here is the canonical theta structure which induces an isomorphism



=



X H C

2

Going down Since U a id any lift of U a to a linear map U a V V

PV

2

satises U a I for a nonzero C The map U a gives the action of an element

t u l H on V The explicit formula for the action of H on V see shows that if

 +

a the map U a has two eigenspaces V in V Their pro jectivizations PV and PV will

a a a

b e called the eigenspaces of U a The signs are not dened intrinsically any element in an

eigenspace of U a is xed by U a

g 1

The eigenspaces of U a have dimension they are pro jectivizations of linear spaces

1

g g 1

The action of H H on V induces an action of a similar group of dimension

g

2



H on the spaces V As in Theorem and section this allows us to dene for

g 1

a

H and the corresp onding theta structure H L H maps and similar ones

g 1 g 1

with a sign

a+ + a+ + a+ a+

X PV H PV

g 1

a a

December

Boundary comp onents The images of H in the eigenspaces nicely t in the

g 1

holes of the image of H in PV in fact we have disjoint union

g

 

a+ a



H H H H

g g aH C f0g g 1 g 1

2g

a a

H may still intersect The H are images the b oundary the various

g 1 g 1

comp onents of the Satake compactication of A It is known that vG

g

+

a+

H H PV

g 1 g

a

Intersection Let again a H C fg and the corresp onding a X We

consider

+

X PV PV

a

+

the intersection of a Kummer variety with an eigenspace of U a Since p PV PV i

a a

U ap p and since U a x x a we get

+

x PV PV x a x

a a

K X Thus Now assume that is indecomp osable then X

+

x PV PV x a x or x a x

a a

The rst condition is imp ossible but the second gives x a so x is a p oint of order The

2g

number of p oints x with x a is the dierence of any two such is a p oint of order two

1

2g 2g 1

p oints in the Kummer variety K and each eigenspace PV gets These map to

a

2

1

2g 1 2(g 1)

p oints

2

+ 2(g 1)

We conclude that X PV is a set of p oints for indecomp osable s Each

a

of these sets is an orbit of the group H Since each eigenspace also contains a b oundary

g 1

comp onent of H we introduce the following denition

g

Denition of SchottkyJung relations A pair X x with x X fg satises

the SchottkyJung relations if for some y X with y x we have

+

a+

y H PV

g 1

a

+

here a x and PV is the eigenspace which contains y

a

This condition dep ends only on the p oint x X X not on the choice of or y In fact

+

PV is one of the two eigenspaces of translation by x on PV cf The Heisenberg group

a

a+ a

H or maps it to H p ermutes the y s while it stabilizes

g 1 g 1

Examples

We insp ect the SchottkyJung relations for low genus In the cases g the

answer is easy In case g we nd a more interesting situation however

December

g The ab elian variety X CZ Z is an The line bundle

 2

O O with O X the origin The Schrodinger representation of H C Z L

as sets on V is given by see

     

t t t

t t t

t t t



the matrices are with resp ect to the basis of functions The eigenspaces PV of these elements

1

P they are resp ectively are p oints in PV

i i

2

with i

If we denote the image of the origin by a b

X PV a b



then using the equivariance of for the action of X and H C we nd that

1 1

t b a

2 2

and similarly the images of the other twotorsion p oints can b e determined

1 1 1

a b b a

2 2 2

X the image of X consists of distinct p oints Therefore a and b Since X

are nonzero Then we can write these image p oints as

1 1

ab

1

The elliptic curve X can b e recovered as the cover of P ramied in these p oints thus

an ane equation for X is

2 4 2 2 2

X y x x

Given any p oint PV which is not one of the six eigenspaces we get in this way an

elliptic curve which is isomorphic to an X and

By denition so we get

1

H P f eigenspaces g

1



In fact in this case A A H The eigenspaces PV corresp ond to to the

1 1 1

a

a

H b oundary comp onents so formally each is a

0



We recall that we have shown that the inverse image under of a p oint PV PV is a

a

p oint of order four in X for any H Thus as moves in H the images of these p oints

g 1

remain xed Since the image of any p oint of of order is a b oundary comp onent all pairs

X x satisfy the SchottkyJung relations

December

g Let H b e indecomp osable Then maps X onto its Kummer surface K

2

in PV

3

P X K PV

The surface K has degree and its singular consists of p oints the images of the twotorsion

p oints

+ 1 3

Each eigenspace PV P is now a pro jective line we get such lines in P These

a

lines can b e found easily as in the g case and they form a quite interesting symmetric

conguration We just observe that each line is a copy of H and thus has marked p oints

1

on it Through each such p oint there pass other lines and the lines only intersect in such

p oints

Since the Kummer surface has degree it will meet each eigenline in p oints which are

the images of p oints of order This shows that any pair X x with H satises the

2

SchottkyJung relations

0

0 0

K K for some H Then X X X In case is decomp osable X

1

1 1 3

P P which is embedded as a smo oth quadric in P There are quadrics which arise as

0

image of a decomp osable X one can recover the pair from the image of the origin of X

1 1

in P P Any two of these quadrics intersect in lines these are eigenlines

As in the g case the image of H is the complement of the eigenspaces and is isomorphic

2

to the mo duli space

H PV f eigenspaces g A

2 g

The space PV is isomorphic to the Satake compactication of A The image in PV of

g

the lo cus of the p erio d matrices of Riemann surfaces is

0

M J PV f quadrics g

2

2

these quadrics are the images of decomp osable ab elian varieties It can b e shown that the

universal cover of M is the Teichmuller space T

2 2

The nite group G is the symmetry of the conguration of lines and

2 2

quadrics it sits in an exact sequence

4

ZZ G S

6

where S is the symmetric group the is related to the Weierstrass p oints on a genus two

6

4 

curve The action of the subgroup ZZ on PV coincides with the action of H C on PV

+

This subgroup xes the quadrics and xes the pairs of eigenlines but can p ermute PV and

a

PV The group S p ermutes the pairs of lines like it p ermutes the unordered subsets

6

a

with two elements fi j g f g it p ermutes the quadrics like the unordered

pairs of complementary unordered subsets with three elements f fi j k g fl n mg g with

fi j k g fl n mg f g

December

+ 3 7

g In this case each eigenspace PV P is a linear subspace of PV P We

a

+

in p oints From know that for indecomp osable H the threefold X intersects PV

3

a

+

a+

H Since any p oint of order maps to an the g example we know that PV

2

a

a+

eigenspace its image lies in a b oundary comp onent H Thus again we get for free

2

that for all H and x X of order two the pair X x satises the SchottkyJung

3

relations

7

The fold H P has degree its dening equation is given in GG

3

+

g Now a p oint y of order maps to an eigenspace PV of dimension and may

a

a+

or may not lie on the fold H We will discuss this situation in the next lectures

3

Remark For g the indeomp osable ab elian varieties are all Jacobians For a

Jacobian X of any dimension g we know from that X H contains a surface

g

1

and that this surface contains all the p oints of order these are Since is indecop osable

4

+ +

a+

H PV H it is X PV consists of images of p oints of order As

g g 1

a a

clear that for any choice of p erio d matrix for the Jacobian and any x X fg the pair

X x satises the SchottkyJung relations We sketch another pro of in

Schottky Lo ci

We dene two lo ci algebraic subsets of the quasi pro jective variety A of imp ortance

g

for the study of the SchottkyJung SJ relations These were introduced by Donagi who showed

that there is an interesting dierence b etween them when g The rst one is

big

S f A nH x X fgst X x satises the SJ relations g

g g g

g

For a given ppav X we ask for at least one p oint of order two a such that X a satises the

SchottkyJung relations We can also ask for the ppavs X such that for all a X fg we

have the SchottkyJung relations

smal l

S f A nH X x satises the SJ relations x X fg g

g g g

g

smal l

The Schottky lo cus S as dened in vG coincides with S

g

g

We already observed the following result in other pro ofs were given by Schottky and

Jung Rauch and Farkas Fay and Mumford

Theorem Let j M A b e the closure of the lo cus of Jacobians j M in A

g g g g

Then

smal l big

j M S S

g

g g

December

smal l

Scketch of pro of of Theorem Since S is closed it suces to show

g

smal l

Let C b e a Riemann surface of genus g choose a H with X J acC j M S

g g

g

0 

H C H C Z and let x X fg We will construct a ppav P of dimension g

1 x

such that there exists a H with P X and with a x

g 1 x

a+ +

X PV

a

2g 2

a+

H which implies the SchottkyJung relations if one of the p oints on the right is in

g 1

then all of them are To construct P one constructs rst an unramied double cover of the

x

Riemann surface

0

P ic C corresp onds to a line bundle L on C with The p oint of order two x J C

L L O As in we get a curve C L the square ro ot of The bundle pro jection

x

L C restricts to an unramied map C C Thus the genus of C is g The

x x x

0

Prym variety of the covering is dened to b e kerN m J C J C with N m the map

x x

0

induced by and stands for connected comp onent of the origin in this case kerN m has

x

two comp onents

Another way to construct the unramied cover C C is to use the theory of covering

x x

spaces and fundamental groups Note that

1

X H C ZH C Z H C ZH C Z

1 1 1 1

2

Using the intersection form on H C Z the p oint x denes a co dimension one subspace

1

?

x in this F vector space

2

?

x fb X xb mo d g

ab

The surjective Hurewich map C C H C Z comp osed with the mo d map

1 1 1

1 ?

gives a surjection C X Then G x is a subgroup of index two of

1 x

C The quotient of the universal cover of C by G is a unramied cover of C and this

1 x

is C moreover G C Now P is dened as ab ove

x x 1 x x

Next one studies the theta functions on the Jacobian of C and restricts them to the image

x

of J C in J C and to P The main result is the SchottkyJung prop ortionalities which

x x

for suitable p erio d matrices of J C and of P state that x PV coincides with

x

+

PV PV

a

1

g

C Then These prop ortionalities are derived in C p with x

2

1

g

y x C corresp onds to a p oint in X with y x and thus maps to an eigenspace of

2

U x Explicit computations and elementary manipulations show

(

0

g

2

y

(0 )

1 0 0

g

2

y y with

y

(1 )

g

2

were we mix our notation on the left with Clemens notation on the right and we to ok p q

in Clemens formula That formula the SchottkyJung prop ortionalities shows

0

0

0

1 0 0

December

0 g 1

where the index runs over ZZ In our notation we have

0

0

0

To get our geometrical interpretation of the SchottkyJung prop ortionalities note that the

+

which contains y is dened by X if The natural co ordinates eigenspace PV

1

a

on this eigenspace are the X which are induced by the X on PV Thus the natural map

 (0 )

a+ +

H ! PV  PV

g 1

a

is given by

(

if

( ) 1

g

2

g

with c 7! c

2(Z2Z)

if

1

a+

Thus we see that y where 2 H is a p erio d matrix of the Prym variety P 2

g 1 x

Results

Here are the three main results on the Schottky lo ci The rst shows that the Schottky

problem is solved in genus For essentially trivial reasons for g  the big Schottky lo cus

is really bigger than the small Schottky lo cus and thus cannot coincide with the Jacobi lo cus

The second surprising result is of Donagi who shows that there is in g also an interesting

dierence b etween the big and small Schottky lo cus Finally one can show that for any g the

j M Schottky lo cus is lo cally equal to

g

We observe that for g it is not known if j M and S coincide For example we do

g g

not even know if they coincide for g Donagis example shows that this may not b e so easy

to decide

Theorem Schottky Igusa Freitag We have

big

smal l

S S

4

4

and this lo cus is dened by one explicitly known mo dular form of weight for S p Z

4

Moreover Ig Fr

smal l

j M S

4

4

smal l big

Theorem For g  we have S 6 S

g g

Pro of Let 2 H J and let 2 J Then the isomorphism class of X X  X in

4 4 g 4

smal l big 0

A is not in S but it is in S For a a 2 X the pair X a do es not satisfy the

g

g g

00

SchottkyJung relations but any pair X a will satisfy the SchottkyJung relations 2

December

4

Theorem Donagi D Let Y P b e a smo oth cubic threefold Let J Y

21 

H Y H Y Z b e its intermediate Jacobian J Y is a ppav of dimension Then J Y

3

big

J J Y is not decomp osable ie is not a pro duct of lower dimensional ppavs but J Y S

5

5

Pro of The theta divisor of J Y has just one singular p oint B Since the theta divisor

of a Jacobian has a singular lo cus of dimension g and a decomp osable ppav has

dim S ing g we get the rst two statements For the most interesting result

big

J Y S one uses the following relation discovered by Donagi

5

Let C b e Riemann surface and let a b J C with E a b where E J C

X J C ZZ is the Weil pairing which induced by the p olarization Let P

(a) a

? ?

a a with a as in X b e the corresp onding Pryms One can prove that P P

a b

(b)

the pro of of and we have a similar result for b Thus we get p oints b P and a P

a b

Donagi proves that one can choose p oints u P with u b and v P with v a such

a b

that



b+ a + + +

u PV PV PV

a

b

(a) (b)

+

etc strictly sp eaking we should write PV

(a)

g 2

+

2 1 +

P is an eigenspace b H the Heisenberg group acting The space PV PV

g 1

a b

+

+

on PV and also for an a in the copy of H acting on PV Then we have the natural map

g 1

a b



a b+ + +

H PV PV

g 2

a

b

Donagis relation implies if P b satises the SchottkyJung relations then also P a satisifes

a b



 

b+ a +

a b+ a b+

them b ecause u H implies H

g 2 g 2

(a) (b)

Next Donagi chooses a genus curve C a plane quintic and a b in such a way that

P J D for some genus curve D and P is the intermediate Jacobian of a cubic threefold

a b

Thus P b obviously satises the SchottkyJung relations and one concludes that so do es

a

P a 2

b

smal l

The lo cus S is an algebraic subset of A and thus it is a nite union of irreducible

g

g

comp onents One would like to show that it has only one comp onent and that this comp onent

is j M The next theorem shows that is the case The pro of which we sketch in do esnt

g

give much information on the existence of other comp onents

smal l

Theorem van Geemen The variety j M is an irreducible comp onent of S

g

g

Equations for the Schottky lo ci

We show how one expresses the SchottkyJung relations in terms of mo dular forms

these are used in the pro of of

1

g  1

n m C for some n m For a H C we can represent a a X by

2

1

g

f g Let b n m then b represents a p oint b X with b a We have

4

December

+

seen that b lies in an eigenspace of a lets call that one PV Then X a satises the

a

SchottkyJung relations i

+

b H PV

g 1

a

H PV will b e denoted by I so I The ideal of the pro jective variety

g g g

g

contains all the homogeous p olynomials which are zero on that va C X

f0 1g

a+

riety Using the Heisenberg group actions I can b e identied with the ideal I of

g 1

g 1

+

a+

H PV for any a H C Then X a satises the SchottkyJung relations

g 1

a

if and only if

a+

P P b P I P homogeneous

a

g 1

For every a H C and every homogeneous P I we get a holomorphic function

g 1

P H C Since the co ordinate functions of b are mo dular forms of half

a g

integral weight the holomorphic functions P are also mo dular forms So we see that the

a

Schottky lo ci can b e dened by mo dular forms and that

smal l

S f A nH P a H C fg and all homogeneous P I g

g g g a g 1

g

In general we know very little ab out I In the rst nontrivial case g we know that I

g 3

is generated by one p olynomial F of degree This implies that in genus we get a mo dular

form F of weight for each a H C Schottky already proved that all these mo dular

a

forms are the same and that they are not identically zero Much later Igusa and Freitag proved

that the zero lo cus of this mo dular form is exactly J It is known that I contains elements of

4 4

degree

Pro of of Theorem We give a rough sketch of the pro of of Theorem In

fact to simplify matters we will assume that we have an universal family of ab elian varieties

over A and this family as well as A are smo oth To justify the arguments given here one has

g g

to use level structures We will also make some other simplifying assumptions but we b elieve

that we still convey the main idea of the pro of

smal l

Since S is a quasipro jective variety it is a nite union of irreducible comp onents

g

smal l

S Z Z Z

1 2 n

g

For simplicity we write

J j M A

g g g

smal l

As J S and J is irreducible it is the closure of the image of the Teichmuller space it

g g

g

must b e contained in at least one of the Z lets say J Z If dim Z g dim J then

i g 1 1 g

since b oth are irreducible J Z So the theorem follows if we prove that dim Z g

g 1 1

We will use induction on g to prove this the case g or g b eing trivial

A is the disjoint union of A and A It is We recall that the Satake compactication

g g g 1

wellknown that J the closure of J in A intersects A in J It is not hard to show that

g g g g 1 g 1

something similar happ ens with the small Schottky lo cus

smal l

smal l

J A J S A S

g g 1 g 1 g 1

g 1 g

December

smal l smal l

Z \ A then Z  S Since J  Z we get J  Z  S The Let Z

1 g 1 g 1 g 1

g 1 g 1

1 1 1

smal l

implies that induction hypothesis that J is an irreducible comp onent of S J is an

g 1 g 1

g 1

A in A is very large it is irreducible comp onent of Z Unfortunately the co dimension of

g 1 g

1

1 1

g g g g g Therefore we cannot get a go o d estimate of dim Z from this fact

1

2 2

Igusas compactication We consider another compactication of A introduced

g

by Igusa Let

A ! A

g g

1

A along its b oundary A The inverse image A of A is a b e the blow up of

g g 1 g 1 g 1

divisor on A Let Z b e the closure of Z in A and let Z b e an irreducible comp onent of

g 1 1 g

1

1

Z \ A mapping onto the irreducible comp onent J of Z

1 g 1 g 1

1

Assuming there are smo oth p oints of A in Z we get

g

1

dim Z dim Z dimZ

1 1

1

If we can show that the map

Z ! J

~

g 1

1

jZ

1

has a b er of dimension  then it follows that dim Z  g g and thus

1

dim Z g g as desired

1

The inverse image of the op en subset A  A under the map is the universal family

g 1 g 1

1

of ab elian varieties over A note we simplify here Thus we may identify X

g 1

g

Since Z  S the closure of the small Schotky lo cus in A it suces to show that

1 g

smal l

g

dimS \ X 

smal l

smal l

Recall that S is dened by the mo dular forms P with P 2 I We need to know

a g 1

g

1

the intersection of the closure in A of the zero lo cus of P with X

g a

For this we consider a p erio d matrix

z

g 1

2 H 2 H z 2 C 2 H

g g 1 1 1

z

1

for any z we can nd with I m such that 2 H This matrix moves

1 1 g

2 i

1

to the b oundary p oint 2 A if we let I m ! 1 equivalently q e ! here

g 1 1

The mo dular form P has a q expansion

a

X

n

P z q P

n a

nZ

0

The functions z 7! P z are theta functions on X The intersection of the zero lo cus of

n

P with X is given by zeros of P z where k is the smallest integer for which P z

a k n

is not identically zero

The co ordinates b of b have the q expansion

b z q H O T

  1 g 1

December

when Since P is a p olynomial we get the following formula

1 g

X

P

z H O T P P q

  a 

X





Since the p olynomials P I are zero on the image of H we have P

g 1 g 1 

P

P

We have seen in that z is in and moreover if is a

  00



X



smo oth p oint on H then is spanned by these functions where P runs over I

g 1 00 g 1

Putting everything together we get

smal l 1

S F fz X z g

00

g

Thus it suces to show that for the p erio d matrix of a genus g curve we have

dim F Fortunately this was proved by Welters and thus the pro of is complete

1 1

Remark One can see why J J C is C C J C J C If one

g

degenerates a genus g curve to a curve with a no de the normalization C of the no dal curve has

genus g and the inverse image of the no de are two p oints p q on C Following the mo duli

p oint of the genus g curve we arrive at the p oint J C A in the Satake compactication

g 1

1

A and at the p oint p q F J C J C in Igusas compactication A

g J (C ) g

Remark Recall that in case g the lo cus W of intermediate Jacobians of cubic

big

threefolds are contained in S W in the Satake compactication intersects the Its closure

5

b oundary A in j M Consider now the closure W A of this lo cus in Igusas compacti

4 4 5

cation and the blow down map W W Let JC j M b e a general Jacobian then one

4

1 1 1

nds that JC W consists of the p oints g h which are exactly the exceptional

3 3

p oints in F the subvariety dened by see

JC 00

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December

CGV M Cornalba X GomezMont A Verjovsky Lectures on Riemann Surfaces World

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Math 157

December

Dipartimento di Matematica

Universita di Torino

Via Carlo Alb erto

Torino

Italy geemendmunitoit