Introduction

xix

forms for T0(k). Part of the extra difficulties involves the presence of tor-

sion in these groups. We need more detailed analysis, in this and subsequent

chapters, to handle these richer groups.

Chapter 4 is a systematic study of theta identities. Theta constant

identities are interesting for several reasons. One reason is their inherent

elegance and symmetry. There is something tantalizingly beautiful about

the identities of Jacobi, for example,

" 0 '

0

(O,r)02

" 0 "

0

" 0 '

1

(O,r)02

" 0 "

1

(z,r)+92

' 1 "

0 _

(O,r)02

" 1 "

0

or its restriction to z = 0 (known as the Jacobi quartic identity)

" 0 "

0

(o,r) =

e4

' o"

I

(0,r)+64

' 1 "

0

Aside from the inherent beauty of the form there is a combinatorial content

to the identity. It relates the number of representations of an integer as a

sum of four squares to its representation as a sum of four triangular numbers.

This is of course just the beginning of a chapter. As one delves deeper into

the theory, one finds more and more beautiful identities with more and more

combinatorial content.

We present four distinct ways of constructing such identities. In two

of theses methods, we use the classical technique of constructing finite di-

mensional linear spaces of theta functions or modular forms or functions on

certain Riemann surfaces and use linear algebra and the simple notions of

independence and dependence. We present another newer technique which

makes use of the fact that we can use theta functions to construct elliptic

functions and the fact that the sum of the residues of an elliptic function

in a period parallelogram vanishes. This technique is very powerful and

succeeds in giving a rather large set of identities. The main idea here is to

construct the correct elliptic function, which turns out to be more an art

than a science, that leads to an interesting identity. The fourth method for

constructing identities uses uniformizations of Riemann surfaces. In a very

nontrivial sense the next two chapters are also studies of theta identities,

this time of a very special sort with a very special purpose.

In Chapter 5 we turn to the congruences discovered by Ramanujan for

the partition function and show how they follow in a rather simple way from

function theory on the appropriate Riemann surfaces. The main ingredient

is the construction of the same function in more than one way. Some of

the constructions involve averaging operators. It turns out that the aver-

aging processes produce in some cases constant functions. We study the