Introduction

xvn

Riemann surface lead to theta identities. It is an open problem to deter-

mine uniformizations of all four punctured spheres by the methods described

above.

The theta constants which appear in our constructions of the uniformiz-

ing functions for H

2

/r(3) are closely related with the formulae used by Euler

and Ramanujan in the theory of partitions. Specifically, we note

that3

(o,T ) = exp(^)*£ £

(-irx3-^

=

e^Qx^ii(i-xn,oo

n=—oo n = l

where x = exp

(^z)-

Continuing in this direction, we discover that uni-

formizations of the Riemann surfaces

H2/r(£;)

involve functions which ap-

pear in the Jacobi triple product. We give a function theoretic proof of this

famous formula and then generalize it to the quintuple and septuple product

identities, explaining along the way why the formulae obtained are natural

from the point of view of the theory of N-ih order theta functions. The

highlights of the book are systematic studies of theta constant identities,

uniformizations of surfaces represented by subgroups of the modular group,

partition identities and Fourier series coefficients of automorphic functions,

and identities involving the a-function and Fourier series coefficients of au-

tomorphic forms. More detailed information on the contents of each of the

chapters follows.

In Chapter 1 we explain the genesis of the modular group in our theory.

This group appears naturally when one classifies compact Riemann surfaces

of genus one (elliptic curves) up to conformal equivalence. We discuss the

generators of this group, find all the fixed points of elements of this group

and describe some of the subgroups we shall need in the sequel. Almost

everything we do in this chapter is well known and covered in a standard

course on complex variables. We describe the structure of the Riemann

surfaces

M2/G

for subgroups G of PSL(2, Z). In order to show that this well

known and elementary material has nontrivial consequences, we use this

theory to show that factors of integers of the form

iV2

+ 1 are always the

sums of two squares, and we give a geometric criterion for

N2

+ 1 to be a

prime number. The result is that

iV2

+ 1 is prime if and only if the portion in

the upper half plane of the straight line joining the origin to the point N +1

in the complex plane intersects the orbit of i under PSL(2, Z) in exactly two

points, namely N + i and ^ t \ - We have included some of the function

theoretic prerequisites in this chapter. However, most of the prerequisites

will be described when needed. In general, we provide full definitions of

all concepts. We do not repeat proofs or arguments readily available in

3 The reader may at this point conclude that the 77-function is a disguised theta constant with

a rational characteristic. It will also become obvious that the prime 3 plays a special role in our

drama.