XV111

Introduction

other books, but do reproduce, usually in modified form, proofs from many

research papers. The bibliography of relevant books (after the last chapter)

is followed by a set of bibliographical notes containing an (incomplete) list

of research and expository notes on the material covered by this volume.

In Chapter 2 we define the theta functions and theta constants with

r — i

characteristics and specialize to rational characteristics of the form fy

L T J

with rrij

mf

and k

integers4

of the same parity in order to construct a cor-

respondence between equivalence classes of sets of characteristics and the

punctures on the surface

H2/r(A:).

In this chapter we derive a most impor-

tant property of the theta functions and theta constants, the transformation

formula (a significant generalization of known transformation rules) for the

action of PSL(2,Z) on the upper half plane, and we give a function theo-

retic proof of the Jacobi triple product formula and some generalizations.

The transformation formula allows us to use theta functions to construct

modular and cusp forms for subgroups of PSL(2, Z). The function theoretic

proof of the Jacobi triple product formula yields new proofs of important

identities of Jacobi and Euler that are needed for our presentation of par-

tition theory in Chapter 5. We construct theta constant identities which

turn out to agree with discoveries of Ramanujan. Our derivations of these

identities are on the one hand quite natural, and on the other hand lead to

simpler expressions of the equivalent identities discovered by Ramanujan in

the sense that they do not involve irrationalities (extracting roots of single

valued functions) until they are artificially introduced. It appears that the

theta constants which we use are a lot richer than the ones that Ramanujan

had at his disposal.

Chapter 3, in a sense, contains the most important material of the book.

In it we construct automorphic forms and functions for the principal con-

gruence subgroups and some related groups. The theory we describe is

particularly well suited for the study of

H2/r(/c),

and we obtain holomor-

phic mappings of these Riemann surfaces into projective spaces of rather

low dimensions. Some interesting geometry and topology emerges as we ob-

serve connections of the principal congruence subgroups with the Platonic

solids. This phenomenon first occurs for k = 3, 4 and 5. In these cases,

r/r(fc) = PSL(2,Z/C) are the symmetry groups of the regular tetrahedron,

octahedron and icosahedron, respectively. This suggests a relation between

the images of these curves in the projective space and the regular solids and

leads to a generalization of the regular solids based on curves of (some) pos-

itive genera. While our development is most suited for the groups r(fc), for

many of the most important applications we need to construct automorphic

4Assume, unless otherwise stated, for these introductory remarks that k is a (positive) prime.