XVI

Introduction

The main actors in our presentation are genus one theta functions and

theta

constants1

(including the classical rj-function), the modular group V —

PSL(2, Z), and some of the Riemann surfaces that arise as quotients of the

action of finite index subgroups of T on

H2.

We are particularly interested

in the principal congruence subgroups T(k) and the related subgroups T0(k)

for (usually small) primes k. Some very interesting combinatorial identities

follow from the function theory on these

surfaces.2

Theta functions and theta constants with integral characteristics are

classical objects intimately connected with the principal congruence sub-

group of level 2, T(2). This theory is well understood and has as one of its

consequences the theorem of Picard: every entire function which omits two

values is constant. As is well known, the basic ingredients in the proof of

Picard's theorem are that the holomorphic universal covering of the sphere

punctured at three points is the upper half plane and that its fundamental

group is T(2). We use theta constants with even integral characteristics

to construct the universal covering map, and in this way obtain, without

using the general uniformization theorem, the hyperbolicity of the three

times punctured sphere. The universal covering map is constructed here as

a quotient of fourth powers of any two of the three theta constants. We

noticed that in this construction the three even characteristics correspond

in a natural way to the three punctures on H

2

/r(2), and we began to won-

der about natural generalizations. In this book, we present the answer to

these inquiries. We uniformize the Riemann surfaces

H2/r(fc)

using theta

constants with special rational characteristics, and establish a one-to-one

(almost canonical) correspondence between the punctures on

M2/T(k)

and

certain equivalence classes of characteristics. For example, the four punc-

tures on H

2

/r(3) correspond to the characteristics

' ' '

Furthermore, the Riemann surface is uniformized by a quotient of cubes

of any two of the four corresponding theta constants. Similar, but obvi-

ously more complicated, expressions uniformize the surfaces represented by

the higher level congruence groups. Multiple uniformizations of the same

1

Thus the #-functions we study, #[x](C

r)

depend on three variables: a characteristic x G l

2

;

a variable ( G C ; and a parameter T E H 2 , the upper half plane. Fixing the variable £ = 0 yields

the family of theta constants, an abuse of notation since these are holomorphic functions on H 2 ;

as functions of the local coordinates q — e2lx%T these are classically known as g-series. We will

use the symbol x for the local variable, since tradition in (parts of complex analysis) reserves the

letter q for the weight of an automorphic form.

2

What is interesting is clearly in the eyes of the beholder. The identities we discuss are

obviously interesting to us. The reader must decide whether or not to share our enthusiasm.