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.
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