Chapter 1

The modular group

and elliptic function

theory

This chapter contains mostly well known material on the modular group

PSL(2, Z) and elliptic curves along with some relatively new material on

once punctured tori. We present in §4 a more or less classical treatment of

function theory on elliptic curves (a euclidean theory) as a model for sub-

sequent work. It is followed by an outline of the equivalent (noneuclidean)

theory on once punctured tori using Poincare series to illustrate another

approach to some of the problems we will encounter in this book. The most

important fact for our subsequent chapters from this material is that the

sum of the residues of an elliptic function is zero. We introduce and de-

scribe the geometry of several families of subgroups of the modular group

that will be needed in the sequel. This section (§6) establishes much of

the notation that will be used throughout our presentation. The chapter

ends with the elementary application to primality testing discussed in the

introduction.

Most of the material in this chapter can be found in the references sup-

plied. Much of the material is presented as background, a guide to different

approaches, and to establish notation. As pointed out above, an exception

is made in subsections of §4, where complete arguments are provided.

1

http://dx.doi.org/10.1090/gsm/037/01