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
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