This chapter introduces modular functions and forms, a subject central to the remainder of the book. Some earlier parts of this chapter are beautifully covered in .
Section 2.1 supplies the underlying geometry, but can be skimmed on a first reading. In spite of this background material, the theory of modular forms and functions discussed in Sections 2.2 and 2.3 will probably appear as somewhat arbitrary to the uninitiated reader. Section 2.4.1 addresses some of this apparent artificiality, by developing the broader context of automorphic forms.
As explained in the introductory chapter, Moonshine involves unexpected occurrences of modularity. The modularity of Moonshine functions follows from Zhu's Theorem (Theorem 5.3.8). However, the complexity of the underlying mathematics begs the question: Can modularity be established in a more elementary way? The simplest example of Moonshine involves theta functions. Hence we explore the limits and potentials of four classical strategies for proving the modularity of theta functions: Poisson summation, Dirichlet series, the heat kernel and representations of Heisenberg groups (Sections 2.2.3, 2.3.1, 2.3.4 and 2.4.2, respectively).
Moonshine has really only been worked out in genus 1, but conformal field theory tells us that there is an analogue for every genus (Section 6.3.1). It will be much more complicated, but it will be more rewarding because the number theoretic side is much less developed. In otherwords, we will find traces of, for example, the Monster in automorphic forms for the higher mapping class groups Γg,n and Sp2n(ℤ). We include Sections 2.1.4 and 2.3.5 in anticipation of this most natural and significant future development.