3 - Modular forms for SL(2; Z)

Summary

In Chapter 2, the dependence of elliptic functions on the points in the torus was studied for a fixed lattice. In this chapter, it is the dependence on the lattice that will be investigated. The modular group SL(2,Z) is introduced as the group of automorphisms of the lattice, and its generators, elliptic points, and cusps are identified. The hyperbolic geometry of the Poincaré upper half plane is reviewed, and the fundamental domain for SL(2,Z) is constructed. Modular forms and cusp forms are defined and shown to form a polynomial ring. They are related to holomorphic Eisenstein series, the discriminant function, the Dedekind eta-function, and the j-function and are expressed in terms of Jacobi theta-functions. The Fourier and Poincaré series representations of Eisenstein series are analyzed as well.