Some fundamental results on modular forms

The purpose of this paper is to give complete proofs of several fundamental results about modular forms. Modular forms are complex functions with certain analytic properties, and that transform nicely under a certain group of transformations of the complex upper half plane. It turns out that modular forms can be used to study number theory, by investigating the coefficients in series expansions of them. In fact, using modular forms we can discover and prove things in number theory where a direct proof might not be obvious. To help the reader get a feel for this, we will give an example; all the terms used here are defined in the paper. An important class of modular forms, called Eisenstein series, have expansions that involve the divisor sum σr(n) = ∑ d|n d . Another modular form, called the modular discriminant, has a series expansion that involves the Ramanujan tau function, τ(n). Modular forms comprise vector spaces whose dimensions we can explicitly determine. Using this information one can prove the congruence τ(n) ≡ σ11(n) (mod 691).