Modüler formlar arasındaki denklikler

Modular forms admit Fourier expansions whose coefficients encode a lot of interesting arithmetic information. One fruitful method of studying these coefficients is through studying their congruences modulo primes l or powers of l. This theory can be traced back to the work of Ramanujan, and was developed by Serre, Swinnerton-Dyer, and Katz, among others. In this thesis, we will study Serre and Swinnerton-Dyer's theory of modular forms modulo l. We will examine some applications of this theory, including Jochnowitz's proof of finiteness of systems of Hecke eigenvalues modulo l. We will also give a brief exposition of Serre's theory of l-adic modular forms.