Congruences between modular forms

Abstract

This master s thesis is intended to give a presentation of the theory of congruences between the Fourier coecients of modular forms. In order to do that we introduce the reader to the basic theory of modular forms from the beginning and we study the structure of their Fourier coecients in di↵erent ways using Hecke operators. Then we start the theory of congruences finding some of them by classical methods of Number Theory. After that, we introduce the advances made by Swinnerton-Dyer in the study of congruences using l-adic representations and the generalisation by Ken Ono. Finally, we explain the papers by Hida and Ribet in two chapters giving some conditions for the existence of congruences using the associated L-functions and decomposing the space of modular forms.