Different versions of the modularity theorem

Abstract

One of the most interesting results in number theory is the proof of the Modu larity Theorem. The Modularity Theorem has many different versions. The geometric version states that there is a surjective morphism between elliptic curves and modular curves over the field of rational numbers. The arithmetic version states that there is a relation between elliptic curves over the field of rational numbers and modular forms. In this thesis, we will give an outline of a proof of the fact that the geometric version of the Modularity Theorem implies the arithmetic version.