Summary:
The purpose of this master thesis is to study Modular forms and Elliptic
curves. Both modular forms and elliptic curves are important areas of modern
number theory. The aim of this work is to study the deeper relationship between
these two number theoretic structures, as expressed by the Modularity theorem.
The structure of work is as follows: in the first chapter develops the basic
theory of algebraic tools needed for the study of elliptic curves, while the
second chapter studies the classical theory of geometry and arithmetic of
elliptic curves. In particular, we consider the problem of determining the
structure of E(K) for a given body K.
In the third chapter we enter the analytic part of theory. Here, the central
point of study are the upper complex half-plane and modular curves. In the
fourth chapter we define modular functions and modular forms and examine the
structure of these spaces, as well as the normal operators acting on them.
Furthermore, we develop the theory of moduli interpretation.
Finally, in the last chapter, we describe the Modularity theorem by a few
different versions.
