Elliptic Curves, Modular Forms and p-adic Heights

Abstract

The aim of this thesis is to provide an introduction to the study of elliptic curves and modular forms over general commutative rings or schemes. We will recall a few aspects of the classical theory of these objects (over the complex numbers) while placing emphasis on the geometric picture. Moreover, we will formulate the theory of elliptic curves in the modern language of algebraic geometry following the work of Katz and Mazur. In addition, we provide an application of p−adic modular forms to the theory of p−adic heights on elliptic curves.