Abstract

In this thesis we study a paper by Barry Mazur ([11]) about deforming Galois representations. In particular we will prove that, if $\bar{\rho}: \Pi \rightarrow \mathrm{GL}_N(k)$ is an absolutely irreducible residual representation, a universal deformation ring $R=R(\Pi,k,\bar{\rho})$ and a universal deformation $\boldsymbol{\rho}$ of $\bar{\rho}$ to $R$ exist. This result is part of the proof of the modularity conjecture.
The modularity conjecture is of great importance since it states a connection between modular forms and elliptic curves over $\Q$, providing a great tool to study the arithmetic properties of those elliptic curves. Andrew Wiles studied the conjecture as a part of the more general problem of relating two-dimensional Galois representations and modular forms and used [11] to complete his construction.
To better understand the proof of Mazur, we will analyze in detail the paper of Michael Schlessinger ([13]). This article, which is focused on functors over Artin rings, provides a criterion for a functor to be pro-representable. Moreover, it gives the definition of a "hull", which is a weaker property than pro-representability.