Comparing Galois representations and the Faltings-Serre-Livné method

Abstract

In 1984, Jean-Pierre Serre, based on the ideas of Gerd Faltings, explained in his course at the Collège de France a method for comparing irreducible $\ell$-adic Galois representations. This method would later be anointed as the Faltings-Serre method by the mathematical community. In 1987 Ron Livné gave an algorithm to compare the case of 2-dimensional 2-adic Galois representations with even trace. In 2008 Gabriel Chênevert generalised it erasing the condition on the traces. In this thesis we are going to draw on his work to explore and formalise Serre’s ideas. In addition, we are going to collect some examples from Serre himself in the case of 2-dimensional 2-adic Galois representations from elliptic curves to understand the use of it and we are going to explain them in detail. Finally, we are also going to study Livné’s approach and give an example of this as well.