On the equivalence of 3-adic Galois representations

Throughout this thesis, we develop theory and the algorithms that lead to an effective method to study the equivalence of two-dimensional 3-adic Galois representations attached to number fields. In order to reach our goal we need three steps: recognising the determinant characters, determining the residual or mod3 representations, and finally proving that the representations agree modulo 3k for any positive integer k. We are able to achieve this using only a finite amount of information coming from the representations. We start with a method that allows us to recognise any one-dimensional Galois representation of any number field K that is unramified outside a given finite set S of primes of K. Afterwards, we extend the methods developed by Argaez- Garcia and Cremona to determine 2-dimensional black box Galois representations of K, unramified outside a given finite set S, whose image lies in GL2(F3). If such representations are irreducible, we can also prove if they are equivalent over GL2(F3). Moreover, due to recent results in modularity lifting, these two methods have an impact on solving modularity problems. Furthermore, starting from two 2-dimensional 3-adic Galois representations of K unramified outside the same S, which we proved by the previous results to have the same determinant character and equivalent residual representations, we are able to prove whether they are equivalent over GL2(Z3) by checking their traces at finitely many places. Finally, since we are able to achieve each step by just computing characteristic polynomials of Frobenius elements of GK, the absolute Galois group of K, at a suitable and computable finite set of primes of K, all our theoretical result are actually effective. That is, we can, and we did, implement them as algorithms that return a precise answer in a finite (and reasonable) amount of time. An application of the algorithms developed is to prove modularity of elliptic curves. We address this studying Galois representations attached to Bianchi modular forms and elliptic curves defined over imaginary quadratic fields of class number one.