Black box Galois representations
The Serre-Faltings-Livné method concerns the comparison of two 2-dimensional 2-adic Galois representations of the absolute Galois group $G_K$ of a number field $K$. Assuming that both are known to be unramified outside the same set $S$ of primes of $K$ and that they have the same determinant, the method provides a finite set $T$ of primes of $K$, disjoint from $S$ and depending only on $K$ and $S$, such that if the two representations have the same trace at the Frobenius elements of all the primes in $T$ then they are isomorphic. When we describe a Galois representation with data $K$, $S$ as a Black Box, we mean that the only information available to us about it is the trace and determinant of Frobenius at any given prime not in $S$; in these terms, the SFL method allows us to determine whether two black box representations are isomorphic. In this talk we consider black box representations from a slightly wider perspective, and consider how much information we can obtain about such a representation by asking only a finite number of questions: information such as residual reducibility, the determinant character, and the size and structure of the isogeny class of the representation. This is joint work with Alejandro Argaez.