I will present some results in the literature that answer the following two kinds of questions
about Galois representations:
(1) Let K be a field and S be a finite set of places of K. Let E be a topological field. Under what
conditions can one prove that there are only finitely many isomorphism classes of representations of the Galois group $G_K,S$ over E?
(2) In the above setting, one may pose a weaker problem: when can one prove that two such Galois
representations are isomorphic?