Faltings Serre method

The starting point of this method is Falting’s article in which he proves the Mordell-Weil theorem. He remarked and Serre turned it into a working method, the fact that the equivalence of two λ-adic representations is something that can be basically determined on some finite extension of the base field (even though the representations might not factor through a finite quotient). Let Oλ be the ring of integers in a finite extension of Ql and consider two semisimple representations ρ1, ρ2 : G→ GLn(Oλ) of a given group G (we will focus mostly on Galois group, but will stick for now in this generality). We will be interested in the image M of Oλ[G] in Mn(Oλ) ×Mn(Oλ) by the linear map induced from ρ1× ρ2 and δ(G) the image of G in the quotient (M/λM)×. δ(G) will be called the deviation group of the pair (ρ1, ρ2) . We shall prove that the group δ(G) is a finite quotient of G, unramified outside Ram(ρ1)∪Ram(ρ2) and has the property that for any Σ ⊂ G surjecting onto δ(G) we have that