Comparison of semi-simplifications of Galois representations

AbstractLet Γ be the absolute Galois group of a global field. Let ρ1 and ρ2 be two p-adic, finite dimensional representations of Γ. Then there exists a finite number of primes q such that if the characteristic polynomials of ρ1(Frobq) and ρ2(Frobq) are equal, then ρ1 and ρ2 have isomorphic semi-simplifications and so the same L-functions. We give a method to compute a sufficient list of primes, based on the ramification and the dimension of the representations. We use then the result to prove a conjecture by B. van Geemen and J. Top