Comparison of semi-simplifications of Galois representations

Abstract

Let G be the absolute Galois group of a global field. Let r1 and r2 be two p-adic, finite dimensional representations of G. Then there exists a finite number of primes q such that if the characteristic polynomials of r1(Frob_q) and r2(Frob_q) are equal then r1 and r2 have isomorphic semi-semplifications 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 apply then the result to an article of B. van Geemen and J. Top where they described two representations and conjectured that they had isomorphic semisemplification.