On Congruences of Galois Representations of Number Fields

Abstract

We give a criterion for two $\ell$-adic Galois representations of an algebraic number field to be isomorphic when restricted to a decomposition group, in terms of the global representations mod $\ell$. This is applied to prove a generalization of a conjecture of Rasmussen-Tamagawa under a semistablity condition, extending some results of one of the authors. It is also applied to prove a congruence result on the Fourier coefficients of modular forms.