  We give a criterion for two l-adic Galois representations of an algebraic
number field to be isomorphic when restricted to a decomposition group, in
terms of the global representations mod l. 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.
