On congruences of Galois representations of number fields

TL;DR

This paper establishes criteria for when two l-adic Galois representations of number fields are locally isomorphic based on their mod l reductions, with applications to conjectures and modular form coefficients.

Contribution

It provides a new criterion for local isomorphism of Galois representations using global mod l data and extends existing conjectures and results in the field.

Findings

Proved a criterion for local isomorphism of Galois representations

Generalized a conjecture of Rasmussen-Tamagawa under semistability

Derived a congruence relation for Fourier coefficients of modular forms

Abstract

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.