An effective isomorphy criterion for mod ℓ Galois representations

In this paper, we consider mod ℓ Galois representations of Q. In particular, we develop an effective criterion to decide whether or not two mod ℓ Galois representations Q are isomorphic. The proof uses methods from Khare-Wintenbergerʼs recent theorem on Serreʼs conjecture along with theorems by Sturm and Kohnen.