An effective isomorphy criterion for mod $\ell$ Galois representations

Yuuki Takai

Published 2010-09-30, updated 2010-10-14Version 2

In this paper, we consider mod $\ell$ Galois representations of $\mathbb{Q}$. In particular, we obtain an effective criterion to distinguish two semisimple 2-dimensional, odd mod $\ell$ Galois representations up to isomorphism. Serre's conjecture (Khare-Wintenberger's theorem), Sturm's theorem, and its modification by Kohnen are used in our proof.