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

TL;DR

This paper introduces an effective criterion to distinguish two semisimple 2-dimensional odd mod  Galois representations over q, utilizing Serre's conjecture, Sturm's theorem, and Kohnen's modification.

Contribution

It provides a new, practical method to identify isomorphism classes of mod  Galois representations, advancing the understanding of their classification.

Findings

Developed an effective isomorphy criterion for mod  Galois representations.

Applied Serre's conjecture, Sturm's theorem, and Kohnen's modification in the proof.

Enhanced tools for distinguishing Galois representations in number theory.

Abstract

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.