An irreducibility criterion for group representations, with arithmetic applications

TL;DR

This paper establishes a criterion for determining when an integral group representation remains irreducible after reduction, with applications to deformations of Galois representations linked to modular forms.

Contribution

It introduces a new irreducibility criterion based on reductions at prime ideals, applicable to universal deformations of residual Galois representations.

Findings

Irreducibility criterion for integral group representations.

Application to universal deformations of Galois representations.

Results relevant for modular forms of weight at least 2.

Abstract

We prove a criterion for the irreducibility of an integral group representation \rho over the fraction field of a noetherian domain R in terms of suitably defined reductions of \rho at prime ideals of R. As applications, we give irreducibility results for universal deformations of residual representations, with a special attention to universal deformations of residual Galois representations associated with modular forms of weight at least 2.