Lifting irreducible Galois representations

TL;DR

This paper investigates conditions under which irreducible mod p Galois representations, valued in reductive groups, can be lifted to geometric or p-adic representations, with results depending on the nature of the number field and local conditions.

Contribution

It establishes new criteria for lifting irreducible Galois representations over number fields, including totally real fields, under various local and global conditions.

Findings

Odd representations over totally real fields admit geometric lifts.

Representations over general fields admit p-adic lifts if locally realizable.

Lifting results depend on ramification and multiplicity conditions.

Abstract

We study irreducible mod p representations, valued in general reductive groups, of the Galois group of a number field. When the number field is totally real, we show that odd representations satisfying local ramification hypotheses and a certain multiplicity-free condition on the adjoint representation admit geometric lifts. For general number fields, we show without any oddness or multiplicity condition that the representation admits a p-adic lift if it does everywhere locally.