Modularity of $\operatorname{PGL}_2(\mathbb{F}_p)$-representations over totally real fields

TL;DR

This paper investigates a Serre-type modularity conjecture for projective Galois representations over totally real fields, proving new cases for representations over F_5 using automorphy lifting theorems.

Contribution

It extends the modularity conjecture to F_5 representations over totally real fields and proves new cases leveraging automorphy lifting theorems.

Findings

Proves new cases of the conjecture for F_5 representations.

Utilizes automorphy lifting theorems over CM fields.

Advances understanding of Galois representations over totally real fields.

Abstract

We study an analogue of Serre's modularity conjecture for projective representations $\overline{\rho}: \operatorname{Gal}(\overline{K} / K) \rightarrow \operatorname{PGL}_2(k)$, where $K$ is a totally real number field. We prove new cases of this conjecture when $k = \mathbb{F}_5$ by using the automorphy lifting theorems over CM fields established in previous work of the authors.