Companion Forms in Parallel Weight One

TL;DR

This paper establishes a criterion for when mod p Galois representations come from mod p Hilbert modular forms of parallel weight one, using modularity lifting and geometric methods, with implications for Serre's and Artin's conjectures.

Contribution

It proves a companion forms theorem for parallel weight one Hilbert modular forms over totally real fields, expanding understanding of Galois representations and modularity.

Findings

Provides a sufficient criterion for mod p Galois representations to originate from weight one forms.

Demonstrates that Serre's conjecture implies Artin's conjecture over totally real fields.

Introduces new techniques combining modularity lifting and geometric methods.

Abstract

Let $p>2$ be prime, and let $F$ be a totally real field in which $p$ is unramified. We give a sufficient criterion for a mod $p$ Galois representation to arise from a mod $p$ Hilbert modular form of parallel weight one, by proving a "companion forms" theorem in this case. The techniques used are a mixture of modularity lifting theorems and geometric methods. As an application, we show that Serre's conjecture for $F$ implies Artin's conjecture for totally odd two-dimensional representations over $F$.