Companion Forms in Parallel Weight One

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 $F$.

12 pages

References in corpus (6)

• On the automorphy of $l$-adic Galois representations with small residual image
• Adequate subgroups
• Companion forms for unitary and symplectic groups
• Congruences between Hilbert modular forms: constructing ordinary lifts
• Serre weights for rank two unitary groups
• Potential automorphy and change of weight

Cited by in corpus (5)

• A Serre weight conjecture for geometric Hilbert modular forms in characteristic p
• Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p
• Mod $p$ Hilbert modular forms of parallel weight one: the ramified case
• Artin representations for GSp_4 attached to real analytic Siegel cusp forms of weight (2,1)
• Modularity of certain mod $p^n$ Galois representations