Mod $p$ Hilbert modular forms of parallel weight one: the ramified case

TL;DR

This paper extends the understanding of mod p Hilbert modular forms of weight one over totally real fields, proving a key part of Serre's conjecture under certain conditions.

Contribution

It generalizes previous results to all totally real fields, establishing a link between unramified, p-distinguished Galois representations and modular forms of parallel weight one.

Findings

Proves the existence of mod p Hilbert modular forms of weight one for a broad class of Galois representations.

Resolves the weight one case of Serre's conjecture for totally real fields under mild hypotheses.

Establishes a connection between Galois representations and modular forms in the ramified case.

Abstract

We generalize the main result of arXiv:1206.6631 [math.NT] to all totally real fields. In other words, for $p>2$ prime, we prove (under a mild Taylor-Wiles hypothesis) that if a modular representation is unramified and $p$-distinguished at all places above $p$, then it arises from a mod $p$ Hilbert modular form of parallel weight one. This (mostly) resolves the weight one part of Serre's conjecture for totally real fields.