A Serre weight conjecture for geometric Hilbert modular forms in characteristic $p$

Abstract

Let $p$ be a prime and $F$ a totally real field in which $p$ is unramified. We consider mod $p$ Hilbert modular forms for $F$, defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to $p$ in characteristic $p$. For a mod $p$ Hilbert modular Hecke eigenform of arbitrary weight (without parity hypotheses), we associate a two-dimensional representation of the absolute Galois group of $F$, and we give a conjectural description of the set of weights of all eigenforms from which it arises. This conjecture can be viewed as a “geometric” variant of the “algebraic” Serre weight conjecture of Buzzard–Diamond–Jarvis, in the spirit of Edixhoven's variant of Serre's original conjecture in the case $F=\mathbb{Q}$. We develop techniques for studying the set of weights giving rise to a fixed Galois representation, and prove results in support of the conjecture, including cases of partial weight 1.