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

TL;DR

This paper proposes a geometric version of the Serre weight conjecture for mod p Hilbert modular forms over totally real fields, associating Galois representations to eigenforms and describing their possible weights.

Contribution

It introduces a conjectural geometric framework for understanding weights of mod p Hilbert modular forms and supports it with partial results and techniques.

Findings

Established a conjectural description of weights for Galois representations

Developed techniques to analyze weight sets for fixed Galois representations

Proved results supporting the conjecture, including cases of partial weight one

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 = 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 one.