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 = 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.

arXiv comment74 pages, corrected reference in v3 (section 11.4), published in JEMS (online)

References in corpus (11)

• On Serre's conjecture for mod l Galois representations over totally real fields
• The weight part of Serre's conjecture for GL(2)
• An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod-p cohomology of GL(n,Z)
• Minimal weights of Hilbert modular forms in characteristic p
• Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p
• Companion Forms in Parallel Weight One
• Galois representations and torsion in the coherent cohomology of Hilbert modular varieties
• Serre weights and wild ramification in two-dimensional Galois representations
• Explicit Serre weights for two-dimensional Galois representations
• A mod p Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties: Splicing and dicing
• Automorphy of some residually S_5 Galois representations

Cited by in corpus (7)

• The cone of minimal weights for mod $p$ Hilbert modular forms
• A mod p Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties: Splicing and dicing
• Geometric weight-shifting operators on Hilbert modular forms in characteristic p
• Compactifications of Iwahori-level Hilbert modular varieties
• Unramifiedness of weight one Hilbert Hecke algebras
• Kodaira-Spencer isomorphisms and degeneracy maps on Iwahori-level Hilbert modular varieties: the saving trace
• Plectic Galois action on CM points and connected components of Hilbert modular varieties