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

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.Comment: Revised introduction, updated references, 71 page

On Serre's conjecture for mod l Galois representations over totally real fields

In 1987 Serre conjectured that any mod l ("ell", not "1") two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where l is unramified. The hard work is in formulating an analogue of the "weight" part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a "mod l Langlands philosophy". Using ideas of Emerton and Vigneras, we formulate a mod l local-global principle for the group D^*, where D is a quaternion algebra over a totally real field, split above l and at 0 or 1 infinite places, and show how it implies the conjecture.Comment: Version 5: Our Shimura variety conventions differ from Emerton's (see footnote 4) and so (a) some chi's needed to be changed to chi^{-1}'s in section 4 and (b) our Eichler-Shimura relation needed fixing too (also in section 4)

Compactifications of Iwahori-level Hilbert modular varieties

We study minimal and toroidal compactifications of p-integral models of Hilbert modular varieties. We review the theory in the setting of Iwahori level at primes over p, and extend it to certain finer level structures. We also prove extensions to compactifications of recent results on Iwahori-level Kodaira–Spencer isomorphisms and cohomological vanishing for degeneracy maps. Finally we apply the theory to study q-expansions of Hilbert modular forms, especially the effect of Hecke operators at primes over p over general base rings

Formes modulaires de Hilbert modulo p et valeurs d'extensions galoisiennes

Let F be a totally real field, v an unramified place of F dividing p and rho a continuous irreducible two-dimensional mod p representation of G_F such that the restriction of rho to G_{F_v} is reducible and sufficiently generic. If rho is modular (and satisfies some weak technical assumptions), we show how to recover the corresponding extension between the two characters of G_{F_v} in terms of the action of GL_2(F_v) on the cohomology mod p.Comment: in French, to appear in Annales Scientifiques de l'Ecole Normale Superieur

The cone of minimal weights for mod pp Hilbert modular forms

We prove that all mod $p$ Hilbert modular forms arise via multiplication by generalized partial Hasse invariants from forms whose weight falls within a certain minimal cone. This answers a question posed by Andreatta and Goren, and generalizes our previous results which treated the case where $p$ is unramified in the totally real field. Whereas our previous work made use of deep Jacquet-Langlands type results on the Goren-Oort stratification (not yet available when $p$ is ramified), here we instead use properties of the stratification at Iwahori level which are more readily generalizable to other Shimura varieties

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