On the weights of mod p Hilbert modular forms

Abstract

We prove many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod p Hilbert modular forms, by making use of modularity lifting theorems and computations in p-adic Hodge theory.

Crystalline extensions and the weight part of Serre’s conjecture

“…Conjectural descriptions of the set of Serre weights were made in increasing generality in [Buzzard et al 2010;Schein 2008;, and cases of these conjectures were proved in [Gee 2011;Gee and Savitt 2011a]. Most recently, significant progress was made towards completely establishing the conjecture for rank-two unitary groups in [Barnet-Lamb et al 2011].…”

“…Unfortunately, this does not seem to be the case, because in general the connection between being modular of some Serre weight and having a potentially Barsotti-Tate lift of some type is less direct. We expect that our methods could be used to reprove the results of Section 3.4 of [Gee 2011], but we do not see how to extend them to cover the unramified case completely. In particular, we are unsure as to when the equality L flat = L crys holds in general.…”

F_p-représentations semi-stables

“…Dans les deux cas, on montre que l'on obtient encore des pylonets et que la restriction de T st aux objets maximaux correspondants est à nouveau exacte et pleinement fidèle. Soulignons que ces variantes interviennent de façon cruciale dans [18] pour étudier certains problèmes de modularité de représentations galoisiennes liés à la généralisation par Buzzard, Diamond et Jarvis de la conjecture de modularité de Serre (voir [8] pour l'énoncé de cette généralisation). Bien que n'étant pas logiquement nécessaire, il nous semble que le cadre théorique fourni par cet article éclaire de façon spectaculaire les calculs de [18], §3.4 (voir aussi [9] à ce sujet).…”

Rank two Breuil modules: Basic structures

“…(See for example [1] and [5].) Let k be a finite extension of F p of degree r, W (k) the ring of Witt vectors.…”

“…5) [g] : M → M are additive bijections for each g ∈ G, preserving Fil κ−1 M, commuting with the φ κ−1 -, N-, and E-actions, and satisfying [g 1 ] • [g 2 ] = [g 1 g 2 ] for all g 1 , g 2 ∈ G, and [1] is the identity map. Furthermore, if a ∈ k ⊗ F p E, m ∈ M, then dd,K /F is equivalent to the category of finite flat group schemes over O K together with an E-action and descent data on the generic fiber from K to F (this equivalence depends on π ).…”