A de Rham weight part of Serre's conjecture and generalized mod $p$ BGG decompositions
Martin Ortiz
TL;DR
The paper proposes a geometric analogue of the weight part of Serre's conjecture using de Rham cohomology of Shimura varieties and proves the equivalence with the classical weight part in generic non-Eisenstein cases for a unitary $(2,1)$ SH variety with $G_{\mathbb{Q}_p}=GL_3$. It develops generalized mod $p$ BGG decompositions in a parabolic category O setting, and shows how these filtrations induce explicit, affine-Weyl-reflection–driven decompositions of de Rham cohomology that control modular weights via theta linkage maps. The results provide a framework to compare de Rham and étale realizations and to deduce weight entailments from middle-degree concentration, with concrete GL$_3$ and GSp$_4$ cases and vanishing statements for coherent cohomology, aided by prismatic and Hodge-theory techniques. In particular, for GL$_3$ the middle-degree concentration yields explicit criteria linking cohomological nontrivialities to Serre weights, while for GSp$_4$ the conjectural concentration allows analogous entailments and a detailed understanding of Breuil–Mezard-type structures in the modular weight problem. The work thus connects geometric, representation-theoretic, and automorphic data to advance the Serre weight program in higher rank settings.
Abstract
We propose the use of de Rham cohomology of special fibers of Shimura varieties to formulate a geometric version of the weight part of Serre's conjecture. We conjecture that this formulation is equivalent to the one using Serre weights and the étale cohomology of Shimura varieties. We prove this equivalence for generic weights and generic non-Eisenstein eigensystems for a compact $U(2,1)$ Shimura variety such that $G_{\mathbb{Q}_p}=GL_3$. We do this by proving a generic concentration in middle degree of mod $p$ de Rham cohomology with coefficients. In turn, we prove this generic concentration by constructing generalized mod $p$ BGG decompositions for de Rham cohomology. After applying the results from our companion paper, this reduces to computing some BGG-like resolutions in a certain mod $p$ version of category $\mathcal{O}$, which is the main content of the article. In the $GSp_4$ case we also compute some explicit BGG decompositions, and assuming the generic concentration in middle degree of de Rham cohomology we obtain an improvement on the main result of arxiv:2410.09602.