A de Rham weight part of Serre's conjecture and generalized mod $p$ BGG decompositions

TL;DR

This paper introduces a geometric approach to the weight part of Serre's conjecture using de Rham cohomology of Shimura varieties, proves equivalence with existing formulations in generic cases, and constructs generalized mod p BGG decompositions.

Contribution

It formulates a geometric version of the weight part of Serre's conjecture, proves its equivalence to classical formulations in generic cases, and develops generalized mod p BGG decompositions for de Rham cohomology.

Findings

Proved equivalence of geometric and classical formulations in generic weights.

Constructed generalized mod p BGG decompositions for de Rham cohomology.

Computed explicit BGG decompositions in the GSp_4 case.

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 \'etale 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.