On the \'etale cohomology of Hilbert modular varieties with torsion coefficients

TL;DR

This paper investigates the étale cohomology of Hilbert modular varieties with torsion coefficients, establishing vanishing theorems in the generic case and exploring the structure beyond, with applications to the p-adic local Langlands correspondence.

Contribution

It extends methods from unitary Shimura varieties to Hilbert modular varieties, proving a vanishing theorem and analyzing cohomology beyond the generic case, avoiding trace formula computations.

Findings

Cohomology with torsion coefficients is concentrated in the middle degree in the generic case.

Bounds are obtained for degrees where torsion cohomology can be non-zero.

The p-adic local Langlands correspondence appears in the completed homology under certain conditions.

Abstract

We study the \'etale cohomology of Hilbert modular varieties, building on the methods introduced for unitary Shimura varieties in [CS17, CS19]. We obtain the analogous vanishing theorem: in the "generic" case, the cohomology with torsion coefficients is concentrated in the middle degree. We also probe the structure of the cohomology beyond the generic case, obtaining bounds on the range of degrees where cohomology with torsion coefficients can be non-zero. The proof is based on the geometric Jacquet--Langlands functoriality established by Tian--Xiao and avoids trace formula computations for the cohomology of Igusa varieties. As an application, we show that, when $p$ splits completely in the totally real field and under certain technical assumptions, the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ occurs in the completed homology of Hilbert modular varieties.