On the étale cohomology of Hilbert modular varieties with torsion coefficients

2021/07/21 by Ana Caraiani, Matteo Tamiozzo, Caraiani, Ana +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2107.10081

openalex publication_date 2021/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the étale 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 GL₂(ℚ_p) occurs in the completed homology of Hilbert modular varieties.

Cited by

• On the non-generic part of cohomology of compact unitary Shimura varieties of signature (1,n)
• The Asai--Flach Euler system in p-adic families

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