On the étale cohomology of Hilbert modular varieties with torsion coefficients

2023/09/18 by Ana Caraiani, Matteo Tamiozzo · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Geometry and complex manifolds

paper · pdf · doi:10.1112/s0010437x23007431

Abstract

We study the étale cohomology of Hilbert modular varieties, building on the methods introduced by Caraiani and Scholze for unitary Shimura varieties. 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 and 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 GL₂(\mathbb Q_p) occurs in the completed homology of Hilbert modular varieties.

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• 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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