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 $\mathrm {GL}_2(\mathbb {Q}_p)$ occurs in the completed homology of Hilbert modular varieties.
Subject
Algebra and Number Theory
Cited by
4 articles.
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1. Hirzebruch–Zagier classes and rational elliptic curves over quintic fields;Mathematische Zeitschrift;2024-08-09
2. On -adic -functions for Hilbert modular forms;Memoirs of the American Mathematical Society;2024-06
3. Endoscopy on SL2-eigenvarieties;Journal für die reine und angewandte Mathematik (Crelles Journal);2024-05-28
4. Congruence modules in higher codimension and zeta lines in Galois cohomology;Proceedings of the National Academy of Sciences;2024-04-19