On the étale cohomology of Hilbert modular varieties with torsion coefficients
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 nonzero. The proof is based on the geometric JacquetLanglands functoriality established by TianXiao 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 DOI:
 10.48550/arXiv.2107.10081
 arXiv:
 arXiv:2107.10081
 Bibcode:
 2021arXiv210710081C
 Keywords:

 Mathematics  Number Theory
 EPrint:
 Minor corrections