p-adic étale cohomology of period domains

We compute the p -torsion and p -adic étale cohomologies with compact support of period domains over local fields in the case of basic isocrystals for quasi-split reductive groups. As in the cases of ℓ -torsion or ℓ -adic coefficients, ℓ ≠ p , considered by Orlik, the results involve generalized Ste...

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Bibliographic Details
Published in:Mathematische annalen 2021-10, Vol.381 (1-2), p.105-180
Main Authors: Colmez, Pierre, Dospinescu, Gabriel, Hauseux, Julien, Nizioł, Wiesława
Format: Article
Language:English
Subjects:
Algebraic Geometry
Domains
Homology
Mathematics
Mathematics and Statistics
Number Theory
Representation Theory
Representations
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Summary:We compute the p -torsion and p -adic étale cohomologies with compact support of period domains over local fields in the case of basic isocrystals for quasi-split reductive groups. As in the cases of ℓ -torsion or ℓ -adic coefficients, ℓ ≠ p , considered by Orlik, the results involve generalized Steinberg representations. For the p -torsion case, we follow the method used by Orlik in his computations of the ℓ -torsion étale cohomology using as a key new ingredient the computation of Ext groups between mod p generalized Steinberg representations of p -adic groups. For the p -adic case, we do not use Huber’s definition of étale cohomology with compact support as Orlik did since it seems to give spaces that are much too big; instead we use continuous étale cohomology with compact support.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-020-02139-6