Galois irreducibility implies cohomology freeness for KHT Shimura varieties
Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 199-232.

Given a KHT Shimura variety with an action of its unramified Hecke algebra 𝕋, we proved in [7], see also [12] for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal 𝔪 of 𝕋, happen to be free. In this work, we obtain the same result for 𝔪 such that its associated Galois 𝔽 ¯ -representation ρ 𝔪 ¯ is irreducible, under the hypothesis that [F(exp(2iπ/):F]>d, where F is the reflex field, d the dimension of the KHT Shimura variety and the residual characteristic.

Étant donnée une variété de Shimura unitaire de type KHT de dimension relative d-1 sur son corps reflex F et munie de l’action de son algèbre de Hecke 𝕋 non ramifiée, nous prouvons dans  [7], voir aussi [12] pour les autres variétés de Shimura de type PEL, que ses groupes de ¯ -cohomologie localisés en un idéal maximal générique 𝔪 de 𝕋, sont libres. Dans ce travail, sous l’hypothèse que [F(exp(2iπ/):F]>d, nous montrons le même résultat pour 𝔪 tel que sa 𝔽 ¯ -représentation galoisienne associée, ρ 𝔪 ¯, est irréductible.

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DOI: 10.5802/jep.216
Classification: 11F70, 11F80, 11F85, 11G18, 20C08
Keywords: Shimura varieties, torsion in the cohomology, maximal ideal of the Hecke algebra, localized cohomology, Galois representation
Mot clés : Variétés de Shimura, torsion dans la cohomologie, algèbres de Hecke, localisation de la cohomologie, représentations galoisiennes

Pascal Boyer 1

1 Université Sorbonne Paris Nord, LAGA, CNRS, UMR 7539 99 avenue J.-B. Clément, F-93430 Villetaneuse, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Pascal Boyer. Galois irreducibility implies cohomology freeness for KHT Shimura varieties. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 199-232. doi : 10.5802/jep.216. https://jep.centre-mersenne.org/articles/10.5802/jep.216/

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