# SUR LA TORSION DANS LA COHOMOLOGIE DES VARI ETES DE SHIMURA DE KOTTWITZ-HARRIS-TAYLOR

Abstract : When the level at $l$ of a Shimura variety of Kottwitz-Harris-Taylor is not maximal, its cohomology with coefficients in a $\overline{\mathbb Z}_l$-local system isn't in general torsion free. In order to prove torsion freeness results of the cohomology, we localize at a maximal ideal $\mathfrak m$ of the Hecke algebra. We then prove a result of torsion freeness resting either on $\mathfrak m$ itself or on the Galois representation $\overline \rho_{\mathfrak m}$ associated to it. Concerning the torsion, in a rather restricted case than \cite{scholze-cara}, we prove that the torsion doesn't give new Satake parameters systems by showing that each torsion cohomology class can be raised in the free part of the cohomology of a Igusa variety.
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https://hal.archives-ouvertes.fr/hal-01137273
Contributor : Pascal Boyer <>
Submitted on : Monday, October 31, 2016 - 6:28:16 PM
Last modification on : Thursday, February 7, 2019 - 5:48:40 PM

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torsion-localise3.pdf
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• HAL Id : hal-01137273, version 2
• ARXIV : 1503.03303

### Citation

Pascal Boyer. SUR LA TORSION DANS LA COHOMOLOGIE DES VARI ETES DE SHIMURA DE KOTTWITZ-HARRIS-TAYLOR. Journal de l'Institut de Mathématiques de Jussieu, 2016. ⟨hal-01137273v2⟩

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