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SUR LA TORSION DANS LA COHOMOLOGIE DES VARIÉTÉS DE SHIMURA DE KOTTWITZ-HARRIS-TAYLOR
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 18 / Issue 3 / May 2019
- Published online by Cambridge University Press:
- 30 March 2017, pp. 499-517
- Print publication:
- May 2019
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- Article
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(Torsion in the cohomology of Kottwitz–Harris–Taylor Shimura varieties) 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 
$\overline{\unicode[STIX]{x1D70C}}_{\mathfrak{m}}$ associated to it. Concerning the torsion, in a rather restricted case than Caraiani and Scholze (« On the generic part of the cohomology of compact unitary Shimura varieties », Preprint, 2015), 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.
