Sur la torsion dans la cohomologie des vari\'et\'es de Shimura de Kottwitz-Harris-Taylor

TL;DR

This paper investigates torsion phenomena in the cohomology of certain Shimura varieties, establishing conditions under which the cohomology is torsion free and analyzing the nature of torsion classes.

Contribution

It provides new torsion freeness results for the cohomology of Kottwitz-Harris-Taylor Shimura varieties by localizing at maximal ideals and relating torsion classes to Igusa varieties.

Findings

Torsion freeness can be achieved under conditions on the maximal ideal or associated Galois representation.

Torsion classes do not introduce new Satake parameters in certain restricted cases.

Torsion classes can be lifted to free cohomology classes in Igusa varieties.

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 the work of Caraiani-Scholze, 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.