La $\mathrm{Z}_l$-cohomologie du mod\`ele de Deligne-Carayol est sans torsion

TL;DR

This paper proves that the $ ext{Z}_l$-cohomology of the Deligne-Carayol model of certain Shimura varieties is torsion-free, extending previous work on vanishing cycles and cohomology sheaves in algebraic geometry.

Contribution

It establishes the freeness of cohomology sheaves for the Deligne-Carayol model, providing a key step towards understanding torsion phenomena in Shimura varieties.

Findings

Cohomology sheaves are free, with no torsion.

Main theorem of Berkovich on vanishing cycles is applied.

Foundation for future study of torsion in Shimura variety cohomology.

Abstract

This article is the $\mathrm{Z}_l$-version of my paper "Monodromie du faisceau pervers des cycles \'evanescents de quelques vari\'et\'es de Shimura simples" in Invent. Math. 2009 vol 177 pp. 239-280, where we study the vanishing cycles of some unitary Shimura variety. The aim is to prove that the cohomology sheaves of this complexe are free so that, thanks to the main theorem of Berkovich on vanishing cycles, we can deduce that the $\mathrm{Z}_l$-cohomology of the model of Deligne-Carayol is free. There will be a second article which will be the $\mathrm{Z}_l$ version of my paper "Conjecture de monodromie-poids pour quelques vari\'t\'es de Shimura unitaires" in Compositio vol 146 part 2, pp. 367-403. The aim of this second article will be to study the torsion of the cohomology groups of these Shimura varieties.