This article is the $mathrm{Z}_l$-version of my paper Monodromie du faisceau pervers des cycles evanescents de quelques varietes 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 varites 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.
The principal aim of this paper is to construct torsion cohomology classes in the initial terms of a spectral sequence computing the cohomology of a Kottwitz-Harris-Taylor Shimura variety. Beside we produce some global congruences between automorphic representations.
In the geometric situation of some simple unitary Shimura varieties studied by Harris and Taylor, I have built two filtrations of the perverse sheaf of vanishing cycles. The graduate of the first are the $p$-intermediate extension of some local Harri
s-Taylors local systems, while for the second, obtained by duality, they are the $p+$-intermediate extensions. In this work, we describe the difference between these $p$ and $p+$ intermediate extension. Precisely, we show, in the case where the local system is associated to an irreducible cuspidal representation whose reduction modulo $l$ is supercuspidal, that the two intermediate extensions are the same. Otherwise, if the reduction modulo $l$ is just cuspidal, we describe the $l$-torsion of their difference.
The principal result of this work is the freeness in the $ overline{mathbb Z}_l$-cohomology of the Lubin-Tate tower. The strategy is of global nature and relies on studying the filtration of stratification of the perverse sheaf of vanishing cycles of
some Shimura varieties of Kottwitz-Harris-Taylor types, whose graduates can be explicited as some intermediate extension of some local system constructed in the book of Harris andTaylor. The crucial point relies on the study of the difference between such extension for the two classical $t$-structures $p$ and $p+$. The main ingredients use the theory of derivative for representations of the mirabolic group.
We define and study new filtrations called of stratification of a perverse sheaf on a scheme; beside the cases of the weight or monodromy filtrations, these filtrations are available whatever are the ring of coefficients. We illustrate these construc
tions in the geometric situation of the simple unitary Shimura varieties of Harris and Taylors book for the perverse sheaves of Harris-Taylor and the complex of vanishing cycles, introduced and studied in my 2009 paper at inventiones. In the situation studied in loc. cit., we show how to use these filtrations to simplify the principal step of this paper; the cases of finite field or ring of integer of a local field will be studied in the next published paper.
We study the reduction modulo $l$ of some elliptic representations; for each of these representations, we give a particular lattice naturally obtained by parabolic induction in giving the graph of extensions between its irreducible sub-quotient of it
s reduction modulo $l$. The principal motivation for this work, is that these lattices appear in the cohomology of Lubin-Tate towers.
