On the $\bar{\mathbb F}_l$-cohomology of a simple unitary Shimura variety

Abstract

We study the torsion cohomology classes of Shimura varieties of type Kottwitz-Harris-Taylor and we show that " up to an arbitrary place " one can raise them to an automorphic representation. In application, to any mod $l$ system of Hecke eigenvalues appearing in the $\bar{\mathbb F}_l$-cohomology of a Shimura's variety of Kottwitz-Harris-Taylor type, we associate a $\bar{\mathbb F}_l$-Galois representation which Frobenius eigenvalues are given by Hecke's. Compared to the highly more general construction of Scholze, we gain both the simplicity of the proof and the control at places ramified and at those dividing $l$.