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

TL;DR

This paper investigates torsion cohomology classes in certain Shimura varieties, demonstrating how they relate to automorphic representations and Galois representations, with a focus on simplifying proofs and controlling ramified places.

Contribution

It provides a new, simpler approach to associating Galois representations to torsion classes in the cohomology of Kottwitz-Harris-Taylor Shimura varieties, improving control at ramified places.

Findings

Torsion classes can be lifted to automorphic representations.

Construction of Galois representations from mod l cohomology classes.

Enhanced control at ramified and l-dividing places.

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 $l$.