On the cohomology of some simple Shimura varieties with bad reduction

TL;DR

This paper analyzes the Galois representations in the $l$-adic cohomology of certain Shimura varieties with bad reduction, extending previous results to more general levels and confirming theoretical predictions.

Contribution

It generalizes prior work by Reimann and Kottwitz to arbitrary levels at $p$, providing a comprehensive description of cohomology for these Shimura varieties.

Findings

Determined Galois representations in the cohomology of specific Shimura varieties.

Extended previous results to arbitrary levels at $p$.

Confirmed the Langlands-Kottwitz predicted cohomology structure.

Abstract

We determine the Galois representations inside the $l$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at $p$, and confirm the expected description of the cohomology due to Langlands and Kottwitz.