On the $l$-adic cohomology of some $p$-adically uniformized Shimura varieties

TL;DR

This paper investigates the $l$-adic cohomology of certain $p$-adically uniformized Shimura varieties, confirming aspects of the Langlands-Kottwitz conjecture in new cases involving Galois representations.

Contribution

It explicitly determines Galois representations in the $l$-adic cohomology of specific Shimura varieties with uniformization by Drinfeld spaces, extending previous results.

Findings

Confirmed Langlands-Kottwitz's description of cohomology in new cases

Identified Galois representations within the cohomology

Analyzed cohomology at split places with uniformization by Drinfeld spaces

Abstract

We determine the Galois representations inside the $l$-adic cohomology of some unitary Shimura varieties at split places where they admit uniformization by finite products of Drinfeld upper half spaces. Our main results confirm Langlands-Kottwitz's description of the cohomology of Shimura varieties in new cases.