ON THE -ADIC COHOMOLOGY OF SOME -ADICALLY UNIFORMIZED SHIMURA VARIETIES

Abstract

We determine the Galois representations inside the$\ell$-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.