On the Cohomology of Some Simple Shimura Varieties With Bad Reduction

Abstract

Abstract. We determine the Galois representations inside the -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.

On the -Adic Cohomology of Some -Adically Uniformized Shimura Varieties

“…The case of n = 2 in [26] but for arbitrary p (with the above requirement on local cocharacters at ramified places) will be a typical example. We will treat this case and the related quaternionic Shimura varieties in [42].…”

“…In [42] we shall use our results for the test functions to describe the cohomology of quaternionic and related unitary Shimura varieties at ramified places, see section 7 for more details.…”

“…However, as the reader can see in this introduction, our general purpose is trying to prove results for as many cases as possible. Our theory of test functions developed in sections 4-6 by Scholze's method, will be used in [42] in an essential way to prove results for some other Shimura varieties.…”