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

Abstract

We determine the Galois representations inside the -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.

On the Cohomology of Some Simple Shimura Varieties With Bad Reduction

“…This paper is a continuation of our work [28]. Based on the methods and results established in [27] and [28], we determine the Galois representations inside the -adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places.…”

“…Here as [16], [19] and [28], we use the original approach of Kottwitz as in [8] to get suitable combinatorial description of the points over finite fields. As in [27] and [28], having the description of the set of points of reduction modulo p on these varieties, the crucial ingredient is to define some suitable test functions at p which will appear in the trace formula when analyzing the cohomology. There are two cases.…”

“…• For the unitary case, these functions are already available: if D is split at a place ν above p, the factor at ν was defined in [26] (the EL case n = 2); if D is ramified at a place ν above p, the factor at ν was defined in [28] (for n = 2) which in turn was based the approach of Scholze [25,26] by deformation spaces of p-divisible groups. • For the quaternionic case, one needs just to take the Galois twist into consideration and the method of Scholze applies.…”

Note on weight-monodromy conjecture for $p$-adically uniformized varieties

“…We may apply Theorem 2.5 to the unitary Shimura varieties appearing in [RZ96, Theorem 6.50]. By the same method as in [She14,§3], one can compute the ℓ-adic cohomology of them using Theorem 2.5 (i). This considerably simplifies the proof of the main result of [She14]; the study on test functions in [She14, §4-7] is no longer needed.…”

“…By the same method as in [She14,§3], one can compute the ℓ-adic cohomology of them using Theorem 2.5 (i). This considerably simplifies the proof of the main result of [She14]; the study on test functions in [She14, §4-7] is no longer needed. The local Hasse-Weil zeta functions of such Shimura varieties can be computed directly.…”