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

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.…”

“…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. The main results generalize the work of Reimann [19,22] and Kottwitz [12] in this setting to arbitrary levels at p ( = ), and confirm the expected description of the cohomology due to Langlands and Kottwitz.…”

“…In [27] Scholze and Shin have solved this problem for some compact unitary Shimura varieties at (ramified) split places such that the related local reductive groups are products of Weil restriction of GL n . In [28] we have also solved the case of Shimura varieties which admit p-adic uniformization by finite products of Drinfeld upper half spaces. There the local reductive groups have some factors as the multiplicative group of a central division algebra of invariant 1 n .…”

“…There the local reductive groups have some factors as the multiplicative group of a central division algebra of invariant 1 n . Both [27] and [28] allow the level structures at p to be arbitrary.…”

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

“…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.…”