$l$-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients

Abstract

Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum of Hodge type. Let $p$ be an odd prime such that ${\mathsf{G}}_{{\mathbb{Q}}_p}$ splits after a tamely ramified extension and $p\nmid |{\pi }_1({\mathsf{G}}^{\mathrm{der}})|$. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and ${\mathsf{G}}_{{\mathbb{Q}}_p}$ is either unramified or residually split, we prove the generalisation of Mantovan's formula for the $l$-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction modulo $p$ of the Kisin-Pappas integral model.

60 pages