$l$-adic \'etale cohomology of Shimura varieties of Hodge type with non-trivial coefficients

TL;DR

This paper generalizes Mantovan's formula for the $l$-adic cohomology of Shimura varieties of Hodge type with non-trivial coefficients, under certain conditions on the prime $p$ and the group $ extsf{G}$.

Contribution

It extends Mantovan's formula to a broader class of Shimura varieties, incorporating non-trivial coefficients and analyzing the Newton stratification geometry.

Findings

Generalization of Mantovan's formula for $l$-adic cohomology.

New results on Newton stratification geometry.

Conditions under which the formula applies are clarified.

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 $p\nmid |\pi_1(\mathsf{G}^{\rm der})|$. 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.