Mod p isogeny classes on Shimura varieties with parahoric level structure

Abstract

We study the special fiber of the integral model for Shimura varieties of Hodge type with parahoric level structure recently constructed by Kisin and Pappas. We show that when the group at $p$ is residually split, the points in the mod $p$ isogeny classes have the form predicted by the Langlands–Rapoport conjecture. We also verify most of the He–Rapoport axioms for these integral models without the residually split assumption. This allows us to prove that all Newton strata are nonempty for these models. The verification of the axioms in full is reduced to a question on the connected components of affine Deligne–Lusztig varieties.