Mod-$p$ isogeny classes on Shimura varieties with parahoric level structure

TL;DR

This paper investigates the structure of mod p isogeny classes on Shimura varieties with parahoric level, confirming predictions of the Langlands Rapoport conjecture and establishing non-emptiness of Newton strata.

Contribution

It proves the form of points in mod p isogeny classes for residually split groups and verifies key axioms, extending results to non-residually split cases.

Findings

Points in mod p isogeny classes match predictions for residually split groups.

Most He-Rapoport axioms are verified for these models.

All Newton strata are shown to be non-empty.

Abstract

We study the special fiber of the integral models for Shimura varieties of Hodge type with parahoric level structure constructed by Kisin and Pappas in [KP]. We show that when the group is residually split, the points in the mod $p$ isogeny classes have the form predicted by the Langlands Rapoport conjecture in [LR]. 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 non-empty for these models.