Mod $p$ points on Shimura varieties of parahoric level (with an appendix by Rong Zhou)

Abstract

We study the mod $p$-points of the Kisin--Pappas integral models of Shimura varieties of Hodge type with parahoric level. We show that if the group is quasi-split, then every isogeny class contains the reduction of a CM point, proving a conjecture of Kisin--Madapusi-Pera--Shin. We furthermore show that the mod $p$ isogeny classes are of the form predicted by the Langlands--Rapoport conjecture if either the Shimura variety is proper or if the group at $p$ is unramified. The main ingredient in our work is a global argument that allows us to reduce the conjecture to the case of very special parahoric level. This case is dealt with in the appendix by Rong Zhou. As a corollary to our arguments, we determine the connected components of Ekedahl--Oort strata.

Near final version, to appear in Forum of Mathematics, Pi. v4 is a significantly revised version of v3 with improved main results

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• Geometrization of the local Langlands correspondence
• The stable trace formula for Shimura varieties of abelian type
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• On the ordinary Hecke orbit conjecture
• Monodromy and Irreducibility of Igusa Varieties
• Igusa Stacks and the Cohomology of Shimura Varieties
• Connectedness of affine Deligne-Lusztig varieties for unramified groups