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

We study the mod $p$-points of the Kisin-Pappas integral models of abelian type Shimura varieties with parahoric level structure. We show that if the group is quasi-split and unramified, then the mod $p$ isogeny classes are of the form predicted by the Langlands-Rapoport conjecture (c.f. Conjecture 9.2 of arXiv:math/0205022). We prove the same results for quasi-split and tamely ramified groups when their Shimura varieties are proper. The main innovation in this work is a global argument that allows us to reduce the conjecture to the case of a very special parahoric, which is handled in the appendix. This way we avoid the complicated local problem of understanding connected components of affine Deligne-Lusztig varieties for general parahoric subgroups. Along the way, we give a simple irreducibility criterion for Ekedahl-Oort and Kottwitz-Rapoport strata.