Mod p points on shimura varieties of parahoric level

We study the mod F<overline>(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.