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We prove the isogeny property for special fibres of integral canonical modelsof compact Shimura varieties of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type. Theapproach used also shows that many crystalline cycles on abelian varieties overfinite fields which are specializations of Hodge cycles, are algebraic. Thesetwo results have many applications. First, we prove a variant of theconditional Langlands--Rapoport conjecture for these special fibres. Second,for certain isogeny sets we prove a variant of the unconditionalLanglands--Rapoport conjecture (like for many basic loci). Third, we prove thatintegral canonical models of compact Shimura varieties of Hodge type that areof $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type, are closed subschemes ofintegral canonical models of Siegel modular varieties.

On the Tate and Langlands--Rapoport conjectures for special fibres of integral canonical models of Shimura varieties of abelian type