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

TL;DR

This paper proves the isogeny property for special fibers of integral canonical models of certain Shimura varieties, demonstrating algebraicity of crystalline cycles and advancing the Langlands--Rapoport conjecture in these contexts.

Contribution

It establishes the isogeny property and algebraicity of crystalline cycles for specific Shimura varieties, and proves variants of the Langlands--Rapoport conjecture.

Findings

Proved the isogeny property for special fibers of certain Shimura varieties.

Showed many crystalline cycles are algebraic.

Established that integral canonical models are closed subschemes of Siegel modular varieties.

Abstract

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