Systems of Hecke eigenvalues on subschemes of Shimura varieties

TL;DR

This paper demonstrates that systems of Hecke eigenvalues in the coherent cohomology of mod p Shimura varieties are consistent across various subschemes, including noncompact and nonclosed ones, revealing a unified structure.

Contribution

It establishes the equivalence of Hecke eigenvalue systems across different subschemes of Shimura varieties, extending known results to noncompact and nonclosed cases.

Findings

Hecke eigenvalues are consistent across all subschemes of Shimura varieties.

Results apply to both compact and noncompact Shimura varieties.

Includes cases like Ekedahl-Oort strata and central leaves.

Abstract

We show that the systems of Hecke eigenvalues that appear in the coherent cohomology with coefficients in automorphic line bundles of any mod $p$ abelian type compact Shimura variety at hyperspecial level are the same as those appearing in any Hecke-equivariant closed subscheme. We also prove analogous results for noncompact Shimura varieties or nonclosed subschemes, such as Ekedahl-Oort strata, length strata and central leaves.