Cycles on splitting models of Shimura varieties

TL;DR

This paper constructs new Hecke correspondences between special fibers of PEL type Shimura varieties with bad reduction, leading to advances in geometric Langlands and Tate conjecture verification.

Contribution

It introduces exotic Hecke correspondences for Shimura varieties with bad reduction and develops splitting models to study their geometry and related conjectures.

Findings

Constructed new Hecke correspondences for bad reduction cases.

Verified instances of the Tate conjecture for special fibers.

Developed splitting models for local shtukas and affine Deligne-Lusztig varieties.

Abstract

We construct exotic Hecke correspondences between the special fibers of different PEL type Shimura varieties, when the local groups are restrictions of scalars of unramified groups. In particular, the local groups themselves are not necessarily unramified, and the Shimura varieties can have bad reduction. By adapting the methods of Xiao-Zhu in the case of good reduction, we use this to construct new instances of the geometric Jacquet-Langlands correspondence, including a motivic refinement, and verify generic instances of the Tate conjecture for the special fibers of these Shimura varieties at very special level. Our main tool is to resolve the integral models by the splitting models of Pappas-Rapoport. We also define splitting versions of the moduli stacks of local shtukas and affine Deligne-Lusztig varieties, and study their geometry.