Cycles on splitting models of Shimura varieties

I will explain how to construct exotic Hecke correspondences between the special fibers of different PEL type Shimura varieties, at possibly ramified primes. These can be used to construct new geometric realizations of the Jacquet-Langlands correspondence, as well as verify generic instances of the Tate conjecture for the special fibers of these Shimura varieties, generalizing the work of Xiao-Zhu in the unramified case. The key is to resolve the integral models by the splitting models of Pappas-Rapoport.

In the first part of the talk, I will recall the splitting models of Shimura varieties, and explain how they can be used to construct exotic Hecke correspondences. By using a part of the categorical local Langlands correspondence, I will then deduce geometric realizations of the Jacquet-Langlands correspondence.
In the second part of the talk, I will introduce splitting versions of the affine Deligne-Lusztig varieties, and explain how they can be applied to the Tate conjecture for the special fibers of certain Shimura varieties.