Abstract

The goal of this talk is to explain joint work in progress with Jack Sempliner on the construction of "exotic" Hecke correspondences between the mod p fibers of different Shimura varieties of Hodge type. Our work generalizes forthcoming work of Xiao-Zhu; our results cover the new situation where the groups underlying the two different Shimura varieties are allowed to be to be non-isomorphic at p. As a consequence of our main results, we obtain exotic isomorphisms of Igusa varieties in the style of Caraiani-Tamiozzo which lead to geometric incarnations of the Jacquet--Langlands correspondence.