Number Theory Seminar

Over the past decade there has been a great deal of progress by Helm, Tian-Xiao, and finally Xiao-Zhu on the topic of exceptional Hecke correspondences between Shimura varieties in characteristic p. I will explain a new approach (building on that of Xiao-Zhu) to constructing exotic Hecke correspondences purely in the world of p-adic geometry and the geometry of Igusa stacks, which allows one to relate the cohomology of different Shimura varieties in a great deal of generality and with no restrictions on the level at p. Time permitting, I will discuss how these results, in tandem with some results of Fargues-Scholze, allow one to construct certain cases of a cohomological Jacquet-Langlands correspondence between many Shimura varieties of abelian type. All results are joint with Pol van Hoften.