2022 Fall POSTECH-PMI NUMBER THEORY SEMINAR

• Speaker : Alex Youcis​(University of Tokyo, Japan)
• Time : 2022.10.06.(Thu) 16:00 ~ 18:00
• Online Streaming(Zoom)

Title : A prismatic realization functor on integral Shimura varieties of abelian type

Abstract : Shimura varieties are certain types of algebro-geometric objects attached to a reductive group $G$. Their geometry has been important in a wide range of applications; from differential geometry to number theory. That said, one of their most profound usages is in the Langlands program where they, or more particularly their etale cohomology, is expected to contain the global Langlands correspondence. Shimura varieties should be moduli spaces of certain motives with $G$-structure, and the various cohomology theories applied to these Shimura varieties should be closely related to the realizations of these motives in those cohomology theories.

In the recent work of Bhatt and Scholze, most cohomology theories considered by number theorists have been shown to be avatars of a single unifying cohomology theory: prismatic cohomology. In this talk I will discuss ongoing work with Naoki Imai and Hiroki Kato on constructing a 'prismatic realization functor' on certain integral models of Shimura varieties of abelian type. This realization functor should be thought of as producing the 'prismatic cohomology' realization of the motives with $G$ structure that these Shimura varieties parameterize. We will then talk about applications of this result to the etale cohomology of such Shimura varieties.