MS Seminar (Mathematics - String Theory)

Speaker: Joseph Muller (Sorbonne University)
Title: On the cohomology of the unramified PEL unitary Rapoport-Zink space of signature (1,n−1)
Date (JST): Mon, Nov 28, 2022, 13:30 - 15:00
Place: Hybrid
Abstract: Rapoport-Zink (RZ) spaces are moduli spaces which classify the deformations of a p-divisible group with additional structures. It is equipped with compatible actions of p-adic and Galois groups, and their cohomology is believed to play a role in the local Langlands program. So far, the cohomology of RZ spaces is entirely known only in the cases of the Lubin-Tate tower and of the Drinfeld space; in particular both of them are RZ spaces of EL type. In this talk, we consider the basic unramified PEL unitary RZ space with signature (1,n−1) at hyperspecial level. In 2011, Vollaard and Wedhorn proved that its special fiber is stratified by generalized Deligne-Lusztig varieties, whose incidence relations mimic the combinatorics of the Bruhat-Tits building of a unitary group. We compute the cohomology of these strata and we draw
some consequences on the cohomology of the RZ space, such as its non admissibility. When n=3,4 we deduce an automorphic description of the cohomology of the basic locus in the corresponding Shimura variety at hyperspecial level via p-adic uniformization.