Geometry around unramified CM points

Abstract

We compute an explicit contribution of supercuspidal representations arising from Yu's construction to the middle \'etale cohomology of local Shimura varieties. For the proof, we construct special affinoids in local Shimura varieties whose reductions are isomorphic to Heisenberg Deligne-Lusztig varieties and then apply positive-depth Deligne-Lusztig theory. Along the way, we propose a stacky analogue of special affinoids using the stack of local shtukas and observe that the `general canonical subgroup problem'can be solved purely on the stack of local shtukas, which yields the corresponding claim on Shimura varieties via the Hodge-Tate period map.