Supercuspidal part of the mod l cohomology of GU(1,n - 1)-Shimura varieties

Abstract

Let l be a prime. In this paper we are concerned with GU(1,n - 1)-type Shimura varieties with arbitrary level structure at l and investigate the part of the cohomology on which G(ℚ[subscript p]) acts through mod l supercuspidal representations, where p ≠ l is any prime such that G(ℚ[subscript p]) is a general linear group. The main theorem establishes the mod l analogue of the local-global compatibility. Our theorem also encodes a global mod l Jacquet–Langlands correspondence in that the cohomology is described in terms of mod l automorphic forms on some compact inner form of G.