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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  p  ) acts through mod  l  supercuspidal representations, where  p  ≠  l  is any prime such that  G (ℚ  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 .

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