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

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  • Sug Woo Shin
    Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA; and Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea

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

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