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The Hodge conjecture and arithmetic quotients of complex balls

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Acta Mathematica

Abstract

Let S be a closed Shimura variety uniformized by the complex n-ball associated with a standard unitary group. The Hodge conjecture predicts that every Hodge class in \({H^{2k} (S, \mathbb{Q})}\), \({k=0,\dots, n}\), is algebraic. We show that this holds for all degrees k away from the neighborhood \({\bigl]\tfrac13n,\tfrac23n\bigr[}\) of the middle degree. We also prove the Tate conjecture for the same degrees as the Hodge conjecture and the generalized form of the Hodge conjecture in degrees away from an interval (depending on the codimension c of the subvariety) centered at the middle dimension of S. These results are derived from a general theorem that applies to all Shimura varieties associated with standard unitary groups of any signature. The proofs make use of Arthur’s endoscopic classification of automorphic representations of classical groups. As such our results rely on the stabilization of the trace formula for the (disconnected) groups \({GL (N) \rtimes \langle \theta \rangle}\) associated with base change.

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Authors and Affiliations

  1. Sorbonne Universités, UPMC Université Paris 06, Institut de Mathématiques de Jussieu–Paris Rive Gauche, UMR 7586, CNRS, Université Paris Diderot, Sorbonne Paris Cité, FR-75005, Paris, France

    Nicolas Bergeron

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, U.S.A.

    John Millson

  3. CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, Sorbonne Universités, UPMC Université Paris 06, Université Paris Diderot, Sorbonne Paris Cité, FR-75005, Paris, France

    Colette Moeglin

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  1. Nicolas Bergeron
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  2. John Millson
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  3. Colette Moeglin
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Correspondence to Nicolas Bergeron.

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In memory of Raquel Maritza Gilbert, beloved wife of the second author.

N. B. is a member of the Institut Universitaire de France.

J. M. was partially supported by NSF grant DMS-1206999.

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Bergeron, N., Millson, J. & Moeglin, C. The Hodge conjecture and arithmetic quotients of complex balls. Acta Math 216, 1–125 (2016). https://doi.org/10.1007/s11511-016-0136-2

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