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A geometric Jacquet-Langlands correspondence for U(2) Shimura varieties

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Abstract

Let G be a unitary group over ℚ, associated to a CM-field F with totally real part F +, with signature (1, 1) at all the archimedean places of F +. Under certain hypotheses on F +, we show that Jacquet-Langlands correspondences between certain automorphic representations of G and representations of a group G′ isomorphic to G except at infinity can be realized in the cohomology of Shimura varieties attached to G and G′.

We obtain these Jacquet-Langlands correspondences by studying the bad reduction of a Shimura variety X attached to G at a prime p for which X has “Γ0(p)” level structure, and construct a “Deligne-Rapoport” model for X. The irreducible components of the special fiber of this model have a global structure that can be explicitly described in terms of Shimura varieties X′ for unitary groups G′ isomorphic to G away from infinity.

The weight spectral sequence of Rapoport-Zink then yields an expression for certain pieces of the weight filtration on the étale cohomology of X in terms of the cohomology of X′. This identifies a piece of this weight filtration with a space of algebraic modular forms for G′. This result implies certain cases of the Jacquet-Langlands correspondence for G and G′ in terms of a canonical map between spaces of arithmetic interest, rather than simply as an abstract bijection between isomorphism classes of representations.

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  1. Department of Mathematics, University of Texas at Austin, Ustin, TX, 78712, USA

    David Helm

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  1. David Helm
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Helm, D. A geometric Jacquet-Langlands correspondence for U(2) Shimura varieties. Isr. J. Math. 187, 37–80 (2012). https://doi.org/10.1007/s11856-011-0162-x

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