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

Abstract

Let G be a unitary group over the rationals, 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 maximal parabolic level structure. We construct a "Deligne-Rapoport" model for X and show that the irreducible components of its special fiber have a global structure that can be explicitly described in terms of Shimura varieties X' for unitary groups G' isomorphic to G except at infinity. The weight spectral sequence of Rapoport-Zink then yields an expression for certain pieces of the weight filtration on the etale cohomology of X in terms of the cohomology of a suitable X'. This identifies a piece of this weight filtration with a space of algebraic modular forms for G'. A consequence is certain cases of the Jacquet-Langlands correspondence between G and G' in terms of a canonical isomorphism between spaces of arithmetic interest, rather than simply as an abstract bijection between isomorphism classes of representations.