Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties

Abstract

Let G be a unitary group over a totally real field, and X a Shimura variety for G. For certain primes p of good reduction for X, we construct cycles on the characteristic p fiber of X. These cycles are defined as the loci on which the Verschiebung map has small rank on particular pieces of the Lie algebra of the universal abelian variety on X. The geometry of these cycles turns out to be closely related to Shimura varieties for a different unitary group G', which is isomorphic to G at finite places but not isomorphic to G at archimedean places. More precisely, each such cycle has a natural desingularization, and this desingularization is "almost" isomorphic to a scheme parametrizing certain subbundles of the Lie algebra of the universal abelian variety over a Shimura variety X' arising from G'. We exploit this relationship to construct an injection of the etale cohomology of X' into that of X. This yields a geometric construction of "Jacquet-Langlands transfers" of automorphic representations of G' to automorphic representations of G.