TOWARDS A GEOMETRIC JACQUET-LANGLANDS CORRESPONDENCE FOR UNITARY SHIMURA VARIETIES

Let G be a unitary group over a totally real field and let X be a Shimura variety associated to G For certain primes p of good reduction for X, we construct cycles X-tau 0 (1) 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 all finite places but not isomorphic to G at archimedean places More precisely, each cycle X-tau 0 1 has a natural desingularization (X) over tilde (tau 0 1) which is almost isomorphic to a scheme parameterizing certain subbundles of the Lie algebra of the universal abelian variety over a Shimura variety X' associated to 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