Let be a unitary group over a totally real field, and let be a Shimura variety associated to . For certain primes of good reduction for , we construct cycles on the characteristic fiber of . 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 . The geometry of these cycles turns out to be closely related to Shimura varieties for a different unitary group , which is isomorphic to at all finite places but not isomorphic to at archimedean places. More precisely, each cycle has a natural desingularization , which is almost isomorphic to a scheme parameterizing certain subbundles of the Lie algebra of the universal abelian variety over a Shimura variety associated to . We exploit this relationship to construct an injection of the étale cohomology of into that of . This yields a geometric construction of Jacquet-Langlands transfers of automorphic representations of to automorphic representations of .
"Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties." Duke Math. J. 155 (3) 483 - 518, 1 December 2010. https://doi.org/10.1215/00127094-2010-061