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 $X_{\tau_0,i}$ 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 $G^{\prime}$ , which is isomorphic to $G$ at all finite places but not isomorphic to $G$ at archimedean places. More precisely, each cycle $X_{\tau_0,i}$ has a natural desingularization ${\tilde X}_{\tau_0,i}$ , which is almost isomorphic to a scheme parameterizing certain subbundles of the Lie algebra of the universal abelian variety over a Shimura variety $X^{\prime}$ associated to $G^{\prime}$ . We exploit this relationship to construct an injection of the étale cohomology of $X^{\prime}$ into that of $X$ . This yields a geometric construction of Jacquet-Langlands transfers of automorphic representations of $G^{\prime}$ to automorphic representations of $G$ .