Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties

Abstract

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 $G$.