A geometric Jacquet-Langlands correspondence for U(2) Shimura varieties

Let G be a unitary group over Q, associated to a CM-field F with totally real part F, with signature (1, 1) at all the archimedean places of F. For certain primes p, and X a Shimura variety associated to G with good reduction at p, we construct a stratification of the characteristic p fiber of X , whose closed strata are indexed by subsets S of the archimedean places of F. Certain closed strata XS turn out to be closely related to Shimura varieties for different unitary groups. For each suitable S, we find a unitary groupG, isomorphic toG at nonarchimedean places but not isomorphic to G at certain archimedean places. We then construct a natural morphism from XS to a certain (P )bundle over the characteristic p fiber of a Shimura variety X ′ associated to G. This morphism is finite and purely inseparable, and is therefore an isomorphism on étale cohomology. We also construct an analogue of the Deligne-Rapoport model for Shimura varieties for G with “Γ0(p)”level structure. Given such a Shimura variety X , we break its characteristic p fiber into a union of smooth subschemes Y , each of which is a disjoint union of irreducible components of the model. For each such Y , we construct a (P)-bundle Z over a closed stratum of X , and a finite, purely inseparable morphism from Y to Z. This, together with the aforementioned result on the closed strata of X , identifies each such Y as a product of familiar P-bundles on a unitary Shimura variety X , up to a finite, purely inseparable morphism (essentially a Frobenius twist). Finally, we illustrate, in the case where F is real quadratic and p is inert in F, how the above geometric results can be used to deduce a “Jacquet-Langlands correspondence” relating the étale cohomology of Shimura varieties for two distinct unitary groups G and G. 2000 MSC Classification: 11G18