Intersection theory on Shimura surfaces

Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of an Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla-Rapoport-Yang. We will prove results in a higher dimensional setting: on the integral model of a Shimura surface we consider the intersection of an embedded Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersection numbers to Fourier coefficients of a Hilbert modular form of half-integral weight.