  Kudla has proposed a general program to relate arithmetic intersection
multiplicities of special cycles on Shimura varieties to Fourier coefficients
of 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. In the present paper we prove results in a
higher dimensional setting. On the integral model of a Shimura surface we
consider the intersection of a Shimura curve with a codimension two cycle of
complex multiplication points, and relate the intersection to certain cycles
classes constructed by Kudla-Rapoport-Yang. As a corollary we deduce that our
intersection multiplicities appear as Fourier coefficients of a Hilbert modular
form of half-integral weight.
