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This is the third of a series of papers relating intersections of specialcycles on the integral model of a Shimura surface to Fourier coefficients ofHilbert modular forms. More precisely, we embed the Shimura curve over Qassociated to a rational quaternion algebra into the Shimura surface associatedto the base change of the quaternion algebra to a real quadratic field. Afterextending the associated moduli problems over Z we obtain an arithmeticthreefold with a embedded arithmetic surface, which we view as a cycle ofcodimension one. We then construct a family, indexed by totally positivealgebraic integers in the real quadratic field, of codimension two cycles(complex multiplication points) on the arithmetic threefold. The intersectionmultiplicities of the codimension two cycles with the fixed codimension onecycle are shown to agree with the Fourier coefficients of a (very particular)Hilbert modular form of weight 3/2. The results are higher dimensional variantsof results of Kudla-Rapoport-Yang, which relate intersection multiplicities ofspecial cycles on the integral model of a Shimura curve to Fourier coefficientsof a modular form in two variables.

Intersection theory on Shimura surfaces II