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We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature $$(n-1,1)$$ . We construct an arithmetic theta lift from harmonic Maass forms of weight $$2-n$$ to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form $$f$$ a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of $$f$$ . Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of $$f$$ against a CM cycle, and (2) the central derivative of the convolution $$L$$ -function of a weight $$n$$ cusp form (depending on $$f$$ ) and the theta function of a positive definite hermitian lattice of rank $$n-1$$ . When specialized to the case $$n=2$$ , this result can be viewed as a variant of the Gross-Zagier formula for Shimura curves associated to unitary groups of signature $$(1,1)$$ . The proof relies on, among other things, a new method for computing improper arithmetic intersections. [ABSTRACT FROM AUTHOR]

This is the third of a series of papers relating intersections of special cycles on the integral model of a Shimura surface to Fourier coefficients of Hilbert modular forms. More precisely, we embed the Shimura curve over ℚ associated to a rational quaternion algebra into the Shimura surface associated to the base change of the quaternion algebra to a real quadratic field. After extending the associated moduli problems over ℤ we obtain an arithmetic threefold with a embedded arithmetic surface, which we view as a cycle of codimension one. We then construct a family, indexed by totally positive algebraic integers in the real quadratic field, of codimension two cycles (complex multiplication points) on the arithmetic threefold. The intersection multiplicities of the codimension two cycles with the fixed codimension one cycle are shown to agree with the Fourier coefficients of a (very particular) Hilbert modular form of weight 3/2. The results are higher dimensional variants of results of Kudla-Rapoport-Yang, which relate intersection multiplicities of special cycles on the integral model of a Shimura curve to Fourier coefficients of a modular form in two variables. [ABSTRACT FROM AUTHOR]

Given a weight two modular form f with associated p-adic Galois representation V f , for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V f by taking the Kummer images of Heegner points on the modular abelian variety attached to f. We show that these classes can be interpolated as f varies in a Hida family and construct an Euler system of big Heegner points for Hida’s universal ordinary deformation of V f . We show that the specialization of this big Euler system to any form in the Hida family is nontrivial, extending results of Cornut and Vatsal from modular forms of weight two and trivial character to all ordinary modular forms, and propose a horizontal nonvanishing conjecture for these cohomology classes. The horizontal nonvanishing conjecture implies, via the theory of Euler systems, a conjecture of Greenberg on the generic ranks of Selmer groups in Hida families. [ABSTRACT FROM AUTHOR]