We construct ;;generalized Heegner cycles;; on a variety fibered over a Shimura curve, defined over a number field. We show that their images under the p-adic Abel-Jacobi map coincide with the values (outside the range of interpolation) of a p-adic L-function which interpolates special values of the Rankin-Selberg convolution of a fixed newform f and a theta-series attached to an unramified Hecke character of an imaginary quadratic field. This generalizes previous work of Bertolini, Darmon, and Prasanna, which demonstrated a similar result in the case of modular curves. Our main tool is the theory of Serre-Tate coordinates, which yields p-adic expansions of modular forms at CM points, replacing the role of q-expansions in computations on modular curves.
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Generalized Heegner Cycles, Shimura Curves, and Special Values of p-ADIC L-Functions.