Heegner cycles and derivatives of p-adic L-functions

Let f be an even weight k >= 2 modular form on a p-adically uniformizable Shimura curve for a suitable Gamma(0)-type level structure. Let K/Q be an imaginary quadratic field, satisfying Heegner conditions assuring that the sign appearing in the functional equation of the complex L-function of f/K is negative. We may attach to f, or rather a deformation of it, a p-adic L-function of the weight variable kappa, also depending on K. Our main result is a formula relating the derivative of this p-adic L-function at kappa = k to the Abel-Jacobi images of so-called Heegner cycles.