We relate the @math -adic heights of generalized Heegner cycles to the derivative of a @math -adic @math -function attached to a pair @math , where @math is an ordinary weight @math newform and @math is an unramified imaginary quadratic Hecke character of infinity type @math , with @math . This generalizes the @math -adic Gross-Zagier formula in the case @math due to Perrin-Riou (in weight two) and Nekov 'a r (in higher weight).
In this article, we study @math -adic torus periods for certain @math -adic valued functions on Shimura curves coming from classical origin. We prove a @math -adic Waldspurger formula for these periods, generalizing the recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic @math -adic @math -function of Rankin-Selberg type. At a character of positive weight, the @math -adic @math -function interpolates the central critical value of the complex Rankin-Selberg @math -function. Its value at a Dirichlet character, which is outside the range of interpolation, essentially computes the corresponding @math -adic torus period. In this paper, we deduce the vanishing of Selmer groups for the Rankin-Selberg convolution of a cusp form with a theta series of higher weight from the nonvanishing of the associated @math -value, thus establishing the rank 0 case of the Bloch-Kato conjecture in these cases. Our methods are based on the connection between Heegner cycles and @math -adic @math -functions, building upon recent work of Bertolini, Darmon and Prasanna, and on an extension of Kolyvagin's method of Euler systems to the anticyclotomic setting. In the course of the proof, we also obtain a higher weight analogue of Mazur's conjecture (as proven in weight 2 by Cornut-Vatsal), and as a consequence of our results, we deduce from Nekovar's work a proof of the parity conjecture in this setting. Let @math be a primitive Hilbert modular form of parallel weight @math and level @math for the totally real field @math , and let @math be a rational prime coprime to @math . If @math is ordinary at @math and @math is a CM extension of @math of relative discriminant @math prime to @math , we give an explicit construction of the @math -adic Rankin-Selberg @math -function @math . When the sign of its functional equation is @math , we show, under the assumption that all primes @math are principal ideals of @math which split in @math , that its central derivative is given by the @math -adic height of a Heegner point on the abelian variety @math associated with @math . This @math -adic Gross--Zagier formula generalises the result obtained by Perrin-Riou when @math and @math satisfies the so-called Heegner condition. We deduce applications to both the @math -adic and the classical Birch and Swinnerton-Dyer conjectures for @math . 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 L-p(f, chi) which interpolates special values of the Rankin-Selberg convolution of a fixed newform f and a theta-series theta(chi) attached to an unramified Hecke character of an imaginary quadratic field K. 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. Let p be a prime number and let E be an elliptic curve defined over ℚ of conductor N. Let K be an imaginary quadratic field with discriminant prime to pN such that all prime factors of N split in K. B. Perrin-Riou established the p-adic Gross-Zagier formula that relates the first derivative of the p-adic L-function of E over K to the p-adic height of the Heegner point for K when E has good ordinary reduction at p. In this article, we prove the p-adic Gross-Zagier formula of E for the cyclotomic ℤ p -extension at good supersingular prime p. Our result has an application for the full Birch and Swinnerton-Dyer conjecture. Suppose that the analytic rank of E over ℚ is 1 and assume that the Iwasawa main conjecture is true for all good primes and the p-adic height pairing is not identically equal to zero for all good ordinary primes, then our result implies the full Birch and Swinnerton-Dyer conjecture up to bad primes. In particular, if E has complex multiplication and of analytic rank 1, the full Birch and Swinnerton-Dyer conjecture is true up to a power of bad primes and 2.