CM cycles on Shimura curves, and p-adic L-function

Let f be a modular form of weight k >= 2 and level N, let K be a quadratic imaginary field and assume that there is a prime p exactly dividing N. Under certain arithmetic conditions on the level N and the field K, one can attach to this data a p-adic L-function L-p (f, K, s), as done by Bertolini-Darmon-Iovita-Spiess in [Teitelbaum's exceptional zero conjecture in the anticyclotomic setting, Amer. J. Math. 124 (2002), 411-449]. In the case of p being inert in K, this analytic function of a p-adic variable s vanishes in the critical range s = 1, ... , k - 1, and one may be interested in the values of its derivative in this range. We construct, for k >= 4, a Chow motive endowed with a distinguished collection of algebraic cycles which encode these values, via the p-adic Abel-Jacobi map. Our main result generalizes the result obtained by Iovita and Spiess in [Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms, Invent. Math. 1 5 4 (2003), 333-384], which gives a similar formula for the central value s = k/2. Even in this case our construction is different from the one found by Iovita and Spiess.