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

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- Spie β 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 Spieβ in [Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms, Invent. Math. 154 (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 Spieβ.