Modular p-adic L-functions attached to real quadratic fields and arithmetic applications

Let f ∈ Sk0+2(Γ0(Np)) be a normalized N-new eigenform with p ∤ N and such that ap2 ≠ pk0+1 and ordp(ap) < k0 + 1. By Coleman's theory, there is a p-adic family of eigenforms whose weight k0 + 2 specialization is f. Let K be a real quadratic field and let ψ be an unramified character of Gal(K̅ /K). Under mild hypotheses on the discriminant of K and the factorization of N, we construct a p-adic L-function ℒ/K,ψ interpolating the central critical values of the Rankin L-functions associated to the base change to K of the specializations of in classical weight, twisted by ψ. When the character ψ is quadratic, ℒ/K,ψ factors into a product of two Mazur-Kitagawa p-adic L-functions. If, in addition, has p-new specialization in weight k0 + 2, then under natural parity hypotheses we may relate derivatives of each of the Mazur-Kitagawa factors of ℒ/K,ψ at k0 to Bloch–Kato logarithms of Heegner cycles. On the other hand the derivatives of our p-adic L-functions encodes the position of the so called Darmon cycles.