Quaternionic families of Heegner points and p -adic L -Functions

Following up on a previous article by the authors, which studies the interpolation of certain anticyclotomic p-adic L-functions associated to quaternionic modular forms in a Hida family, we extend the work of F. Castella on the interpolation and specialization of big Heegner points to the quaternionic setting. We prove an explicit reciprocity law relating the big p-adic L-function to the big Heegner points in this quaternionic setting. We deduce arithmetic results for the Selmer group of Hida’s big Galois representation and its specialization over an imaginary quadratic field satisfying a relaxed Heegner hypothesis.