p-adic Abel-Jacobi map and p-adic Gross-Zagier formula for Hilbert modular forms

We compute the p-adic Abel-Jacobi map of the product of a Hilbert modular surface and a modular curve at a null-homologous (modified) embedding of the modular curve in this product, evaluated on differentials associated to a Hilbert cuspidal form f of weight (2,2) and a cuspidal form of weight 2. We generalize this computation to suitable null-homological cycles in the fibre products of the universal families on the surface and the curve, evaluated at differentials associated to f and g of higher weights. We express the values of the p-adic Abel-Jacobi map at these weights in terms of a p-adic L- function associated to a Hida family of Hilbert modular forms and a Hida family of cuspidal forms. Our function is a Hilbert modular analogue of the p- adic L-function defined by Darmon and Rotger.