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

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