A p-adic arithmetic inner product formula

Fix a prime number p and let E/F be a CM extension of number fields in which p splits relatively. Let π be an automorphic representation of a quasi-split unitary group of even rank with respect to E/F such that π is ordinary above p with respect to the Siegel parabolic subgroup. We construct the cyclotomic p-adic L-function of π, and show, under certain conditions, that if its order of vanishing at the trivial character is 1, then the rank of the Selmer group of the Galois representation of E associated with π is at least 1. Furthermore, under a certain modularity hypothesis, we use special cycles on unitary Shimura varieties to construct some explicit elements in the Selmer group called Selmer theta lifts; and we prove a precise formula relating their p-adic heights to the derivative of the p-adic L-function. In parallel to Perrin-Riou's p-adic analogue of the Gross--Zagier formula, our formula is the p-adic analogue of the arithmetic inner product formula recently established by Chao~Li and the second author.