A p-adic arithmetic inner product formula

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