A $p$ -adic arithmetic inner product formula

Abstract

Fix a prime number \(p\) and let \(E/F\) be a CM extension of number fields in which \(p\) splits relatively. Let \(\pi \) be an automorphic representation of a quasi-split unitary group of even rank with respect to \(E/F\) such that \(\pi \) is ordinary above \(p\) with respect to the Siegel parabolic subgroup. We construct the cyclotomic \(p\) -adic \(L\) -function of \(\pi \) , 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 \(p\) -adic \(L\) -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 \(E\) associated with \(\pi \) ; in particular, the rank of this Selmer group is at least 1. In fact, we prove a precise formula relating the \(p\) -adic heights of Selmer theta lifts 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.