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 $π$ 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.