A $p$-adic arithmetic inner product formula

TL;DR

This paper constructs a cyclotomic p-adic L-function for automorphic representations of unitary groups, relating its derivatives to Selmer groups and special cycles on Shimura varieties, extending p-adic Gross--Zagier type formulas.

Contribution

It develops a new p-adic L-function for unitary groups and establishes a formula connecting its derivatives to Selmer groups via special cycles.

Findings

Construction of the cyclotomic p-adic L-function for automorphic representations.

Proof that the order of vanishing at the trivial character influences Selmer group rank.

Explicit relation between p-adic heights of special cycles and derivatives of p-adic L-functions.

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 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 $\pi$ 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.