The Gan-Gross-Prasad period of Klingen Eisenstein families over unitary groups

TL;DR

This paper computes Gan-Gross-Prasad period integrals of Klingen Eisenstein series over unitary groups, relates them to special L-values, and constructs a p-adic L-function, advancing the understanding of Eisenstein series in Iwasawa theory.

Contribution

It provides explicit formulas for Gan-Gross-Prasad periods of Klingen Eisenstein series and develops their p-adic interpolation, introducing a new p-adic L-function over unitary groups.

Findings

Gan-Gross-Prasad period integrals relate to special Rankin-Selberg L-values.

Constructs a p-adic L-function of Rankin-Selberg type.

Establishes groundwork for proving p-primitivity of Klingen Eisenstein series.

Abstract

In this article, we compute the Gan-Gross-Prasad period integral of Klingen Eisenstein series over the unitary group $\mathrm{U}(m+1, n+1)$ with a cuspidal automorphic form over $\mathrm{U}(m+1, n)$, and show that it is related to certain special Rankin-Selberg $L$-values. We $p$-adically interpolate these Gan-Gross-Prasad period integrals as the Klingen Eisenstein series and the cuspidal automorphic form vary in Hida families. As a byproduct, we obtain a $p$-adic $L$-function of Rankin-Selberg type over $\mathrm{U}(m,n) \times \mathrm{U}(m+1, n)$. The ultimate motivation is to show the $p$-primitive property of Klingen Eisenstein series over unitary groups, by computing such Gan-Gross-Prasad period integrals, and this article is a starting point of this project. The $p$-primitivity of Eisenstein series is an essential property in the automorphic method in Iwasawa theory.