Refined global Gross–Prasad conjecture on special Bessel periods and Böcherer’s conjecture

Abstract

In this paper we pursue the refined global Gross–Prasad conjecture for Bessel periods formulated by Yifeng Liu in the case of special Bessel periods for $\mathrm{SO}(2n+1)\times \mathrm{SO}(2)$. Recall that a Bessel period for $\mathrm{SO}(2n+1)\times \mathrm{SO}(2)$ is called special when the representation of $\mathrm{SO}(2)$ is trivial. Let $\pi$ be an irreducible cuspidal tempered automorphic representation of a special orthogonal group of an odd-dimensional quadratic space over a totally real number field $F$ whose local component ${\pi }_v$ at any archimedean place $v$ of $F$ is a discrete series representation. Let $E$ be a quadratic extension of $F$ and suppose that the special Bessel period corresponding to $E$ does not vanish identically on $\pi$. Then we prove the Ichino–Ikeda type explicit formula conjectured by Liu for the central value $L(1/2,\pi )L(1/2,\pi \times {\chi }_E)$, where ${\chi }_E$ denotes the quadratic character corresponding to $E$. Our result yields a proof of Böcherer’s conjecture on holomorphic Siegel cusp forms of degree two which are Hecke eigenforms.