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

### Masaaki Furusawa

Osaka City University, Japan### Kazuki Morimoto

Kobe University, Japan

## 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.

## Cite this article

Masaaki Furusawa, Kazuki Morimoto, Refined global Gross–Prasad conjecture on special Bessel periods and Böcherer’s conjecture. J. Eur. Math. Soc. 23 (2021), no. 4, pp. 1295–1331

DOI 10.4171/JEMS/1034