A proof of the refined Gan--Gross--Prasad conjecture for non-endoscopic Yoshida lifts

TL;DR

This paper proves a specific case of the refined Gan--Gross--Prasad conjecture relating Bessel periods to central L-values for Yoshida lifts obtained via automorphic induction from GL(2) over quadratic extensions.

Contribution

It establishes the conjecture for Yoshida lifts from GL(2) over quadratic extensions, extending previous results to new cases.

Findings

Verified the conjecture for Yoshida lifts from quadratic extensions

Connected Bessel periods with central L-values in this setting

Extended prior results to non-endoscopic automorphic forms

Abstract

We prove a precise formula relating the Bessel period of certain automorphic forms on ${\rm GSp}_{4}(\mathbb{A}_{F})$ to a central $L$-value. This is a special case of the refined Gan--Gross--Prasad conjecture for the groups $({\rm SO}_{5},{\rm SO}_{2})$ as set out by Ichino--Ikeda and Liu. This conjecture is deep and hard to prove in full generality; in this paper we succeed in proving the conjecture for forms lifted, via automorphic induction, from ${\rm GL}_{2}(\mathbb{A}_{E})$ where $E$ is a quadratic extension of $F$. The case where $E=F\times F$ has been previously dealt with by Liu.