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

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.