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

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.