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 ${\mathrm{G}\mathrm{S}\mathrm{p}}_4({\mathbb{A}}_F)$ 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$E$isaquadraticextensionof$F$. The case where $E=F\times F$ has been previously dealt with by Liu.