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

Abstract

Abstract We prove a precise formula relating the Bessel period of certain automorphic forms on GSp 4 ⁢ ( 𝔸 F ) ${\mathrm{GSp}_{4}(\mathbb{A}_{F})}$ to a central L-value. This is a special case of the refined Gan–Gross–Prasad conjecture for the groups ( SO 5 , SO 2 ) ${(\mathrm{SO}_{5},\mathrm{SO}_{2})}$ as set out by Ichino–Ikeda [12] and Liu [14]. 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 GL 2 ⁢ ( 𝔸 E ) ${\mathrm{GL}_{2}(\mathbb{A}_{E})}$ where E is a quadratic extension of F. The case where E = F × F ${E=F\times F}$ has been previously dealt with by Liu [14].