A proof of the refined Gan-Gross-Prasad conjecture for non-endoscopic Yoshida lifts. (English) Zbl 1433.11055
Summary: We prove a precise formula relating the Bessel period of certain automorphic forms on \(\mathrm{GSp}_4(\mathbb{A}_F)\) to a central \(L\)-value. This is a special case of the refined Gan-Gross-Prasad conjecture for the groups \((\mathrm{SO}_5,\mathrm{SO}_2)\) as set out by A. Ichino and T. Ikeda [Geom. Funct. Anal. 19, No. 5, 1378–1425 (2010; Zbl 1216.11057)] and Y. Liu [J. Reine Angew. Math. 717, 133–194 (2016; Zbl 1404.11065)]. 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 \(\mathrm{GL}_2(\mathbb{A}_E)\) where \(E\) is a quadratic extension of \(F\). The case where \(E=F\times F\) has been previously dealt with by Liu [loc. cit.].
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
Keywords:
automorphic period integrals; special values of \(L\)-functions; the refined Gan-Gross-Prasad conjectureReferences:
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