Pullbacks of Saito--Kurokawa lifts and a central value formula

Abstract

We prove an explicit central value formula for a family of complex $L$-series of degree $6$ for ${\mathrm{GL}}_2\times {\mathrm{GL}}_3$ which arise as factors of certain Garret--Rankin triple product $L$-series associated with modular forms. Our result generalizes a previous formula of Ichino involving Saito--Kurokawa lifts, and as an application, we prove Deligne's conjecture stating the algebraicity of the central values of the considered $L$-series up to the relevant periods.

59 pages, typos and minor issues corrected, To appear in Documenta Mathematica