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

TL;DR

This paper derives an explicit formula for the central values of certain degree 6 L-series related to modular forms, generalizing previous results and confirming Deligne's conjecture on their algebraicity.

Contribution

It provides a new explicit central value formula for complex L-series arising from Saito--Kurokawa lifts, extending prior work and verifying algebraicity conjectures.

Findings

Derived an explicit central value formula for degree 6 L-series.

Generalized Ichino's formula involving Saito--Kurokawa lifts.

Proved algebraicity of central L-values up to periods.

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.