Aprameyo Pal (Universität Duisburg-Essen): A central value formula of degree 6 complex $L$-series and arithmetic applications

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 about the algebraicity of the central values of the considered $L$-series up to the relevant periods. I would also include some other arithmetic applications towards subconvexity problem, construction of associated $p$-adic $L$- function etc.