Pullback formulae for nearly holomorphic Saito-Kurokawa lifts

TL;DR

This paper derives explicit pullback formulae for nearly holomorphic Saito-Kurokawa lifts, extending Ichino's work to higher levels and weights, with applications to automorphic L-functions and the Gan-Gross-Prasad conjecture.

Contribution

It generalizes pullback formulae for Saito-Kurokawa lifts to higher levels and weights, providing new tools for automorphic forms and L-functions.

Findings

Explicit pullback formulae for nearly holomorphic Saito-Kurokawa lifts.

Verification of Deligne's conjecture for certain automorphic L-values.

Examples supporting the refined Gan-Gross-Prasad conjecture in non-tempered cases.

Abstract

We give explicit pullback formulae for nearly holomorphic Saito-Kurokawa lifts restrict to product of upper half-plane against with product of elliptic modular forms. We generalize the formula of Ichino to modular forms of higher level and free the restriction on weights. The explicit formulae provide non-trivial examples for the refined Gan-Gross-Prasad conjecture for $({\rm SO}_5,{\rm SO}_4)$ in the non-tempered cases. As an application, we obtain Deligne's conjecture for critical values of certain automorphic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$. We also expect to apply our pullback formulae to construct two-variables $p$-adic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$ in the future.