Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass

TL;DR

This paper analyzes the spectral decomposition of Saito-Kurokawa lifts of square-free levels, relating their pullbacks to central L-values, and explores their non-vanishing and $L^2$-mass distribution.

Contribution

It provides a complete spectral decomposition of these lifts, links pullback projections to central L-values, and formulates a conjecture for their $L^2$-mass based on heuristics.

Findings

Main term of $L^2$-mass matches heuristics on average over $f$

Proves non-vanishing conditions for pullbacks

Relates pullback projections to central $L$-values

Abstract

We obtain the full spectral decomposition of the pullback of a Saito-Kurokawa (SK) newform $F$ of odd, square-free level; and show that the projections onto the elements $\mathbf g \otimes \mathbf g$ of an arithmetically orthogonalized old-basis are either zero or whose squares are given by the certain $\mathrm{GL}(3)\times \mathrm{GL}(2)$ central $L$-values $L(f\otimes \mathrm{sym}^2 g, \frac{1}{2})$, where $F$ is the lift of the $\mathrm{GL}(2)$ newform $f$ and $g$ is the newform underlying $\mathbf g$. Based on this, we work out a conjectural formula for the $L^2$-mass of the pullback of $F$ via the CFKRS heuristics, which becomes a weighted average (over $g$) of the central $L$-values. We show that on average over $f$, the main term predicted by the above heuristics matches with the actual main term. We also provide several results and sufficient conditions that ensure the non-vanishing of the pullbacks.