Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass
Pramath Anamby, Soumya Das
TL;DR
The paper develops a complete spectral analysis of pullbacks of Saito–Kurokawa lifts at odd square-free levels, and expresses the diagonal old-class contributions in terms of central L-values L(f⊗sym^2 g, 1/2) within a GL3×GL2 framework. It introduces an explicit coset-representative setup and a Fourier–Jacobi unfolding strategy to derive pullback formulae, culminating in a precise L^2-mass formula N(F_f) as a weighted average of central L-values. The authors prove a uniform central-value relation for old-classes, establish non-vanishing criteria and average-nonvanishing results, and formulate a norm conjecture predicting the size and positivity of pullbacks; an appendix verifies the main average-term against the predicted main term. Collectively, the work ties pullback spectra to central L-values across the level aspect, providing both conditional and unconditional non-vanishing results and informing broader GGP-type phenomena in higher-rank settings.
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.