Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909610100654080 |
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| author | Anamby, Pramath Das, Soumya |
| author_facet | Anamby, Pramath Das, Soumya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08660 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass Anamby, Pramath Das, Soumya Number Theory Primary 11F46, 11F50, 11F67, Secondary 11F37 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. |
| title | Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass |
| topic | Number Theory Primary 11F46, 11F50, 11F67, Secondary 11F37 |
| url | https://arxiv.org/abs/2505.08660 |