{
  "id": "2505.08660",
  "version": "v1",
  "published": "2025-05-13T15:22:37.000Z",
  "updated": "2025-05-13T15:22:37.000Z",
  "title": "Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass",
  "authors": [
    "Pramath Anamby",
    "Soumya Das"
  ],
  "comment": "Comments are most welcome",
  "categories": [
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-13T15:22:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F46",
      "11F50",
      "11F67",
      "11F37"
    ],
    "keywords": [
      "square-free level",
      "saito-kurokawa lifts",
      "full spectral decomposition",
      "actual main term",
      "non-vanishing"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}