{
  "id": "2206.02192",
  "version": "v1",
  "published": "2022-06-05T14:34:55.000Z",
  "updated": "2022-06-05T14:34:55.000Z",
  "title": "Jacobi forms, Saito-Kurokawa lifts, their Pullbacks and sup-norms on average",
  "authors": [
    "Pramath Anamby",
    "Soumya Das"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We formulate a precise conjecture about the size of the $L^\\infty$-mass of the space of Jacobi forms on $\\mathbb H_n \\times \\mathbb C^{g \\times n}$ of matrix index $S$ of size $g$. This $L^\\infty$-mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such $n,g,S$ and prove the upper bound in the $k$ aspect when $n=1$, $g \\ge 1$. When $n=1$ and $g=1$, we make a more refined study of the sizes of the index-(old and) new spaces, the latter via the Waldspurger's formula. Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central $L$-values $L(1/2, f \\otimes \\chi_D)$ with $f$ varying over newforms of level a prime $p$ and even weight $k$ as $k,p \\to \\infty$ and $D$ being (explicitly) polynomially bounded by $k,p$. Here $\\chi_D$ is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight $k$) is $k^{5/2}$ by three different methods (with or without the use of central $L$-values), and show that the size of their pullbacks to the diagonally embedded $\\mathbb H \\times \\mathbb H$ is $k^2$. In an appendix, the same question is answered for the pullbacks of the whole space $S^2_k$, the size here being $k^3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-05T14:34:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F46",
      "11F50",
      "11F66",
      "11F11",
      "11F30",
      "11F37"
    ],
    "keywords": [
      "saito-kurokawa lifts",
      "jacobi forms",
      "real quadratic dirichlet character",
      "power saving asymptotic formula",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}