{
  "id": "2206.02190",
  "version": "v1",
  "published": "2022-06-05T14:26:13.000Z",
  "updated": "2022-06-05T14:26:13.000Z",
  "title": "Bounds for the Bergman kernel and the sup-norm of holomorphic Siegel cusp forms",
  "authors": [
    "Soumya Das",
    "Hariram Krishna"
  ],
  "comment": "39 pp",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove `polynomial in $k$' bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree $n$ and weight $k$. When $n=1,2$ our bounds agree with the conjectural bounds on the aforementioned size, while the lower bounds match for all $n \\ge 1$. For an $L^2$-normalised Siegel cusp form $F$ of degree $2$, our bound for its sup-norm is $O_\\epsilon (k^{9/4+\\epsilon})$. Further, we show that in any compact set $\\Omega$ (which does not depend on $k$) contained in the Siegel fundamental domain of $\\mathrm{Sp}(2, \\mathbb Z)$ on the Siegel upper half space, the sup-norm of $F$ is $O_\\Omega(k^{3/2 - \\eta})$ for some $\\eta>0$, going beyond the `generic' bound in this setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-05T14:26:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F46",
      "11F30"
    ],
    "keywords": [
      "holomorphic siegel cusp forms",
      "bergman kernel",
      "siegel upper half space",
      "lower bounds match",
      "normalised siegel cusp form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}