{
  "id": "2309.15255",
  "version": "v1",
  "published": "2023-09-26T20:27:59.000Z",
  "updated": "2023-09-26T20:27:59.000Z",
  "title": "Distribution of the successive minima of the Petersson norm on cusp forms",
  "authors": [
    "Souparna Purohit"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $\\Gamma \\subseteq \\text{PSL}_2(\\mathbb{Z})$ be a finite index subgroup. Let $\\mathscr{X}(\\Gamma)$ be a regular proper model of the modular curve associated with $\\Gamma$, and let $\\overline{\\mathscr{L}}^{\\otimes k}$ be the logarithmically singular metrized line bundle on $\\mathscr{X}(\\Gamma)$ associated to modular forms of level $\\Gamma$ and weight $12k$, endowed with the Petersson metric. For each $k \\geq 1$, the sub-lattice $\\mathscr{S}_k \\subseteq H^0(\\mathscr{X}(\\Gamma), \\mathscr{L}^{\\otimes k})$ of integral cusp forms of level $\\Gamma$ and weight $12k$ is a euclidean lattice with respect to the Petersson norm. In this paper, we describe the distribution of the successive minima of the $\\mathscr{S}_k$ as $k \\to \\infty$, generalizing the work of Chinburg, Guignard, and Soul\\'{e} which addressed the case $\\Gamma = \\text{PSL}_2(\\mathbb{Z})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-26T20:27:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "petersson norm",
      "successive minima",
      "distribution",
      "integral cusp forms",
      "logarithmically singular metrized line bundle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}