{
  "id": "2403.13383",
  "version": "v1",
  "published": "2024-03-20T08:25:29.000Z",
  "updated": "2024-03-20T08:25:29.000Z",
  "title": "Size of discriminants of periodic geodesics of the modular surface",
  "authors": [
    "François Maucourant"
  ],
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Pick a random matrix $\\gamma$ in $\\Gamma={\\rm SL}(2,\\mathbb{Z})$. Denote by $\\mathcal{O}_K$ the Dedekind ring generated by its eigenvalues, and let $\\Delta_K$, $\\Delta_\\gamma$ and $\\Delta = {\\rm Tr}(\\gamma)^2-4$ be the respective discriminant of the rings $\\mathcal{O}_K$, the multiplier ring $M(2,\\mathbb{Z})\\cap \\mathbb{Q}[\\gamma]$ and $\\mathbb{Z}[\\gamma]$. We show that their ratios admit probability limit distributions. In particular, 42% of the elements of $\\Gamma$ have a fundamental discriminant, and $\\mathbb{Z}[\\gamma]$ is a ring of integers with probability 32%.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-20T08:25:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "periodic geodesics",
      "modular surface",
      "ratios admit probability limit distributions",
      "random matrix",
      "fundamental discriminant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}