{
  "id": "math/0411484",
  "version": "v3",
  "published": "2004-11-22T14:46:23.000Z",
  "updated": "2005-07-27T10:59:34.000Z",
  "title": "The number of $S_4$ fields with given discriminant",
  "authors": [
    "Juergen Klueners"
  ],
  "comment": "new version",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that the number of quartic $S_4$--extensions of the rationals of given discriminant $d$ is $O_\\eps(d^{1/2+\\eps})$ for all $\\eps>0$. For a prime number $p$ we derive that the dimension of the space of octahedral modular forms of weight 1 and conductor $p$ or $p^2$ is bounded above by $O(p^{1/2}\\log(p)^2)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-07-27T10:59:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11R16",
      "11R32"
    ],
    "keywords": [
      "discriminant",
      "octahedral modular forms",
      "prime number",
      "extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....11484K"
    }
  }
}