{
  "id": "2110.06260",
  "version": "v2",
  "published": "2021-10-12T18:17:32.000Z",
  "updated": "2022-10-06T17:34:00.000Z",
  "title": "On Kitaoka's conjecture and lifting problem for universal quadratic forms",
  "authors": [
    "Vítězslav Kala",
    "Pavlo Yatsyna"
  ],
  "comment": "8 pages, to appear in Bull. LMS",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal (namely, those that have a short basis in a suitable sense). Along the way we give a general construction of a universal form of rank bounded by $D(\\log D)^{d-1}$, where $d$ is the degree of $K$ over $\\mathbb Q$ and $D$ is its discriminant. Furthermore, for any fixed degree we prove (weak) Kitaoka's conjecture that there are only finitely many totally real number fields with a universal ternary quadratic form.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-10-06T17:34:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E12",
      "11E20",
      "11E25",
      "11H06",
      "11R04"
    ],
    "keywords": [
      "universal quadratic forms",
      "kitaokas conjecture",
      "lifting problem",
      "totally real number field",
      "positive definite quadratic form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}