{
  "id": "2206.00412",
  "version": "v1",
  "published": "2022-06-01T11:23:11.000Z",
  "updated": "2022-06-01T11:23:11.000Z",
  "title": "Quaternary quadratic forms with prime discriminant",
  "authors": [
    "Jeremy Rouse",
    "Katherine Thompson"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on the Petersson norm $\\langle C, C \\rangle$ of the cuspidal part of the theta series of $Q$. We derive an upper bound on $\\langle C, C \\rangle$ that depends on the smallest positive integer not represented by the dual form $Q^{*}$. In addition, we give a non-trivial upper bound on the sum of the integers $n$ excepted by $Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-01T11:23:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E20",
      "11F27",
      "11F30",
      "11E12"
    ],
    "keywords": [
      "prime discriminant",
      "positive-definite quaternary quadratic form",
      "non-trivial upper bound",
      "explicit lower bound",
      "petersson norm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}