{
  "id": "2112.01615",
  "version": "v2",
  "published": "2021-12-02T21:46:34.000Z",
  "updated": "2023-03-22T22:31:03.000Z",
  "title": "The average number of integral points on the congruent number curves",
  "authors": [
    "Stephanie Chan"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that the total number of non-torsion integral points on the elliptic curves $\\mathcal{E}_D:y^2=x^3-D^2x$, where $D$ ranges over positive squarefree integers less than $N$, is $O( N(\\log N)^{-1/4+\\epsilon})$. The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the $2$-Selmer group of the curves in this family.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-03-22T22:31:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11N45",
      "11R45"
    ],
    "keywords": [
      "congruent number curves",
      "average number",
      "integral binary quartic forms",
      "non-torsion integral points",
      "positive squarefree integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}