{
  "id": "2302.03986",
  "version": "v1",
  "published": "2023-02-08T10:51:01.000Z",
  "updated": "2023-02-08T10:51:01.000Z",
  "title": "Rational Points of some genus $3$ curves from the rank $0$ quotient strategy",
  "authors": [
    "Tony Ezome",
    "Brice Miayoka Moussolo",
    "Régis Freguin Babindamana"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In 1922, Mordell conjectured that the set of rational points on a smooth curve $C$ over $\\mathbb{Q}$ with genus $g \\ge 2$ is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of $\\#C(\\mathbb{Q})$ by following Chabauty's approach which considers the special case when the Jacobian variety of $C$ has Mordell-Weil rank $< g$. In 2006, Stoll improved the Coleman's bound. Balakrishnan with her co-authors in [1] implemented the Chabauty-Coleman method to compute the rational points of genus $3$ hyperelliptic curves. Then, Hashimoto and Morrison [8] did the same work for Picard curves. But it happens that this work has not yet been done for all genus 3 curves. In this paper, we describe an algorithm to compute the complete set of rational points $C(\\mathbb{Q})$ for any genus $3$ curve $C/\\mathbb{Q}$ that is a degree-$2$ cover of a genus $1$ curve whose Jacobian has rank $0$. We implemented this algorithm in Magma, and we ran it on approximately $40, 000$ curves selected from databases of plane quartics and genus $3$ hyperellitic curves. We discuss some interesting examples, and we exhibit curves for which the number of rational points meets the Stoll's bound",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-08T10:51:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quotient strategy",
      "rational points meets",
      "smooth curve",
      "hyperellitic curves",
      "plane quartics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}