{
  "id": "1711.06242",
  "version": "v1",
  "published": "2017-11-16T18:37:34.000Z",
  "updated": "2017-11-16T18:37:34.000Z",
  "title": "Methods for constructing elliptic and hyperelliptic curves with rational points",
  "authors": [
    "Kirti Joshi"
  ],
  "comment": "14 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number field equipped with a rational point, (resp. with two rational points) of infinite order over the given number field, and elliptic curves over the rationals with two rational points over `simplest cubic fields.' I also provide hyperelliptic curves of genus exceeding any given number over any given number fields with points (over the given number field) which span a subgroup of rank at least $g$ in the group of rational points of the Jacobian of this curve. I also provide a method of constructing hyperelliptic curves over rational function fields with rational points defined over field extensions with large finite simple Galois groups, such as the Mathieu group $M_{24}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-16T18:37:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G10",
      "11G30",
      "14H52"
    ],
    "keywords": [
      "rational point",
      "hyperelliptic curves",
      "constructing elliptic",
      "number field",
      "large finite simple galois groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}