{
  "id": "math/9912041",
  "version": "v3",
  "published": "1999-12-06T08:07:24.000Z",
  "updated": "1999-12-16T03:00:18.000Z",
  "title": "Elliptic Curves from Sextics",
  "authors": [
    "Mutsuo Oka"
  ],
  "comment": "A remark is added. (after replace mistake). 16 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $\\mathcal N$ be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space ${\\mathcal N}/G$ is one-dimensional and consists of two components, ${\\mathcal N}_{torus}/G$ and ${\\mathcal N}_{gen}/G$. By quadratic transformations, they are transformed into one-parameter families $C_s$ and $D_s$ of cubic curves respectively. We study the Mordell-Weil torsion groups of cubic curves $C_s$ over $\\bfQ$ and $D_s$ over $\\bfQ(\\sqrt{-3})$ respectively. We show that $C_{s}$ has the torsion group $\\bf Z/3\\bf Z$ for a generic $s\\in \\bf Q$ and it also contains subfamilies which coincide with the universal families given by Kubert with the torsion groups $\\bf Z/6\\bf Z$, $\\bf Z/6\\bf Z+\\bf Z/2\\bf Z$, $\\bf Z/9\\bf Z$ or $\\bf Z/12\\bf Z$. The cubic curves $D_s$ has torsion $\\bf Z/3\\bf Z+\\bf Z/3\\bf Z$ generically but also $\\bf Z/3\\bf Z+\\bf Z/6\\bf Z$ for a subfamily which is parametrized by $ \\bf Q(\\sqrt{-3}) $.",
  "revisions": [
    {
      "version": "v3",
      "updated": "1999-12-16T03:00:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H45",
      "14H20",
      "14H52"
    ],
    "keywords": [
      "elliptic curves",
      "quotient moduli space",
      "mordell-weil torsion groups",
      "universal families",
      "quadratic transformations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math.....12041O"
    }
  }
}