{
  "id": "1509.00528",
  "version": "v1",
  "published": "2015-09-02T00:10:16.000Z",
  "updated": "2015-09-02T00:10:16.000Z",
  "title": "Torsion subgroups of rational elliptic curves over the compositum of all cubic fields",
  "authors": [
    "Harris B. Daniels",
    "Alvaro Lozano-Robledo",
    "Filip Najman",
    "Andrew V. Sutherland"
  ],
  "comment": "31 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E/\\mathbb{Q}$ be an elliptic curve and let $\\mathbb{Q}(3^\\infty)$ be the compositum of all cubic extensions of $\\mathbb{Q}$. In this article we show that the torsion subgroup of $E(\\mathbb{Q}(3^\\infty))$ is finite and determine 20 possibilities for its structure, along with a complete description of the $\\overline{\\mathbb{Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many $\\overline{\\mathbb{Q}}$-isomorphism classes of elliptic curves, and a complete list of $j$-invariants for each of the 4 that do not.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-02T00:10:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11R21",
      "12F10",
      "14H52"
    ],
    "keywords": [
      "rational elliptic curves",
      "torsion subgroup",
      "cubic fields",
      "compositum",
      "isomorphism classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150900528D"
    }
  }
}