{
  "id": "1710.05228",
  "version": "v1",
  "published": "2017-10-14T19:53:44.000Z",
  "updated": "2017-10-14T19:53:44.000Z",
  "title": "Torsion subgroups of rational elliptic curves over the compositum of all extensions of generalized $D_4$-type",
  "authors": [
    "Harris B. Daniels"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E/\\mathbb{Q}$ be an elliptic curve and let $\\mathbb{Q}(D_4^\\infty)$ be the compositum of all extensions of $\\mathbb{Q}$ whose Galois closure has Galois group isomorphic to a subdirect product of a finite number of transitive subgroups of $D_4$. In this article we prove that the torsion subgroup of $E(\\mathbb{Q}(D_4^\\infty))$ is finite and determine the 24 possibility for its structure. We also give a complete classification of the elliptic curves that have each possible torsion structure in terms of their $j$-invariants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-14T19:53:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11R21",
      "12F10",
      "14H52"
    ],
    "keywords": [
      "rational elliptic curves",
      "torsion subgroup",
      "extensions",
      "compositum",
      "galois group isomorphic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}