{
  "id": "1808.00029",
  "version": "v1",
  "published": "2018-07-31T19:02:45.000Z",
  "updated": "2018-07-31T19:02:45.000Z",
  "title": "On 5-torsion of CM elliptic curves",
  "authors": [
    "Laura Paladino"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathcal{E}$ be an elliptic curve defined over a number field $K$. Let $m$ be a positive integer. We denote by ${\\mathcal{E}}[m]$ the $m$-torsion subgroup of $\\mathcal{E}$ and by $K_m:=K({\\mathcal{E}}[m])$ the number field obtained by adding to $K$ the coordinates of the points of ${\\mathcal{E}}[m]$. We describe the fields $K_5$, when $\\mathcal{E}$ is a CM elliptic curve defined over $K$, with Weiestrass form either $y^2=x^3+bx$ or $y^2=x^3+c$. In particular we classify the fields $K_5$ in terms of generators, degrees and Galois groups. Furthermore we show some applications of those results to the Local-Global Divisibility Problem, to modular curves and to Shimura curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-31T19:02:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cm elliptic curve",
      "number field",
      "local-global divisibility problem",
      "shimura curves",
      "torsion subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}