{
  "id": "2106.04579",
  "version": "v1",
  "published": "2021-06-08T15:09:40.000Z",
  "updated": "2021-06-08T15:09:40.000Z",
  "title": "On 7 division fields of CM elliptic curves",
  "authors": [
    "Jessica Alessandrì",
    "Laura Paladino"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1808.00029",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathcal{E}$ be a CM elliptic curve defined over a number field $K$, with Weiestrass form $y^3=x^3+bx$ or $y^2=x^3+c$. For every positive integer $m$, we denote by ${\\mathcal{E}}[m]$ the $m$-torsion subgroup of ${\\mathcal{E}}$ and by $K_m:=K({\\mathcal{E}}[m])$ the $m$-th division field, i.e. the extension of $K$ obtained by adding to it the coordinates of the points in ${\\mathcal{E}}[m]$. We classify all the fields $K_7$ in terms of generators, degrees and Galois groups. We also show some applications to the Local-Global Divisibility Problem and to modular curves and Shimura curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-08T15:09:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cm elliptic curve",
      "local-global divisibility problem",
      "th division field",
      "weiestrass form",
      "shimura curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}