{
  "id": "1109.5859",
  "version": "v2",
  "published": "2011-09-27T12:35:21.000Z",
  "updated": "2013-01-29T15:55:44.000Z",
  "title": "Small Height and Infinite Non-Abelian Extensions",
  "authors": [
    "Philipp Habegger"
  ],
  "comment": "Added new corollary on the structure of the group $E(F)$ and corrected some typos in version 2",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E$ be an elliptic curve defined over the rationals without complex multiplication. The field $F$ generated by all torsion points of $E$ is an infinite, non-abelian Galois extension of the rationals which has unbounded, wild ramification above all primes. We prove that the absolute logarithmic Weil height of an element of $F$ is either zero or bounded from below by a positive constant depending only on $E$. We also show that the N\\'eron-Tate height has a similar gap on $E(F)$ and use this to determine the structure of the group $E(F)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-29T15:55:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G50",
      "11G05",
      "14H52",
      "14G40"
    ],
    "keywords": [
      "infinite non-abelian extensions",
      "small height",
      "absolute logarithmic weil height",
      "non-abelian galois extension",
      "torsion points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.5859H"
    }
  }
}