{
  "id": "0811.3109",
  "version": "v3",
  "published": "2008-11-19T20:30:17.000Z",
  "updated": "2008-12-10T21:22:26.000Z",
  "title": "Specializations of elliptic surfaces, and divisibility in the Mordell-Weil group",
  "authors": [
    "Patrick Ingram"
  ],
  "comment": "Introduction re-written, and minor additions. The results of the paper are largely unchanged, with one small observation added",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic) degree at most $D$ over $k$, such that $\\ell^n Q=P_t$, for some $n\\geq 1$. The bound obtained depends only on $\\ell$, the surface and section in question, $D$, and the degree $[k(t):k]$; that is, it is uniform across all fibres of bounded degree. In special cases, we obtain more specific, in some instances sharp, bounds.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-12-10T21:22:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G05",
      "14J27"
    ],
    "keywords": [
      "elliptic surface",
      "mordell-weil group",
      "specializations",
      "divisibility",
      "number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.3109I"
    }
  }
}