{
  "id": "math/0604107",
  "version": "v2",
  "published": "2006-04-05T14:02:20.000Z",
  "updated": "2006-04-06T01:40:40.000Z",
  "title": "Heegner points and the rank of elliptic curves over large extensions of global fields",
  "authors": [
    "Florian Breuer",
    "Bo-Hae Im"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let k be a global field, $\\bar{k}$ a separable closure of k, and $G_k$ the absolute Galois group $\\Gal(\\bar{k}/k)$ of $\\bar{k}$ over k. For every g in $G_k$, let $\\bar{k}^g$ be the fixed subfield of $\\bar{k}$ under g. Let E/k be an elliptic curve over k. We show that for each g in $G_k$, the Mordell-Weil group $E(\\bar{k}^g)$ has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametrized by a Drinfeld modular curve, and secondly when k is a totally real number field and E/k is parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points on E defined over ring class fields.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-04-06T01:40:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05"
    ],
    "keywords": [
      "heegner points",
      "global field",
      "elliptic curve",
      "large extensions",
      "totally real number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......4107B"
    }
  }
}