{
  "id": "0709.0132",
  "version": "v1",
  "published": "2007-09-02T22:01:37.000Z",
  "updated": "2007-09-02T22:01:37.000Z",
  "title": "On the index of the Heegner subgroup of elliptic curves",
  "authors": [
    "Carlos Castano-Bernard"
  ],
  "comment": "13 pages, 2 tables and 1 figure",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let E be an elliptic curve of conductor N and rank one over Q. So there is a non-constant morphism X+0(N) --> E defined over Q, where X+0(N) = X0(N)/wN and wN is the Fricke involution of the modular curve X+0(N). Under this morphism the traces of the Heegner points of X+0(N) map to rational points on E. In this paper we study the index I of the subgroup generated by all these traces on E(Q). We propose and also discuss a conjecture that says that if N is prime and I > 1, then either the number of connected components of the real locus X+0(N)(R) is greater than 1 or (less likely) the order S of the Tate-Safarevich group is non-trivial. This conjecture is backed by computations performed on each E that satisfies the above hypothesis in the range N < 129999. This paper was prepared for the proceedings of the Conference on Algorithmic Number Theory, Turku, May 8-11, 2007. We tried to make the paper as self contained as possible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-02T22:01:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic curve",
      "heegner subgroup",
      "algorithmic number theory",
      "non-constant morphism",
      "tate-safarevich group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.0132C"
    }
  }
}