{
  "id": "0801.1054",
  "version": "v2",
  "published": "2008-01-07T16:39:29.000Z",
  "updated": "2020-01-14T09:47:36.000Z",
  "title": "Conjectural estimates on the Mordell-Weil and Tate-Shafarevich groups of an abelian variety",
  "authors": [
    "Andrea Surroca Ortiz"
  ],
  "comment": "38 pages. Submitted version. This version improves substantially, in many ways, the unpublished previous version arXiv:0801.1054. In particular, Masser's lower bound for non-torsion points is replaced by Bosser-Gaudron's lower bound. The analytic estimates on the L-function, as well as the part relying the Tamagawa numbers and the Faltings' height were also substantially modified",
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the N\\'eron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators of the free part of the Mordell-Weil group. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-01-07T16:39:29.000Z",
      "title": "On some conjectures on the Mordell-Weil and the Tate-Shafarevich groups of an abelian variety",
      "abstract": "We consider an abelian variety defined over a number field. We give conditonal bounds for the order of its Tate-Shafarevich group, as well as bounds for the N\\'eron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals.",
      "comment": "22 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2020-01-14T09:47:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "11G40",
      "14G05",
      "11G50"
    ],
    "keywords": [
      "tate-shafarevich group",
      "abelian variety",
      "non-torsion rational points",
      "conditonal bounds",
      "elliptic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0801.1054S"
    }
  }
}