{
  "id": "0908.3823",
  "version": "v2",
  "published": "2009-08-26T15:52:49.000Z",
  "updated": "2009-10-22T18:04:24.000Z",
  "title": "Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero",
  "authors": [
    "Amod Agashe"
  ],
  "comment": "The first version has a mistake (the result in the appendix may be incorrect). Version 2 and later correct this mistake",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $E$ be an optimal elliptic curve over $\\Q$ of conductor $N$ having analytic rank zero, i.e., such that the $L$-function $L_E(s)$ of $E$ does not vanish at $s=1$. Suppose there is another optimal elliptic curve over $\\Q$ of the same conductor $N$ whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to $E$ modulo an integer $r$. The theory of visibility then shows that under certain additional hypotheses, $r$ divides the product of the order of the Shafarevich-Tate group of $E$ and the orders of the arithmetic component groups of $E$. We extract an explicit integer factor from the the Birch and Swinnerton-Dyer conjectural formula for the product mentioned above, and under some hypotheses similar to the ones made in the situation above, we show that $r$ divides this integer factor. This provides theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture in the analytic rank zero case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-10-22T18:04:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G40",
      "14G10"
    ],
    "keywords": [
      "swinnerton-dyer conjecture",
      "optimal elliptic curve",
      "visibility",
      "analytic rank zero case",
      "swinnerton-dyer conjectural formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.3823A"
    }
  }
}