{
  "id": "math/0406330",
  "version": "v3",
  "published": "2004-06-16T18:52:36.000Z",
  "updated": "2005-04-06T18:26:28.000Z",
  "title": "Low-lying zeros of families of elliptic curves",
  "authors": [
    "Matthew P. Young"
  ],
  "comment": "v2: Enhanced exposition, 56 pages. v3: One reference added and one sentence changed in the paragraph following Corollary 3.4",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the low-lying zeros of various interesting families of elliptic curve L-functions. One application is an upper bound on the average analytic rank of the family of all elliptic curves. The upper bound obtained is less than two, which implies that a positive proportion of elliptic curves over the rationals have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. These results are conditional on the Generalized Riemann Hypothesis.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-04-06T18:26:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26"
    ],
    "keywords": [
      "low-lying zeros",
      "upper bound",
      "elliptic curve l-functions",
      "average analytic rank",
      "algebraic rank equal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6330Y"
    }
  }
}