{
  "id": "math/0303369",
  "version": "v1",
  "published": "2003-03-28T21:37:01.000Z",
  "updated": "2003-03-28T21:37:01.000Z",
  "title": "Moments of the rank of elliptic curves",
  "authors": [
    "Siman Wong"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Fix an elliptic curve $E/\\Q$, and assume the generalized Riemann hypothesis for the $L$-function $ L(E_D, s) $ for every quadratic twist $E_D$ of $E$ by $D\\in\\Z$. We combine Weil's explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of $E_D$. It follows from this that, for any unbounded increasing function $f$ on $\\R$, the analytic rank and (assuming in addition the Birch-Swinnerton-Dyer conjecture) the number of integral points of $E_D$ are less than $f(D)$ for almost all $D$. We also derive an upper bound for the density of low-lying zeros of $L(E_D, s)$ which is compatible with the random matrix models of Katz and Sarnak.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-03-28T21:37:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11M41",
      "11G40"
    ],
    "keywords": [
      "elliptic curve",
      "analytic rank",
      "random matrix models",
      "weils explicit formula",
      "asymptotic upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......3369W"
    }
  }
}