{
  "id": "1403.7108",
  "version": "v1",
  "published": "2014-03-27T16:06:47.000Z",
  "updated": "2014-03-27T16:06:47.000Z",
  "title": "A conditional determination of the average rank of elliptic curves",
  "authors": [
    "Daniel Fiorilli"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Under a hypothesis which is slightly stronger than the Riemann Hypothesis for elliptic curve $L$-functions, we show that both the average analytic rank and the average algebraic rank of elliptic curves in families of quadratic twists are exactly $\\frac 12$. As a corollary we obtain that under this last hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all curves in our family, and that asymptotically one half of these curves have algebraic rank $0$, and the remaining half $1$. We also prove an analogous result in the family of all elliptic curves. A way to interpret our results is to say that nonreal zeros of elliptic curve $L$-functions in a family have a direct influence on the average rank in this family. Results of Katz-Sarnak and of Young constitute a major ingredient in the proofs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-27T16:06:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G40",
      "11M41"
    ],
    "keywords": [
      "elliptic curve",
      "average rank",
      "conditional determination",
      "swinnerton-dyer conjecture holds",
      "average analytic rank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.7108F"
    }
  }
}