{
  "id": "1106.3099",
  "version": "v1",
  "published": "2011-06-15T20:50:27.000Z",
  "updated": "2011-06-15T20:50:27.000Z",
  "title": "Experimental Data for Goldfeld's Conjecture over Function Fields",
  "authors": [
    "Salman Baig",
    "Chris Hall"
  ],
  "comment": "26 pages, 8 figures, 11 tables, submitted to Experimental Mathematics",
  "categories": [
    "math.NT"
  ],
  "abstract": "This paper presents empirical evidence supporting Goldfeld's conjecture on the average analytic rank of a family of quadratic twists of a fixed elliptic curve in the function field setting. In particular, we consider representatives of the four classes of non-isogenous elliptic curves over F_q(t) with (q,6)=1 possessing two places of multiplicative reduction and one place of additive reduction. The case of q=5 provides the largest data set as well as the most convincing evidence that the average analytic rank converges to 1/2, which we also show is a lower bound following an argument of Kowalski. The data was generated via explicit computation of the L-function of these elliptic curves, and we present the key results necessary to implement an algorithm to efficiently compute the L-function of non-isotrivial elliptic curves over F_q(t) by realizing such a curve as a quadratic twist of a pullback of a `versal' elliptic curve. We also provide a reference for our open-source library ELLFF, which provides all the necessary functionality to compute such L-functions, and additional data on analytic rank distributions as they pertain to the density conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-06-15T20:50:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G40",
      "11Y35"
    ],
    "keywords": [
      "function field",
      "elliptic curve",
      "experimental data",
      "evidence supporting goldfelds conjecture",
      "quadratic twist"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.3099B"
    }
  }
}