{
  "id": "math/0609144",
  "version": "v4",
  "published": "2006-09-05T14:49:52.000Z",
  "updated": "2007-11-26T04:48:52.000Z",
  "title": "Sato--Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height",
  "authors": [
    "William D. Banks",
    "Igor E. Shparlinski"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We obtain asymptotic formulae for the number of primes $p\\le x$ for which the reduction modulo $p$ of the elliptic curve $$ \\E_{a,b} : Y^2 = X^3 + aX + b $$ satisfies certain ``natural'' properties, on average over integers $a$ and $b$ with $|a|\\le A$ and $|b| \\le B$, where $A$ and $B$ are small relative to $x$. Specifically, we investigate behavior with respect to the Sato--Tate conjecture, cyclicity, and divisibility of the number of points by a fixed integer $m$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2007-11-26T04:48:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11L40",
      "14H52"
    ],
    "keywords": [
      "elliptic curve",
      "divisibility statistics",
      "small height",
      "sato-tate conjecture",
      "asymptotic formulae"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9144B"
    }
  }
}