{
  "id": "1511.02388",
  "version": "v1",
  "published": "2015-11-07T19:07:27.000Z",
  "updated": "2015-11-07T19:07:27.000Z",
  "title": "Orders of reductions of elliptic curves with many and few prime factors",
  "authors": [
    "Lee Troupe"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we investigate extreme values of $\\omega(E(\\mathbb{F}_p))$, where $E/\\mathbb{Q}$ is an elliptic curve with complex multiplication and $\\omega$ is the number-of-distinct-prime-divisors function. For fixed $\\gamma > 1$, we prove that \\[ \\#\\{p \\leq x : \\omega(E(\\mathbb{F}_p)) > \\gamma\\log\\log x\\} = \\frac{x}{(\\log x)^{2 + \\gamma\\log\\gamma - \\gamma + o(1)}}. \\] The same result holds for the quantity $\\#\\{p \\leq x : \\omega(E(\\mathbb{F}_p)) < \\gamma\\log\\log x\\}$ when $0 < \\gamma < 1$. The argument is worked out in detail for the curve $E : y^2 = x^3 - x$, and we discuss how the method can be adapted for other CM elliptic curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-07T19:07:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11N36",
      "11G05"
    ],
    "keywords": [
      "prime factors",
      "reductions",
      "cm elliptic curves",
      "complex multiplication",
      "number-of-distinct-prime-divisors function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151102388T"
    }
  }
}