{
  "id": "1811.10149",
  "version": "v1",
  "published": "2018-11-26T02:39:12.000Z",
  "updated": "2018-11-26T02:39:12.000Z",
  "title": "The average number of subgroups of elliptic curves over finite fields",
  "authors": [
    "Corentin Perret-Gentil"
  ],
  "comment": "40 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof \\`a la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most 2. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivi\\'c--Zhai and Blomer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-26T02:39:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "average number",
      "elliptic curve",
      "finite abelian groups",
      "cyclic subgroups",
      "arithmetic progressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}