{
  "id": "0710.0574",
  "version": "v1",
  "published": "2007-10-02T16:35:09.000Z",
  "updated": "2007-10-02T16:35:09.000Z",
  "title": "Combinatorial Aspects of Elliptic Curves II: Relationship between Elliptic Curves and Chip-Firing Games on Graphs",
  "authors": [
    "Gregg Musiker"
  ],
  "comment": "24 pages, 2 figures, part of author's Ph.D. Thesis, presented at FPSAC 2007",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let q be a power of a prime and E be an elliptic curve defined over F_q. In \"Combinatorial aspects of elliptic curves\" [17], the present author examined a sequence of polynomials which express the N_k's, the number of points on E over the field extensions F_{q^k}, in terms of the parameters q and N_1 = #E(F_q). These polynomials have integral coefficients which alternate in sign, and a combinatorial interpretation in terms of spanning trees of wheel graphs. In this sequel, we explore further ramifications of this connection. In particular, we highlight a relationship between elliptic curves and chip-firing games on graphs by comparing the groups structures of both. As a coda, we construct a cyclic rational language whose zeta function is dual to that of an elliptic curve.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-10-02T16:35:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G07",
      "05C25"
    ],
    "keywords": [
      "combinatorial aspects",
      "chip-firing games",
      "relationship",
      "cyclic rational language",
      "integral coefficients"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.0574M"
    }
  }
}