{
  "id": "1904.12393",
  "version": "v1",
  "published": "2019-04-28T22:34:33.000Z",
  "updated": "2019-04-28T22:34:33.000Z",
  "title": "Primitive divisors of elliptic divisibility sequences over function fields with constant j-invariant",
  "authors": [
    "Bartosz Naskręcki",
    "Marco Streng"
  ],
  "comment": "28 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the sequence $D_1$, $D_2, \\ldots$ of denominators of multiples $P$, $2P,\\ldots$ of $P$ is a strong divisibility sequence in the sense that $\\gcd(D_m, D_n) = D_{\\gcd(m,n)}$. This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences. A number $N$ is called a Zsigmondy bound of the sequence if each term $D_{n}$ with $n>N$ presents a new prime factor. The optimal uniform Zsigmondy bound for the genus-zero sequences over $\\mathbf{Q}$ is $30$ by Bilu-Hanrot-Voutier, 2000, but finding such a bound remains an open problem in genus one, both over $\\mathbf{Q}$ and over function fields. We prove that the optimal Zsigmondy bound for ordinary elliptic divisibility sequences over function fields is $2$ if the $j$-invariant is constant. In the supersingular case, we give a complete classification of which terms can and cannot have a new prime factor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-28T22:34:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11B39",
      "14H52",
      "11G07",
      "11B83"
    ],
    "keywords": [
      "function fields",
      "constant j-invariant",
      "primitive divisors",
      "optimal zsigmondy bound",
      "prime factor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}