{
  "id": "math/0402016",
  "version": "v1",
  "published": "2004-02-02T14:22:06.000Z",
  "updated": "2004-02-02T14:22:06.000Z",
  "title": "Common Divisors of Elliptic Divisibility Sequences over Function Fields",
  "authors": [
    "Joseph H. Silverman"
  ],
  "journal": "Manuscripta Math. 114 (2004), no. 4, 431--446. (MR2081943)",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R in E(k(T)), write x_R=A_R/D_R^2 with relatively prime polynomials A_R(T) and D_R(T) in k[T]. The sequence {D_{nR}) for n \\ge 1 is called the ``elliptic divisibility sequence of R.'' Let P,Q in E(k(T)) be independent points. We conjecture that deg (gcd(D_{nP},D_{mQ})) is bounded for m,n \\ge 1, and that gcd(D_{nP},D_{nQ}) = gcd(D_{P},D_{Q}) for infinitely many n \\ge 1. We prove these conjectures in the case that j(E) is in k. More generally, we prove analogous statements with k(T) replaced by the function field of any curve and with P and Q allowed to lie on different elliptic curves. If instead k is a finite field of characteristic p, and again assuming that j(E) is in k, we show that deg (gcd(D_{nP},D_{nQ})) > n + O(sqrt{n}) for infinitely many n satisfying gcd(n,p) = 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-02-02T14:22:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61",
      "11G35"
    ],
    "keywords": [
      "elliptic divisibility sequence",
      "common divisors",
      "elliptic curve",
      "rational function field",
      "finite field"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......2016S"
    }
  }
}