{
  "id": "math/0011102",
  "version": "v6",
  "published": "2000-11-15T19:19:30.000Z",
  "updated": "2003-05-02T16:25:53.000Z",
  "title": "Analogues of Lehmer's conjecture in positive characteristic",
  "authors": [
    "Amilcar Pacheco"
  ],
  "comment": "revised version",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $C$ be a smooth projective irreducible curve defined over a finite field $\\mathbb{F}_q$ and $K=\\mathbb{F}_q(C)$. Let $A\\subset K$ be the ring of functions regular outside a fixed place $\\infty$ of $K$. Let $\\phi:A\\to\\text{End}(\\mathbb{G}_a)$ be a Drinfeld $A$-module of rank $r$ defined over a finite extension $L$ of $K$ and $\\hat{h}_{\\phi}$ its canonical height. Given a non-torsion point $\\alpha$ of $\\phi$ of degree $d$ over $K$, we prove that $\\hat{h}_{\\phi}(\\alpha)\\ge 1/d$. A similar statement is proved for the canonical height of a point of infinite order of a non-constant semi-stable elliptic curve defined over $K$, with the absolute constant 1 replaced by a constant depending on the elliptic curve.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2003-05-02T16:25:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lehmers conjecture",
      "positive characteristic",
      "semi-stable elliptic curve",
      "projective irreducible curve",
      "functions regular outside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math.....11102P"
    }
  }
}