{
  "id": "math/0509600",
  "version": "v2",
  "published": "2005-09-26T14:35:45.000Z",
  "updated": "2006-07-18T15:28:56.000Z",
  "title": "Ordinary elliptic curves of high rank over $\\bar F_p(x)$ with constant j-invariant II",
  "authors": [
    "Claus Diem",
    "Jasper Scholten"
  ],
  "comment": "14 pages, new version",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We show that for all odd primes $p$, there exist ordinary elliptic curves over $\\bar{\\mathbb{F}}_p(x)$ with arbitrarily high rank and constant $j$-invariant. This shows in particular that there are elliptic curves with arbitrarily high rank over these fields for which the corresponding elliptic surface is not supersingular. The result follows from a theorem which states that for all odd prime numbers $p$ and $\\ell$, there exists a hyperelliptic curve over $\\bar{\\mathbb{F}}_p$ of genus $(\\ell-1)/2$ whose Jacobian is isogenous to the power of one ordinary elliptic curve.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-07-18T15:28:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G20",
      "14H40",
      "14H52"
    ],
    "keywords": [
      "ordinary elliptic curve",
      "constant j-invariant",
      "arbitrarily high rank",
      "odd prime numbers",
      "hyperelliptic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9600D"
    }
  }
}