{
  "id": "math/0408153",
  "version": "v4",
  "published": "2004-08-11T16:06:42.000Z",
  "updated": "2005-06-07T20:36:10.000Z",
  "title": "Root numbers and ranks in positive characteristic",
  "authors": [
    "B. Conrad",
    "K. Conrad",
    "H. Helfgott"
  ],
  "comment": "40 pages; last version; to appear in Adv. Math",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "For a global field K and an elliptic curve E_eta over K(T), Silverman's specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all but finitely many t in P^1(K). If this inequality is strict for all but finitely many t, the elliptic curve E_eta is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K = kappa(u) over any finite field kappa with odd characteristic, we construct an explicit 2-parameter family E_{c,d} of non-isotrivial elliptic curves over K(T) (depending on arbitrary c, d in kappa^*) such that, under the parity conjecture, each E_{c,d} has elevated rank.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2005-06-07T20:36:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G40"
    ],
    "keywords": [
      "positive characteristic",
      "root numbers",
      "elevated rank",
      "non-isotrivial elliptic curve",
      "parity conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......8153C"
    }
  }
}