{
  "id": "math/0608593",
  "version": "v1",
  "published": "2006-08-24T00:23:34.000Z",
  "updated": "2006-08-24T00:23:34.000Z",
  "title": "Points of Low Height on Elliptic Curves and Surfaces, I: Elliptic surfaces over P^1 with small d",
  "authors": [
    "Noam D. Elkies"
  ],
  "comment": "15 pages; some lines in the TeX source are commented out with \"%\" to meet the 15-page limit for ANTS proceedings",
  "journal": "Lecture Notes in Computer Science 4076 (proceedings of ANTS-7, 2004; F.Hess, S.Pauli, and M.Pohst, ed.), 287--301",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "For each of n=1,2,3 we find the minimal height h^(P) of a nontorsion point P of an elliptic curve E over C(T) of discriminant degree d=12n (equivalently, of arithmetic genus n), and exhibit all (E,P) attaining this minimum. The minimal h^(P) was known to equal 1/30 for n=1 (Oguiso-Shioda) and 11/420 for n=2 (Nishiyama), but the formulas for the general (E,P) were not known, nor was the fact that these are also the minima for an elliptic curve of discriminant degree 12n over a function field of any genus. For n=3 both the minimal height (23/840) and the explicit curves are new. These (E,P) also have the property that that mP is an integral point (a point of naive height zero) for each m=1,2,...,M, where M=6,8,9 for n=1,2,3; this, too, is maximal in each of the three cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-24T00:23:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G50",
      "11G05"
    ],
    "keywords": [
      "elliptic curve",
      "elliptic surfaces",
      "low height",
      "minimal height",
      "discriminant degree 12n"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8593E"
    }
  }
}