{
  "id": "1310.1772",
  "version": "v1",
  "published": "2013-10-07T13:17:34.000Z",
  "updated": "2013-10-07T13:17:34.000Z",
  "title": "Rational points on some Fermat curves and surfaces over finite fields",
  "authors": [
    "Jose Felipe Voloch",
    "Michael E. Zieve"
  ],
  "comment": "8 pages",
  "doi": "10.1142/S1793042113500954",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We give an explicit description of the F_{q^i}-rational points on the Fermat curve u^{q-1}+v^{q-1}+w^{q-1}=0 for each i=1,2,3. As a consequence, we observe that for any such point (u,v,w), the product uvw is a cube in F_{q^i}. We also describe the F_{q^2}-rational points on the Fermat surface u^{q-1}+v^{q-1}+w^{q-1}+x^{q-1}=0, and show that the product of the coordinates of any such points is a square.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-07T13:17:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "14G15"
    ],
    "keywords": [
      "fermat curve",
      "finite fields",
      "rational points",
      "explicit description",
      "product uvw"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.1772V"
    }
  }
}