{
  "id": "math/0501181",
  "version": "v2",
  "published": "2005-01-12T12:13:48.000Z",
  "updated": "2006-03-30T09:44:42.000Z",
  "title": "Chow groups and higher congruences for the number of rational points on proper varieties over finite fields",
  "authors": [
    "Najmuddin Fakhruddin"
  ],
  "comment": "Added a section containing applications to low degree intersections in homogenous spaces",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Given a proper family of varieties over a smooth base, with smooth total space and general fibre, all over a finite field k with q elements, we show that a finiteness hypothesis on the Chow groups, CH_i, i=0,1,...,r, of the fibres in the family leads to congruences mod q^{r+1} for the number of rational points in all the fibres over k-rational points of the base. These hypotheses on the Chow groups are expected to hold for families of low degree intersections in many Fano varieties leading to a broad generalisation of the theorem of Ax--Katz, as well as results of the author and C. S. Rajan. As an unconditional application, we give an asymptotic generalisation of the Ax--Katz theorem to low degree intersections in a large class of homogenous spaces.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-03-30T09:44:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chow groups",
      "finite field",
      "rational points",
      "proper varieties",
      "higher congruences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......1181F"
    }
  }
}