{
  "id": "0808.2169",
  "version": "v1",
  "published": "2008-08-15T17:02:18.000Z",
  "updated": "2008-08-15T17:02:18.000Z",
  "title": "Étale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields",
  "authors": [
    "Sudhir R. Ghorpade",
    "Gilles Lachaud"
  ],
  "comment": "42 pages; corrected, revised and updated version of a paper published earlier",
  "journal": "Mosc. Math. J. 2 (2002), 589--631 and Mosc. Math. J. 9 (2009), 431-438.",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang-Weil inequality. Moreover, we prove the Lang-Weil inequality for affine as well as projective varieties with an explicit description and a bound for the constant appearing therein. We also prove a conjecture of Lang and Weil concerning the Picard varieties and \\'etale cohomology spaces of projective varieties. The general inequality for complete intersections may be viewed as a more precise version of the estimates given by Hooley and Katz. The proof is primarily based on a suitable generalization of the Weak Lefschetz Theorem to singular varieties together with some Bertini-type arguments and the Grothendieck-Lefschetz Trace Formula. We also describe some auxiliary results concerning the \\'etale cohomology spaces and Betti numbers of projective varieties over finite fields and a conjecture along with some partial results concerning the number of points of projective algebraic sets over finite fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-08-15T17:02:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G25",
      "14F20",
      "14G15",
      "14M10"
    ],
    "keywords": [
      "finite field",
      "singular varieties",
      "etale cohomology spaces",
      "projective varieties",
      "lang-weil inequality"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0808.2169G"
    }
  }
}