{
  "id": "0912.1685",
  "version": "v1",
  "published": "2009-12-09T08:19:20.000Z",
  "updated": "2009-12-09T08:19:20.000Z",
  "title": "Zeta function factorisation, Dwork hypersurfaces, hypergeometric hypersurfaces",
  "authors": [
    "Philippe Goutet"
  ],
  "comment": "22 pages, submitted for publication",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathbb{F}_q$ be a finite field with $q$ elements, $\\psi$ a non-zero element of $\\mathbb{F}_q$, and $n$ an integer $\\geq 3$ prime to $q$. The aim of this article is to show that the zeta function of the projective variety over $\\mathbb{F}_q$ defined by $X_\\psi \\colon x_1^n+...+x_n^n - n \\psi x_1... x_n=0$ has, when $n$ is prime and $X_\\psi$ is non singular (i.e. when $\\psi^n \\neq 1$), an explicit decomposition in factors coming from affine varieties of odd dimension $\\leq n-4$ which are of hypergeometric type. The method we use consists in counting separately the number of points of $X_\\psi$ and of some varieties of the preceding type and then compare them. This article answers, at least when $n$ is prime, a question asked by D. Wan in his article \"Mirror Symmetry for Zeta Functions\".",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-09T08:19:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G10",
      "11G25",
      "14G15"
    ],
    "keywords": [
      "zeta function factorisation",
      "hypergeometric hypersurfaces",
      "dwork hypersurfaces",
      "non-zero element",
      "mirror symmetry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.1685G"
    }
  }
}