{
  "id": "1010.1754",
  "version": "v1",
  "published": "2010-10-08T18:19:47.000Z",
  "updated": "2010-10-08T18:19:47.000Z",
  "title": "Functional equations for zeta functions of $\\mathbb{F}_1$-schemes",
  "authors": [
    "Oliver Lorscheid"
  ],
  "comment": "4 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "For a scheme $X$ whose $\\mathbb F_q$-rational points are counted by a polynomial $N(q)=\\sum a_iq^i$, the $\\mathbb{F}_1$-zeta function is defined as $\\zeta(s)=\\prod(s-i)^{-a_i}$. Define $\\chi=N(1)$. In this paper we show that if $X$ is a smooth projective scheme, then its $\\mathbb{F}_1$-zeta function satisfies the functional equation $\\zeta(n-s) = (-1)^\\chi \\zeta(s)$. We further show that the $\\mathbb{F}_1$-zeta function $\\zeta(s)$ of a split reductive group scheme $G$ of rank $r$ with $N$ positive roots satisfies the functional equation $\\zeta(r+N-s) = (-1)^\\chi ( \\zeta(s) )^{(-1)^r}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-08T18:19:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G25",
      "14G10",
      "14G15",
      "20G15"
    ],
    "keywords": [
      "functional equation",
      "split reductive group scheme",
      "zeta function satisfies",
      "rational points",
      "smooth projective scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.1754L"
    }
  }
}