{
  "id": "0806.0044",
  "version": "v2",
  "published": "2008-05-31T00:12:33.000Z",
  "updated": "2008-06-09T19:39:45.000Z",
  "title": "The Riemann Hypothesis for Function Fields over a Finite Field",
  "authors": [
    "Machiel van Frankenhuijsen"
  ],
  "comment": "30 pages, 2 figures all \\o's are now \\mathcal{O}",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We discuss Enrico Bombieri's proof of the Riemann hypothesis for curves over a finite field. Reformulated, it states that the number of points on a curve $\\C$ defined over the finite field $\\F_q$ is of the order $q+O(\\sqrt{q})$. The first proof was given by Andr\\'e Weil in 1942. This proof uses the intersection of divisors on $\\C\\times\\C$, making the application to the original Riemann hypothesis so far unsuccessful, because $\\spec\\Z\\times\\spec\\Z=\\spec\\Z$ is one-dimensional. A new method of proof was found in 1969 by S. A. Stepanov. This method was greatly simplified and generalized by Bombieri in 1973. Bombieri's method uses functions on $\\C\\times\\C$, again precluding a direct translation to a proof of the original Riemann hypothesis. However, the two coordinates on $\\C\\times\\C$ have different roles, one coordinate playing the geometric role of the variable of a polynomial, and the other coordinate the arithmetic role of the coefficients of this polynomial. The Frobenius automorphism of $\\C$ acts on the geometric coordinate of $\\C\\times\\C$. In the last section, we make some suggestions how Nevanlinna theory could provide a model of $\\spec\\Z\\times\\spec\\Z$ that is two-dimensional and carries an action of Frobenius on the geometric coordinate.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-06-09T19:39:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "11R58",
      "14G15",
      "30D35"
    ],
    "keywords": [
      "finite field",
      "function fields",
      "original riemann hypothesis",
      "geometric coordinate",
      "enrico bombieris proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.0044V"
    }
  }
}