{
  "id": "math/0111105",
  "version": "v3",
  "published": "2001-11-09T00:49:36.000Z",
  "updated": "2002-06-09T15:36:23.000Z",
  "title": "Distribution of the traces of Frobenius on elliptic curves over function fields",
  "authors": [
    "Amilcar Pacheco"
  ],
  "comment": "11 pages, replaced version, minor correction on the degree of the j-map",
  "journal": "Acta Arithmetica, 106.3 (2003), 255-263",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let C be a smooth irreducible projective curve defined over a finite field $\\mathbb{F}_{q}$ of q elements of characteristic p>3 and $K=\\mathbb{F}_{q}(C)$ its function field and $\\phi_{\\mathcal{E}}:\\mathcal{E}\\to C$ the minimal regular model of $\\mathbf{E}/K$. For each $P\\in C$ denote $\\mathcal{E}_P=\\phi^{-1}_{\\mathcal{E}}(P)$. The elliptic curve $E/K$ has good reduction at $P\\in C$ if and only if $\\mathcal{E}_P$ is an elliptic curve defined over the residue field $\\kappa_P$ of $P$. This field is a finite extension of $\\mathbb{F}_q$ of degree $\\deg(P)$. Let $t(\\mathcal{E}_P)=q^{\\deg(P)}+1-#\\mathcal{E}_P(\\kappa_P)$ be the trace of Frobenius at P. By Hasse-Weil's theorem (cf. [10, Chapter V, Theorem 2.4]), $t(\\mathcal{E}_P)$ is the sum of the inverses of the zeros of the zeta function of $\\mathcal{E}_P$. In particular, $|t(\\mathcal{E}_P)|\\le 2q^{\\deg(P)}$. Let $C_0\\subset C$ be the set of points of C at which $E/K$ has good reduction and $C_0(\\mathbb{F}_{q^k})$ the subset of $\\mathbb{F}_{q^k}$-rational points of $C_0$. We discuss the following question. Let $k\\ge 1$ and t be integers and suppose $|t|\\le 2q^{k/2}$. Let $\\pi(k,t)=#\\{P\\in C_0(\\mathbb{F}_{q^k}) | t(\\mathcal{E}_P)=t\\}$. How big is $\\pi(k,t)$?",
  "revisions": [
    {
      "version": "v3",
      "updated": "2002-06-09T15:36:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05"
    ],
    "keywords": [
      "elliptic curve",
      "function field",
      "irreducible projective curve",
      "distribution",
      "minimal regular model"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....11105P"
    }
  }
}