{
  "id": "math/0211315",
  "version": "v1",
  "published": "2002-11-20T11:23:51.000Z",
  "updated": "2002-11-20T11:23:51.000Z",
  "title": "On the distribution of the of Frobenius elements on elliptic curves over function fields",
  "authors": [
    "Amilcar Pacheco"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $C$ be a smooth projective curve over $\\mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $\\phi:\\mathcal{E}\\to C$ its minimal regular model. For each $P\\in C$ such that $E$ has good reduction at $P$, i.e., the fiber $\\mathcal{E}_P=\\phi^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $\\mathcal{E}_P$ over the residue field $\\kappa_P$ of $P$ are of the form $q_P^{1/2}e^{i\\theta_P},q_{P}e^{-i\\theta_P}$, where $q_P=q^{\\deg(P)}$ and $0\\le\\theta_P\\le\\pi$. The goal of this note is to determine given an integer $B\\ge 1$, $\\alpha,\\beta\\in[0,\\pi]$ the number of $P\\in C$ where the reduction of $E$ is good and such that $\\deg(P)\\le B$ and $\\alpha\\le\\theta_P\\le\\beta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-11-20T11:23:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "function field",
      "frobenius elements",
      "distribution",
      "nonconstant elliptic curve",
      "minimal regular model"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.4064/aa106-3-4",
      "journal": "Acta Arithmetica",
      "year": 2003,
      "volume": 106,
      "number": 3,
      "pages": 255
    },
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003AcAri.106..255P"
    }
  }
}