{
  "id": "math/0412368",
  "version": "v5",
  "published": "2004-12-18T19:18:03.000Z",
  "updated": "2005-04-01T07:47:42.000Z",
  "title": "The A-module Structure Induced by a Drinfeld A-module over a Finite Field",
  "authors": [
    "Mohamed Ahmed Mohamed saadbouh"
  ],
  "comment": "23",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $\\Phi $ be a Drinfeld $\\mathbf{F}_{q}[T]$-module of rank 2, over a finite field $L$, a finite extension of $n$ degrees for a finite field with $q$ elements $% \\mathbf{F}_{q}$. Let $P_{\\Phi}(X)=$ $X^{2}-cX+\\mu P^{m}$ ($c$ an element of $% \\mathbf{F}_{q}[T]$ and $\\mu $ a no null element of $\\mathbf{F}_{q}$, $m$ the degree of the extension $L$ over the field $\\mathbf{F}_{q}[T]/P$, $P$ is a $\\mathbf{F}%_{q}[T]$-characteristic of $L$ and $d$ the degree of the polynomial $P$) the characteristic polynomial, of the Frobenius $F$ of $L$. We will interested to the structure of finite $\\mathbf{F}_{q}[T]$-module $L^{\\Phi}$ deduct by $\\Phi $ over $L$ and will proof our main result, the analogue of Deuring theorem for the elliptic curves : Let $M=\\frac{\\mathbf{F}%_{q}[T]}{I_{1}}\\oplus \\frac{\\mathbf{F}_{q}[T]}{I_{2}}$, where $I_{1}=(i_{1})$,$% I_{2}=(i_{2})$ ($i_{1}$, $i_{2}$ two polynomials of $\\mathbf{F}_{q}[T]$%) and such that : $i_{2}\\mid (c-2)$. Then there exists an ordinary Drinfeld $\\mathbf{F}_{q}[T]$% -module $\\Phi $ over $L$ of rank 2, such that : $% L^{\\Phi}$ $\\simeq M$. We finish by a statistic about the cyclicity of such structure $L^{\\Phi}$, and we prove that is cyclic only for the trivial extensions of $\\mathbf{F}_{q}$.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2005-04-01T07:47:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J32"
    ],
    "keywords": [
      "finite field",
      "a-module structure",
      "drinfeld a-module",
      "finite extension",
      "trivial extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....12368S"
    }
  }
}