{
  "id": "1406.0949",
  "version": "v2",
  "published": "2014-06-04T06:44:39.000Z",
  "updated": "2015-01-19T06:46:31.000Z",
  "title": "Algebraic tori revisited",
  "authors": [
    "Ming-chang Kang"
  ],
  "comment": "Section 4 is re-written completely to correct an erroneous proof in the first version. Shizuo Endo is added as a coauthor",
  "categories": [
    "math.NT",
    "math.GR"
  ],
  "abstract": "Let $K/k$ be a finite Galois extension and $\\pi = \\fn{Gal}(K/k)$. An algebraic torus $T$ defined over $k$ is called a $\\pi$-torus if $T\\times_{\\fn{Spec}(k)} \\fn{Spec}(K)\\simeq \\bm{G}_{m,K}^n$ for some integer $n$. The set of all algebraic $\\pi$-tori defined over $k$ under the stably isomorphism form a semigroup, denoted by $T(\\pi)$. We will give a complete proof of the following theorem due to Endo and Miyata \\cite{EM5}. Theorem. Let $\\pi$ be a finite group. Then $T(\\pi)\\simeq C(\\Omega_{\\bm{Z}\\pi})$ where $\\Omega_{\\bm{Z}\\pi}$ is a maximal $\\bm{Z}$-order in $\\bm{Q}\\pi$ containing $\\bm{Z}\\pi$ and $C(\\Omega_{\\bm{Z}\\pi})$ is the locally free class group of $\\Omega_{\\bm{Z}\\pi}$, provided that $\\pi$ is isomorphic to the following four types of groups : $C_n$ ($n$ is any positive integer), $D_m$ ($m$ is any odd integer $\\ge 3$), $C_{q^f}\\times D_m$ ($m$ is any odd integer $\\ge 3$, $q$ is an odd prime number not dividing $m$, $f\\ge 1$, and $(\\bm{Z}/q^f\\bm{Z})^{\\times}=\\langle \\bar{p}\\rangle$ for any prime divisor $p$ of $m$), $Q_{4m}$ ($m$ is any odd integer $\\ge 3$, $p\\equiv 3 \\pmod{4}$ for any prime divisor $p$ of $m$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-04T06:44:39.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-01-19T06:46:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E08",
      "11R33",
      "20C10",
      "11R29"
    ],
    "keywords": [
      "algebraic torus",
      "algebraic tori",
      "odd integer",
      "prime divisor",
      "finite galois extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.0949K"
    }
  }
}