{
  "id": "1404.0308",
  "version": "v1",
  "published": "2014-04-01T16:45:31.000Z",
  "updated": "2014-04-01T16:45:31.000Z",
  "title": "Birational classification of fields of invariants for groups of order $128$",
  "authors": [
    "Akinari Hoshi"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $G$ be a finite group acting on the rational function field $\\mathbb{C}(x_g : g\\in G)$ by $\\mathbb{C}$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\\in G$. Noether's problem asks whether the invariant field $\\mathbb{C}(G)=k(x_g : g\\in G)^G$ is rational (i.e. purely transcendental) over $\\mathbb{C}$. Saltman and Bogomolov, respectively, showed that for any prime $p$ there exist groups $G$ of order $p^9$ and of order $p^6$ such that $\\mathbb{C}(G)$ is not rational over $\\mathbb{C}$ by showing the non-vanishing of the unramified Brauer group: $Br_{nr}(\\mathbb{C}(G))\\neq 0$. For $p=2$, Chu, Hu, Kang and Prokhorov proved that if $G$ is a 2-group of order $\\leq 32$, then $\\mathbb{C}(G)$ is rational over $\\mathbb{C}$. Chu, Hu, Kang and Kunyavskii showed that if $G$ is of order 64, then $\\mathbb{C}(G)$ is rational over $\\mathbb{C}$ except for the groups $G$ belonging to the two isoclinism families $\\Phi_{13}$ and $\\Phi_{16}$. Bogomolov and B\\\"ohning's theorem claims that if $G_1$ and $G_2$ belong to the same isoclinism family, then $\\mathbb{C}(G_1)$ and $\\mathbb{C}(G_2)$ are stably $\\mathbb{C}$-isomorphic. We investigate the birational classification of $\\mathbb{C}(G)$ for groups $G$ of order 128 with $Br_{nr}(\\mathbb{C}(G))\\neq 0$. Moravec showed that there exist exactly 220 groups $G$ of order 128 with $Br_{nr}(\\mathbb{C}(G))\\neq 0$ forming 11 isoclinism families $\\Phi_j$. We show that if $G_1$ and $G_2$ belong to $\\Phi_{16}, \\Phi_{31}, \\Phi_{37}, \\Phi_{39}, \\Phi_{43}, \\Phi_{58}, \\Phi_{60}$ or $\\Phi_{80}$ (resp. $\\Phi_{106}$ or $\\Phi_{114}$), then $\\mathbb{C}(G_1)$ and $\\mathbb{C}(G_2)$ are stably $\\mathbb{C}$-isomorphic with $Br_{nr}(\\mathbb{C}(G_i))\\simeq C_2$. Explicit structures of non-rational fields $\\mathbb{C}(G)$ are given for each cases including also the case $\\Phi_{30}$ with $Br_{nr}(\\mathbb{C}(G))\\simeq C_2\\times C_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-01T16:45:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12F12",
      "13A50",
      "14E08",
      "14F22",
      "16K50",
      "20C10"
    ],
    "keywords": [
      "birational classification",
      "isoclinism family",
      "rational function field",
      "noethers problem asks",
      "explicit structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.0308H"
    }
  }
}