{
  "id": "2309.16187",
  "version": "v1",
  "published": "2023-09-28T06:17:18.000Z",
  "updated": "2023-09-28T06:17:18.000Z",
  "title": "Rationality problem for norm one tori for $A_5$ and ${\\rm PSL}_2(\\mathbb{F}_8)$ extensions",
  "authors": [
    "Akinari Hoshi",
    "Aiichi Yamasaki"
  ],
  "comment": "23 pages. arXiv admin note: substantial text overlap with arXiv:2302.06231; text overlap with arXiv:1811.01676, arXiv:1910.01469, arXiv:1811.02145, arXiv:1210.4525",
  "categories": [
    "math.AG",
    "math.NT",
    "math.RA"
  ],
  "abstract": "We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $T=R^{(1)}_{K/k}(\\mathbb{G}_m)$ of $K/k$ whose Galois closures $L/k$ are $A_5\\simeq {\\rm PSL}_2(\\mathbb{F}_4)$ and ${\\rm PSL}_2(\\mathbb{F}_8)$ extensions. In particular, we prove that $T$ is stably $k$-rational for $G={\\rm Gal}(L/k)\\simeq {\\rm PSL}_2(\\mathbb{F}_{8})$, $H={\\rm Gal}(L/K)\\simeq (C_2)^3$ and $H\\simeq (C_2)^3\\rtimes C_7$ where $C_n$ is the cyclic group of order $n$. Based on the result, we conjecture that $T$ is stably $k$-rational for $G\\simeq {\\rm PSL}_2(\\mathbb{F}_{2^d})$, $H\\simeq (C_2)^d$ and $H\\simeq (C_2)^d\\rtimes C_{2^d-1}$. Some other cases $G\\simeq A_n$, $S_n$, ${\\rm GL}_n(\\mathbb{F}_{p^d})$, ${\\rm SL}_n(\\mathbb{F}_{p^d})$, ${\\rm PGL}_n(\\mathbb{F}_{p^d})$, ${\\rm PSL}_n(\\mathbb{F}_{p^d})$ and $H\\lneq G$ are also investigated for small $n$ and $p^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-28T06:17:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E72",
      "12F20",
      "13A50",
      "14E08",
      "20C10",
      "20G15"
    ],
    "keywords": [
      "rationality problem",
      "extensions",
      "cyclic group",
      "complete answer",
      "galois closures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}