{
  "id": "2210.09119",
  "version": "v1",
  "published": "2022-10-17T14:11:32.000Z",
  "updated": "2022-10-17T14:11:32.000Z",
  "title": "Hasse norm principle for $M_{11}$ extensions",
  "authors": [
    "Akinari Hoshi",
    "Kazuki Kanai",
    "Aiichi Yamasaki"
  ],
  "comment": "19 pages. arXiv admin note: text overlap with arXiv:2003.08253",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $k$ be a field and $T$ be an algebraic $k$-torus. In 1969, over a global field $k$, Voskresenskii proved that there exists an exact sequence $0\\to A(T)\\to H^1(k,{\\rm Pic}\\,\\overline{X})^\\vee\\to Sha(T)\\to 0$ where $A(T)$ is the kernel of the weak approximation of $T$, $Sha(T)$ is the Shafarevich-Tate group of $T$, $X$ is a smooth $k$-compactification of $T$, ${\\rm Pic}\\,\\overline{X}$ is the Picard group of $\\overline{X}=X\\times_k\\overline{k}$ and $\\vee$ stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus $T=R^{(1)}_{K/k}(G_m)$ of $K/k$, $Sha(T)=0$ if and only if the Hasse norm principle holds for $K/k$. We determine $H^1(k,{\\rm Pic}\\, \\overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ when the Galois group ${\\rm Gal}(L/k)$ of the Galois closure $L/k$ of $K/k$ is isomorphic to the Mathieu group $M_{11}$ of degree $11$. We also give a necessary and sufficient condition for the Hasse norm principle for such extensions $K/k$ with ${\\rm Gal}(L/k)\\simeq M_{11}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-17T14:11:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E72",
      "12F20",
      "13A50",
      "14E08",
      "20C10",
      "20G15"
    ],
    "keywords": [
      "extensions",
      "hasse norm principle holds",
      "global field",
      "exact sequence",
      "picard group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}