{
  "id": "2403.07659",
  "version": "v1",
  "published": "2024-03-12T13:49:30.000Z",
  "updated": "2024-03-12T13:49:30.000Z",
  "title": "The power operation in the Galois cohomology of a reductive group over a number field",
  "authors": [
    "Mikhail Borovoi",
    "Zinovy Reichstein"
  ],
  "comment": "52 pages. Comments are welcome!",
  "categories": [
    "math.NT",
    "math.AG",
    "math.GR",
    "math.RT"
  ],
  "abstract": "For a connected reductive group $G$ over a local or global field $K$, we define a *diamond* (or *power*) operation $$(\\xi, n)\\mapsto \\xi^{\\Diamond n}\\, \\colon\\ H^1(K,G)\\times {\\mathbb Z}\\to H^1(K,G)$$ of raising to power $n$ in the Galois cohomology pointed set (this operation is new when $K$ is a number field). We show that this operation has many functorial properties. When $G$ is a torus, the ponted set $H^1(K,G)$ has a natural group structure, and our new operation coincides with the usual power operation $(\\xi,n)\\mapsto \\xi^n$. For a cohomology class $\\xi$ in $H^1(K,G)$, we define the period ${\\rm per}(\\xi)$ to be the greatest common divisor of $n\\ge 1$ such that $\\xi^{\\Diamond n}=1$, and the index ${\\rm ind}(\\xi)$ to be the greatest common divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting $\\xi$. We show that ${\\rm per}(\\xi)$ divides ${\\rm ind}(\\xi)$, but they need not be equal. However, ${\\rm per}(\\xi)$ and ${\\rm ind}(\\xi)$ have the same set of prime factors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-12T13:49:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E72",
      "20G10",
      "20G20",
      "20G25",
      "20G30"
    ],
    "keywords": [
      "number field",
      "reductive group",
      "greatest common divisor",
      "usual power operation",
      "natural group structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}