{
  "id": "1903.01929",
  "version": "v1",
  "published": "2019-03-05T16:37:50.000Z",
  "updated": "2019-03-05T16:37:50.000Z",
  "title": "Uniform boundedness for Brauer groups of forms in positive characteristic",
  "authors": [
    "Emiliano Ambrosi"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $k$ be a finitely generated field of characteristic $p>0$ and $X$ a smooth and proper scheme over $k$. Recent works of Cadoret, Hui and Tamagawa show that, if $X$ satisfies the $\\ell$-adic Tate conjecture for divisors for every prime $\\ell\\neq p$, the Galois invariant subgroup $Br(X_{\\overline k})[p']^{\\pi_1(k)}$ of the prime-to-$p$ torsion of the geometric Brauer group of $X$ is finite. The main result of this note is that, for every integer $d\\geq 1$, there exists a constant $C:=C(X,d)$ such that for every finite field extension $k \\subseteq k'$ with $[k':k]\\leq d$ and every $(\\overline k/k')$-form $Y$ of $X$ one has $|(Br(Y\\times_{k'}\\overline k)[p']^{\\pi_1(k')}|\\leq C$. The theorem is a consequence of general results on forms of compatible systems of $\\pi_1(k)$-representations and it extends to positive characteristic a recent result of Orr and Skorobogatov in characteristic zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-05T16:37:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive characteristic",
      "uniform boundedness",
      "galois invariant subgroup",
      "adic tate conjecture",
      "geometric brauer group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}