{
  "id": "1112.1676",
  "version": "v3",
  "published": "2011-12-07T19:58:36.000Z",
  "updated": "2012-07-24T16:20:22.000Z",
  "title": "Dimensions of group schemes of automorphisms of truncated Barsotti--Tate groups",
  "authors": [
    "Ofer Gabber",
    "Adrian Vasiu"
  ],
  "comment": "52 pages. Final version as close to the galley proofs as possible. To appear in IMRN",
  "categories": [
    "math.NT",
    "math.AG",
    "math.RT"
  ],
  "abstract": "Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of characteristic $p>0$. Let $n_D$ be the smallest non-negative integer such that $D$ is determined by $D[p^{n_D}]$ within the class of $p$-divisible groups over $k$ of the same codimension $c$ and dimension $d$ as $D$. We study $n_D$, lifts of $D[p^m]$ to truncated Barsotti--Tate groups of level $m+1$ over $k$, and the numbers $\\gamma_D(i):=\\dim(\\pmb{Aut}(D[p^i]))$. We show that $n_D\\le cd$, $(\\gamma_D(i+1)-\\gamma_D(i))_{i\\in\\Bbb N}$ is a decreasing sequence in $\\Bbb N$, for $cd>0$ we have $\\gamma_D(1)<\\gamma_D(2)<...<\\gamma_D(n_D)$, and for $m\\in\\{1,...,n_D-1\\}$ there exists an infinite set of truncated Barsotti--Tate groups of level $m+1$ which are pairwise non-isomorphic and lift $D[p^m]$. Different generalizations to $p$-divisible groups with a smooth integral group scheme in the crystalline context are also proved.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-07-24T16:20:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "11G18",
      "14F30",
      "14G35",
      "14L05",
      "14L15",
      "14L30",
      "20G15"
    ],
    "keywords": [
      "truncated barsotti-tate groups",
      "divisible group",
      "automorphisms",
      "smooth integral group scheme",
      "infinite set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.1676G"
    }
  }
}