{
  "id": "math/0608032",
  "version": "v2",
  "published": "2006-08-01T18:30:48.000Z",
  "updated": "2007-10-08T21:13:49.000Z",
  "title": "Level m stratifications of versal deformations of p-divisible groups",
  "authors": [
    "Adrian Vasiu"
  ],
  "comment": "35 pages. Accepted (in final form) for publication in J. Alg. Geom",
  "journal": "J. Alg. Geom. 17 (2008), no. 4, 599-641",
  "doi": "10.1090/S1056-3911-08-00495-5",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d,m$ be positive integers. Let $D$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. Let $\\scrD$ be a versal deformation of $D$ over a smooth $k$-scheme $\\scrA$ which is equidimensional of dimension $cd$. We show that there exists a reduced, locally closed subscheme $\\grs_D(m)$ of $\\scrA$ that has the following property: a point $y\\in\\scrA(k)$ belongs to $\\grs_D(m)(k)$ if and only if $y^*(\\scrD)[p^m]$ is isomorphic to $D[p^m]$. We prove that $\\grs_D(m)$ is {\\it regular and equidimensional} of {\\it dimension} $cd-\\dim(\\pmb{\\text{Aut}}(D[p^m]))$. We give a proof of {\\it Traverso's formula} which for $m>>0$ computes the codimension of $\\grs_D(m)$ in $\\scrA$ (i.e., $\\dim(\\pmb{\\text{Aut}}(D[p^m]))$) in terms of the Newton polygon of $D$. We also provide a criterion of when $\\grs_D(m)$ satisfies the {\\it purity property} (i.e., it is an affine $\\scrA$-scheme). Similar results are proved for {\\it quasi Shimura $p$-varieties of Hodge type} that generalize the special fibres of good integral models of Shimura varieties of Hodge type in unramified mixed characteristic $(0,p)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-10-08T21:13:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E57",
      "11G10",
      "11G18",
      "11G25",
      "14F30",
      "14G35",
      "14L05",
      "14L15",
      "14R20",
      "20G25"
    ],
    "keywords": [
      "versal deformation",
      "p-divisible groups",
      "hodge type",
      "stratifications",
      "traversos formula"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8032V"
    }
  }
}