{
  "id": "math/0607268",
  "version": "v4",
  "published": "2006-07-11T17:40:18.000Z",
  "updated": "2009-10-19T16:48:49.000Z",
  "title": "Reconstructing $p$-divisible groups from their truncations of small level",
  "authors": [
    "Adrian Vasiu"
  ],
  "comment": "32 pages. Final version identical with the galley proofs (modulo style). Paper dedicated to the memory of Angela Vasiu. To appear in Comment. Math. Helv",
  "journal": "Comment. Math. Helv. 85 (2010), no. 1, 165--202",
  "doi": "10.4171/CMH/192",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$. Let $n_D$ be the smallest non-negative integer for which the following statement holds: if $C$ is a $p$-divisible group over $k$ of the same codimension and dimension as $D$ and such that $C[p^{n_D}]$ is isomorphic to $D[p^{n_D}]$, then $C$ is isomorphic to $D$. To the Dieudonn\\'e module of $D$ we associate a non-negative integer $\\ell_D$ which is a computable upper bound of $n_D$. If $D$ is a product $\\prod_{i\\in I} D_i$ of isoclinic $p$-divisible groups, we show that $n_D=\\ell_D$; if the set $I$ has at least two elements we also show that $n_D\\le\\max\\{1,n_{D_i},n_{D_i}+n_{D_j}-1|i,j\\in I, j\\neq i\\}$. We show that we have $n_D\\Le 1$ if and only if $\\ell_D\\Le 1$; this recovers the classification of minimal $p$-divisible groups obtained by Oort. If $D$ is quasi-special, we prove the Traverso truncation conjecture for $D$. If $D$ is $F$-cyclic, we compute explicitly $n_D$. Many results are proved in the general context of latticed $F$-isocrystals with a (certain) group over $k$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2009-10-19T16:48:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E57",
      "11G10",
      "11G18",
      "11G25",
      "14F30",
      "14G35",
      "14L05",
      "14L30",
      "20G25"
    ],
    "keywords": [
      "divisible group",
      "small level",
      "traverso truncation conjecture",
      "isomorphic",
      "smallest non-negative integer"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7268V"
    }
  }
}