{
  "id": "math/0007201",
  "version": "v1",
  "published": "2000-07-01T00:00:00.000Z",
  "updated": "2000-07-01T00:00:00.000Z",
  "title": "Newton polygons and formal groups: Conjectures by Manin and Grothendieck",
  "authors": [
    "Frans Oort"
  ],
  "comment": "24 pages, published version, abstract added in migration",
  "journal": "Ann. of Math. (2) 152 (2000), no. 1, 183-206",
  "categories": [
    "math.AG"
  ],
  "abstract": "We consider p-divisible groups (also called Barsotti-Tate groups) in characteristic p, their deformations, and we draw some conclusions. For such a group we can define its Newton polygon (abbreviated NP). This is invariant under isogeny. For an abelian variety (in characteristic p) the Newton polygon of its p-divisible group is ``symmetric''. In 1963 Manin conjectured that conversely any symmetric Newton polygon is ``algebroid''; i.e., it is the Newton polygon of an abelian variety. This conjecture was shown to be true and was proved with the help of the ``Honda-Serre-Tate theory''. We give another proof. Grothendieck showed that Newton polygons ``go up'' under specialization: no point of the Newton polygon of a closed fiber in a family is below the Newton polygon of the generic fiber. In 1970 Grothendieck conjectured the converse: any pair of comparable Newton polygons appear for the generic and special fiber of a family. This was extended by Koblitz in 1975 to a conjecture about a sequence of comparable Newton polygons. We prove these conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-07-01T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conjecture",
      "formal groups",
      "grothendieck",
      "abelian variety",
      "p-divisible group"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......7201O"
    }
  }
}