Purity results for pp-divisible groups and Abelian schemes over regular bases of mixed characteristic

  • Adrian Vasiu

    P. O. Box 6000, Department of Fakultät für Mathematik Mathematical Sciences Universität Bielefeld Binghamton University P.O. Box 100 131 Binghamton, New York 13902- D-33 501 Bielefeld 6000 Germany U.S.A
  • Thomas Zink

    P. O. Box 6000, Department of Fakultät für Mathematik Mathematical Sciences Universität Bielefeld Binghamton University P.O. Box 100 131 Binghamton, New York 13902- D-33 501 Bielefeld 6000 Germany U.S.A
Purity results for $p$-divisible groups and Abelian schemes over regular bases of mixed characteristic cover
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Abstract

Let pp be a prime. Let (R,\idealm)(R,\ideal{m}) be a regular local ring of mixed characteristic (0,p)(0,p) and absolute index of ramification ee. We provide general criteria of when each abelian scheme over \SpecR\idealm\Spec R\setminus{\ideal{m}} extends to an abelian scheme over \SpecR\Spec R. We show that such extensions always exist if elep1ele p-1, exist in most cases if pleele2p3ple ele 2p-3, and do not exist in general if e2p2e\ge 2p-2. The case elep1ele p-1 implies the uniqueness of integral canonical models of Shimura varieties over a discrete valuation ring OO of mixed characteristic (0,p)(0,p) and index of ramification at most p1p-1. This leads to large classes of examples of Néron models over OO. If p>2p>2 and index p1p-1, the examples are new.

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Adrian Vasiu, Thomas Zink, Purity results for pp-divisible groups and Abelian schemes over regular bases of mixed characteristic. Doc. Math. 15 (2010), pp. 571–599

DOI 10.4171/DM/307