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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Order $p$ automorphisms of the open disc of a $p$-adic field
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by Barry Green and Michel Matignon PDF
J. Amer. Math. Soc. 12 (1999), 269-303 Request permission

Abstract:

Let $k$ be an algebraically closed field of characteristic $p>0,$ $W(k)$ the ring of Witt vectors and $R$ a complete discrete valuation ring dominating $W(k)$ and containing $\zeta ,$ a primitive $p$-th root of unity. Let $\pi$ denote a uniformizing parameter for $R.$ We study order $p$ automorphisms of the formal power series ring $R[[Z]],$ which are defined by a series \begin{equation*}\sigma (Z)=\zeta Z(1+a_{1}Z+\cdots +a_{i}Z^{i}+\cdots )\in R[[Z]].\end{equation*} The set of fixed points of $\sigma$ is denoted by $F_{\sigma }$ and we suppose that they are $K$-rational and that $|F_{\sigma }|=m+1$ for $m\geq 0.$ Let ${\mathcal {D}}^{o}$ be the minimal semi-stable model of the $p$-adic open disc over $R$ in which $F_{\sigma }$ specializes to distinct smooth points. We study the differential data that can be associated to each irreducible component of the special fibre of ${\mathcal {D}}^{o}.$ Using this data we show that if $m<p$, then the fixed points are equidistant, and that there are only a finite number of conjugacy classes of order $p$ automorphisms in $\operatorname {Aut}_{R}(R[[Z]])$ which are not the identity $\operatorname {mod} (\pi ).$
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Additional Information
  • Barry Green
  • Affiliation: Department of Mathematics, University of Stellenbosch, Stellenbosch, 7602, South Africa
  • MR Author ID: 76490
  • Email: bwg@land.sun.ac.za
  • Michel Matignon
  • Affiliation: Mathématiques Pures de Bordeaux, UPRS-A 5467, C.N.R.S Université de Bordeaux I, 351, cours de la Libération 33405 – Talence, Cedex, France
  • Email: matignon@math.u-bordeaux.fr
  • Received by editor(s): November 25, 1997
  • Received by editor(s) in revised form: June 24, 1998
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 12 (1999), 269-303
  • MSC (1991): Primary 14G20, 14L27; Secondary 14D15, 14E22
  • DOI: https://doi.org/10.1090/S0894-0347-99-00284-2
  • MathSciNet review: 1630112