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Order $p$ automorphisms of the open disc
of a $p$-adic field

Authors: Barry Green and Michel Matignon
Journal: J. Amer. Math. Soc. 12 (1999), 269-303
MSC (1991): Primary 14G20, 14L27; Secondary 14D15, 14E22
MathSciNet review: 1630112
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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

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

Keywords: Order $p$ automorphisms of open $p$-adic discs, fixed points, Hurwitz data, Lubin-Tate formal groups, semi-stable models, degeneration of $\mu _{p}$-torsors, automorphism conjugacy classes
Received by editor(s): November 25, 1997
Received by editor(s) in revised form: June 24, 1998
Article copyright: © Copyright 1999 American Mathematical Society