### Order p automorphisms of the open disc of a p-adic field, by Barry Green, Michel Matignon

This paper has now appeared in the Journal of the AMS, Vol 12 no. 1
(January 1999), 269-303, and so the dvi version has been removed.
Let $k$ be an algebraically closed field of characteristic $p>0,$
$W(k)$ the ring of Witt vectors and $R$ be 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
$$\sigma(Z)=\zeta Z(1+a_1Z+\cdots+a_iZ^i+\cdots)\in R[\![Z]\!].$$
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 ${\cal D}^o$ 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 ${\cal D}^o.$
Using this data we show that if $m$ is less than $p$ the fixed points are
equidistant, and that there are only a finite number of
conjugacy classes of order $p$ automorphisms in
${\rm Aut}_{R}(R[\![Z]\!])$ which are not the identity
$\mod\,(\pi).$

Barry Green, Michel Matignon <bwg@land.sun.ac.za, matignon@math.u-bordeaux.fr>