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>