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

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 $\operatorname {mod} (\pi ).$