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

Abstract

Let k k be an algebraically closed field of characteristic p > 0 , p>0, W ( k ) W(k) the ring of Witt vectors and R R a complete discrete valuation ring dominating W ( k ) W(k) and containing ζ , \zeta , a primitive p p -th root of unity. Let π \pi denote a uniformizing parameter for R . R. We study order p p automorphisms of the formal power series ring R [ [ Z ] ] , R[[Z]], which are defined by a series σ ( Z ) = ζ Z ( 1 + a 1 Z + ⋯ + a i Z i + ⋯ ) ∈ R [ [ Z ] ] . \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 σ \sigma is denoted by F σ F_{\sigma } and we suppose that they are K K -rational and that | F σ | = m + 1 |F_{\sigma }|=m+1 for m ≥ 0. m\geq 0. Let D o {\mathcal {D}}^{o} be the minimal semi-stable model of the p p -adic open disc over R R in which F σ 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 D o . {\mathcal {D}}^{o}. Using this data we show that if m > p m>p , then the fixed points are equidistant, and that there are only a finite number of conjugacy classes of order p p automorphisms in Aut R ⁡ ( R [ [ Z ] ] ) \operatorname {Aut}_{R}(R[[Z]]) which are not the identity mod ⁡ ( π ) . \operatorname {mod} (\pi ).