ORDER p AUTOMORPHISMS OF THE OPEN DISC OF A p -ADIC FIELD

Let k be an algebraically closed field of characteristic p > 0, and let R be a complete discrete valuation ring dominating the ring of Witt vectors W (k). Let π denote a uniformizing parameter of R, and let R be the integral closure in a fixed algebraic closure of K := Fr(R). Throughout the paper we shall assume that R contains a prescribed primitive p-th root of unity, which we denote by ζ. In our previous paper [G-M], we were concerned with liftings of G-Galois covers of proper smooth curves over k to G-Galois covers of curves over R. There we showed that this problem is of local nature and so the crucial study is that of Gcovers R[[Z]]/R[[Z]] which induce G-covers k[[z]]/k[[z]] mod π, i.e. determination of the automorphism groups G of k[[z]] which can be lifted to automorphism groups of R[[Z]]. Another weaker question is to ask what the finite groups which occur as subgroups of AutRR[[Z]] with no inertia at (π) are. We were able to show that the local lifting for p-cyclic covers is always possible; the key point was to produce enough automorphisms of order p of R[[Z]] which are p-powers. This confirmed our conviction that the objects to be studied are automorphisms of order p, which is the aim of this paper. Now we describe the content of the paper. In §II we explain that this study is that of R-automorphisms σ ∈ AutRR[[Z]] with series representation σ(Z) = ζZ(1 + a1Z + · · ·+ amZ + · · · ) ∈ R[[Z]], such that the p-th iterate σ(Z) = Z. Our programme is to classify such automorphisms up to a change of parameter. Such an automorphism σ acts naturally on D := Spec R[[Z]] giving the following geometric data: let Fσ be the set of points fixed by σ, i.e. the roots in the maximal ideal of R of the series σ(Z) − Z. It is shown that they are simple roots and in the sequel we shall only consider those σ for which Fσ = {Z0, ..., Zm} is nonempty and for convenience the Zi are R-rational. We can attach Hurwitz data, Hσ = (h0, ..., hm) ∈ (Z/pZ), to the fixed points of an order p automorphism, which gives the list of exponents of ζ which occur in the series representation of the action of σ on the tangent space of the fixed points.