Let k be an algebraically closed field of positive characteristic p > 0 and C → P 1 k a p-cyclic cover of the projective line ramified in exactly one point. We are interested in the p-Sylow subgroups of the full automorphism group Aut k C. We prove that for curves C with genus 2 or higher, these groups are exactly the extensions of a p-cyclic group by an elementary abelian p-group. The main tool is an efficient algorithm to compute the p-Sylow subgroups of Aut k C starting from an Artin-Schreier equation for the cover C → P 1 k . We also characterize curves C with genus g C 2 and a p-groupOur methods rely on previous work by Stichtenoth whose approach we have adopted.

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 ).

Let k be an algebraically closed field of characteristic p > 0 and C a connected nonsingular projective curve over k with genus g ≥ 2. This paper continues our study of big actions, that is, pairs (C, G) where G is a p-subgroup of the k-automorphism group of C such that |G|/g > 2 p/( p−1). If G 2 denotes the second ramification group of G at the unique ramification point of the cover C → C/G, we display necessary conditions on G 2 for (C, G) to be a big action, which allows us to pursue the classification of big actions.Our main source of examples comes from the construction of curves with many rational points using ray class field theory for global function fields, as initiated by J.-P. Serre and continued by Lauter and by Auer. In particular, we obtain explicit examples of big actions with G 2 abelian of large exponent.