Automorphism groups for p-cyclic covers of the affine line

Abstract

let k be an algebraically closed field of positive characteristic p > 0 and $c \to {\mathbb p}^1_k$ 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 $c \to {\mathbb p}^1_k$ . we also characterize curves c with genus $g_c\geq 2$ and a p -group action $g\subset \text{aut}_k c$ such that $2p/(p-1) and $4/(p-1)^2\leq |g|/g_c^2$ . our methods rely on previous work by stichtenoth whose approach we have adopted.