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 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-group action G Aut k C such that 2p/(p -1) < |G|/g C and 4/(p -1) 2 |G|/g 2 C . Our methods rely on previous work by Stichtenoth whose approach we have adopted.