Automorphisms groups for $p$-cyclic covers of the affine line

Abstract

Let $k$ be an algebraically closed field of positive characteristic $p>0$ and $C\to {\mathbb{P}}_k^1$ a $p$-cyclic cover of the projective line ramified in exactly one point. We are interested in the $p$-part of the full automorphism group $Au{t}_{k}C$. First we prove that these groups are exactly the extra-special $p$-groups and groups G which are subgroups of an extra-special group E such that $Z(E)\subseteq G$. The paper also describes an efficient algorithm to compute the $p$-part of $\Aut_k C$ starting from an Artin-Schreier equation for the cover $C\to {\mathbb{P}}_k^1$. The interest for these objects initially came from the study of the stable reduction of $p$-cyclic covers over the $p$-adics. There the covers $C\to {\mathbb{P}}_k^1$ naturally arise and their automorphism groups play a major role in understanding the arithmetic monodromy. Our methods rely on previous work by Stichtenoth whose approach we have adopted.