Smooth curves having a large automorphism $p$-group in characteristic $p\gt 0$

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$ and $C$ a connected nonsingular projective curve over $k$ with genus $g\ge 2$. This paper continues our study of big actions, that is, pairs $\left(C,G\right)$ where $G$ is a $p$-subgroup of the $k$-automorphism group of $C$ such that $\left|G\right|∕g>2p∕\left(p-1\right)$. If $G_2$ denotes the second ramification group of $G$ at the unique ramification point of the cover $C\to C∕G$, we display necessary conditions on $G_2$ for $\left(C,G\right)$ 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.