On smooth curves endowed with a large automorphism $p$-group in characteristic $p>0$

TL;DR

This paper investigates the structure of automorphism groups of smooth curves in characteristic p, providing conditions on ramification groups for large automorphism p-groups and constructing explicit examples using class field theory.

Contribution

It establishes necessary conditions on the second ramification group for big actions and constructs explicit examples with large abelian ramification groups using class field theory.

Findings

Necessary conditions on G_2 for big actions.

Explicit examples of big actions with large abelian G_2.

Construction of curves with many rational points.

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$ and $C$ a connected nonsingular projective curve over $k$ with genus $g \geq 2$. This paper continues the work begun by Lehr and Matignon, namely the study of "big actions", i.e. the pairs $(C,G)$ where $G$ is a $p$-subgroup of the $k$-automorphism group of $C$ such that$\frac{|G|}{g} >\frac{2 p}{p-1}$. 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 $(C,G)$ 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 followed by Lauter and Auer. In particular, we obtain explicit examples of big actions with $G_2$ abelian of large exponent.