Large $3$-groups of automorphisms of algebraic curves in characteristic $3$

TL;DR

This paper classifies large p-subgroups of automorphisms of algebraic curves in characteristic 3, revealing specific ramification and field extension properties depending on the p-rank of the curve.

Contribution

It establishes new bounds and characterizes the structure of automorphism groups of algebraic curves in characteristic 3, extending previous results to larger automorphism subgroups.

Findings

If |S| > 2(g-1), then either the p-rank is zero with a unique ramification point, or p=3, the curve is general, and the automorphism group attains Nakajima's bound.

In the second case, the function field is an unramified Galois extension of a specific genus 2 curve.

The paper completes the classification for large automorphism p-groups in characteristic 3.

Abstract

Let $S$ be a $p$-subgroup of the $\K$-automorphism group $\aut(\cX)$ of an algebraic curve $\cX$ of genus $\gg\ge 2$ and $p$-rank $\gamma$ defined over an algebraically closed field $\mathbb{K}$ of characteristic $p\geq 3$.In this paper we prove that if $|S|>2(\gg-1)$ then one of the following cases occurs. \begin{itemize} \item[(i)] $\gamma=0$ and the extension $\K(\cX)/\K(\cX)^S$ completely ramifies at a unique place, and does not ramify elsewhere. \item[(ii)] $\gamma>0$, $p=3$, $\cX$ is a general curve, $S$ attains the Nakajima's upper bound $3(\gamma-1)$ and $\K(\cX)$ is an unramified Galois extension of the function field of a general curve of genus $2$ with equation $Y^2=cX^6+X^4+X^2+1$ where $c\in\K^*$. \end{itemize} Case (i) was investigated by Stichtenoth, Lehr, Matignon, and Rocher.