Algebraic curves with a large cyclic automorphism group

TL;DR

This paper classifies algebraic curves with large cyclic automorphism groups in positive characteristic, extending a known complex case classification to a broader setting with additional challenges like wild ramification.

Contribution

It provides a classification of curves with cyclic automorphism groups of order at least 2g+1 in positive characteristic, complementing the complex case results.

Findings

Classified curves with large cyclic automorphism groups in positive characteristic

Extended Irokawa-Sasaki classification to positive characteristic

Addressed challenges from wild ramification

Abstract

The study of algebraic curves $\cX$ with numerous automorphisms in relation to their genus $g(\cX)$ is a well-established area in Algebraic Geometry. In 1995, Irokawa and Sasaki \cite{Sasaki} gave a complete classification of curves over $\mathbb{C}$ with an automorphism of order $N \geq 2g(\mathcal{X}) + 1$. Precisely, such curves are either hyperelliptic with $N=2g(\cX)+2$ with $g(\cX)$ even, or are quotients of the Fermat curve of degree $N$ by a cyclic group of order $N$. Such a classification does not hold in positive characteristic $p$, the curve with equation $y^2=x^p-x$ being a well-studied counterexample. This paper successfully classifies curves with a cyclic automorphism group of order $N$ at least $2g(\mathcal{X}) + 1$ in positive characteristic $p \neq 2$, offering the positive characteristic counterpart to the Irokawa-Sasaki result. The possibility of wild ramification in positive characteristic has presented a few challenges to the investigation.