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Algebraic curves with a large cyclic automorphism group

Arianna Dionigi, Massimo Giulietti, Marco Timpanella

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.

Algebraic curves with a large cyclic automorphism group

Abstract

The study of algebraic curves with numerous automorphisms in relation to their genus is a well-established area in Algebraic Geometry. In 1995, Irokawa and Sasaki \cite{Sasaki} gave a complete classification of curves over with an automorphism of order . Precisely, such curves are either hyperelliptic with with even, or are quotients of the Fermat curve of degree by a cyclic group of order . Such a classification does not hold in positive characteristic , the curve with equation being a well-studied counterexample. This paper successfully classifies curves with a cyclic automorphism group of order at least in positive characteristic , 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.

Paper Structure

This paper contains 6 sections, 26 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Background and Preliminary Results
  3. Proof of Theorem \ref{['MAIN']} when $p$ divides $N$
  4. The case $N=p^h$
  5. The case $N=p^hm$ with $m>1$ and $(p,m)=1$
  6. Proof of Theorem \ref{['MAIN']} when $p$ does not divide $N$

Key Result

Theorem 1

Let $p \geq 3$, and let $\mathcal{X}$ be a curve of genus $g(\mathcal{X}) \geq 2$ with a cyclic automorphism group $G$ of order $N \geq 2g(\mathcal{X}) + 1$. Then, up to birational equivalence, one of the following holds:

Theorems & Definitions (44)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Proposition 6
  • Theorem 7: silverman2009, HKT
  • Proposition 8
  • proof
  • Proposition 9
  • ...and 34 more