Algebraic curves with automorphism groups of large prime order

TL;DR

This paper classifies algebraic curves with automorphism groups of large prime order, specifically for primes $q = g+1$ and $q=2g+1$, and characterizes their automorphism groups.

Contribution

It provides a classification of $(g+1)$-curves and fully characterizes automorphism groups for $q=2g+1$ and $q=g+1$, extending previous results.

Findings

Classified $(g+1)$-curves.

Characterized automorphism groups for $q=2g+1$ and $q=g+1$.

Partial results for $q=g$ and $q=g-1$.

Abstract

Let $\mathcal{X}$ be an algebraic curve of genus $g$ defined over an algebraically closed field $K$ of characteristic $p \geq 0$, and $q$ a prime dividing $|\mbox{Aut}(\mathcal{X})|$. We say that $\mathcal{X}$ is a $q$-curve. Homma proved that either $q \leq g+1$ or $q = 2g+1$, and classified $(2g+1)$-curves. In this note, we classify $(g+1)$-curves, and fully characterize the automorphism groups of $q$-curves for $q= 2g+1, g+1$. We also give some partial results on $q$-curves for $q = g, g-1$.