Algebraic curves with automorphism groups of large prime order

Abstract

Let $\mathcal{X}$ be an algebraic curve of genus $g$ defined over an algebraically closed field $K$ of characteristic $p\ge 0$, and $q$ a prime dividing $|\mbox{Aut}(\mathcal{X})|$. We say that $\mathcal{X}$ is a $q$-curve. Homma proved that either $q\le g+1$ 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 $q=g,g-1$.