Large automorphism groups of ordinary curves of even genus in odd characteristic

Abstract

Let $\mathcal{X}$ be a (projective, non-singular, geometrically irreducible) curve of even genus $g(\mathcal{X}) \geq 2$ defined over an algebraically closed field $K$ of odd characteristic $p$. If the $p$-rank $\gamma(\mathcal{X})$ equals $g(\mathcal{X})$, then $\mathcal{X}$ is \emph{ordinary}. In this paper, we deal with \emph{large} automorphism groups $G$ of ordinary curves of even genus. We prove that $|G| < 821.37g(\mathcal{X})^{7/4}$. The proof of our result is based on the classification of automorphism groups of curves of even genus in positive characteristic, see \cite{giulietti-korchmaros-2017}. According to this classification, for the exceptional cases ${\rm Aut}(\mathcal{X}) \cong {\rm Alt}_7$ and ${\rm Aut}(\mathcal{X}) \cong \rm{M}_{11}$ we show that the classical Hurwitz bound $|{\rm Aut}(\mathcal{X})| < 84(g(\mathcal{X})-1)$ holds, unless $p=3$, $g(\mathcal{X})=26$ and ${\rm Aut}(\mathcal{X}) \cong \rm{M}_{11}$; an example for the latter case being given by the modular curve $X(11)$ in characteristic $3$.