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Let $\mathcal{X}$ be a (projective, non-singular, geometrically irreducible)curve of even genus $g(\mathcal{X}) \geq 2$ defined over an algebraicallyclosed 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 theclassification of automorphism groups of curves of even genus in positivecharacteristic, see \cite{giulietti-korchmaros-2017}. According to thisclassification, for the exceptional cases ${\rm Aut}(\mathcal{X}) \cong {\rmAlt}_7$ and ${\rm Aut}(\mathcal{X}) \cong \rm{M}_{11}$ we show that theclassical 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 $3$.

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