Large automorphism groups of ordinary curves in characteristic $2$

Abstract

Let $\mathcal{X}$ be a (projective, non-singular, irreducible) curve of even genus $g(\mathcal{X})\ge 2$ defined over an algebraically closed field $K$ of characteristic $p$. If the $p$-rank $\gamma (\mathcal{X})$ equals $g(\mathcal{X})$, then $\mathcal{X}$ is ordinary. In this paper, we deal with large automorphism groups $G$ of ordinary curves. Under the hypotheses that $p=2$, $g(\mathcal{X})$ is even and $G$ is solvable, we prove that $|G|<35(g(\mathcal{X})+1{)}^{3/2}$.