Ordinary algebraic curves with many automorphisms in positive characteristic

Abstract

Let $\mathcal{X}$ be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus $\mathfrak{g}\left(\mathcal{X}\right)\ge 2$ defined over an algebraically closed field $\mathbb{K}$ of odd characteristic $p$. Let $Aut\left(\mathcal{X}\right)$ be the group of all automorphisms of $\mathcal{X}$ which fix $\mathbb{K}$ elementwise. For any solvable subgroup $G$ of $Aut\left(\mathcal{X}\right)$ we prove that $\left|G\right|\le 34{\left(\mathfrak{g}\left(\mathcal{X}\right)+1\right)}^{3∕2}$. There are known curves attaining this bound up to the constant $34$. For $p$ odd, our result improves the classical Nakajima bound $\left|G\right|\le 84\left(\mathfrak{g}\left(\mathcal{X}\right)-1\right)\mathfrak{g}\left(\mathcal{X}\right)$ and, for solvable groups $G$, the Gunby–Smith–Yuan bound $\left|G\right|\le 6\left(\mathfrak{g}{\left(\mathcal{X}\right)}^2+12\sqrt{21}\mathfrak{g}{\left(\mathcal{X}\right)}^{3∕2}\right)$ where $\mathfrak{g}\left(\mathcal{X}\right)>c{p}^2$ for some positive constant $c$.