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