Ordinary algebraic curves with many automorphisms in positive characteristic

Abstract

Let $\mathcal{X}$ be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus $g(\mathcal{X})\ge 2$ 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 $|G|\le 34(g(\mathcal{X})+1{)}^{3/2}$. There are known curves attaining this bound up to the constant $34$. For $p$ odd, our result improves the classical Nakajima bound $|G|\le 84(g(\mathcal{X})-1)g(\mathcal{X})$, and, for solvable groups $G$, the Gunby-Smith-Yuan bound $|G|\le 6(g(\mathcal{X}{)}^2+12\sqrt{21}g(\mathcal{X}{)}^{3/2})$ where $g(\mathcal{X})>c{p}^2$ for some positive constant $c$.

References in corpus (1)

• On Abelian Automorphism Groups of Mumford Curves

Cited by in corpus (2)

• Transcendency Degree One Function Fields Over a Finite Field with Many Automorphisms
• Generalized Artin-Mumford curves over finite fields