Ordinary algebraic curves with many automorphisms in positive characteristic

TL;DR

This paper establishes a new upper bound on the size of solvable automorphism groups of ordinary algebraic curves in positive characteristic, improving classical bounds and matching known extremal examples.

Contribution

It proves a sharper bound for solvable automorphism groups of ordinary curves, extending and refining previous classical bounds in positive characteristic.

Findings

Bound |G| ≤ 34(g+1)^{3/2} for solvable G

Improves Nakajima's classical bound for odd p

Matches known extremal curves with the bound

Abstract

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 $c$.