#### Vol. 13, No. 1, 2019

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Ordinary algebraic curves with many automorphisms in positive characteristic

### Gábor Korchmáros and Maria Montanucci

Vol. 13 (2019), No. 1, 1–18
##### Abstract

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

##### Keywords
algebraic curves, algebraic function fields, positive characteristic, automorphism groups
Primary: 14H37
Secondary: 14H05