Algebra & Number Theory

Ordinary algebraic curves with many automorphisms in positive characteristic

Abstract

Let $X$ be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus $g(X)≥2$ defined over an algebraically closed field $K$ of odd characteristic $p$. Let $Aut(X)$ be the group of all automorphisms of $X$ which fix $K$ elementwise. For any solvable subgroup $G$ of $Aut(X)$ we prove that $|G|≤34(g(X)+1)3∕2$. There are known curves attaining this bound up to the constant $34$. For $p$ odd, our result improves the classical Nakajima bound $|G|≤84(g(X)−1)g(X)$ and, for solvable groups $G$, the Gunby–Smith–Yuan bound $|G|≤6(g(X)2+1221g(X)3∕2)$ where $g(X)>cp2$ for some positive constant $c$.

Article information

Source
Algebra Number Theory, Volume 13, Number 1 (2019), 1-18.

Dates
Revised: 18 October 2018
Accepted: 20 November 2018
First available in Project Euclid: 27 March 2019

https://projecteuclid.org/euclid.ant/1553652019

Digital Object Identifier
doi:10.2140/ant.2019.13.1

Mathematical Reviews number (MathSciNet)
MR3917914

Zentralblatt MATH identifier
07041705

Subjects
Primary: 14H37: Automorphisms

Citation

Korchmáros, Gábor; Montanucci, Maria. Ordinary algebraic curves with many automorphisms in positive characteristic. Algebra Number Theory 13 (2019), no. 1, 1--18. doi:10.2140/ant.2019.13.1. https://projecteuclid.org/euclid.ant/1553652019

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