The automorphism groups of a family of maximal curves
type: Item
instance of: scholarly article
title: The automorphism groups of a family of maximal curves
zbMATH Open document ID: 1285.11094
author: Beth Malmskog
author: Rachel Pries
author: Robert Guralnick
published in: Journal of Algebra
publication date: 2013-01-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.3952
review text: Let \(q=p^h\) be a prime power, and let \(n\geq 3\) be an odd integer. The generalized GK curve \(\mathcal C_n\) is a maximal curve defined over \(\mathbb F_q^{2n}\) by the equations \[ \mathcal H_q:~x^q+x-y^{q+1}=0, ~~ \mathcal X_n:~y^{q^2}-y-z^m. \] In this paper, the authors determine the automorphism group of this curve \(\mathcal C_n\). As it turns out, the automorphism group is a semidirect product \(\Gamma\) of the form \(Q\times\Sigma\), where \(\Sigma\) is a cyclic group of order \((q^n+1)(q-1)\) and \(Q\) is a non-abelian group of order \(q^3\). Their proof uses ideas from ramification theory and a new structural result about automorphism groups of curves in characteristic \(p\). It is worth noting that \textit{C. Güneri} et al. [Adv. Geom. 13, No. 2, 369--380 (2013; Zbl 1285.11093)] determined the automorphism group of the same curve using a different approach based on Riemann-Roch theory.
reviewed by: Kit Ho Mak
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14H37
zbMATH DE Number: 6125721
zbMATH Keywords: Weil bound
zbMATH Keywords: maximal curve
zbMATH Keywords: automorphism
zbMATH Keywords: ramification
MaRDI profile type: Publication
OpenAlex ID: W2073911879
arXiv ID: 1105.3952
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DOI: 10.1016/J.JALGEBRA.2012.03.036
description: scientific article