The curve \(y^n=x^\ell(x^m+1)\) over finite fields. II
type: Item
instance of: scholarly article
title: The curve \(y^n=x^\ell(x^m+1)\) over finite fields. II
zbMATH Open document ID: 1476.11092
author: Saeed Tafazolian
author: Fernando Torres
publication date: 2021-08-19T00:00:00Z
review text: When \(X\) is a (projective, smooth, absolutely irreducible) curve of genus \(g\) over \(\mathbb{F}_q\), Hasse-Weil bound shows that \(\#X(\mathbb{F}_q) \leq 1+q+ 2g \sqrt{q}\). When \(q\) is a square, curves whose number of rational points reaches this bound are called maximal and have been studied extensively and partially classified. The present article deals with maximal curves of the form \(y^n= x^{\ell} (x^m+1)\) with certain conditions on \(n,\ell\) and \(m\) with respect to \(q\), divided into cases which have been handled in previous papers by the same authors and the present one. In order to show that the curve is maximal, the authors use the classical trick to reveal them as explicit sub-covers of curves which are known to be maximal (mainly the Hermitian curve). For Part I see [the authors, ibid. 19, No. 2, 263--268 (2019; Zbl 1478.11090)].
reviewed by: Christophe Ritzenthaler
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11M38
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 7383266
zbMATH Keywords: finite field
zbMATH Keywords: maximal curve
zbMATH Keywords: Kummer extension
MaRDI profile type: Publication
full work available at URL: advgeom-2021-0017
OpenAlex ID: W3176462672
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DOI: 10.1515/ADVGEOM-2021-0017
published in: advg
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description: scientific article; zbMATH DE number 7383266