Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic
type: Item
instance of: scholarly article
title: Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic
zbMATH Open document ID: 1206.14044
author: Huah Chu
author: Yingpu Deng
author: Tse-Chung Yang
publication date: 2010-11-17T00:00:00Z
full work available at URL: 10.pdf
review text: Any hyperelliptic curve \(C\) of genus \(g\), having a rational Weierstrass point, admits a non-singular affine model of the type: \(v^2+h(u)v=f(u)\), where \(h(u),f(u)\) are polynomials of respective degree \(\deg h\leq g\), \(\deg f=2g+1\). For \(C\) given by such a model, P. Lockhart characterized the isomorphism classes of pointed curves \((C,P)\), where \(P\) is the Weierstrass point of \(C\) at infinity, in terms of explicit plane coordinate transformations preserving the shape of the model [\textit{P. Lockhart}, Trans. Am. Math. Soc. 342, No. 2, 729--752 (1994; Zbl 0815.11031)]. The paper under review is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves of genus \(4\) over finite fields of characteristic \(2\), by counting the number of Lockhart's transformations. For finite fields of odd characteristic the number of (pointed and non-pointed) hyperelliptic curves of arbitrary genus \(g\) is counted in [the reviewer, Adv. Math. 221, No. 3, 774--787 (2009; Zbl 1214.11078)].
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 11G20
zbMATH DE Number: 5816549
zbMATH Keywords: hyperelliptic curves
zbMATH Keywords: Weierstrass points
zbMATH Keywords: isomorphism classes
reviewed by: Enric Nart
MaRDI profile type: Publication
OpenAlex ID: W2093205513
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DOI: 10.1007/S10114-010-7186-8
published in: Acta Mathematica Sinica, English Series
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description: scientific article; zbMATH DE number 5816549