On the simplicity of Jacobians for hyperelliptic curves of genus 2 over the field of rational numbers with torsion points of high order
type: Item
instance of: scholarly article
title: On the simplicity of Jacobians for hyperelliptic curves of genus 2 over the field of rational numbers with torsion points of high order
zbMATH Open document ID: 1278.14045
DOI: 10.1134/S1064562413030216
author: V. S. Zhgun
author: M. M. Petrunin
author: V. P. Platonov
published in: Doklady Mathematics
publication date: 2013-11-11T00:00:00Z
review text: In two 2012 papers of \textit{V. P. Platonov} and \textit{M. M. Petrunin} [Dokl. Math. 85, No. 2, 286--288 (2012); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 443, No. 6, 664--667 (2012; Zbl 1319.11041); Dokl. Math. 86, No. 2, 642--643 (2012); translation from Dokl. Akad. Nauk. 446, No. 3, 263--264 (2012; Zbl 1345.11045)], several families of genus \(2\) curves over \(\mathbb{Q}\) were given, whose Jacobians have \(\mathbb{Q}\)-rational torsion points of order \(m\) covering all values \(1\leq m\leq 30\). In the reviewed paper, the authors find out which of these Jacobians are absolutely simple. For certain curves they use a method of \textit{F. Leprévost} [J. Théor. Nombres Bordx. 7, No. 1, 283--306 (1995; Zbl 0864.14017)] based on the properties of the characteristic polynomial of the Frobenius endomorphism for the reduction of the curve over a certain finite field. In some cases where this method does not lead to a conclusion, the presence of zero divisors in the algebra of endomorphisms allows to conclude that the Jacobian is not absolutely simple.
reviewed by: Enric Nart
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 6225257
zbMATH Keywords: hyperelliptic curves
zbMATH Keywords: genus two curves
zbMATH Keywords: torsion points
zbMATH Keywords: Jacobians
MaRDI profile type: Publication
cites work: New orders of torsion points in Jacobians of curves of genus 2 over the rational number field
cites work: On the torsion problem in Jacobians of curves of genus 2 over the rational number field
cites work: A new local-global principle for quadratic function fields
cites work: Arithmetic of quadratic fields and torsion in Jacobians
cites work: Jacobians of some curves of genus 2: torsion and simplicity
full work available at URL: s1064562413030216
OpenAlex ID: W2052980309
description: scientific article; zbMATH DE number 6225257