On certain torsion subgroups of Jacobians of hyperelliptic curves of genus \(g \geq 1\)
type: Item
instance of: scholarly article
title: On certain torsion subgroups of Jacobians of hyperelliptic curves of genus \(g \geq 1\)
zbMATH Open document ID: 0872.14016
DOI: 10.1007/BF02678180
author: Franck Leprévost
published in: Manuscripta Mathematica
publication date: 1997-04-08T00:00:00Z
full work available at URL: https://eudml.org/doc/156251
review text: Given two integers \(d,r\geq 1\) and an abelian variety \(A\) of dimension \(g\) defined over a number field \(K\) of degree \(d\) the strong boundedness conjecture asks for a bound for the torsion subgroup \(A(K)_{\text{tor}}\) of the group of rational points \(A(K)\) of \(A\) in terms of \(g\) and \(d\). In the case where \(g=1\) this conjecture was established by \textit{B. Mazur} [ Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 33-186 (1978; Zbl 0394.14008)], \textit{S. Kamienny} [Invent. Math. 109, No. 2, 221-229 (1992; Zbl 0773.14016)], \textit{S. Kamienny} and \textit{B. Mazur} [ in: Columbia Univ. Number Theory Semin., New York 1992, Astérisque 228, 81-100 (1995; Zbl 0846.14012)] and \textit{D. Abramovich} [ibid., Astérisque 228, 5-17 (1995; Zbl 0846.14013)]. For \(g=1\) and any \(d\geq 1\) this result was established by \textit{L. Merel} [Invent. Math. 124, No. 1-3, 437-449 (1996)]. However, in the case \(g\geq 2\), for no \(d\geq 1\) this conjecture has been verified. In this paper the author addresses this question for the Jacobian varieties of hyperelliptic curves defined over the field of rational numbers. More precisely, he shows that there exists a family of hyperelliptic curves of genus \(g\) defined over \(\mathbb Q\) whose Jacobian varieties have a rational point of order \(2g(2g+1)\). For \(l=2g^2+5g+5\), he constructs a family of genus \(g\) hyperelliptic curves defined over \(\mathbb Q\) such that their Jacobian varieties have a rational point of order \(l\), \(l/2\) or \(l/4\). Another family of hyperelliptic curves defined over \(\mathbb Q\) (which does not belong to any of the previous ones) is constructed such that the Jacobian varieties of the curves of the family have a rational point of order \(l\).
reviewed by: Amílcar Pacheco
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 980716
zbMATH Keywords: genus
zbMATH Keywords: hyperelliptic curves
zbMATH Keywords: strong boundedness conjecture
zbMATH Keywords: group of rational points
zbMATH Keywords: Jacobian varieties of hyperelliptic curves
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 980716