Sequences of Jacobian varieties with torsion divisors of quadratic order
type: Item
instance of: scholarly article
title: Sequences of Jacobian varieties with torsion divisors of quadratic order
zbMATH Open document ID: 1188.11034
author: Roger D. Patterson
author: Alfred J. van der Poorten
author: Hugh C. Williams
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2009-11-18T00:00:00Z
review text: In this paper the authors deal with the rational torsion structure of abelian varieties of dimension \(> 1\). Using the continuous fraction expansion over function fields, they study the hyperelliptic curves \(C_n\) defined by the equation \[ d^2Y^2 = (qrf^n+(mf^k-l)/q)^2+4lrf^n \;\;(n = 1,2,\ldots) \] where each term on the right is a nontrivial polynomial in \(X\) with \(f\) irreducible, \(r\), \(l\) and \(m\) squarefree, and \(d\) chosen such that \(Y^2\) is squarefree, and so that \(\gcd (qr, ml) =1\), \(\gcd (f, qrml) = 1\), \(\gcd (m,l) = 1\), and \(q|(mf^k-l)\). The main result of the paper is that the divisor at infinity of the Jacobian of \(C_n\) is torsion of quadratic order in \(n\).
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 5635557
zbMATH Keywords: Jacobian variety
zbMATH Keywords: rational torsion
zbMATH Keywords: torsion divisor
zbMATH Keywords: continued fraction expansion
zbMATH Keywords: hyperelliptic curves
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.7169/facm/1229696580
OpenAlex ID: W2044406154
DOI: 10.7169/FACM/1229696580
description: scientific article; zbMATH DE number 5635557