Sequences of Jacobian varieties with torsion divisors of quadratic order
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 \(n\).
scientific article; zbMATH DE number 5635557