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Sequences of Jacobian varieties with torsion divisors of quadratic order. (English) Zbl 1188.11034

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 \(n\).

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G20 Curves over finite and local fields
14H40 Jacobians, Prym varieties
14H45 Special algebraic curves and curves of low genus
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