## 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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