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Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Tomasz Jędrzejak (2013)

Acta Arithmetica

Consider the families of curves C n , A : y ² = x + A x and C n , A : y ² = x + A where A is a nonzero rational. Let J n , A and J n , A denote their respective Jacobian varieties. The torsion points of C 3 , A ( ) and C 3 , A ( ) are well known. We show that for any nonzero rational A the torsion subgroup of J 7 , A ( ) is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to J 7 , A ( ) [ 2 ] (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for J 3 , A (A ≠ 4) and J 5 , A . We also almost...

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