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Constructing distinct curves with isomorphic Jacobians. (English) Zbl 0842.14019
Summary: We show that the hyperelliptic curves \(y^2= x^5+ x^3+ x^2- x-1\) and \(y^2= x^5- x^3+ x^2- x-1\) over the field with three elements are not geometrically isomorphic, and yet they have isomorphic Jacobian varieties. Furthermore, their Jacobians are absolutely simple. We present a method for constructing further such examples. We also present two curves of genus three, one hyperelliptic and one a plane quartic, that have isomorphic absolutely simple Jacobians.

MSC:
14H40 Jacobians, Prym varieties
14H52 Elliptic curves
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