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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
##### Keywords:
distinct curves with isomorphic Jacobians
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