Abstract :
We show that the hyperelliptic curvesy2=x5+x3+x2−x−1 andy2=x5−x3+x2−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
