Constructing distinct curves with isomorphic Jacobians

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. (C) 1996 Academic Press, Inc.