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We construct examples of number fields which are not isomorphic but for whichtheir idele class groups are isomorphic. We also construct examples ofprojective algebraic curves which are not isomorphic but for which theirJacobian varieties are isomorphic. Both are constructed using an example ingroup theory provided by Leonard Scott of a finite group $G$ and subgroups$H_1$ and $H_2$ which are not conjugate in $G$ but for which the $G$-module${\mathbb Z}[G/H_1]$ is isomorphic to ${\mathbb Z}[G/H_2]$.

A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians