A refined notion of arithmetically equivalent number fields, and curves   with isomorphic Jacobians
  We construct examples of number fields which are not isomorphic but for which their idele class groups are isomorphic. We also construct examples of projective algebraic curves which are not isomorphic but for which their Jacobian varieties are isomorphic. Both are constructed using an example in group 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 ${\mathbb Z}[G/H_2]$. 
Dipendra Prasad
math.NT math.AG