  Given a pair of number fields with isomorphic rings of adeles, we construct
bijections between objects associated to the pair. For instance we construct an
isomorphism of Brauer groups that commutes with restriction. We additionally
construct bijections between central simple algebras, maximal orders, various
Galois cohomology sets, and commensurability classes of arithmetic lattices in
simple, inner algebraic groups. We show that under certain conditions, lattices
corresponding to one another under our bijections have the same covolume and
pro-congruence completion. We also make effective a finiteness result of Prasad
and Rapinchuk.
