Corresponding Abelian Extensions of Integrally Equivalent Number Fields

Integrally equivalent number fields admit a natural correspondence of abelian extensions. It is natural to ask which aspects of the extensions' arithmetic are invariant under this bijection. We show that corresponding extensions share many of the same features enjoyed by arithmetically equivalent fields, despite not generally being arithmetically equivalent themselves. In the process of elaborating on this, we observe that arithmetically equivalent fields have the same odd K-groups, and we prove a simple lemma allowing for quick proofs of theorems characterizing Kronecker and weak Kronecker equivalence. We also extend a group cohomological result of Arapura et. al. and conclude with a geometric application.