Corresponding Abelian extensions of integrally equivalent number fields

Abstract

Extensive work has been done to determine necessary and sufficient conditions for a bijective correspondence of abelian extensions of number fields to force an isomorphism of the base fields. However, explicit examples of correspondences over non-isomorphic fields are rare. Integrally equivalent number fields admit an induced correspondence of abelian extensions. Studying this correspondence using idelic class field theory and linear algebra, we show that the corresponding extensions share features similar to those of arithmetically equivalent fields, and yet they are not generally weakly Kronecker equivalent. We also extend a group cohomological result of Arapura-Katz-McReynolds-Solapurkar and present geometric and arithmetic applications.