Corresponding Abelian Extensions of Integrally Equivalent Number Fields

TL;DR

This paper explores the correspondence of abelian extensions between integrally equivalent number fields, revealing similarities to arithmetically equivalent fields and extending cohomological results with applications in geometry and arithmetic.

Contribution

It provides a detailed analysis of abelian extension correspondences in integrally equivalent fields, including new insights and extensions of existing cohomological results.

Findings

Extensions share features with arithmetically equivalent fields

Extensions are not generally weakly Kronecker equivalent

Extended a group cohomological result with new applications

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 et. al. and present geometric and arithmetic applications.