The Hasse norm principle for abelian extensions

TL;DR

This paper investigates the distribution of abelian extensions of number fields that fail the Hasse norm principle, classifying groups based on the proportion of such failures and analyzing related local-global principles.

Contribution

It provides a classification of finite abelian groups for which a positive proportion of extensions fail the Hasse norm principle, using harmonic analysis techniques.

Findings

Classified abelian groups with positive failure proportion

Counted abelian extensions with local conditions

Connected failures to weak approximation for norm one tori

Abstract

We study the distribution of abelian extensions of bounded discriminant of a number field $k$ which fail the Hasse norm principle. For example, we classify those finite abelian groups $G$ for which a positive proportion of $G$-extensions of $k$ fail the Hasse norm principle. We obtain a similar classification for the failure of weak approximation for the associated norm one tori. These results involve counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which we achieve using tools from harmonic analysis, building on work of Wright.