A note on the Hasse norm principle

TL;DR

This paper proves the existence of a density for finite abelian group extensions satisfying the Hasse norm principle when ordered by discriminant, extending previous results by removing certain cyclicity assumptions.

Contribution

It establishes the existence of the density of $A$-extensions satisfying the Hasse norm principle without the cyclicity restriction on $A/A[ ext{prime}]$, generalizing earlier work.

Findings

Density of $A$-extensions satisfying the Hasse norm principle exists.

The result holds when extensions are ordered by discriminant.

Extends previous work by removing cyclicity assumptions.

Abstract

Let $A$ be a finite, abelian group. We show that the density of $A$-extensions satisfying the Hasse norm principle exists, when the extensions are ordered by discriminant. This strengthens earlier work of Frei--Loughran--Newton \cite{FLN}, who obtained a density result under the additional assumption that $A/A[\ell]$ is cyclic with $\ell$ denoting the smallest prime divisor of $\# A$.