The Hasse norm principle for abelian extensions

@article{Frei2015TheHN,
  title={The Hasse norm principle for abelian extensions},
  author={Christopher Frei and Christopher Daniel Rachel Loughran and Christopher Daniel Rachel Newton},
  journal={American Journal of Mathematics},
  year={2015},
  volume={140},
  pages={1639 - 1685}
}
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… Expand
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