The Hasse norm principle for abelian extensions
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.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 arXiv:
 arXiv:1508.02518
 Bibcode:
 2015arXiv150802518F
 Keywords:

 Mathematics  Number Theory;
 11R37 (primary);
 11R45;
 43A70;
 14G05;
 20G30 (secondary)
 EPrint:
 40 pages