The Hasse norm principle for abelian extensions

Christopher Frei, Daniel Loughran, Rachel Newton
2018 American Journal of Mathematics  
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,
more » ... itions imposed, which we achieve using tools from harmonic analysis, building on work of Wright.
doi:10.1353/ajm.2018.0048 fatcat:lwypc3dtsjehhpff32t6dbwiaq