The Hasse norm principle for abelian extensions

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