The Hasse norm principle for abelian extensions

Christopher Frei, Daniel Loughran, Rachel Newton

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We study the distribution of abelian extensions of bounded discriminantof a number eld k which fail the Hasse norm principle. For example,we classify those nite abelian groups G for which a positive proportion ofG-extensions of k fail the Hasse norm principle. We obtain a similar classi cation for the failure of weak approximation for the associated norm onetori. These results involve counting abelian extensions of bounded discriminantwith in nitely many local conditions imposed, which we achieve usingtools from harmonic analysis, building on work of Wright.
Original languageEnglish
Pages (from-to)1639-1685
Number of pages47
JournalAmerican Journal of Mathematics
Issue number6
Publication statusPublished - 1 Dec 2018

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