The Hasse norm principle for abelian extensions

Christopher Frei, Daniel Loughran, Rachel Newton

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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.
Original languageEnglish
Pages (from-to)1639-1685
Number of pages47
JournalAmerican Journal of Mathematics
Volume140
Issue number6
Early online date20 Nov 2018
DOIs
Publication statusPublished - 31 Dec 2018

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