### Abstract

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 language | English |
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Pages (from-to) | 1639-1685 |

Number of pages | 47 |

Journal | American Journal of Mathematics |

Volume | 140 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Dec 2018 |

## Cite this

Frei, C., Loughran, D., & Newton, R. (2018). The Hasse norm principle for abelian extensions.

*American Journal of Mathematics*,*140*(6), 1639-1685. https://doi.org/10.1353/ajm.2018.0048