Number fields with prescribed norms

Christopher Frei, Daniel Loughran, Rachel Newton

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We study the distribution of extensions of a number field k with fixed abelian Galois group G, from which a given finite set of elements of k are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for 100% of G-extensions of k, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.
Original languageEnglish
Pages (from-to)138-181
Number of pages44
JournalCommentarii Mathematici Helvetici
Issue number1
Publication statusPublished - 14 Apr 2022

Bibliographical note

Funding Information:
Acknowledgements. Our work on this project began at Michael Stoll’s workshop Rational Points 2017 held at Franken-Akademie Schloss Schney. Substantial progress was made when the third author visited the other two at the University of Manchester, and also at the workshop Rational and Integral Points via Analytic and Geometric Methods organised by Tim Browning, Ulrich Derenthal and Cecília Salgado at Hotel Hacienda Los Laureles, Oaxaca. We are very grateful to the organisers of both workshops, to the funding bodies, and to the staff at all three places for providing us with excellent working conditions. We thank the anonymous referee for a meticulous reading of an earlier draft of this paper and for several suggestions that helped improve it. The first-named author is supported by EPSRC-grant EP/T01170X/2. The second-named author is supported by EPSRC grant EP/R021422/1 and UKRI Future Leaders Fellowship MR/V021362/1. The third-named author is supported by EPSRC grant EP/S004696/1 and UKRI Future Leaders Fellowship MR/T041609/1.

Publisher Copyright:
© 2022 Swiss Mathematical Society


  • Hasse norm principle
  • class field theory
  • harmonic analysis
  • rational points on varieties

ASJC Scopus subject areas

  • General Mathematics


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