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Weak approximation on the norm one torus

Part of: Multiplicative number theory Linear algebraic groups and related topics Exponential sums and character sums Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  06 May 2024

P. Koymans Open the ORCID record for P. Koymans [Opens in a new window]  and
N. Rome Open the ORCID record for N. Rome [Opens in a new window]
P. Koymans
Affiliation:
Institute for Theoretical Studies, ETH Zurich, 8006 Zurich, Switzerland peter.koymans@eth-its.ethz.ch
N. Rome
Affiliation:
Graz University of Technology, Institute of Analysis and Number Theory, Kopernikusgasse 24/II, 8010 Graz, Austria rome@tugraz.at
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Abstract

For any abelian group $A$, we prove an asymptotic formula for the number of $A$-extensions $K/\mathbb {Q}$ of bounded discriminant such that the associated norm one torus $R_{K/\mathbb {Q}}^1 \mathbb {G}_m$ satisfies weak approximation. We are also able to produce new results on the Hasse norm principle and to provide new explicit values for the leading constant in some instances of Malle's conjecture.

Keywords

weak approximationnorm one torusHasse norm principleMalle's conjecturearithmetic statisticslarge sieve

MSC classification

Primary: 11N45: Asymptotic results on counting functions for algebraic and topological structures 14G05: Rational points 11L40: Estimates on character sums 20G30: Linear algebraic groups over global fields and their integers

Information

Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 6 , June 2024 , pp. 1304 - 1348
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Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

Dedicated to Adèle Koymans-Funcken

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