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Explicit methods for the Hasse norm principle and applications to An and Sn extensions

Part of: Algebraic number theory: global fields Forms and linear algebraic groups Linear algebraic groups and related topics Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  22 April 2021

ANDRÉ MACEDO  and
RACHEL NEWTON
ANDRÉ MACEDO
Affiliation:
University of Reading, Department of Mathematics and Statistics, Pepper Lane, Whiteknights, Reading RG6 6AX, UK e-mails: c.a.v.macedo@pgr.reading.ac.uk, r.d.newton@reading.ac.uk
RACHEL NEWTON
Affiliation:
University of Reading, Department of Mathematics and Statistics, Pepper Lane, Whiteknights, Reading RG6 6AX, UK e-mails: c.a.v.macedo@pgr.reading.ac.uk, r.d.newton@reading.ac.uk
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Abstract

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Let K/k be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for K/k and the defect of weak approximation for the norm one torus \[R_{K/k}^1{\mathbb{G}_m}\] . We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of K/k has symmetric or alternating Galois group.

MSC classification

Primary: 14G05: Rational points
Secondary: 11E72: Galois cohomology of linear algebraic groups 11R37: Class field theory 20G30: Linear algebraic groups over global fields and their integers
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 3 , May 2022 , pp. 489 - 529
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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