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THE PROPORTION OF FAILURES OF THE HASSE NORM PRINCIPLE

Part of: Homological methods (field theory) Arithmetic algebraic geometry Algebraic number theory: global fields Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  22 January 2016

T. D. Browning  and
R. Newton
T. D. Browning
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. email [email protected]
R. Newton
Affiliation:
Institut des Hautes Études Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, France email [email protected]
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Abstract

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.

MSC classification

Primary: 11G35: Varieties over global fields
Secondary: 11R37: Class field theory 12G05: Galois cohomology 14G05: Rational points
Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 337 - 347
Copyright
Copyright © University College London 2016 

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References

Bartels, H.-J., Zur Arithmetik von Konjugationsklassen in algebraischen Gruppen. J. Algebra 70 1981, 179–199.Google Scholar
Bartels, H.-J., Zur Arithmetik von Diedergruppenerweiterungen. Math. Ann. 256 1981, 465–473.Google Scholar
Bhargava, M., A positive proportion of plane cubics fail the Hasse principle, Preprint, 2014,arXiv:1402.1131.Google Scholar
de la Bretèche, R. and Browning, T. D., Contre-exemples au principe de Hasse pour certains tores coflasques. J. Théor. Nombres Bordeaux 26 2014, 25–44.CrossRefGoogle Scholar
Cassels, J. W. S. and Fröhlich, A., Algebraic Number Theory (Proc. Instructional Conf. Brighton, 1965), Academic (1967).Google Scholar
Colliot-Thélène, J.-L. and Kunyavskiǐ, B. È., Groupe de Brauer non ramifié des espaces principaux homogènes de groupes linéaires. J. Ramanujan Math. Soc. 13 1998, 37–49.Google Scholar
Gurak, S., On the Hasse norm principle. J. Reine Angew. Math. 299/300 1978, 16–27.Google Scholar
Hochschild, G. and Nakayama, T., Cohomology in class field theory. Ann. of Math. (2) 55 1952, 348–366.CrossRefGoogle Scholar
Hürlimann, W., On algebraic tori of norm type. Comment. Math. Helv. 59 1984, 539–549.Google Scholar
Hürlimann, W., H 3 and rational points on biquadratic bicyclic norm forms. Arch. Math. (Basel) 47 1986, 113–116.CrossRefGoogle Scholar
Loughran, D., The number of varieties in a family which contain a rational point, Preprint, 2014,arXiv:1310.6219.Google Scholar
Manin, Y. I., Le groupe de Brauer–Grothendieck en géométrie diophantienne. In Actes du Congrès International des Mathématiciens (Nice, 1970), Gauthier-Villars (Paris, 1971), 401–411.Google Scholar
Neukirch, J., Algebraic Number Theory, Springer (1991).Google Scholar
Neukirch, J., Schmidt, A. and Wingberg, K., Cohomology of Number Fields (Grundlehren der Mathematischen Wissenschaften 323 ), Springer (2008).Google Scholar
Odoni, R. W. K., The Farey density of norm subgroups of global fields. I. Mathematika 20 1973, 155–169.Google Scholar
Razar, M. J., Central and genus class fields and the Hasse norm theorem. Compos. Math. 35 1977, 281–298.Google Scholar
Sansuc, J.-J., Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327 1981, 12–80.Google Scholar
Serre, J.-P., Lectures on N X (p), CRC Press (Boca Raton, FL, 2012).Google Scholar
Wei, D., The unramified Brauer group of norm one tori. Adv. Math. 254 2014, 642–663.CrossRefGoogle Scholar