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Mathematische Annalen

, Volume 361, Issue 3–4, pp 1021–1042 | Cite as

Universal torsors and values of quadratic polynomials represented by norms

  • Ulrich DerenthalEmail author
  • Arne Smeets
  • Dasheng Wei
Article

Abstract

Let \(K/k\) be an extension of number fields, and let \(P(t)\) be a quadratic polynomial over \(k\). Let \(X\) be the affine variety defined by \(P(t) = N_{K/k}(\mathbf {z})\). We study the Hasse principle and weak approximation for \(X\) in three cases. For \([K:k]=4\) and \(P(t)\) irreducible over \(k\) and split in \(K\), we prove the Hasse principle and weak approximation. For \(k=\mathbb {Q}\) with arbitrary \(K\), we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For \([K:k]=4\) and \(P(t)\) irreducible over \(k\), we determine the Brauer group of smooth proper models of \(X\). In a case where it is non-trivial, we exhibit a counterexample to weak approximation.

Mathematics Subject Classification

14G05 (11D57, 14F22) 

Notes

Acknowledgments

The first named author was supported by Grant DE 1646/2–1 of the Deutsche Forschungsgemeinschaft and grant 200021_124737/1 of the Schweizer Nationalfonds. The second named author was supported by a PhD fellowship of the Research Foundation—Flanders (FWO). The third named author was supported by National Key Basic Research Program of China (Grant No. 2013CB834202) and National Natural Science Foundation of China (Grant Nos. 11371210 and 11321101). This collaboration was supported by the Center for Advanced Studies of LMU München. We thank T. D. Browning, J.-L. Colliot-Thélène, C. Demarche and B. Kunyavskiĭ for useful discussions and remarks. Finally, we thank the referee for his suggestions for improvement.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für Algebra, Zahlentheorie und Diskrete MathematikLeibniz Universität HannoverHannoverGermany
  2. 2.Departement WiskundeKU LeuvenLeuvenBelgium
  3. 3.Département de MathématiquesUniversité Paris-Sud 11OrsayFrance
  4. 4.Academy of Mathematics and System Science, CASBeijingPeople’s Republic of China

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