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.
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References
Browning, T.D., Heath-Brown, D.R.: Quadratic polynomials represented by norm forms. Geom. Funct. Anal. 22(5), 1124–1190 (2012)
Bourbaki, N.: Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et Anneaux Semi-Simples. Actualités Sci. Ind., vol. 1261. Hermann, Paris (1958)
Colliot-Thélène, J.-L.: Points rationnels sur les fibrations. In: Higher Dimensional Varieties and Rational Points (Budapest, 2001). Bolyai Soc. Math. Stud., vol. 12, pp. 171–221. Springer, Berlin (2003)
Colliot-Thélène, J.-L., Harari, D., Skorobogatov, A. N.: Valeurs d’un polynôme à une variable représentées par une norm. In: Number Theory and Algebraic Geometry. London Math. Soc. Lecture Note Ser., vol. 303, pp. 69–89. Cambridge Univ. Press, Cambridge (2003)
Colliot-Thélène, J.-L., Salberger, P.: Arithmetic on some singular cubic hypersurfaces. Proc. London Math. Soc. (3) 58(3), 519–549 (1989)
Colliot-Thélène, J.-L., Sansuc, J.J.: La descente sur les variétés rationnelles. II. Duke Math. J. 54(2), 375–492 (1987)
Colliot-Thélène, J.-L., Sansuc, J.J.: Principal homogeneous spaces under flasque tori: applications. J. Algebra 106(1), 148–205 (1987)
Colliot-Thélène, J.-L., Sansuc, J.-J., Swinnerton-Dyer, P.: Intersections of two quadrics and Châtelet surfaces. I. J. Reine Angew. Math. 373, 37–107 (1987)
Colliot-Thélène, J.-L., Sansuc, J.-J., Swinnerton-Dyer, P.: Intersections of two quadrics and Châtelet surfaces. II. J. Reine Angew. Math. 374, 72–168 (1987)
Colliot-Thélène, J.-L., Swinnerton-Dyer, P.: Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties. J. Reine Angew. Math. 453, 49–112 (1994)
Colliot-Thélène, J.-L., Skorobogatov, A.N.: Descent on fibrations over \({\bf P}^1_k\) revisited. Math. Proc. Cambridge Philos. Soc. 128(3), 383–393 (2000)
Colliot-Thélène, J.-L., Skorobogatov, A.N., Swinnerton-Dyer, P.: Rational points and zero-cycles on fibred varieties: Schinzel’s hypothesis and Salberger’s device. J. Reine Angew. Math. 495, 1–28 (1998)
Fouvry, E., Iwaniec, H.: Gaussian primes. Acta Arith. 79(3), 249–287 (1997)
Harari, D.: Flèches de spécialisations en cohomologie étale et applications arithmétiques. Bull. Soc. Math. France 125(2), 143–166 (1997)
Hasse, H.: Die Normenresttheorie relativ-Abelscher Zahlkörper als Klassenkörpertheorie im Kleinen. J. Reine Angew. Math. 162, 145–154 (1930)
Heath-Brown, D.R., Skorobogatov, A.N.: Rational solutions of certain equations involving norms. Acta Math. 189(2), 161–177 (2002)
Kunyavskiĭ, B.È.: Arithmetic properties of three-dimensional algebraic tori. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 116, 102–107 (1982)
Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften, vol. 323, 2nd edn. Springer, Berlin (2008)
Sansuc, J.-J.: Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327, 12–80 (1981)
Sansuc, J.-J.: Le principe de Hasse normique dans le cas diédral non galoisien. Unpublished (1981)
Jones, M.S.: A note on a theorem of Heath-Brown and Skorobogatov. Q. J. Math. 64(4), 1239–1251 (2013)
Skorobogatov, A.N.: Beyond the Manin obstruction. Invent. Math. 135(2), 399–424 (1999)
Skorobogatov, A.N.: Torsors and Rational Points. Cambridge Tracts in Mathematics, vol. 144. Cambridge University Press, Cambridge (2001)
Schindler, D., Skorobogatov, A.N.: Norms as products of linear polynomials. J. Lond. Math. Soc. (2) 89(2), 559–580 (2014)
Wei, D.: On the equation \(N_{K/k}(\Xi )=P(t)\). Proc. Lond. Math. Soc. (2014). doi:10.1112/plms/pdu035
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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Derenthal, U., Smeets, A. & Wei, D. Universal torsors and values of quadratic polynomials represented by norms. Math. Ann. 361, 1021–1042 (2015). https://doi.org/10.1007/s00208-014-1106-7
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DOI: https://doi.org/10.1007/s00208-014-1106-7