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Rational points on cubic hypersurfaces that split off a form. With an appendix by J.-L. Colliot-Thélène

Part of: Forms and linear algebraic groups Diophantine equations

Published online by Cambridge University Press:  15 February 2010

T. D. Browning
T. D. Browning*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK (email: [email protected])
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Abstract

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Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over ℚ. We show that X(ℚ) is non-empty provided that the cubic form defining X can be written as the sum of two forms that share no common variables.

Keywords

Hasse principlecubic hypersurfacesHardy–Littlewood circle method

MSC classification

Secondary: 11D72: Equations in many variables 11E76: Forms of degree higher than two
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 853 - 885
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Baker, R. C., Diagonal cubic equations, II, Acta Arith. 53 (1989), 217–250.CrossRefGoogle Scholar
[2]Birch, B. J., Davenport, H. and Lewis, D. J., The addition of norm forms, Mathematika 9 (1962), 75–82.CrossRefGoogle Scholar
[3]Browning, T. D., Counting rational points on cubic hypersurfaces, Mathematika 54 (2007), 93–112.CrossRefGoogle Scholar
[4]Browning, T. D. and Heath-Brown, D. R., Integral points on cubic hypersurfaces, in Analytic number theory: essays in honour of Klaus Roth (Cambridge University Press, Cambridge, 2009), 75–90.Google Scholar
[5]Browning, T. D. and Heath-Brown, D. R., Rational points on quartic hypersurfaces, J. Reine Angew. Math. 629 (2009), 37–88.Google Scholar
[6]Bruce, J. W. and Wall, C. T. C., On the classification of cubic surfaces, J. Lond. Math. Soc. (2) 19 (1979), 245–256.CrossRefGoogle Scholar
[7]Cayley, A., A memoir on cubic surfaces, Phil. Trans. Roy. Soc. 159 (1869), 231–326.Google Scholar
[8]Colliot-Thélène, J.-L., The Brauer–Manin obstruction for complete intersections of dimension ≥3, appendix to [PV04].Google Scholar
[9]Colliot-Thélène, J.-L. and Salberger, P., Arithmetic on some singular cubic hypersurfaces, Proc. London Math. Soc. (3) 58 (1989), 519–549.CrossRefGoogle Scholar
[10]Colliot-Thélène et, J.-L. and Sansuc, J.-J., La descente sur les variétés rationnelles, II, Duke Math. J. 54 (1987), 375–492.CrossRefGoogle Scholar
[11]Colliot-Thélène, J.-L., Sansuc, J.-J. and Swinnerton-Dyer, P., Intersections of two quadrics and Châtelet surfaces. II, J. Reine Angew. Math. 374 (1987), 72–168.Google Scholar
[12]Coray, D. F., Algebraic points on cubic hypersurfaces, Acta Arith. 30 (1976), 267–296.CrossRefGoogle Scholar
[13]Coray, D. F., Cubic hypersurfaces and a result of Hermite, Duke Math. J. 54 (1987), 657–670.CrossRefGoogle Scholar
[14]Coray, D. F., Lewis, D. J., Shepherd-Barron, N. and Swinnerton-Dyer, P., Cubic threefolds with six double points, Number Theory in Progress, vol. 1 (Zakopane-Kościelisko, 1997) (de Gruyter, Berlin, 1999), 63–74.Google Scholar
[15]Coray, D. F. and Tsfasman, M. A., Arithmetic on singular Del Pezzo surfaces, Proc. London Math. Soc. (3) 57 (1988), 25–87.CrossRefGoogle Scholar
[16]Davenport, H., Cubic forms in thirty-two variables, Philos. Trans. R. Soc. Lond. Ser. A 251 (1959), 193–232.Google Scholar
[17]Davenport, H., Cubic forms in sixteen variables, Philos. Trans. R. Soc. London. Ser. A 272 (1963), 285–303.Google Scholar
[18]Davenport, H., Analytic methods for diophantine equations and diophantine inequalities, second edition (Cambridge University Press, Cambridge, 2005).CrossRefGoogle Scholar
[19]Fowler, J., A note on cubic equations, Math. Proc. Cambridge Philos. Soc. 58 (1962), 165–169.CrossRefGoogle Scholar
[20]Franke, J., Manin, Y. I. and Tschinkel, Y., Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), 421–435.CrossRefGoogle Scholar
[21]Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA2). Augmenté d’un exposé de Michèle Raynaud, in Séminaire de Géométrie Algébrique du Bois Marie, 1962, Documents Mathématiques (Paris), vol. 4 (Société Mathématique de France, Paris, 2005).Google Scholar
[22]Harris, J., Algebraic geometry (Springer, Berlin, 1992).CrossRefGoogle Scholar
[23]Heath-Brown, D. R., Cubic forms in ten variables, Proc. London. Math. Soc. (3) 47 (1983), 225–257.CrossRefGoogle Scholar
[24]Heath-Brown, D. R., Cubic forms in 14 variables, Invent. Math. 170 (2007), 199–230.CrossRefGoogle Scholar
[25]Hooley, C., On nonary cubic forms, J. Reine Angew. Math. 386 (1988), 32–98.Google Scholar
[26]Hooley, C., On nonary cubic forms. II, J. Reine Angew. Math. 415 (1991), 95–165.Google Scholar
[27]Kollár, J., Unirationality of cubic hypersurfaces, J. Inst. Math. Jussieu 1 (2002), 467–476.CrossRefGoogle Scholar
[28]Lewis, D. J., Diophantine problems: solved and unsolved, Number theory and applications (Kluwer, Dordrecht, 1989), 103–121.Google Scholar
[29]Manin, Y. I., Cubic forms, North-Holland Mathematical Library, second edition, vol. 4, (North-Holland, Amsterdam, 1986).Google Scholar
[30]Poonen, B. and Voloch, J. F., Random diophantine equations, in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progress in Mathematics, vol. 226 (Birkhäuser, Boston, MA, 2004), 175–184.CrossRefGoogle Scholar
[31]Schläfli, L., On the distribution of surfaces of the third order into species, Philos. Trans. R. Soc. 153 (1864), 193–247.Google Scholar
[32]Segre, B., On arithmetical properties of singular cubic surfaces, J. London Math. Soc. (2) 19 (1944), 84–91.CrossRefGoogle Scholar
[33]Segre, C., Suella varieta cubica con dieci punti doppi dello spazio a quattro dimensioni, Atta della R. Accademia della Scienae di Torino 22 (1886–87), 791–801.Google Scholar
[34]Skolem, T., Einige Bemerkungen über die Auffindung der rationalen Punkte auf gewissen algebraischen Gebilden, Math. Z. 63 (1955), 295–312.CrossRefGoogle Scholar
[35]Vaughan, R. C., The Hardy–Littlewood method, second edition (Cambridge University Press, Cambridge, 1997).CrossRefGoogle Scholar
[36]Wooley, T., On Weyl’s inequality, Hua’s lemma and exponential sums over binary forms, Duke Math. J. 100 (1999), 373–423.CrossRefGoogle Scholar