Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T17:30:23.309Z Has data issue: false hasContentIssue false
  • English
  • Français

MANY CUBIC SURFACES CONTAIN RATIONAL POINTS

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  29 November 2017

T. D. Browning
T. D. Browning*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. email t.d.browning@bristol.ac.uk
Get access

Abstract

Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.

MSC classification

Primary: 14G05: Rational points
Secondary: 11G05: Elliptic curves over global fields 11G35: Varieties over global fields 14G25: Global ground fields
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 818 - 839
Copyright
Copyright © University College London 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bhargava, M., A positive proportion of plane cubics fail the Hasse principle. Preprint, 2014, arXiv:1402.1131.Google Scholar
Bhargava, M., Cremona, J. and Fisher, T., The proportion of plane cubic curves over ℚ that everywhere locally have a point. Int. J. Number Theory 12 2016, 1077–1092.Google Scholar
Bhargava, M., Elkies, N. and Shnidman, A., The average size of the 3-isogeny Selmer groups of elliptic curves $y^{2}=x^{3}+k$ . Preprint, 2016, arXiv:1610.05759.Google Scholar
Bright, M., Obstructions to the Hasse principle in families. Preprint, 2016, arXiv:1607.01303.Google Scholar
Bright, M., Browning, T. D. and Loughran, D., Failures of weak approximation in families. Compos. Math. 152 2016, 1435–1475.Google Scholar
Cassels, J. W. S. and Guy, M. J. T., On the Hasse principle for cubic surfaces. Mathematika 13 1966, 111–120.Google Scholar
Colliot-Thélène, J.-L. and Sansuc, J.-J., La descente sur les variétés rationnelles. In Journées de géométrie algébrique d’Angers (Juillet 1979), Sijthoff and Noordhoff (Alphen aan den Rijn and Germantown, MD, 1980), 223–237.Google Scholar
Davenport, H., On the class-number of binary cubic forms I and II. J. Lond. Math. Soc. 26 1951, 183–198.Google Scholar
Davenport, H. and Heilbronn, H., On the density of discriminants of cubic fields II. Proc. R. Soc. Lond. Ser. A 322 1971, 405–420.Google Scholar
Gross, B. H. and Zagier, D., Heegner points and derivatives of L-series. Invent. Math. 84 1986, 225–320.CrossRefGoogle Scholar
Harpaz, Y. and Skorobogatov, A., Hasse principle for Kummer varieties. Algebra Number Theory 10 2016, 813–841.CrossRefGoogle Scholar
Heath-Brown, D. R. and Moroz, B. Z., On the representation of primes by cubic polynomials in two variables. Proc. Lond. Math. Soc. (3) 88 2004, 289–312.Google Scholar
Jahnel, J., Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society (Providence, RI, 2014).Google Scholar
Kolyvagin, V. A., Euler systems. In The Grothendieck Festschrift, Vol. II (Progress in Mathematics 87), Birkhäuser (Boston, MA, 1990), 435–483.Google Scholar
Kriz, D. and Li, C., Goldfeld’s conjecture and congruences between Heegner points. Preprint, 2016, arXiv:1609.06687.Google Scholar
Manin, Y. I., Le groupe de Brauer–Grothendieck en géométrie diophantienne. In Actes du Congrès International des Mathématiciens (Nice, 1970), Vol. 1, Gauthier-Villars (Paris, 1971), 401–411.Google Scholar
Satgé, P., Un analogue du calcul de Heegner. Invent. Math. 87 1987, 425–439.Google Scholar
Segre, B., On arithmetical properties of singular cubic surfaces. J. Lond. Math. Soc. 19 1944, 84–91.Google Scholar
Silverman, J. H., The Arithmetic of Elliptic Curves, Springer (1986).CrossRefGoogle Scholar
Swinnerton-Dyer, P., The Brauer group of cubic surfaces. Math. Proc. Cambridge Philos. Soc. 113 1993, 449–460.CrossRefGoogle Scholar
Swinnerton-Dyer, P., The solubility of diagonal cubic surfaces. Ann. Sci. Éc. Norm. Supér. (4) 34 2001, 891–912.Google Scholar