Geometric and Functional Analysis

, Volume 26, Issue 5, pp 1449–1482 | Cite as

Fibrations with few rational points

  • D. LoughranEmail author
  • A. Smeets


We study the problem of counting the number of varieties in families which have a rational point. We give conditions on the singular fibres that force very few of the varieties in the family to contain a rational point, in a precise quantitative sense. This generalises and unifies existing results in the literature by Serre, Browning–Dietmann, Bright–Browning–Loughran, Graber–Harris–Mazur–Starr, et al.

Mathematics Subject Classification

14G05 14D10 11N36 11G35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bha14.
    M. Bhargava. A positive proportion of plane cubics fail the Hasse principle. arXiv:1402.1131.
  2. BGW15.
    M. Bhargava, B. Gross. and X. Wang, Pencils of quadrics and the arithmetic of hyperelliptic curves. Journal of the American Mathematical Society, to appear. arXiv:1310.7692.
  3. BCFJK15.
    M. Bhargava, J.E. Cremona, N.G. Jones, T. Fisher, and J.P. Keating. What is the probability that a random integral quadratic form in n variables has an integral zero? International Mathematics Research Notices, to appear. arXiv:1502.05992
  4. BCF16.
    Bhargava M., Cremona J.E., Fisher T.: The proportion of plane cubic curves over \({\mathbb{Q}}\) that everywhere locally have a point. International Journal of Number Theory 12(04), 1077–1092 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bou81.
    N. Bourbaki. Éléments de mathématique. Algèbre. Chapitres 4 à 7. In: Lecture Notes in Mathematics, Vol. 864. Masson, Paris (1981).Google Scholar
  6. BBL16.
    Bright M., Browning T., Loughran D.: Failures of weak approximation in families. Compositio Mathematica 152(71), 1435–1475 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. BD09.
    Browning T., Dietmann R.: Solubility of Fermat equations. Quadratic Forms–Algebra, Arithmetic, and Geometry, Contemporary Mathematics 493, 99–106 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. BN15.
    Browning T., Newton R.: The proportion of failures of the Hasse norm principle. Mathematika 62, 337–347 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. BD14.
    Brüdern J., Dietmann R.: Random Diophantine equations, I. Advances in Mathematics 256, 18–45 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. BLR90.
    S. Bosch, W. Lütkebohmert, and M. Raynaud. Néron Models. Springer, New York (1990).Google Scholar
  11. CT14.
    Colliot-Thélène J.-L.: Groupe de Brauer non ramifié d’espaces homogènes de tores. Journal de Théorie des Nombres de Bordeaux 26, 69–83 (2014)CrossRefGoogle Scholar
  12. CTSSD97.
    Colliot-Thélène J.-L., Skorobogatov A., Swinnerton-Dyer P.: Double fibres and double covers: paucity of rational points. Acta Arithmetica 79, 113–135 (1997)MathSciNetzbMATHGoogle Scholar
  13. Del80.
    Deligne P.: La conjecture de Weil, II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  14. DM15.
    Dietmann R., Marmon O.: Random Thue and Fermat equations. Acta Arithmetica 167, 189–200 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Eke91.
    Ekedahl T.: An infinite version of the Chinese remainder theorem. Comment. Math. Univ. St. Paul. 40, 53–59 (1991)MathSciNetzbMATHGoogle Scholar
  16. Fal84.
    G. Faltings. Complements to Mordell. In: Rational Points (Bonn, 1983/1984), Aspects Math., E6. Vieweg, Braunschweig (1984), pp. 203–227.Google Scholar
  17. Fu184.
    W. Fulton. Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 2. Springer-Verlag, Berlin (1984).Google Scholar
  18. FH91.
    W. Fulton, and J. Harris. Representation Theory. A First Course. Graduate Texts in Mathematics, Vol. 129. Springer-Verlag, New York (1991).Google Scholar
  19. GHMS04.
    T. Graber, J. Harris, B. Mazur and J. Starr. Jumps in Mordell-Weil rank and arithmetic surjectivity. In: Arithmetic of Higher-Dimensional Algebraic Varieties (Palo Alto, CA, 2002). Progr. Math., Vol. 226. Birkhäuser Boston, Boston, MA (2004), pp. 141–147.Google Scholar
  20. Gro66.
    A. Grothendieck. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math., 28 (1966).Google Scholar
  21. Guo95.
    Guo C.R.: On solvability of ternary quadratic forms. Proceedings of the London Mathematical Society 70(3), 241–263 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  22. HW16.
    Harpaz Y., Wittenberg O.: On the fibration method for zero-cycles and rational points. Annals of Mathematics 183, 229–295 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Har77.
    R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics, Vol. 52. Springer-Verlag, New York (1977).Google Scholar
  24. Hoo93.
    Hooley C.: On ternary quadratic forms that represent zero. Glasgow Mathematical Journal 35, 13–23 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Hoo07.
    Hooley C.: On ternary quadratic forms that represent zero. II. Journal für die reine und angewandte Mathematik 602, 179–225 (2007)MathSciNetzbMATHGoogle Scholar
  26. Kol90.
    V.A. Kolyvagin. Euler systems. In: The Grothendieck Festschrift, Vol. II, Progr. Math., Vol. 87. Birkhäuser Boston, Boston, MA (1990), pp. 435–483.Google Scholar
  27. Knu71.
    D. Knutson. Algebraic Spaces, Lecture Notes in Mathematics, Vol. 203. Springer-Verlag, Berlin-New York (1971).Google Scholar
  28. LL01.
    Q. Liu and D. Lorenzini. Special fibers of Néron models and wild ramification. Journal für die reine und angewandte Mathematik, 532 (2001), 179–222.Google Scholar
  29. Lou13.
    D. Loughran. The number of varieties in a family which contain a rational point. Journal of the European Mathematical Society, to appear. arXiv:1310.6219.
  30. OS98.
    Ono K., Skinner C.: Non-vanishing of quadratic twists of modular L-functions. Inventiones Mathematicae 134, 651–660 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  31. PV04.
    B. Poonen, and P. Voloch. Random Diophantine equations. Progr. Math. In: Arithmetic of Higher-Dimensional Algebraic Varieties, Vol. 226. Birkhäuser Boston, Boston, MA (2004), pp. 175–184.Google Scholar
  32. Pop14.
    F. Pop. Little Survey on Large Fields. In: A. Campillo, F.V. Kuhlmann and B. Teissier (eds.) Valuation Theory in Interaction, EMS Series of Congress Reports (2014), pp. 432–463.Google Scholar
  33. RS15.
    Rams S., Schütt M.: 64 lines on smooth quartic surfaces. Mathematische Annalen 362, 679–698 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Rom11.
    M. Romagny. Composantes connexes et irréductibles en familles. Manuscripta Mathematica, 136 (2011), 1–32.Google Scholar
  35. Sch79.
    S. Schanuel. Heights in number fields. Bulletin de la Société mathématique de France, 107 (1979), 433–449.Google Scholar
  36. Ser65.
    J.-P. Serre. Zeta and L functions. In: Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ.), Vol. 1963. Harper & Row, New York (1965), pp. 82–92.Google Scholar
  37. Ser90.
    J.-P. Serre. Spécialisation des éléments de \({{\rm Br_2}(\mathbb{Q}(T_1,\ldots,T_n))}\), Comptes Rendus de l’Académie des Sciences—Series. I Mathematica, 311 (1990), 397–402.Google Scholar
  38. Ser97a.
    J.-P. Serre. Galois Cohomology, corrected ed. Springer, New York (1997a).Google Scholar
  39. Ser97b.
    J.-P. Serre. Lectures on the Mordell-Weil Theorem, Third ed. Aspects of Mathematics. Friedr. Vieweg & Sohn, Braunschweig (1997b).Google Scholar
  40. Ser03.
    Serre J.-P.: On a theorem of Jordan. Bulletin of the American Mathematical Society 40, 429–440 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  41. Ser08.
    J.-P. Serre. Topics in Galois theory. Second edition. Research Notes in Mathematics, Vol. 1. A K Peters, Ltd., Wellesley, MA (2008).Google Scholar
  42. Ser12.
    J.-P. Serre. Lectures on \({N_X(p)}\). Chapman & Hall/CRC Research Notes in Mathematics, Vol. 11. CRC Press, Boca Raton, FL (2012).Google Scholar
  43. Sil94.
    J.H Silverman. Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, Vol. 151. Springer-Verlag, New York (1994).Google Scholar
  44. Sko96.
    Skorobogatov A.: Descent on fibrations over the projective line. American Journal of Mathematics 118, 905–923 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  45. Sko01.
    A. Skorobogatov. Torsors and Rational Points. Cambridge University press, Cambridge (2001).Google Scholar
  46. Sko15.
    A. Skorobogatov. Descent on toric fibrations. In: Arithmetic and Geometry. LMS Lecture Note Series, Vol. 420. Cambridge University Press, Cambridge (2015), pp. 422–435.Google Scholar
  47. Sof16.
    Sofos E.: Serre’s problem on the density of isotropic fibres in conic bundles. Proceedings of the London Mathematical Society 113, 1–28 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  48. Ten95.
    G. Tenenbaum. Introduction to Analytic and Probabilistic Number Theory. Cambridge University press, Cambridge (1995).Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.School of MathematicsUniversity of ManchesterManchesterUK
  2. 2.Radboud University Nijmegen, IMAPPNijmegenThe Netherlands
  3. 3.Departement Wiskunde KU LeuvenLeuvenBelgium

Personalised recommendations