Skip to main content
SpringerLink
Log in

Counting rational points on cubic curves

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a combination of the “determinant method” with an m-descent on the curve.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bombieri E, Pila J. The number of integral points on arcs and ovals. Duke Math J, 1989, 59: 337–357

    Article  MATH  MathSciNet  Google Scholar 

  2. Davenport H. Indefinite quadratic forms in many variables (II). Proc London Math Soc (3), 1958, 8: 109–126

    Article  MATH  MathSciNet  Google Scholar 

  3. David S. Points de petite hauteur sur les courbes elliptiques. J Number Theory, 1997, 64: 104–129

    Article  MATH  MathSciNet  Google Scholar 

  4. Ellenberg J, Venkatesh A. On uniform bounds for rational points on nonrational curves. Int Math Res Not, 2005, 35: 2163–2181

    Article  MathSciNet  Google Scholar 

  5. Heath-Brown D R. Counting rational points on cubic surfaces. Astérisque, 1998, 251: 13–30

    MathSciNet  Google Scholar 

  6. Heath-Brown D R. The density of rational points on curves and surfaces. Ann of Math (2), 2002, 155: 553–595

    Article  MATH  MathSciNet  Google Scholar 

  7. Kawamata Y. A generalization of Kodaira-Ramanujam’s vanishing theorem. Math Ann, 1982, 261: 43–46

    Article  MATH  MathSciNet  Google Scholar 

  8. Kodaira K. On a differential-geometric method in the theory of analytic stacks. Proc Nat Acad Sci USA, 1953, 39: 1268–1273

    Article  MATH  MathSciNet  Google Scholar 

  9. Mazur B. Modular curves and the Eisenstein ideal. Inst Hautes études Sci Publ Math No, 1977, 47: 33–186

    Article  MATH  MathSciNet  Google Scholar 

  10. Mestre J F. Courbes elliptiques et formules explicites. In: Seminar on Number Theory, Paris, 1981-82, Progr Math, 38. Boston, MA: Birkhäuser, 1983, 179–187

    Google Scholar 

  11. Viehweg E. Vanishing theorems. J Reine Angew Math, 1982, 335: 1–8

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, UK

    Roger Heath-Brown & Damiano Testa

Authors
  1. Roger Heath-Brown
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Damiano Testa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roger Heath-Brown.

Additional information

Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heath-Brown, R., Testa, D. Counting rational points on cubic curves. Sci. China Math. 53, 2259–2268 (2010). https://doi.org/10.1007/s11425-010-4037-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-010-4037-0

Keywords

MSC(2000)

Search

Navigation