Skip to main content
SpringerLink
Log in

Conjectured Diophantine Estimates on Elliptic Curves

  • Chapter
Arithmetic and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 35))

Abstract

Let A be an elliptic curve defined over the rational numbers Q. Mordell’s theorem asserts that the group of points A(Q) is finitely generated. Say {P 1,..., P r } is a basis of A(Q) modulo torsion. Explicit upper bounds for the heights of elements in such a basis are not known. The purpose of this note is to conjecture such bounds for a suitable basis. Indeed, R⊗A(Q) is a vector space over R with a positive definite quadratic form given by the Néron-Tate height: if A is defined by the equation

$${y^2} = {x^3} + ax + b,a,b \in {\text{Z}}$$

, and P = (x, y) is a rational point with x = c/d written as a fraction in lowest form, then one defines the x-height

$${h_x} = {\text{log max}}\left( {\left| c \right|,\left| d \right|} \right)$$

.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. BIRCH, S. CHOWLA, M. HALL, A. SCHINZEL, On the difference x3–y2, Norske Vid. Selsk. Forrh. 38 (1965), pp. 65–69.

    MathSciNet  MATH  Google Scholar 

  2. B. BIRCH and N. STEPHENS, The parity of the rank of the Mordell-Weil group, Topology 5 (1966) pp. 295–299.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. W. CASSELS, The rational solutions of the Diophantine equation Y2 = X3 — D, Acta Math. 82 (1950) pp. 243–273. Addenda and corrigenda to the above, Acta Math. 84 (1951), p. 299.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. DAVENPORT, On f 3 (t) — g 2 (t), Kon. Norsk Vid. Selsk. For. Bd. 38 Nr. 20 (1965) pp. 86–87.

    MathSciNet  MATH  Google Scholar 

  5. V. A. DEMJANENKO, Estimate of the remainder term in Tate ‘s formula, Mat. Zam. 3 (1968), pp. 271–278.

    MathSciNet  Google Scholar 

  6. N. I. FELDMAN, An effective refinement of the exponent in Liouville’s theorem, Izv. Akad. Nauk 35 (1971) pp. 973990, AMS Transl. (1971), pp. 985–1002.

    Google Scholar 

  7. M. HALL, The diophantine equation x3–y2 = k, Computers in Number Theory, Academic Press (1971), pp. 173–198.

    Google Scholar 

  8. S. LANG, Elliptic curves: diophantine analysis, Springer Verlag, 1978.

    Google Scholar 

  9. S. LANG, On the zeta function of number fields, Invent. Math. 12 (1971), pp. 337–345.

    MATH  Google Scholar 

  10. J. MANIN, Cyclotomic fields and modular curves, Russian Mathematical Surveys Vol. 26 No. 6, Published by the London Mathematical Society, Macmillan Journals Ltd, 1971

    Google Scholar 

  11. H. MONTGOMERY, Extreme values of the Riemann zeta function, Comment. Math. Helvetici 52 (1977), pp. 511–518.

    Article  MATH  Google Scholar 

  12. V. D. PODSYPANIN, On the equation x 3 = y2 + Az 6,Math. Sbornik 24 (1949), pp. 391–403 (See also Cassels’ corrections in [Ca]).

    Google Scholar 

  13. W. SCHMIDT, Thue’s equation over function fields, J. Australian Math. Soc. (A) 25 (1978) pp. 385–422.

    Article  MATH  Google Scholar 

  14. E. SELMER, The diophantine equation axa + by 3 + cz 3, Acta Math. (1951), pp. 203–362.

    Google Scholar 

  15. E. SELMER, Ditto, Completion of the tables, Acta. Math. (1954), pp. 191–197.

    Google Scholar 

  16. C. L. SIEGEL, Abschätzung von Einheiten, Nachr. Wiss. Göttingen (1969), pp. 71–86.

    Google Scholar 

  17. J. SILVERMAN, Lower bound for the canonical height on elliptic curves, Duke Math. J. Vol. 48 No. 3 (1981), pp. 633–648.

    MathSciNet  MATH  Google Scholar 

  18. J. SILVERMAN, Integer points and the rank of Thue Elliptic curves, Invent. Math. 66 (1962), pp. 395–404.

    Google Scholar 

  19. J. SILVERMAN, Heights and the Specialization map for families of abelian varieties,to appear.

    Google Scholar 

  20. H. STARK, Effective estimates of solutions of some Diophantine Euations, Acta. Arith. 24 (1973), pp. 251–259.

    MATH  Google Scholar 

  21. N. STEPHENS, The diophantine equation X 3 +Y 3 = DZ 3 and the conjectures of Birch-Swinnerton Dyer, J. reine angew. Math. 231 (1968), pp. 121–162.

    MathSciNet  MATH  Google Scholar 

  22. J. TATE, The arithmetic of elliptic curves, Invent. Math. (1974), pp. 179–206.

    Google Scholar 

  23. J. TATE, Algorithm for determining the Type of a Singular Fiber in an Elliptic Pencil, Modular Functions of One Variable IV, Springer Lecture Notes 476 (Antwerp Conference) (1972), pp. 33–52.

    MathSciNet  Google Scholar 

  24. E. C. TITCHMARSH, The theory of the Riemann zeta function, Oxford Clarendon Press, 1951.

    Google Scholar 

  25. H. ZIMMER, On the difference of the Weil height and the NéronTate height, Math. Z. 147 (1976), pp. 35–51.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yale University, Box 2155, Yale Station, New Haven, Connecticut, 06520, USA

    Professor Serge Lang

Authors
  1. Professor Serge Lang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematics Department, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Michael Artin

  2. Mathematics Department, Harvard University, 02138, Cambridge, MA, USA

    John Tate

Rights and permissions

Reprints and permissions

Copyright information

© 1983 Springer Science+Business Media New York

About this chapter

Cite this chapter

Lang, S. (1983). Conjectured Diophantine Estimates on Elliptic Curves. In: Artin, M., Tate, J. (eds) Arithmetic and Geometry. Progress in Mathematics, vol 35. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-9284-3_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-9284-3_7

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-0-8176-3132-1

  • Online ISBN: 978-1-4757-9284-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation