Conjectured Diophantine Estimates on Elliptic Curves

  • Serge Lang
Part of the Progress in Mathematics book series (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, RA(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
$${h_x} = {\text{log max}}\left( {\left| c \right|,\left| d \right|} \right)$$
.

Keywords

Zeta Function Elliptic Curve Elliptic Curf Riemann Zeta Function Riemann Hypothesis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [B-C-II-S]
    B. BIRCH, S. CHOWLA, M. HALL, A. SCHINZEL, On the difference x3–y2, Norske Vid. Selsk. Forrh. 38 (1965), pp. 65–69.MathSciNetMATHGoogle Scholar
  2. [B-S]
    B. BIRCH and N. STEPHENS, The parity of the rank of the Mordell-Weil group, Topology 5 (1966) pp. 295–299.MathSciNetMATHCrossRefGoogle Scholar
  3. [Ca]
    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.MathSciNetMATHCrossRefGoogle Scholar
  4. [Da]
    H. DAVENPORT, On f 3 (t) — g 2 (t), Kon. Norsk Vid. Selsk. For. Bd. 38 Nr. 20 (1965) pp. 86–87.MathSciNetMATHGoogle Scholar
  5. [De]
    V. A. DEMJANENKO, Estimate of the remainder term in Tate ‘s formula, Mat. Zam. 3 (1968), pp. 271–278.MathSciNetGoogle Scholar
  6. [Fe]
    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. [H]
    M. HALL, The diophantine equation x3–y2 = k, Computers in Number Theory, Academic Press (1971), pp. 173–198.Google Scholar
  8. [L 1]
    S. LANG, Elliptic curves: diophantine analysis, Springer Verlag, 1978.Google Scholar
  9. [L 2]
    S. LANG, On the zeta function of number fields, Invent. Math. 12 (1971), pp. 337–345.MATHGoogle Scholar
  10. [M]
    J. MANIN, Cyclotomic fields and modular curves, Russian Mathematical Surveys Vol. 26 No. 6, Published by the London Mathematical Society, Macmillan Journals Ltd, 1971Google Scholar
  11. [Mo]
    H. MONTGOMERY, Extreme values of the Riemann zeta function, Comment. Math. Helvetici 52 (1977), pp. 511–518.MATHCrossRefGoogle Scholar
  12. [Po]
    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. [Sch]
    W. SCHMIDT, Thue’s equation over function fields, J. Australian Math. Soc. (A) 25 (1978) pp. 385–422.MATHCrossRefGoogle Scholar
  14. [Se 1]
    E. SELMER, The diophantine equation axa + by 3 + cz 3, Acta Math. (1951), pp. 203–362.Google Scholar
  15. [Se 2]
    E. SELMER, Ditto, Completion of the tables, Acta. Math. (1954), pp. 191–197.Google Scholar
  16. [Sie]
    C. L. SIEGEL, Abschätzung von Einheiten, Nachr. Wiss. Göttingen (1969), pp. 71–86.Google Scholar
  17. [Sil 1]
    J. SILVERMAN, Lower bound for the canonical height on elliptic curves, Duke Math. J. Vol. 48 No. 3 (1981), pp. 633–648.MathSciNetMATHGoogle Scholar
  18. [Sil 2]
    J. SILVERMAN, Integer points and the rank of Thue Elliptic curves, Invent. Math. 66 (1962), pp. 395–404.Google Scholar
  19. [Sil 3]
    J. SILVERMAN, Heights and the Specialization map for families of abelian varieties,to appear.Google Scholar
  20. [St]
    H. STARK, Effective estimates of solutions of some Diophantine Euations, Acta. Arith. 24 (1973), pp. 251–259.MATHGoogle Scholar
  21. [Ste]
    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.MathSciNetMATHGoogle Scholar
  22. [Ta 1]
    J. TATE, The arithmetic of elliptic curves, Invent. Math. (1974), pp. 179–206.Google Scholar
  23. [Ta 2]
    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.MathSciNetGoogle Scholar
  24. [Ti]
    E. C. TITCHMARSH, The theory of the Riemann zeta function, Oxford Clarendon Press, 1951.Google Scholar
  25. [Zi]
    H. ZIMMER, On the difference of the Weil height and the NéronTate height, Math. Z. 147 (1976), pp. 35–51.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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