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Compositio Mathematica

, Volume 110, Issue 3, pp 335–367 | Cite as

On Mordell's Equation

  • J. Gebel
  • A. Pethö
  • H. G. Zimmer
Article

Abstract

In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordell's Equation y2=x3+k for 0 ≠ k ∈ Z and draw some conclusions from our numerical findings. In fact we solve Mordell's Equation in Z for all integers k within the range 0 < | k | ≤ 10 000 and partially extend the computations to 0 < | k | ≤ 100 000. For these values of k, the constant in Hall's conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.

elliptic curve Mordell-Weil group rank torsion integral points L-series Birch and Swinnerton-Dyer conjecture height regulator elliptic logarithm 

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References

  1. [BSt]
    Birch, B. J. and Stephens, N. M.: The parity of the rank of the Mordell-Weil group. Topology5 (1966) 295-299.Google Scholar
  2. [BSD]
    Birch, B. J. and Swinnerton-Dyer, H. P. F.: Notes on elliptic curves I and II. J. Reine Angew. Math.212 (1963) 7-15, 218 (1965) 79-208.Google Scholar
  3. [Br]
    Brumer, A.: The average rank of elliptic curves I. Invent. Math. 109 (1992) 445-472.Google Scholar
  4. [BMG]
    Brumer, A. and McGuiness, O.: The behavior of the Mordell-Weil group of elliptic curves. Bull. Amer. Math. Soc. (N. S.)23 (1990) 375-382.Google Scholar
  5. [Ca]
    Cassels, J. W. S.: An introduction to the geometry of numbers. Die Grundlehren der mathematischen Wissenschaften, Band 99, Springer-Verlag, Berlin und New York.Google Scholar
  6. [CW]
    Coates, J. and Wiles, A.: On the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 39 (1977) 233-251.Google Scholar
  7. [Cr]
    Cremona, J. E.: Algorithms formodular elliptic curves. Cambridge University Press (1992).Google Scholar
  8. [Dav]
    David, S.: Minorations de formes linéaires de logarithmes elliptiques. Mémoires de la SocietéMathématique de France, Numéro 62, Nouvelle série 1995. Supplément au Bulletin de la S. M. F. Tome 123, 1995, fascicule 3.Google Scholar
  9. [Dan]
    Danilov, L. V.: The diophantine equation y 2- x 3 = k and a conjecture of M. Hall. (Russian), Mat. Zametki32 (1982) 273-275. Corr. 36 (1984) 457-458. Engl. transl.: Math. Notes32 617-618, 36 726.Google Scholar
  10. [EEPSS]
    Ellison, W. J., Ellison, F., Pesek, F., Stahl, C. E. and Stall, D. S.: The diophantine equation y 2 + k = x 3. J. Number Theory2 (1970) 310-321.Google Scholar
  11. [Fu]
    Fueter, R.: Über kubische diophantische Gleichungen. Comm. Math. Helv.2 (1930) 69-89.Google Scholar
  12. [G]
    Gebel, J.: Bestimmung aller ganzen und S-ganzen Punkte auf elliptischen Kurven über den rationalen Zahlen mit Anwendung auf die Mordellschen Kurven. PhD Thesis, Universität des Saarlandes, Saarbrücken 1996.Google Scholar
  13. [GPZ1]
    Gebel, J., Pethő, A. and Zimmer, H. G.: Computing integral points on elliptic curves. Acta Arith. 68 (1994) 171-192.Google Scholar
  14. [GPZ2]
    Gebel, J., Pethő, A. and Zimmer, H. G.: Computing S-integral points on elliptic curves. Algorithmic Number Theory, Ed. H. Cohen in Proceedings ANTS-II, Bordeaux 1996, Lecture Notes in Comp. Sci., Vol. 1122 (1996) 157-171, Springer-Verlag.Google Scholar
  15. [GZi]
    Gebel, J. and Zimmer, H. G.: Computing the Mordell-Weil group of an elliptic curve over ℚ. CRM Proc. and Lect. Notes4 (1994) 61-83.Google Scholar
  16. [GM]
    Gordon, B. and Mohanty, S. P.: On a theorem of Delaunay and some related results. Pacific J. Math. 68 (1977) 399-409.Google Scholar
  17. [Gra]
    Grayson, D. R.: The arithmetic-geometric mean. Arch. Math. 52 (1989) 507-512.Google Scholar
  18. [Gre]
    Greenberg, R.: On the Birch and Swinnerton-Dyer conjecture. Invent. Math. 72 (1983) 241-265.Google Scholar
  19. [GZa]
    Gross, B. and Zagier, D.: Heegner points and derivatives of L-series. Invent. Math. 84 (1986) 225-320.Google Scholar
  20. [Ha]
    Hall, M.: The diophantine equation x 3 - y 2 = k. Computers in Number Theory, A. O. L. Atkin and B. J. Birch eds., Academic Press (1971) 173-198.Google Scholar
  21. [Ko1]
    Kolyvagin, V. A.: Finiteness of E (ℚ) and III for a subclass of Weil curves. (Russian) Izv. Acad. Nauk USSR52 (1988) 522-540.Google Scholar
  22. [Ko2]
    Kolyvagin, V. A.: Euler Systems. The Grothendieck Festschrift, 2 Progr. in Math. 87 (1990) 474-499, Birkhäuser Boston.Google Scholar
  23. [LJB]
    Lal, M., Jones, M. F. and Blundon, W. J.: Numerical solutions of x 3 - y 2 = k. Math. Comp. 20 (1966), 322-325.Google Scholar
  24. [La]
    Lang, S.: Conjectured diophantine estimates on elliptic curves. Progr. in Math. 35 (1983) 155-171, Birkhäuser, Basel.Google Scholar
  25. [LLL]
    Lenstra, A. K., Lenstra, H. W. and Lovász, L.: Factoring polynomials with rational coeffi-cients. Math. Ann. 261 (1982) 515-534.Google Scholar
  26. [LF]
    London, J. and Finkelstein, M.: On Mordell's Equation x 3 - y 2 = k. Bowling Green State University, Bowling Green, Ohio (1973).Google Scholar
  27. [Ma]
    Manin, Y. I.: Cyclotomic fields and modular curves. Russian Math. Surveys26(6) (1971) 7-78.Google Scholar
  28. [Me]
    Mestre, J. F.: Formules explicites et minorations de conducteurs de variétés algébriques. Compos. Math. 58 (1986) 209-232.Google Scholar
  29. [Mo]
    Mordell, L. J.: Diophantine equations. Academic Press (1969) 238-254.Google Scholar
  30. [Ru1]
    Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 64 (1981) 455-470.Google Scholar
  31. [Ru2]
    Rubin, K.: The work of Kolyvagin on the arithmetic of elliptic curves. Arith. of Compl. Manifolds, Proc., Lect. Notes in Math., Vol. 1399 (1988) 128-136, Springer-Verlag, Berlin und New York.Google Scholar
  32. [Si]
    Silverman, J. H.: The difference between the Weil height and the canonical height on elliptic curves. Math. Comp. 55 (1990) 723-743.Google Scholar
  33. [Sp]
    Sprindžuk, V. G.: Classical diophantine equations. Lect. Notes in Math., Vol. 1559 (1993) 113, Springer-Verlag, Berlin und New York.Google Scholar
  34. [Sta]
    Stark, H. M.: Effective estimates of solutions of some diophantine equations, Acta Arith. 24 (1973) 251-259.Google Scholar
  35. [Ste]
    Steiner, R. P.: On Mordell's Equation x 3 - y 2 = k: a problem of Stolarsky. Math. Comp. 46 (1986) 703-714.Google Scholar
  36. [SM]
    Steiner, R. P. and Mohanty, S. P.: On Mordell's Equation x 3 - y 2 = k. Indian J. Pure Appl. Math. 22 (1991) 13-21.Google Scholar
  37. [StT]
    Stewart, C. L. and Top, J.: On ranks of twists of elliptic curves and power-free values of binary forms. J. Amer. Math. Soc. 8 (1995) 943-973.Google Scholar
  38. [STz]
    Stroeker, R. J. and Tzanakis, N.: Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. Acta Arith. 67 (1994), 177-196.Google Scholar
  39. [dW]
    de Weger, B. M. M.: Algorithms for diophantine equations. Ph. D. Thesis, Centrum voor Wiskunde en Informatica, Amsterdam (1987).Google Scholar
  40. [Ta]
    Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. Modular Functions in One Variable IV, Lect. Notes in Math., Vol. 476 (1975) 33-52, Springer-Verlag, Berlin und New York.Google Scholar
  41. [Za]
    Zagier, D.: Large integral points on elliptic curves. Math. Comp. 48 (1987) 425-436.Google Scholar
  42. [ZK]
    Zagier, D. and Kramarz, G.: Numerical investigations related to the L-series of certain elliptic curves. J. Indian Math. Soc. 52 (1987) 51-60, (Ramanujan Centennary volume).Google Scholar
  43. [Zi1]
    Zimmer, H. G.: On Manin's conditional algorithm. Bull. Soc. Math. France, Mémoire 49-50 (1977) 211-224.Google Scholar
  44. [Zi2]
    Zimmer, H. G.: Generalization of Manin's conditional algorithm. SYMSAC 76. Proc. ACM Sympos. Symbolic Alg. Comp., Yorktown Heights, N. Y. (1976) 285-299.Google Scholar
  45. [Zi3]
    Zimmer, H. G.: A limit formula for the canonical height of an elliptic curve and its application to height computations. Number Theory. Ed. R. A. Mollin, W. de Gruyter Verlag, Berlin and New York (1990) 641-659.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • J. Gebel
    • 1
  • A. Pethö
    • 2
  • H. G. Zimmer
    • 3
  1. 1.Department of Computer Science and EngineeringConcordia UniversityMontréalCanada
  2. 2.Laboratory of InformaticsUniversity of MedicineDebrecenHungary
  3. 3.Fachbereich 9 MathematikUniversität des SaarlandesSaarbrücken;Germany

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