Download PDF - Computing integral points on elliptic curvesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Computing integral points on elliptic curves

Authors 1, 2, 1

Affiliations

  1. Fachbereich 9 Mathematik, Universität des Saarlandes, D-66041 Saarbrücken, Germany
  2. Laboratory for Informatics, University Medical School, Nagyerdei Krt. 98, H-4028 Debrecen, Hungary

Bibliography

  1. [B] A. Baker, The Diophantine equation y²=ax³+bx²+cx+d, J. London Math. Soc. 43 (1968), 1-9.
  2. [D] S. David, Minorations de formes linéaires de logarithmes elliptiques, manuscript, Paris, 1993.
  3. [F] G. Frey, L-series of elliptic curves: results, conjectures and consequences, in: Proc. Ramanujan Centenn. Internat. Conf., Annamalainagar, December 1987, 31-43.
  4. [GPP] I. Gaàl, A. Pethő and M. Pohst, On the resolution of index form equations in biquadratic number fields II, J. Number Theory 38 (1991), 35-51.
  5. [GSch] I. Gaàl and N. Schulte, Computing all power integral bases of cubic number fields II, Math. Comp. 53 (1989), 689-696.
  6. [G] F. R. Gantmacher, The Theory of Matrices I, Chelsea, New York, N.Y., 1977.
  7. [GZ] J. Gebel and H. G. Zimmer, Computing the Mordell-Weil group of an elliptic curve over ℚ, in: Elliptic Curves and Related Topics, H. Kisilevsky and M. Ram Murty (eds.), CRM Proceedings and Lecture Notes, Amer. Math. Soc., Providence, RI, 1994, 61-83.
  8. [Gr] D. R. Grayson, The arithmetic-geometric mean, Arch. Math. (Basel) 52 (1989), 507-512.
  9. [HSi] A. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), 419-450.
  10. [L1] S. Lang, Diophantine approximation on toruses, Amer. J. Math. 86 (1964), 521-533.
  11. [L2] S. Lang, Elliptic Functions, Addison-Wesley, Reading, 1973.
  12. [L3] S. Lang, Elliptic Curves; Diophantine Analysis, Grundlehren Math. Wiss. 231, Springer, Berlin, 1978.
  13. [L4] S. Lang, Conjectured diophantine estimates on elliptic curves, in: Progr. Math. 35, Birkhäuser, Basel, 1983, 155-171.
  14. [LLL] A. K. Lenstra, H. W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515-534.
  15. [M] Yu. I. Manin, Cyclotomic fields and modular curves, Russian Math. Surveys 26 (6) (1971), 7-78.
  16. [Mz] B. Mazur, Rational points on modular curves, in: Modular Functions of One Variable V, Lecture Notes in Math. 601, Springer, Berlin, 1977, 107-148.
  17. [Me] J.-F. Mestre, Formules explicites et minorations de conducteurs de variétés algébriques, Compositio Math. 58 (1986), 209-232.
  18. [PS] A. Pethő und R. Schulenberg, Effektives Lösen von Thue Gleichungen, Publ. Math. Debrecen 34 (1987), 189-196.
  19. [PdW] A. Pethő and B. M. M. de Weger, Product of prime powers in binary recurrence sequences, Part I: The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation, Math. Comp. 47 (1986), 713-727.
  20. [Sch] W. Schmidt, Integer points on curves of genus 1, Compositio Math. 81 (1992), 33-59.
  21. [S] C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. (1929), 1-41.
  22. [Si1] J. H. Silverman, A quantitative version of Siegel's theorem, J. Reine Angew. Math. 378 (1981), 60-100.
  23. [Si2] J. H. Silverman, The difference between the Weil height and the canonical height on elliptic curves, Math. Comp. 55 (1990), 723-743.
  24. [SM] SIMATH, Manual, Saarbrücken, 1993.
  25. [St] R. P. Steiner, On Mordell's equation y²-k = x³. A problem of Stolarsky, Math. Comp. 46 (1986), 703-714.
  26. [ST] R. J. Stroeker and N. Tzanakis, Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67 (1994), 177-196.
  27. [TdW1] N. Tzanakis and B. M. M. de Weger, On the practical solution of the Thue equation, J. Number Theory 31 (1989), 99-132.
  28. [TdW2] N. Tzanakis and B. M. M. de Weger, How to explicitly solve a Thue-Mahler equation, Compositio Math. 84 (1992), 223-288.
  29. [dW] B. M. M. de Weger, Algorithms for diophantine equations, Ph.D. Thesis, Centrum voor Wiskunde en Informatica, Amsterdam, 1987.
  30. [Za] D. Zagier, Large integral points on elliptic curves, Math. Comp. 48 (1987), 425-436.
  31. [Zs] H. Zassenhaus, On Hensel factorization, I, J. Number Theory 1 (1969), 291-311.
  32. [Zi1] H. G. Zimmer, On the difference between the Weil height and the Néron-Tate height, Math. Z. 147 (1976), 35-51.
  33. [Zi2] H. G. Zimmer, On Manin's conditional algorithm, Bull. Soc. Math. France Mém. 49-50 (1977), 211-224.
  34. [Zi3] H. G. Zimmer, Generalization of Manin's conditional algorithm, in: SYMSAC 76, Proc. ACM Sympos. Symbolic Alg. Comp., Yorktown Heights, N.Y., 1976, 285-299.
  35. [Zi4] H. G. Zimmer, Computational aspects of the theory of elliptic curves, in: Number Theory and Applications, R. A. Mollin (ed.), Kluwer, 1989, 279-324.
  36. [Zi5] H. G. Zimmer, A limit formula for the canonical height of an elliptic curve and its application to height computations, in: Number Theory, R. A. Mollin (ed.), W. de Gruyter, Berlin, 1990, 641-659.
Acta Arithmetica
1994/68/2
Export to PBN, Opens in new tab
Pages:
171-192
Main language of publication
English
Received
1993-12-09
Published
1994
Exact and natural sciences