Computing integral points on elliptic curves

J. Gebel; A. Pethő; H. G. Zimmer

Acta Arithmetica (1994)

  • Volume: 68, Issue: 2, page 171-192
  • ISSN: 0065-1036

How to cite

top

J. Gebel, A. Pethő, and H. G. Zimmer. "Computing integral points on elliptic curves." Acta Arithmetica 68.2 (1994): 171-192. <http://eudml.org/doc/206653>.

@article{J1994,
author = {J. Gebel, A. Pethő, H. G. Zimmer},
journal = {Acta Arithmetica},
keywords = {Mordell-Weil group; height; LLL-reduced basis; elliptic logarithms; linear forms in elliptic logarithms; Néron-Tate height; naive height},
language = {eng},
number = {2},
pages = {171-192},
title = {Computing integral points on elliptic curves},
url = {http://eudml.org/doc/206653},
volume = {68},
year = {1994},
}

TY - JOUR
AU - J. Gebel
AU - A. Pethő
AU - H. G. Zimmer
TI - Computing integral points on elliptic curves
JO - Acta Arithmetica
PY - 1994
VL - 68
IS - 2
SP - 171
EP - 192
LA - eng
KW - Mordell-Weil group; height; LLL-reduced basis; elliptic logarithms; linear forms in elliptic logarithms; Néron-Tate height; naive height
UR - http://eudml.org/doc/206653
ER -

References

top
  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. Zbl0726.11023
  5. [GSch] I. Gaàl and N. Schulte, Computing all power integral bases of cubic number fields II, Math. Comp. 53 (1989), 689-696. Zbl0677.10013
  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. Zbl0809.14024
  8. [Gr] D. R. Grayson, The arithmetic-geometric mean, Arch. Math. (Basel) 52 (1989), 507-512. Zbl0686.14040
  9. [HSi] A. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), 419-450. Zbl0657.14018
  10. [L1] S. Lang, Diophantine approximation on toruses, Amer. J. Math. 86 (1964), 521-533. Zbl0142.29601
  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. Zbl0488.12001
  15. [M] Yu. I. Manin, Cyclotomic fields and modular curves, Russian Math. Surveys 26 (6) (1971), 7-78. Zbl0266.14012
  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. Zbl0623.10011
  20. [Sch] W. Schmidt, Integer points on curves of genus 1, Compositio Math. 81 (1992), 33-59. Zbl0747.11026
  21. [S] C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. (1929), 1-41. Zbl56.0180.05
  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. Zbl0729.14026
  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. Zbl0601.10011
  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. Zbl0805.11026
  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. Zbl0657.10014
  28. [TdW2] N. Tzanakis and B. M. M. de Weger, How to explicitly solve a Thue-Mahler equation, Compositio Math. 84 (1992), 223-288. Zbl0773.11023
  29. [dW] B. M. M. de Weger, Algorithms for diophantine equations, Ph.D. Thesis, Centrum voor Wiskunde en Informatica, Amsterdam, 1987. Zbl0625.10013
  30. [Za] D. Zagier, Large integral points on elliptic curves, Math. Comp. 48 (1987), 425-436. Zbl0611.10008
  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. Zbl0303.14003
  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. Zbl0738.14020

Citations in EuDML Documents

top
  1. Andrej Dujella, A parametric family of elliptic curves
  2. R. J. Stroeker, On the sum of consecutive cubes being a perfect square
  3. Emanuel Herrmann, Attila Pethö, S -integral points on elliptic curves - Notes on a paper of B. M. M. de Weger
  4. J. R. Merriman, S. Siksek, N. P. Smart, Explicit 4-descents on an elliptic curve
  5. Hayder Raheem Hashim, Szabolcs Tengely, Lucas sequences and repdigits
  6. Benjamin M. M. de Weger, S -integral solutions to a Weierstrass equation
  7. Roelof J. Stroeker, Benjamin M. M. de Weger, Solving elliptic diophantine equations: the general cubic case
  8. N. Tzanakis, Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations
  9. Sinnou David, Minorations de formes linéaires de logarithmes elliptiques
  10. Éric Gaudron, Formes linéaires de logarithmes effectives sur les variétés abéliennes

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.