On Mordell's equation

Compositio Mathematica

Volumn 110, Issue 3, 1998, Pages 335-367

  Gebel, J ^a   Petho A ^b   Zimmer, H G ^c  

^a Gina Cody School of Engineering and Computer Science   (Canada)

^b UNIVERSITY OF DEBRECEN   (Hungary)

^c SAARLAND UNIVERSITY   (Germany)

Author keywords

Birch Swinnerton Dyer conjecture; Elliptic curve; Elliptic logarithm; Height; Integral points; L series; Mordell Weil group; Rank; Regulator; Torsion

Indexed keywords

EID: 0000253128     PISSN: 0010437X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1000281602647     Document Type: Article

References (48)

1. Birch, B. J. and Stephens, N. M.: The parity of the rank of the Mordell-Weil group. Topology 5 (1966) 295-299.
2. 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.
3. 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.
4. Brumer, A.: The average rank of elliptic curves I. Invent. Math. 109 (1992) 445-472.
5. 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.
6. Cassels, J. W. S.: An introduction to the geometry of numbers. Die Grundlehren der mathematischen Wissenschaften, Band 99, Springer-Verlag, Berlin und New York.
7. Coates, J. and Wiles, A.: On the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 39 (1977) 233-251.
8. Cremona, J. E.: Algorithms for modular elliptic curves. Cambridge University Press (1992).
9. David, S.: Minorations de formes linéaires de logarithmes elliptiques. Mémoires de la Societé Mathématique de France, Numéro 62, Nouvelle série 1995. Supplément au Bulletin de la S.M.F. Tome 123, 1995, fascicule 3.
10. 3 = k and a conjecture of M. Hall. (Russian), Mat. Zametki 32 (1982) 273-275. Corr. 36 (1984) 457-458.
11. 3 = k and a conjecture of M. Hall. (Russian), Mat. Zametki 32 (1982) 273-275. Corr. 36 (1984) 457-458.
12. Engl. transl.: Math. Notes 32 617-618, 36 726.
13. 3. J. Number Theory 2 (1970) 310-321.
14. Fueter, R.: Über kubische diophantische Gleichungen. Comm. Math. Helv. 2 (1930) 69-89.
15. 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.
16. Gebel, J., Petho, A. and Zimmer, H. G.: Computing integral points on elliptic curves. Acta Arith. 68 (1994) 171-192.
17. Gebel, J., Petho, 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.
18. Gebel, J. and Zimmer, H. G.: Computing the Mordell-Weil group of an elliptic curve over ℚ. CRM Proc. and Lect. Notes 4 (1994) 61-83.
19. Gordon, B. and Mohanty, S. P.: On a theorem of Delaunay and some related results. Pacific J. Math. 68 (1977) 399-409.
20. Grayson, D. R.: The arithmetic-geometric mean. Arch. Math. 52 (1989) 507-512.
21. Greenberg, R.: On the Birch and Swinnerton-Dyer conjecture. Invent. Math. 72 (1983) 241-265.
22. Gross, B. and Zagier, D.: Heegner points and derivatives of L-series. Invent. Math. 84 (1986) 225-320.
23. 2 = k. Computers in Number Theory, A. O. L. Atkin and B. J. Birch eds., Academic Press (1971) 173-198.
24. Kolyvagin, V. A.: Finiteness of E(1ℚ) and III for a subclass of Weil curves. (Russian) Izv. Acad. Nauk USSR 52 (1988) 522-540.
25. Kolyvagin, V. A.: Euler Systems. The Grothendieck Festschrift, 2 Progr. in Math. 87 (1990) 474-499, Birkhäuser Boston.
26. 2 = k. Math. Comp. 20 (1966), 322-325.
27. Lang, S.: Conjectured diophantine estimates on elliptic curves. Progr. in Math. 35 (1983) 155-171, Birkhäuser, Basel.
28. Lenstra, A. K., Lenstra, H. W. and Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261 (1982) 515-534.
29. 3. Bowling Green State University, Bowling Green, Ohio (1973).
30. Manin, Y. I.: Cyclotomic fields and modular curves. Russian Math. Surveys 26(6) (1971) 7-78.
31. Mestre, J. F.: Formules explicites et minorations de conducteurs de variétés algébriques. Compos. Math. 58 (1986) 209-232.
32. Mordell, L. J.: Diophantine equations. Academic Press (1969) 238-254.
33. Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 64 (1981) 455-470.
34. 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.
35. Silverman, J. H.: The difference between the Weil height and the canonical height on elliptic curves. Math. Comp. 55 (1990) 723-743.
36. Sprindžuk, V. G.: Classical diophantine equations. Lect. Notes in Math., Vol. 1559 (1993) 113, Springer-Verlag, Berlin und New York.
37. Stark, H. M.: Effective estimates of solutions of some diophantine equations, Acta Arith. 24 (1973) 251-259.
38. 3: a problem of Stolarsky. Math. Comp. 46 (1986) 703-714.
39. 3. Indian J. Pure Appl. Math. 22 (1991) 13-21.
40. 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.
41. Stroeker, R. J. and Tzanakis, N.: Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. Acta Arith. 67 (1994), 177-196.
42. de Weger, B. M. M.: Algorithms for diophantine equations. Ph.D. Thesis, Centrum voor Wiskunde en Informatica, Amsterdam (1987).
43. 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.
44. Zagier, D.: Large integral points on elliptic curves. Math. Comp. 48 (1987) 425-436.
45. 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).
46. Zimmer, H. G.: On Manin's conditional algorithm. Bull. Soc. Math. France, Mémoire 49-50 (1977) 211-224.
47. Zimmer, H. G.: Generalization of Manin's conditional algorithm. SYMSAC 76. Proc. ACM Sympos. Symbolic Alg. Comp., Yorktown Heights, N.Y. (1976) 285-299.
48. 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.