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Computation of the Néron-Tate height on elliptic curves


Authors: Heinz M. Tschöpe and Horst G. Zimmer
Journal: Math. Comp. 48 (1987), 351-370
MSC: Primary 14G25; Secondary 11D25, 11Y50, 14K15
DOI: https://doi.org/10.1090/S0025-5718-1987-0866121-6
MathSciNet review: 866121
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Abstract: Using Néron's reduction theory and a method of Tate, we develop a procedure for calculating the local and global Néron-Tate height on an elliptic curve over the rationals. The procedure is illustrated by means of two examples of Silverman and is then applied to calculate the global Néron-Tate height of a series of rank-one curves of Bremner-Cassels and of a series of rank-two curves of Selmer. In the latter case, the regulator is also computed, and a conjecture of S. Lang is investigated numerically. In dealing with the arithmetic of elliptic curves E over a global field K, the task arises of computing the Néron-Tate height on the group $ E(K)$ of rational points of E over K. Solving this task in an efficient manner is important, for instance, in view of calculations concerning the Birch and Swinnerton-Dyer conjecture (see [2]) or of the conjectures of Serge Lang [6]. The purpose of this note is to suggest a procedure for performing the necessary calculations.


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  • [1] A. Bremner & J. W. S. Cassels, "On the equation $ {Y^2} = {X^2}(X + p)$," Math. Comp., v. 42, 1984, pp. 257-264. MR 726003 (85f:11017)
  • [2] J. P. Buhler, B. H. Gross & D. B. Zagier, "On the conjecture of Birch and Swinnerton-Dyer for an ellliptic curve of rank 3," Math. Comp., v. 44, 1985, pp. 473-481. MR 777279 (86g:11037)
  • [3] J. W. S. Cassels, "Diophantine equations with special reference to elliptic curves," J. London Math. Soc., v. 41, 1966, pp. 193-291. MR 0199150 (33:7299)
  • [4] H. G. Folz, Ein Beschränktheitssatz für die Torsion von 2-defizienten elliptischen Kurven über algebraischen Zahlkörpern, Ph.D. Thesis, Saarbrücken, 1985.
  • [5] S. Lang, Elliptic Curves: Diophantine Analysis, Springer-Verlag, Berlin and New York, 1978. MR 518817 (81b:10009)
  • [6] S. Lang, "Conjectured Diophantine estimates on elliptic curves," Progr. Math., v. 35, 1983, pp. 155-171. MR 717593 (85d:11024)
  • [7] S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, Berlin and New York, 1983. MR 715605 (85j:11005)
  • [8] E. Selmer, "The Diophantine equation $ a{x^3} + b{y^3} + c{z^3}$," Acta Math., v. 85, 1951, pp. 203-362. MR 0041871 (13:13i)
  • [9] J. H. Silverman, "Lower bound for the canonical height on elliptic curves," Duke Math. J., v. 48, 1981, pp. 633-648. MR 630588 (82k:14043)
  • [10] J. T. Tate, "The arithmetic of elliptic curves," Invent. Math., v. 23, 1974, pp. 179-206. MR 0419359 (54:7380)
  • [11] J. T. Tate, "Algorithm for finding the type of a singular fibre in an elliptic pencil," in Modular Functions of One Variable IV, Lecture Notes in Math., vol. 476, Springer-Verlag, Berlin and New York, 1975, pp. 33-52. MR 0393039 (52:13850)
  • [12] J. T. Tate, Letter to J.-P. Serre, Oct. 1, 1979.
  • [13] H. G. Zimmer, "On the difference of the Weil height and the Néron-Tate height," Math. Z., v. 147, 1976, pp. 35-51. MR 0419455 (54:7476)
  • [14] H. G. Zimmer, "Quasifunctions on elliptic curves over local fields," J. Reine Angew. Math., v. 307/308, 1979, pp. 221-246. MR 534221 (80g:14024)
  • [15] H. G. Zimmer, "Torsion points on elliptic curves over a global field," Manuscripta Math., v. 29, 1979, pp. 119-145. MR 545037 (81a:14018)
  • [16] Modular Functions of One Variable IV (B. J. Birch & W. Kuyk, eds.), Lecture Notes in Math., vol. 476, Springer-Verlag, Berlin and New York, 1975. MR 0376533 (51:12708)

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DOI: https://doi.org/10.1090/S0025-5718-1987-0866121-6
Article copyright: © Copyright 1987 American Mathematical Society

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