Computation of the NéronTate height on elliptic curves
Authors:
Heinz M. Tschöpe and Horst G. Zimmer
Journal:
Math. Comp. 48 (1987), 351370
MSC:
Primary 14G25; Secondary 11D25, 11Y50, 14K15
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 NéronTate 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 NéronTate height of a series of rankone curves of BremnerCassels and of a series of ranktwo 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 NéronTate height on the group 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 SwinnertonDyer 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 ," Math. Comp., v. 42, 1984, pp. 257264. MR 726003 (85f:11017)
 [2]
 J. P. Buhler, B. H. Gross & D. B. Zagier, "On the conjecture of Birch and SwinnertonDyer for an ellliptic curve of rank 3," Math. Comp., v. 44, 1985, pp. 473481. MR 777279 (86g:11037)
 [3]
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 [4]
 H. G. Folz, Ein Beschränktheitssatz für die Torsion von 2defizienten elliptischen Kurven über algebraischen Zahlkörpern, Ph.D. Thesis, Saarbrücken, 1985.
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 [6]
 S. Lang, "Conjectured Diophantine estimates on elliptic curves," Progr. Math., v. 35, 1983, pp. 155171. MR 717593 (85d:11024)
 [7]
 S. Lang, Fundamentals of Diophantine Geometry, SpringerVerlag, Berlin and New York, 1983. MR 715605 (85j:11005)
 [8]
 E. Selmer, "The Diophantine equation ," Acta Math., v. 85, 1951, pp. 203362. MR 0041871 (13:13i)
 [9]
 J. H. Silverman, "Lower bound for the canonical height on elliptic curves," Duke Math. J., v. 48, 1981, pp. 633648. MR 630588 (82k:14043)
 [10]
 J. T. Tate, "The arithmetic of elliptic curves," Invent. Math., v. 23, 1974, pp. 179206. MR 0419359 (54:7380)
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 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, SpringerVerlag, Berlin and New York, 1975, pp. 3352. 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 NéronTate height," Math. Z., v. 147, 1976, pp. 3551. MR 0419455 (54:7476)
 [14]
 H. G. Zimmer, "Quasifunctions on elliptic curves over local fields," J. Reine Angew. Math., v. 307/308, 1979, pp. 221246. MR 534221 (80g:14024)
 [15]
 H. G. Zimmer, "Torsion points on elliptic curves over a global field," Manuscripta Math., v. 29, 1979, pp. 119145. MR 545037 (81a:14018)
 [16]
 Modular Functions of One Variable IV (B. J. Birch & W. Kuyk, eds.), Lecture Notes in Math., vol. 476, SpringerVerlag, Berlin and New York, 1975. MR 0376533 (51:12708)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198708661216
PII:
S 00255718(1987)08661216
Article copyright:
© Copyright 1987
American Mathematical Society
