Computing heights on elliptic curves
Author:
Joseph H. Silverman
Journal:
Math. Comp. 51 (1988), 339358
MSC:
Primary 11G05; Secondary 11D25, 11Y40, 14G25, 14K15
MathSciNet review:
942161
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Abstract 
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Abstract: We describe how to compute the canonical height of points on elliptic curves. Tate has given a rapidly converging series for Archimedean local heights over R. We describe a modified version of Tate's series which also converges over C, and give an efficient procedure for calculating local heights at nonArchimedean places. In this way we can calculate heights over number fields having complex embeddings. We also give explicit estimates for the tail of our series, and present several examples.
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J. H. Silverman, The NéronTate Height on Elliptic Curves, Ph.D. thesis, Harvard, 1981.
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B. L. van der Waerden, Algebra, 7th ed., Ungar, New York, 1970.
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534221 (80g:14024), http://dx.doi.org/10.1515/crll.1979.307308.221
 [1]
 J. P. Buhler, B. H. Gross & D. B. Zagier, "On the conjecture of Birch and SwinnertonDyer for an elliptic curve of rank 3," Math. Comp., v. 44, 1985, pp. 473481. MR 777279 (86g:11037)
 [2]
 D. Cox & S. Zucker, "Intersection numbers of sections of elliptic surfaces," Invent. Math., v. 53, 1979, pp. 144. MR 538682 (81i:14023)
 [3]
 P. Deligne, "Courbes Elliptiques: Formulaire (d'après J. Tate)," in Modular Functions of One Variable IV, Lecture Notes in Math., vol. 476, SpringerVerlag, Berlin and New York, 1975, pp. 5374. MR 0387292 (52:8135)
 [4]
 B. H. Gross, "Local heights on curves," in Arithmetic Geometry, SpringerVerlag, Berlin and New York, 1986, pp. 327340. MR 861983
 [5]
 B. H. Gross & D. B. Zagier, "Heegner points and derivatives of Lseries," Invent. Math., v. 84, 1986, 225320. MR 833192 (87j:11057)
 [6]
 S. Lang, Elliptic Curves: Diophantine Analysis, SpringerVerlag, Berlin and New York, 1978. MR 518817 (81b:10009)
 [7]
 S. Lang, Fundamentals of Diophantine Geometry, SpringerVerlag, Berlin and New York, 1983. MR 715605 (85j:11005)
 [8]
 M. Laska, "An algorithm for finding a minimal Weierstrass equation for an elliptic curve," Math. Comp., v. 38, 1982, pp. 257260. MR 637305 (84e:14033)
 [9]
 D. Masser & G. Wüstholz, "Fields of large transcendence degree generated by values of elliptic functions," Invent. Math., v. 72, 1983, pp. 407464. MR 704399 (85g:11060)
 [10]
 J. H. Silverman, The NéronTate Height on Elliptic Curves, Ph.D. thesis, Harvard, 1981.
 [11]
 J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Text 106, Springer, New York, 1986. MR 817210 (87g:11070)
 [12]
 J. H. Silverman, "A quantitative version of Siegel's theorem," J. Reine Angew. Math., v. 378, 1987, pp. 60100. MR 895285 (89g:11047)
 [13]
 J. H. Silverman, Elliptic Curve Calculator v. 5.05, a program for the Apple Macintosh computer, 1987.
 [14]
 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)
 [15]
 J. T. Tate, Letter to J.P. Serre, Oct. 1, 1979.
 [16]
 H. M. Tschöpe & H. G. Zimmer, "Computation of the NéronTate height on elliptic curves," Math. Comp., v. 48, 1987, pp. 351370. MR 866121 (87m:14025)
 [17]
 B. L. van der Waerden, Algebra, 7th ed., Ungar, New York, 1970.
 [18]
 D. B. Zagier, "Large integral points on curves," Math. Comp., v. 48, 1987, pp. 425436. MR 866125 (87k:11062)
 [19]
 H. G. Zimmer, "Quasifunctions on elliptic curves over local fields," J. Reine Angew. Math., v. 307/308, 1979, pp. 221246; "Corrections and remarks concerning quasifunctions on elliptic curves," J. Reine Angew. Math., v. 343, 1983, pp. 203211. MR 534221 (80g:14024)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809421614
PII:
S 00255718(1988)09421614
Article copyright:
© Copyright 1988 American Mathematical Society
