The difference between the Weil height and the canonical height on elliptic curves

Author:
Joseph H. Silverman

Journal:
Math. Comp. **55** (1990), 723-743

MSC:
Primary 11G05; Secondary 14G25

DOI:
https://doi.org/10.1090/S0025-5718-1990-1035944-5

MathSciNet review:
1035944

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Estimates for the difference of the Weil height and the canonical height of points on elliptic curves are used for many purposes, both theoretical and computational. In this note we give an explicit estimate for this difference in terms of the *j*-invariant and discriminant of the elliptic curve. The method of proof, suggested by Serge Lang, is to use the decomposition of the canonical height into a sum of local heights. We illustrate one use for our estimate by computing generators for the Mordell-Weil group in three examples.

**[1]**A. Bremner and J. W. S. Cassels,*On the equation*, Math. Comp.**42**(1982), 257-264. MR**726003 (85f:11017)****[2]**A. Brumer and K. Kramer,*The rank of elliptic curves*, Duke Math. J.**44**(1977), 715-743. MR**0457453 (56:15658)****[3]**J. P. Buhler, B. H. Gross, and D. B. Zagier,*On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank*3, Math. Comp.**44**(1985), 473-481. MR**777279 (86g:11037)****[4]**V. A. Dem'janenko,*An estimate of the remainder term in Tate's formula*, Mat. Zametki**3**(1968), 271-278. (Russian) MR**0227166 (37:2751)****[5]**M. Hindry and J. H. Silverman,*The canonical height and integral points on elliptic curves*, Invent. Math.**93**(1988), 419-450. MR**948108 (89k:11044)****[6]**S. Lang,*Elliptic curves*:*Diophantine analysis*, Springer-Verlag, New York, 1978. MR**518817 (81b:10009)****[7]**-,*Elliptic functions*, 2nd ed., Springer-Verlag, New York, 1987. MR**890960 (88c:11028)****[8]**-,*Conjectured Diophantine estimates on elliptic curves*, Progr. Math.**35**(1983), 155-171. MR**717593 (85d:11024)****[9]**J. H. Silverman,*Heights and the specialization map for families of abelian varieties*, J. Reine Angew. Math.**342**(1983), 197-211. MR**703488 (84k:14033)****[10]**-,*The arithmetic of elliptic curves*, Graduate Texts in Math., vol. 106, Springer-Verlag, New York, 1986. MR**817210 (87g:11070)****[11]**-,*Computing heights on elliptic curves*, Math. Comp.**51**(1988), 339-358. MR**942161 (89d:11049)****[12]**D. B. Zagier,*Large integral points on curves*, Math. Comp.**48**(1987), 425-436. MR**866125 (87k:11062)****[13]**H. Zimmer,*On the difference of the Weil height and the Néron-Tate height*, Math. Z.**174**(1976), 35-51. MR**0419455 (54:7476)**

Retrieve articles in *Mathematics of Computation*
with MSC:
11G05,
14G25

Retrieve articles in all journals with MSC: 11G05, 14G25

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1990-1035944-5

Article copyright:
© Copyright 1990
American Mathematical Society