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

Silverman, Joseph H.

doi:10.1090/S0025-5718-1990-1035944-5

The difference between the Weil height and the canonical height on elliptic curves
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Math. Comp. 55 (1990), 723-743 Request permission

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.

References

A. Bremner and J. W. S. Cassels, On the equation $Y^{2}=X(X^{2}+p)$, Math. Comp. 42 (1984), no. 165, 257–264. MR 726003, DOI 10.1090/S0025-5718-1984-0726003-4
Armand Brumer and Kenneth Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), no. 4, 715–743. MR 457453
Joe P. Buhler, Benedict H. Gross, and Don B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank $3$, Math. Comp. 44 (1985), no. 170, 473–481. MR 777279, DOI 10.1090/S0025-5718-1985-0777279-X
V. A. Dem′janenko, An estimate of the remainder term in Tate’s formula, Mat. Zametki 3 (1968), 271–278 (Russian). MR 227166
M. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), no. 2, 419–450. MR 948108, DOI 10.1007/BF01394340
Serge Lang, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 231, Springer-Verlag, Berlin-New York, 1978. MR 518817
Serge Lang, Elliptic functions, 2nd ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. With an appendix by J. Tate. MR 890960, DOI 10.1007/978-1-4612-4752-4
Serge Lang, Conjectured Diophantine estimates on elliptic curves, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 155–171. MR 717593
Joseph H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. MR 703488, DOI 10.1515/crll.1983.342.197
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Joseph H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), no. 183, 339–358. MR 942161, DOI 10.1090/S0025-5718-1988-0942161-4
Don Zagier, Large integral points on elliptic curves, Math. Comp. 48 (1987), no. 177, 425–436. MR 866125, DOI 10.1090/S0025-5718-1987-0866125-3
Horst Günter Zimmer, On the difference of the Weil height and the Néron-Tate height, Math. Z. 147 (1976), no. 1, 35–51. MR 419455, DOI 10.1007/BF01214273