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The difference between the Weil height and the canonical height on elliptic curves. (English) Zbl 0729.14026
If E: \(y^ 2=x^ 3+Ax+B\) is an elliptic curve over a number field K then for each point \(P\in E(K)\) one has the Weil height h(P) and the canonical height \(\hat h(P)\). It is known that the difference \(\hat h(P)- {1/2}h(P)\) is bounded. This paper gives explicit upper and lower bounds for this difference in terms of the coefficients A and B. This improves earlier bounds by Dem’janenko and Zimmer. A practical application is related to the determination of generators of the Mordell-Weil group E(K). The author illustrates this with three examples.

MSC:
14H52 Elliptic curves
14G05 Rational points
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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