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Computation of the Néron-Tate height on elliptic curves. (English) Zbl 0611.14028

The authors show an explicit method to calculate the local and global Néron-Tate height on the group E(K) of rational points on an elliptic curve E (given by a generalized Weierstraß equation \(y^ 2+a_ 1xy+a_ 3y=x^ 3+a_ 2x^ 2+a_ 4x+a_ 6\) with \(a_ i\in K)\) over the rational number field \(K={\mathbb{Q}}\). - The procedure is developed by using the reduction theory of Néron and a method of Tate.
In the second part of the paper the procedure is illustrated by verifying two height calculations of Silverman as well as by calculating the global Néron-Tate height of Bremner-Cassels’ rank-one curves and Selmer’s rank-two curves. Thereby S. Lang’s conjecture about lower bounds for the Néron-Tate height on nontorsion points in E(Q) is also examined.
Reviewer: E.Ederle

MSC:

14H45 Special algebraic curves and curves of low genus
14H52 Elliptic curves
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
14G05 Rational points
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