Tschöpe, Heinz M.; Zimmer, Horst G. Computation of the Néron-Tate height on elliptic curves. (English) Zbl 0611.14028 Math. Comput. 48, 351-370 (1987). 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 Cited in 5 Documents 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 Keywords:group of rational points; Néron-Tate height; elliptic curve; nontorsion points PDFBibTeX XMLCite \textit{H. M. Tschöpe} and \textit{H. G. Zimmer}, Math. Comput. 48, 351--370 (1987; Zbl 0611.14028) Full Text: DOI