Computig heights on elliptic curves. (English) Zbl 0656.14016

An efficient algorithm for computing the global canonical height \(\hat h\) on the group of rational points E(K) of an elliptic curve E over a number field K is given. The algorithm proceeds by calculating local canonical heights and adding them up with appropriate multiplicities.
At non-Archimedean places v of K, the local canonical height \(\hat h_ v\) depends on the reduction behaviour of E modulo v, and explicit formulae can be given for \(\hat h_ v\), where the one holding in the case of multiplicative reduction modulo v goes back already to the author’s dissertation.
At Archimedean places v of K, a fast-converging series, due to Tate in case K has real completion \(K_ v\cong {\mathbb{R}}\) and modified by the author in case K has complex completion \(K_ v\cong {\mathbb{C}}\), is used for calculating the local height \(\hat h_ v.\)
Tate’s series actually is independent of whether or not the place v of K is Archimedean or non-Archimedean. The author derives an error estimate for the tail of Tate’s series which is similar to the one obtained by the reviewer [“A limit formula for the canonical height on an elliptic curve and its application to height computations” (to appear in Proc. First Conf. CNTA, Banff, Canada 1988)]. The error estimate is based on a resultant result (proposition 3.1 on page 347) of the same type as the one in an earlier paper of the reviewer [Math. Z. 147, 35-51 (1976; Zbl 0303.14003), formulas (7) and (8) on page 44].
Reviewer: H.G.Zimmer


14H25 Arithmetic ground fields for curves
14G05 Rational points
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
11D25 Cubic and quartic Diophantine equations
14H52 Elliptic curves
Full Text: DOI


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