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A limit formula for the canonical height of an elliptic curve and its application to height computations. (English) Zbl 0738.14020
Number theory, Proc. 1st Conf. Can. Number Theory Assoc., Banff/Alberta (Can.) 1988, 641-659 (1990).
[For the entire collection see Zbl 0689.00005.]
Let $$\mathbb{K}$$ be an algebraic number field of finite degree over $$\mathbb{Q}$$, and let $$E/\mathbb{K}$$ be an elliptic curve. It is well known that the canonical global height $$\hat h$$ on the group of points $$E(\mathbb{K})$$ can be given by the limit formula $$\hat h(P)=\lim_{m\to\infty} h(n^ mP)/n^{2m}$$, for any fixed integer $$n>1$$. Here $$h$$ denotes the ordinary global height function. This paper presents a limit formula for the canonical logal height $$\hat h_{\mathfrak p}$$ in case of an elliptic curve $$E$$ defined over a local field $$K_{\mathfrak p}$$. The actual formula is somewhat involved, so it is not reproduced here. The proof is deferred to a forthcoming paper, but a brief outline is given, as well as sharp estimates of the difference between the canonical local height and the so-called modified local height (modified with respect to the ordinary local height) on which the proof is based. The estimates are stronger than those previously obtained by the author. The final section discusses the computation of the canonical global height.

##### MSC:
 14H52 Elliptic curves 11G05 Elliptic curves over global fields 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14Q05 Computational aspects of algebraic curves 11G07 Elliptic curves over local fields