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The canonical height of an algebraic point on an elliptic curve. (English) Zbl 0973.11062
Let $$K$$ be an algebraic number field of degree $$d$$ with ring of integers $$O_K$$. Let $$E$$ be an elliptic curve defined over $$K$$ given by an equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ with $$a_1,\ldots,a_6\in K$$. Let $$\Delta_E$$ denote the discriminant of $$E$$. Denote by $$\widehat{h}$$ the global canonical height of $$E$$. It is well-known that $$\widehat{h}$$ is a positive definite quadratic form on $$E(K)/E(K)_{\text{tors}}$$. J. H. Silverman [Math. Comput. 51, 339–358 (1988; Zbl 0656.14016)] gave an algorithm which for a given point $$Q\in E(K)$$ computes an approximation to $$\widehat{h}(Q)$$. His algorithm is based on the decomposition of $$\widehat{h}$$ into local canonical heights. Silverman’s method gives a very accurate approximation but it has the disadvantage that it requires the factorization of $$\Delta_E$$ into prime ideal factors. This may considerably slow down the algorithm if $$\Delta_E$$ has large norm. In another paper, J. H. Silverman [Math. Comput. 66, 787–805 (1997; Zbl 0898.11021)] published a modification of his algorithm which in many cases, (e.g., if $$\Delta_E$$ is square-free) but not yet in all cases eliminates the necessity to factor $$\Delta_E$$.
In the present paper the authors suggest another method to compute an approximation of $$\widehat{h}(Q)$$ which completely avoids the factorization of $$\Delta_E$$. Let $$\overline{K}$$ denote the algebraic closure of $$K$$. Using a suitable recurrence relation one may define a sequence of so-called division polynomials $$\psi_n\in O_K[x,y]$$ with the property that the zeros of $$\psi_n$$ lying on $$E(\overline{K})$$ are precisely the points in $$E(\overline{K})$$ of order dividing $$n$$. Let $$T$$ be the set consisting of the infinite prime on $$\mathbb{Q}$$ and of the prime numbers dividing $$N_{K/ \mathbb Q}(\Delta_E)$$. For $$Q=(x_Q,y_Q)\in E(K)$$, put $$F_n(Q):= \prod_{p\in T} |N_{K/ \mathbb{Q}}(\psi_n(x_Q,y_Q))|_p$$. Then the authors show that $$\widehat{h}(Q)=\lim_{n\to\infty}{1\over n^2}\log F_n(Q)$$. Their proof uses estimates by S. David on linear forms in elliptic logarithms. The authors argue that in order to compute $$F_n(Q)$$ no factorization of $$\Delta_E$$ is required. Further they argue that for $$n$$ not too large $$F_n(Q)$$ gives an approximation to $$\widehat{h}(Q)$$ which is not as good as Silverman’s, but sufficiently strong for many applications. In the last section of their paper they give some numerical examples.

##### MSC:
 11G07 Elliptic curves over local fields 11G50 Heights
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