The canonical height of an algebraic point on an elliptic curve
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 \(E(K)/E(K)_{\text{tors}}\). \textit{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, \textit{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.
scientific article; zbMATH DE number 1550782