The canonical height of an algebraic point on an elliptic curve.

*(English)*Zbl 0973.11062Let \(K\) 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.

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.

Reviewer: Jan-Hendrik Evertse (Leiden)