Heegner points on modular elliptic curves
Let \(E\) be a modular elliptic curve over \(\mathbb{Q}\), of conductor \(N_s\) with a parametrization \(\varphi: (X_0(N)\to E\). Let \(x_1\in X_0(N) (\mathbb{C})\) be a complex point on \(X_0 (N)\), depending on a choice of an imaginary quadratic field \(K\) of class number \(h_K\), an integral ideal \(\kappa\vartriangleleft O_K\) satisfying \(O_K/ \kappa \simeq \mathbb{Z}/N\mathbb{Z}\), and the ideal class of \(K\). Let \(K_1\) be the Hilbert class field of \(K\), and let \(y_1: =\varphi (x_1)\in E(K_1)\) be the corresponding Heegner point on \(E\). The following result is a generalization of \textit{P. Kurchanov}'s Proposition [Math. USSR, Sb. 19, 320-324 (1974); translation from Mat. Sb., Nov. Ser. 90(132), No. 2, 320-324 (1973; Zbl 0252.14008)]. Theorem 1.1. If \(h_K> \deg (\varphi)\), then the point \(y_1\) has infinite order. Let \(y_K: =\text{Tr}_{K_1/K} (y_1) \in E(K)\). It is known that if \(y_K\) has infinite order, then \(L'(E/K,1)\neq 0\) (Gross and Zagier), and \(\text{rank} E(K) =1\), \(\#\text{ Ш} (E/K) <\infty\) (Kolyvagin). The author proves the following result. Theorem 1.2. If \(y_K\) is a torsion point, then \(y_K\in E(\mathbb{Q})\).
scientific article; zbMATH DE number 1022273
scientific article