Finiteness of \(E({\mathbb{Q}})\) and Ш\((E,{\mathbb{Q}})\) for a subclass of Weil curves. (English. Russian original) Zbl 0662.14017

Kolyvagin, V. A.

Math. USSR, Izv. 32, No. 3, 523-541 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 3, 522-540 (1988).

Let E be an elliptic curve defined over the field of rational numbers \({\mathbb{Q}}\); \(L(E,{\mathbb{Q}},s)=\sum^{\infty}_{n=1}a_ nn^{-s} \) be a canonical L-function of E over \({\mathbb{Q}}\) \((a_ n\) is a natural number); K be an imaginary quadratic extension of \({\mathbb{Q}}\) with discriminant \(\Delta <0\) and \(\Delta \equiv a^ 2\quad (mod\quad 4N)\) where N is a natural number. Let H be the Hilbert’s class field of K; \(y\in E(H)\) be Heegner’s points of a weak Weil parametrization \(X_ N\to E(X_ N)\) \((X_ N\) be a modular curve over \({\mathbb{Q}}).\)
The author proves: if L(E,\({\mathbb{Q}},1)\neq 0\) and \(N_{H/K}(y)\) have an infinite order than E(\({\mathbb{Q}})\) and the Shafarevich-Tate group are finite.

Reviewer: S.Kotov

MSC:

14H25	Arithmetic ground fields for curves
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G05	Rational points
11R11	Quadratic extensions
14H52	Elliptic curves
14H45	Special algebraic curves and curves of low genus

Keywords:

Weil curves; elliptic curve; L-function; Shafarevich-Tate group