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