# zbMATH — the first resource for mathematics

Kolyvagin’s work on modular elliptic curves. (English) Zbl 0743.14021
$$L$$-functions and arithmetic, Proc. Symp., Durham/UK 1989, Lond. Math. Soc. Lect. Note Ser. 153, 235-256 (1991).
[For the entire collection see Zbl 0718.00005.]
Let $$E$$ be a modular elliptic curve over $$\mathbb{Q}$$ and let $$K$$ be an imaginary quadratic field. With the theorem by the author and D. B. Zagier [Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)] one can compute the height of a Heegner point $$y_ K\in E(K)$$ by the derivative of the $$L$$-series of the elliptic curve $$E$$ at the critical point. If the Heegner point $$y_ K$$ has infinite order, then the conjecture of Birch and Swinnerton-Dyer implies that the group $$E(K)$$ has rank 1; furthermore the order of the Tate-Shafarevich group $$\text Ш(E/K)$$ is predicted. Indeed in this case A.V.Kolyvagin proved that $$E(K)$$ has rank 1, that $$\text Ш(E/K)$$ is a finite group, and he determined a multiple of its order. In this note the author explains the ideas of Kolyvagin’s proof. He sketches a slightly weaker result to avoid technical difficulties.
Reviewer: H.-G.Rück (Essen)

##### MSC:
 14H52 Elliptic curves 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G35 Modular and Shimura varieties 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture