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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)

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