Travaux de Kolyvagin et Rubin. (The works of Kolyvagin and Rubin).

*(French)*Zbl 0735.14017
Sémin. Bourbaki, Vol. 1989/90, 42ème année, Astérisque 189-190, Exp. No. 717, 69-106 (1990).

[For the entire collection see Zbl 0722.00001.]

This is a survey of results of V. A. Kolyvagin (from 1988) and K. Rubin (from 1987).

Let \(E\) be an elliptic curve over \(\mathbb{Q}\) parametrized by modular functions. It is known that the group \(E(F)\) of points of \(E\) rational over \(F\) has finite rank. However, it is difficult to compute the rank. Using an idea of Heegner, Birch constructed points of \(E\) rational over certain Abelian extensions of an imaginary quadratic field. It was not known whether they generated a subgroup of finite index and, if yes, whether that index was connected with the cardinality of the Shafarevich- Tate group. There was no example of an elliptic curve with a finite Shafarevich-Tate group found before the works of Kolyvagin and Rubin.

Results of Kolyvagin and Rubin answer this type of questions. Also, they permit to give an elementary proof to the main conjecture of Iwasawa for \(\mathbb{Q}\) (proved by Mazur and Wiles) and prove the main conjecture of Coates-Wiles for imaginary quadratic fields.

This is a survey of results of V. A. Kolyvagin (from 1988) and K. Rubin (from 1987).

Let \(E\) be an elliptic curve over \(\mathbb{Q}\) parametrized by modular functions. It is known that the group \(E(F)\) of points of \(E\) rational over \(F\) has finite rank. However, it is difficult to compute the rank. Using an idea of Heegner, Birch constructed points of \(E\) rational over certain Abelian extensions of an imaginary quadratic field. It was not known whether they generated a subgroup of finite index and, if yes, whether that index was connected with the cardinality of the Shafarevich- Tate group. There was no example of an elliptic curve with a finite Shafarevich-Tate group found before the works of Kolyvagin and Rubin.

Results of Kolyvagin and Rubin answer this type of questions. Also, they permit to give an elementary proof to the main conjecture of Iwasawa for \(\mathbb{Q}\) (proved by Mazur and Wiles) and prove the main conjecture of Coates-Wiles for imaginary quadratic fields.

Reviewer: B.M.Schein (Fayetteville)

##### MSC:

14G05 | Rational points |

14H52 | Elliptic curves |

11G05 | Elliptic curves over global fields |

14-03 | History of algebraic geometry |

14G25 | Global ground fields in algebraic geometry |