Euler systems and exact formulas in number theory.

*(English)*Zbl 0871.11044This article is an introduction to Kolyvagin’s method of Euler systems and their applications to number theory. The article applies this concept to elliptic curves \(E\) given by the equation \(y^2=x^3-q^2x\), for a prime \(q\equiv 7\) modulo 8 and the real quadratic fields \(F=\mathbb{Q}(\sqrt{q})\), for a prime \(q\equiv 1\) modulo 4. The author points out the analogy between rational points, Heegner points and the Tate-Shafarevich group of \(E\) on the one hand and units, cyclotomic units and the ideal class group of \({\mathcal O}_F\) on the other hand. These are two of very few known examples for Euler systems.

Using Euler systems Kolyvagin showed that the order of the Tate-Shafarevich group of \(E\) divides the square of the order of \(E(\mathbb{Q})\) modulo the subgroup generated by the Heegner point and the torsion points. Assuming the Birch Swinnerton-Dyer conjecture these two numbers should coincide. The author sketches a proof based on Euler systems, following Thaine and Kolyvagin, for the fact that the ideal class number of the ring \({\mathcal O}_F\) divides the order of the units in \({\mathcal O}_F\) modulo the cyclotomic units. Although in this case the equality of both numbers is classically known, this Euler system proof works more directly, because the theory of \(L\)-functions is not needed.

Using Euler systems Kolyvagin showed that the order of the Tate-Shafarevich group of \(E\) divides the square of the order of \(E(\mathbb{Q})\) modulo the subgroup generated by the Heegner point and the torsion points. Assuming the Birch Swinnerton-Dyer conjecture these two numbers should coincide. The author sketches a proof based on Euler systems, following Thaine and Kolyvagin, for the fact that the ideal class number of the ring \({\mathcal O}_F\) divides the order of the units in \({\mathcal O}_F\) modulo the cyclotomic units. Although in this case the equality of both numbers is classically known, this Euler system proof works more directly, because the theory of \(L\)-functions is not needed.

Reviewer: J.Kramer (Berlin)

##### MSC:

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11R29 | Class numbers, class groups, discriminants |

11G05 | Elliptic curves over global fields |

11R34 | Galois cohomology |