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Introduction to Kolyvagin systems. (English) Zbl 1081.11069

Burns, David (ed.) et al., Stark’s conjectures: recent work and new directions. Papers from the international conference on Stark’s conjectures and related topics, Johns Hopkins University, Baltimore, MD, USA, August 5–9, 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3480-0/pbk). Contemporary Mathematics 358, 207-221 (2004).
V. Kolyvagin [The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 435–483 (1990; Zbl 0742.14017)] introduced the method of Euler systems for giving upper bounds for certain Selmer groups. An important step is the construction of what he calls derivative classes, which are used to produce the bounds. In [Mem. Am. Math. Soc. 799 (2004; Zbl 1055.11041)], B. Mazur and K. Rubin defined Kolyvagin systems, which are cohomology classes satisfying certain relations and which include the derivative classes defined by Kolyvagin. These Kolyvagin systems are more powerful in many ways than Euler systems and sometimes yield exact formulas rather than upper bounds for the orders of Selmer groups. Moreover, Kolyvagin systems exist in some situations where Euler systems are not known to exist.
The present paper gives an introduction to Kolyvagin systems in one of the simplest cases, namely, where the Selmer group is a Galois eigenspace of the ideal class group of a prime cyclotomic field and the Kolyvagin system is derived from the Euler system constructed from cyclotomic units. The reader is referred to the paper of Mazur and Rubin [op. cit.] for several of the proofs.
For the entire collection see [Zbl 1052.11003].

MSC:

11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants
11R42 Zeta functions and \(L\)-functions of number fields
11R34 Galois cohomology
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F80 Galois representations
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