## Euler systems, Iwasawa theory, and Selmer groups.(English)Zbl 0993.11033

From the introduction: V. A. Kolyvagin discovered the method of Euler system, and used it to analyze ideal class groups of certain cyclotomic fields and Selmer groups of elliptic curves. K. Rubin used the method of Euler system to obtain a new proof of Iwasawa’s main conjecture and a proof of the main conjecture for imaginary quadratic fields [V. A. Kolyvagin, Math. USSR, Izv. 32, No. 3, 523-541 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 3, 522-540 (1988; Zbl 0662.14017); The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 435-483 (1990; Zbl 0742.14017); K. Rubin, Appendix to S. Lang, Cyclotomic fields I and II, Graduate Texts in Math. 121, 397-420 (1990; Zbl 0704.11038); Invent. Math. 103, 25-68 (1991; Zbl 0737.11030)]. It is vaguely believed that once a nice Euler system is discovered, we can analyze certain étale cohomology groups, and “Selmer groups”, which are generalizations of ideal class groups and of Selmer groups of elliptic curves. This paper is an attempt to prove the truth of this belief.
In this paper, we show that once a nice Euler system of a $$p$$-adic representation of $$\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})$$ is given (see Proposition 1.1 for the meaning of “an Euler system for a $$p$$-adic representation of $$\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})$$”), then we can prove finiteness theorems for the second étale cohomology $$H_{\text{ét}}^2$$ (Theorem 13.3) and for the Selmer group (Theorem 13.2) of the Galois representation, and can prove a part of an analogue of Iwasawa’s main conjecture (Theorem 0.8) of the Galois representation.
Similar results were obtained also by B. Perrin-Riou [Ann. Inst. Fourier 48, 1231-1307 (1998; Zbl 0930.11078)] and by K. Rubin [Lond. Math. Soc. Lect. Note Ser. 254, 351-367 (1998; Zbl 0952.11016)] independently. The results of this paper will be used in [K. Kato, $$p$$-adic Hodge theory and special values of zeta functions of elliptic cusp forms (in preparation)] to develop the Iwasawa theory of elliptic cusp forms and Iwasawa theory of elliptic curves without complex multiplication. Results of this paper on $$H_{\text{ét}}^2$$ and Selmer groups are obtained under the assumption that we are given a nice Euler system, and how to find an Euler system is a difficult problem. In the paper to appear, we will actually find nice Euler systems for two-dimensional Galois representations associated to elliptic cusp forms. These Euler systems come from Beilinson’s elements in $$K_2$$ of modular curves.

### MSC:

 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R23 Iwasawa theory 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 19F15 Symbols and arithmetic ($$K$$-theoretic aspects) 11F85 $$p$$-adic theory, local fields
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### References:

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