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.


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
Full Text: DOI


[1] ARTIN, M. AND VERDIER, J., Seminar on etale cohomology ofnumber fields, Wood Hole (1964).
[2] BEILINSON, A., Higher regulators and values of L-functions, J. Soviet Math., 30 (1985 2036-2070. · Zbl 0588.14013
[3] BLOCH, S. AND KATO, K., l-functions and Tamagawa numbers of motives, The Gro thendieck Festschrift, vol.1, Birkhauser, 1990, 334-400. · Zbl 0768.14001
[4] BERTHELOT, P AND OGUS, A., Notes on Crystalline Cohomology, Princeton Univ. Press, 1978 · Zbl 0383.14010
[5] BOURBAKI, N., Groupes et Algebras de Lie, Elements de mathematique, 16, Hermann, 1960 · Zbl 0329.17002
[6] CARTAN, H. AND EILENBERG, S., Homological Algebra, Princeton Math. Ser., 19, Princeto Univ. Press, 1956. · Zbl 0075.24305
[7] DELIGNE, P., Laconjecture de Weil, I, Inst. Hautes Etudes Sci. Publ. Math., 43(1974), 273-307; II, ibid., 52 (1980), 137-252 · Zbl 0287.14001
[8] FONTAINE, J. -M., Sur certains types derepresentations /7-adiques dugroupe de Galois d’u corps local: constructuion d’un anneau de Barsotti-Tate, Ann. of Math., 115 (1982), 529-577 · Zbl 0544.14016
[9] JANNSEN, U., On the /-adic cohomology of varieties over number fields and its Galoi cohomology, Galois Group over Q, Springer, 1989, 315-360. · Zbl 0703.14010
[10] KATO, K., Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions vi BdR, Lecture Notes in Math., 1553, Springer, 1993, 50-163. · Zbl 0815.11051
[11] KATO, K., /?-adic Hodge theory and special values of zeta functions of elliptic cusp forms, i preparation.
[12] KOLYVAGIN, V. A., Fimteness of E(Q) and UI(E/Q) for a class of Weil curves, Izv. Acad Nauk SSSR, 52 (1988), 522-540, 670-671. · Zbl 0662.14017
[13] KOLYVAGIN, V A., Euler systems, The Grothendieck Festschrift, vol. 2, Birkhauser, 199 435-483. · Zbl 0742.14017
[14] LAZARD, M., Groupes analytiques /?-adiques, Inst. Hautes Etudes Sci. Publ. Math., 2 (1965), 1-219. · Zbl 0139.02302
[15] MAZUR, B., Notes on etale cohomology of number fields, Ann. Sci. Ecole. Norm. Sup., (1973), 521-556. · Zbl 0282.14004
[16] MAZUR, B. AND WILES, A., Class fields of abelian extensions of Q, Invent. Math., 7 (1984), 179-330. · Zbl 0545.12005
[17] PERRIN-RIOU, B., Systemes d’Euler /?-adique et theoe d’lwasawa,
[18] RUBIN, K., The main conjecture, Appendix to Lang, S., Cyclotomic Fields I and II, Grad Texts in Math., 121, Spnger, 1990, 397-420. · Zbl 0704.11038
[19] RUBIN, K., The ”main conjecture” of Iwasawa theory for imaginary quadratic fields, Invent. Math., 103 (1991), 25-68 · Zbl 0737.11030
[20] RUBIN, K., Stark units and Kolyvagin’s ”Euler systems”, J. Reine Angew. Math., 42 (1992), 141-154. · Zbl 0752.11045
[21] RUBIN, K., Euler systems and modular elliptic curves, London Math. Soc. Lecture Not Ser., 254, Cambridge Univ. Press, Cambridge, 1998, 351-367. · Zbl 0952.11016
[22] SERRE, J. -P., Corps Locaux, Hermann, 1960
[23] SERRE, J. -P., Cohomologie Galoisienne, Lecture Notes in Math., 5, Spnger, 1964
[24] SERRE, J. -P., Sur les groupes de congruences des varietes, Izv. Akad. Nauk SSSR, 28 (1964 3-18, II, ibid., 35 (1971), 731-735. · Zbl 0128.15601
[25] SERRE, J. -P., Abelian /-adic Representations and Elliptic Curves, Benjamin, 1968
[26] TATE, J., On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Sem Bourbaki, 1965-66, n?306, Bemjamin, 1966. · Zbl 0199.55604
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.