Euler system for Galois deformations
Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 113-146.

In this paper, we develop the Euler system theory for Galois deformations. By applying this theory to the Beilinson-Kato Euler system for Hida’s nearly ordinary modular deformations, we prove one of the inequalities predicted by the two-variable Iwasawa main conjecture. Our method of the proof of the Euler system theory is based on non-arithmetic specializations. This gives a new simplified proof of the inequality between the characteristic ideal of the Selmer group of a Galois deformation and the ideal associated to a Euler system even in the case of d p -extensions already treated by Kato, Perrin-Riou, Rubin.

Dans cet article, on obtient la généralisation de la théorie du système d’Euler pour les déformations galoisiennes. Si on applique ce résultat aux système d’Euler de Beilinson- Kato, on prouve une des inégalités prévues par la conjecture principale d’Iwasawa à deux variables. La clef de notre démonstration est l’utilisation de spécialisations non- arithmétiques. Notre méthode donne une nouvelle preuve plus simple de l’inégalité entre l’idéal caractéristique du groupe de Selmer d’une déformation galosienne et l’idéal associé à un système d’Euler, y compris dans le cas des -extensions déjà traité par Kato, Perrin-Riou, Rubin.

DOI: 10.5802/aif.2091
Classification: 11G40, 11R23, 11R34, 11F80, 11F33
Keywords: Euler system, Hida theory, Iwasawa Main conjecture
Mot clés : système d'Euler, théorie de Hida, conjecture principale d'Iwasawa
Ochiai, Tadashi 1

1 Osaka University, Department of Mathematics, 1-16 Machikaneyama, Toyonaka, Osaka 560-0043 (Japon)
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Ochiai, Tadashi. Euler system for Galois deformations. Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 113-146. doi : 10.5802/aif.2091. http://archive.numdam.org/articles/10.5802/aif.2091/

[BK] S. Bloch; K. Kato L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift I, Progress in Math., Volume 86 (1990), pp. 333-400 | MR | Zbl

[Bo] N. Bourbaki Eléments de mathématique, Algèbre commutative, Chapitre 5--7, 1985 | MR | Zbl

[Bu] J.M. Fontaine (éd.) Périodes p-adiques, Séminaire de Bures (1988) (Astérisque), Volume 223 (1994)

[De1] P. Deligne Formes modulaires et représentations -adiques, Séminaire Bourbaki 355, p. 139-172, 179, Springer Verlag, 1969 | Numdam

[De2] P. Deligne Valeurs des fonctions L et périodes d'intégrales (Proc. Sympos. Pure Math.), Volume XXXIII, Part 2 (1979), pp. 247-289 | Zbl

[Fl] M. Flach A generalisation of Cassels-Tate pairing, J. reine angew. Math., Volume 412 (1990), pp. 113-127 | MR | Zbl

[Gr1] R. Greenberg Iwasawa theory for p-adic representations, Advanced studies in Pure Math., Volume 17 (1987), pp. 97-137 | MR | Zbl

[Gr2] R. Greenberg Iwasawa theory for p-adic deformations of motives, Proceedings of Symposia in Pure Math., Volume 55 (1994) no. 2, pp. 193-223 | MR | Zbl

[GS] R. Greenberg; G. Stevens p-adic L-functions and p-adic periods of modular forms, Invent. Math., Volume 111 (1993) no. 2, pp. 407-447 | MR | Zbl

[Hi1] H. Hida Galois representations into GL 2 ( p [[X]]) attached to ordinary cusp forms, Invent. Math., Volume 85 (1986), pp. 545-613 | DOI | MR | Zbl

[Hi2] H. Hida Elementary theory of L-functions and Eisenstein series, London Math. Society Student Texts, 26, Cambridge University Press, 1993 | MR | Zbl

[Ka1] K. Kato; Springer Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via B dR , I (Lecture Notes in Math.), Volume 1553 (1993), pp. 50-163 | Zbl

[Ka2] K. Kato Series of lectures on Iwasawa main conjectures for modular elliptic curves (1998) (given at Tokyo University)

[Ka3] K. Kato p-adic Hodge theory and values of zeta functions of modular forms (Preprint)

[Ka4] K. Kato Euler systems, Iwasawa theory, and Selmer groups, Kodai Math. J., Volume 22 (1999) no. 3, pp. 313-372 | DOI | MR | Zbl

[Ki] K. Kitagawa On standard p-adic L-functions of families of elliptic cusp forms, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture, p.81-110, Contemp. Math., 165, Amer. Math. Soc., 1994 | MR | Zbl

[Ma] H. Matsumura Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1986 | MR | Zbl

[Mi] J.S. Milne Arithmetic duality theorems,, Perspectives in Math. 1, Academic Press, 1986 | MR | Zbl

[MR] B. Mazur; K. Rubin Kolyvagin systems (2001) (Preprint) | MR

[MT] B. Mazur; J. Tilouine Représentations galoisiennes, différentielles de Kähler et ``conjectures principales", Inst. Hautes Études Sci. Publ. Math., Volume 71 (1990), pp. 65-103 | DOI | Numdam | MR | Zbl

[MW1] B. Mazur; A. Wiles Class fields of abelian extensions of , Invent. Math., Volume 76 (1984) no. 2, pp. 179-330 | DOI | MR | Zbl

[MW2] B. Mazur; A. Wiles On p-adic analytic families of Galois representations, Compos. Math., Volume 59 (1986) no. 2, pp. 231-264 | Numdam | MR | Zbl

[NSW] J. Neukirch; A. Schmidt; K. Wingberg Cohomology of number fields, Grundlehren Math. Wiss., 323, Springer-Verlag, Berlin, 2000 | MR | Zbl

[Oc1] T. Ochiai Control theorem for Greenberg's Selmer groups for Galois deformations, J. Number Theory, Volume 88 (2001), pp. 59-85 | DOI | MR | Zbl

[Oc2] T. Ochiai A generalization of the Coleman map for Hida deformations, Amer. J. Math., Volume 125 (2003), pp. 849-892 | DOI | MR | Zbl

[Oc3] T. Ochiai On the two-variable Iwasawa Main conjecture for Hida deformations (in preparation)

[Pe] B. Perrin-Riou Systèmes d'Euler p-adiques et théorie d'Iwasawa, Ann. Inst. Fourier, Volume 48 (1998) no. 5, pp. 1231-1307 | DOI | Numdam | MR | Zbl

[Ru1] K. Rubin The ``main conjectures" of Iwasawa theory for imaginary quadratic fields, Invent. Math., Volume 103 (1991) no. 1, pp. 25-68 | DOI | MR | Zbl

[Ru2] K. Rubin Euler systems, Annals Math. Studies, 147, 2000 | MR | Zbl

[Se] J.-P. Serre Cohomologie galoisienne, 5th ed. (Lecture Notes in Math.), Volume 5 (1994) | Zbl

[Ta] J. Tate Relations between K 2 and Galois cohomology, Invent. Math., Volume 36 (1976), pp. 257-274 | DOI | MR | Zbl

[Wi] A. Wiles On λ-adic representations associated to modular forms, Invent. Math., Volume 94 (1988), pp. 529-573 | DOI | MR | Zbl

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