Euler system for Galois deformations

[Système d'Euler pour les déformations galoisiennes]

Ochiai, Tadashi

Annales de l'Institut Fourier, Tome 55 (2005), p. 113-146 / Harvested from Numdam

Résumé

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 $\mathbb {Z}$-extensions déjà traité par Kato, Perrin-Riou, Rubin.

Abstract

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 $\mathbb {Z}^{d_p}$-extensions already treated by Kato, Perrin-Riou, Rubin.

Détail

Publié le : 2005-01-01

MR 2141691 | Zbl 02162466 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2091
Classification:  11G40,  11R23,  11R34,  11F80,  11F33
Mots clés: système d'Euler, théorie de Hida, conjecture principale d'Iwasawa

Bibliographie

[BK] S. Bloch; K. Kato $L$-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift I, Progress in Math., Tome 86 (1990), pp. 333-400 | MR 1086888 | Zbl 0768.14001

[Bo] N. Bourbaki Eléments de mathématique, Algèbre commutative, Chapitre 5--7 (1985) | MR 722608 | Zbl 0547.13002

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

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

[De2] P. Deligne Valeurs des fonctions $L$ et périodes d'intégrales, Amer. Math. Soc. (Proc. Sympos. Pure Math.) Tome XXXIII, Part 2 (1979), pp. 247-289 | Zbl 0449.10022

[Fl] M. Flach A generalisation of Cassels-Tate pairing, J. reine angew. Math., Tome 412 (1990), pp. 113-127 | MR 1079004 | Zbl 0711.14001

[Gr1] R. Greenberg Iwasawa theory for $p$-adic representations, Advanced studies in Pure Math., Tome 17 (1987), pp. 97-137 | MR 1097613 | Zbl 0739.11045

[Gr2] R. Greenberg Iwasawa theory for $p$-adic deformations of motives, Proceedings of Symposia in Pure Math., Tome 55 (1994) no. 2, pp. 193-223 | MR 1265554 | Zbl 0819.11046

[GS] R. Greenberg; G. Stevens $p$-adic $L$-functions and $p$-adic periods of modular forms, Invent. Math., Tome 111 (1993) no. 2, pp. 407-447 | MR 1198816 | Zbl 0778.11034

[Hi1] H. Hida Galois representations into ${\rm GL}_2({\Bbb Z}_p[[X]])$ attached to ordinary cusp forms, Invent. Math., Tome 85 (1986), pp. 545-613 | Article | MR 848685 | Zbl 0612.10021

[Hi2] H. Hida Elementary theory of $L$-functions and Eisenstein series, Cambridge University Press, London Math. Society Student Texts, Tome 26 (1993) | MR 1216135 | Zbl 0942.11024

[Ka1] K. Kato; Springer Lectures on the approach to Iwasawa theory for Hasse-Weil $L$-functions via $B_{\rm dR}$, I (Lecture Notes in Math.) Tome 1553 (1993), pp. 50-163 | Zbl 0815.11051

[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) | MR 2104361 | Zbl 02123616

[Ka4] K. Kato Euler systems, Iwasawa theory, and Selmer groups, Kodai Math. J., Tome 22 (1999) no. 3, pp. 313-372 | Article | MR 1727298 | Zbl 0993.11033

[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, Amer. Math. Soc., Contemp. Math., Tome 165 (1994) | MR 1279604 | Zbl 0841.11028

[Ma] H. Matsumura Commutative ring theory, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Tome 8 (1986) | MR 879273 | Zbl 0603.13001

[Mi] J.S. Milne Arithmetic duality theorems,, Academic Press, Perspectives in Math. 1 (1986) | MR 881804 | Zbl 05081843

[MR] B. Mazur; K. Rubin Kolyvagin systems (2001) (Preprint) | MR 2031496

[MT] B. Mazur; J. Tilouine Représentations galoisiennes, différentielles de Kähler et ``conjectures principales", Inst. Hautes Études Sci. Publ. Math., Tome 71 (1990), pp. 65-103 | Article | Numdam | MR 1079644 | Zbl 0744.11053

[MW1] B. Mazur; A. Wiles Class fields of abelian extensions of ${\Bbb Q}$, Invent. Math., Tome 76 (1984) no. 2, pp. 179-330 | Article | MR 742853 | Zbl 0545.12005

[MW2] B. Mazur; A. Wiles On $p$-adic analytic families of Galois representations, Compos. Math., Tome 59 (1986) no. 2, pp. 231-264 | Numdam | MR 860140 | Zbl 0654.12008

[NSW] J. Neukirch; A. Schmidt; K. Wingberg Cohomology of number fields, Springer-Verlag, Berlin, Grundlehren Math. Wiss., Tome 323 (2000) | MR 1737196 | Zbl 0948.11001

[Oc1] T. Ochiai Control theorem for Greenberg's Selmer groups for Galois deformations, J. Number Theory, Tome 88 (2001), pp. 59-85 | Article | MR 1825991 | Zbl 01598949

[Oc2] T. Ochiai A generalization of the Coleman map for Hida deformations, Amer. J. Math., Tome 125 (2003), pp. 849-892 | Article | MR 1993743 | Zbl 1057.11048

[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, Tome 48 (1998) no. 5, pp. 1231-1307 | Article | Numdam | MR 1662231 | Zbl 0930.11078

[Ru1] K. Rubin The ``main conjectures" of Iwasawa theory for imaginary quadratic fields, Invent. Math., Tome 103 (1991) no. 1, pp. 25-68 | Article | MR 1079839 | Zbl 0737.11030

[Ru2] K. Rubin Euler systems, Annals Math. Studies, Tome 147 (2000) | MR 1749177 | Zbl 0977.11001

[Se] J.-P. Serre Cohomologie galoisienne, 5th ed., Springer-Verlag (Lecture Notes in Math.) Tome 5 (1994) | Zbl 0812.12002

[Ta] J. Tate Relations between $K_2$ and Galois cohomology, Invent. Math., Tome 36 (1976), pp. 257-274 | Article | MR 429837 | Zbl 0359.12011

[Wi] A. Wiles On $\lambda$-adic representations associated to modular forms, Invent. Math., Tome 94 (1988), pp. 529-573 | Article | MR 969243 | Zbl 0664.10013