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.
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 -extensions already treated by Kato, Perrin-Riou, Rubin.
Keywords: Euler system, Hida theory, Iwasawa Main conjecture
Mot clés : système d'Euler, théorie de Hida, conjecture principale d'Iwasawa
@article{AIF_2005__55_1_113_0, author = {Ochiai, Tadashi}, title = {Euler system for {Galois} deformations}, journal = {Annales de l'Institut Fourier}, pages = {113--146}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {55}, number = {1}, year = {2005}, doi = {10.5802/aif.2091}, mrnumber = {2141691}, zbl = {02162466}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2091/} }
TY - JOUR AU - Ochiai, Tadashi TI - Euler system for Galois deformations JO - Annales de l'Institut Fourier PY - 2005 SP - 113 EP - 146 VL - 55 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2091/ DO - 10.5802/aif.2091 LA - en ID - AIF_2005__55_1_113_0 ER -
Ochiai, Tadashi. Euler system for Galois deformations. Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 113-146. doi : 10.5802/aif.2091. http://www.numdam.org/articles/10.5802/aif.2091/
[BK] -functions and Tamagawa numbers of motives, in The Grothendieck Festschrift I, Progress in Math., Volume 86 (1990), pp. 333-400 | MR | Zbl
[Bo] Eléments de mathématique, Algèbre commutative, Chapitre 5--7, 1985 | MR | Zbl
[Bu] Périodes -adiques, Séminaire de Bures (1988) (Astérisque), Volume 223 (1994)
[De1] Formes modulaires et représentations -adiques, Séminaire Bourbaki 355, p. 139-172, 179, Springer Verlag, 1969 | Numdam
[De2] Valeurs des fonctions et périodes d'intégrales (Proc. Sympos. Pure Math.), Volume XXXIII, Part 2 (1979), pp. 247-289 | Zbl
[Fl] A generalisation of Cassels-Tate pairing, J. reine angew. Math., Volume 412 (1990), pp. 113-127 | MR | Zbl
[Gr1] Iwasawa theory for -adic representations, Advanced studies in Pure Math., Volume 17 (1987), pp. 97-137 | MR | Zbl
[Gr2] Iwasawa theory for -adic deformations of motives, Proceedings of Symposia in Pure Math., Volume 55 (1994) no. 2, pp. 193-223 | MR | Zbl
[GS] -adic -functions and -adic periods of modular forms, Invent. Math., Volume 111 (1993) no. 2, pp. 407-447 | MR | Zbl
[Hi1] Galois representations into attached to ordinary cusp forms, Invent. Math., Volume 85 (1986), pp. 545-613 | DOI | MR | Zbl
[Hi2] Elementary theory of -functions and Eisenstein series, London Math. Society Student Texts, 26, Cambridge University Press, 1993 | MR | Zbl
[Ka1] Lectures on the approach to Iwasawa theory for Hasse-Weil -functions via , I (Lecture Notes in Math.), Volume 1553 (1993), pp. 50-163 | Zbl
[Ka2] Series of lectures on Iwasawa main conjectures for modular elliptic curves (1998) (given at Tokyo University)
[Ka3] -adic Hodge theory and values of zeta functions of modular forms (Preprint)
[Ka4] Euler systems, Iwasawa theory, and Selmer groups, Kodai Math. J., Volume 22 (1999) no. 3, pp. 313-372 | DOI | MR | Zbl
[Ki] On standard -adic -functions of families of elliptic cusp forms, in -adic monodromy and the Birch and Swinnerton-Dyer conjecture, p.81-110, Contemp. Math., 165, Amer. Math. Soc., 1994 | MR | Zbl
[Ma] Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1986 | MR | Zbl
[Mi] Arithmetic duality theorems,, Perspectives in Math. 1, Academic Press, 1986 | MR | Zbl
[MR] Kolyvagin systems (2001) (Preprint) | MR
[MT] 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] Class fields of abelian extensions of , Invent. Math., Volume 76 (1984) no. 2, pp. 179-330 | DOI | MR | Zbl
[MW2] On -adic analytic families of Galois representations, Compos. Math., Volume 59 (1986) no. 2, pp. 231-264 | Numdam | MR | Zbl
[NSW] Cohomology of number fields, Grundlehren Math. Wiss., 323, Springer-Verlag, Berlin, 2000 | MR | Zbl
[Oc1] Control theorem for Greenberg's Selmer groups for Galois deformations, J. Number Theory, Volume 88 (2001), pp. 59-85 | DOI | MR | Zbl
[Oc2] A generalization of the Coleman map for Hida deformations, Amer. J. Math., Volume 125 (2003), pp. 849-892 | DOI | MR | Zbl
[Oc3] On the two-variable Iwasawa Main conjecture for Hida deformations (in preparation)
[Pe] Systèmes d'Euler -adiques et théorie d'Iwasawa, Ann. Inst. Fourier, Volume 48 (1998) no. 5, pp. 1231-1307 | DOI | Numdam | MR | Zbl
[Ru1] The ``main conjectures" of Iwasawa theory for imaginary quadratic fields, Invent. Math., Volume 103 (1991) no. 1, pp. 25-68 | DOI | MR | Zbl
[Ru2] Euler systems, Annals Math. Studies, 147, 2000 | MR | Zbl
[Se] Cohomologie galoisienne, 5th ed. (Lecture Notes in Math.), Volume 5 (1994) | Zbl
[Ta] Relations between and Galois cohomology, Invent. Math., Volume 36 (1976), pp. 257-274 | DOI | MR | Zbl
[Wi] On -adic representations associated to modular forms, Invent. Math., Volume 94 (1988), pp. 529-573 | DOI | MR | Zbl
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