Euler system for Galois deformations

Ochiai, Tadashi

Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 113-146.

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.

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.

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