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Théorie d’Iwasawa classique et de l’algèbre de Hecke ordinaire. (Classical Iwasawa theory and ordinary Hecke algebra theory). (French) Zbl 0663.12008
Let $$p$$ be a prime and embed $$\bar\mathbb{Q}$$ into the completion $$C_p$$ of the algebraic closure of $$\mathbb{Q}_p$$. Let $$\rho$$ denote the complex conjugation. Let $$M$$ be an imaginary quadratic field and $$\lambda$$ a Grössencharacter of $$M$$ having conductor $$f$$. According to Weil, one can associate with $$\lambda$$ a character $$\lambda: \text{Gal}(\bar\mathbb{Q}/M) \to \mathbb{C}^*_p$$. Define $$\lambda ^\rho$$ by $$\lambda^\rho(\sigma) = \lambda (\rho \sigma \rho^{-1})$$, and let $$\kappa$$ be the $$p$$-prime part of the character $$\lambda/\lambda^\rho$$. Let $$F/M$$ be the ray class field of conductor $$pN{\mathfrak f}$$, and let $$M^-_\infty$$ denote the anticyclotomic $$\mathbb{Z}^p$$-extension of $$M$$. Suppose that $$p$$ is decomposed in $$M$$, say $$(p) = {\mathfrak p}{\mathfrak p}^\rho$$. Let $$X$$ denote the Galois group over $$M^-_\infty F$$ of the maximal abelian $$p$$-extension of $$M^-_\infty F$$ which is unramified outside $$p$$ considered as $$\mathbb{Z}_p [[\text{Gal}(M^-_\infty) F / F)]]$$-module. The author investigates the congruence module $$H$$ introduced by H. Hida [Ann. Sci. Ec. Norm. Super., IV. Ser. 19, 231-273 (1986; Zbl 0607.10022)], and shows that under certain restrictive conditions the characteristic series of $$H$$ divides the characteristic series of a twisted version of the kappa-part of the module $$X$$.
Reviewer: V.Ennola

##### MSC:
 11R18 Cyclotomic extensions 11R37 Class field theory 11R52 Quaternion and other division algebras: arithmetic, zeta functions
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