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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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