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On certain semisimple Iwasawa modules. (Sur quelques modules d’Iwasawa semi-simples.) (French) Zbl 0869.11084
Let \(p\) be a prime number. Let \(K_\infty/K\) be a \(\mathbb{Z}_p\)-extension of number fields and let \(\gamma\) be a generator of the Galois group. Let \(K^c\) be the cyclotomic \(\mathbb{Z}_p\)-extension of \(K\). The authors define a pro-\(p\)-extension \(L/K\) to be cyclotomically ramified if all the places of \(K\) above \(p\) are infinitely ramified in \(L/K\) and the extension \(K^cL/K^c\) is everywhere unramified. The dimension of the set of cyclotomically ramified \(\mathbb{Z}_p\)-extensions of a number field is determined in some cases. Let \(\mathcal C_{K_\infty}\) be the Galois group of the maximal unramified abelian \(p\)-extension of \(K_\infty\). It is a module over \(\mathbb{Z}_p[[T]]\), where \(T\) acts as \(\gamma-1\). When the minimal polynomial of this module is not divisible by \(T^2\), the module is said to be algebraically semisimple at \(\gamma-1\). This concept was studied by R. Greenberg [Invent. Math. 21, 117-124 (1973; Zbl 0268.12004)], L. Federer and B. Gross [Invent. Math. 62, 443-457 (1980; Zbl 0468.12005)], and also by J. Carroll and H. Kisilevsky and by the first author. The \(\mathbb{Z}_p\)-extension is said to be arithmetically semisimple if the places above \(p\) are finitely decomposed in \(K_\infty/K\) and if the maximal unramified abelian \(p\)-extension of \(K_\infty\) in which all primes above \(p\) split completely is finite. The authors prove arithmetic semisimplicity implies algebraic semisimplicity. For the cyclotomic \(\mathbb{Z}_p\)-extension of \(K\), arithmetic semisimplicity is a consequence of the generalized conjecture of Gross.
Suppose now that \(K\) is imaginary abelian, the exponent of the Galois group of \(K/\mathbb{Q}\) divides \(p-1\), and the maximal real subfield of \(K\) has at least two places above \(p\). To each imaginary irreducible \(p\)-adic character of Gal(\(K/\mathbb{Q}\)), there is associated an absolutely Galois \(\mathbb{Z}_p\)-extension of \(K\). The authors prove that such an extension is either cyclotomically ramified and arithmetically semisimple or is neither cyclotomically ramified nor arithmetically semisimple.

11R23 Iwasawa theory
Full Text: Numdam EuDML
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