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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.

##### MSC:
 11R23 Iwasawa theory
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##### References:
 [1] Brumer, A. : On the units of algebraic number fields . Mathematika 14 (1967), 121-124. · Zbl 0171.01105 · doi:10.1112/S0025579300003703 [2] Buchmann, J. and Sands, J. : Leopoldt’s conjecture in parametrized families . Proc. Amer. Math. Soc. 104 (1988), 43-49. · Zbl 0663.12011 · doi:10.2307/2047458 [3] Carroll, J. and Kisilevsky, H. : On Iwasawa’s \lambda invariant for certain Zp-extension . Acta Arithmética 40 (1981), 1-8. · Zbl 0496.12005 · eudml:205794 [4] Carroll, J. and Kisilevsky, H. : On the Iwasawa invariants of certain Z p-extensions . Compositio Math. 49 (1983), 217-229. · Zbl 0517.12003 · numdam:CM_1983__49_2_217_0 · eudml:89611 [5] Federer, L.J. and Gross, B.H. , (Appendix by W. Sinnot ): Regulators and Iwasawa modules . Invent. Math. 62 (1981), 443-457. · Zbl 0468.12005 · doi:10.1007/BF01394254 · eudml:142783 [6] Greenberg, R. : On a certain l-adic representation . Invent. Math. 21 (1973), 117-124. · Zbl 0268.12004 · doi:10.1007/BF01389691 · eudml:142221 [7] Iwasawa, K. : On Zl-extensions of algebraic number fields . Ann. of math. 98 (1973),243-326. · Zbl 0285.12008 · doi:10.2307/1970784 [8] Jaulent, J.-F. : Théorie d’Iwasawa des tours métabéliennes . Sém. Th. Nombres Bordeaux (1980/81), exp. n^\circ 21. · Zbl 0482.12007 · eudml:182096 [9] Jaulent, J.-F. : Sur la théorie des genres dans les tours métabéliennes . Sém. Th. Nombres Bordeaux (1981/82), exp. n^\circ 24. · Zbl 0504.12007 · eudml:182125 [10] Jaulent, J.-F. : L’arithmétique des l-extensions (Thèse d’Etat) . Pub. Math. Fac. Sci. Besançon Théor. Nombres 1984-85 and 1985/86, fasc. 1 (1986), 1-349. · Zbl 0601.12002 [11] Kisilevsky, H. : Some non-semi-simple Iwasawa modules . Composito Math. 49 (1983), 399-404. · Zbl 0517.12004 · numdam:CM_1983__49_3_399_0 · eudml:89616
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