*Cohomology of units and Z_2-torsion of the cyclotomic Z_2-extension of some CM fields*

**Ali Mouhib **

Univ. Mohammed Ben Abdellah, Fes LSI,

Sciences and Engineering Laboratory Polydisciplinary Faculty of Taza B.P 1223 Taza-Gare Morocco.

Pages 93-107 | Received 26 November 2022, Accepted 14 February 2023, Published 16 June 2023

**Abstract**

It is well known from the results of Ferrero and Kida that the \mathbb{Z}_2-torsion part of the unramified abelian Iwasawa module X_{\infty}(K) of any imaginary quadratic number field K is trivial or cyclic of order 2. In this article, we will determine an infinite family of imaginary multiquadratic number fields, in which the \mathbb{Z}_2-torsion of the Iwasawa module X_{\infty} is of arbitrary large rank, giving also the exact value of the rank of X_{\infty}. Also, we will compute the first and the second cohomology groups of units of the cyclotomic \mathbb{Z}_2-extension of some CM fields K. Hence, as application, using the Iwasawa Riemann-Hurwitz formula, we obtain the Iwasawa \lambda_2(K) invariant of the cyclotomic \mathbb{Z}_2-extension of K in terms of the Iwasawa \lambda_2 invariant of the cyclotomic \mathbb{Z}_2-extension of some totally real subfield of K, thus giving an alternative proof of the previous results obtained by Kida.

**Keywords**: Class group, Unit group, Capitulation problem, \mathbb{Z}_2-extension.

**MSC numbers**: 11R29, 11R32, 11R37, 11R23.

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