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We show that the cyclotomic conjecture on the characteristic polynomial of T-ramified S-split Iwasawa modules introduced in a previous paper and satisfied by abelian fields governs the Z{\ell}-rank of the submodule of fixed points for all finite disjoint sets S and T of places.Last, in the CM-case we prove that the weak and the strong versions of the cyclotomic conjecture both are equivalent to the conjunction of the classical conjectures of Leopoldt and Gross-Kuz'min., in French language

We discuss the triviality of some classical Iwasawa modules in connection with the notion of $\ell$-rationality for totally $\ell$-adic number fields., Comment: in French language

We correct the faulty formulas given in a previous article and we compute the defect group for the Iwasawa $\lambda$ invariants attached to the S-ramified T-decomposed a belian pro-{\ell}-extensions on the Z{\ell}-cyclotomic extensionof a number field. As a consequence, we extend the results of Itoh, Mizusawa and Ozaki on tamely ramified Iwasawa modules for the cyclotomic Z{\ell}-extension of abelian fields., Comment: in French

We show that Riemann-Hurwitz-style translation formulas obtained by Kuz'min, Kida, Iwasawa, Wingberg et alii for the lambda invariant attached to certain Iwasawa moduli in cyclotomic Z{\ell}-extension of number fields are essentially equivalent. More precisely, we prove that all these formulas,including those stated in terms of representations, resul tidentically for purely algebraic reasons from the arithmetic computation of a suitable Herbrand quotient which it suffices to carry out in the cyclic case of prime degree {\ell}., Comment: in French. Le texte est la mise au format LATEX de l'article dactylographi{\'e} original paru aux Publications Math{\'e}matiques de Besan\c{c}on en 1986. Il n'en diff{\`e}re que par la correction de diverses coquilles,par l'harmonisation de quelques notations avec celles des articles ult{\'e}rieurs (notamment l'inversion de position de S et T) ainsi que par la num{\'e}rotation des th{\'e}or{\`e}mes

Given a real abelian field F with group G and an odd prime number {\ell}, we define the circular subgroup of the pro-{\ell}-group of logarithmic units and we show that for any Galois morphism $\rho$ from the pro-{\ell}-group of logarithmic units to Z{\ell} [G ], the image of the circular subgroup annihilates the {\ell}-group of logarithmic classes. We deduce from this a proof of a logarithmic version of Solomon conjecture., Comment: in French

For any odd prime number $\ell$ and any abelian number field F containing the $\ell$-th roots of unity, we show that the Stickelberger ideal annihilates the imaginary component of the $\ell$-group of logarithmic classes and that its reflection annihilates the real componen of the Bertrandias-Payan module. As a consequence we obtain a very simple proof of annihilation results for the so-called wild {\'e}tale $\ell$-kernels of F ., Comment: in French

We investigate the group of universal norms attached to the cyclotomic Z {\ell}-tower of a totally real number field in connection with Grenberg's conjecture on Iwasawa invariants of such a field., Comment: in French. Acta Arithmetica, Instytut Matematyczny PAN, A para{\^i}tre

We use logarithmic {\ell}-class groups to take a new view on Greenberg's conjecture about Iwasawa {\ell}-invariants of a totally real number field K. By the way we recall and complete some classical results. Under Leopoldt's conjecture, we prove that Greenberg's conjecture holds if and only if the logarithmic classes of K principalize in the cyclotomic Z{\ell}-extensions of K. As an illustration of our approach, in the special case where the prime {\ell} splits completely in K, we prove that the sufficient condition introduced by Gras just asserts the triviality of the logarithmic class group of K.Last, in the abelian case, we provide an explicit description of the circular class groups in connexion with the so-called weak conjecture., Comment: in French

We use $\ell$-adic class field theory to take a new view on cyclotomic norms and Leopoldt or Gross generalized conjectures. By the way we recall and complete some classical results. We illustrate the logarithmic approach by various numerical examples and counter-examples obtained with PARI., Comment: in French

International audience; We define l-adic analogs of classical Weil numbers in connexion both with complex or l-adic imbeddings of number fields and real or l-adic absolute values. As an application we give some consequences related to the Iwasawa theory of cyclotomic towers.

We compute the Galois group of the maximal 2-ramified pro-2-extension of a 2-rational number field

We present an algorithm for the computation of logarithmic l-class groups of number fields. Our principal motivation is the effective determination of the l-rank of the wild kernel in the K-theory of number fields.

The text is a synthetic presentation of the state of the knowledge about the capitulation for the class-groups of numbers fields, shortly before the demonstration by Suzuki of the main conjecture on this question., Comment: in French, Journal de Th\'eorie des Nombres de Bordeaux, Soci\'et\'e Arithm\'etique de Bordeaux, 1987

Peu après la mise en ligne de cette note, Georges Gras m'a fait observer que la preuve du Théorème principal est inexacte, l'extension construite n'étant pas décomposée en chacune des places au-dessus de l, mais seulement en l'une d'elles. Ce peut être un exercice intéressant que découvrir pourquoi !; We prove that number fields with arbitrary degree but weak ramification satisfy the Leopoldt conjecture on the l-adic rank of the group of units