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We establish a logarithmic version of the classical result of Artin-Furw{\"a}ngler on the principalization ofideal classes in the Hilbert class-field by applying the group theoretic description of the transfert map to logarithmic class-groups of degree 0., Comment: in French language

We are interested in classical and logarithmic imaginary classes of abelian number fields in connection with Iwasawa theory. For any given odd prime ${\ell}$ and any imaginary abelian number field K, we compute the isotypic components of the logarithmic ${\ell}$-class group of K associated to imaginary irreducible ${\ell}$-adic characters of its Galois group and we extend to this situation the Gold criterium on lambda invariants. We first study the semi-simple case, thus the non semi-simple case in connection with transition formulas of genus for supercircular fields., Comment: in French language

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., Comment: 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 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

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 language, Annales des sciences math{\'e}matiques du Qu{\'e}bec, A para{\^i}tre

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

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

We formulate a general conjecture on the characteristic polynomials of S-decomposed T-ramified Iwasawa modules over the cyclotomic Z {\ell}-extension of a number field. We show that this conjecture is equivalent to the conjunctions of the classical conjectures of Leopoldt and of Gross-Kuz'min. We so extend a result of semi-simplicity of Greenberg and, by the way, an isomorphism of Kuz'min., Comment: in French

We extend to logarithmic class groups the results on abelian principalization of tame ray class groups of a number field obtained in a previous article., Comment: in French

Building on Bosca's method, we extend to tame ray class groups the results on capitulation of ideals of a number field by composition with abelian extensions of a subfield first studied by Gras. More precisely, for every extension of number fields K/k, where at least one infinite place splits completely, and every squarefree divisor m of K, we prove that there exist infinitely many abelian extensions F/k such that the ray class group mod m of K capitulates in KF. As a consequence we generalize to tame ray class groups the results of Kurihara on triviality of class groups for maximal abelian pro-extensions of totally real number fields., Comment: in French

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 compute the Z-rank of the subgroup of elements of the multiplicative group of a number field K that are norms from every finite level of the cyclotomic Z{\ell}-extension of K. Thus we compare its {\ell}-adification with the group of logarithmic units of K. By the way we point out an easy proof of the Gross-Kuz'min conjecture for {\ell}-undecomposed extensions of abelian fields., Comment: in French

On the basis of a previous work, we elaborate a new description of the Kummer radical associated to the first layers of $\mathbb{Z}_\ell$--extensions of a number fields K, by using inverse limits for the norm maps in the cyclotomic $\mathbb{Z}_\ell$-extension. Our main result contains, as an obvious consequence, the inclusions provided by Soogil Seo in a set of papers. By the same way we also give in the last section a similar description of the Tate kernel for universal symbols in $K_2(K)$., Comment: in French

We compute the capitulation kernel for the module of Bertrandias-Payan in an arbitrary $\ell$-extension of number fields which satisfies the Leopoldt conjecture. As a consequence we relate the existence of extensions with trivial such module to the classical problem of locally cyclotomic towers., 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

We study a logarithmic version of the classical result of Artin-Furw{\"a}ngler on principalization of ideal classes in the Hilbert class-field by applying the group theoretic description of the transfert map to logarithmic class-groups of degree 0., Comment: in French

Let L/K be a 2-birational CM-extension of a totally real 2-rational number field. We characterize in terms of tame ramification totally real 2-extensions K'/K such that the compositum L'= LK' is still 2-birational. In case the 2-extensions K'/K is linearly disjoint from the cyclotomic Z2-extension Kc/K, we prove that K'/K is at most quadratic. In the other hand we construct infinite towers of such 2-extensions.

In this paper we clarify some asymptotic formulas given by Jaulent-Maire, which relate orders of finite quotients of S-infinitesimal T-classes l-groups $Cl^S_T(K_n)$ associated to finite layers $K_n$ of a Zl-extension $K_\infty/K$ over a number field to the structural invariants of the Iwasawa module $X^S_T:=\varprojlim \Cl^S_T(K_n)$. We especially show that the lambda invariant $\tilde\lambda^S_T$ of those quotients sensibly differs from the structural invariant $\lambda^S_T$, and we illustrate this fact with explicit examples, where it can be made as large as desired, positive or negative.

We compute the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field., Comment: This Note is motivated by the paper ``Galois 2-extensions unramified outside 2'' of J. Jossey. We bring into focus some classical technics on abelian $\ell$-ramification which considerably simplify proofs in such subjects; for instance, the main Theorem 2, due to J-F. Jaulent, generalizes the purpose of Jossey's paper

We give the complete proof of a conjecture of Georges Gras which claims that, for any extension $K/k$ of number fields in which at least one infinite place is totally split, every ideal $I$ of $K$ principalizes in the compositum $Kk^{ab}$ of $K$ with the maximal abelian extension $k^{ab}$ of $k$

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

We present an algorithm for computing the 2-group of the positive divisor classes of a number field F in case F has exceptional dyadic places. As an application, we compute the 2-rank of the wild kernel WK2(F) in K2(F) for such number fields.

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.

We study the l-part of the the wild \'etale kernels WK2i(F) of an arbitary number field F for a given prime l in connection with the logarithmic l-class groups. From the logarithmic arithmetic we deduce rank formulas, periodicity and reflection theorems, triviality characterizations and various consequences.

We present an algorithm for computing the 2-group of narrow logarithmic divisor classes of degree 0 for number fields F. As an application, we compute in some cases the 2-rank of the wild kernel WK2(F).

We extend to convenient finite quotients of a noetherian Lambda-module the classical result of K. Iwasawa giving the asymptotic expression of the l-part of the number of ideal class groups in Zl-extensions of number fields. Then, in the arithmetic context, we compute the three characters associated by this way to the l-groups of T-infinitesimal S-classes in the cyclotomic tower and relate them to the classical invariants and the decomposition characters associated to the finite sets of places S and T. A main tool in this study is the so-called Spiegelungssatz of Georges Gras, which exchanges (wild or tame) ramification and decomposition. The main results of this arithmetical part extend those we obtained with Christian Maire in a previous article. The most intricate study of the wild contribution of the sets S and T involves a generalization of a classical result of R. Greenberg on the genus theory of cyclotomic towers.

By using the logarithmic approach of the classical kernels for the K2 of number fields, we compute the 2-rank of the wild kernel WK2(F) and the 2-rank of the subgroup of infinite heigh elements in K2(F) in terms of positive class groups for any number field F.

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 construct a compact topological group Rl which contains both the real additive group R and the l-adic one Ql (for a given prime number l) as dense subgroups; thus we study some of its properties. This construction gives an arithmetic description of the so-called l-adic solenoid classically defined in terms of foliations.

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