Généralisation d’un théorème de Greenberg

We formulate a general conjecture on the characteristic polynomials of S-decomposed T-ramified Iwasawa modules over the cyclotomic \(\mathbb {Z}_\ell \)-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.     

Approximation d’un nombre réel par des unités algébriques

 In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection .     

Nouvelle démonstration d’un théorème de P. A. Smith

Sans résumé A Monsieur H. Hopf, à l’occasion de son soixantième anniversaire     

Approximation d’un nombre réel par des unités algébriques

 In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection .

(Received 14 March 2000; in revised form 16 November 2000)     

Toute forme modérément ramifiée d’un polydisque ouvert est triviale

Let k be a complete, non-Archimedean field and let X be a k-analytic space. Assume that there exists a finite, tamely ramified extension L of k such that X _L is isomorphic to an open polydisc over L ; we prove that X is itself isomorphic to an open polydisc over k. The proof consists in using the graded reduction (a notion which is due to Temkin) of the algebra of functions on X, together with some graded counterparts of classical commutative algebra results : Nakayama’s lemma, going-up theorem, basic notions about étale algebras, etc.     

Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif

This work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group, defined over the field of algebraic numbers . Let G be such a group. Let W be a hyperplane of the tangent space at the origin of G, defined over , and u be a complex point of this tangent space, such that the image of u by the exponential map of the Lie group G(ℂ) is an algebraic point. Then we obtain a lower bound for the distance between u and W⊗ℂ, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane W. The proof rests on Baker’s method and Hirata’s reduction as well as a new arithmetic argument (Chudnovsky’s process of variable change) which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof.

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Mathematics Subject Classification (2000) 11J86, 11J20, 14L10