Détermination géométrique de la systole des groupes de triangles

In this Note we give an explicit formula for the length of the shortest geodesic loop for hyperbolic spheres with three singularities of order greater than 3.     

Rationalité des séries de poincaré et des fonctions zêta motiviques

Soient k un corps de caractéristique 0 et X une k[[t]]-variété (éventuellement singulière) plate, purement de dimension relative d. Nous prouvons la rationalité des séries de Poincaré motiviques et de fonctions Zêta dIgusa motiviques, associées à X, à laide de lintégration motivique, du théorème de désingularisation plongée dHironaka, de la théorie des modèles de Néron faibles pour les schémas formels et dun théorème délimination des quantificateurs en théorie des modèles.Revised version: 17 March 2004     

Groupes géométriquement finis d’automorphismes d’arbres et approximation diophantienne dans les arbres

We introduce a class of non uniform tree lattices, analogous to the fundamental group of geometrically finite negatively curved manifolds. This class contains all the algebraic examples. Given such a tree lattice, we develop, as in [HP2] in the manifold case, a theory of diophantine approximation of points in the boundary of the tree by the parabolic fixed points. We compute approximation constants for some algebraic examples. All results are expressed intrinsically in terms of a natural geodesic flow for a graph of groups. DMA, UMR 8553 CNRS, École Normale Supérieure, 45 rue dUlm, 75230 Paris Cedex 05, France. e-mail:Frederic.Paulin@ens.fr Mathematics Subject Classification (2000): 20E08, 11J61, 20G25, 11J06     

Dégénérescence de sous-groupes discrets des groupes de Lie semi-simples

We prove that a sequence of faithful and discrete (but not necessarily with finite covolume) representations of a finitely generated group Γ in a semisimple Lie group, that goes to infinity, converges to an isometric action of Γ on an affine building, with controlled germ of apartment stabilizers.     

Approximants de Padé et séries hypergéométriques équilibrées

In this paper, we present and solve some very general new Padé approximant problems, whose solutions can be expressed with hypergeometric series. These series appear in the proofs of the irrationality of ζ(3), of infinitely many ζ(2n+1), and in essentially all results of this kind in the literature. We also prove two new Diophantine results with this method.     

Géométrie des espaces de Banach ayant des approximations de l'identité contractantes

Let a, c ≥ 0 and let B be a compact set of scalars. We introduce property M^* (a, B, c) of Banach spaces X which is a geometric property of Banach spaces generalizing property (M^*) due to Kalton. Using M^*(a, B, c) with max ¦B¦ + c > 1, we characterize intrinsically a large class of shrinking approximations of the identity, including those related to M-, u-, and h-ideals of compact operators. We also show that the existence of these approximations of the identity is separably determined. As an application, we study ideals of compact and approximable operators. In particular, this provides an alternative unified and easier approach to the theories of M-, u-, and h-ideals of compact operators.     

L'influence de la géométrie de Bolyai-Lobatchevsky sur le développement de la géométrie

Sans résumé     

Dualité de Spanier–Whitehead en géométrie algébrique

In this Note, we study finitely presented objects in the stable ${\mathbb{A}}^1$-homotopy category introduced by Morel and Voevodsky and we use a theorem by Voevodsky and Ayoub on the four functors to get Spanier–Whitehead duality in ${\mathbb{A}}^1$-homotopy theory. To cite this article: J. Riou, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Irrationalité de certaines sommes de séries

Résumé Let with |q| > 1, and a be a rational number such that a ² is not equal to for . In this note, we prove that the sum is irrational.     

Complexité bilinéaire de la multiplication dans des petits corps finis

In this Note, we improve a result of Shokrollahi who has applied the algorithm of D.V. Chudnovsky and G.V. Chudnovsky to algebraic curves of genus one. More precisely, from the Abelian group structure of the set of rational points on elliptic curves, we show that, if $\frac{1}{2}q+1<n?\frac{1}{2}(q+1+\mathrm{?}(q))$, then the bilinear complexity of the multiplication in all extensions ${\mathbb{F}}_{q^n}$ is equal to 2n. To cite this article: J. Chaumine, C. R. Acad. Sci. Paris, Ser. I 343 (2006).