Représentations <Emphasis Type="Italic">p</Emphasis>-adiques de GL<Subscript>2</Subscript>(L) et Catégories Dérivées

We construct some locally ℚ _p -analytic representations of GL₂(L), L a finite extension of ℚ _p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel’d’s upper half space has a structure of rank 2 admissible filtered (φ, N)-module. Finally, we prove that this filtered module is associated, via Fontaine’s theory, to the initial Galois representation.     

Principe de bloch et estimations de la métrique de Kobayashi des domaines de<Emphasis Type="Bold">C</Emphasis><Superscript>2</Superscript>

Nous ramenons l'existence d'estimations optimales pour la métrique de Kobayashi dans les domaines pseudoconvexes de type fini deC ² à un principe de Bloch asymptotique. Nous établissons ce principe en combinant la méthode de renormalisation utilisée par Gromov dans le contexte des applications harmoniques aux techniques de dilatation des coordonnées. Cecifournit une preuve totalement élémentaire d'un résultat de Catlin particulièrement utile dans l'étude des questions de prolongement et de rigidité d'applications holomorphes.

     

Diagrammes de Diamond et (<Emphasis Type="Italic">φ</Emphasis>, Γ)-modules

Let ρ be a 2-dimensional continuous semi-simple generic representation of Gal(̅ℚ _p /ℚ _p ) over ̅F _p . The modulo p Langlands correspondence for GL₂(ℚ _p ) defined in [5], as realized in [9], can be reformulated as a quite simple recipee giving back the (φ, Γ)-module of the dual of ρ starting from the “Diamond diagram” associated to ρ. Let F be a finite unramified extension of ℚ _p and ρ a 2-dimensional continuous semi-simple generic representation of Gal(̅ℚ _p /F) over ̅F _p . When one formally extends this recipee to the Diamond diagrams associated to ρ in [6], we show that one essentially finds the (φ, Γ)-module of the tensor induction from F to ℚ _p of the dual of ρ.     

Sous-groupes discrets de <Emphasis Type="Italic">PU</Emphasis>(2,1) engendrés par <Emphasis Type="Italic">n</Emphasis> réflexions complexes et Déformation

Let PU(2,1) be the group of holomorphic isometries in the hyperbolic complex plane and let G _n be a sub-group of PU(2,1) which is generated by n complex reflections with respect to complex lines in . Under certain conditions, we prove that G _n is discrete. We construct representations ρ of the fundamental group Γ _g of the compact surface Σ _g of genus g, into PU(2,1), we prove they are discrete, faithful and we compute the dimension their deformation space.      

L'inégalité de Bohr pour les séries de Dirichlet

We extend to the setting of Dirichlet series previous results of Bohr for Taylor series in one variable, themselves generalized by Paulsen, Popescu and Singh or extended to several variables by Aizenberg, Boas and Khavinson. We show in particular that, if $f(s)={\sum }_{n=1}^{\infty }{a}_n{n}^{?s}$, with $6f{6}_{\infty }:={\sup }_{\mathfrak{R}s>0}|f(s)|<\infty$, then ${\sum }_{n=1}^{\infty }|a_n|n^{\mathrm{?2}}?6f{6}_{\infty }$ and even slightly better, and ${\sum }_{n=1}^{\infty }|a_n|n^{?1/2}?C6f{6}_{\infty }$, C being an absolute constant. To cite this article: R. Balasubramanian et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Solutions globales d'énergie infinie de l'équation de Navier–Stokes 2D

We study in this Note the solutions of the 2D Navier–Stokes equations with initial data in ?BMO. For $u{|}_{t=0}$ in the closure of the Schwartz class, we obtain the existence and uniqueness of a global solution, and besides an estimate on its norm in ?BMO. To cite this article: P. Germain, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Sur la densité d'état de l'opérateur de Schrödinger quasi-périodique unidimensionnel

We prove two results on the density of states of the discrete one dimensional quasi-periodic Schrödinger equation with an analytic potential and Diophantine frequencies in the perturbed regime. On the one hand, we prove that this function has the behavior of a Hölder-$\frac{1}{2}$ function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. To cite this article: S. Hadj Amor, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Propriétés de l'EMV de rupture de processus de Poisson

This Note concerns the estimation of the parameter of the intensity of an inhomogeneous spatial Poisson process by the maximum likelihood method. We consider an intensity admitting a discontinuity along a closed curve depending of an unknown parameter. The established results show the consistency, the convergence in law and the convergence of the moments of the maximum likelihood estimator.     

Paradoxe de Klein pour l'équation de Klein–Gordon chargée : superradiance et opérateur de diffusion

We develop the scattering theory for the charged Klein–Gordon equation on ${\mathbb{R}}_t\times {\mathbb{R}}_x$, when the electrostatic potential $A(x)$ has different asymptotics $a^{\pm }$ as $x\to \pm \infty$. In this case, the conserved energy is not positive definite (Klein Paradox). We construct the spectral representation for the harmonic equation, and we establish the existence of a Scattering Operator the symbol of which has a norm strictly larger than 1, for the frequencies in $(a^{?}\text{,}{a}^{+})$. These results can be applied to the DeSitter–Reissner–Nordstrøm metric, to justify the notion of superradiance of the charged black-holes. To cite this article: A. Bachelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Un théorème de finitude pour le groupe de Chow des zéro-cycles d’un groupe algébrique linéaire sur un corps <Emphasis Type="Italic">p</Emphasis>-adique

Zusammenfassung Sei X eine glatte Kompaktifizierung einer zusammenhängenden linearen Gruppe über einem Körper k. Die Chowgruppe der nulldimensionalen Zyklen von X vom Grad Null ist eine Torsionsgruppe. Wir zeigen: wenn k ein p-adischer Körper ist, dann ist der prim-zu-p-Anteil dieser Gruppe endlich.      

Propriétés d'indépendance algébrique de valeurs de séries de Hecke–Mahler

Here we characterize in a complete and explicit way the relations of algebraic dependence over $\mathbb{Q}$ of complex values of Hecke–Mahler series at algebraic points of the multiplicative group ${\mathbb{G}}_{\mathrm{m}}^2(\mathbb{C})$. Our result contains previous results of Loxton and van der Poorten, Mahler, and Masser. To cite this article: F. Pellarin, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Inégalités d'oracle pour l'estimation d'une densité de probabilité

We study the problem of the nonparametric estimation of a probability density in $L_2(\mathbb{R})$. Expressing the mean integrated squared error in the Fourier domain, we show that it is close to the mean squared error in the Gaussian sequence model. Then, applying a modified version of Stein's blockwise method, we obtain a linear monotone oracle inequality and a kernel oracle inequality. As a consequence, the proposed estimator is sharp minimax adaptive (i.e. up to a constant) on a scale of Sobolev classes of densities. To cite this article: Ph. Rigollet, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Quincunx multiresolution analysis for <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℚ<Stack><Subscript>2</Subscript><Superscript>2</Superscript></Stack>)

With an eye on applications in quantum mechanics and other areas of science, much work has been done to generalize traditional analytic methods to p-adic systems. In 2002 the first paper on p-adic wavelets was published. Since then p-adic wavelet sets, multiresolution analyses, and wavelet frames have all been introduced. However, so far all constructions have involved dilations by p. This paper presents the first construction of a p-adic wavelet system with a more general matrix dilation, laying the foundation for further work in this direction.     

Sur une généralisation de la notion de V-variété

We consider a topological space which is locally isomorphic to the quotient of R^k by the action of a discrete group and we call it quasifold of dimension k. Quasifolds generalize manifolds and orbifolds and represent the natural framework for performing symplectic reduction with respect to the induced action of any Lie subgroup, compact or not, of a torus. We define quasitori, Hamiltonian actions of quasitori and the moment mapping for symplectic quasifolds, and we show that every simple convex polytope, rational or not, is the image of the moment mapping for the action of a quasitorus on a quasifold.     

Cohomologie des groupes linéaires,K-théorie de Milnor et groupes de Witt

We construct some remarkable classes in the cohomology of linear groups.